关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The objective of this paper is to give a comprehensive introduction to applied cryptography with an engineer or computer scientist in mind. The emphasis is on the knowledge needed to create practical systems which supports integrity, confidentiality, or authenticity. Topics covered includes an introduction to the concepts in cryptography, attacks against cryptographic systems, key use and handling, random bit generation, encryption modes, and message authentication codes. Recommendations on algorithms and further reading is given in the end of the paper. This paper should make the reader able to build, understand and evaluate system descriptions and designs based on the cryptographic components described in the paper.

[서평]「Applied Cryptography」

Astrophysical magnetic fields and nonlinear dynamo theory

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Methods Of Modern Mathematical Physics

The work presents a short proof of localization under the conditions of either strong disorder (λ > λ0) or extreme energies for a wide class of self adjoint operators with random matrix elements, acting inl
2 spaces. A prototypical example is the discrete Schrodinger operatorH=−Δ+U
0(x)+λV

 x
 onZ

 d
 ,d≧1, withU
0(x) a specified background potential and {V

 x
 } generated as random variables. The general results apply to operators with −Δ replaced by a non-local self adjoint operatorT whose matrix elements satisfy: ∑
 y
 |T

 x,y
 |
 S
 ≦Const., uniformly inx, for somes<1. Localization means here that within a specified energy range the spectrum ofH is of the pure-point type, or equivalently — the wave functions do not spread indefinitely under the unitary time evolution generated byH. The effect is produced by strong disorder in either the potential or in the off-diagonal matrix elementsT

 x, y
 . Under rapid decay ofT

 x, y
 , the corresponding eigenfunctions are also proven to decay exponentially. The method is based on resolvent techniques. The central technical ideas include the use of low moments of the resolvent kernel, i.e. <|G

 E
 
(x, y)|
 s
 > withs small enough (<1) to avoid the divergence caused by the distribution's Cauchy tails, and an effective use of the simple form of the dependence ofG

 E
 
(x, y) on the individual matrix elements ofH in elucidating the implications of the fundamental equation (H−E)G

 E
 (x,x
0)=δ
 x,x0
 . This approach simplifies previous derivations of localization results, avoiding the small denominator difficulties which have been hitherto encountered in the subject. It also yields some new results which include localization under the following sets of conditions: i) potentials with an inhomogeneous non-random partU
0
(x), ii) the Bethe lattice, iii) operators with very slow decay in the off-diagonal terms (T
x,y≈1/|x−y|(d+e)), and iv) localization produced by disordered boundary conditions.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104253939

Localization at large disorder and at extreme energies: an elementary derivation

Parabolic Anderson Problem and Intermittency

This paper studies the asymptotic behaviour as of the transition density of a diffusion process on a smooth manifold. The results are stated in the language of differential geometry. We look at applications of the asymptotic formulae to the local structure of diffusion processes and to spectral theory.

http://iopscience.iop.org/article/10.1070/RM1975v030n01ABEH001400/pdf

Diffusion processes and riemannian geometry

The objective of this paper is a mathematically rigorous investigation of intermittency and related questions intensively studied in different areas of physics, in particular in hydrodynamics. On a qualitative level, intermittent random fields are distinguished by the appearance of sparsely distributed sharp “peaks” which give the main contribution to the formation of the statistical moments. The paper deals with the Cauchy problem (∂/∂t)u(t,x)=Hu(t, x), u(0,x)=t
0(x) ≥ 0, (t, x) ∈ ℝ+ × ℤ
 d
 , for the Anderson HamiltonianH = κΔ + ξ(·), ξ(x),x ∈ ℤd where is a (generally unbounded) spatially homogeneous random potential. This first part is devoted to some basic problems. Using percolation arguments, a complete answer to the question of existence and uniqueness for the Cauchy problem in the class of all nonnegative solutions is given in the case of i.i.d. random variables. Necessary and sufficient conditions for intermittency of the fieldsu(t,·) ast→∞ are found in spectral terms ofH. Rough asymptotic formulas for the statistical moments and the almost sure behavior ofu(t,x) ast→∞ are also derived.

/pdf/parabolic-problems-for-the-anderson-model-i-intermittency-yd7wl5eyno.pdf

Parabolic problems for the Anderson model. I. Intermittency and related topics

Some specific structures in which a growing quantity reaches record high values typically arise for instabilities in random media. Despite the rarity of these concentrations, they dominate the integral characteristics of the growing quantity (the mean value, the mean square value, etc.). The appearance of such structures is called intermittency.The geometric properties of intermittent structures depend strongly on whether the growing quantity is a scalar or a vector. The scalar case is illustrated here by the example of an instability which arises in problems of chemical kinetics. The vector case is illustrated by the problem of the selfexcitation of a magnetic field in a random flow of a conducting fluid.

Stanislav Molchanov

Papers

Localization at large disorder and at extreme energies: an elementary derivation

Parabolic Anderson Problem and Intermittency

Diffusion processes and riemannian geometry

Parabolic problems for the Anderson model. I. Intermittency and related topics

Intermittency in random media