A relation between Bernoulli numbers

We give a p-adic proof of a certain new relation between the Bernoulli numbers B_k, similar to Euler's formula Σ_k=2^m?2(_k^m)B_kB_m?k = ?(m+1)B_m, m ≥ 4.     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

On symmetrical <Emphasis Type="Italic">q</Emphasis>-extensions of the 2-dimensional grid

Properties of symmetrical q-extensions of grids are investigated. A criterion is obtained for a set of symmetrical q-extensions of the 2-dimensional grid Λ² to be finite. This criterion is used to prove, in particular, that the set of all Aut ₀(Λ²)-symmetrical q-extensions of the grid Λ2 is finite for any prime q. The list of all Aut ₀(Λ²)-symmetrical 3-extensions of the grid Λ² is obtained.     

On Gibbs measures of P-adic Potts model on the cayley tree

We consider a nearest-neighbor p-adic Potts (with q ≥ 2 spin values and coupling constant J ? _p) model on the Cayley tree of order k ≥ 1. It is proved that a phase transition occurs at k = 2, q ? p and p ≥ 3 (resp. q ? 2², p = 2). It is established that for p-adic Potts model at k ≥ 3 a phase transition may occur only at q ? p if p ≥ 3 and q ? 2² if p = 2.     

On the relation of Cesàro summability to generalized (F α, q) summability

Standard special cases of the sequence-to-functionF (a, q)-transform of Meir4 admit of a more general, essentially more refined, characterization as theF ^k (a, q)-transform of the sequel, adapted from one of Faulhaber’s.1 The theorem proved,viz., that (F _{a, q} ^k) summability for sequences, corresponding to the latter transform, includes Cesàro summability of a positive integer order with a certain rapidity, applies to the standard special cases of (F _{α, q}) and (F _{a, q} ^k) summabilities which are, in famiiar notation, summabilities(E _p), (T _α), (S _β), (V _a) and (B _{α, γ}) whose further special case (B_{1, 1}) is Borel summability. The special cases of (V_1/2) and Borel summabilities go back to Hyslop.3     

On a matrix trace inequality due to Ando,Hiai and Okubo

Ando et al. have proved that inequality \(\Re \mathfrak{e}trA^{p_1 } B^{q_1 \ldots } A^{p_k } B^{q_k } \leqslant trA^{p_1 + \ldots + p_k } B^{q_1 + \ldots + q_k }p\) is valid for all positive semidefinite matrices A,B and those nonnegative real numbers p₁, q₁,..., p_k, q_k which satisfy certain additional conditions. We give an example to show that this inequality is not valid for all collections of p₁, q₁,..., p_k, q_k ≥ 0. We also study related trace inequalities.     

Classification of p-adic 6-dimensional filiform Leibniz algebras by solutions of x q = a

We study the p-adic equation x ^q = a over the field of p-adic numbers. We construct an algorithm which gives a solvability criteria in the case of q = p ^m and present a computer program to compute the criteria for any fixed value of m ≤ p ? 1. Moreover, using this solvability criteria for q = 2; 3; 4; 5; 6, we classify p-adic 6-dimensional filiform Leibniz algebras.     

Sharp Estimates of p-Adic Hardy and Hardy-Littlewood-Pólya Operators

In this paper we get the sharp estimates of the p-adic Hardy and Hardy-Littlewood-Pólya operators on Lq(|x|αpdx). Also, we prove that the commutators generated by the p-adic Hardy operators(Hardy-Littlewood-Pólya operators) and the central BMO functions are bounded on Lq(|x|αpdx), more generally, on Herz spaces.     

Non-orientable fundamental surfaces in lens spaces

We give a concrete example of an infinite sequence of (pn,qn)-lens spaces L(pn,qn) with natural triangulations T(pn,qn) with pn tetrahedra such that L(pn,qn) contains a certain non-orientable closed surface which is fundamental with respect to T(pn,qn) and of minimal crosscap number among all closed non-orientable surfaces in L(pn,qn) and has n−2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L(pn,qn). Actually, we can set p₀=0, q₀=1, pk+1=3pk+2qk and qk+1=pk+qk.     

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes [p-1, k]p with p a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3.     

Hochschild cohomology ring modulo nilpotence of a one point extension of a quiver algebra defined by two cycles and a quantum-like relation

We consider a one point extension algebra B of a quiver algebra A _q over a field k defined by two cycles and a quantum-like relation depending on a nonzero element q in k. We determine the Hochschild cohomology ring of B modulo nilpotence and show that if q is a root of unity, then B is a counterexample to Snashall-Solberg’s conjecture.     

A <Emphasis Type="Italic">p</Emphasis>-adic hard-core model with three states on a Cayley tree

We examine the p-adic hard-core model with three states on a Cayley tree. Translationinvariant and periodic p-adic Gibbs measures are studied for the hard-core model for k = 2. We prove that every p-adic Gibbs measure is bounded for p ≠ 2. We show in particular that there is no strong phased transition for a hard-core model on a Cayley tree of order k.     

On the complexity of some problems related to graph extensions

The computational complexity of problems related to the construction of k-extensions of graphs is studied. It is proved that the problems of recognizing vertex and edge k-extensions are NP-complete. The complexity of recognizing irreducible, minimal, and exact vertex and edge k-extensions is considered.     

p-Adic invariant summation of some p-adic functional series

We consider summation of some finite and infinite functional p-adic series with factorials. In particular, we are interested in the infinite series which are convergent for all primes p, and have the same integer value for an integer argument. In this paper, we present rather large class of such p-adic functional series with integer coefficients which contain factorials. By recurrence relations, we constructed sequence of polynomials A _k (n; x) which are a generator for a few other sequences also relevant to some problems in number theory and combinatorics.     

A note on p-adic q-ζ-functions II

We show that p-adic q-ζ-function constructed by Koblitz [7] (see also D?browski [4]) can be obtained as Γ-transform of some p-adic measure coming from Lubin–Tate formal group.     

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ? _p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ? _p . By using this symmetry of fermionic p-adic invariant integral on ? _p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ? _p .     

Irrationality of some p-adic L-values

We give a proof of the irrationality of p-adic zeta-values ξp（κ） for p = 2, 3 and κ = 2,3.Such results were recently obtained by Calegari as an application of overconvergent p-adic modular forms. In this paper we present an approach using classical continued fractions discovered by Stieltjes. In addition we show the irrationality of some other p-adic L-series values, and values of the p-adic Hurwitz zeta-function.     

The annihilator of Bergman space

LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetB _p(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromL _p(D) ontoB _p(D), 1<p<∞. With this we have the decompositionL _p(D)=B _p(D)⊕B _q ^⊥ (D,p ^–1+q ^–=1, 1<p< ∞. Here, we show that the annihilatorB _q ^⊥ (D) is the space of allL _p-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onL _p(D). Especially, we show that , whereI is the identity operator and ? is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toB _q ^⊥ (D) are studied.     

A Series of New Congruences for Bernoulli Numbers and Eisenstein Series

We prove congruences of shape E_k+h≡E_k·E_h (mod N) modulo powers N of small prime numbers p, thereby refining the well-known Kummer-type congruences modulo these p of the normalized Eisenstein series E_k. The method uses Serre's theory of Iwasawa functions and p-adic Eisenstein series; it presents a rather general procedure to find and verify such congruences with a modest amount of numerical calculation.     

p-Adic valued quantization

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ? _p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ? _p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.