Authors:Y.V. Lavrenyuk; A.S. Oliynyk Abstract: For an arbitrary odd prime $p$, we consider groups of all $p$-automata and all finite $p$-automata. We construct minimal generating sets in both the groups of all $p$-automata and its subgroup of finite $p$-automata. The key ingredient of the proof is the lifting technique, which allows the construction of a minimal generating set in a group provided a minimal generating set in its abelian quotient is given. To find the required quotient, the elements of the groups of $p$-automata and finite $p$-automata are presented in terms of tableaux introduced by L. Kaloujnine. Using this presentation, a natural homomorphism on the additive group of all infinite sequences over the field $\mathbb{Z}_p$ is defined and examined. PubDate: Sat, 30 Dec 2023 12:03:55 +000

Authors:M.R. Kuryliak; O.B. Skaskiv Abstract: Let $(\lambda_n)$ be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series $F(z)=\sum\limits_{n=0}^{+\infty} a_ne^{z\lambda_n}$, $z\in\mathbb{C}$, we denote $G_{\mu}(F),$ $G_{c}(F),$ $G_{a}(F)$ the domains of the existence, of the convergence and of the absolute convergence of maximal term $\mu(z,F)=\max\big\{ a_n e^{\Re(z\lambda_n)} : n\geq 0\big\}$, respectively. It is well known that $G_\mu(F), G_a(F)$ are convex domains. Let us denote $\mathcal{N}_1(z):=\{n : \Re(z\lambda_n)>0\}$, $\mathcal{N}_2(z):=\{n : \Re(z\lambda_n)<0\}$ and \[\alpha^{(1)}(\theta) :=\varliminf\limits_{\genfrac{}{}{0pt}{2}{n\to +\infty}{n\in\mathcal{N}_1(e^{i\theta})}}\frac{-\ln a_n }{\Re(e^{i\theta}\lambda_n)},\qquad \alpha^{(2)}(\theta) :=\varlimsup\limits_{\genfrac{}{}{0pt}{2}{n\to +\infty}{n\in\mathcal{N}_2(e^{i\theta})}}\frac{-\ln a_n }{\Re(e^{i\theta}\lambda_n)}.\] Assume that $a_n\to 0$ as $n\to +\infty$. In the article, we prove the following statements. $1)$ If $\alpha^{(2)}(\theta)<\alpha^{(1)}(\theta)$ for some $\theta\in [0,\pi)$ then \[\big\{te^{i\theta} : t\in (\alpha^{(2)}(\theta),\alpha^{(1)}(\theta))\big\}\subset G_\mu(F)\] as well as \[\big\{te^{i\theta} : t\in (-\infty,\alpha^{(2)}(\theta))\cup (\alpha^{(1)}(\theta),+\infty)\big\}\cap G_\mu(F)=\emptyset.\] $2)$ $G_\mu(F)=\bigcup\limits_{\theta\in [0,\pi)}\{z=te^{i\theta} : t\in (\alpha^{(2)}(\theta),\alpha^{(1)}(\theta))\}.$ $3)$ If $h:=\varliminf\limits_{n\to +\infty}\frac{-\ln a_n }{\ln n}\in (1,+\infty)$, then \[\Big(\frac{h}{h-1}\cdot G_a(F)\Big)\supset G_\mu(F)\supset G_c(F).\] If $h=+\infty$ then $G_a(F)=G_c(F)=G_\mu(F)$, therefore $G_c(F)$ is also a convex domain. PubDate: Sat, 30 Dec 2023 11:21:25 +000

Authors:H.A. Ganie; B.A. Rather, M. Aouchiche Abstract: Several matrices are associated with graphs in order to study their properties. In such a study, researchers are interested in the spectra of the matrix under consideration, therefore, the properties are called spectral properties, with reference to the matrix. One of the interesting and hard problems in the spectral study of graphs is to order the graphs based on some spectral graph invariant, like the spectral radius, the second smallest eigenvalue, the energy, etc. Due to hardness of this problem it has been considered in the literature for small classes of graphs. Here we continue this study and add some more classes of graphs which can be ordered on the basis of spectral graph invariants. In this article, we study spectral properties of trees of diameter three, called double stars, and their complements through their reciprocal distance Laplacian eigenvalues. We give ordering of these graphs based on their reciprocal distance Laplacian spectral radius, on their second smallest reciprocal distance Laplacian eigenvalue, and on their reciprocal distance Laplacian energy. PubDate: Sat, 30 Dec 2023 09:29:43 +000

Authors:V.F. Babenko; V.V. Babenko, O.V. Kovalenko, N.V. Parfinovych Abstract: We obtain a sharp Nagy type inequality in a metric space $(X,\rho)$ with measure $\mu$ that estimates the uniform norm of a function using its $\ \cdot\ _{H^\omega}$-norm determined by a modulus of continuity $\omega$, and a seminorm that is defined on a space of locally integrable functions. We consider charges $\nu$ that are defined on the set of $\mu$-measurable subsets of $X$ and are absolutely continuous with respect to $\mu$. Using the obtained Nagy type inequality, we prove a sharp Landau-Kolmogorov type inequality that estimates the uniform norm of a Radon-Nikodym derivative of a charge via a $\ \cdot\ _{H^\omega}$-norm of this derivative, and a seminorm defined on the space of such charges. We also prove a sharp inequality for a hypersingular integral operator. In the case $X={\mathbb R}_+^m\times {\mathbb R}^{d-m}$, $0\le m\le d$, we obtain inequalities that estimate the uniform norm of a mixed derivative of a function using the uniform norm of the function and the $\ \cdot\ _{H^\omega}$-norm of its mixed derivative. PubDate: Thu, 28 Dec 2023 09:10:46 +000

Authors:O. Bezushchak Abstract: We describe derivations of finitary Mackey algebras over fields of characteristics not equal to $2.$ We prove that an arbitrary derivation of an associative finitary Mackey algebra or one of the Lie algebras $\mathfrak{sl}_{\infty}(V W)$, $\mathfrak{o}_{\infty}(f)$ is an adjoint operator of an element in the corresponding Mackey algebra. It provides a description of the derivations of all algebras in the Baranov-Strade classification of finitary simple Lie algebras. The proof is based on N. Jacobson's result on derivations of associative algebras of linear transformations of an infinite-dimensional vector space and the results on Herstein's conjectures. PubDate: Thu, 28 Dec 2023 08:53:52 +000

Authors:M. Aijaz; K. Rani, S. Pirzada Abstract: Let $R$ be a commutative ring with unity $1\ne 0$. In this paper, we completely describe the vertex and the edge chromatic number of the compressed zero divisor graph of the ring of integers modulo $n$. We find the clique number of the compressed zero divisor graph $\Gamma_E(\mathbb Z_n)$ of $\mathbb Z_n$ and show that $\Gamma_E(\mathbb Z_n)$ is weakly perfect. We also show that the edge chromatic number of $\Gamma_E(\mathbb Z_n)$ is equal to the largest degree proving that $\Gamma_E(\mathbb Z_n)$ resides in class 1 family of graphs. PubDate: Tue, 26 Dec 2023 08:04:15 +000

Authors:N.V. Prevysokova Abstract: The paper deals with the factorization of the matrices of discrete wavelet transform based on the Galois functions of different orders. It is used the known method of factorization of the matrices of the discrete Haar transform. Factorized matrices of transforms are presented in the form of a product of sparse matrices. This representation is the basis for building fast transforms algorithms. PubDate: Mon, 25 Dec 2023 09:55:41 +000

Authors:V.M. Simulik Abstract: The generalized Dirac equation related to 7-component space-time with one time coordinate and six space coordinates has been introduced. Three 8-component Dirac equations have been derived from the same 256-dimensional Clifford-Dirac matrix algebra. Corresponding Clifford-Dirac algebra is considered in the Pauli-Dirac representation of $8 \times 8$ gamma matrices. It is proved that this matrix algebra over the field of real numbers has 256-dimensional basis and it is isomorphic to geometric $\textit{C}\ell^{\texttt{R}}$(1,7) algebra. The corresponding gamma matrix representation of 45-dimensional $\mathrm{SO}(1,9)$ algebra is derived and the way of its generalization to the $\mathrm{SO}(m,n)$ algebra is demonstrated. The Klein-Gordon equation in 7-component space-time is considered as well. The way of corresponding consideration of the Maxwell equations and of equations for an arbitrary spin is indicated. PubDate: Sun, 24 Dec 2023 09:23:44 +000

Authors:O.O. Zavarzina Abstract: It is known that if any function acting from precompact metric space to itself increases the distance between some pair of points then it must decrease distance between some other pair of points. We show that this is not the case for quasi-metric spaces. After that, we present some sufficient conditions under which the previous property holds true for hereditarily precompact quasi-metric spaces. PubDate: Mon, 18 Dec 2023 07:57:22 +000

Authors:A.A. Basumatary; D.J. Sarma, B.C. Tripathy Abstract: The intention of this article is to define the concept of weakly $M$-preopen function in biminimal structure spaces. Several properties of this function have been established and its relationship with some other notions related to $M$-preopen sets in biminimal spaces have been investigated. PubDate: Sun, 17 Dec 2023 15:14:24 +000

Authors:L.H. Gallardo Abstract: We give a lower bound of the degree and the number of distinct prime divisors of the index of special perfect polynomials. More precisely, we prove that $\omega(d) \geq 9$, and $\deg(d) \geq 258$, where $d := \gcd(Q^2,\sigma(Q^2))$ is the index of the special perfect polynomial $A := p_1^2 Q^2$, in which $p_1$ is irreducible and has minimal degree. This means that $ \sigma(A)=A$ in the polynomial ring ${\mathbb{F}}_2[x]$. The function $\sigma$ is a natural analogue of the usual sums of divisors function over the integers. The index considered is an analogue of the index of an odd perfect number, for which a lower bound of $135$ is known. Our work use elementary properties of the polynomials as well as results of the paper [J. Théor. Nombres Bordeaux 2007, 19 (1), 165$-$174]. PubDate: Mon, 11 Dec 2023 07:49:29 +000

Authors:A. Şahin; O. Alagöz Abstract: In this paper, we consider the class of mappings satisfying $(CSC)$-condition. Further, we prove the strong and $\triangle$-convergence theorems of the $JF$-iteration process for this class of mappings in Hadamard spaces. At the end, we give a numerical example to show that the $JF$-iteration process is faster than some well known iterative processes. Our results improve and extend the corresponding recent results of the current literature. PubDate: Sun, 10 Dec 2023 09:12:42 +000

Authors:H.A. Erdem; A. Uçum, K. İlarslan, Ç. Camcı Abstract: In the theory of curves in Euclidean $3$-space, it is well known that a curve $\beta $ is said to be a Bertrand curve if for another curve $\beta^{\star}$ there exists a one-to-one correspondence between $\beta $ and $\beta^{\star}$ such that both curves have common principal normal line. These curves have been studied in different spaces over a long period of time and found wide application in different areas. In this article, the conditions for a timelike curve to be Bertrand curve are obtained by using a new approach in contrast to the well-known classical approach for Bertrand curves in Minkowski $3$-space. Related examples that meet these conditions are given. Moreover, thanks to this new approach, timelike, spacelike and Cartan null Bertrand mates of a timelike general helix have been obtained. PubDate: Tue, 05 Dec 2023 13:47:09 +000

Authors:O.V. Fedunyk-Yaremchuk; S.B. Hembars'ka, K.V. Solich Abstract: We obtain the exact-order estimates of approximation of the Nikol'skii-Besov-type classes $B^{\Omega}_{\infty,\theta}$ of periodic functions of several variables with a given function $\Omega(t)$ of a special form by using linear operators satisfying certain conditions. The approximation error is estimated in the metric of the space $L_{\infty}$. The obtained estimates of the considered approximation characteristic, in addition to independent interest, can be used to establish the lower bounds of the orthowidths of the corresponding functional classes. PubDate: Tue, 05 Dec 2023 13:25:53 +000

Authors:D.I. Bodnar; O.S. Bodnar, I.B. Bilanyk Abstract: Truncation error bounds for branched continued fractions of the special form are established. These fractions can be obtained by fixing the values of variables in branched continued fractions with independent variables, which is an effective tool for approximating complex functions of two variables. The main result is a two-dimensional analog of the theorem considered in [SCIAM J. Numer. Anal. 1983, 20 (3), 1187$-$1197] for van Vleck's continued fractions. For its proving, the $\mathcal{C}$-figure convergence and estimates of the difference between approximants of fractions in an angular domain are significantly used. In comparison with the previously established results, the elements of a branched continued fraction of the special form can tend to zero at a certain rate. An example of the effectiveness of using a two-dimensional analog of van Vleck's theorem is considered. PubDate: Tue, 21 Nov 2023 11:09:10 +000

Authors:W. Ramírez; D. Bedoya, A. Urieles, C. Cesarano, M. Ortega Abstract: In this paper, we introduce the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomials, their numbers, and their relationship with the Riemann zeta function. We also derive the Apostol-type generalizations to obtain some of their algebraic and differential properties. We introduce generalized $U$-Bernoulli, $U$-Euler and $U$-Genocchi polynomial Pascal-type matrix. We deduce some product formulas related to this matrix. Furthermore, we establish some explicit expressions for the $U$-Bernoulli, $U$-Euler, and $U$-Genocchi polynomial matrices, which involves the generalized Pascal matrix. PubDate: Tue, 21 Nov 2023 10:54:10 +000

Authors:T. Goy; M. Shattuck Abstract: In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schröder and Fine number sequences. Algebraic proofs of these results can be given which make use of Trudi's formula and the generating function of the associated sequence of determinants. We also provide direct arguments of our results that utilize various counting techniques, among them sign-changing involutions, on combinatorial structures related to classes of lattice paths enumerated by the Schröder and Fine numbers. As a consequence of our results, we obtain some new formulas for the Schröder and Catalan numbers as well as for some additional sequences from the OEIS in terms of determinants of certain Hessenberg-Toeplitz matrices. PubDate: Mon, 20 Nov 2023 13:16:01 +000

Authors:I.V. Burtnyak; Yu.Yu. Chopyuk, S.I. Vasylyshyn, T.V. Vasylyshyn Abstract: In this work, we investigate algebras of block-symmetric and weakly symmetric polynomials and analytic functions on complex Banach spaces of Lebesgue measurable functions, for which the $p$th power of the absolute value is Lebesgue integrable, where $p\in[1,+\infty),$ and Lebesgue measurable essentially bounded functions on $[0,1].$ We construct generating systems of algebras of all weakly symmetric continuous complex-valued polynomials on these spaces. Also we establish conditions under which sets of weakly symmetric analytic functions are algebras. PubDate: Tue, 14 Nov 2023 11:56:24 +000

Authors:A. Bougoutaia; A. Belacel, R. Macedo, H. Hamdi Abstract: In this article, we establish new relationships involving the class of Cohen positive strongly $p$-summing multilinear operators. Furthermore, we introduce a new class of multilinear operators on Banach lattices, called positive Cohen weakly nuclear multilinear operators. We establish a Pietsch domination-type theorem for this new class of multilinear operators. As an application, we show that every positive Cohen weakly $p$-nuclear multilinear operator is positive Dimant strongly $p$-summing and Cohen positive strongly $p$-summing. We conclude with a tensor representation of our class. PubDate: Tue, 14 Nov 2023 10:28:00 +000

Authors:N. Elsharkawy; C. Cesarano, R.I. Dmytryshyn, A. Elsharkawy Abstract: In this paper, we study the equiform Bishop formulae for the equiform timelike curves in 3-dimensional Minkowski space where the equiform timelike spherical curves are defined according to the equiform Bishop frame. We establish a necessary and sufficient condition for an equiform timelike curve to be an equiform timelike spherical curve. Furthermore, we give certain characterizations of equiform spherical curves in 3-dimensional Minkowski space, which are timelike with an equiform spacelike principal normal vector. PubDate: Sun, 29 Oct 2023 19:04:31 +000

Authors:M.M. Osypchuk Abstract: In the paper, the transition probability density of an isotropic $\alpha$-stable stochastic process in a finite dimensional Euclidean space is considered. The results of applying pseudo-differential operators with respect spatial variables to this function are estimated from the both side: above and below. Operators in the consideration are defined by the symbols $ \lambda ^\varkappa$ and $\lambda \lambda ^{\varkappa-1}$, where $\varkappa$ is some constant. The first operator with negative sign is fractional Laplacian and the second one multiplied by imaginary unit is fractional gradient. PubDate: Thu, 19 Oct 2023 13:24:13 +000

Authors:E. Zikkos Abstract: Recently D.T. Stoeva proved that if two Bessel sequences in a separable Hilbert space $\mathcal H$ are biorthogonal and one of them is complete in $\mathcal H$, then both sequences are Riesz bases for $\mathcal H$. This improves a well known result where completeness is assumed on both sequences. In this note we present an alternative proof of Stoeva's result which is quite short and elementary, based on the notion of Riesz-Fischer sequences. PubDate: Sun, 10 Sep 2023 06:35:11 +000

Authors:K.N. Soltanov Abstract: In this article, the existence of the spectrum (the eigenvalues) for the nonlinear continuous operators acting in the Banach spaces is investigated. For the study this question it is used a different approach that allows the studying of all eigenvalues of a nonlinear operator relative to another nonlinear operator. Here we show that in nonlinear operators case it is necessary to seek the spectrum of the given nonlinear operator relative to another nonlinear operator satisfying certain conditions. The different examples, for which eigenvalues can be found, are provided. Moreover, the nonlinear problems including parameters are studied. PubDate: Thu, 07 Sep 2023 15:16:16 +000

Authors:O.V. Gutik; O.B. Popadiuk Abstract: We study the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$, which is introduced in the paper [Visnyk Lviv Univ. Ser. Mech.-Mat. 2020, 90, 5-19 (in Ukrainian)], in the case when the ${\omega}$-closed family $\mathscr{F}_n$ generated by the set $\{0,1,\ldots,n\}$. We show that the Green relations $\mathscr{D}$ and $\mathscr{J}$ coincide in $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$, the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is isomorphic to the semigroup $\mathscr{I}_\omega^{n+1}(\overrightarrow{\mathrm{conv}})$ of partial convex order isomorphisms of $(\omega,\leqslant)$ of the rank $\leqslant n+1$, and $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ admits only Rees congruences. Also, we study shift-continuous topologies on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$. In particular, we prove that for any shift-continuous $T_1$-topology $\tau$ on the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ every non-zero element of $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is an isolated point of $(\boldsymbol{B}_{\omega}^{\mathscr{F}_n},\tau)$, $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ admits the unique compact shift-continuous $T_1$-topology, and every $\omega_{\mathfrak{d}}$-compact shift-continuous $T_1$-topology is compact. We describe the closure of the semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ in a Hausdorff semitopological semigroup and prove the criterium when a topological inverse semigroup $\boldsymbol{B}_{\omega}^{\mathscr{F}_n}$ is $H$-closed in the class of Hausdorff topological semigroups. PubDate: Wed, 09 Aug 2023 07:01:09 +000

Authors:S. Nykorovych; O. Nykyforchyn Abstract: In two ways we introduce metrics on the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric on a fixed set, and prove that the obtained metrics are compact and topologically equivalent. To achieve this, we give a characterization of the sets being the hypographs of the mentioned pseudoultrametrics, and apply Hausdorff metric to their family. It is proved that the uniform convergence metric is a limit case of metrics defined via hypographs. It is shown that the set of all pseudoultrametrics, not exceeding a given compact pseudoultrametric, with the induced topology is a Lawson compact Hausdorff upper semilattice. PubDate: Thu, 03 Aug 2023 19:02:00 +000