A Note on Complex <Emphasis Type="Italic">p</Emphasis>-Adic Exponential Fields

In this paper we apply Ax-Schanuel’s Theorem to the ultraproduct of p-adic fields in order to get some results towards algebraic independence of p-adic exponentials for almost all primes p.     

Cell decomposition for semibounded p-adic sets

We study a reduct ${\mathcal{L}_*}$ of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the ${\mathcal{L}_*}$ -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, ${\mathcal{L}_*}$ ) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From this we can derive quantifier-elimination, and give a characterization of definable functions. In particular, we conclude that multiplication can only be defined on bounded sets, and we consider the existence of definable Skolem functions.     

On the character of certain tilting modules

Let G be a semisimple group over an algebraically closed field of characteristic p > 0. We give a (partly conjectural) closed formula for the character of many indecomposable tilting rational G-modules assuming that p is large.     

The local lifting problem for <Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript>

For a prime p, a cyclic-by-p group G and a G-extension L|K of complete discrete valuation fields of characteristic p with algebraically closed residue field, the local lifting problem asks whether the extension L|K lifts to characteristic zero. In this paper, we characterize D₄-extensions of fields of characteristic two, determine the ramification breaks of (suitable) D₄- extensions of complete discrete valuation fields of characteristic two, and solve the local lifting problem in the affirmative for every D₄-extension of complete discrete valuation fields of characteristic two with algebraically closed residue field; that is, we show that D₄ is a local Oort group for the prime 2.     

<Emphasis Type="Italic">p</Emphasis>-Adic Analogue of the Porous Medium Equation

We consider a nonlinear evolution equation for complex-valued functions of a real positive time variable and a p-adic spatial variable. This equation is a non-Archimedean counterpart of the fractional porous medium equation. Developing, as a tool, an \(L^1\)-theory of Vladimirov’s p-adic fractional differentiation operator, we prove m-accretivity of the appropriate nonlinear operator, thus obtaining the existence and uniqueness of a mild solution. We give also an example of an explicit solution of the p-adic porous medium equation.     

Algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups

We study algebraically and verbally closed subgroups and retracts of finitely generated nilpotent groups. A special attention is paid to free nilpotent groups and the groups UT _n (Z) of unitriangular (n×n)-matrices over the ring Z of integers for arbitrary n. We observe that the sets of retracts of finitely generated nilpotent groups coincides with the sets of their algebraically closed subgroups. We give an example showing that a verbally closed subgroup in a finitely generated nilpotent group may fail to be a retract (in the case under consideration, equivalently, fail to be an algebraically closed subgroup). Another example shows that the intersection of retracts (algebraically closed subgroups) in a free nilpotent group may fail to be a retract (an algebraically closed subgroup) in this group. We establish necessary conditions fulfilled on retracts of arbitrary finitely generated nilpotent groups. We obtain sufficient conditions for the property of being a retract in a finitely generated nilpotent group. An algorithm is presented determining the property of being a retract for a subgroup in free nilpotent group of finite rank (a solution of a problem of Myasnikov). We also obtain a general result on existentially closed subgroups in finitely generated torsion-free nilpotent with cyclic center, which in particular implies that for each n the group UT _n (Z) has no proper existentially closed subgroups.     

Simultaneous approximation problems of <Emphasis Type="Italic">p</Emphasis>-adic numbers and <Emphasis Type="Italic">p</Emphasis>-adic knapsack cryptosystems - Alice in <Emphasis Type="Italic">p</Emphasis>-adic numberland

In this paper we construct the multi-dimensional p-adic approximation lattices by using simultaneous approximation problems (SAP) of p-adic numbers and we estimate the l _∞ norm of the p-adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p-adic lattices, we propose a p-adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p-adic numberland.     

<Emphasis Type="Italic">e</Emphasis>-Principal numberings

We prove the existence of the computable families of finite sets and general recursive functions with no e-principal numbering. We give a series of examples of e-degrees such that the p-degrees of their computable numberings include no top p-degree.     

Weak and strong estimates for rough Hausdorff type operator defined on <Emphasis Type="Italic">p</Emphasis>-adic linear space

For rough Hausdorff type operator defined on p-adic linear space Q _p ⁿ and its commutator with symbol from Lipschitz space, we give sufficient conditions of their boundedness from one Lorentz space into another.     

On the Classification of <Emphasis Type="Italic">p</Emphasis>-Adic UHF Banach Algebras

For any prime number p let Ω_p be the p-adic counterpart of the complex numbers C. In this paper we investigate the class of p-adic UHF Banach algebras. A p-adic UHF Banach algebra is any unital p-adic Banach algebra A of the form \(A = \overline {U{M_n}} \), where (M_n) is an increasing sequence of p-adic Banach subalgebras of M such that each M_n is generated over Ω_p by an algebraic system of matrix units {e _ij ^{( n)} | 1 ≤ i, j ≤ p_n }. The main result is that the supernatural number associated to a p-adic TUHF Banach algebra is an invariant of the algebra.     

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.     

An inequality for values of Koksma’s functions of two algebraically dependent <Emphasis Type="Italic">p</Emphasis>-adic numbers

We establish a p-adic version of the inequalities linking values of Koksma’s functions of two algebraically dependent transcendental numbers and show that in a very special, but nontrivial case these inequalities become equalities.     

Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form

Let p ≥ 3 be a prime and F a totally real number field. Let f be a Hilbert cuspidal eigenform of parallel weight 2, trivial Nebentypus and ordinary at p. It is possible to construct a p-adic L function which interpolates the complex L-function associated with the symmetric square representation of f. This p-adic L-function vanishes at s = 1 even if the complex L-function does not. Assuming p inert and f Steinberg at p, we give a formula for the p-adic derivative at s = 1 of this p-adic L-function, generalizing unpublished work of Greenberg and Tilouine. Under some hypotheses on the conductor of f we prove a particular case of a conjecture of Greenberg on trivial zeros.     

Model of <Emphasis Type="Italic">p</Emphasis>-Adic Random Walk in a Potential

We consider the p-adic random walk model in a potential which can be viewed as a generalization of p-adic random walk models used for describing protein conformational dynamics. This model is based on the Kolmogorov-Feller equations for the distribution function defined on the field of p-adic numbers in which the transition rate depends on ultrametric distance between the transition points as well as on function of potential violating the spatial homogeneity of a random process. This equation which will be called the equation of p-adic random walk in a potential, is equivalent to the equation of p-adic random walk with modified measure and reaction source. With a special choice of a power-law potential the last equation is shown to have an exact analytic solution. We find the analytic solution of the Cauchy problem for such equation with an initial condition, whose support lies in the ring of integer p-adic numbers.We also examine the asymptotic behaviour of the distribution function for large times. It is shown that in the limit t→∞ the distribution function tends to the equilibrium solution according to the law, which is bounded from above and below by power laws with the same exponent. Our principal conclusion is that the introduction of a potential in the model of p-adic random walk conserves the power-law behaviour of relaxation curves for large times.     

Representations of the Witt Lie Superalgebra with <Emphasis Type="Italic">p</Emphasis>-character of Height One

Let W(n) be the Witt Lie superalgebra over an algebraically closed field k of characteristic p>2. We study the representations of W(n),n≥3 with p-character of height one. We prove that the Kac module always has a unique simple quotient and all simple modules are given in this way. In fact, most Kac modules are simple. We will determine the sufficient and necessary conditions such that the Kac module is simple. Moreover, composition factors of each Kac module can be determined. It is proved that the number of composition factors in each Kac module is at most 2.     

Divisibility properties of hyperharmonic numbers

We extend Wolstenholme’s theorem to hyperharmonic numbers. Then, we obtain infinitely many congruence classes for hyperharmonic numbers using combinatorial methods. In particular, we show that the numerator of any hyperharmonic number in its reduced fractional form is odd. Then we give quantitative estimates for the number of pairs (n, r) lying in a rectangle where the corresponding hyperharmonic number \({ h_n^{(r)} }\) is divisible by a given prime number p. We also provide p-adic value lower bounds for certain hyperharmonic numbers. It is an open problem that given a prime number p, there are only finitely many harmonic numbers h _n which are divisible by p. We show that if we go to the higher levels r ≥  2, there are infinitely many hyperharmonic numbers \({ h_n^{(r)} }\) which are divisible by p. We also prove a finiteness result which is effective.     

Generalized <Emphasis Type="Italic">p</Emphasis>-Adic Fourier Transform and Estimates of Integral Modulus of Continuity in Terms of This Transform

We consider a new class of functions on the p-adic linear space ? _p ⁿ for which a Fourier transform can be defined.We prove equalities of Parseval type, an inversion formula and a sufficient condition for a function to be represented as this Fourier transform. Also we give a sharp estimate of the L²(? _p ⁿ ) modulus of continuity in terms of Fourier transform generalizing the result of S. S. Platonov in the case n = 1. Finally we prove a generalization of this result and its converse for L^q(? _p ⁿ ) with appropriate q.     

The <Emphasis Type="Italic">p</Emphasis>-adic weighted Hardy-Cesàro operators on weighted Morrey-Herz space

In this article we give necessary and sufficient conditions for the boundedness of the weighted Hardy-Cesà ro operators which is associated to the parameter curve γ(t, x) = γ(t)x defined by \({U_{\psi ,\gamma }}f\left( x \right) = \int {\left( {\gamma \left( t \right)x} \right)} \psi \left( t \right)dt\) on the weighted Morrey-Herz space over the p-adic field. Especially, the corresponding operator norms are established in each case. These results actually extend those of K. S. Rim and J. Lee [27] and of the authors [9]. Moreover, the sufficient conditions of boundedness of commutators of p-adic weighted Hardy-Cesàro operator with symbols in the Lipschitz space on the weighted Morrey-Herz space are also established.     

Measure-Preservation and the Existence of a Root of <Emphasis Type="Italic">p</Emphasis>-Adic 1-Lipschitz Functions in Mahler’s Expansion

In this paper, we provide necessary and sufficient conditions for 1-Lipschitz functions that are uniformly differentiable mod p on ?_p to be measure-preserving, in Mahler’s expansion, and show that these conditions can be modified to guarantee the existence of a root of p-adic Lipschitz functions.     

Non-Archimedean White Noise,Pseudodifferential Stochastic Equations,and Massive Euclidean Fields

We construct p-adic Euclidean random fields \(\varvec{\Phi }\) over \(\mathbb {Q}_{p}^{N}\), for arbitrary N, these fields are solutions of p-adic stochastic pseudodifferential equations. From a mathematical perspective, the Euclidean fields are generalized stochastic processes parametrized by functions belonging to a nuclear countably Hilbert space, these spaces are introduced in this article, in addition, the Euclidean fields are invariant under the action of certain group of transformations. We also study the Schwinger functions of \(\varvec{\Phi }\).