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Sergio Albeverio
Sergio Albeverio (born 17 January 1939) is a Swiss mathematician and mathematical physicist working in numerous fields of mathematics and its applications. In particular he is known for his work in probability theory, analysis (including infinite dimensional, non-standard, and stochastic analysis), mathematical physics, and in the areas algebra, geometry, number theory, as well as in applications, from natural to social-economic sciences. He initiated (with Raphael Høegh-Krohn) a systematic mathematical theory of Feynman path integrals and of infinite dimensional Dirichlet forms and associated stochastic processes (with applications particularly in quantum mechanics, statistical mechanics and quantum field theory). He also gave essential contributions to the development of areas such as ''p''-adic functional and stochastic analysis as well as to the singular perturbation theory for differential operators. Other important contributions concern constructive quantum field the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lugano
Lugano (, , ; lmo, label=Ticinese dialect, Ticinese, Lugan ) is a city and municipality in Switzerland, part of the Lugano District in the canton of Ticino. It is the largest city of both Ticino and the Italian-speaking southern Switzerland. Lugano has a population () of , and an urban agglomeration of over 150,000. It is the List of cities in Switzerland, ninth largest Swiss city. The city lies on Lake Lugano, at its largest width, and, together with the adjacent town of Paradiso, Switzerland, Paradiso, occupies the entire bay of Lugano. The territory of the municipality encompasses a much larger region on both sides of the lake, with numerous isolated villages. The region of Lugano is surrounded by the Lugano Prealps, the latter extending on most of the Sottoceneri region, the southernmost part of Ticino and Switzerland. Both western and eastern parts of the municipality share an international border with Italy. Described as a market town since 984, Lugano was the object of con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Perturbation Theory
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to zero. More precisely, the solution cannot be uniformly approximated by an asymptotic expansion :\varphi(x) \approx \sum_^N \delta_n(\varepsilon) \psi_n(x) \, as \varepsilon \to 0. Here \varepsilon is the small parameter of the problem and \delta_n(\varepsilon) are a sequence of functions of \varepsilon of increasing order, such as \delta_n(\varepsilon) = \varepsilon^n. This is in contrast to regular perturbation problems, for which a uniform approximation of this form can be obtained. Singularly perturbed problems are generally characterized by dynamics operating on multiple scales. Several classes of singular perturbations are outlined below. The term "singular perturbation" was coined in the 1940s by Kurt Otto Friedrichs and Wolfgang R. Wasow. Methods of analysis A perturbed problem whose solution can be approximated on the wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Analysis
In mathematics, ''p''-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers. The theory of complex-valued numerical functions on the ''p''-adic numbers is part of the theory of locally compact groups. The usual meaning taken for ''p''-adic analysis is the theory of ''p''-adic-valued functions on spaces of interest. Applications of ''p''-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of ''p''-adic functional analysis and spectral theory. In many ways ''p''-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of ''p''-adic numbers is much simpler. Topological vector spaces over ''p''-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule. Stochastic processes have applications in many disciplines such as biology, chemistry, ecology, neuroscience, physics, image processing, signal processing, control theory, information theory, computer science, cryptography and telecommunications. Furthermore, seemingly random changes in financial markets have motivated the extensive use of stochastic processes in finance. Applications and the study of phenomena have in turn inspired the proposal of new stochastic processes. Examples of such stochastic processes include the Wiener process or Brownian motion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Form
In potential theory (the study of harmonic function) and functional analysis, Dirichlet forms generalize the Laplacian (the mathematical operator on scalar fields). Dirichlet forms can be defined on any measure space, without the need for mentioning partial derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds, for example, fractals. The benefit on these spaces is that one can do this without needing a gradient operator, and in particular, one can even weakly define a "Laplacian" in this manner if starting with the Dirichlet form. Definition When working on \mathbb^n , the "classical" Dirichlet form is given by: \mathcal(u, v ) = \int_ \nabla u(x) \cdot \nabla v(x) \; dx where one often discusses \mathcal(u) := \mathcal(u, u) = \, \nabla u\, _2^2 which is often referred to as the "energy" of the function u(x). More generally, a Dirichlet form is a Markovian closed symmetric form on an ''L''2-space.Fukushima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Path Integrals
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals (for interactions of a certain type, these are ''coordina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raphael Høegh-Krohn
Jan Raphael Høegh-Krohn (10 February 1938 – 24 January 1988) was a Norwegian mathematician. He finished his Ph.D. in 1966, titled ''On Partly Gentle Perturbation with Application to Perturbation by Annihilation-Creation Operator,'' under the supervision of Kurt Friedrichs at the New York University. He authored over 150 papers and is known for the discovery of a fundamental duality in relativistic quantum statistical mechanics Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a ... by representing the basic correlation functions in terms of a certain stochastic process, now known as the Høegh-Krohn process. Books * Albeverio, Sergio; Gesztesy, Friedrich; Høegh-Krohn, Raphael; Holden, Helge: Solvable models in quantum mechanics. Texts and Monographs in Physics. Springer-Verlag, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Economic Sciences
Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes what's viewed as basic elements in the economy, including individual agents and markets, their interactions, and the outcomes of interactions. Individual agents may include, for example, households, firms, buyers, and sellers. Macroeconomics analyzes the economy as a system where production, consumption, saving, and investment interact, and factors affecting it: employment of the resources of labour, capital, and land, currency inflation, economic growth, and public policies that have impact on these elements. Other broad distinctions within economics include those between positive economics, describing "what is", and normative economics, advocating "what ought to be"; between economic theory and applied economics; between rational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Science
Social science is one of the branches of science, devoted to the study of societies and the relationships among individuals within those societies. The term was formerly used to refer to the field of sociology, the original "science of society", established in the 19th century. In addition to sociology, it now encompasses a wide array of academic disciplines, including anthropology, archaeology, economics, human geography, linguistics, management science, communication science and political science. Positivist social scientists use methods resembling those of the natural sciences as tools for understanding society, and so define science in its stricter modern sense. Interpretivist social scientists, by contrast, may use social critique or symbolic interpretation rather than constructing empirically falsifiable theories, and thus treat science in its broader sense. In modern academic practice, researchers are often eclectic, using multiple methodologies (for instan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Science
Natural science is one of the branches of science concerned with the description, understanding and prediction of Nature, natural Phenomenon, phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer review and repeatability of findings are used to try to ensure the validity of scientific advances. Natural science can be divided into two main branches: list of life sciences, life science and Outline of physical science, physical science. Life science is alternatively known as biology, and physical science is subdivided into branches: physics, chemistry, earth science, and astronomy. These branches of natural science may be further divided into more specialized branches (also known as fields). As empirical sciences, natural sciences use tools from the formal sciences, such as mathematics and logic, converting information about nature into measurements which can be explained as clear statements of the "laws of science, laws of nature". Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
