完全非线性偏微分方程解的Gevrey微局部正则性

本文中，我们首先简要回顾了Gevrey类中的仿微分运算，然后考察了相关的完全非线性偏微分方程的象征的一些性质。作为应用，我们得到解在椭圆点附近的Gevrey微局部正则性。     

Polynomial Quantization on Rank One Para-Hermitian Symmetric Spaces

We construct the polynomial quantization on the space G/H where G=SL(n,R),H=GL(n–1,R). It is a variant of quantization in the spirit of Berezin. In our case covariant and contravariant symbols are polynomials on G/H. We introduce a multiplication of covariant symbols, establish the correspondence principle, study transformations of symbols (the Berezin transform) and of operators. We write a full asymptotic decomposition of the Berezin transform.     

On the number of full levels in tries

We study the asymptotic distribution of the fill‐up level in a binary trie built over n independent strings generated by a biased memoryless source. The fill‐up level is the number of full levels in a tree. A level is full if it contains the maximum allowable number of nodes (e.g., in a binary tree level k can have up to 2^k nodes). The fill‐up level finds many interesting applications, e.g., in the internet IP lookup problem and in the analysis of level compressed tries (LC tries). In this paper, we present a complete asymptotic characterization of the fill‐up distribution. In particular, we prove that this distribution concentrates on one or two points around the most probably value k = ?log_1/qn ? log log log n + 1 + log log(p/q)?, where p > q = 1 ? p is the probability of generating the more likely symbol (while q = 1 ? p is the probability of the less likely symbol). We derive our results by analytic methods such as generating functions, Mellin transform, the saddle point method, and analytic depoissonization. We also present some numerical verification of our results. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Quadratic transformations and Guillera’s formulas for 1/π2

We prove two new series of Ramanujan type for 1/π².     

Functional analysis proofs of Abel's theorems

We give alternative proofs to the classical theorems of Abel, using the concept of Berezin symbol.

     

Simultaneous Pseudo-Timestepping for PDE-Model Based Optimization Problems

In this paper we present a new method for the solution of optimization problems with PDE constraints. It is based on simultaneous pseudo-time stepping for evolution equations. The new method can be viewed as a continuous reduced SQP method in the sense that it uses a preconditioner derived from that method. The reduced Hessian in the preconditioner is approximated by a pseudo-differential operator, whose symbol can be investigated analytically. We apply our method to a boundary control model problem. The new optimization method needs 3.2-times the overall computational effort of the solution of simulation problem alone.     

A public-key cryptosystem utilizing cyclotomic fields

While it is well-known that the RSA public-key cryptosystem can be broken if its modulusN can be factored, it is not known whether there are other ways of breaking RSA. This paper presents a public-key scheme which necessarily requires knowledge of the factorization of its modulus in order to be broken. Rabin introduced the first system whose security is equivalent to the difficulty of factoring the modulus. His scheme is based on squaring (cubing) for encryption and extracting square (cube) roots for decryption. This introduces a 14 (19) ambiguity in the decryption. Various schemes which overcome this problem have been introduced for both the quadratic and cubic case. We generalize the ideas of Williams' cubic system to larger prime exponents. The cases of higher prime order introduce a number of problems not encountered in the quadratic and cubic cases, namely the existence of fundamental units in the underlying cyclotomic field, the evaluation of higher power residue symbols, and the increased difficulty of Euclidean division in the field.