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Том 10, № 1 (2018)

Год: 2018
Статей: 6
URL: https://journals.rcsi.science/2070-0466/issue/view/12521

Review Articles

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Bendikov A.

Аннотация

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) − f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (−L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (−L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (−L^J,D).

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p-Adic Numbers, Ultrametric Analysis and Applications. 2018;10(1):1-11

1-11

(Eng)

(JATS XML)

Research Articles

New Applications of the p-Adic Nevanlinna Theory

Escassut A., An T.

Аннотация

Let IK be an algebraically closed field of characteristic 0 complete for an ultrametric absolute value. Following results obtained in complex analysis, here we examine problems of uniqueness for meromorphic functions having finitely many poles, sharing points or a pair of sets (C.M. or I.M.) defined either in the whole field IK or in an open disk, or in the complement of an open disk. Following previous works in C, we consider functions fⁿ(x)f^m(ax + b), gⁿ(x)g^m(ax + b) with |a| = 1 and n ≠ m, sharing a rational function and we show that f/g is a n + m-th root of 1 whenever n + m ≥ 5. Next, given a small function w, if n, m ∈ IN are such that |n − m|_∞ ≥ 5, then fⁿ(x)f^m(ax + b) − w has infinitely many zeros. Finally, we examine branched values for meromorphic functions fⁿ(x)f^m(ax + b).

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p-Adic Numbers, Ultrametric Analysis and Applications. 2018;10(1):12-31

12-31

(Eng)

(JATS XML)

On Integrable Delta Functions on the Levi-Civita Field

Flynn D., Shamseddine K.

Аннотация

In this paper, we develop a theory of integrable delta functions on the Levi-Civita field R as well as on R² and R³ with similar properties to the one-dimensional, two-dimensional and three-dimensional Dirac Delta functions and which reduce to them when restricted to points in R, R² and R³, respectively. First we review the recently developed Lebesgue-like measure and integration theory over R, R² and R³. Then we introduce delta functions on R, R² and R³ that are integrable in the context of the aforementioned integration theory; and we study their properties and some applications.

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p-Adic Numbers, Ultrametric Analysis and Applications. 2018;10(1):32-56

32-56

(Eng)

(JATS XML)

Non-Archimedean Pseudodifferential Operators and Feller Semigroups

Torresblanca-Badillo A., Zúñiga-Galindo W.

Аннотация

In this article we study a large class of non-Archimedean pseudodifferential operators whose symbols are negative definite functions.We prove that these operators extend to generators of Feller semigroups. In order to study these operators, we introduce a new class of anisotropic Sobolev spaces, which are the natural domains for the operators considered here.We also study the Cauchy problem for certain pseudodifferential equations.

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p-Adic Numbers, Ultrametric Analysis and Applications. 2018;10(1):57-73

57-73

(Eng)

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Short Communications

On a Congruence Involving Generalized Fibonomial Coefficients

Trojovský P.

Аннотация

Let (F_n)_n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as

\({\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_F} = \frac{{{F_{n - k + 1}} \cdots {F_{n - 1}}{F_n}}}{{{F_1} \cdots {F_k}}}\)

. In 2013, Marques, Sellers and Trojovský proved that if p is a prime number such that p ≡ ±1 (mod 5), then p∤\({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_F}\) for all integers a ≥ 1. In 2010, in particular, Kilic generalized the Fibonomial coefficients for

\({\left[ {\begin{array}{*{20}{c}} n \\ k \end{array}} \right]_{F,m}} = \frac{{{F_{\left( {n - k + 1} \right)m}} \cdots {F_{\left( {n - 1} \right)m}}{F_{nm}}}}{{{F_m} \cdots {F_{km}}}}\)

. In this note, we generalize Marques, Sellers and Trojovský result to prove, in particular, that if p ≡ ±1 (mod 5), then \({\left[ {\begin{array}{*{20}{c}} {{p^{a + 1}}} \\ {{p^a}} \end{array}} \right]_{F,m}} \equiv 1\) (mod p), for all a ≥ 0 and m ≥ 1.

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p-Adic Numbers, Ultrametric Analysis and Applications. 2018;10(1):74-78

74-78