🌐 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.
📚 @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
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.
📚 @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 📚
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.
📚 @topmathlearn 📚
#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.
📚 @topmathlearn 📚
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.
📚 @topmathlearn 📚
#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.
📚 @topmathlearn 📚
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.
📚 @topmathlearn 📚
#آمار_احتمال
❤4
🌐 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.
📚 @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
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.
📚 @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
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
🌐 Data Science (علوم داده)
Data Science is a combination of multiple disciplines that uses statistics, data analysis, and machine learning to analyze data and extract knowledge and meaningful insights and develop strategies from it for business and industry. By the late 1990s, the term "data science" had gained broader recognition. In 2001, American computer scientist and professor William S. Cleveland outlined a broader vision for statistics that shifted from the traditional theoretical one to a more applied, data-centric focus. This way, a new field would emerge that integrates elements of machine learning, visualization, and computing. The emergence of big data in the early 21st century truly cemented data science as the discipline for working with and making sense of complex, large-scale information. Data Science aims at gaining insights about complex real-world effects through information from existing datasets. Modern data-centric approaches combine deep foundations in Statistics and Applied Mathematics with state-of-the-art algorithms and provide a basis for Computer Science, Artificial Intelligence (AI), and Machine Learning. Data-enabled discoveries, permitted only due to recent methods and the advent of modern computing power, accelerate innovation across the Sciences and Engineering and bridge together distant fields, giving rise to Information Engineering and Bioinformatics. Complex data problems require a mathematical disciplinary interplay to develop new theories and address interdisciplinary questions. For example, in recent years, there have been many new theoretical developments combining ideas from topology and geometry with statistical and machine learning methods, for data analysis, visualization, and dimensionality reduction. Applications range from classification and clustering in fields such as action recognition, handwriting analysis, natural language processing, and biology, to the analysis of complex systems.
📚 @topmathlearn 📚
The term data scientist was coined as recently as 2008, when companies first realized the need for data professionals skilled in organizing and analyzing massive amounts of data. Data scientists gather, organize, clean, and analyze data much like data analysts, but they are more forward-looking and prediction-oriented. They use the data to build machine learning models and use the resulting information to develop data-driven solutions to difficult challenges across a variety of industries. Combining tools, methods, and technology such as data analysis/modeling, human-machine interaction and algorithms, data scientists ask and answer questions like what happened, why did it happen, what will happen, and how can the results be extrapolated and used for planning and decision-making. Nowadays, there is an overwhelming amount of data generated and effective data scientists masterfully identify relevant questions, collect data from a multitude of different data sources, organize the information, translate results into solutions, and communicate their findings in a way that positively affects business decisions. These skills are now required in almost all industries, causing skilled data scientists to be increasingly valuable to companies. Data science continues to evolve as one of the most promising and in-demand career paths for skilled professionals with an aptitude for research, programming, mathematics, and computers. Today, successful data professionals understand they must advance past the traditional skills of analyzing large amounts of data, data mining, and programming skills. To uncover useful intelligence for their organizations, data scientists must master the full spectrum of the data science life cycle and possess a level of flexibility and understanding to maximize returns at each phase of the process.
📚 @topmathlearn 📚
#علوم_داده
Data Science is a combination of multiple disciplines that uses statistics, data analysis, and machine learning to analyze data and extract knowledge and meaningful insights and develop strategies from it for business and industry. By the late 1990s, the term "data science" had gained broader recognition. In 2001, American computer scientist and professor William S. Cleveland outlined a broader vision for statistics that shifted from the traditional theoretical one to a more applied, data-centric focus. This way, a new field would emerge that integrates elements of machine learning, visualization, and computing. The emergence of big data in the early 21st century truly cemented data science as the discipline for working with and making sense of complex, large-scale information. Data Science aims at gaining insights about complex real-world effects through information from existing datasets. Modern data-centric approaches combine deep foundations in Statistics and Applied Mathematics with state-of-the-art algorithms and provide a basis for Computer Science, Artificial Intelligence (AI), and Machine Learning. Data-enabled discoveries, permitted only due to recent methods and the advent of modern computing power, accelerate innovation across the Sciences and Engineering and bridge together distant fields, giving rise to Information Engineering and Bioinformatics. Complex data problems require a mathematical disciplinary interplay to develop new theories and address interdisciplinary questions. For example, in recent years, there have been many new theoretical developments combining ideas from topology and geometry with statistical and machine learning methods, for data analysis, visualization, and dimensionality reduction. Applications range from classification and clustering in fields such as action recognition, handwriting analysis, natural language processing, and biology, to the analysis of complex systems.
📚 @topmathlearn 📚
The term data scientist was coined as recently as 2008, when companies first realized the need for data professionals skilled in organizing and analyzing massive amounts of data. Data scientists gather, organize, clean, and analyze data much like data analysts, but they are more forward-looking and prediction-oriented. They use the data to build machine learning models and use the resulting information to develop data-driven solutions to difficult challenges across a variety of industries. Combining tools, methods, and technology such as data analysis/modeling, human-machine interaction and algorithms, data scientists ask and answer questions like what happened, why did it happen, what will happen, and how can the results be extrapolated and used for planning and decision-making. Nowadays, there is an overwhelming amount of data generated and effective data scientists masterfully identify relevant questions, collect data from a multitude of different data sources, organize the information, translate results into solutions, and communicate their findings in a way that positively affects business decisions. These skills are now required in almost all industries, causing skilled data scientists to be increasingly valuable to companies. Data science continues to evolve as one of the most promising and in-demand career paths for skilled professionals with an aptitude for research, programming, mathematics, and computers. Today, successful data professionals understand they must advance past the traditional skills of analyzing large amounts of data, data mining, and programming skills. To uncover useful intelligence for their organizations, data scientists must master the full spectrum of the data science life cycle and possess a level of flexibility and understanding to maximize returns at each phase of the process.
📚 @topmathlearn 📚
#علوم_داده
❤4
🌐 Mathematics Education (آموزش ریاضی)
Mathematics Education is the field of knowledge and research that deals with theories, practices, policies, curriculum, and issues about the teaching and learning of mathematics meaningfully and considers how people think about mathematics, how they learn it, and how they teach it. One of its major goals is to develop theories of learning and teaching that will be foundational for strategies and curricular materials that improve the conceptual learning and use of mathematics. The field as an academic discipline involves systematic research into aspects of the learning and teaching of mathematics at all levels of education, with a focus on understanding, problem-solving, and the construction of mathematical knowledge and cognitive skills. It also involves designing effective teaching environments and creating innovative assessment methods aligned with the goals of mathematics education. It includes aspects of psychology, sociology, philosophy, and history - all studied through the lens of mathematics, and will be of interest to students who would like to better understand the way human beings think about mathematics and the place of mathematics in society. Mathematics Education concerns the relationship of mathematics and people, including comparisons of mathematics in different cultures, and how mathematics is used to make a difference in the world. Curriculum plays a significant role in mathematics education because it influences what students learn and when, and how well they learn it. Advocates of mathematics education reform often attempt to change classroom practice, and thereby students’ learning, by changing the curriculum. Historically, curriculum conveys what students should learn and, at the same time also serves as an agent for instructional improvement.
📚 @topmathlearn 📚
The field of mathematics education research lies at the intersection of pedagogy and mathematics. As this intersection is quite large, the field is quite broad in scope. Researchers might be interested in how students conceptualize transformations in linear algebra, different approaches to teaching proof by contradiction, curriculum design in an algebra course, the role of technology in middle school classrooms, or the influence of factors such as race or gender on the learning of mathematics. Researchers use a mixture of qualitative (interviews, ethnographies) and quantitative tools (data science, statistics) to explore phenomena from small scales (individual students) to large scales (school districts, states). Overall, the goal is to improve mathematics instruction using the tools of academic research. With such a broad knowledge base, teachers and mathematics enthusiasts always have something new to learn. For some decades now, research in mathematics education has been emerging as a field of study in its own right. After a long struggle for its identity, today one can say that mathematics education has become a full-fledged, interdisciplinary field of research and analysis, aiming at a better understanding of the processes underlying the acquisition and development of mathematical knowledge, skills, beliefs, and attitudes applied to the design of valuable tools and powerful environments for teaching and learning mathematics.
📚 @topmathlearn 📚
#آموزش_ریاضی
Mathematics Education is the field of knowledge and research that deals with theories, practices, policies, curriculum, and issues about the teaching and learning of mathematics meaningfully and considers how people think about mathematics, how they learn it, and how they teach it. One of its major goals is to develop theories of learning and teaching that will be foundational for strategies and curricular materials that improve the conceptual learning and use of mathematics. The field as an academic discipline involves systematic research into aspects of the learning and teaching of mathematics at all levels of education, with a focus on understanding, problem-solving, and the construction of mathematical knowledge and cognitive skills. It also involves designing effective teaching environments and creating innovative assessment methods aligned with the goals of mathematics education. It includes aspects of psychology, sociology, philosophy, and history - all studied through the lens of mathematics, and will be of interest to students who would like to better understand the way human beings think about mathematics and the place of mathematics in society. Mathematics Education concerns the relationship of mathematics and people, including comparisons of mathematics in different cultures, and how mathematics is used to make a difference in the world. Curriculum plays a significant role in mathematics education because it influences what students learn and when, and how well they learn it. Advocates of mathematics education reform often attempt to change classroom practice, and thereby students’ learning, by changing the curriculum. Historically, curriculum conveys what students should learn and, at the same time also serves as an agent for instructional improvement.
📚 @topmathlearn 📚
The field of mathematics education research lies at the intersection of pedagogy and mathematics. As this intersection is quite large, the field is quite broad in scope. Researchers might be interested in how students conceptualize transformations in linear algebra, different approaches to teaching proof by contradiction, curriculum design in an algebra course, the role of technology in middle school classrooms, or the influence of factors such as race or gender on the learning of mathematics. Researchers use a mixture of qualitative (interviews, ethnographies) and quantitative tools (data science, statistics) to explore phenomena from small scales (individual students) to large scales (school districts, states). Overall, the goal is to improve mathematics instruction using the tools of academic research. With such a broad knowledge base, teachers and mathematics enthusiasts always have something new to learn. For some decades now, research in mathematics education has been emerging as a field of study in its own right. After a long struggle for its identity, today one can say that mathematics education has become a full-fledged, interdisciplinary field of research and analysis, aiming at a better understanding of the processes underlying the acquisition and development of mathematical knowledge, skills, beliefs, and attitudes applied to the design of valuable tools and powerful environments for teaching and learning mathematics.
📚 @topmathlearn 📚
#آموزش_ریاضی
❤4
🌐 Mathematical Biology (زیست ریاضی)
Mathematical Biology (also known as Biomathematics or Mathematical and Theoretical Biology) is an interdisciplinary field that uses mathematics to understand biological processes and is concerned with the construction of mathematical models to describe and solve biological problems. The field involves the development of novel mathematical tools to provide insight into biological processes and uses mathematical models and analyses and representations of living organisms to examine the systems that govern the structure, development, and behaviour of and within biological systems. Mathematical Biology relies on a more theoretical approach and analysis to solve problems, rather than using experiments to prove theories like its experimental biology counterpart. Its scope of application ranges from the microscopic level, such as cellular processes and genetic networks, to the macroscopic level, including the dynamics of organisms, populations, ecosystems, and evolutionary biology. Indeed, given its ability to reveal otherwise invisible worlds in all kinds of biological systems, mathematics has been called the "new microscope in biology". To more easily theorize the impacts of environmental factors on living organisms, biologists and mathematicians sometimes combine forces to create mathematical models to identify, understand, and analyze how these factors might come into effect in the real world. In recent decades, an explosive synergy between biology and mathematics has greatly enriched and extended both fields. In turn, biology has stimulated the creation of new realms of mathematics.
📚 @topmathlearn 📚
Computational biology and bioinformatics develop and apply techniques from applied mathematics, statistics, computer science, physics and chemistry to study biological problems, from molecular to macro-evolutionary. This area of study seeks to model, analyze, interpret, and predict various biological phenomena by means of both novel and existing mathematical techniques. This is done in close collaboration with empirical researchers from biological sciences, medicine, engineering, and public health, among others. By drawing insights from biological systems, new directions in mathematics and other areas may emerge. By formulating mathematical models, mathematicians can describe biological systems, predict their behavior under different conditions, and gain insights into their underlying mechanisms. These models can take the form of ordinary and partial differential equations, stochastic processes, statistical models, and computational simulations, allowing for a quantitative understanding of complex biological interactions. The studies of complex social phenomena also belong under the scope of mathematical life sciences and include the modeling of evolution of language, learning, categorization, and human behavior in general.
📚 @topmathlearn 📚
#زیست_ریاضی
Mathematical Biology (also known as Biomathematics or Mathematical and Theoretical Biology) is an interdisciplinary field that uses mathematics to understand biological processes and is concerned with the construction of mathematical models to describe and solve biological problems. The field involves the development of novel mathematical tools to provide insight into biological processes and uses mathematical models and analyses and representations of living organisms to examine the systems that govern the structure, development, and behaviour of and within biological systems. Mathematical Biology relies on a more theoretical approach and analysis to solve problems, rather than using experiments to prove theories like its experimental biology counterpart. Its scope of application ranges from the microscopic level, such as cellular processes and genetic networks, to the macroscopic level, including the dynamics of organisms, populations, ecosystems, and evolutionary biology. Indeed, given its ability to reveal otherwise invisible worlds in all kinds of biological systems, mathematics has been called the "new microscope in biology". To more easily theorize the impacts of environmental factors on living organisms, biologists and mathematicians sometimes combine forces to create mathematical models to identify, understand, and analyze how these factors might come into effect in the real world. In recent decades, an explosive synergy between biology and mathematics has greatly enriched and extended both fields. In turn, biology has stimulated the creation of new realms of mathematics.
📚 @topmathlearn 📚
Computational biology and bioinformatics develop and apply techniques from applied mathematics, statistics, computer science, physics and chemistry to study biological problems, from molecular to macro-evolutionary. This area of study seeks to model, analyze, interpret, and predict various biological phenomena by means of both novel and existing mathematical techniques. This is done in close collaboration with empirical researchers from biological sciences, medicine, engineering, and public health, among others. By drawing insights from biological systems, new directions in mathematics and other areas may emerge. By formulating mathematical models, mathematicians can describe biological systems, predict their behavior under different conditions, and gain insights into their underlying mechanisms. These models can take the form of ordinary and partial differential equations, stochastic processes, statistical models, and computational simulations, allowing for a quantitative understanding of complex biological interactions. The studies of complex social phenomena also belong under the scope of mathematical life sciences and include the modeling of evolution of language, learning, categorization, and human behavior in general.
📚 @topmathlearn 📚
#زیست_ریاضی
❤4
🌐 Mathematical Physics (فیزیک ریاضی)
Mathematical Physics includes the development of new mathematical methods for application to problems in physics, as well as the mathematically rigorous proofs of claims that physicists have only supported with nonrigorous arguments. Mathematics and Physics are traditionally tightly linked subjects, and many historical figures, such as Newton and Gauss, were both physicists and mathematicians. Traditionally Mathematical Physics has been quite closely associated to ideas in calculus, particularly those of differential equations. In recent years however, in part due to the rise of superstring theory, many more branches of mathematics have become major contributors to physics. Mathematical Physics works with topics and concepts such as vector spaces, matrix algebra, differential equations, integral equations, integral transforms, infinite series, and complex variables. It is a very broad topic that, in basic terms, applies mathematical ideas to physics and engineering problems. Mathematical Physics encompasses branches such as statistical mechanics, atomic and molecular physics, electromagnetism, quantum field theory, classical mechanics, and theoretical physics. The formulation of fundamental physical laws has always been tightly interwoven with the deepest mathematics available. This is even more evident today, with geometry providing the foundations of general relativity, gauge theories, string theory, and models of statistical physics. In turn, developments in these fields of physics have been instrumental in driving progress in various areas of mathematics, such as Riemann surface theory, quantum geometry, knot theory, mirror symmetry, representation theory, nonlinear partial differential equations, and differential geometry. Studying Mathematical Physics reveals the hidden intricacies of relativity and quantum theory. The cross-fertilization between mathematics and physics has perhaps never been richer.
📚 @topmathlearn 📚
There is a long history of fruitful exchanges between Probability Theory and Statistical Mechanics and Mathematical Physics. Models of physical systems present challenging and relevant problems to probabilists; ideas and techniques flow back and forth. A general problem is to understand how microscopic noise and local interactions generate random structures on the macroscopic scale, and to describe qualitatively and quantitatively these structures. While the formulation of basic models of physical phenomena, such as percolation, the Ising model, random growth models and interacting particle systems, is completely elementary, their analysis involves a wide array of techniques and concepts: combinatorics and graphs, complex analysis and algebraic curves, operators and representation theory, symmetric functions and integrable systems, Brownian motion, Gaussian fields, stochastic partial differential equations. The fact that current progress toward the understanding of these models involves such diverse ideas is a testimony to the vivacity and depth of the field. Mathematical research in physical modeling focuses on the formulation and analysis of mathematical representations of problems motivated by other branches of science and engineering. In addition to generating novel problems with new computational and analytical challenges, constructing accurate models for complex systems may uncover the need for fundamental extensions to the governing equations.
📚 @topmathlearn 📚
#فیزیک_ریاضی
Mathematical Physics includes the development of new mathematical methods for application to problems in physics, as well as the mathematically rigorous proofs of claims that physicists have only supported with nonrigorous arguments. Mathematics and Physics are traditionally tightly linked subjects, and many historical figures, such as Newton and Gauss, were both physicists and mathematicians. Traditionally Mathematical Physics has been quite closely associated to ideas in calculus, particularly those of differential equations. In recent years however, in part due to the rise of superstring theory, many more branches of mathematics have become major contributors to physics. Mathematical Physics works with topics and concepts such as vector spaces, matrix algebra, differential equations, integral equations, integral transforms, infinite series, and complex variables. It is a very broad topic that, in basic terms, applies mathematical ideas to physics and engineering problems. Mathematical Physics encompasses branches such as statistical mechanics, atomic and molecular physics, electromagnetism, quantum field theory, classical mechanics, and theoretical physics. The formulation of fundamental physical laws has always been tightly interwoven with the deepest mathematics available. This is even more evident today, with geometry providing the foundations of general relativity, gauge theories, string theory, and models of statistical physics. In turn, developments in these fields of physics have been instrumental in driving progress in various areas of mathematics, such as Riemann surface theory, quantum geometry, knot theory, mirror symmetry, representation theory, nonlinear partial differential equations, and differential geometry. Studying Mathematical Physics reveals the hidden intricacies of relativity and quantum theory. The cross-fertilization between mathematics and physics has perhaps never been richer.
📚 @topmathlearn 📚
There is a long history of fruitful exchanges between Probability Theory and Statistical Mechanics and Mathematical Physics. Models of physical systems present challenging and relevant problems to probabilists; ideas and techniques flow back and forth. A general problem is to understand how microscopic noise and local interactions generate random structures on the macroscopic scale, and to describe qualitatively and quantitatively these structures. While the formulation of basic models of physical phenomena, such as percolation, the Ising model, random growth models and interacting particle systems, is completely elementary, their analysis involves a wide array of techniques and concepts: combinatorics and graphs, complex analysis and algebraic curves, operators and representation theory, symmetric functions and integrable systems, Brownian motion, Gaussian fields, stochastic partial differential equations. The fact that current progress toward the understanding of these models involves such diverse ideas is a testimony to the vivacity and depth of the field. Mathematical research in physical modeling focuses on the formulation and analysis of mathematical representations of problems motivated by other branches of science and engineering. In addition to generating novel problems with new computational and analytical challenges, constructing accurate models for complex systems may uncover the need for fundamental extensions to the governing equations.
📚 @topmathlearn 📚
#فیزیک_ریاضی
❤4
🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃
👩💻🧑💻 دوستان گرامی، برای معرفی گرایشها و زیرشاخههای مختلف رشته ریاضی به هشتگهای زیر مراجعه کنید.
✅ لازم به ذکر است که توضیحات هر شاخه و زیرشاخه بهمنظور یادگیری اصطلاحات و کلمات کلیدی و درک بهتر آنها به زبان انگلیسی میباشد.
#جبر
#ترکیبیات
#نظریه_اعداد
#نظریه_نمایش
#منطق_ریاضی
#توپولوژی
#توپولوژی_جبری
#آنالیز
#معادلات_دیفرانسیل_جزئی
#هندسه
#هندسه_دیفرانسیل
#هندسه_جبری
#ریاضی_کاربردی
#آمار
#ریاضی_مالی
#علوم_کامپیوتر
#علوم_داده
#آموزش_ریاضی
#زیست_ریاضی
#فیزیک_ریاضی
📚 @topmathlearn 📚
🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃
👩💻🧑💻 دوستان گرامی، برای معرفی گرایشها و زیرشاخههای مختلف رشته ریاضی به هشتگهای زیر مراجعه کنید.
✅ لازم به ذکر است که توضیحات هر شاخه و زیرشاخه بهمنظور یادگیری اصطلاحات و کلمات کلیدی و درک بهتر آنها به زبان انگلیسی میباشد.
#جبر
#ترکیبیات
#نظریه_اعداد
#نظریه_نمایش
#منطق_ریاضی
#توپولوژی
#توپولوژی_جبری
#آنالیز
#معادلات_دیفرانسیل_جزئی
#هندسه
#هندسه_دیفرانسیل
#هندسه_جبری
#ریاضی_کاربردی
#آمار
#ریاضی_مالی
#علوم_کامپیوتر
#علوم_داده
#آموزش_ریاضی
#زیست_ریاضی
#فیزیک_ریاضی
📚 @topmathlearn 📚
🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃🌸🍃