sciMath.algebra
description::
× generic: mathAppliedNo.
· "Algebra (from Arabic الجبر (al-jabr) 'reunion of broken parts,[1] bonesetting'[2]) is the study of variables and the rules for manipulating these variables in formulas;[3] it is a unifying thread of almost all of mathematics.[4]
Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields. Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory.
The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist."
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Algebra]
name::
* McsEngl.algebra-sciMath,
* McsEngl.mathAlgebra,
* McsEngl.sciMath.005-algebra,
* McsEngl.sciMath.algebra,
====== langoGreek:
* McsElln.άλγεβρα!η!=mathAlgebra,
GENERIC-SPECIFIC-TREE of mathAlgebra
generic-tree-of-mathAlgebra::
* mathAppliedNo,
* ... entity,
* McsEngl.mathAlgebra'generic-tree,
specific-tree-of-mathAlgebra::
* groups,
* rings,
* fields,
* linear algebra,
* abstract algebra,
* McsEngl.mathAlgebra.specific-tree,
mathAlgebra.abstract
description::
· "In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures.[1] Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Presently, the term "abstract algebra" is typically used for naming courses in mathematical education, and is rarely used in advanced mathematics.
Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.
Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the variety of groups."
[{2023-08-18 retrieved} https://en.wikipedia.org/wiki/Abstract_algebra]
name::
* McsEngl.abstract-algebra,
* McsEngl.mathAlgebra.abstract,
* McsEngl.modern-algebra,
mathAlgebra.elementary
description::
· "Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers,[1] whilst algebra introduces variables (quantities without fixed values).[2]
This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
It is typically taught to secondary school students and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as"
[{2023-08-18 retrieved} https://en.wikipedia.org/wiki/Elementary_algebra]
name::
* McsEngl.elementary-algebra,
* McsEngl.mathAlgebra.elementary,
mathAlgebra.linear
description::
· "Linear algebra is the branch of mathematics concerning linear equations such as:
\( a_{1}x_{1}+\cdots +a_{n}x_{n}=b,\)
linear maps such as:
\( (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},\)
and their representations in vector spaces and through matrices.[1][2][3]
Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point."
[{2023-08-18 retrieved} https://en.wikipedia.org/wiki/Linear_algebra]
name::
* McsEngl.linear-algebra,
* McsEngl.mathAlgebraLinear,
* McsEngl.mathAlgebra.linear,
====== langoGreek:
* McsElln.γραμμική-άλγεβρα!η!=mathAlgebraLinear,
evoluting of mathAlgebraLinear
description::
"evolution of linear-algebra:
The evolution of linear algebra can be traced back to ancient civilizations, but it has undergone significant development and refinement over the centuries. Here is a brief overview of the key milestones in the evolution of linear algebra:
1. Ancient Origins:
- The earliest recorded use of linear equations and systems can be found in ancient Babylonian mathematics, around 2000 BCE. They used methods for solving linear equations and made some early contributions to the subject.
2. Greek Mathematics:
- Ancient Greek mathematicians, like Euclid and Diophantus, made contributions to number theory and geometry, which are closely related to linear algebra. They worked on problems involving proportions and ratios.
3. Renaissance and Early Modern Period:
- During the Renaissance, mathematicians like René Descartes and Pierre de Fermat developed coordinate systems and introduced the concept of vectors in geometry. These ideas laid the foundation for modern linear algebra.
4. Development of Matrix Notation:
- The concept of a matrix, although not formalized as we know it today, emerged in the late 17th century with the work of Japanese mathematician Seki Takakazu. His work involved methods similar to matrix multiplication.
- In the 19th century, British mathematician Arthur Cayley introduced matrix notation and began to study matrices more formally.
5. Vector Spaces and Abstract Algebra:
- In the late 19th and early 20th centuries, mathematicians like David Hilbert and Emmy Noether developed the theory of abstract vector spaces and vector spaces over fields. This abstract approach is a fundamental part of modern linear algebra.
6. Matrix Theory:
- In the early 20th century, matrix theory became more formalized and widely used, particularly in the study of linear transformations and systems of linear equations. Matrices became a fundamental tool in various scientific and engineering disciplines.
7. Eigenvectors and Eigenvalues:
- The study of eigenvectors and eigenvalues, which are fundamental in linear algebra, gained prominence in the early 20th century. These concepts have applications in a wide range of fields, including physics, engineering, and computer science.
8. Numerical Linear Algebra:
- With the advent of computers in the mid-20th century, numerical linear algebra gained significance. Algorithms for solving systems of linear equations, finding eigenvalues, and other linear algebra computations were developed.
9. Applications in Modern Science and Technology:
- Linear algebra plays a crucial role in various scientific and technological fields, such as computer graphics, machine learning, cryptography, quantum mechanics, and data analysis.
10. Ongoing Development:
- Linear algebra continues to evolve with the development of new mathematical techniques, applications in emerging fields, and advancements in computational tools.
The evolution of linear algebra is ongoing, as it remains a foundational mathematical subject with ever-expanding applications in various domains of science, engineering, and technology. The development of more advanced algorithms and the integration of linear algebra with other mathematical disciplines continue to drive its evolution."
[{2023-10-20 retrieved} https://chat.openai.com/c/40c537ae-12b9-4024-8325-14a3e4155b23]
name::
* McsEngl.evoluting-of-mathAlgebraLinear,
* McsEngl.mathAlgebraLinear'evoluting,
mathAlgebra.sigma
description::
· "In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a nonempty collection Σ of subsets of X closed under complement, countable unions, and countable intersections. The ordered pair \((X,\Sigma)\) is called a measurable space.
The σ-algebras are a subset of the set algebras; elements of the latter only need to be closed under the union or intersection of finitely many subsets, which is a weaker condition.[1]
The main use of σ-algebras is in the definition of measures; specifically, the collection of those subsets for which a given measure is defined is necessarily a σ-algebra. This concept is important in mathematical analysis as the foundation for Lebesgue integration, and in probability theory, where it is interpreted as the collection of events which can be assigned probabilities. Also, in probability, σ-algebras are pivotal in the definition of conditional expectation.
In statistics, (sub) σ-algebras are needed for the formal mathematical definition of a sufficient statistic,[2] particularly when the statistic is a function or a random process and the notion of conditional density is not applicable."
[{2023-08-18 retrieved} https://en.wikipedia.org/wiki/%CE%A3-algebra]
name::
* McsEngl.mathAlgebra.sigma,
* McsEngl.sigma-algebra,
mathAlgebra.universal
description::
· "Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of study."
[{2023-08-18 retrieved} https://en.wikipedia.org/wiki/Universal_algebra]
name::
* McsEngl.general-algebra,
* McsEngl.mathAlgebra.universal,
* McsEngl.universal-algebra,
sciMath.analysis
description::
× generic: mathAppliedNo.
"Mathematical analysis, also known simply as analysis, is a branch of mathematics that focuses on the study of limits, continuity, derivatives, integrals, sequences, and series. It provides rigorous techniques for studying the behavior of functions and their properties. Analysis is a fundamental area of mathematics and is essential for understanding various mathematical concepts and applications in science, engineering, and other fields.
Mathematical analysis can be classified into several subfields:,
* Real Analysis: Real analysis deals with functions of real numbers, their properties, and the properties of real numbers themselves. It includes topics like limits, continuity, differentiation, integration, sequences, and series of real numbers.,
* Complex Analysis: Complex analysis is concerned with the study of functions of complex numbers. It explores properties of complex functions, such as holomorphic functions, complex integration, and the behavior of functions in the complex plane.,
* Functional Analysis: Functional analysis focuses on the study of vector spaces of functions and their properties. It deals with spaces of functions, linear operators, and concepts like normed spaces, Banach spaces, and Hilbert spaces.,
* Measure Theory: Measure theory provides a rigorous foundation for defining and measuring sets and functions. It's the basis for understanding concepts like Lebesgue integration and probability theory.,
* Differential Equations: While differential equations can also be seen as a separate area, they heavily involve analysis techniques. Differential equations involve studying functions and their derivatives to model and solve various real-world problems.,
* Harmonic Analysis: Harmonic analysis deals with the representation and decomposition of functions or signals into sinusoidal components. It's often used in signal processing, Fourier analysis, and wavelet analysis.,
* Nonlinear Analysis: Nonlinear analysis studies functions that are not necessarily linear. It explores the behavior of nonlinear equations, systems of equations, and their solutions.,
* Variational Analysis: Variational analysis deals with optimization problems and functionals. It studies how to find the optimal value of a functional given certain constraints.,
* Complex Dynamics: Complex dynamics studies the behavior of iterated functions, especially in the complex plane. It often involves the study of fractals and chaotic behavior.,
* Numerical Analysis: While largely computational, numerical analysis involves using mathematical techniques to approximate solutions to problems that might not have analytical solutions. It often relies on analysis principles to ensure the accuracy and convergence of numerical methods.,
These subfields are interconnected and provide the tools and theories necessary to understand the properties of functions, study their behavior, and solve mathematical problems.",
[{2023-08-25 retrieved} https://chat.openai.com/?model=text-davinci-002-render-sha]
· "Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.[1][2]
These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space)."
[{2023-08-16 retrieved} https://en.wikipedia.org/wiki/Mathematical_analysis]
Mathematical analysis, which mathematicians refer to simply as analysis, is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series,[1] and analytic functions. These theories are often studied in the context of real numbers, complex numbers, and real and complex functions. Analysis may be conventionally distinguished from geometry. However, theories of analysis can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or, more specifically, distance (a metric space).
[http://en.wikipedia.org/wiki/Mathematical_analysis]
name::
* McsEngl.analysis!⇒mathAnalysis,
* McsEngl.mathAnalysis,
* McsEngl.mathAnalysis!=science.math.analysis,
* McsEngl.mathematical-analysis!⇒mathAnalysis,
* McsEngl.sciMath.021-analysis!⇒mathAnalysis,
* McsEngl.sciMath.analysis!⇒mathAnalysis,
* McsEngl.science.math.analysis!⇒mathAnalysis,
====== langoGreek:
* McsElln.μαθηματική-ανάλυσις!η!=mathAnalysis,
descriptionLong::
"mathematical analysis topics
Mathematical analysis is the branch of mathematics that deals with limits and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. It is one of the foundational branches of mathematics, and it has applications in many other fields, such as physics, engineering, and economics.
Some of the main topics in mathematical analysis include:
* Real analysis: The study of the real numbers and real-valued functions.
* Complex analysis: The study of the complex numbers and complex-valued functions.
* Measure theory: The study of measures, which are functions that assign a non-negative real number to certain subsets of a set.
* Functional analysis: The study of vector spaces and linear operators.
* Harmonic analysis: The study of functions using Fourier transforms.
* Differential equations: The study of equations that relate an unknown function to its derivatives.
Here are some specific examples of topics in mathematical analysis:
* Limits: The concept of a limit is essential for calculus and many other areas of mathematics.
* Continuity: A continuous function is one whose graph can be drawn without lifting the pen from the paper.
* Differentiation: The derivative of a function measures its rate of change.
* Integration: The integral of a function represents the area under its graph.
* Infinite series: An infinite series is a sum of infinitely many terms.
* Analytic functions: Analytic functions are complex-valued functions that have derivatives of all orders.
Mathematical analysis is a challenging but rewarding subject. It provides a deep understanding of the foundations of mathematics and the tools to solve a wide variety of problems.
If you are interested in learning more about mathematical analysis, there are many resources available online and in libraries. You can also find many courses on mathematical analysis at universities and colleges."
[{2023-10-03 retrieved} https://bard.google.com/chat/6eeac4dbe9b6e921]
WHOLE-PART-TREE of mathAnalysis
whole-tree-of-mathAnalysis::
* mathAppliedNo,
* ... Sympan,
* McsEngl.mathAnalysis'whole-tree,
part-tree-of-mathAnalysis::
* ,
* McsEngl.mathAnalysis'part-tree,
GENERIC-SPECIFIC-TREE of mathAnalysis
generic-tree-of-mathAnalysis::
* mathAppliedNo,
* ... entity,
* McsEngl.mathAnalysis'generic-tree,
specific-tree-of-mathAnalysis::
* calculus,
* complex analysis,
* differential equations,
* functional analysis,
* harmonic analysis,
* measure theory,
* numerical analysis,
* real analysis,
* scalar analysis,
* tensor analysis,
* vector analysis,
* McsEngl.mathAnalysis.specific-tree,
mathAnalysis.measure-theory
description::
"In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.
The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Ιmile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathιodory, and Maurice Frιchet, among others."
[{2023-10-04 retrieved} https://en.wikipedia.org/wiki/Measure_(mathematics)]
name::
* McsEngl.mathAnalysis.measure-theory,
* McsEngl.measure-theory,
* McsEngl.sciMath.028-measure-theory,
* McsEngl.sciMath.measure-theory,
mathAnalysis.numerical-analysis
description::
"numerical analysis overview
Numerical analysis is the study of methods for solving mathematical problems using numerical approximation. Numerical methods are used in all sectors of engineering and the physical sciences, as well as in the life and social sciences, health, business, and even the arts.
Numerical analysis is concerned with two main areas:
* Developing new numerical methods: This involves designing algorithms that are efficient and accurate, and that can be used to solve a wide range of problems.
* Analyzing the stability and convergence of numerical methods: This involves understanding how errors propagate through numerical methods, and how to choose the best parameters for a given problem.
Some of the most common numerical methods include:
* Interpolation and approximation: These methods are used to approximate the values of a function at points where they are not known.
* Numerical integration: This method is used to approximate the value of a definite integral.
* Numerical differentiation: This method is used to approximate the value of the derivative of a function at a point.
* Numerical solution of ordinary differential equations: This method is used to approximate the solution of an ordinary differential equation.
* Numerical solution of partial differential equations: This method is used to approximate the solution of a partial differential equation.
Numerical analysis is a rapidly growing field, as new algorithms are constantly being developed and existing algorithms are being improved. Numerical methods are also becoming increasingly important in many areas of science and engineering, as they allow us to solve problems that would be too complex or time-consuming to solve using analytical methods.
Here are some examples of how numerical analysis is used in the real world:
* Weather forecasting: Numerical methods are used to solve the complex equations that govern the weather, and to produce accurate forecasts of temperature, precipitation, and other weather conditions.
* Space exploration: Numerical methods are used to design and guide spacecraft, and to predict their trajectories.
* Financial modeling: Numerical methods are used to model the behavior of financial markets and to develop trading strategies.
* Medical imaging: Numerical methods are used to process medical images, such as MRI and CT scans, and to produce detailed images of the human body.
* Climate modeling: Numerical methods are used to model the climate system and to predict the effects of climate change.
Numerical analysis is a powerful tool that can be used to solve a wide range of problems in science and engineering. It is an essential part of the education of many scientists and engineers, and it is used in research and development across many industries."
[{2023-10-04 retrieved} https://bard.google.com/chat/44e202dfff831409]
name::
* McsEngl.numerical-analysis,
* McsEngl.mathAnalysis.numerical,
* McsEngl.mathAnalysisNumerical,
* McsEngl.sciMath.027-numerical-analysis,
* McsEngl.sciMath.numerical-analysis,
====== langoGreek:
* McsElln.αριθμητική-ανάλυση!η!=mathAnalysisNumerical,
sciMath.calculus
description::
· "Calculus[nb 1] is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.[1]
Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz.[2][3] Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, and social science.[4]"
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Calculus]
name::
* McsEngl.calculus-sciMath,
* McsEngl.mathCalc,
* McsEngl.mathCalc!=math.calculus,
* McsEngl.sciMath.006-calculus,
* McsEngl.sciMath.calculus,
====== langoGreek:
* McsElln.απειροστικός-λογισμός!ο!=mathCalc,
PARENT-CHILD-TREE of mathCalc
description::
· analysis is a-child of calculus.
* McsEngl.mathAnalysis;;mathCalc,
* McsEngl.mathCalc;mathAnalysis,
name::
* McsEngl.mathCalc'parent-child-tree,
mathCalc.differential
description::
The process of finding the derivative [Παράγωγο] of a given function is called differentiation. The study of derivatives is called the differential calculus.
[Richardson, 1966, 349 #cptResource451#]
===
Differential calculus is concerned with rates of change.
"Calculus," Microsoft(R) Encarta(R) 97 Encyclopedia. (c) 1993-1996 Microsoft Corporation. All rights reserved.
name::
* McsEngl.calcDifferential,
* McsEngl.differential-calculus,
* McsEngl.mathCalc.differential,
* McsEngl.sciMath.036-differential-calculus,
* McsEngl.sciMath.differential-calculus,
====== langoGreek:
* McsElln.διαφορικός-λογισμός!ο!=calcDifferential,
derivative of calcDifferential
description::
· the-derivative-of-function \(f(x)\) is the-function \(f'(x) = \lim\limits_{ℎ \to 0} \frac{f(x+ℎ)−f(x)}{ℎ}\).
· the-function \(f(x) = x\ sin(x^2)\ +1\) has derivative the-function \(f'(x) = sin(x^2)\ +2x^2\ cos(x^2)\). Here you see the-derivative at different points of the-differentiable-function.
[{2023-08-17 retrieved} https://en.wikipedia.org/wiki/File:Tangent_function_animation.gif]
name::
* McsEngl.Mathderivative,
* McsEngl.Mathderivative!=math-derivative,
* McsEngl.derivative-function-of-a-function!⇒Mathderivative,
* McsEngl.calcDifferential'derivative!⇒Mathderivative,
* McsEngl.math-derivative!⇒Mathderivative,
====== langoGreek:
* McsElln.παράγωγη-συνάρτηση-μιας-συνάρτησης!=Mathderivative,
descriptionLong::
· "what is the derivative of function:
The derivative of a function measures how the function's output value changes as its input value changes. In other words, it gives you the rate of change or slope of the function at a particular point. Mathematically, if you have a function f(x), its derivative f′(x) with respect to x is defined as the limit of the difference quotient:
\(f'(x) = \lim_{ℎ \to 0} \frac{f(x+ℎ)−f(x)}{ℎ}\)
This limit represents the instantaneous rate of change of the function at the point x. Geometrically, it's the slope of the tangent line to the graph of the function at that point.
The derivative can also be interpreted as the function that gives you the slope of the tangent line at each point on the graph. If the derivative is positive, the function is increasing at that point; if it's negative, the function is decreasing; and if it's zero, the function has a horizontal tangent.
There are various rules and techniques for finding derivatives of different types of functions, such as the power rule, chain rule, product rule, and quotient rule, among others. The study of derivatives is a fundamental concept in calculus and is used extensively in many areas of mathematics, science, and engineering to analyze rates of change, optimization problems, and more."
[{2023-08-17 retrieved} https://chat.openai.com/?model=text-davinci-002-render-sha]
===
· "what is the derivative of function:
The derivative of a function is a measure of its rate of change. It can be calculated using the following formula:
\(f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)
where f(x) is the function and f'(x) is its derivative.
In other words, the derivative of a function at a point is the slope of the line tangent to the function's graph at that point.
The derivative of a function can be used to find many important properties of the function, such as its increasing/decreasing intervals, its maximum/minimum points, and its points of inflection.
For example, the derivative of the function f(x) = x^2 is f'(x) = 2x. This means that the slope of the line tangent to the graph of f(x) = x^2 is 2x at any point.
The derivative of a function can also be used to solve problems involving motion, economics, and other fields.
In the Python code you provided, the function derivative() calculates the derivative of the function f at the point x. The code uses the formula for the derivative, and it takes a small value h as an input to approximate the limit.
The output of the code is 4.0001000000078335, which is the derivative of the function f(x) = x^2 at the point x = 2."
[{2023-08-17 retrieved} https://bard.google.com/]
Webapp of Mathderivative
description::
* free with steps: https://www.derivative-calculator.net/,
* https://www.wolframalpha.com/calculators/derivative-calculator/,
* https://openomnia.com/derivative-calculator,
name::
* McsEngl.Mathderivative'Webapp,
* McsEngl.Webapp.Mathderivative,
* McsEngl.Webapp.derivative-function-finder,
mathCalc.integral
description::
· "Integral calculus
The branch of mathematics in which the notion of an integral, its properties and methods of calculation are studied. Integral calculus is intimately related to differential calculus, and together with it constitutes the foundation of mathematical analysis. The origin of integral calculus goes back to the early period of development of mathematics and it is related to the method of exhaustion developed by the mathematicians of Ancient Greece (cf. Exhaustion, method of). This method arose in the solution of problems on calculating areas of plane figures and surfaces, volumes of solid bodies, and in the solution of certain problems in statistics and hydrodynamics. It is based on the approximation of the objects under consideration by stepped figures or bodies, composed of simplest planar figures or special bodies (rectangles, parallelopipeds, cylinders, etc.). In this sense, the method of exhaustion can be regarded as an early method of integration. The greatest development of the method of exhaustion in the early period was obtained in the works of Eudoxus (4th century B.C.) and especially Archimedes (3rd century B.C.). Its subsequent application and perfection is associated with the names of several scholars of the 15th–17th centuries.
The fundamental concepts and theory of integral and differential calculus, primarily the relationship between differentiation and integration, as well as their application to the solution of applied problems, were developed in the works of P. de Fermat, I. Newton and G. Leibniz at the end of the 17th century. Their investigations were the beginning of an intensive development of mathematical analysis. The works of L. Euler, Jacob and Johann Bernoulli and J.L. Lagrange played an essential role in its creation in the 18th century. In the 19th century, in connection with the appearance of the notion of a limit, integral calculus achieved a logically complete form (in the works of A.L. Cauchy, B. Riemann and others). The development of the theory and methods of integral calculus took place at the end of 19th century and in the 20th century simultaneously with research into measure theory (cf. Measure), which plays an essential role in integral calculus."
[{2023-08-17 retrieved} https://encyclopediaofmath.org/wiki/Integral_calculus]
name::
* McsEngl.calcIntegral,
* McsEngl.integral-calculus,
* McsEngl.mathCalc.integral,
* McsEngl.sciMath.037-integral-calculus,
* McsEngl.sciMath.integral-calculus,
====== langoGreek:
* McsElln.ολοκληρωτικός-λογισμός!ο!=calcIntegral,
integral of calcIntegral
description::
"In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus,[a] the other being differentiation. Integration started as a method to solve problems in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity. Today integration is used in a wide variety of scientific fields.
The integrals enumerated here are called definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are positive while areas below are negative. Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function when its antiderivative is known; differentiation and integration are inverse operations.
Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable.
Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space."
[{2023-10-08 retrieved} https://en.wikipedia.org/wiki/Integral]
name::
* McsEngl.calcIntegral'integral,
* McsEngl.integral-of-calcIntegral,
====== langoGreek:
* McsElln.ολοκλήρωμα!το!=integral,
mathCalc.stochastic
description::
"Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created and started by the Japanese mathematician Kiyosi Itô during World War II.
The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modeling Brownian motion as described by Louis Bachelier in 1900 and by Albert Einstein in 1905 and other physical diffusion processes in space of particles subject to random forces. Since the 1970s, the Wiener process has been widely applied in financial mathematics and economics to model the evolution in time of stock prices and bond interest rates.
The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes, but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines). The Stratonovich integral can readily be expressed in terms of the Itô integral, and vice versa. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Itô's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than Rn. The dominated convergence theorem does not hold for the Stratonovich integral; consequently it is very difficult to prove results without re-expressing the integrals in Itô form."
[{2023-10-08 retrieved} https://en.wikipedia.org/wiki/Stochastic_calculus]
name::
* McsEngl.calcStochastic,
* McsEngl.mathCalc.stochastic,
* McsEngl.sciMath.038-stochastic-calculus,
* McsEngl.sciMath.stochastic-calculus,
sciMath.category-theory
description::
"overview of category-theory:
Category theory is a branch of mathematics that studies abstract relationships between mathematical structures. It was introduced in the 1940s by Samuel Eilenberg and Saunders Mac Lane, and has since become a powerful tool in many areas of mathematics, including abstract algebra, topology, and computer science.
A category consists of two collections: objects and arrows. Objects can be anything, such as sets, groups, or topological spaces. Arrows represent relationships between objects, and they can be composed just like functions. For example, if we have a category of sets, the objects could be sets and the arrows could be functions between sets.
There are two basic axioms that a category must satisfy:
1. For every object $A$, there is an identity arrow from $A$ to itself.
2. Arrow composition is associative.
These axioms may seem simple, but they allow us to prove many powerful results about categories. For example, one important theorem is the Yoneda lemma, which states that every functor from a category to the category of sets is uniquely determined by its values on the objects of the category.
Category theory is a very abstract subject, but it has many applications in concrete areas of mathematics and computer science. For example, category theory is used in topology to study homotopy theory, which is a branch of mathematics that studies the continuous deformations of shapes. Category theory is also used in computer science to study programming languages and type systems.
Here are some examples of categories:
* The category of sets, where the objects are sets and the arrows are functions between sets.
* The category of groups, where the objects are groups and the arrows are group homomorphisms.
* The category of topological spaces, where the objects are topological spaces and the arrows are continuous maps between topological spaces.
* The category of vector spaces, where the objects are vector spaces and the arrows are linear transformations between vector spaces.
Category theory is a powerful tool for understanding the relationships between different mathematical structures. It is also a very abstract subject, which can make it difficult to learn. However, there are many resources available to help beginners learn category theory, and it is a very rewarding subject to study."
[{2023-10-13 retrieved} https://bard.google.com/chat/c10c0b82d38a297f]
name::
* McsEngl.category-theory,
* McsEngl.sciMath.047-category-theory,
* McsEngl.sciMath.category-theory,
evoluting of category-theory
description::
"evolution of category-theory:
Category theory is a branch of mathematics that was developed in the mid-20th century and has since become an important and foundational area of study in pure mathematics. Its evolution can be summarized in several key stages:
1. Prehistory (Late 19th and early 20th centuries): Category theory didn't emerge in isolation but was influenced by developments in various mathematical fields. Algebraic structures, such as groups, rings, and fields, were being studied, and the idea of abstracting common structures from these algebraic objects began to take shape.
2. Category Theory Emerges (1940s - 1950s): The origins of category theory can be traced back to the work of various mathematicians, including Samuel Eilenberg and Saunders Mac Lane, who published their foundational paper "General Theory of Natural Equivalences" in 1945. This paper introduced the concept of a "category" as a framework for understanding mathematical structures. Eilenberg and Mac Lane's work was inspired by algebraic topology and led to the development of homological algebra.
3. Early Development (1950s - 1960s): Category theory gained traction in the 1950s and 1960s as mathematicians began to appreciate its power and generality. Researchers like Alexander Grothendieck and Jean-Pierre Serre used category theory to develop new ideas in algebraic geometry and homological algebra. This period saw the expansion of category theory into various branches of mathematics.
4. Category Theory in Logic (1960s - 1970s): Category theory found applications in logic, particularly through the work of William Lawvere. He introduced topos theory, which connected category theory with mathematical logic, providing a framework for studying the foundations of mathematics.
5. Category Theory in Computer Science (1970s - 1980s): Category theory started to find applications in computer science and programming languages. Researchers like Dana Scott and Christopher Strachey used category theory to model computation, leading to the development of denotational semantics.
6. Continued Development and Applications (1980s - Present): Category theory continued to evolve, finding applications in diverse fields, including physics, linguistics, and even economics. It has become a unifying framework for abstracting and understanding mathematical structures and relationships between them.
7. Impact on Mathematics: Category theory has significantly influenced various branches of mathematics, including algebra, topology, and mathematical logic. It has provided a common language for mathematicians to communicate and explore relationships between different areas of mathematics.
8. Modern Developments: Category theory is an active area of research with ongoing developments. It has led to the exploration of higher categories, such as 2-categories and n-categories, which provide a more refined way of studying relationships between mathematical structures. The development of new concepts and tools within category theory continues to shape the field and its applications.
Category theory's evolution has been characterized by its adaptability and versatility, making it an important tool for organizing and understanding complex mathematical structures across various disciplines. Its impact on mathematics and other fields continues to grow, making it an integral part of contemporary mathematical research."
[{2023-10-13 retrieved} https://chat.openai.com/c/46408d70-c172-474e-b89b-1c5799ba067a]
name::
* McsEngl.evoluting-of-category-theory,
* McsEngl.category-theory'evoluting,
sciMath.combinatorics
description::
× generic: mathDiscrete,
"Combinatorics is a branch of mathematics that focuses on counting, arranging, and selecting objects or elements from a set, often with an emphasis on their configurations and relationships. It can be broadly classified into several subfields:
* Enumerative Combinatorics: This branch is concerned with counting the number of possible configurations or arrangements of objects. It deals with problems like counting permutations (arrangements) and combinations (selections) of elements from a set.,
* Graph Theory: Graph theory is a subfield of combinatorics that studies the properties and relationships of graphs, which consist of vertices (nodes) connected by edges. Graph theory has applications in various areas, including computer science, social networks, and optimization.,
* Combinatorial Designs: Combinatorial designs involve constructing arrangements of elements that satisfy specific criteria. For example, Latin squares, block designs, and error-correcting codes fall under this category.,
* Generating Functions: Generating functions are used to represent sequences of numbers as formal power series. They are a powerful tool in combinatorial analysis for solving problems related to counting and arrangement.,
* Combinatorial Geometry: This field combines combinatorics and geometry to study geometric properties that are determined by discrete structures, such as points, lines, and polygons. Problems like tiling, coloring, and arrangements of geometric objects fall into this category.,
* Partially Ordered Sets (Posets): Posets involve the study of sets with a partial order relation. They find applications in various areas, including algebra, computer science, and optimization.,
* Ramsey Theory: Ramsey theory deals with the emergence of order within large disordered structures. It focuses on the existence of structured substructures within larger structures, even when randomness is involved.,
* Extremal Combinatorics: This area examines the extreme values that certain combinatorial structures can achieve while satisfying specific conditions. It often involves finding the largest or smallest possible configurations given certain constraints.,
* Algebraic Combinatorics: Algebraic techniques are applied to combinatorial problems. This involves using tools from algebra, such as group theory and linear algebra, to study combinatorial structures.,
* Probabilistic Combinatorics: This field combines combinatorial concepts with probability theory to analyze random structures and their properties. It involves studying the likelihood of specific outcomes in combinatorial situations.,
These are just a few examples of the various subfields within combinatorics. Combinatorics has applications in computer science, cryptography, statistics, optimization, and various other scientific and engineering disciplines.",
[{2023-08-25 retrieved} https://chat.openai.com/?model=text-davinci-002-render-sha]
name::
* McsEngl.combinatorics!⇒mathCombinatorics,
* McsEngl.mathCombinatorics,
* McsEngl.mathCombinatorics!=science.math.combinatorics,
* McsEngl.sciMath.022-combinatorics!⇒mathCombinatorics,
* McsEngl.science.math.combinatorics!⇒mathCombinatorics,
sciMath.computational
description::
"computational mathematics overview
Computational mathematics is a branch of mathematics that focuses on using numerical methods, algorithms, and computer simulations to solve mathematical problems, analyze mathematical models, and make predictions. It plays a crucial role in various fields such as science, engineering, finance, and computer science. Here's an overview of computational mathematics:
1. Numerical Analysis: Numerical analysis is at the heart of computational mathematics. It involves the development and analysis of algorithms for solving mathematical problems that are difficult or impossible to solve analytically. Numerical methods are used to approximate solutions to problems involving calculus, linear algebra, differential equations, and more.
2. Approximation Theory: Approximation theory deals with finding approximate solutions to mathematical problems. It includes techniques for approximating functions, data interpolation, and fitting curves to data points.
3. Linear Algebra: Linear algebra is fundamental to many computational mathematics techniques. Matrix operations, eigenvalue problems, and solutions to linear systems of equations are essential in various applications, including computer graphics, data analysis, and optimization.
4. Differential Equations: Differential equations describe dynamic systems and change over time. Computational mathematics provides methods for solving ordinary and partial differential equations numerically, which is crucial in physics, engineering, and many other fields.
5. Optimization: Optimization involves finding the best solution from a set of possible solutions. Computational mathematics is used to develop optimization algorithms for various applications, such as optimizing financial portfolios, network routing, and machine learning models.
6. Statistics and Data Analysis: Computational mathematics plays a significant role in statistics and data analysis. It includes techniques for data visualization, hypothesis testing, regression analysis, and machine learning algorithms for predictive modeling.
7. Numerical Simulation: Numerical simulation involves creating computer models to simulate real-world phenomena. Computational mathematics is used to design and run simulations in fields such as physics, engineering, climate science, and biology.
8. Computational Complexity: Computational mathematics also deals with the study of computational complexity, which assesses the resources (time and memory) required to solve mathematical problems. This is essential for evaluating the efficiency of algorithms and determining if a problem is tractable or intractable.
9. Software Development: Developing numerical software libraries and tools is a critical aspect of computational mathematics. These libraries provide pre-built functions and algorithms that researchers, engineers, and scientists can use to solve mathematical problems efficiently.
10. High-Performance Computing: Computational mathematics often involves high-performance computing (HPC) techniques to solve complex problems that require substantial computational resources. Supercomputers, parallel computing, and distributed computing are used to tackle large-scale simulations and calculations.
11. Applications: Computational mathematics has a wide range of applications, including solving physical and engineering problems, optimizing supply chains, predicting weather patterns, simulating biological processes, and developing artificial intelligence algorithms.
In summary, computational mathematics is a multidisciplinary field that bridges mathematics, computer science, and various scientific and engineering disciplines. It empowers researchers and practitioners to solve complex problems, make data-driven decisions, and model real-world phenomena with the aid of computational tools and techniques."
[{2023-10-05 retrieved} https://chat.openai.com/c/0d0d7049-a861-4e04-b5c8-5f744aaf2d5e]
name::
* McsEngl.computational-mathematics,
* McsEngl.sciMath.030-computational,
* McsEngl.sciMath.computational,
descriptionLong::
"Computational mathematics is an area of mathematics devoted to the interaction between mathematics and computer computation.[1]
A large part of computational mathematics consists roughly of using mathematics for allowing and improving computer computation in areas of science and engineering where mathematics are useful. This involves in particular algorithm design, computational complexity, numerical methods and computer algebra.
Computational mathematics refers also to the use of computers for mathematics itself. This includes mathematical experimentation for establishing conjectures (particularly in number theory), the use of computers for proving theorems (for example the four color theorem), and the design and use of proof assistants.
Areas of computational mathematics
Computational mathematics emerged as a distinct part of applied mathematics by the early 1950s. Currently, computational mathematics can refer to or include:
* Computational science, also known as scientific computation or computational engineering
* Solving mathematical problems by computer simulation as opposed to analytic methods of applied mathematics
* Numerical methods used in scientific computation, for example numerical linear algebra and numerical solution of partial differential equations
* Stochastic methods,[2] such as Monte Carlo methods and other representations of uncertainty in scientific computation
* The mathematics of scientific computation,[3][4] in particular numerical analysis, the theory of numerical methods
* Computational complexity
* Computer algebra and computer algebra systems
* Computer-assisted research in various areas of mathematics, such as logic (automated theorem proving), discrete mathematics, combinatorics, number theory, and computational algebraic topology
* Cryptography and computer security, which involve, in particular, research on primality testing, factorization, elliptic curves, and mathematics of blockchain
* Computational linguistics, the use of mathematical and computer techniques in natural languages
* Computational algebraic geometry
* Computational group theory
* Computational geometry
* Computational number theory
* Computational topology
* Computational statistics
* Algorithmic information theory
* Algorithmic game theory
* Mathematical economics, the use of mathematics in economics, finance and, to certain extents, of accounting.
* Experimental mathematics"
[{2023-10-06 retrieved} https://en.wikipedia.org/wiki/Computational_mathematics]
sciMath.cryptography
description::
× generic: mathDiscrete,
"The parent field of cryptography is mathematics.
Cryptography is a branch of mathematics that studies the principles and practices of secure communication in the presence of third parties called adversaries. It encompasses a wide range of topics, including cryptographic algorithms, cryptographic protocols, cryptographic analysis, and cryptographic applications.
Cryptography is based on the principle of using mathematical algorithms to transform information into a format that is unreadable to unauthorized individuals. This process is called encryption. The encrypted information can then be transmitted or stored securely. To make the information readable again, it must be decrypted using the same mathematical algorithm and the appropriate cryptographic key.
Many cryptographic algorithms are based on complex mathematical problems, such as the difficulty of factoring large numbers or the difficulty of finding discrete logarithms. These problems are believed to be difficult to solve even with the most powerful computers available today.
Cryptography is used in a wide variety of applications, including secure communication, data protection, digital signatures, and financial transactions. It is an essential tool for protecting information in the digital age.
Here are some specific examples of how mathematics is used in cryptography:
* Number theory: Number theory is the study of the properties of numbers. It is used in many cryptographic algorithms, such as RSA, Diffie-Hellman, and Elliptic curve cryptography.
* Algebra: Algebra is the study of mathematical symbols and the rules for manipulating them. It is used in many cryptographic algorithms, such as DES and AES.
* Probability and statistics: Probability and statistics are the study of randomness and uncertainty. They are used in many cryptographic algorithms, such as random number generators and digital signatures.
Cryptography is a challenging but rewarding field. It requires a deep understanding of mathematics and computer science. However, the rewards are great, as cryptography is used to protect our most valuable information."
[{2023-10-03 retrieved} https://bard.google.com/chat/6eeac4dbe9b6e921]
name::
* McsEngl.cryptography,
* McsEngl.sciMath.026-cryptography,
* McsEngl.sciMath.cryptography,
evoluting of cryptography
description::
"evolution of cryptography:
Cryptography is the practice of protecting information from unauthorized access, use, disclosure, disruption, modification, or destruction. It is the art and science of creating secret codes and ciphers to protect communications and data.
The evolution of cryptography has been driven by the need to protect increasingly sensitive information and the development of increasingly sophisticated computing technologies.
Early cryptography
The earliest forms of cryptography were simple substitution ciphers, in which each letter in the original message is replaced with another letter according to a secret key. For example, in a Caesar cipher, each letter is shifted three places to the right.
Substitution ciphers were relatively easy to break, even in ancient times. To improve security, cryptographers developed more complex ciphers, such as polyalphabetic ciphers and transposition ciphers.
Modern cryptography
In the 20th century, the development of computers led to a revolution in cryptography. New encryption algorithms were developed that were much more difficult to break than traditional ciphers.
One of the most important developments in modern cryptography was the invention of public key cryptography. Public key cryptography uses two different keys, a public key and a private key. The public key can be shared with anyone, but the private key must be kept secret.
Public key cryptography is used in a wide variety of applications, such as secure communication, digital signatures, and electronic commerce.
Cryptography in the digital age
Cryptography is essential for protecting our data in the digital age. It is used to secure our online communications, protect our financial information, and keep our personal data safe.
Some of the most common cryptographic algorithms used today include:
* AES (Advanced Encryption Standard)
* RSA (Rivest–Shamir–Adleman)
* ECC (Elliptic Curve Cryptography)
These algorithms are used to protect a wide range of data, from email messages to credit card transactions.
The future of cryptography
Cryptography is a rapidly evolving field. Researchers are constantly developing new and more sophisticated encryption algorithms.
One of the most exciting areas of research is quantum cryptography. Quantum cryptography uses the principles of quantum mechanics to create encryption algorithms that are unbreakable by any known computer.
Quantum cryptography is still in its early stages of development, but it has the potential to revolutionize the way we protect our data in the future.
Conclusion
Cryptography has played a vital role in human history, from protecting military secrets to securing online commerce. As technology continues to evolve, so too will cryptography. New and more sophisticated encryption algorithms will be developed to protect our data in the digital age."
[{2023-10-13 retrieved} https://bard.google.com/chat/54eab205159ec38b]
name::
* McsEngl.evoluting-of-cryptography,
* McsEngl.cryptography'evoluting,
sciMath.game-theory
description::
"Game theory is the study of mathematical models of strategic interactions among rational agents.[1] It has applications in all fields of social science, as well as in logic, systems science and computer science. The concepts of game theory are used extensively in economics as well.[2] The traditional methods of game theory addressed two-person zero-sum games, in which each participant's gains or losses are exactly balanced by the losses and gains of other participants. In the 21st century, the advanced game theories apply to a wider range of behavioral relations; it is now an umbrella term for the science of logical decision making in humans, animals, as well as computers.
Modern game theory began with the idea of mixed-strategy equilibria in two-person zero-sum game and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was followed by the 1944 book Theory of Games and Economic Behavior, co-written with Oskar Morgenstern, which considered cooperative games of several players.[3] The second edition of this book provided an axiomatic theory of expected utility, which allowed mathematical statisticians and economists to treat decision-making under uncertainty. Therefore, it is evident that game theory has evolved over time with consistent efforts of mathematicians, economists and other academicians.[citation needed]
Game theory was developed extensively in the 1950s by many scholars. It was explicitly applied to evolution in the 1970s, although similar developments go back at least as far as the 1930s. Game theory has been widely recognized as an important tool in many fields. As of 2020, with the Nobel Memorial Prize in Economic Sciences going to game theorists Paul Milgrom and Robert B. Wilson, fifteen game theorists have won the economics Nobel Prize. John Maynard Smith was awarded the Crafoord Prize for his application of evolutionary game theory."
[{2023-10-06 retrieved} https://en.wikipedia.org/wiki/Game_theory]
name::
* McsEngl.game-theory,
* McsEngl.sciMath.034-game-theory,
* McsEngl.sciMath.game-theory,
descriptionLong::
"Game theory is a branch of mathematics and economics that studies the strategic interactions between rational decision-makers, often referred to as "players" or "agents." It provides a framework for analyzing and understanding how individuals or entities make decisions when their actions affect not only their own outcomes but also the outcomes of others involved in the same situation. Game theory has applications in various fields, including economics, political science, biology, and computer science.
Here are some key concepts and components of game theory:
1. Players: Game theory typically involves two or more players, each of whom makes choices or decisions that impact the overall outcome of the game.
2. Strategies: Players choose from a set of possible strategies, which represent their courses of action. The strategies available to each player can be finite or infinite, depending on the specific game.
3. Payoffs: Players receive payoffs or outcomes as a result of their chosen strategies and the strategies chosen by others. Payoffs represent the players' preferences or utility, and they can be represented numerically.
4. Normal Form and Extensive Form: Game theory can be represented in two main forms. The normal form (also known as the strategic form) lists all possible combinations of strategies and their associated payoffs in a matrix. The extensive form represents the game as a tree, showing the sequence of moves and decisions made by players over time.
5. Zero-Sum vs. Non-Zero-Sum: Games can be categorized as zero-sum or non-zero-sum. In a zero-sum game, one player's gain is always matched by another player's loss. Non-zero-sum games allow for the possibility of mutually beneficial outcomes.
6. Nash Equilibrium: A fundamental concept in game theory is the Nash equilibrium, named after John Nash. It is a set of strategies in which no player can improve their payoff by unilaterally changing their strategy, assuming all other players' strategies remain unchanged.
7. Dominant Strategy: A dominant strategy is a strategy that always yields a higher payoff for a player, regardless of the strategies chosen by other players.
8. Cooperative Games: In some cases, players may choose to cooperate and form coalitions to achieve better outcomes. Cooperative game theory studies how players can distribute the gains from cooperation.
9. Game Applications: Game theory is applied in various real-world scenarios, such as pricing strategies in economics, negotiation tactics in business, voting behavior in politics, and evolutionary biology to model animal behavior.
10. Examples: Some classic examples of games in game theory include the Prisoner's Dilemma, the Battle of the Sexes, and the Ultimatum Game.
Game theory helps analyze decision-making in situations where there is interdependence among actors and can provide insights into optimal strategies and potential outcomes. It has numerous practical applications and continues to be an essential tool in understanding and modeling strategic interactions."
[{2023-10-06 retrieved} https://chat.openai.com/c/998cbf83-a349-4a17-8a86-d5c067a6e6ea]
evoluting of game-theory
description::
"evolution of the game theory:
The evolution of game theory as a field of study has been a fascinating journey with contributions from various disciplines and scholars over time. Here is a brief overview of the key milestones in the development and evolution of game theory:
1. Early Roots (18th-19th Century): Game theory's origins can be traced back to the 18th-century work of mathematicians and economists like Daniel Bernoulli and Antoine Cournot, who explored concepts related to decision-making, probability, and strategic interactions. However, the formalization of game theory began in the 19th century with the work of mathematician Émile Borel, who introduced the concept of games with mixed strategies.
2. Early Game Theory Models (20th Century): Game theory took significant strides in the early 20th century, particularly with the groundbreaking work of mathematician John von Neumann and economist Oskar Morgenstern. Their book "Theory of Games and Economic Behavior" (1944) laid the foundation for modern game theory. They introduced concepts like utility theory, zero-sum games, and the minimax theorem, which were critical in advancing the field.
3. Evolution of Nash Equilibrium (1950s): John Nash, a mathematician, made a groundbreaking contribution with his development of Nash equilibria in the 1950s. He demonstrated that in non-cooperative games, there exists at least one set of strategies where no player has an incentive to unilaterally change their strategy, thus forming a stable point of equilibrium. Nash's work earned him the Nobel Prize in Economics in 1994.
4. Evolutionary Game Theory (1970s): The 1970s saw the emergence of evolutionary game theory, which applied principles from game theory to biology and evolution. Researchers like John Maynard Smith and George R. Price used game theory to explain the evolution of altruism, cooperation, and other behaviors in the context of natural selection.
5. Behavioral Game Theory (1980s-1990s): Behavioral economics and experimental studies led to a growing interest in understanding how individuals deviate from purely rational decision-making. Researchers like Robert Axelrod explored the evolution of cooperation in repeated games, while others focused on real-world applications of game theory in fields such as politics and negotiation.
6. Game Theory Applications (Contemporary): Game theory continues to find applications in various fields, including economics, political science, biology, psychology, computer science, and more. It is used to analyze complex interactions in market competition, voting systems, climate change negotiations, and even artificial intelligence and machine learning.
7. Computational Advances: Advances in computational power and techniques have allowed researchers to analyze more complex game situations and conduct simulations that provide insights into strategic decision-making.
8. Nobel Prizes: Several Nobel Prizes in Economics have been awarded for work related to game theory, including those to John Nash, Reinhard Selten, Robert Aumann, and Alvin E. Roth.
The evolution of game theory has been marked by a continuous interplay between mathematics, economics, psychology, biology, and other disciplines. It has grown from its early foundations into a multidisciplinary field that offers valuable insights into strategic decision-making, cooperation, conflict resolution, and the dynamics of complex systems in various contexts. As our understanding of human behavior and the interactions between rational agents continues to evolve, game theory remains a powerful tool for analysis and prediction."
[{2023-10-06 retrieved} https://chat.openai.com/c/998cbf83-a349-4a17-8a86-d5c067a6e6ea]
name::
* McsEngl.evoluting-of-game-theory,
* McsEngl.game-theory'evoluting,
sciMath.geometry
description::
× generic: mathAppliedNo.
· "Geometry (from Ancient Greek γεωμετρία (geōmetrνa) 'land measurement'; from γῆ (gκ) 'earth, land', and μέτρον (mιtron) 'a measure')[citation needed] is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures.[1] Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a geometer. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry,[a] which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts.[2]
Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics.[3] Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained unsolved for several centuries.
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically, that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries without the parallel postulate (non-Euclidean geometries) can be developed without introducing any contradiction. The geometry that underlies general relativity is a famous application of non-Euclidean geometry.
Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry (also known as combinatorial geometry), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits the concept of angle and distance, finite geometry that omits continuity, and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined."
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Geometry]
name::
* McsEngl.geometry-sciMath,
* McsEngl.mathGeometry,
* McsEngl.mathGeometry!=math.geometry,
* McsEngl.sciMath.007-geometry,
* McsEngl.sciMath.geometry,
GENERIC-SPECIFIC-TREE of mathGeometry
generic-tree-of-mathGeometry::
* mathAppliedNo,
* ... entity,
* McsEngl.mathGeometry'generic-tree,
specific-tree-of-mathGeometry::
* Euclidean geometry,
* non-Euclidean geometry,
* differential geometry,
* algebraic geometry,
* McsEngl.mathGeometry.specific-tree,
mathGeometry.analytic
description::
· "In mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry.
Analytic geometry is used in physics and engineering, and also in aviation, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including algebraic, differential, discrete and computational geometry.
Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations. That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor–Dedekind axiom."
[{2023-08-19 retrieved} https://en.wikipedia.org/wiki/Analytic_geometry]
name::
* McsEngl.analytic-geometry,
* McsEngl.mathGeometry.analytic,
mathGeometry.synthetic
description::
· "Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms.
The term "synthetic geometry" has been coined only after the 17th century, and the introduction by Renι Descartes of the coordinate method, which was called analytic geometry. So the term "synthetic geometry" was introduced to refer to the older methods that were, before Descartes, the only known ones."
[{2023-08-19 retrieved} https://en.wikipedia.org/wiki/Synthetic_geometry]
name::
* McsEngl.mathGeometry.synthetic,
* McsEngl.synthetic-geometry,
sciMath.graph-theory (link)
sciMath.logic
description::
× generic: mathAppliedNo.
"Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories. Results of Kurt Gφdel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics) rather than trying to find theories in which all of mathematics can be developed."
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Mathematical_logic]
name::
* McsEngl.logic//sciMath!⇒mathLogic,
* McsEngl.logicMath!⇒mathLogic,
* McsEngl.mathLogic,
* McsEngl.mathLogic!⇒science.math.logic,
* McsEngl.mathematical-logic!⇒mathLogic,
* McsEngl.sciMath.009-logic!⇒mathLogic,
* McsEngl.sciMath.logic!⇒mathLogic,
evoluting of mathLogic
description::
"evolution of mathematical logic:
The evolution of mathematical logic is a fascinating journey that spans centuries and has significantly impacted the foundations of mathematics and philosophy. Here's a brief overview of some key developments in the history of mathematical logic:
1. Ancient Greece:
- The roots of mathematical logic can be traced back to ancient Greece, particularly with the works of philosophers like Pythagoras and Euclid. They developed foundational ideas in geometry and number theory, which can be seen as early forms of logical reasoning.
2. Aristotle (384-322 BCE):
- Aristotle made significant contributions to logic, particularly in his works "Organon." He developed a system of syllogistic logic, which formed the basis for deductive reasoning for centuries. His work laid the groundwork for propositional and predicate logic.
3. Middle Ages:
- During the Middle Ages, the works of Aristotle were preserved and studied by scholars in the Islamic world and Europe. This period saw the development of commentaries and interpretations of Aristotle's logical works.
4. Early Modern Period:
- The Renaissance and the Enlightenment brought renewed interest in logic. Scholars like René Descartes, Gottfried Wilhelm Leibniz, and George Boole made significant contributions. Leibniz, in particular, is known for developing a binary system of mathematical logic and the idea of a universal characteristic (a universal language of symbols).
5. George Boole (1815-1864):
- Boole is often considered the father of modern mathematical logic. He introduced symbolic algebraic methods for representing and manipulating logical statements, which led to the development of Boolean algebra. His work is foundational for digital circuit design and computer science.
6. Augustus De Morgan (1806-1871):
- De Morgan made important contributions to symbolic logic and set theory, and he introduced De Morgan's Laws, which describe the relationships between logical operations (AND, OR, NOT).
7. Gottlob Frege (1848-1925):
- Frege's work on predicate logic and his book "Begriffsschrift" (Concept Script) is considered a turning point in the development of modern formal logic. He introduced the concept of quantification and developed a notation that influenced later logicians, including Bertrand Russell and Alfred North Whitehead.
8. Principia Mathematica:
- Bertrand Russell and Alfred North Whitehead collaborated on "Principia Mathematica," a monumental three-volume work published between 1910 and 1913. They aimed to derive all of classical mathematics from a small number of logical axioms using symbolic notation.
9. Kurt Gödel (1906-1978):
- Gödel's incompleteness theorems, proved in the early 20th century, showed that there are inherent limitations in formal mathematical systems, and no single system can capture all mathematical truth. His work had a profound impact on the philosophy of mathematics and logic.
10. Alan Turing (1912-1954):
- Turing is known for his pioneering work in computer science and the development of the Turing machine, which laid the foundation for the theory of computation. This work had a significant impact on the intersection of logic, mathematics, and computing.
11. Modern Logic:
- Since the mid-20th century, mathematical logic has continued to evolve, with developments in modal logic, model theory, set theory, and non-classical logics. Researchers have explored the limits and applications of formal logic in various fields, including artificial intelligence, linguistics, and philosophy.
The evolution of mathematical logic has been marked by a continuous refinement of symbolic notation, the development of new logical systems, and deep philosophical exploration of the nature of mathematical truth and reasoning. It has played a crucial role in shaping the way we understand and use mathematics and has influenced a wide range of disciplines."
[{2023-10-12 retrieved} https://chat.openai.com/c/e73a5f5b-397b-4612-98c0-db6f187eab2e]
name::
* McsEngl.evoluting-of-mathLogic,
* McsEngl.mathLogic'evoluting,
GENERIC-SPECIFIC-TREE of mathLogic
generic-tree-of-mathLogic::
* mathAppliedNo,
* mathLogic'generic,
specific-tree-of-mathLogic::
* Propositional logic,
* predicate logic,
* modal logic,
* Model theory,
* Proof theory,
* Set theory,
* Type theory,
* Recursion theory,
* Theory of Computation,
* mathLogic.specific,
mathLogic.computability
description::
"Computability logic (CoL) is a research program and mathematical framework for redeveloping logic as a systematic formal theory of computability, as opposed to classical logic which is a formal theory of truth. It was introduced and so named by Giorgi Japaridze in 2003.[1]
In classical logic, formulas represent true/false statements. In CoL, formulas represent computational problems. In classical logic, the validity of a formula depends only on its form, not on its meaning. In CoL, validity means being always computable. More generally, classical logic tells us when the truth of a given statement always follows from the truth of a given set of other statements. Similarly, CoL tells us when the computability of a given problem A always follows from the computability of other given problems B1,...,Bn. Moreover, it provides a uniform way to actually construct a solution (algorithm) for such an A from any known solutions of B1,...,Bn.
CoL formulates computational problems in their most general – interactive sense. CoL defines a computational problem as a game played by a machine against its environment. Such a problem is computable if there is a machine that wins the game against every possible behavior of the environment. Such a game-playing machine generalizes the Church-Turing thesis to the interactive level. The classical concept of truth turns out to be a special, zero-interactivity-degree case of computability. This makes classical logic a special fragment of CoL. Thus CoL is a conservative extension of classical logic. Computability logic is more expressive, constructive and computationally meaningful than classical logic. Besides classical logic, independence-friendly (IF) logic and certain proper extensions of linear logic and intuitionistic logic also turn out to be natural fragments of CoL.[2][3] Hence meaningful concepts of "intuitionistic truth", "linear-logic truth" and "IF-logic truth" can be derived from the semantics of CoL.
CoL systematically answers the fundamental question of what can be computed and how; thus CoL has many applications, such as constructive applied theories, knowledge base systems, systems for planning and action. Out of these, only applications in constructive applied theories have been extensively explored so far: a series of CoL-based number theories, termed "clarithmetics", have been constructed[4][5] as computationally and complexity-theoretically meaningful alternatives to the classical-logic-based Peano arithmetic and its variations such as systems of bounded arithmetic.
Traditional proof systems such as natural deduction and sequent calculus are insufficient for axiomatizing nontrivial fragments of CoL. This has necessitated developing alternative, more general and flexible methods of proof, such as cirquent calculus.[6][7]"
[{2023-10-13 retrieved} https://en.wikipedia.org/wiki/Computability_logic]
name::
* McsEngl.computability-logic,
* McsEngl.mathLogic.computability,
* McsEngl.sciMath.046computability-logic,
* McsEngl.sciMath.computability-logic,
mathLogic.intuitionistic
description::
"overview of intuitionistic-logic:
Intuitionistic logic, also known as constructive logic, is a branch of mathematical logic and philosophy of mathematics that was developed as a response to classical (or classical) logic. It was first formulated by the Dutch mathematician and philosopher L.E.J. Brouwer in the early 20th century. Intuitionistic logic is distinct from classical logic in several key ways:
1. Rejection of the Law of Excluded Middle: In classical logic, the Law of Excluded Middle (LEM) states that for any proposition, either the proposition or its negation is true, and there are no other possibilities. In contrast, intuitionistic logic rejects LEM. In intuitionistic logic, a proposition is only considered true if there is constructive evidence or a proof for it. If there is no such evidence, the proposition is neither true nor false; it is considered "undetermined."
2. Rejection of Double Negation Elimination: In classical logic, the principle of double negation elimination allows you to infer a statement from its double negation (¬¬A implies A). Intuitionistic logic does not accept this principle. In intuitionistic logic, you cannot necessarily infer a statement from its double negation without additional proof.
3. Emphasis on Constructive Proofs: Intuitionistic logic places a strong emphasis on constructive reasoning and the idea that mathematical truths must be established through constructive proofs. This means that to assert the existence of an object, one must provide a method or a construction to obtain it.
4. Intuitionistic Truth Table: The truth table for intuitionistic logic is different from classical logic. In classical logic, the truth values are either true or false, but in intuitionistic logic, there is a third value, "undetermined," which indicates the lack of evidence for a proposition.
5. No Use of Proof by Contradiction: In classical logic, proof by contradiction is a common method, where you assume the opposite of what you want to prove and then derive a contradiction. Intuitionistic logic restricts the use of proof by contradiction. If you want to prove ¬A, you must provide direct evidence for it rather than relying on a contradiction.
Intuitionistic logic has applications in various areas of mathematics, particularly in constructive mathematics and computer science, where the focus on constructive reasoning aligns with the design and verification of computer programs. It also has philosophical implications, as it challenges traditional views of mathematical truth and the nature of logical reasoning.
Overall, intuitionistic logic offers an alternative perspective on logic and mathematics that differs significantly from classical logic, emphasizing the constructive and evidence-based nature of mathematical reasoning."
[{2023-10-13 retrieved} https://chat.openai.com/c/1041e5d0-3360-4776-9b8f-35a0e7b602dc]
name::
* McsEngl.constructive-logic,
* McsEngl.intuitionistic-logic,
* McsEngl.logicIntuitionistic,
* McsEngl.mathLogic.intuitionistic,
* McsEngl.sciMath.044-intuitionistic,
* McsEngl.sciMath.intuitionistic,
mathLogic.lambda-calculus
description::
"Lambda calculus is a formal system in mathematical logic and computer science for expressing computation based on function abstraction and application. It was introduced by mathematician Alonzo Church in the 1930s as a way to explore the foundations of mathematics and computation. Lambda calculus serves as a foundation for functional programming languages and plays a fundamental role in the theory of computation.
The key elements of lambda calculus are:
1. Lambda Abstraction: Lambda calculus uses the symbol λ (lambda) to denote function abstraction. A lambda abstraction defines an anonymous function with a formal parameter. For example, λx.x represents a function that takes an argument x and returns x.
2. Function Application: In lambda calculus, you can apply a function to an argument. Application is denoted by placing the function and argument next to each other without any separators. For example, (λx.x) y represents the application of the function λx.x to the argument y, resulting in y.
3. Variables: Variables are used to represent values that can be passed as arguments or returned as results from functions. Variables are typically single letters, such as x, y, or z.
4. Reduction: Lambda calculus uses reduction rules to simplify expressions. The most common reduction rule is called beta reduction, which involves substituting the argument into the body of a function where its formal parameter occurs. For example, (λx.x) y reduces to y.
Lambda calculus is a simple and minimalistic formalism, but it has the expressive power to represent arbitrary computations. It is Turing complete, meaning that any computation that can be carried out by a Turing machine can also be expressed in lambda calculus. This makes it a fundamental tool in the study of computability and the foundations of programming languages.
Functional programming languages like Lisp, Haskell, and ML are influenced by lambda calculus and use its principles to define functions and perform computations. Lambda calculus also plays a role in type theory, formal semantics of programming languages, and the study of program correctness."
[{2023-10-08 retrieved} https://chat.openai.com/c/ce3ac68f-1498-4dda-84c1-d50e10d6ce3b]
name::
* McsEngl.lambda-calculus:mathLogic,
* McsEngl.mathLogic.lambda-calculus,
* McsEngl.sciMath.039-lambda-calculus,
* McsEngl.sciMath.lambda-calculus,
* McsEngl.λ-calculus,
====== langoGreek:
* McsElln.λογισμός-λάμδα,
mathLogic.linear
description::
"overview of linear-logic:
Linear logic is a type of logic that was developed in the 1980s by Jean-Yves Girard as a refinement of classical and intuitionistic logic. It has applications in various fields, including computer science, linguistics, and philosophy. Linear logic differs from classical logic in its treatment of propositions and the way it handles the concept of truth. Here is an overview of some key concepts and features of linear logic:
1. Propositions: In linear logic, propositions are treated as resources that can be used and consumed. Unlike classical logic, where propositions are considered true or false and can be freely duplicated or discarded, linear logic introduces the idea of "linear propositions." Linear propositions represent resources that can be used exactly once. This is in contrast to classical propositions, which can be used repeatedly without restriction.
2. Connectives: Linear logic has a different set of logical connectives, including "tensor," "par," "with," "lollipop," and "bang." These connectives are designed to capture the notion of resource management and the controlled use of propositions.
- Tensor (⊗): This represents the combination or conjunction of two resources.
- Par (⊕): This represents a choice between two resources.
- With (&): This represents the simultaneous availability of two resources.
- Lollipop (⊸): This represents the implication or the use of one resource to obtain another.
- Bang (!): This indicates that a resource is persistent and can be used multiple times.
3. Weakening and Contraction: Linear logic introduces weakening (the ability to discard a resource) and contraction (the ability to duplicate a resource) as controlled operations. These operations are not freely allowed but must be explicitly managed in a proof.
4. Proofs: Proofs in linear logic are constructed to account for the linear use of resources. This requires careful management of resources and the explicit tracking of how they are used and combined. Proofs in linear logic are often structured differently from classical logic proofs and are more suitable for modeling processes and resource management.
5. Applications: Linear logic has found applications in various areas, including computer science (e.g., in programming language semantics, type theory, and resource-aware computation), linguistics (for modeling meaning and discourse), and philosophy (for addressing issues related to the nature of truth and the use of language).
6. Substructural Logic: Linear logic is a substructural logic, which means that it modifies the structural rules governing logical operations. In addition to weakening and contraction, linear logic often uses a limited form of exchange, where the order of propositions matters.
Overall, linear logic provides a formal framework for reasoning about resource allocation and management, making it valuable in fields where resource consumption and interaction are essential aspects of the analysis, such as computer science and linguistics."
[{2023-10-13 retrieved} https://chat.openai.com/c/2cfe2719-7823-473a-8f84-20d2b2446b09]
name::
* McsEngl.linear-logic,
* McsEngl.mathLogic.linear,
* McsEngl.sciMath.045-linear-logic,
* McsEngl.sciMath.linear-logic,
mathLogic.model-theory
description::
"In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).[1] The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954.[2] Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics. This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane".[3] The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic."
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Model_theory]
name::
* McsEngl.mathModelth,
* McsEngl.mathModel_theory,
* McsEngl.model-theory-sciMath,
* McsEngl.logicModel_theory,
* McsEngl.sciMath.011-model-theory,
* McsEngl.sciMath.model-theory,
evoluting of logicModel_theory
description::
"evolution of model-theory:
The evolution of model theory is a complex and multifaceted development in the field of mathematical logic and the broader area of mathematics. Model theory is concerned with the relationships between mathematical structures and the first-order logic expressions that describe them. It has undergone significant developments and refinements since its inception. Here's a brief overview of the key stages in the evolution of model theory:
1. Early Foundations:
- The origins of model theory can be traced back to the late 19th and early 20th centuries, with the work of mathematicians like Gottlob Frege and Alfred North Whitehead and Bertrand Russell in the development of formal logic and set theory.
- Early efforts, such as Alfred Tarski's work on elementary geometry and the quantifier elimination for real closed fields, laid the foundation for modern model theory.
2. Tarski's Contributions:
- Alfred Tarski is often regarded as one of the founding figures of model theory. His work on the theory of models and the concept of elementary equivalence was groundbreaking.
- Tarski's most famous result, the Tarski-Vaught test, provided a criterion for determining when a structure can be expanded to a larger structure while preserving elementary equivalence.
3. Stability Theory:
- Around the mid-20th century, the Israeli mathematician Saharon Shelah introduced the concept of stability theory.
- Stability theory focused on classifying first-order theories based on their model-theoretic properties, including the notion of stability and the number of types.
- This development greatly advanced our understanding of first-order theories and their structural properties.
4. Classification of Theories:
- Shelah and his contemporaries made significant progress in classifying first-order theories into different "stability classes."
- This classification helped mathematicians understand the complexity and richness of first-order logic.
5. Model Theory of Fields:
- Model theory has been extensively applied to algebraic structures like fields. Model theorists like Anand Pillay and H. Dugald Macpherson made significant contributions to the model theory of fields, including the study of algebraically closed fields, differentially closed fields, and more.
6. Shelah's Categoricity Theorem:
- Saharon Shelah's work on categoricity results for complete first-order theories was another major milestone. He developed the notion of "superstability" and showed that for certain theories, categoricity implies stability.
7. Geometric Model Theory:
- Geometric model theory, developed by Boris Zilber and others, is a more recent subfield that applies model theory to the study of algebraic and geometric structures.
- This approach has had a profound impact on the model theory of fields and motivic integration, among other areas.
8. Contemporary Research:
- Model theory continues to evolve, with contemporary research spanning a wide range of topics, including o-minimality, finite model theory, and applications to algebraic geometry and number theory.
The evolution of model theory has been characterized by a combination of foundational work and the exploration of specific mathematical structures and concepts. It has led to insights into the relationships between logic, algebra, and geometry and has found applications in various branches of mathematics. This field remains an active and dynamic area of research with ongoing developments and discoveries."
[{2023-10-12 retrieved} https://chat.openai.com/c/b034b780-2da9-442c-a7a7-6890bfba3341]
name::
* McsEngl.logicModel_theory'evolution,
mathLogic.probabilistic-logic
description::
"Probabilistic logic (also probability logic and probabilistic reasoning) involves the use of probability and logic to deal with uncertain situations. Probabilistic logic extends traditional logic truth tables with probabilistic expressions. A difficulty of probabilistic logics is their tendency to multiply the computational complexities of their probabilistic and logical components. Other difficulties include the possibility of counter-intuitive results, such as in case of belief fusion in Dempster–Shafer theory. Source trust and epistemic uncertainty about the probabilities they provide, such as defined in subjective logic, are additional elements to consider. The need to deal with a broad variety of contexts and issues has led to many different proposals."
[{2023-10-08 retrieved} https://en.wikipedia.org/wiki/Probabilistic_logic]
name::
* McsEngl.logicPobabilistic,
* McsEngl.mathLogic.probabilistic-logic,
* McsEngl.probabilistic-logic,
* McsEngl.sciMath.040-probabilistic-logic,
* McsEngl.sciMath.probabilistic-logic,
evoluting of logicPobabilistic
description::
"Probabilistic logic, also known as probabilistic reasoning or probabilistic inference, is a field that combines elements of probability theory and formal logic to handle uncertainty in reasoning and decision-making. The evolution of probabilistic logic has been shaped by developments in both probability theory and artificial intelligence. Here's a brief overview of its evolution:
1. Early Roots:
- Probability theory has a long history dating back to the 17th century, with contributions from mathematicians like Blaise Pascal and Pierre-Simon Laplace.
- Early philosophers and logicians, such as Thomas Bayes, made connections between probability and logic.
2. Bayesian Probability and Logic:
- Thomas Bayes's work on conditional probability and Bayes' theorem laid the foundation for probabilistic reasoning. Bayes' theorem is central to Bayesian probability, which is a key component of probabilistic logic.
- In the 20th century, Bayesian probability was further developed and applied to various fields, including statistics and artificial intelligence.
3. Early AI and Uncertainty:
- In the early days of artificial intelligence (AI), researchers faced challenges in representing and reasoning with uncertainty.
- Probability theory was integrated into AI to deal with uncertain information. This led to the development of probabilistic models for reasoning, such as Bayesian networks.
4. Probabilistic Logic Programming:
- Probabilistic logic programming languages, such as Prolog-based languages extended with probability distributions, began to emerge in the late 20th century.
- Systems like PRISM, ICL, and ProbLog allowed for the representation and inference of probabilistic knowledge in a logical framework.
5. Markov Logic Networks:
- Markov Logic Networks (MLNs), introduced by Domingos and Richardson in the mid-2000s, combine first-order logic and probabilistic graphical models (Markov networks).
- MLNs enable the integration of logic rules with uncertain probabilities, making them a powerful tool for various AI applications.
6. Statistical Relational Learning (SRL):
- SRL is an interdisciplinary field that combines statistical and logical reasoning to handle uncertainty in relational data.
- SRL approaches often involve the use of probabilistic graphical models and first-order logic for knowledge representation and inference.
7. Deep Probabilistic Models:
- Recent advances in deep learning have led to the development of deep probabilistic models, such as Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs) with probabilistic extensions.
- These models combine deep neural networks with probabilistic reasoning to handle uncertainty in complex data.
8. Probabilistic Programming Languages:
- Probabilistic programming languages like Pyro, Stan, and Edward have gained popularity, allowing researchers and developers to build probabilistic models using a programming language and perform probabilistic inference.
The evolution of probabilistic logic has been marked by a growing interest in combining formal logic with probability theory to reason in uncertain and complex domains. This has found applications in fields like AI, machine learning, natural language processing, robotics, and more, where handling uncertainty is crucial for making informed decisions and predictions."
[{2023-10-08 retrieved} https://chat.openai.com/c/bb196a8d-2afc-48b3-86bd-81c74c6d5695]
name::
* McsEngl.evoluting-of-logicPobabilistic,
* McsEngl.logicPobabilistic'evoluting,
mathLogic.proof-theory
description::
proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Model_theory]
"Proof theory is a major branch[1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy."
[{2023-08-26 retrieved} https://en.wikipedia.org/wiki/Proof_theory]
name::
* McsEngl.logicProof_theory,
* McsEngl.mathLogic.proof-theory!=logicProof_theory,
* McsEngl.mathProof_theo!⇒logicProof_theory,
* McsEngl.mathProof_theo!=science.math.logic.proof-theory,
* McsEngl.science.math.logic.proof-theory!⇒logicProof_theory,
evoluting of logicProof_theory
description::
"evolution of proof-theory:
The evolution of proof theory is a fascinating journey through the development of formal systems and techniques for reasoning about the validity of mathematical statements. Proof theory has deep roots in the history of mathematics and has evolved significantly over the centuries. Here's a brief overview of its evolution:
1. Ancient and Classical Antiquity:
- The earliest recorded use of deductive reasoning can be traced back to ancient Greek philosophers like Euclid, who formulated axiomatic systems for geometry in his work "Elements." This laid the foundation for rigorous proofs in mathematics.
2. Medieval Period:
- During the Middle Ages, scholars like Thomas Aquinas used deductive reasoning to integrate Aristotelian philosophy with Christian theology.
3. 17th and 18th Centuries:
- The work of mathematicians like René Descartes and Pierre de Fermat led to the development of coordinate geometry and algebraic methods for proving geometric theorems. This period marked the transition from geometric to algebraic proof techniques.
4. 19th Century:
- Mathematicians such as Augustus De Morgan and George Boole made significant contributions to symbolic logic and algebraic methods for reasoning about mathematical statements. This laid the groundwork for modern logic and proof theory.
5. Late 19th and Early 20th Centuries:
- The work of Gottlob Frege and Bertrand Russell in the late 19th and early 20th centuries laid the foundation for formal logic systems, including predicate logic and set theory. These systems provided a rigorous framework for representing and analyzing mathematical statements.
6. Early 20th Century:
- David Hilbert's program aimed to put mathematics on a secure, axiomatic foundation. This led to the development of proof theory, particularly in the form of axiomatic set theory and first-order logic. Kurt Gödel's incompleteness theorems, published in the 1930s, had a profound impact on proof theory, highlighting the limits of formal systems.
7. Mid-20th Century:
- The development of formal proof calculi and systems, such as natural deduction and the sequent calculus, by logicians like Gerhard Gentzen, greatly advanced proof theory. These calculi provided precise rules for deriving conclusions from premises and were used in the development of automated theorem provers.
8. Late 20th Century to Present:
- Proof theory has continued to evolve with the development of more sophisticated proof assistants and interactive theorem provers, such as Coq, Isabelle, and Lean. These tools allow mathematicians and computer scientists to formally verify complex mathematical proofs and programs.
9. Beyond Mathematics:
- Proof theory has also found applications in computer science, where it plays a crucial role in formal verification and the development of reliable software and hardware systems.
The evolution of proof theory reflects the ongoing pursuit of formalizing and mechanizing mathematical reasoning. It has had a profound impact on various fields, including mathematics, logic, philosophy, and computer science, and continues to be an active area of research and development."
[{2023-10-12 retrieved} https://chat.openai.com/c/ea122d37-8be7-4207-9cf1-9ab6ee042771]
name::
* McsEngl.evoluting-of-logicProof_theory,
* McsEngl.logicProof_theory'evoluting,
mathLogic.propositional
description::
"overview of propositional-logic:
Propositional logic, also known as sentential logic or propositional calculus, is a fundamental branch of mathematical logic that deals with the manipulation and analysis of propositions or statements. It forms the foundation of more complex logical systems and is used in various fields, including mathematics, philosophy, computer science, and artificial intelligence. Here's an overview of propositional logic:
1. Propositions: Propositions are the basic units of propositional logic. They are declarative statements that can be either true or false but not both. Propositions are typically represented by letters (e.g., p, q, r) or symbols.
2. Logical Connectives: Propositional logic uses logical connectives to combine propositions and build more complex statements. The main logical connectives include:
- Conjunction (AND, ∧): The conjunction of two propositions is true only if both propositions are true.
- Disjunction (OR, ∨): The disjunction of two propositions is true if at least one of the propositions is true.
- Negation (NOT, ¬): Negation inverts the truth value of a proposition. If a proposition is true, its negation is false, and vice versa.
- Implication (→): Implication represents "if-then" relationships. It is true unless the antecedent is true and the consequent is false.
- Biconditional (↔): Biconditional, or equivalence, is true when both propositions have the same truth value. It can be thought of as "if and only if."
3. Truth Tables: Truth tables are used to represent the possible truth values of compound propositions for all combinations of truth values of their constituent propositions. Truth tables help in determining the validity of logical arguments and evaluating complex expressions.
4. Logical Laws and Identities: Propositional logic is governed by a set of logical laws and identities that dictate how the logical connectives interact. Some well-known laws include the commutative, associative, and distributive laws, as well as De Morgan's laws.
5. Tautology and Contradiction: A tautology is a proposition that is always true, regardless of the truth values of its constituent propositions. A contradiction is a proposition that is always false. Tautologies and contradictions are used to analyze the validity of arguments and simplify logical expressions.
6. Inference and Deduction: Propositional logic allows for reasoning and deduction. Given a set of premises and logical rules, you can derive conclusions using inference rules like modus ponens, modus tollens, and hypothetical syllogism.
7. Applications: Propositional logic has numerous practical applications, including in computer science (programming, artificial intelligence), formal reasoning, mathematics, and philosophy. It provides a foundation for constructing and analyzing complex logical systems.
8. Limitations: Propositional logic has limitations, as it deals only with the truth values of propositions and does not capture the nuances of natural language. It cannot handle quantifiers (e.g., "for all" and "there exists") and other complex forms of reasoning found in predicate logic and higher-order logics.
Overall, propositional logic is a powerful and foundational system for representing and manipulating simple statements and making logical inferences. It serves as a building block for more advanced logical systems used in various fields."
[{2023-10-12 retrieved} https://chat.openai.com/c/82f59a9e-5ee3-4aa6-8d1e-23ee9a5ebb7b]
name::
* McsEngl.logicPropositional,
* McsEngl.mathLogic.propositional,
* McsEngl.propositional-calculus,
* McsEngl.propositional-logic,
* McsEngl.sciMath.043-ropositional-logic,
* McsEngl.sciMath.propositional-logic,
* McsEngl.sentential-logic,
mathLogic.set-theory
description::
· "Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and the Burali-Forti paradox), various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice.[1] Besides its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science (such as in the theory of relational algebra), philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals."
[{2023-08-16 retrieved} https://en.wikipedia.org/wiki/Set_theory]
name::
* McsEngl.mathLogic.set-theory,
* McsEngl.mathSettheo,
* McsEngl.set-theory!⇒mathSettheo,
* McsEngl.sciMath.017-set-theory,
* McsEngl.sciMath.set-theory,
set of mathSettheo
description::
· "A set is the mathematical model for a collection of different[1] things;[2][3][4] a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets.[5] The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements.[6]
Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.[5]"
[{2023-08-21 retrieved} https://en.wikipedia.org/wiki/Set_(mathematics)]
name::
* McsEngl.Mathsset,
* McsEngl.Mathsset!=math-set,
* McsEngl.Mathsset!=mset,
* McsEngl.mathSettheo'set!=Mathsset,
* McsEngl.mset!=Mathsset,
* McsEngl.set.math!⇒Mathsset,
* McsEngl.setMath!⇒Mathsset,
cardinality of Mathsset
description::
· the-number of its elements.
name::
* McsEngl.Mathsset'cardinality,
* McsEngl.cardinality-of-Mathsset,
Mathsset.empty
description::
· "In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero.[1] Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.
Any set other than the empty set is called non-empty.
In some textbooks and popularizations, the empty set is referred to as the "null set".[1] However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set."
[{2023-08-21 retrieved} https://en.wikipedia.org/wiki/Empty_set]
· Texcode: \O, \emptyset, \varnothing
· Htmlcode: ∅
· Unicode: 8709 U+2205 ∅ EMPTY-SET
· Unicode: 10672 U+29B0 ⦰ REVERSED-EMPTY-SET
name::
* McsEngl.Mathsset.empty {} ∅,
* McsEngl.empty-Mathsset,
* McsEngl.msetEmpty {} ∅,
* McsEngl.∅!=Mathsset,
* McsEngl.⦰!=Mathsset,
Mathsset.singleton
description::
· "In mathematics, a singleton, also known as a unit set[1] or one-point set, is a set with exactly one element. For example, the set \(\{0\}\) is a singleton whose single element is 0."
name::
* McsEngl.msetSingleton,
* McsEngl.singleton//sciMath,
* McsEngl.one-point-set//sciMath,
* McsEngl.unit-set//sciMath,
Mathsset.field
description::
· "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics.
The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements."
[{2023-08-21 retrieved} ]
name::
* McsEngl.Mathsset.field!⇒msetField,
* McsEngl.field//Mathsset,
* McsEngl.msetField,
Cartesian-product of mathSettheo
description::
· "In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted AΧB, is the set of all ordered pairs (a, b) where a is in A and b is in B.[1] In terms of set-builder notation, that is
\[A\times B=\{(a,b)\ |\ a\in A\ and\ b\in B\}.[2][3]\]
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows Χ columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).[4]
One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. An ordered pair is a 2-tuple or couple. More generally still, one can define the Cartesian product of an indexed family of sets.
The Cartesian product is named after Renι Descartes,[5] whose formulation of analytic geometry gave rise to the concept, which is further generalized in terms of direct product."
[{2023-08-21 retrieved} https://en.wikipedia.org/wiki/Cartesian_product]
name::
* McsEngl.Cartesian-product,
* McsEngl.mathSettheo'Cartesian-product,
evoluting of mathSettheo
description::
"evolution of set-theory:
Set theory is a foundational branch of mathematical logic and has evolved significantly since its inception in the late 19th century. Here is a brief overview of the key developments in the evolution of set theory:
1. Georg Cantor (late 19th century):
- Georg Cantor is considered the father of set theory. He developed the fundamental concepts of sets and their cardinality, including the concept of different sizes of infinity.
- Cantor's work led to the discovery of countable and uncountable infinities and the controversial but influential Continuum Hypothesis.
2. Ernst Zermelo and Abraham Fraenkel (early 20th century):
- Zermelo introduced axiomatic set theory, which laid down a set of axioms for set theory, known as Zermelo-Fraenkel set theory (ZF).
- Abraham Fraenkel added the Axiom of Replacement to Zermelo's axioms, resulting in Zermelo-Fraenkel set theory with the Axiom of Replacement (ZFC), which is the most widely accepted foundation for mathematics.
3. Axiomatic Set Theory (early to mid-20th century):
- The development of ZFC and other set theories (e.g., von Neumann-Bernays-Gödel set theory) solidified the axiomatic basis of set theory, providing a rigorous framework for mathematics.
4. Set Theory and Paradoxes:
- The early 20th century also saw the discovery of set-theoretic paradoxes, such as Russell's paradox. These paradoxes raised questions about the consistency and completeness of set theory.
- Various efforts, including the development of axiomatic systems like ZFC, were made to resolve these paradoxes.
5. Axiom of Choice:
- The Axiom of Choice, which allows for the arbitrary selection of elements from a set, has been a subject of debate and study. It was added to ZFC, creating ZFC with the Axiom of Choice (ZFC+AC), which is the standard set theory used in mathematics.
6. Independence Results (mid to late 20th century):
- Work by Paul Cohen and Kurt Gödel showed that some set-theoretic questions, like the Continuum Hypothesis, cannot be resolved within ZFC alone. This led to the concept of independence results, demonstrating that the truth of certain propositions cannot be decided using the ZFC axioms.
7. Set Theory and Its Applications:
- Set theory has found applications in various areas of mathematics, including topology, analysis, and algebra. It plays a fundamental role in defining the structure of mathematical objects and the foundations of mathematics.
8. Contemporary Developments (21st century):
- Set theory continues to evolve, with researchers exploring variations of set theory, such as constructive set theory and alternative set theories beyond ZFC.
- Modern developments also include advances in forcing and large cardinal axioms, which have deepened our understanding of set-theoretic independence and consistency.
The evolution of set theory reflects the ongoing quest for a solid foundation of mathematics and the exploration of the nature of sets, infinity, and mathematical truth. While ZFC remains the most widely accepted foundation, the study of alternative set theories and the resolution of open questions continue to be active areas of research in contemporary mathematics."
[{2023-10-12 retrieved} https://chat.openai.com/c/efe777bd-d8c2-4942-8812-4b546d638f71]
name::
* McsEngl.evoluting-of-mathSettheo,
* McsEngl.mathSettheo'evoluting,
mathLogic.subjective
description::
"Subjective logic is a type of probabilistic logic that explicitly takes epistemic uncertainty and source trust into account. In general, subjective logic is suitable for modeling and analysing situations involving uncertainty and relatively unreliable sources.[1][2][3] For example, it can be used for modeling and analysing trust networks and Bayesian networks.
Arguments in subjective logic are subjective opinions about state variables which can take values from a domain (aka state space), where a state value can be thought of as a proposition which can be true or false. A binomial opinion applies to a binary state variable, and can be represented as a Beta PDF (Probability Density Function). A multinomial opinion applies to a state variable of multiple possible values, and can be represented as a Dirichlet PDF (Probability Density Function). Through the correspondence between opinions and Beta/Dirichlet distributions, subjective logic provides an algebra for these functions. Opinions are also related to the belief representation in Dempster–Shafer belief theory.
A fundamental aspect of the human condition is that nobody can ever determine with absolute certainty whether a proposition about the world is true or false. In addition, whenever the truth of a proposition is expressed, it is always done by an individual, and it can never be considered to represent a general and objective belief. These philosophical ideas are directly reflected in the mathematical formalism of subjective logic."
[{2023-10-08 retrieved} https://en.wikipedia.org/wiki/Subjective_logic]
name::
* McsEngl.mathLogic.subjective,
* McsEngl.sciMath.042-subjective-logic,
* McsEngl.sciMath.subjective-logic,
* McsEngl.subjective_logic,
evoluting of subjective_logic
description::
"evolution of subjective logic:
Subjective logic is a mathematical framework for handling uncertainty and belief in a principled way. It was developed by Audun Jøsang, a Norwegian computer scientist, and has its roots in Bayesian probability theory and Dempster-Shafer theory of evidence. The evolution of subjective logic can be traced through several key developments:
1. Bayesian Probability Theory: Subjective logic builds upon Bayesian probability theory, which itself has a long history dating back to the 18th century. Bayesian probability provides a mathematical framework for updating beliefs based on evidence, making it a foundational concept for subjective logic.
2. Dempster-Shafer Theory of Evidence: In the 1960s and 1970s, Dempster-Shafer theory extended traditional Bayesian probability theory to handle uncertainty and evidence in a more versatile way. It introduced the notion of belief functions, which can represent degrees of belief in a proposition even when there is incomplete or conflicting evidence. This laid the groundwork for subjective logic.
3. Development of Subjective Logic: Audun Jøsang began working on subjective logic in the late 1990s. He aimed to create a more intuitive and practical framework for handling uncertainty and belief, particularly in situations where it is challenging to obtain precise probabilities. Subjective logic extends Dempster-Shafer theory by introducing the concept of opinions, which are subjective beliefs about the probability of an event. Opinions consist of two parts: the belief mass, which represents the degree of belief in an event, and the plausibility mass, which represents the degree of disbelief in the complement of the event.
4. Applications and Refinements: Subjective logic has found applications in various fields, including information fusion, decision support systems, and artificial intelligence. Researchers have continued to refine and extend the theory, developing more sophisticated algorithms for reasoning with opinions and handling complex belief structures.
5. Integration with Other Theories: Subjective logic has been integrated with other uncertainty modeling frameworks, such as fuzzy logic and possibility theory, to create hybrid approaches that can handle different types of uncertainty more effectively.
6. Challenges and Critiques: Like any framework, subjective logic has faced challenges and critiques. Some researchers have raised concerns about its computational complexity and the difficulty of assigning opinions in practice. However, ongoing research aims to address these issues and improve the practicality of the approach.
Overall, the evolution of subjective logic reflects the ongoing effort to develop more robust and practical methods for reasoning under uncertainty and managing subjective beliefs in various applications, particularly in the context of artificial intelligence and decision support systems."
[{2023-10-08 retrieved} https://chat.openai.com/c/7d922a98-d89d-4773-a817-b28dff450db1]
name::
* McsEngl.evoluting-of-subjective_logic,
* McsEngl.subjective_logic'evoluting,
mathLogic.type-theory
description::
"Type theory is a branch of mathematical logic that studies the relationship between types of data and the operations that can be performed on them. In computer science, type theory is used to design and implement programming languages that are more reliable and secure.
A type system is a set of rules that govern the use of types in a programming language. The rules define what types of values can be assigned to variables, what types of expressions can be evaluated, and what types of functions can be called.
Type theory provides a formal foundation for type systems. It allows us to reason about the correctness of programs by proving that they satisfy certain properties, such as the absence of type errors.
There are many different type theories, but some of the most common ones include:
* Simple type theory: This is the simplest type theory, and it is based on the idea that every value has a single type.
* Hindley-Milner type system: This type system is more powerful than simple type theory, and it allows for the recursive definition of types.
* Calculus of constructions: This type system is based on the idea of inductive types, which can be used to represent data structures such as lists and trees.
Type theory is a rapidly growing field of research, and it is being used to develop new programming languages, verification tools, and mathematical foundations.
Here are some of the benefits of using type theory in programming languages:
* Reliability: Type systems can help to prevent errors in programs by ensuring that the types of values are consistent.
* Security: Type systems can help to prevent security vulnerabilities in programs by preventing the execution of code with the wrong type.
* Expressiveness: Type systems can help to make programs more expressive by allowing programmers to write code that is more concise and easier to understand.
* Verifiability: Type systems can be used to verify the correctness of programs by proving that they satisfy certain properties.
Type theory is a powerful tool that can be used to improve the reliability, security, expressiveness, and verifiability of programming languages. It is a rapidly growing field of research, and it is likely to play an increasingly important role in the development of future programming languages.
[{2023-08-27 retrieved} https://bard.google.com/]
name::
* McsEngl.logicType_theory,
* McsEngl.mathLogic.type-theory!⇒logicType_theory,
* McsEngl.mathType_theo!⇒logicType_theory,
* McsEngl.mathType_theo!=science.math.logic.type-theory,
* McsEngl.sciMath.023-type-theory!⇒logicType_theory,
* McsEngl.sciMath.type-theory!⇒logicType_theory,
* McsEngl.science.math.logic.type-theory!⇒logicType_theory,
descriptionLong::
"Type theory is a foundational framework in computer science, logic, and mathematics that deals with classifying and organizing objects into distinct categories called "types." It provides a formal system for specifying and reasoning about the structure of data, functions, and their interactions. Type theory is utilized in programming languages, proof systems, and other formal systems to ensure correctness, safety, and rigor in software development and logical reasoning.
Here are some key concepts and aspects of type theory:
* Types and Values: In type theory, everything has a type. Types classify values and determine the operations that can be performed on them. For example, in a programming language, integers, strings, and booleans are different types of values.
* Type Checking: Type checking is the process of verifying that operations performed on values are consistent with their types. This helps catch errors at compile-time rather than at runtime, leading to more robust and reliable software.
* Strong Typing: Type theory often enforces strong typing, which means that operations between values of incompatible types are disallowed. This can prevent unintended behavior and enhance program safety.
* Static vs. Dynamic Typing: Programming languages can implement type systems either statically (type checking is done at compile-time) or dynamically (type checking is done at runtime). Statically typed languages tend to catch more errors early, while dynamically typed languages offer more flexibility.
* Type Inference: Some type systems support type inference, where the type of an expression is automatically deduced by the compiler or interpreter. This can reduce the need for explicit type annotations.
* Polymorphism: Polymorphism allows values or functions to work with different types in a uniform manner. There are different forms of polymorphism, such as parametric polymorphism (generics) and ad-hoc polymorphism (overloading).
* Dependent Types: Dependent types are a more advanced feature of type theory where types can depend on values. This allows for precise specifications and more expressive type systems that can catch subtle errors.
* Type Hierarchy: In many type systems, types are organized in a hierarchy. Subtyping allows one type to be considered a subtype of another, enabling more flexible type relationships.
* Type Equivalence: Type theory distinguishes between different forms of type equivalence, such as structural equivalence (based on the structure of types) and nominal equivalence (based on names).
* Type Theory in Logic and Mathematics: Type theory has been used to develop formal logical systems, such as the theory of types developed by Russell and Whitehead. It's also connected to constructive mathematics and plays a role in formal proof development.
* Programming Language Paradigms: Different programming language paradigms, like functional programming and object-oriented programming, incorporate various aspects of type theory to varying degrees.
* Higher-Order Types: Type theory often allows the creation of higher-order types, where types themselves can be manipulated as values. This is a key concept in functional programming.
Prominent examples of type theories include the simply typed lambda calculus, System F, Martin-Lφf type theory, and various type systems used in programming languages like Haskell, ML, Scala, and dependent type systems like Coq and Agda.
In summary, type theory provides a systematic way to classify and reason about data and functions in various contexts, ensuring correctness and reliability in programming, logic, and mathematics. It's a foundational concept that underpins many aspects of modern computer science and formal reasoning."
[{2023-08-27 retrieved} https://chat.openai.com/?model=text-davinci-002-render-sha]
evoluting of logicType_theory
description::
"evolution of type-theory:
Type theory is a formal system in the field of mathematical logic and computer science that deals with the classification of objects and their relationships based on their types. It has evolved over the years, with significant contributions from various researchers. Here's an overview of the evolution of type theory:
1. Simple Type Theory (STT):
- Developed in the early 20th century by logicians such as Bertrand Russell and Alfred North Whitehead.
- This is the foundation of modern type theory.
- It introduced the idea of classifying objects into types, and it sought to resolve foundational issues in mathematics, such as the Russell Paradox.
2. Higher-Order Logic (HOL):
- HOL extended simple type theory to allow for functions and predicates over types.
- It played a significant role in the development of automated theorem proving systems and formal methods in computer science.
3. Intuitionistic Type Theory (ITT):
- Proposed by Swedish logician Per Martin-Löf in the 1970s.
- It is also known as constructive type theory.
- Unlike classical logic, ITT is based on intuitionistic logic, which is more closely related to mathematical constructivism. This has implications for the philosophy of mathematics.
4. Dependent Type Theory (DTT):
- Developed by Martin-Löf as an extension of ITT.
- It introduced the idea of dependent types, where the type of an object can depend on the value of another object.
- Dependent types are a fundamental concept in modern type theory and have applications in programming languages (like Agda and Idris) and formal verification.
5. NuPRL and the Cambridge LCF System:
- Various proof assistants and theorem provers, like NuPRL and the Cambridge LCF system, have been developed based on type theory.
- These systems allow for formal verification of software and mathematical proofs.
6. Homotopy Type Theory (HoTT):
- HoTT is an extension of type theory that connects it with homotopy theory from algebraic topology.
- It suggests a new way of looking at types and equality, where types are topological spaces.
- This has sparked significant interest in the intersection of type theory and higher-dimensional category theory.
7. Practical Application in Programming Languages:
- Modern programming languages like Haskell, Scala, and Rust incorporate ideas from type theory. For example, they have type systems that use concepts like parametric polymorphism and dependent types.
8. Continued Research and Development:
- Type theory continues to evolve with ongoing research in various areas. Researchers are working on combining aspects of different type theories to create more expressive and practical systems for formal reasoning, software verification, and mathematics.
The evolution of type theory has not only had a significant impact on the foundations of mathematics and formal logic but also on the design of programming languages and the development of tools for formal verification and theorem proving in computer science."
[{2023-10-12 retrieved} https://chat.openai.com/c/448c7295-3cc2-4105-90f6-fb28b94b6143]
name::
* McsEngl.evoluting-of-logicType_theory,
* McsEngl.logicType_theory'evoluting,
sciMath.mathematical-economics
description::
"Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods.[1][2] Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.[3]
Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics.[4] Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.[5]
Broad applications include:
* optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker
* static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing
* comparative statics as to a change from one equilibrium to another induced by a change in one or more factors
* dynamic analysis, tracing changes in an economic system over time, for example from economic growth.[2][6][7]
* Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics.[8][7]
This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics."
[{2023-10-06 retrieved} https://en.wikipedia.org/wiki/Mathematical_economics]
name::
* McsEngl.mathematical-economics,
* McsEngl.sciMath.032-mathematical-economics,
* McsEngl.sciMath.mathematical-economics,
sciMath.mathematical-finance
description::
"Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require advanced quantitative techniques: derivatives pricing on the one hand, and risk and portfolio management on the other.[1] Mathematical finance overlaps heavily with the fields of computational finance and financial engineering. The latter focuses on applications and modeling, often by help of stochastic asset models, while the former focuses, in addition to analysis, on building tools of implementation for the models. Also related is quantitative investing, which relies on statistical and numerical models (and lately machine learning) as opposed to traditional fundamental analysis when managing portfolios.
French mathematician Louis Bachelier's doctoral thesis, defended in 1900, is considered the first scholarly work on mathematical finance. But mathematical finance emerged as a discipline in the 1970s, following the work of Fischer Black, Myron Scholes and Robert Merton on option pricing theory. Mathematical investing originated from the research of mathematician Edward Thorp who used statistical methods to first invent card counting in blackjack and then applied its principles to modern systematic investing.[2]
The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory that is involved in financial mathematics. While trained economists use complex economic models that are built on observed empirical relationships, in contrast, mathematical finance analysis will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. See: Valuation of options; Financial modeling; Asset pricing. The fundamental theorem of arbitrage-free pricing is one of the key theorems in mathematical finance, while the Black–Scholes equation and formula are amongst the key results.[3]
Today many universities offer degree and research programs in mathematical finance."
[{2023-10-06 retrieved} https://en.wikipedia.org/wiki/Mathematical_finance]
name::
* McsEngl.mathematical-finance,
* McsEngl.sciMath.033-mathematical-finance,
* McsEngl.sciMath.mathematical-finance,
sciMath.mathematical-modeling
description::
"A mathematical model is an abstract description of a concrete system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in applied mathematics and in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences[1] (such as economics, psychology, sociology, political science). It can also be taught as a subject in its own right.[2]"
[{2023-09-28 retrieved} https://en.wikipedia.org/wiki/Mathematical_model]
name::
* McsEngl.mathematical-modeling,
* McsEngl.sciMath.025-mathematical-modeling,
* McsEngl.sciMath.mathematical-modeling,
mathematical-model (link)
sciMath.mathematical-optimization
description::
"Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives.[1] It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering[2] to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.[3]
In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding "best available" values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains."
[{2023-09-28 retrieved} https://en.wikipedia.org/wiki/Mathematical_optimization]
name::
* McsEngl.mathematical-optimization,
* McsEngl.optimization//sciMath,
* McsEngl.sciMath.024-mathematical-optimization,
* McsEngl.sciMath.mathematical-optimization,
sciMath.matrix-theory
description::
· "Matrix theory is a branch of mathematics which focuses on the study of matrices. Initially a sub-branch of linear algebra, it has grown to cover subjects related to graph theory, algebra, combinatorics, and statistics as well.
[http://en.wikipedia.org/wiki/Matrix_theory]"
name::
* McsEngl.mathMatrix_theo,
* McsEngl.mathMatrix_theo!=science.math.matrix-theory,
* McsEngl.matrix-theory-sciMath,
* McsEngl.sciMath.010-matrix-theory,
* McsEngl.sciMath.matrix-theory,
* McsEngl.science.math.matrix-theory!⇒mathMatrix_theo,
matrix of mathMatrix_theo
description::
· Mathsmatrix is a-table of math-objects arranged in rows and columns.
name::
* McsEngl.Mathsmatrix,
* McsEngl.Mathsmatrix!=matrix//sciMath,
* McsEngl.matrix//sciMath!⇒Mathsmatrix,
tensor of sciMath
description::
· Mathstensor is a-multidimensional matrix.
===
· "In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, ...), general relativity (stress–energy tensor, curvature tensor, ...) and others. In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one location to another. This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the absolute differential calculus. The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.[1]"
[{2023-08-19 retrieved} https://en.wikipedia.org/wiki/Tensor]
name::
* McsEngl.Mathstensor,
* McsEngl.Mathstensor!=tenosr//sciMath,
* McsEngl.sciMath'tensor!⇒Mathstensor,
* McsEngl.tensor//sciMath!⇒Mathstensor,
sciMath.statistics
description::
· "Statistics (from German: Statistik, orig. "description of a state, a country")[1][2] is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.[3][4][5] In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.[6]"
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Statistics]
name::
* McsEngl.mathStat,
* McsEngl.mathStat!=science.math.statistics,
* McsEngl.statistics-sciMath!⇒mathStat,
* McsEngl.sciMath.012-statistics!⇒mathStat,
* McsEngl.sciMath.statistics!⇒mathStat,
* McsEngl.science.math.statistics!=mathStat,
mathStat.probability-theory
description::
· "Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event.
Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.
As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics or sequential estimation. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics. [2]"
[{2023-08-16 retrieved} https://en.wikipedia.org/wiki/Probability_theory]
name::
* McsEngl.mathStat.probability-theory,
* McsEngl.probability-calculus,
* McsEngl.probability-theory,
* McsEngl.sciMath.018-probability-theory,
* McsEngl.sciMath.probability-theory,
* McsEngl.statProbabilityth,
mathStat.decision-theory
description::
"Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory and analytic philosophy concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical consequences to the outcome.[1]
There are three branches of decision theory:
1. Normative decision theory: Concerned with the identification of optimal decisions, where optimality is often determined by considering an ideal decision-maker who is able to calculate with perfect accuracy and is in some sense fully rational.
2. Prescriptive decision theory: Concerned with describing observed behaviors through the use of conceptual models, under the assumption that those making the decisions are behaving under some consistent rules.
3.Descriptive decision theory: Analyzes how individuals actually make the decisions that they do.
Decision theory is a broad field from management sciences and is an interdisciplinary topic, studied by management scientists, medical researchers, mathematicians, data scientists, psychologists, biologists,[2] social scientists, philosophers[3] and computer scientists.
Empirical applications of this theory are usually done with the help of statistical and discrete mathematical approaches from computer science."
[{2023-10-06 retrieved} https://en.wikipedia.org/wiki/Decision_theory]
name::
* McsEngl.decision-theory,
* McsEngl.sciGovernance.006-decision-theory,
* McsEngl.sciGovernance.decision-theory,
* McsEngl.mathStat.decision-theory,
* McsEngl.sciMath.019-decision-theory,
* McsEngl.sciMath.decision-theory,
mathStat.mathematical-statistics
description::
"Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory.[1][2]"
[{2023-10-08 retrieved} https://en.wikipedia.org/wiki/Mathematical_statistics]
name::
* McsEngl.mathStat.mathematical-statistics,
* McsEngl.mathematical-statistics,
* McsEngl.sciMath.035-mathematical-statistics,
* McsEngl.sciMath.mathematical-statistics,
sciMath.topology
description::
× generic: mathAppliedNo.
· "In mathematics, topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. The following are basic examples of topological properties: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non-intersecting circles.
The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Kφnigsberg problem and polyhedron formula are arguably the field's first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century; although, it was not until the first decades of the 20th century that the idea of a topological space was developed."
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Topology]
name::
* McsEngl.mathTopology,
* McsEngl.topology-sciMath,
* McsEngl.sciMath.013-topology,
* McsEngl.sciMath.topology,
GENERIC-SPECIFIC-TREE of mathTopology
generic-tree-of-mathTopology::
* mathAppliedNo,
* ... entity,
* McsEngl.mathTopology'generic-tree,
specific-tree-of-mathTopology::
* point-set topology,
* algebraic topology,
* differential topology,
* McsEngl.mathTopology.specific-tree,
sciMath.trigonometry
description::
· "Trigonometry (from Ancient Greek τρίγωνον (trνgōnon) 'triangle', and μέτρον (mιtron) 'measure')[1] is a branch of mathematics concerned with relationships between angles and ratios of lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.[2] The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine.[3]
Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation.[4]
Trigonometry is known for its many identities. These trigonometric identities[5][6] are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation.[7]"
[{2023-08-15 retrieved} https://en.wikipedia.org/wiki/Trigonometry]
name::
* McsEngl.trigonometry-sciMath,
* McsEngl.sciMath.014-trigonometry,
* McsEngl.sciMath.trigonometry,
sciMath.possibility-theory
description::
"Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise."
[{2023-10-08 retrieved} https://en.wikipedia.org/wiki/Possibility_theory]
name::
* McsEngl.possibility_theory,
* McsEngl.sciMath.041-possibility-theory,
* McsEngl.sciMath.possibility-theory,
evoluting of possibility_theory
description::
"evolution of posibility theory
Possibility theory is a mathematical framework used for reasoning under uncertainty, which is closely related to probability theory but allows for more nuanced handling of uncertainty. The evolution of possibility theory can be traced through its key developments and the contributions of various researchers. Here is an overview of the evolution of possibility theory:
1. Fuzzy Sets Theory (1965 - 1970s): The precursor to possibility theory was fuzzy sets theory, introduced by Lotfi Zadeh in the mid-1960s. Fuzzy sets allow for gradual membership degrees, as opposed to traditional set theory where an element is either in or out of a set. This provided a foundation for dealing with uncertainty and imprecision.
2. Possibility Theory (1978): Possibility theory was formally introduced by E. H. Mamdani and S. Assilian in a seminal paper titled "An Experiment in Linguistic Synthesis with a Fuzzy Logic Controller" in 1978. They extended fuzzy set theory to introduce the concept of possibility distributions, which represent the degree to which an element belongs to a fuzzy set.
3. Dempster-Shafer Theory (1980): Although not a direct development of possibility theory, Dempster-Shafer theory (also known as evidence theory) is another framework for reasoning under uncertainty. It was introduced by Glenn Shafer in 1976 and refined with contributions from Arthur P. Dempster in 1980. It provides a different way to model uncertainty by using belief functions and combining evidence from multiple sources. While different from possibility theory, it shares the goal of handling uncertainty.
4. Generalizations and Applications (1980s - Present): Since its inception, possibility theory has seen numerous generalizations and applications in various fields. Researchers have developed mathematical tools for dealing with possibility distributions, such as possibility measures and necessity measures. Applications have been found in areas like decision support systems, artificial intelligence, control systems, and pattern recognition.
5. Hybrid Approaches (1990s - Present): Researchers have also explored hybrid approaches that combine possibility theory with other uncertainty frameworks, such as probability theory and Dempster-Shafer theory, to take advantage of their respective strengths. These hybrid systems aim to provide more comprehensive models for handling different types of uncertainty.
6. Continued Research and Development: Research in possibility theory and related areas continues to evolve. New techniques and methods are being developed to address more complex and practical problems involving uncertainty and imprecision.
Overall, the evolution of possibility theory is closely tied to the broader field of uncertainty modeling and reasoning. It has provided valuable tools for addressing real-world problems where traditional probability theory may be insufficient due to the inherent uncertainty and imprecision present in many domains."
name::
* McsEngl.evoluting-of-possibility_theory,
* McsEngl.possibility_theory'evoluting,
