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The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.

A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.

The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.

A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.

In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.

The quadratic Vieta formulas (x, y) ↦ (u, v) = (x + y, xy) in the complex geometry define a two-fold branched covering ℂ2 → ℂ2 ramified over the parabola u 2 = 4v. Thinking about topics considered in Arnold’s paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ: A 2 → A 2 defined by the Vieta formulas (x, y) ↦ (u, v) = (x + y, xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate the simplest finite rings A = ℤ m and A = ℤ k ×ℤ m . Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.

In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).

The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for $$n \ge 3$$ any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine ( $$n \ge 2$$ ) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and Semrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions ( $$n\ge 2$$ ) is automatically an n-isometry.

The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.

The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.

In this paper, we study the joint Laplace transform and probability generating functions of two pairs of random variables: (1) the two-sided first-exit time and the number of claims by this time; (2) the two-sided smooth exit-recovery time and its associated number of claims. The joint transforms are expressed in terms of the so-called doubly-killed scale function that is defined in this paper. We also find explicit expressions for the joint density function of the two-sided first-exit time and the number of claims by this time. Numerical examples are presented for exponential claims.

This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $$\Psi $$ on domains or manifolds $$\Gamma $$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ , provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces $$b^\alpha _{p,q}(\nabla )$$ of Besov type is developed.

In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .

In this paper we present several natural q-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some related analytical results and ask for combinatorial interpretations.

The author in [7] conjectured the following inequality; If a and b are nonnegative real numbers with a + b = 1/2, then the inequality 1/2 ≤ a(2b)k+ b(2a)k ≤ 1 holds for 0 ≤ k ≤ 1. In this paper, we shall prove the conjecture affirmatively and give the upper and lower estimation of the power exponential functions ab + ba for the nonnegative real numbers a and b with a + b = 2. Moreover, we pose some inequalities with power exponential functions.

In this paper some new refinements of the discrete Jensen’s inequality are obtained in real vector spaces. The idea comes from some former refinements determined by cyclic permutations. We essentially generalize and extend these results by using permutations of finite sets and bijections of the set of positive numbers. We get refinements of the discrete Jensen’s inequality for infinite convex combinations in Banach spaces. Similar results are rare. Finally, some applications are given on different topics.

In the paper, the author generalizes several formulas and series identities involving the Catalan numbers and establishes several new formulas and series identities involving the Catalan-Qi numbers and q-Catalan-Qi numbers.

Domination game (Bresar et al. in SIAM J Discrete Math 24:979–991, 2010) and total domination game (Henning et al. in Graphs Comb 31:1453–1462 (2015) are by now well established games played on graphs by two players, named Dominator and Staller. In this paper, Z-domination game, L-domination game, and LL-domination game are introduced as natural companions of the standard domination games. Versions of the Continuation Principle are proved for the new games. It is proved that in each of these games the outcome of the game, which is a corresponding graph invariant, differs by at most one depending whether Dominator or Staller starts the game. The hierarchy of the five domination games is established. The invariants are also bounded with respect to the (total) domination number and to the order of a graph. Values of the three new invariants are determined for paths up to a small constant independent from the length of a path. Several open problems and a conjecture are listed. The latter asserts that the L-domination game number is not greater than 6 / 7 of the order of a graph.

In this paper, we characterize power bounded weighted composition operators on weighted Bergman spaces of strongly pseudoconvex bounded domains in $${\mathbb {C}}^n$$. Also, we introduce the notion of power bounded below operators, then, for $$\alpha >0$$, we characterize power bounded below composition operators on $${\mathcal {D}}_\alpha $$, the weighted Dirichlet space on the unit disk of the complex plane.

The aim of this paper is to study the operator (ddc▪)q ∧ T on some classes of plurisubharmonic (psh) functions, which are not necessary bounded, where T is a positive closed current of bidimension (q, q) on an open set Ω of ℂn. The author introduces two classes $${\cal F}_p^T\left( {\rm{\Omega }} \right)$$ and $${\cal E}_p^T\left( {\rm{\Omega }} \right)$$ and shows first that they belong to the domain of definition of the operator (ddc▪)q ∧ T. Then the author proves that all functions that belong to these classes are CT-quasi-continuous and that the comparison principle is valid for them.

In this paper, the author gives the discrete criteria and Jorgensen inequalities of subgroups for the special linear group on F((t)) in two and higher dimensions.

In this paper, the authors generalize the concept of asymptotically almost negatively associated random variables from the classic probability space to the upper ex- pectation space. Within the framework, the authors prove some different types of Rosen- thal’s inequalities for sub-additive expectations. Finally, the authors prove a strong law of large numbers as the application of Rosenthal’s inequalities.

This paper deals with the function u which satisfies △ku = 0, where k ≥ 2 is an integer. Such a function u is called a polyharmonic function. The author gives an upper bound of the measure of the nodal set of u, and shows some growth property of u.

This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers.

An association scheme is said to be T-amorphous if it is antisymmetric and any tournament obtained by an appropriate merging of its classes is doubly regular. The goal of this paper is to study basic properties of T-amorphous schemes.

Reed–Muller codes are one of the most well-studied families of codes; however, there are till open problems regarding their structure. Recently a new ring-theoretic approach has emerged that provides a rather intuitive construction of these codes. This approach is centered around the notion of basic Reed–Muller codes. It is known that basic Reed–Muller codes ℳπ(m, k) over a prime field are powers of the radical RS of a corresponding group algebra and over a nonprime field there are no such equalities, except for trivial ones. In this paper, we consider the ideals ℜSℳπ(m, k) that arise while studying the inclusions of the basic codes and radical powers.

In this paper, we discuss our most recent results on the optimal L2 extension problem and Siu’s lemma as applications of the strong openness property of multiplier ideal sheaves obtained by Guan and Zhou.

The n-dimensional integer lattice, denoted by $${{\mathbb {Z}}}^n$$ , is the subset of $${{\mathbb {R}}}^n$$ consisting of those points whose coordinates are all integers. In this expository paper, many concrete, intuitive, and geometric results concerning the integer lattice $${{\mathbb {Z}}}^n$$ are presented, most of them together with new elementary or streamlined proofs. Some of the presented results are new, and others are improved versions of old results.

After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones.

In this paper, we provide a common generalization to the well-known Erdos–Ko–Rado Theorem, Frankl–Wilson Theorem, Alon–Babai–Suzuki Theorem, and Snevily Theorem on set systems with L-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with k-wise L-intersections by Furedi and Sudakov [J. Combin. Theory, Ser. A, 105, 143–159 (2004)]. We will also derive similar results on L-intersecting families of subspaces of an n-dimensional vector space over a finite field F q , where q is a prime power.

Recently, a large deviation multifractal formalism based on histograms of wavelet leader coefficients, compared to some other wavelet-based formalisms, was proved to be efficient for uniform Holder functions. In this paper, we extend this efficiency for non-uniform Holder functions. We first obtain optimal bounds for both wavelet and wavelet leader histograms for all functions in the critical Besov space $$B^{m/t,q}_t({\mathbb {T}})$$ , where $$t,q>0$$ and $${\mathbb {T}}$$ is the unit torus of $$\mathbb {R}^m$$ . We then compute these histograms for quasi-all functions in $$B^{m/t,q}_t({\mathbb {T}})$$ , in the sense of Baire Category. Although, increasing parts of these histograms have increasing visibility, they coincide only if $$00$$ , however wavelet histograms method covers it only if $$0

The Paul Erdős and Andras Gyarfas conjecture states that every graph of minimum degree at least 3 contains a simple cycle whose length is a power of two. In this paper, we prove that the conjecture holds for Cayley graphs on generalized quaternion groups, dihedral groups, semidihedral groups and groups of order \(p^3\).

In this paper, we generalize Rees–Shishikura’s theorem to the class of geometrically finite rational maps.

Let G = (V,E) be a simple graph. A function f: E → {+1,−1} is called a signed cycle domination function (SCDF) of G if Ʃ e∈E(C) f(e) ≥ 1 for every induced cycle C of G. The signed cycle domination number of G is defined as γ′sc(G) = min{Ʃ e∈E f(e)| f is an SCDF of G}. This paper will characterize all maximal planar graphs G with order n ≥ 6 and γ′sc(G) = n.

In the 1960s Cargo and Shisha introduced a metric in a family of quasi-arithmetic means defined on a common interval as the maximal possible difference between these means taken over all admissible vectors with corresponding weights. During the years 2013–2016 we proved that, having two quasi-arithmetic means, we can majorize the distance between them in terms of the Arrow–Pratt index. In this paper we are going to prove that this operator can also be used to establish certain lower bounds of this distance.

In the present paper, we study a Korovkin type approximation theorem in the setting of $$K_{a}$$ -convergence that contains the classical result. We also study the rate of $$K_{a}$$ -convergence and afterwards, we give some concluding remarks.

In this paper, we consider two kinds of 2-Calabi–Yau triangulated categories with finitely many indecomposable objects up to isomorphisms, called An,t = D b (KA(2t+1)(n+1)−3)/τt(n+1)−1[1], where n, t ≥ 1, and Dn,t = D b (KD2t(n+1))/τ(n+1)φ n , where n, t ≥ 1, and φ is induced by an automorphism of D2t(n+1) of order 2. Except the categories An,1, they all contain non-zero maximal rigid objects which are not cluster tilting. An,1 contain cluster tilting objects. We define the cluster complex of An,t (resp. Dn,t) by using the geometric description of cluster categories of type A (resp. type D). We show that there is an isomorphism from the cluster complex of An,t (resp. Dn,t) to the cluster complex of root system of type B n . In particular, the maximal rigid objects are isomorphic to clusters. This yields a result proved recently by Buan–Palu–Reiten: Let $${R_{{A_{n,t}}}}$$ , resp. $${R_{{D_{n,t}}}}$$ , be the full subcategory of An,t, resp. Dn,t, generated by the rigid objects. Then $${R_{{A_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ and $${R_{{D_{n,t}}}} \simeq {R_{{A_{n,1}}}}$$ as additive categories, for all t ≥ 1.

The paper presents a technique for the automatic calculation of Belyi functions for trees with weighted edges. Bibliography: 20 titles.

Given a continuous strictly monotone function $$\varphi $$ defined on an open real interval I and a probability measure $$\mu $$ on the Borel subsets of [0, 1], the Mako–Pales mean is defined by $$\begin{aligned} {\mathcal {M}}_{\varphi ,\mu }(x,y):=\varphi ^{-1}\left( \int ^1_0\varphi (tx+(1-t)y)\, d\mu (t)\right) ,\quad x,y\in I. \end{aligned}$$ Under some conditions on the functions $$\varphi $$ and $$\psi $$ defined on I, the quotient mean is given by $$\begin{aligned} Q_{\varphi ,\psi }(x,y):=\left( \frac{\varphi }{\psi }\right) ^{-1}\left( \frac{\varphi (x)}{\psi (y)}\right) , \quad x,y\in I. \end{aligned}$$ In this paper, we study some invariance of the quotient mean with respect to Mako–Pales means.

Let G be a finite group and σ = {σ i |i∈I} be a partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H is said to be σ-permutable if G possesses a complete Hall σ-set H such that HA x = A x H for all A ∈ H and all x ∈ G. Let H be a subgroup of G. Then we say that: (1) H is σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT = H σG and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G, and H σG is the σ-permutable closure of H, that is, the intersection of all σ-permutable subgroups of G containing H. (2) H is σ-n-embedded in G if there exists a normal subgroup T of G such that HT = H G and H ∩ T ≤ H σG . In this paper, we study the properties of the new embedding subgroups and use them to determine the structure of finite groups.

Let b l (n) denote the number of l-regular partitions of n. In 2012, using the theory of modular forms, Furcy and Penniston presented several infinite families of congruences modulo 3 for some values of l. In particular, they showed that for α, n ≥ 0, b 25 (32α+3 n+2 · 32α+2-1) ≡ 0 (mod 3). Most recently, congruences modulo powers of 5 for c5(n) was proved by Wang, where c N (n) counts the number of bipartitions (λ1,λ2) of n such that each part of λ2 is divisible by N. In this paper, we prove some interesting Ramanujan-type congruences modulo powers of 5 for b25(n), B25(n), c25(n) and modulo powers of 7 for c49(n). For example, we prove that for j ≥ 1, $${c_{25}}\left( {{5^{2j}}n + \frac{{11 \cdot {5^{2j}} + 13}}{{12}}} \right) \equiv 0$$ (mod 5 j+1), $${c_{49}}\left( {{7^{2j}}n + \frac{{11 \cdot {7^{_{2j}}} + 25}}{{12}}} \right) \equiv 0$$ (mod 7 j+1) and b 25 (32α+3 · n+2 · 32α+2-1) ≡ 0 (mod 3 · 52j-1).

Let G be a finite p-group. The automorphism α of a group G is said to be an absolute central automorphism, if for all x ∈ G, x -1 x α ∈ L(G), where L(G) is the absolute center of G. In this paper, we obtain a necessary and sufficient condition that each absolute central automorphism of G fixes the center element-wise.

The main purpose of this paper is to give a complete description of isometry groups on the 4-dimensional simply connected nilpotent Lie groups. We distinguish between two geometrically distinct cases of degenerate and nondegenerate center of the group. Since Walker metrics appear as the underlying structure of neutral signature metrics on the nilpotent Lie groups with degenerate center, we find necessary and sufficient condition for them to locally admit the nilpotent group of isometries.

For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame (WNABF). Under the setting of reducing subspaces of L 2(R d ), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.

The main purpose of this paper is using the analytic method and properties of the classical quadratic Gauss sums to study the computational problem of a hybrid power mean of generalized quadratic Gauss sums and generalized Kloosterman sums and give an exact computational formula for it.

In the paper, we present some general theorems on large deviations of random vectors with cumulants satisfying the generalized Statulevicius condition. The results obtained are applicable in derivation of limit theorems in the scheme of series, including the case where the dimension of the considered random vectors is growing indefinitely.

In this paper, generalized random linear mappings are viewed as random field indexed by random parameters. This point of view leads to questions about sample paths properties. We obtain various necessary and sufficient conditions ensuring the existence of nice sample paths of a generalized random linear mapping. We also give illustrative examples and applications.

The classical Hurwitz theorem claims that there are exactly four normed algebras with division: the real numbers (ℝ), complex numbers (ℂ), quaternions (ℍ), and octonions (𝕆). The length of ℝ as an algebra over itself is zero; the length of ℂ as an ℝ-algebra equals one. The purpose of the present paper is to prove that the lengths of the ℝ-algebras of quaternions and octonions equal two and three, respectively.