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Workshop On Numerical Ranges And Numerical Radii
Workshop on Numerical Ranges and Numerical Radii (WONRA) is a biennial workshop series on numerical ranges and numerical radii which began in 1992. About Numerical ranges and numerical radii are useful in the study of matrix and operator theory. These topics have applications in many subjects in pure and applied mathematics, such as quadratic forms, Banach spaces, dilation theory, control theory, numerical analysis, quantum information science Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in p .... History In the early 1970s, numerical range workshops were organized by Frank Bonsall and John Duncan. More activities were started in early 1990s, including the biennial workshop series, which began in 1992, and special issues devoted to this workshop were published. Workshops Symp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Range
In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n \times n matrix ''A'' is the set :W(A) = \left\ where \mathbf^* denotes the conjugate transpose of the vector \mathbf. The numerical range includes, in particular, the diagonal entries of the matrix (obtained by choosing ''x'' equal to the unit vectors along the coordinate axes) and the eigenvalues of the matrix (obtained by choosing ''x'' equal to the eigenvectors). In engineering, numerical ranges are used as a rough estimate of eigenvalues of ''A''. Recently, generalizations of the numerical range are used to study quantum computing. A related concept is the numerical radius, which is the largest absolute value of the numbers in the numerical range, i.e. :r(A) = \sup \ = \sup_ , \langle Ax, x \rangle, . Properties # The numerical range is the range of the Rayleigh quotient. # (Hausdorff–Toeplitz theorem) The numerical range is convex and compact. # W(\a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Forms
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a fixed field , such as the real or complex numbers, and one speaks of a quadratic form over . If K=\mathbb R, and the quadratic form takes zero only when all variables are simultaneously zero, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology ( intersection forms of four-manifolds), and Lie theory (the Killing form). Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dilation Theory
In mathematics, a dilation is a function f from a metric space M into itself that satisfies the identity :d(f(x),f(y))=rd(x,y) for all points x, y \in M, where d(x, y) is the distance from x to y and r is some positive real number. In Euclidean space, such a dilation is a similarity of the space. Dilations change the size but not the shape of an object or figure. Every dilation of a Euclidean space that is not a congruence has a unique fixed point that is called the ''center of dilation''. Some congruences have fixed points and others do not.. See also * Homothety * Dilation (operator theory) In operator theory, a dilation of an operator ''T'' on a Hilbert space ''H'' is an operator on a larger Hilbert space ''K'', whose restriction to ''H'' composed with the orthogonal projection onto ''H'' is ''T''. More formally, let ''T'' be a boun ... References {{DEFAULTSORT:Dilation (Metric Space) Metric geometry ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Control Theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a desired state, while minimizing any ''delay'', ''overshoot'', or ''steady-state error'' and ensuring a level of control stability; often with the aim to achieve a degree of optimality. To do this, a controller with the requisite corrective behavior is required. This controller monitors the controlled process variable (PV), and compares it with the reference or set point (SP). The difference between actual and desired value of the process variable, called the ''error'' signal, or SP-PV error, is applied as feedback to generate a control action to bring the controlled process variable to the same value as the set point. Other aspects which are also studied are controllability and observability. Control theory is used in control system eng ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Information Science
Quantum information science is an interdisciplinary field that seeks to understand the analysis, processing, and transmission of information using quantum mechanics principles. It combines the study of Information science with quantum effects in physics. It includes theoretical issues in computational models and more experimental topics in quantum physics, including what can and cannot be done with quantum information. The term quantum information theory is also used, but it fails to encompass experimental research, and can be confused with a subfield of quantum information science that addresses the processing of quantum information. Scientific and engineering studies To understand quantum teleportation, quantum entanglement and the manufacturing of quantum computer hardware requires a thorough understanding of quantum physics and engineering. Since 2010s, there has been remarkable progress in manufacturing quantum computers, with companies like Google and IBM investing heavily ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frank Bonsall
Frank Featherstone Bonsall FRS (31 March 1920, Crouch End, London – 22 February 2011, Harrogate) was a British mathematician. Personal life Bonsall was born on 31 March 1920, the youngest son of Wilfred C Bonsall and Sarah Frank. His older brother was Arthur Bonsall. He married Gillian Patrick, a Somerville graduate, in 1947. Bonsall and his wife were keen hill-walkers. He wrote two articles for The Scottish Mountaineering Club on the definition of a Munro. After his retirement, Bonsall and his wife moved to Harrogate. Career Bonsall graduated from Bishop's Stortford College in 1938, and studied at Merton College, Oxford. He served in World War II, in the Corps of Royal Engineers, and in India from 1944 to 1946. He lectured at the University of Edinburgh from 1947 to 1948; was visiting associate professor at Oklahoma State University from 1950 to 1951; taught at Newcastle University, with Werner Wolfgang Rogosinski in the 1950s. He taught at the University of Edinburgh, from 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Conferences
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Theory
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two by three matrix", a "-matrix", or a matrix of dimension . Without further specifications, matrices represent linear maps, and allow explicit computations in linear algebra. Therefore, the study of matrices is a large part of linear algebra, and most properties and operations of abstract linear algebra can be expressed in terms of matrices. For example, matrix multiplication represents composition of linear maps. Not all matrices are related to linear algebra. This is, in particular, the case in graph theory, of incidence matrices, and adjacency matrices. ''This article focuses on matrices related to linear algebra, and, un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
