🌐 Analysis (آنالیز)
Analysis is at the very core of mathematics and covers a wide range of different subjects, studying limit processes, inequalities, integration, differentiation, measure theory, and analytic functions. It is replete with a roster of giants - Newton, Leibnitz, Cauchy, Riemann, Fourier, Hardy, Weierstrass, Gauss, Stokes, Poincaré - and is the original home for topics which have become major branches of mathematics in their own right - topology, differential equations, differential geometry, dynamical systems. Analysis also provides both the theoretical and the practical basis for the mathematics of the sciences and engineering, not to mention other areas in mathematics itself, from analytic number theory to computational mathematics. Applications and connections to other fields, both within and outside mathematics, are numerous. Calculus and the theory of real and complex continuous functions are among the crowning achievements of science. The field of mathematical analysis continues the development of that theory today to give even greater power and generality. Functions are representations of relations between sets, and in particular are useful for representing the changing states of a system: the velocity of a projectile, the frequencies present in a sound signal, the color of a pixel in a digital image, or the prices of a portfolio of stocks. The mathematical field of analysis seeks to formulate methods to analyze quantitatively the change exhibited by the outputs of functions with respect to their inputs, as a way of distilling important information about the underlying systems, such as the way stock prices change over time. Today, analysis forms the foundation of several highly active areas of mathematics, with powerful applications throughout mathematics as well as in the natural sciences and industry.
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Harmonic analysis seeks to decompose functions into their component "harmonics" or waves; that such a decomposition is possible for broad classes of functions was initially discovered by Fourier in the early 1800s, and continues to have applications in a vast swath of mathematics, ranging from image compression methods via "wavelets" to counting integral solutions to Diophantine equations via the Hardy-Littlewood "circle method". The Fourier transform and Fourier series are examples of topics included in harmonic analysis. Applications are extensive in such fields as signal processing, medical imaging and quantum mechanics. Complex analysis studies the behavior of functions on the complex plane and has deep connections to number theory, and in particular to the distribution of prime numbers, via the Riemann zeta function. Real analysis develops a rigorous theory of integration which extends the familiar notions of calculus to a broader class of functions, and in particular provides a foundation for many concepts in probability. Stochastic analysis seeks to understand the behavior of differential equations and dynamical systems whose parameters or inputs are random and unpredictable; this has many applications in the sciences and engineering, and economics.
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#آنالیز
Analysis is at the very core of mathematics and covers a wide range of different subjects, studying limit processes, inequalities, integration, differentiation, measure theory, and analytic functions. It is replete with a roster of giants - Newton, Leibnitz, Cauchy, Riemann, Fourier, Hardy, Weierstrass, Gauss, Stokes, Poincaré - and is the original home for topics which have become major branches of mathematics in their own right - topology, differential equations, differential geometry, dynamical systems. Analysis also provides both the theoretical and the practical basis for the mathematics of the sciences and engineering, not to mention other areas in mathematics itself, from analytic number theory to computational mathematics. Applications and connections to other fields, both within and outside mathematics, are numerous. Calculus and the theory of real and complex continuous functions are among the crowning achievements of science. The field of mathematical analysis continues the development of that theory today to give even greater power and generality. Functions are representations of relations between sets, and in particular are useful for representing the changing states of a system: the velocity of a projectile, the frequencies present in a sound signal, the color of a pixel in a digital image, or the prices of a portfolio of stocks. The mathematical field of analysis seeks to formulate methods to analyze quantitatively the change exhibited by the outputs of functions with respect to their inputs, as a way of distilling important information about the underlying systems, such as the way stock prices change over time. Today, analysis forms the foundation of several highly active areas of mathematics, with powerful applications throughout mathematics as well as in the natural sciences and industry.
📚 @topmathlearn 📚
Harmonic analysis seeks to decompose functions into their component "harmonics" or waves; that such a decomposition is possible for broad classes of functions was initially discovered by Fourier in the early 1800s, and continues to have applications in a vast swath of mathematics, ranging from image compression methods via "wavelets" to counting integral solutions to Diophantine equations via the Hardy-Littlewood "circle method". The Fourier transform and Fourier series are examples of topics included in harmonic analysis. Applications are extensive in such fields as signal processing, medical imaging and quantum mechanics. Complex analysis studies the behavior of functions on the complex plane and has deep connections to number theory, and in particular to the distribution of prime numbers, via the Riemann zeta function. Real analysis develops a rigorous theory of integration which extends the familiar notions of calculus to a broader class of functions, and in particular provides a foundation for many concepts in probability. Stochastic analysis seeks to understand the behavior of differential equations and dynamical systems whose parameters or inputs are random and unpredictable; this has many applications in the sciences and engineering, and economics.
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#آنالیز
❤5
🌐 PDEs & Dynamical Systems
( معادلات دیفرانسیل جزئی و سیستمهای دینامیکی)
The study of Partial Differential Equations (PDEs) is an interdisciplinary field within mathematics and various applied sciences, such as physics, engineering, biology, finance, and beyond. PDEs are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory.
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The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse. Applying analytic methods to partial differential equations (PDEs) allows researchers to study the evolution of a system that is changing in a manner governed by precise constraints (the differential equations in the name); PDEs may represent many important problems in the physical world, such as the diffusion of heat, fluid flow, or quantum mechanics. The formulation of models is grounded in first principles, often drawing from conservation laws and conducted by skilled practitioners. However, a significant challenge arises due to the inherent complexity of the resulting PDEs, rendering them challenging to solve analytically. Consequently, sophisticated numerical algorithms become essential for solving these equations effectively as well as examining the resulting solutions. Developing suitable numerical schemes, in turn, necessitates a clear understanding of the underlying analytical properties of the equations and potential solutions.
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#معادلات_دیفرانسیل_جزئی
#سیستمهای_دینامیکی
( معادلات دیفرانسیل جزئی و سیستمهای دینامیکی)
The study of Partial Differential Equations (PDEs) is an interdisciplinary field within mathematics and various applied sciences, such as physics, engineering, biology, finance, and beyond. PDEs are one of the most fundamental tools for describing continuum phenomena in the sciences and engineering. Early work on PDEs, in the 1700s, was motivated by problems in fluid mechanics, wave motion, and electromagnetism. Since that time, the range of applications of PDEs has expanded rapidly. For example, PDEs are used in mathematical models of weather and climate, in medical imaging technologies, in the design of new composite materials, in models of elementary particle interaction and of the formation of galaxies, in models of cancerous tumor growth or of blood flow in the heart, in simulating semiconductor devices, in models of bacterial colonies, in models of financial markets and asset price bubbles, in describing the flocking behavior of birds and fish. PDEs also have played an important part in the development of other branches of mathematics, including harmonic analysis, differential geometry, probability, optimization and control theory.
📚 @topmathlearn 📚
The phenomena described by PDEs are as complex as the world around us; the mathematical techniques needed to study PDEs are very diverse. Applying analytic methods to partial differential equations (PDEs) allows researchers to study the evolution of a system that is changing in a manner governed by precise constraints (the differential equations in the name); PDEs may represent many important problems in the physical world, such as the diffusion of heat, fluid flow, or quantum mechanics. The formulation of models is grounded in first principles, often drawing from conservation laws and conducted by skilled practitioners. However, a significant challenge arises due to the inherent complexity of the resulting PDEs, rendering them challenging to solve analytically. Consequently, sophisticated numerical algorithms become essential for solving these equations effectively as well as examining the resulting solutions. Developing suitable numerical schemes, in turn, necessitates a clear understanding of the underlying analytical properties of the equations and potential solutions.
📚 @topmathlearn 📚
#معادلات_دیفرانسیل_جزئی
#سیستمهای_دینامیکی
❤5
🌐 Geometry (هندسه)
The modern discipline of geometry is affecting virtually every branch of mathematics and is in a period of great progress. Geometry combines techniques from apparently unconnected areas, such as the use of analytical tools for answering topological questions, and it has strong relations with quantum field theory and string theory. In modern language, the central object of study in geometry is a manifold, which is an object that may have a complicated overall shape, but such that on small scales it looks like ordinary space of a certain dimension. For example, a 1-dimensional manifold is an object such that small pieces of it look like a line, although in general it looks like a curve rather than a straight line. A 2-dimensional manifold, on small scales, looks like a (curved) piece of paper. An n-dimensional manifold likewise looks locally like an ordinary n-dimensional space. Many old problems are being solved using techniques from the area (notably Perelman’s resolution of the Poincaré conjecture) and new directions are being pioneered. An important theme in this area has been the development and use of sophisticated techniques from the theory of PDEs to study natural equations that arise in geometry. Usually, the set of solutions of a system of partial differential equations has the structure of some high dimensional manifold. Understanding the geometry of this manifold often gives new insight into the nature of these solutions, and to the actual phenomenon that is modeled by the differential equations, whether it comes from physics, economics, engineering, or any other quantitative science.
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The Atiyah-Singer index theorem from the 1960s connects the theory of linear PDEs to topology and geometry. The development of tools for nonlinear PDEs in geometry has been slower but has led to many of the most dramatic developments in mathematics including Donaldson's breakthroughs in the theory of four-manifolds using the Yang-Mills equations of high-energy physics. The study of lower dimensional manifolds (dimensions four or less) has particular significance to theoretical physics and has many applications. Floer homology is a mathematically rigorous way of constructing parts of a quantum field theory. Another important and growing area is the mathematics of general relativity. The Lorentz version of the Einstein equations is now at the cutting edge of our hyperbolic PDE technology. One branch of geometric analysis involves the recovery of a function from its integrals over various domains. A well-known application of this idea is Computed Tomography Scanning (CT scans).
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#هندسه
The modern discipline of geometry is affecting virtually every branch of mathematics and is in a period of great progress. Geometry combines techniques from apparently unconnected areas, such as the use of analytical tools for answering topological questions, and it has strong relations with quantum field theory and string theory. In modern language, the central object of study in geometry is a manifold, which is an object that may have a complicated overall shape, but such that on small scales it looks like ordinary space of a certain dimension. For example, a 1-dimensional manifold is an object such that small pieces of it look like a line, although in general it looks like a curve rather than a straight line. A 2-dimensional manifold, on small scales, looks like a (curved) piece of paper. An n-dimensional manifold likewise looks locally like an ordinary n-dimensional space. Many old problems are being solved using techniques from the area (notably Perelman’s resolution of the Poincaré conjecture) and new directions are being pioneered. An important theme in this area has been the development and use of sophisticated techniques from the theory of PDEs to study natural equations that arise in geometry. Usually, the set of solutions of a system of partial differential equations has the structure of some high dimensional manifold. Understanding the geometry of this manifold often gives new insight into the nature of these solutions, and to the actual phenomenon that is modeled by the differential equations, whether it comes from physics, economics, engineering, or any other quantitative science.
📚 @topmathlearn 📚
The Atiyah-Singer index theorem from the 1960s connects the theory of linear PDEs to topology and geometry. The development of tools for nonlinear PDEs in geometry has been slower but has led to many of the most dramatic developments in mathematics including Donaldson's breakthroughs in the theory of four-manifolds using the Yang-Mills equations of high-energy physics. The study of lower dimensional manifolds (dimensions four or less) has particular significance to theoretical physics and has many applications. Floer homology is a mathematically rigorous way of constructing parts of a quantum field theory. Another important and growing area is the mathematics of general relativity. The Lorentz version of the Einstein equations is now at the cutting edge of our hyperbolic PDE technology. One branch of geometric analysis involves the recovery of a function from its integrals over various domains. A well-known application of this idea is Computed Tomography Scanning (CT scans).
📚 @topmathlearn 📚
#هندسه
❤5
Riemannian geometry is the study of manifolds equipped with the additional structure of a Riemannian metric, which is a rule for measuring lengths of curves and angles between tangent vectors. A Riemannian manifold has curvature, and it is precisely this curvature that makes the laws of classical Euclidean geometry, that we learn in elementary school, to be different. For example, the sum of the interior angles of a triangle on a curved Riemannian manifold can be more or less than π if the curvature is positive or negative, respectively. Symplectic geometry is the study of manifolds equipped with an additional structure called a symplectic form. A symplectic form is in some sense (that can be made precise), the opposite of a Riemannian metric, and symplectic manifolds exhibit very different behaviour from Riemannian manifolds. For example, a famous theorem of Darboux says that all symplectic manifolds are “locally” the same, although globally they can be extremely different. Such a theorem is far from true in Riemannian geometry. Symplectic manifolds arise naturally in physical systems from classical mechanics, and are called “phases spaces” in physics. This branch of geometry is very topological in nature. Complex geometry is the study of manifolds which locally “look like” ordinary n-dimensional spaces that are modeled on the complex numbers rather than the real numbers. Because the analysis of holomorphic (or complex-analytic) functions is much more rigid than the real case (for example, not all real smooth functions are real-analytic) there are many fewer “types” of complex manifolds, and there has been more success in (at least partial) classifications. This field is also very closely related to algebraic geometry.
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#هندسه
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#هندسه
❤5
🌐 Differential Geometry (هندسه دیفرانسیل)
Differential geometry is the study of geometry using differential calculus, and uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra. The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. It is a vast subject that has its roots in both the classical theory of curves and surfaces and in the work of Gauss and Riemann motivated by the calculus of variations. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein’s general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is a broad field of mathematics related and with applications to several areas of mathematics, such as algebra, analysis, mathematical physics, partial differential equations, topology and science, such as biology, chemistry, data analysis, engineering, physics. It is not too far-fetched to argue that differential geometry should be in every mathematician’s arsenal.
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While topologists have famously been said to be unable to tell the difference between a donut and a coffee cup (since one can be continuously deformed into the other), geometers definitely care about shape. Differential geometry explores geometric quantities such as curvature and volume, including how such quantities evolve or flow when one continuously deforms a space using specific geometric constraints. Differential geometry has played an essential role in some of the most difficult mathematical problems in history that, at first glance, seem not to even be problems about geometry. The two most well-known examples are the Poincare Conjecture in topology and Fermat’s Last Theorem in number theory.
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#هندسه_دیفرانسیل
Differential geometry is the study of geometry using differential calculus, and uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra. The discipline owes its name to its use of ideas and techniques from differential calculus, though the modern subject often uses algebraic and purely geometric techniques instead. It is a vast subject that has its roots in both the classical theory of curves and surfaces and in the work of Gauss and Riemann motivated by the calculus of variations. It dates back to Newton and Leibniz in the seventeenth century, but it was not until the nineteenth century, with the work of Gauss on surfaces and Riemann on the curvature tensor, that differential geometry flourished and its modern foundation was laid. Over the past one hundred years, differential geometry has proven indispensable to an understanding of the physical world, in Einstein’s general theory of relativity, in the theory of gravitation, in gauge theory, and now in string theory. Differential geometry is a broad field of mathematics related and with applications to several areas of mathematics, such as algebra, analysis, mathematical physics, partial differential equations, topology and science, such as biology, chemistry, data analysis, engineering, physics. It is not too far-fetched to argue that differential geometry should be in every mathematician’s arsenal.
📚 @topmathlearn 📚
While topologists have famously been said to be unable to tell the difference between a donut and a coffee cup (since one can be continuously deformed into the other), geometers definitely care about shape. Differential geometry explores geometric quantities such as curvature and volume, including how such quantities evolve or flow when one continuously deforms a space using specific geometric constraints. Differential geometry has played an essential role in some of the most difficult mathematical problems in history that, at first glance, seem not to even be problems about geometry. The two most well-known examples are the Poincare Conjecture in topology and Fermat’s Last Theorem in number theory.
📚 @topmathlearn 📚
#هندسه_دیفرانسیل
❤5
🌐 Algebraic Geometry (هندسه جبری)
Algebraic geometry is the study of algebraic varieties, which are solution sets of systems of polynomial equations. They are sometimes manifolds but also often have singular points at which they are not smooth. Because they are defined algebraically, there are many more tools available from abstract algebra to study them, and conversely, many questions in pure algebra can be understood better by reformulating the problem in terms of algebraic geometry. Moreover, one can study varieties over any field, not just the real or complex numbers. The objects of study of algebraic geometry are, roughly, the common zeroes of polynomials in one or several variables (algebraic varieties). But because polynomials are so ubiquitous in mathematics, algebraic geometry has always stood at the crossroads of many different fields. Classical questions in algebraic geometry involve the study of particular sets of equations or the geometry of lines and linear spaces. Among the kinds of questions one can ask are enumerative: How many conics in the plane are tangent to a given set of five lines? How many lines are contained in a general surface of degree three in space? Another motivating question has been to try to fit together all algebraic varieties of a given type into a space which is itself an algebraic variety; such spaces are called moduli spaces. Simple examples of this type are projective spaces, which parameterize lines through the origin in a vector space, and their generalizations, Grassmannians, which parameterize linear subspaces of a vector space. In turn, the geometry of a Grassmannian can often be applied to solve an enumerative problem. In other kinds of moduli problems, one attempts to classify all curves, surfaces, or higher dimensional varieties of a certain type; another example is the space of all vector bundles of a given type over a fixed algebraic variety. Then one tries to construct and describe the moduli space of all such objects. Often invariant theory, i.e. the study of all invariant polynomials under the action of a group on a vector space, or a more general algebraic variety, plays a crucial role in the construction.
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Recent developments in high energy physics have led to a host of spectacular results and open problems in complex algebraic geometry. For example, the case where the dimension is one, i.e., the case of algebraic curves, is essentially the study of compact Riemann surfaces. This study has a long history involving calculus, complex analysis, and low dimensional topology. The moduli space of all compact Riemann surfaces has a very rich geometry and enumerative structure, which is an object of much current research, and has surprising connections with fields as diverse as geometric topology in dimensions two and three, nonlinear partial differential equations, and conformal field theory and string theory. Many questions posed by physicists have been solved by using the wealth of techniques developed by algebraic geometers. In turn, physics questions have led to new conjectures and new methods in this very central area of mathematics. Finally, the proposed ten dimensional space-time of string theory involves six very small extra dimensions, which correspond to certain three dimensional algebraic varieties, Calabi-Yau manifolds. These manifolds were already of great interest to mathematicians. Amazing ideas from physics have suggested that Calabi-Yau manifolds come in pairs. The geometry of the so-called mirror manifold of a Calabi-Yau manifold turns out to be connected to classical enumerative questions on the original manifold.
📚 @topmathlearn📚
#هندسه_جبری
Algebraic geometry is the study of algebraic varieties, which are solution sets of systems of polynomial equations. They are sometimes manifolds but also often have singular points at which they are not smooth. Because they are defined algebraically, there are many more tools available from abstract algebra to study them, and conversely, many questions in pure algebra can be understood better by reformulating the problem in terms of algebraic geometry. Moreover, one can study varieties over any field, not just the real or complex numbers. The objects of study of algebraic geometry are, roughly, the common zeroes of polynomials in one or several variables (algebraic varieties). But because polynomials are so ubiquitous in mathematics, algebraic geometry has always stood at the crossroads of many different fields. Classical questions in algebraic geometry involve the study of particular sets of equations or the geometry of lines and linear spaces. Among the kinds of questions one can ask are enumerative: How many conics in the plane are tangent to a given set of five lines? How many lines are contained in a general surface of degree three in space? Another motivating question has been to try to fit together all algebraic varieties of a given type into a space which is itself an algebraic variety; such spaces are called moduli spaces. Simple examples of this type are projective spaces, which parameterize lines through the origin in a vector space, and their generalizations, Grassmannians, which parameterize linear subspaces of a vector space. In turn, the geometry of a Grassmannian can often be applied to solve an enumerative problem. In other kinds of moduli problems, one attempts to classify all curves, surfaces, or higher dimensional varieties of a certain type; another example is the space of all vector bundles of a given type over a fixed algebraic variety. Then one tries to construct and describe the moduli space of all such objects. Often invariant theory, i.e. the study of all invariant polynomials under the action of a group on a vector space, or a more general algebraic variety, plays a crucial role in the construction.
📚 @topmathlearn 📚
Recent developments in high energy physics have led to a host of spectacular results and open problems in complex algebraic geometry. For example, the case where the dimension is one, i.e., the case of algebraic curves, is essentially the study of compact Riemann surfaces. This study has a long history involving calculus, complex analysis, and low dimensional topology. The moduli space of all compact Riemann surfaces has a very rich geometry and enumerative structure, which is an object of much current research, and has surprising connections with fields as diverse as geometric topology in dimensions two and three, nonlinear partial differential equations, and conformal field theory and string theory. Many questions posed by physicists have been solved by using the wealth of techniques developed by algebraic geometers. In turn, physics questions have led to new conjectures and new methods in this very central area of mathematics. Finally, the proposed ten dimensional space-time of string theory involves six very small extra dimensions, which correspond to certain three dimensional algebraic varieties, Calabi-Yau manifolds. These manifolds were already of great interest to mathematicians. Amazing ideas from physics have suggested that Calabi-Yau manifolds come in pairs. The geometry of the so-called mirror manifold of a Calabi-Yau manifold turns out to be connected to classical enumerative questions on the original manifold.
📚 @topmathlearn📚
#هندسه_جبری
❤5
🌐 Applied Mathematics (ریاضی کاربردی)
Applied mathematics is concerned with formulating, analyzing and solving an application problem, whether it arises from physical science or biological science, using mathematical language, theories and techniques. The hallmark of applied mathematics is mathematical modeling, which may be regarded as the art of abstraction that helps us realize the truth. The power of modeling and abstraction has been clearly evidenced by the enormous advancements and progress of physical sciences in the past two centuries. As the solutions to all these application problems hinge on the solutions to their mathematical models, analyzing and solving these problems has been critically important for the resolution of many scientific, engineering, and industrial application problems. However, except in some very simple situations, closed form solutions do not exist even for simple, linear models. As a result, seeking numerical (approximate) solutions becomes the only viable way to find (and to see) the solutions of these mathematical models. The computations involved in solving a mathematical model, in particular, a nonlinear model, are often enormous, which are impossible to carry out by hand. On the other hand, the job is perfect for computers because they are good at number-crunching.
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To realize the potential, a big (and difficult) question which must be answered is how to utilize computers’ superpower of doing arithmetic calculations to compute solutions of mathematical models. The key to this question is to develop reliable and efficient numerical/computational methods and algorithms which can be efficiently implemented on computers. To develop, analyze and implement these enabling methods and algorithms for all kinds of mathematical problems is a scientific field called Computational Mathematics, a sub-field of it, that focuses on partial differential equations (PDEs) related mathematical problems, is known as numerical PDEs, which is one of main focuses in Computational Mathematics. Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.
📚 @topmathlearn 📚
Applied mathematics is concerned with formulating, analyzing and solving an application problem, whether it arises from physical science or biological science, using mathematical language, theories and techniques. The hallmark of applied mathematics is mathematical modeling, which may be regarded as the art of abstraction that helps us realize the truth. The power of modeling and abstraction has been clearly evidenced by the enormous advancements and progress of physical sciences in the past two centuries. As the solutions to all these application problems hinge on the solutions to their mathematical models, analyzing and solving these problems has been critically important for the resolution of many scientific, engineering, and industrial application problems. However, except in some very simple situations, closed form solutions do not exist even for simple, linear models. As a result, seeking numerical (approximate) solutions becomes the only viable way to find (and to see) the solutions of these mathematical models. The computations involved in solving a mathematical model, in particular, a nonlinear model, are often enormous, which are impossible to carry out by hand. On the other hand, the job is perfect for computers because they are good at number-crunching.
📚 @topmathlearn 📚
To realize the potential, a big (and difficult) question which must be answered is how to utilize computers’ superpower of doing arithmetic calculations to compute solutions of mathematical models. The key to this question is to develop reliable and efficient numerical/computational methods and algorithms which can be efficiently implemented on computers. To develop, analyze and implement these enabling methods and algorithms for all kinds of mathematical problems is a scientific field called Computational Mathematics, a sub-field of it, that focuses on partial differential equations (PDEs) related mathematical problems, is known as numerical PDEs, which is one of main focuses in Computational Mathematics. Computational Mathematics involves mathematical research in areas of science and engineering where computing plays a central and essential role. Topics include for example developing accurate and efficient numerical methods for solving physical or biological models, analysis of numerical approximations to differential and integral equations, developing computational tools to better understand data and structure, etc. Computational mathematics is a field closely connected with a variety of other mathematical branches, as for often times a better mathematical understanding of the problem leads to innovative numerical techniques.
📚 @topmathlearn 📚
❤4
The centerpieces of Computational Mathematics are error and speed. Errors are everywhere; they occur in every step of an algorithm (i.e., approximation error) and in every calculation done by a computer (i.e., round-off error). If a beautiful method or algorithm on paper does not have a mechanism to control its error accumulation when implemented on computers, then it is useless. Such an error-control capability is the watershed to distinguish “good” methods and algorithms from “bad” ones. Among good methods and algorithms of the same nature, a faster (in terms of computer execution time) method or algorithm is considered to be a better method or algorithm. Developing good and fast computational methods and algorithms is the heart and soul of Computational Mathematics. Problems in all areas of mathematics, applied science, engineering, economics, medicine and statistics can be posed as mathematical optimization problems. An optimization problem begins with a set of independent variables and often includes conditions or restrictions that define acceptable values of the variables. Such restrictions are known as the constraints of the problem. The other essential component of an optimization problem is a single measure of goodness, termed the objective function, which depends in some way on the variables. The solution of an optimization problem is a set of allowed values of the variables for which the objective function assumes its optimal value. In mathematical terms, this usually involves maximizing or minimizing.
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#ریاضی_کاربردی
📚 @topmathlearn 📚
#ریاضی_کاربردی
❤4
🌐 Statistics & Probability (آمار و احتمال)
Statistics is the science concerned with developing and studying methods for collecting, analyzing, interpreting, and presenting empirical data. Research in statistics finds applicability in virtually all scientific fields and research questions in the various scientific fields motivate the development of new statistical methods and theory. Two fundamental ideas in the field of statistics are uncertainty and variation. Mathematical statistics concerns the logical arguments underlying the justification of statistical methods and inference. Changes in technology are creating an exponential increase in the amount of data available to science and business, but the size and complexity of modern data sets require new mathematical theory. Probability and Stochastic processes are concerned with the mathematical analysis of random phenomena. Probability plays a key role in statistics and has long been an indispensable tool in applied modeling. It is younger than most other areas of Mathematics, as it originated with the study of games of chance by Pascal and Fermat (1654); since then, it has become indispensable for dealing with randomness in almost all branches of science. Although probability began in the early 1600s, the subject as we know it today dates from the twentieth century, when powerful tools from abstract analysis brought new rigor to the field. As scientists study nature in ever greater detail, they increasingly find that random effects play an essential role in almost all denoscriptions of natural phenomena. These new insights enriched probability, and spilled over into many other parts of mathematics.
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Today, in addition to its strong position as a full-fledged area within mathematics and its interactions with analysis, partial differential equations, geometry, combinatorics, number theory, and mathematical physics, probability forms the language of statistics and of the quantitative social sciences. It is essential in the discovery and study of macroscopic regularities that occur when large systems of particles, organisms, or agents, interact according to the laws of physics, biology, or economics; in the study of population genetics and of the genome; in the study of how signals are transmitted through a noisy channel, and then recovered; in the design and analysis of large-scale communication, neural or queueing networks, and of algorithms for combinatorial optimization, computerized tomography, signal processing, pattern recognition, and so on. Probability theory provides the foundation on which signal processing and filtering theory are built in engineering, and it played a critical role in the mathematical attempts to rationalize quantum field theory. Stochastic processes add dynamics to random changes to study the evolution of random systems in time and/or space. Since a great number of phenomena can be best modeled or understood by probabilistic means, probability and stochastic processes provide indispensable tools for applied modeling in a number of fields such as statistical physics, financial mathematics, mathematical biology, data science, machine learning, computational methods, Bayesian statistics, graph theory, and dynamical systems.
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#آمار_احتمال
Statistics is the science concerned with developing and studying methods for collecting, analyzing, interpreting, and presenting empirical data. Research in statistics finds applicability in virtually all scientific fields and research questions in the various scientific fields motivate the development of new statistical methods and theory. Two fundamental ideas in the field of statistics are uncertainty and variation. Mathematical statistics concerns the logical arguments underlying the justification of statistical methods and inference. Changes in technology are creating an exponential increase in the amount of data available to science and business, but the size and complexity of modern data sets require new mathematical theory. Probability and Stochastic processes are concerned with the mathematical analysis of random phenomena. Probability plays a key role in statistics and has long been an indispensable tool in applied modeling. It is younger than most other areas of Mathematics, as it originated with the study of games of chance by Pascal and Fermat (1654); since then, it has become indispensable for dealing with randomness in almost all branches of science. Although probability began in the early 1600s, the subject as we know it today dates from the twentieth century, when powerful tools from abstract analysis brought new rigor to the field. As scientists study nature in ever greater detail, they increasingly find that random effects play an essential role in almost all denoscriptions of natural phenomena. These new insights enriched probability, and spilled over into many other parts of mathematics.
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Today, in addition to its strong position as a full-fledged area within mathematics and its interactions with analysis, partial differential equations, geometry, combinatorics, number theory, and mathematical physics, probability forms the language of statistics and of the quantitative social sciences. It is essential in the discovery and study of macroscopic regularities that occur when large systems of particles, organisms, or agents, interact according to the laws of physics, biology, or economics; in the study of population genetics and of the genome; in the study of how signals are transmitted through a noisy channel, and then recovered; in the design and analysis of large-scale communication, neural or queueing networks, and of algorithms for combinatorial optimization, computerized tomography, signal processing, pattern recognition, and so on. Probability theory provides the foundation on which signal processing and filtering theory are built in engineering, and it played a critical role in the mathematical attempts to rationalize quantum field theory. Stochastic processes add dynamics to random changes to study the evolution of random systems in time and/or space. Since a great number of phenomena can be best modeled or understood by probabilistic means, probability and stochastic processes provide indispensable tools for applied modeling in a number of fields such as statistical physics, financial mathematics, mathematical biology, data science, machine learning, computational methods, Bayesian statistics, graph theory, and dynamical systems.
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#آمار_احتمال
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🌐 Financial Mathematics (ریاضی مالی)
Financial Mathematics is the field of applied mathematics and involves defining problems in finance and providing solutions using the application of mathematical methods to financial problems. It is sometimes referred to as quantitative finance, financial engineering, mathematical finance and computational finance. Financial mathematics is a special area of probability and mathematical statistics focusing on mathematical models of financial markets and, in general, on mathematical areas relevant for the financial (and insurance) industry. Probability has also been central in the study of finance, ever since Bachelier pioneered in 1900 the mathematical study of Brownian motion and understood its significance as a tool for the analysis of financial markets (five years before Einstein developed his physical theory of Brownian motion). The use of mathematics and statistics within the field of finance has been increasing substantially in the past, and such a trend is expected to continue. The discipline combines tools from statistics, probability and stochastic processes, and combines them with economic theory. Financial mathematics is also heavily linked to analysis, numerical mathematics and optimization. Needless to say that it is also heavily related to economics and business studies, as the mathematical modelling of economic markets and agents needs considerable insight into them. As the markets seek to become more efficient, quantitative methods will continue to be adopted. Over the long history of financial markets, the concepts of valuation and pricing, as well as optimizing capital allocation, have been important problems to observe within the capital markets.
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The theory of finance tries to understand how financial markets work, how they can be made more efficient, how they should be regulated, and how they can help manage the risk inherent in various economic activities. Traditionally, investment banks, commercial banks, hedge funds, insurance companies, corporate treasuries, and regulatory agencies apply the methods of financial mathematics to such problems as derivative securities valuation, portfolio structuring, risk management, and scenario simulation. Industries that rely on commodities (e.g., energy, manufacturing) also use financial mathematics. Quantitative analysis has brought efficiency and rigor to financial markets and the investment process and is becoming increasingly important in regulatory concerns. Quantitative Finance was developed as a specialized field within economics to tackle the problems of the valuation of assets and financial instruments, as well as optimizing capital allocation and resources. Over the centuries, fundamental theories about the overall economy and valuation of assets have been developed through mathematical models. Models describe relationships between fundamental variables such as asset prices, market movements, volatility and interest rates. By using quantitative tools, more accurate conclusions can be drawn from the economic variables. These mathematical tools allow us to draw conclusions that can be otherwise difficult to find or not immediately obvious from intuition. A large part of any science is the ability to create testable hypotheses based on a fundamental understanding of the objects of study and prove or contradict the hypotheses through repeatable studies. In this light, mathematics is the language for representing theories and provides tools for testing their validity.
📚 @topmathlearn 📚
#ریاضی_مالی
Financial Mathematics is the field of applied mathematics and involves defining problems in finance and providing solutions using the application of mathematical methods to financial problems. It is sometimes referred to as quantitative finance, financial engineering, mathematical finance and computational finance. Financial mathematics is a special area of probability and mathematical statistics focusing on mathematical models of financial markets and, in general, on mathematical areas relevant for the financial (and insurance) industry. Probability has also been central in the study of finance, ever since Bachelier pioneered in 1900 the mathematical study of Brownian motion and understood its significance as a tool for the analysis of financial markets (five years before Einstein developed his physical theory of Brownian motion). The use of mathematics and statistics within the field of finance has been increasing substantially in the past, and such a trend is expected to continue. The discipline combines tools from statistics, probability and stochastic processes, and combines them with economic theory. Financial mathematics is also heavily linked to analysis, numerical mathematics and optimization. Needless to say that it is also heavily related to economics and business studies, as the mathematical modelling of economic markets and agents needs considerable insight into them. As the markets seek to become more efficient, quantitative methods will continue to be adopted. Over the long history of financial markets, the concepts of valuation and pricing, as well as optimizing capital allocation, have been important problems to observe within the capital markets.
📚 @topmathlearn 📚
The theory of finance tries to understand how financial markets work, how they can be made more efficient, how they should be regulated, and how they can help manage the risk inherent in various economic activities. Traditionally, investment banks, commercial banks, hedge funds, insurance companies, corporate treasuries, and regulatory agencies apply the methods of financial mathematics to such problems as derivative securities valuation, portfolio structuring, risk management, and scenario simulation. Industries that rely on commodities (e.g., energy, manufacturing) also use financial mathematics. Quantitative analysis has brought efficiency and rigor to financial markets and the investment process and is becoming increasingly important in regulatory concerns. Quantitative Finance was developed as a specialized field within economics to tackle the problems of the valuation of assets and financial instruments, as well as optimizing capital allocation and resources. Over the centuries, fundamental theories about the overall economy and valuation of assets have been developed through mathematical models. Models describe relationships between fundamental variables such as asset prices, market movements, volatility and interest rates. By using quantitative tools, more accurate conclusions can be drawn from the economic variables. These mathematical tools allow us to draw conclusions that can be otherwise difficult to find or not immediately obvious from intuition. A large part of any science is the ability to create testable hypotheses based on a fundamental understanding of the objects of study and prove or contradict the hypotheses through repeatable studies. In this light, mathematics is the language for representing theories and provides tools for testing their validity.
📚 @topmathlearn 📚
#ریاضی_مالی
❤4
🌐 Computer Science (علوم کامپیوتر)
Computer Science is the study of computers and computational systems at a deep level, including their theoretical and algorithmic foundations, hardware and software, and their uses for processing information. Computer Science emerged as an independent discipline in the early 1960s, although the electronic digital computer that is the object of its study was invented some two decades earlier. The discipline of Computer Science includes the study of algorithms, data structures and security, computer and network design, modeling data and information processes, and artificial intelligence. Computer scientists define the computational principles that are the basis of all software and use various mathematical algorithms, coding procedures, and their expert programming skills to study computer processes and develop and test new software and software systems. You will use your understanding of mathematical reasoning to reason rigorously about the behaviours of computer programmes and systems.
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Theoretical Computer Science is a natural bridge between mathematics and computer science, and studies computational and algorithmic processes and interactions. This field concerns the foundations of efficient and correct algorithms, and it deals with the tractability of computational problems. Work in this field is mathematical rather than empirical, and offers a unique perspective that excels at exposing and answering key questions about the possibilities and limitations of computation, broadly construed. Theoretical Computer Science comprises two subfields: the theory of algorithms, which involves the design and analysis of computational procedures; and complexity theory, which involves efforts to prove that no efficient algorithms exist in certain cases, and which investigates the classification system for computational tasks. Time, memory, randomness, and parallelism are typical measures of computational effort. The study of Theoretical Computer Science focuses on a deeper understanding of the fundamental principles underlying contemporary information technologies, including non-classical computing systems such as neural networks and quantum computers. In addition to the active acquisition of knowledge of a theoretical and technical nature, special emphasis is placed on the development of abstract thinking.
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#علوم_کامپیوتر
Computer Science is the study of computers and computational systems at a deep level, including their theoretical and algorithmic foundations, hardware and software, and their uses for processing information. Computer Science emerged as an independent discipline in the early 1960s, although the electronic digital computer that is the object of its study was invented some two decades earlier. The discipline of Computer Science includes the study of algorithms, data structures and security, computer and network design, modeling data and information processes, and artificial intelligence. Computer scientists define the computational principles that are the basis of all software and use various mathematical algorithms, coding procedures, and their expert programming skills to study computer processes and develop and test new software and software systems. You will use your understanding of mathematical reasoning to reason rigorously about the behaviours of computer programmes and systems.
📚 @topmathlearn 📚
Theoretical Computer Science is a natural bridge between mathematics and computer science, and studies computational and algorithmic processes and interactions. This field concerns the foundations of efficient and correct algorithms, and it deals with the tractability of computational problems. Work in this field is mathematical rather than empirical, and offers a unique perspective that excels at exposing and answering key questions about the possibilities and limitations of computation, broadly construed. Theoretical Computer Science comprises two subfields: the theory of algorithms, which involves the design and analysis of computational procedures; and complexity theory, which involves efforts to prove that no efficient algorithms exist in certain cases, and which investigates the classification system for computational tasks. Time, memory, randomness, and parallelism are typical measures of computational effort. The study of Theoretical Computer Science focuses on a deeper understanding of the fundamental principles underlying contemporary information technologies, including non-classical computing systems such as neural networks and quantum computers. In addition to the active acquisition of knowledge of a theoretical and technical nature, special emphasis is placed on the development of abstract thinking.
📚 @topmathlearn 📚
#علوم_کامپیوتر
❤4