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Airy Process
The Airy processes are a family of Stationary process, stationary stochastic processes that appear as limit processes in the theory of random growth models and random matrix theory. They are conjectured to be Universality class, universal limits describing the long time, large scale spatial fluctuations of the models in the Kardar–Parisi–Zhang equation#KPZ universality class, (1+1)-dimensional KPZ universality class (Kardar–Parisi–Zhang equation) for many initial conditions (see also KPZ fixed point). The original process Airy2 was introduced in 2002 by the mathematicians Michael Prähofer and Herbert Spohn. They proved that the height function of a model from the (1+1)-dimensional KPZ universality class - the PNG droplet - converges under suitable scaling and initial condition to the Airy2 process and that it is a stationary process with almost surely continuous sample paths. The Airy process is named after the Airy function. The process can be defined through its finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stationary Process
In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. More formally, the joint probability distribution of the process remains the same when shifted in time. This implies that the process is statistically consistent across different time periods. Because many statistical procedures in time series analysis assume stationarity, non-stationary data are frequently transformed to achieve stationarity before analysis. A common cause of non-stationarity is a trend in the mean, which can be due to either a unit root or a deterministic trend. In the case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. With a deterministic trend, the process is called trend-stationary, and shocks have only transitory effects, with the variable tending towards a determin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Surely
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace Class Operator
In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace-class operators generalizes the trace of matrices studied in linear algebra. All trace-class operators are compact operators. In quantum mechanics, quantum states are described by density matrices, which are certain trace class operators. Trace-class operators are essentially the same as nuclear operators, though many authors reserve the term "trace-class operator" for the special case of nuclear operators on Hilbert spaces and use the term "nuclear operator" in more general topological vector spaces (such as Banach spaces). Definition Let H be a separable Hilbert space, \left\_^ an orthonormal basis and A : H \to H a positive bounded linear operator on H. The trace of A is denoted by \operatorname (A) and defined as :\operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tomohiro Sasomoto
Tomohiro is a masculine Japanese given name. Written forms Tomohiro can be written using many different combinations of kanji characters. Some examples: *友弘, "friend, vast" *友広, "friend, wide" *友寛, "friend, generosity" *友博, "friend, doctor" *友大, "friend, big" *友裕, "friend, abundant" *友洋, "friend, ocean" *知弘, "know, vast" *知広, "know, wide" *知寛, "know, generosity" *知博, "know, doctor" *知大, "know, big" *知裕, "know, abundant" *智弘, "intellect, vast" *智広, "intellect, wide" *智寛, "intellect, generosity" *智博, "intellect, doctor" *共弘, "together, vast" *共寛, "together, generosity" *朋弘, "companion, vast" *朋寛, "companion, generosity" *朝弘, "morning/dynasty, vast" *朝広, "morning/dynasty, wide" *朝大, "morning/dynasty, big" *朝洋, "morning/dynasty, ocean" The name can also be written in hiragana ともひろ or katakana is a Japanese syllabary, one component of the Japanese writing system along ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Unitary Ensemble
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the study of properties of random matrices, often as they become large. RMT provides techniques like mean-field theory, diagrammatic methods, the cavity method, or the replica method to compute quantities like traces, spectral densities, or scalar products between eigenvectors. Many physical phenomena, such as the spectrum of nuclei of heavy atoms, the thermal conductivity of a lattice, or the emergence of quantum chaos, can be modeled mathematically as problems concerning large, random matrices. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator (i.e. an operator whose singular values sum up to a finite number). The function is named after the mathematician Erik Ivar Fredholm. Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model. Definition Setup Let H be a Hilbert space and G the set of bounded invertible operators on H of the form I+T, where T is a trace-class operator. G is a group because * The set of trace-class operators is an ideal in the algebra of bounded linear operators, so (I+T)(I+T')-I = T + T' + TT' is trace-class. * (I+T) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-dimensional Distribution
In mathematics, finite-dimensional distributions are a tool in the study of measures and stochastic processes. A lot of information can be gained by studying the "projection" of a measure (or process) onto a finite-dimensional vector space (or finite collection of times). Finite-dimensional distributions of a measure Let (X, \mathcal, \mu) be a measure space. The finite-dimensional distributions of \mu are the pushforward measures f_ (\mu), where f : X \to \mathbb^, k \in \mathbb, is any measurable function. Finite-dimensional distributions of a stochastic process Let (\Omega, \mathcal, \mathbb) be a probability space and let X : I \times \Omega \to \mathbb be a stochastic process. The finite-dimensional distributions of X are the push forward measures \mathbb_^ on the product space \mathbb^ for k \in \mathbb defined by :\mathbb_^ (S) := \mathbb \left\. Very often, this condition is stated in terms of measurable rectangles: :\mathbb_^ (A_ \times \cdots \times A_) := \mathbb \lef ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Airy Function
In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(''x'') and the related function Bi(''x''), are Linear independence, linearly independent solutions to the differential equation \frac - xy = 0 , known as the Airy equation or the Stokes equation. Because the solution of the linear differential equation \frac - ky = 0 is oscillatory for and exponential for , the Airy functions are oscillatory for and exponential for . In fact, the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential). Definitions For real values of , the Airy function of the first kind can be defined by the improper integral, improper Riemann integral: \operatorname(x) = \dfrac\int_0^\infty\cos\left(\dfrac + xt\right)\, dt\equiv \dfrac \lim_ \in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PNG Droplet
Portable Network Graphics (PNG, officially pronounced , colloquially pronounced ) is a raster-graphics file format that supports lossless data compression. PNG was developed as an improved, non-patented replacement for Graphics Interchange Format (GIF). PNG supports palette-based images (with palettes of 24-bit RGB or 32-bit RGBA colors), grayscale images (with or without an alpha channel for transparency), and full-color non-palette-based RGB or RGBA images. The PNG working group designed the format for transferring images on the Internet, not for professional-quality print graphics; therefore, non-RGB color spaces such as CMYK are not supported. A PNG file contains a single image in an extensible structure of ''chunks'', encoding the basic pixels and other information such as textual comments and integrity checks documented in RFC 2083. PNG files have the ".png" file extension and the "image/png" MIME media type. PNG was published as an informational RFC 2083 in March ... [...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 in a probability space, where the index of the family often has the interpretation of time. 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 Ecology () is the natural science of the relationships among living organisms and their Natural environment, environment. Ecology considers organisms at the individual, population, community (ecology), community, ecosystem, and biosphere lev ..., neuroscience, physics, image processing, signal processing, stochastic control, control theory, information theory, computer scien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herbert Spohn
Herbert Spohn (born 1 November 1946) is a German mathematician and mathematical physicist working in kinetic equations, dynamics of stochastic particle systems, hydrodynamic limit, kinetics of growth processes, disordered systems, open quantum systems, dynamics of charged particles coupled to their radiation field, Schrödinger operators, functional integration and stochastic analysis. His PhD was obtained in 1975 at the University of Munich under the supervision of . He is currently (in the year 2021) Emeritus Professor of the Department of Mathematics of the Technical University Munich. He obtained several prizes. In 2011 he was awarded the Dannie Heineman Prize for Mathematical Physics, the Leonard Eisenbud Prize for Mathematics and Physics (AMS) and the Premio Caterina Tomassoni e Felice Pietro Chisesi Prize of University of Roma "La Sapienza". He is Docteur Honoris Causa de L'Université Paris-Dauphine. In 2017, he received the Max Planck Medal of the German Physical Soci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
