Search results for "Mathematic"
showing 10 items of 24974 documents
Random Walk and Diffusion
2014
The concept of random walk as introduced by Einstein is introduced. It is shown that a random walk on a lattice can be descrbed by a difference equation, which becomes a partial differential equation (diffusion equation) in the continuum limit. The equation is solved with the help of Fourier and Laplace transformations.
Bounds for Bessel functions
1989
We establish lower and upper bounds for the Bessel functionJ v (x) and the modified Bessel functionI v(x) of the first kind. Our chief tool is the differential equation satisfied by these functions.
Noether’s International School in Modern Algebra
2020
Pavel Alexandrov and Heinz Hopf met for the first time in Gottingen in the spring of 1926, soon after Alexandrov departed from Blaricum. Hopf had recently taken his doctorate in Berlin under Ludwig Bieberbach and Erhard Schmidt, and his research interests differed sharply from Alexandrov’s work in general topology.
Erzwingt die Quantenmechanik eine drastische Änderung unseres Weltbilds? Gedanken und Experimente nach Einstein, Podolsky und Rosen
1989
Von den Anfangen der Quantenmechanik bis heute gibt es Versuche, sie als statistische Theorie uber Ensembles individueller ‚klassischer’ Systeme zu interpretieren. Die Bedingungen, unter denen Theorien verborgener Parameter zu deterministischen Beschreibungen dieser individuellen Systeme als ‚klassisch’ angesehen werden konnen, wurden von Einstein, Podolsky und Rosen 1935 formuliert: 1. Physikalische Systeme sind im Prinzip separierbar. 2. Zu jeder physikalischen Grose, deren Wert man ohne Storung des betrachteten Systems mit Sicherheit voraussagen kann, existiert ein ihr entsprechendes Element der physikalischen Realitat. Zusammen sind sie, wie Bell 1964 gezeigt hat, prinzipiell unvertragl…
New Types of Jacobian-Free Approximate Riemann Solvers for Hyperbolic Systems
2017
We present recent advances in PVM (Polynomial Viscosity Matrix) methods based on internal approximations to the absolute value function. These solvers only require a bound on the maximum wave speed, so no spectral decomposition is needed. Moreover, they can be written in Jacobian-free form, in which only evaluations of the physical flux are used. This is particularly interesting when considering systems with complex Jacobians, as the relativistic magnetohydrodynamics (RMHD) equations. The proposed solvers have also been extended to the case of approximate DOT (Dumbser-Osher-Toro) methods, which can be regarded as simple and efficient approximations to the classical Osher-Solomon method. Som…
Convergence of Measures
2020
One focus of probability theory is distributions that are the result of an interplay of a large number of random impacts. Often a useful approximation can be obtained by taking a limit of such distributions, for example, a limit where the number of impacts goes to infinity. With the Poisson distribution, we have encountered such a limit distribution that occurs as the number of very rare events when the number of possibilities goes to infinity (see Theorem 3.7). In many cases, it is necessary to rescale the original distributions in order to capture the behavior of the essential fluctuations, e.g., in the central limit theorem. While these theorems work with real random variables, we will a…
A Leibniz variety with almost polynomial growth
2005
Abstract Let F be a field of characteristic zero. In this paper we study the variety of Leibniz algebras V ˜ 1 defined by the identity y 1 ( y 2 y 3 ) ( y 4 y 5 ) ≡ 0 . We give a complete description of the space of multilinear identities in the language of Young diagrams through the representation theory of the symmetric group. As an outcome we show that the variety V ˜ 1 has almost polynomial growth, i.e., the sequence of codimensions of V ˜ 1 cannot be bounded by any polynomial function but any proper subvariety of V ˜ 1 as polynomial growth.
Liftings and extensions of operators in Brownian setting
2020
We investigate the operators T on a Hilbert space H which have 2-isometric liftings S with the property S ∗ S H ⊂ H . We show that such liftings are closely related to some extensions of T, which h...
Solvability of the divergence equation implies John via Poincaré inequality
2014
Abstract Let Ω ⊂ R 2 be a bounded simply connected domain. We show that, for a fixed (every) p ∈ ( 1 , ∞ ) , the divergence equation div v = f is solvable in W 0 1 , p ( Ω ) 2 for every f ∈ L 0 p ( Ω ) , if and only if Ω is a John domain, if and only if the weighted Poincare inequality ∫ Ω | u ( x ) − u Ω | q d x ≤ C ∫ Ω | ∇ u ( x ) | q dist ( x , ∂ Ω ) q d x holds for some (every) q ∈ [ 1 , ∞ ) . This gives a positive answer to a question raised by Russ (2013) in the case of bounded simply connected domains. In higher dimensions similar results are proved under some additional assumptions on the domain in question.
Partial *-Algebras of Operators in a PIP-Space
2009
The family of operators on a pip-space V is endowed with two, possibly different, partial multiplications, where partial means that the multiplication is not defined for any pair A,B of elements of Op(V) but only for certain couples. The two multiplications, to be called strong and weak, give rise to two different structures that coincide in certain situations. In this chapter we will discuss first the structure of Op(V) as partial *-algebra in the sense of [AIT02] and then the possibility of representing an abstract partial *-algebra into Op(V).