On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

A p-adic model of DNA sequence and genetic code

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and the genetic code. In our investigation central role plays an ultrametric p-adic information space whose basic elements are nucleotides, codons and genes. We show that a 5-adicmodel is appropriate for DNA sequence. This 5-adicmodel, combined with 2-adic distance, is also suitable for the genetic code and for amore advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons. The text was submitted by the authors in English. This paper is a slight modification of an article available in the electronic archive form arXiv:qbio. GN/0607018v1 (July 2006). Since that time some other papers on this subject have appeared, e.g. [1], [2].     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

On the equations for p-adic closed and open strings

A survey of several pure mathematical results concerning the boundary-value problems for nonlinear pseudo-differential equation for closed and open strings in d-dimensional flat spacetime is presented. We obtained some results on existence or nonexistence of solutions. In particular, the absence of almost-periodic solutions was shown. We consider also some numerical approaches to the problems. The text was submitted by the author in English.     

Generalized Blueprints and p-adic Buildings I

This paper generalizes the notion of a blueprint as defined by M. Ronan and J. Tits in 1987. These more generalized blueprints can be used to achieve purely combinatorial constructions of affine buildings over p-adics field, depending only on the associated affine diagrams and the structure of the integers inside the appropriate fields.     

An infinite family of p-adic non-Haar wavelet bases and pseudo-differential operators

In this paper an infinite family of new compactly supported non-Haar p-adic wavelet bases in is constructed. We also study the connections between wavelet analysis and spectral analysis of p-adic pseudo-differential operators. A criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator is derived. We prove that these wavelets are eigenfunctions of the fractional operator. Since many p-adic models use pseudo-differential operators (fractional operator), these results can be intensively used in these models. The text was submitted by the authors in English.     

Remarks on lines and minimal rational curves

We determine all of lines in the moduli space M of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section. This work was supported by the Competitive Earmarked Research Grant (Grant No. HKU7025/03P) of the Research Grant Council, Hong Kong     

Multidimensional basis of p-adic wavelets and representation theory

A multidimensional basis of p-adic wavelets is constructed. The relation of the constructed basis to a system of coherent states i.e., orbit of action) for some p-adic group of linear transformations is discussed. We show that the set of products of the vectors from the constructed basis and p-roots of unity is the orbit of the corresponding p-adic group of linear transformations. The text was submitted by the authors in English.     

p-adic超几何函数与Dwork超曲面上的有理点

p-adic超几何函数是经典的Gauss超几何函数在有限域上的模拟,与许多数论问题都有联系.设Fq是q元有限域,λ∈Fq,n为正整数.本文研究了Dwork超曲面Dλ^n:x1^n+x2^n+…+xn^n=nλx1x2…xn及其推广形式上的Fq-有理点,并在n与q(q-1)互素时给出了由p-adic超几何函数表示的各种Fq-有理点个数的公式,从而修正和改进了Barman与Goodson等人的结论.     

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a generic mean zero locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss the relations of this result with the multiresolution wavelet analysis. The text was submitted by the authors in English.     

Non-Archimedean Analogues of Orthogonal and Symmetric Operators and p-adic Quantization

We study orthogonal and symmetric operators in non-Archimedean Hilbert spaces in the connection with p-adic quantization. This quantization describes measurements with finite precision. Symmetric (bounded) operators in the p-adic Hilbert spaces represent physical observables. We study spectral properties of one of the most important quantum operators, namely, the operator of the position (which is represented in the p-adic Hilbert L₂-space with respect to the p-adic Gaussian measure). Orthogonal isometric isomorphisms of p-adic Hilbert spaces preserve precisions of measurements. We study properties of orthogonal operators. It is proved that each orthogonal operator in the non-Archimedean Hilbert space is continuous. However, there exist discontinuous operators with the dense domain of definition which preserve the inner product. There also exist nonisometric orthogonal operators. We describe some classes of orthogonal isometric operators and we study some general questions of the theory of non-Archimedean Hilbert spaces (in particular, general connections between topology, norm and inner product).     

A compactification of the moduli space of twisted holomorphic maps

Given compact symplectic manifold X with a compatible almost complex structure and a Hamiltonian action of S¹ with moment map , and a real number K?0, we compactify the moduli space of twisted holomorphic maps to X with energy ?K. This moduli space parameterizes equivalence classes of tuples (C,P,A,?), where C is a smooth compact complex curve of fixed genus g, P is a principal S¹ bundle over C, A is a connection on P and ? is a section of PS¹_×X satisfying

     

Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.     

The Mumford relations and the moduli of rank three stable bundles

The cohomology ring of the moduli space M(n,d) of semistable bundles of coprime rank n and degree d over a Riemann surface M of genus g 2 has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when n>2. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these dual Mumford relations form a complete set for the rational cohomology ring of M(3,d) and show (Sect. 5) that the Pontryagin ring vanishes in degree 12g-8 and above.     

Gromov-Witten invariants and quantum cohomology

This article is an elaboration of a talk given at an international conference on Operator Theory, Quantum Probability, and Noncommutative Geometry held during December 20–23, 2004, at the Indian Statistical Institute, Kolkata. The lecture was meant for a general audience, and also prospective research students, the idea of the quantum cohomology based on the Gromov-Witten invariants. Of course there are many important aspects that are not discussed here. Dedicated to Professor K B Sinha on the occasion of his 60th birthday     

p-adic Measures of Algebraic Independence for the Values of Ramanujan Functions and Klein Modular Functions

In this paper, we give the p-adic measures of algebraic independence for the values of Ramanujan functions and Klein modular functions at algebraic points.