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On Approximation by Tight Wavelet Frames on the Field of $$p$$ -Adic Numbers P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-03-01
Abstract We discuss the problem on approximation by tight wavelet frames on the field \(\mathbb{Q}_p\) of \(p\) -adic numbers. For tight frames in the field \(\mathbb{Q}p\) , constructed earlier by the authors, we obtain approximation estimates for functions from Sobolev spaces with logarithmic weight.
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Dynamical Systems of Möbius Transformation: Real, $$p$$ -Adic and Complex Variables P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-02-12 E. T. Aliev, U. A. Rozikov
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Hyperstability of the General Linear Functional Equation in Non-Archimedean Banach Spaces P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-02-12 Shujauddin Shuja, Ahmad F. Embong, Nor M. M. Ali
Abstract Let \( X \) be a normed space over \( \mathbb{F} \in\{ \mathbb{R}, \mathbb{C}\} \), \( Y \) be a non-Archimedean Banach space over a non-Archimedean non-trivial field \(\mathbb{K}\) and \(c,d,C,D\) be constants such that, \( c, d \in \mathbb{F}\setminus\{0\} \) and \( C, D \in \mathbb{K}\setminus\{0\} \). In this paper, some preliminaries on non-Archimedean Banach spaces and the concept of
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Results on the Growth of Meromorphic Solutions of some Functional Equations of Painlevé and Schröder Type in Ultrametric Fields P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-02-12 Houda Boughaba, Salih Bouternikh, Tahar Zerzaihi
Abstract Let \(\mathbb{K}\) be a complete ultrametric algebraically closed field of characteristic zero and let \(\mathcal{M}(\mathbb{K})\) be the field of meromorphic functions in all \(\mathbb{K}\). In this paper, we investigate the growth of meromorphic solutions of some difference and \(q\)-difference equations. We obtain some results on the growth of meromorphic solutions when the coefficients
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Estimate for the Intrinsic Square Function on $$p$$ -Adic Herz Spaces with Variable Exponent P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-02-12 Mehvish Sultan, Babar Sultan
Abstract Our aim in this paper is to define \(p\)-adic Herz spaces with variable exponents and prove the boundedeness of \(p\)-adic intrinsic square function in these spaces.
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Public-Key Cryptosystems and Signature Schemes from $$p$$ -Adic Lattices P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-03-01
Abstract In 2018, the longest vector problem and closest vector problem in local fields were introduced, as the \(p\) -adic analogues of the shortest vector problem and closest vector problem in lattices of Euclidean spaces. They are considered to be hard and useful in constructing cryptographic primitives, but no applications in cryptography were given. In this paper, we construct the first signature
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Numerical Solutions of some Nonlinear Integral Equations Arising in the Theory of $$p$$ -Adic Strings and Physical Kinetics P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2024-02-12 Kh. A. Khachatryan, A. Kh. Khachatryan, A. Zh. Narimanyan
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Turing Patterns in a $$p$$ -Adic FitzHugh-Nagumo System on the Unit Ball P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-12-18 L. F. Chacón-Cortés, C. A. Garcia-Bibiano, W. A. Zúñiga-Galindo
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Constructions of Urysohn Universal Ultrametric Spaces P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-12-01
Abstract In this paper, we give new constructions of Urysohn universal ultrametric spaces. We first characterize a Urysohn universal ultrametric subspace of the space of all continuous functions whose images contain the zero, from a zero-dimensional compact Hausdorff space without isolated points into the space of non-negative real numbers equipped with the nearly discrete topology. As a consequence
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Number-Theory Renormalization of Vacuum Energy P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-12-01
Abstract For QFT on a lattice of dimension \(d\geqslant 3\) , the vacuum energy (both bosonic and fermionic) is zero if the Hamiltonian is a function of the square of the momentum, and the calculation of the vacuum energy is performed in the ring of residue classes modulo \(N\) . This fact is related to a problem from number theory about the number of ways to represent a number as a sum of \(d\) squares
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On the Occasion of the Centenary of the Birth of V. S. Vladimirov (1923–2012) P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-11-05 B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich
Abstract In this note, on the occasion of the centenary of the birth, we are remembering Vasilii Sergeevich Vladimirov as an outstanding scientist and his contributions to pure and applied mathematics, especially \(p\)-adic analysis and its applications.
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Analysis on Ultra-Metric Spaces via Heat Kernels P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-11-05 Alexander Grigor’yan
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Coherent States of the $$p$$ -Adic Heisenberg Group and Entropic Uncertainty Relations P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-11-05 Evgeny Zelenov
Abstract Properties of coherent states for the Heisenberg group over a field of \(p\)-adic numbers are investigated. It turns out that coherent states form an orthonormal basis. The family of all such bases is parametrized by a set of self-dual lattices in the phase space. The Wehrl entropy \(S_W\) is considered and its properties are investigated. In particular, it is proved that \(S_W\geq 0\) and
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Modular Nori Motives P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-11-05 N. C. Combe, Yu. I Manin, M. Marcolli
Abstract In a previous article [4], we developed the pioneering Grothendieck approach to the problem of description of the absolute Galois group \(\rm{Gal}(\overline{\bf{Q}}/{\bf{Q}})\) based upon dessins d’enfant. Namely, we replaced in it dessins d’enfant by graphs encoding combinatorics of strata of modular spaces of genus zero \(\overline{M}_{0,n}\), and applied this new category to the study of
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Oscillations in $$p$$ -Adic Diffusion Processes and Simulation of the Conformational Dynamics of Protein P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-11-05 A. Kh. Bikulov, A. P. Zubarev
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Soft Logic as an Extension of Pascal’s Work P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-07-31 Moshe Klein, Oded Maimon
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A Hille-Yosida-Phillips Theorem for Discrete Semigroups on Complete Ultrametric Locally Convex Spaces P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-07-31 Jawad Ettayb
Abstract Let \(E\) be a complete Hausdorff locally convex space over \(\mathbb{C}_{p},\) let \(A\in\mathcal{L}(E)\) such that \((I-\lambda A)^{-1}\) is analytic on its domain. In this paper, we give a necessary and sufficient condition on the resolvent of \(A\) such that \((A^{n})_{n\in\mathbb{N}}\) is equi-continuous.
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$$p$$ -Adic Weaving Multiframelets P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-07-31 Animesh Bhandari, Sudip Mishra, Subenoy Chakraborty
Abstract Frames play significant role as redundant building blocks in distributed signal processing. Getting inspirations from this concept, Bemrose et al. produced the notion of weaving frames in Hilbert space. Weaving frames have useful applications in sensor networks, likewise weaving \(K\)-frames have been proved to be useful during signal reconstructions from the range of a bounded linear operator
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$$L^1$$ -Convergence of Double Vilenkin Series P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-07-31 N. Yu. Agafonova, S. S. Volosivets
Abstract We give necessary and sufficient conditions for \(L^1\) convergence of double Vilenkin series whose coefficients form a double null sequence of bounded variation. Also we study the existence of the special form of Riemann improper integral for sum of this series.
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$$2$$ -Adic 1-Lipschitz Maps-Based Nonlinear Pseudorandom Generators of Arbitrary Rank Having the Longest Period P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-07-31 Alexander Sidorov
Abstract Linear congruential method was one of the first one proposed to generate pseudorandom numbers. However, due to drawbacks arising from linearity nonlinear methods of generating pseudorandom numbers were proposed; however, these methods were mostly nonlinear recurrences of rank 1, i.e., iterations of a univariate map. In this paper we propose a generator which is a recurrence of order \(k\)
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Infinite Series Whose Topology of Convergence Varies From Point to Point P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-07-31 M. C. Siegel
Abstract Let \(p\) and \(q\) be distinct primes, and consider the expression \(S_{p,q}\left(\mathfrak{z}\right)\) defined by the formal series \(\sum_{n=0}^{\infty}q^{\#_{1}\left(\left[\mathfrak{z}\right]_{2^{n}}\right)}/p^{n}\), where \(\mathfrak{z}\) is a \(2\)-adic integer variable, \(\left[\mathfrak{z}\right]_{2^{n}}\) is the integer in \(\left\{ 0,\ldots,2^{n}-1\right\} \) congruent to \(\mathfrak{z}\)
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Number of Zeros of Exponential Polynomials in Zero Residue Characteristic P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-05-10 Alain Escassut
Abstract Let \(\mathbb{L}\) be a complete ultrametric field of residue characteristic \(0\) and let \(F(x)=\sum_{i=1}^kf_i(x)exp(\omega_ix)\), where each \(f_i\in L[x]\), \(\omega_i\in \mathbb{L}\), \(|\omega_i|<1\). The number of zeros of \(F\) in the unit disk is bounded by \(n-1\) where \(n=\sum_{i=1}^k \deg(f_i)+k\).
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On a New Measure on the Levi-Civita Field $$ \mathcal{R} $$ P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-05-10 M. Restrepo Borrero, Vatsal Srivastava, K. Shamseddine
Abstract The Levi-Civita field \( \mathcal{R} \) is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [13], a measure was defined on \( \mathcal{R} \) in terms of the limit of the sums of the lengths of inner and outer covers of a set by countable unions of intervals as those inner
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Ergodicity and Periodic Orbits of $$p$$ -Adic $$(1,2)$$ -Rational Dynamical Systems with Two Fixed Points P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-05-10 I. A. Sattarov, E. T. Aliev
Abstract We consider \((1,2)\)-rational functions given on the field of \(p\)-adic numbers \({\mathbb Q}_p\). In general, such a function has four parameters. We study the case when such a function has two fixed points and show that when there are two fixed points then \((1,2)\)-rational function is conjugate to a two-parametric \((1,2)\)-rational function. Depending on these two parameters we determine
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On the $$p$$ -Adic Properties of $$2$$ -Sected Sums Involving Binomial Coefficients P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-05-10 T. Lengyel
Abstract For a prime \(p\), we determine the \(p\)-adic order of certain lacunary sums involving binomial coefficients. After forming sequences of the sums, we use various techniques, recurrence relations, multisecting ordinary generating functions, periodicity of linear sequences, divisibility sequences, Lucas and their companion sequences, and a \(2\)-adic analytic technique among other methods.
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log-Coulomb Gases in the Projective Line of a $$p$$ -Field P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-05-10 Joe Webster
Abstract This article extends recent results on log-Coulomb gases in a \(p\)-field \(K\) (i.e., a nonarchimedean local field) to those in its projective line \(\mathbb{P}^1(K)\), where the latter is endowed with the \(PGL_2\)-invariant Borel probability measure and spherical metric. Our first main result is an explicit combinatorial formula for the canonical partition function of log-Coulomb gases
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Periodic Points of a $$p$$ -Adic Operator and their $$p$$ -Adic Gibbs Measures P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-03-23 U. A. Rozikov, I. A. Sattarov, A. M. Tukhtabaev
Abstract In this paper we investigate generalized Gibbs measure (GGM) for \(p\)-adic Hard-Core (HC) model with a countable set of spin values on a Cayley tree of order \(k\geq 2\). This model is defined by \(p\)-adic parameters \(\lambda_i\), \(i\in \mathbb N\). We analyze \(p\)-adic functional equation which provides the consistency condition for the finite-dimensional generalized Gibbs distributions
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On the Constructive Solvability of a Two-Dimensional Nonlinear Integral Equation Arising in the Theory of $$p$$ -Adic Strings P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-03-23 Kh. A. Khachatryan, H. S. Petrosyan
Abstract This paper is devoted to the study of existence and uniqueness of the solution of a class of nonlinear two-dimensional integral equations in the plane. Such equations arise in theory of \(p\)-adic strings. In addition, equations of this nature are encountered in mathematical epidemiology. Constructive existence and uniqueness theorems for a positive bounded solution are proved. We also obtain
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A Weaker Smoothness Criterion for the Inverse Function Theorem, the Intermediate Value Theorem, and the Mean Value Theorem in a non-Archimedean Setting P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-03-23 K. Shamseddine, A. Shalev
Abstract We introduce a class of so-called very weakly locally uniformly differentiable (VWLUD) functions at a point of a general non-Archimedean ordered field extension of the real numbers, \(\mathcal{N}\), which is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. This new class of functions is defined by a significantly weaker criterion than
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On the Rate of Approximation by Generalized de la Vallée Poussin Type Matrix Transform Means of Walsh-Fourier Series P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-03-23 I. Blahota, G. Gát
Abstract In this paper, we consider norm convergence issues for matrix-based de la Vallée Poussin-like means of Fourier series for the Walsh system. In the main theorem of the paper, we state a proposition that estimates the difference between the named means above and the corresponding function in norm. The upper estimation is given by and as a function of the modulus of continuity of the function
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On Monogenity of Certain Pure Number Fields Defined by $$x^{2^r\cdot5^s\cdot 7^t}-m$$ P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2023-03-23 O. Kchit, H. Choulli
Abstract Let \(K\) be a pure number field generated by a root of a monic irreducible polynomial \(F(x)=x^{2^r\cdot5^s\cdot 7^t}-m\in \mathbb{Z}[x]\), where \(m\neq \pm 1\) is a square free integer, \(r\), \(s\), and \(t\) are three positive integers. In this paper, we study the monogenity of \(K\). We prove that if \(m\not\equiv 1 (\text{mod }{4})\), \(\overline{m}\not\in\{\pm\overline{1},\pm \overline{7}\}
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On the Magnitude of Fourier Coefficients with Respect to the Character System of $$\mathbb Z_p$$ P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-12-02 S. S. Volosivets, A. N. Mingachev
Abstract We give estimates for the magnitude of Fourier coefficients with respect to the character system of \(p\)-adic integers of functions from generalized Hölder spaces and some fluctuational spaces. In all cases we establish the sharpness of estimates.
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Characterization of $$p$$ -Adic Weighted Central Campanato Spaces via $$p$$ -Adic Hardy Operators Commutator P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-12-02 Teng Ma
Abstract We introduce the \(p\)-adic weighted central Campanato spaces \(\dot{C}^{q,\lambda}(\omega)(\mathbb{Q}_{p}^{n})\) and characterize \(\dot{C}^{q,\lambda}(\omega)(\mathbb{Q}_{p}^{n})\) by the boundedness of the commutators of \(p\)-adic \(n\)-dimensional Hardy operators on \(p\)-adic weighted central Morrey spaces for \(\omega\in A_{1}(\mathbb{Q}_{p}^{n})\). In particular, the characterizations
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Non-Archimedean Quasitriangular Operators and the Invariant Subspace Problem P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-12-02 Azzedine El Asri, Mohammed Babahmed
Abstract In this paper, we are interested in the study of non-Archimedean quasitriangular operators and their relation to the invariant subspace problem.
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Local Well-Posedness of the Cauchy Problem for a $$p$$ -Adic Nagumo-Type Equation P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-12-02 L. F. Chacón-Cortés, C. A. Garcia-Bibiano, W. A. Zúñiga-Galindo
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Radon-Nikodym Theorem with Respect to $$(\rho,q)$$ -Measure on $$\Bbb Z_p$$ P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-12-02 Dongkyu Lim
Abstract Araci et al. introduced a \(p\)-adic \((\rho,q)\)-analogue of the Haar distribution. By means of the distribution, they constructed the \(p\)-adic \((\rho,q)\)-Volkenborn integral. In this paper, by virtue of the Mahler expansion of continuous functions, the author gives the Radon-Nikodym theorem with respect to the \(p\)-adic \((\rho,q)\)-distribution on \(\Bbb Z_p\).
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The Sojourn Time Problem for a $$p$$ -Adic Random Walk and its Applications P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-12-02 A. Kh. Bikulov, A. P. Zubarev
Abstract We consider the problem of the distribution of the sojourn time in a compact set \(\mathbb{Z}_{p}\) in the case of a \(p\)-adic random walk. We rely on the results of our previous studies of the distribution of the first return time for a \(p\)-adic random walk and the results of Takacs on the study of the sojourn time problem for a wide class of random processes. For a \(p\)-adic random walk
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On $$p$$ -Metric Spaces and the $$p$$ -Gromov-Hausdorff Distance P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-08-05 Facundo Mémoli, Zhengchao Wan
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Weighted Inequalities for Commutators of $$p$$ -Adic Hausdorff Operators on Herz Spaces P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-08-05 Tran Luu Cuong, Kieu Huu Dung, Pham Thi Kim Thuy
Abstract In this paper, we establish the boundedness of commutators of \(p\)-adic matrix Hausdorff operators and \(p\)-adic rough Hausdorff operators on the block Herz spaces.
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Note on $$p$$ -Adic Local Functional Equation P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-08-05 Luochen Zhao
Abstract Given primes \(\ell\ne p\), we record here a \(p\)-adic valued Fourier theory on a local field over \(\mathbf{Q}_\ell\), which is developed under the perspective of group schemes. As an application, by substituting rigid analysis for complex analysis, it leads naturally to the \(p\)-adic local functional equation at \(\ell\), which strongly resembles the complex one in Tate’s thesis.
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Some Identities and Congruences for $$q$$ -Stirling Numbers of the Second Kind P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 Bertin Diarra, Hamadoun Maïga, Tongobé Mounkoro
Abstract The subject of this paper is the study of some properties of \(q\)-Stirling numbers of the second kind \(S_q(n,j)\) for \(q\ne 0\) a complex or a \(p\)-adic complex number. In the \(p\)-adic setting, as we known, the Laplace transform plays an important role in the study of some arithmetic sequences. We remind the definition of the Laplace transform of a \(p\)-adic measure and its link with
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Minimum Spanning Paths and Hausdorff Distance in Finite Ultrametric Spaces P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 Evgeniy Petrov
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Non-Lie Subgroups in Lie groups over Local Fields of Positive Characteristic P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 Helge Glöckner
Abstract By Cartan’s Theorem, every closed subgroup \(H\) of a real (or \(p\)-adic) Lie group \(G\) is a Lie subgroup. For Lie groups over a local field \({{\mathbb K}}\) of positive characteristic, the analogous conclusion is known to be wrong. We show more: There exists a \({{\mathbb K}}\)-analytic Lie group \(G\) and a non-discrete, compact subgroup \(H\) such that, for every \({{\mathbb K}}\)-analytic
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Almost Everywhere Convergence of Cesàro-Marczinkiewicz Means of Two-Dimensional Fourier Series on the Group of $$2$$ -Adic Integers P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 György Gát, Gábor Lucskai
Abstract In this paper, we prove the almost everywhere convergence of the \((C,\alpha)\) Marczinkiewicz-means of integrable functions on the group of the two-dimensional \(2\)-adic integers.
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Generalized Volkenborn Integrals Associated with $$p$$ -Adic Distributions and the Bernoulli Numbers P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 Kumi Yasuda
Abstract Our goal is to give a formula representing the Bernoulli numbers by \(p\)-adic distributions. We consider \(p\)-adic distributions on the ring of \(p\)-adic integers which are invariant by rotations around the origin, and define a generalization of the Vokenborn integrals with respect to such distributions. It is shown the generalized Volkenborn integrals of power functions, and of negative
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Rigidity and Unlikely Intersection for Stable $$p$$ -Adic Dynamical Systems P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 Mabud Ali Sarkar, Absos Ali Shaikh
Abstract Berger asked the question “To what extent the preperiodic points of a stable \(p\)-adic power series determines a stable \(p\)-adic dynamical system ?” In this work we have applied the preperiodic points of a stable \(p\)-adic power series in order to determine the corresponding stable \(p\)-adic dynamical system.
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Entropy, Feller Processes and $$p$$ -Adic Analogues of the Scattering Equation P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-05-23 J. Galeano-Peñaloza, Oscar F. Casas-Sánchez, Leonardo F. Chacón-Cortés
Abstract There are several techniques in the classical case for some integro-differential equations involving the concept of entropy to show some properties of the solution. In this work, we deal with the \(p\)-adic scattering equation. We adapt these methods to investigate the convergence of the solutions and their qualitative properties, including mass conservation, regularity and stability. Most
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A Few Remarks on Supercyclicity of Non-Archimedean Linear Operators on $$c_0(\mathbb N)$$ P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-02-04 Farrukh Mukhamedov, Otabek Khakimov, Abdessatar Souissi
Abstract In this paper, we study the hypercyclic, supercyclic and cyclic properties of operators of the form \(I+B_{\bf b}\), where \(B_{\bf b}\) is a weighted backward shift defined on \(c_0(\mathbb N)\). The results are totally different from the real case.
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Giorgio Parisi: The Nobel Prize in Physics 2021 P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-02-04 B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich
Abstract Giorgio Parisi awarded with the Nobel Prize in Physics 2021 “for the discovery of the interplay of disorder and fluctuations in physical systems from atomic to planetary scales”. We are happy to congratulate Professor Parisi with this award and the recognition of his great scientific achievements. G. Parisi is a member of the Editorial Board of this journal – \(p\)-Adic Numbers, Ultrametric
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Non-Archimedean Sendov’s Conjecture P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-02-04 Daebeom Choi, Seewoo Lee
Abstract We prove non-archimedean analogue of Sendov’s conjecure. We also provide complete list of polynomials over an algebraically closed non-archimedean field \(K\) that satisfy the optimal bound in the Sendov’s conjecture.
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A New Class of $$p$$ -Adic Lipschitz Functions and Multidimensional Hensel’s Lemma P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-02-04 Edwin León-Cardenal, John Jaime Rodriguez-Vega, Fausto Bolivar-Barbosa
Abstract In this work we study \(p\)-adic continuous functions in several variables taking values on \(\mathbb{Z}_p\). We describe the orthonormal van der Put base of these functions and study various Lipschitz conditions in several variables, generalizing previous work of Anashin. In particular, we introduce a new class of \(p\)-adic Lipschitz functions and study some of their properties. We also
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A Real-Valued Measure on non-Archimedean Field Extensions of $$\mathbb{R}$$ P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-02-04 Emanuele Bottazzi
Abstract We introduce a real-valued measure \( m _L \) on non-Archimedean ordered fields \(( \mathbb{F} ,<)\) that extend the field of real numbers \(({\mathbb R},<)\). The definition of \( m _L \) is inspired by the Loeb measures of hyperreal fields in the framework of Robinson’s analysis with infinitesimals. The real-valued measure \( m _L \) turns out to be general enough to obtain a canonical measurable
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On Linear Representations of $$p$$ -Adic Heisenberg Groups P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2022-02-04 Bertin Diarra, Tongobé Mounkoro
Abstract Let \(K\) be a complete valued field extension of the field \(\mathbb{Q}_p\) of \(p\)-adic numbers. Let \(\mathcal{D}\) be a closed unitary subring of the valuation ring \(\Lambda_K\) of \(K\). Let \(\mathcal{H}(3 , \mathcal{D})\) be the \(3\)-dimensional Heisenberg group with entries in \(\mathcal{D}\). We shall give continuous linear representations of \(\mathcal{H}(3 , \mathcal{D})\) in
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An Incompatibility Result on non-Archimedean Integration P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2021-11-08 Bottazzi, Emanuele
Abstract We prove that a Riemann-like integral on non-Archimedean extensions of \(\mathbb{R}\) cannot assign an integral to every function whose standard part is measurable and simultaneously satisfy the fundamental theorem of calculus. We also discuss how existing theories of non-Archimedean integration deal with the incompatibility of these conditions.
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A Distributional Proof of $$p$$ -Adic Wiener Tauberian Theorem and Approximation by Translates of a Function P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2021-11-08 Volosivets, S. S.
Abstract Using \(p\)-adic distributions we obtain the uniqueness result for convolution. As a corollary of this result one can obtain \(p\)-adic Wiener Tauberian theorem and Wiener approximation theorem. Also we prove that there is no basis of \(L^1(\mathbb Q^n_p)\) consisting of translates of a function.
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Best Proximity Pairs in Ultrametric Spaces P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2021-11-08 Chaira, Karim, Dovgoshey, Oleksiy, Lazaiz, Samih
Abstract In the present paper, we study the existence of best proximity pair in ultrametric spaces. We show, under suitable assumptions, that the proximinal pair \((A,B)\) has a best proximity pair. As a consequence we generalize a well known best approximation result and we derive some fixed point theorems. Moreover, we provide examples to illustrate the obtained results.
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Weighted Central BMO Type Space Estimates for Commutators of $$p$$ -Adic Hardy-Cesàro Operators P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2021-11-08 Dung, Kieu Huu, Duong, Dao Van, Luan, Tran Nhat
Abstract The aim of this paper is to give some sufficient conditions for the boundedness of commutators of \(p\)-adic Hardy-Cesàro operators with symbols in weighted central BMO type spaces on the Herz spaces, Morrey spaces and Morrey-Herz spaces with both the Muckenhoupt and power weights.
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On $$G_2$$ -Periodic Quasi Gibbs Measures of $$p$$ -Adic Potts Model on a Cayley Tree P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2021-11-08 Tukhtabaev, Akbarkhuja
Abstract In the present paper we study \(G_2\)-periodic \(p\)-adic quasi Gibbs measures for \(p\)-adic Potts model on a Cayley tree of order two. In the case \(q=3\), we prove the occurrence of a phase transition and construct ART quasi Gibbs measures for \(p\)-adic Potts model on a Cayley tree of order \(k\geq3\).
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Some Property of Sets in the Real Line and the Lebesgue Measurability P-Adic Num. Ultrametr. Anal. Appl. Pub Date : 2021-11-08 Chatyrko, Vitalij A.
Abstract In this paper I consider a property of sets in the real line such that every non-empty union of finitely many sets with the property does not contain a set with a positive Lebesgue measure. Selectors of the real numbers \(\mathbb R\) related to any proper dense subgroup of the additive group \((\mathbb R, +)\) as well as cosets of any proper dense subgroup of \((\mathbb R, +)\) possess this