Abstract
We describe the continuous homomorphisms on subalgebras of absolutely convergent series with respect to the character system of (p)-adic integers. Using this characterization we obtain Wiener and Levy type theorems for these subalgebras.
We describe the continuous homomorphisms on subalgebras of absolutely convergent series with respect to the character system of (p)-adic integers. Using this characterization we obtain Wiener and Levy type theorems for these subalgebras.
In this paper, for any nonic number field (K) generated by a root (alpha) of a monic irreducible trinomial (F(x)=x^9+ax^5+b in mathbb{Z}[x]) and for every rational prime (p), we characterize when (p) divides the index of (K). We also describe the prime power decomposition of the index (i(K)). In such a way we give a partial answer of Problem (22) of Narkiewicz [23] for this family of number fields. As an application of our results, if (i(K)neq1), then (K) is not monogenic. We illustrate our results by some computational examples.
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 scheme and public-key encryption cryptosystem based on (p) -adic lattice by proposing a trapdoor function with the norm-orthogonal basis of (p) -adic lattice. These cryptographic schemes have reasonable key size and the signature scheme is efficient, while the encryption scheme works only for short messages, which shows that (p) -adic lattice can be a new alternative to construct cryptographic primitives and well worth studying.
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.
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 hyperstability are presented. Next, the well-known fixed point method [7, Theorem1] is reformulated in non-Archimedean Banach spaces. Using this method, we prove that the general linear functional equation ( h(cx+dy)= Ch(x)+Dh(y) ) is hyperstable in the class of functions ( h:Xrightarrow Y ). In fact, by exerting some natural assumptions on control function ( gamma:X^{2}setminus{0}rightarrow mathbb{R}_{+} ), we show that the map ( h:Xrightarrow Y ) that satisfies the inequality ( lVert h(cx+dy)- Ch(x)-Dh(y)rVert_{ast}leq gamma(x,y) ), is a solution to general linear functional equation for every ( x, y in Xsetminus{0} ). Finally, this paper concludes with some consequences of the results.
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 in such equations are rational functions.
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.
The present work is devoted to finding numerical solutions of two types of nonlinear integral equations on half line with kernels depending on the sum and difference of arguments. These equations arise in various fields of mathematical physics: kinetic theory of gases, theoretical astrophysics, p-adic string theory, etc. The main result of the work is the derivation of an uniform estimate of the norm of difference between two successive approximations of solutions, which plays an important role for the control of the convergence of iterative schemes and number of iterations. The obtained results have been applied to determine numerical solutions of models from different areas of applications.
In this paper we consider function (f(x)={x+aover bx+c}), (where (bne 0), (cne ab), (xne -{cover b})) on three fields: the set of real, (p)-adic and complex numbers. We study dynamical systems generated by this function on each field separately and give some comparison remarks. For real variable case we show that the real dynamical system of the function depends on the parameters ((a,b,c)in mathbb R^3). Namely, we classify the parameters to three sets and prove that: for the parameters from first class each point, for which the trajectory is well defined, is a periodic point of (f); for the parameters from second class any trajectory (under (f)) converges to one of fixed points (there may be up to two fixed points); for the parameters from third class any trajectory is dense in (mathbb R). For the (p)-adic variable we give a review of known results about dynamical systems of function (f). Then using a recently developed method we give simple new proofs of these results and prove some new ones related to trajectories which do not converge. For the complex variables we give a review of known results.
We introduce discrete and (p)-adic continuous versions of the FitzHugh-Nagumo system on the one-dimensional (p)-adic unit ball. We provide criteria for the existence of Turing patterns. We present extensive simulations of some of these systems. The simulations show that the Turing patterns are traveling waves in the (p)-adic unit ball.
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