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A hypergraph $$H=(V,E)$$ has property $$B_k$$ if there exists a 2-coloring of the set $$V$$ such that each edge contains at least $$k$$ vertices of each color. We let $$m_{k,g}(n)$$ and $$m_{k,b}(n)$$ , respectively, denote the least number of edges of an $$n$$ -homogeneous hypergraph without property $$B_k$$ which contains either no cycles of length at least $$g$$ or no two edges intersecting in more than $$b$$ vertices. In the paper, upper bounds for these quantities are given. As a consequence, we obtain results for $$m^{*}_k(n)$$ , i.e., for the least number of edges of an $$n$$ -homogeneous simple hypergraph without property $$B_k$$ . Let $$\Delta(H)$$ be the maximal degree of vertices of a hypergraph $$H$$ . By $$\Delta_k(n,g)$$ we denote the minimal degree $$\Delta$$ such that there exists an $$n$$ -homogeneous hypergraph $$H$$ with maximal degree $$\Delta$$ and girth at least $$g$$ but without property $$B_k$$ . In the paper, an upper bound for $$\Delta_k(n,g)$$ is obtained.

The starting point of the present paper is a result by E.A. Boltarovych (1989) on convergence regions, dealing with branched continued fraction \[\sum_{i_1=1}^N\frac{a_{i(1)}}{1}{\atop+}\sum_{i_2=1}^N\frac{a_{i(2)}}{1}{\atop+}\ldots{\atop+}\sum_{i_n=1}^N\frac{a_{i(n)}}{1}{\atop+}\ldots,\] where $|a_{i(2n-1)}|\le\alpha/N,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ and $|a_{i(2n)}|\le r/(N-1),$ $i_{2n}\ne j_{2n},$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ where $N>1$ and $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions. In the present paper conditions for these regions are replaced by $\sum_{i_1=1}^N|a_{i(1)}|\le\alpha(1-\varepsilon),$ $\sum_{i_{2n+1}=1}^N|a_{i(2n+1)}|\le\alpha(1-\varepsilon),$ $i_p=\overline{1,N},$ $p=\overline{1,2n},$ $n\ge1,$ and for each multiindex $i(2n-1)$ there is a single index $j_{2n},$ $1\le j_{2n}\le N,$ such that $|a_{i(2n-1),j_{2n}}|\ge R$ and $\sum_{i_{2n}\in\{1,2,\ldots,N\}\backslash\{j_{2n}\}}|a_{i(2n)}|\le r,$ $i_p=\overline{1,N},$ $p=\overline{1,2n-1},$ $n\ge1,$ where $\varepsilon,$ $\alpha,$ $r$ and $R$ are real numbers that satisfying certain conditions, and better convergence speed estimates are obtained.

In this study, at first we prove that the existence of best proximity points for cyclic nonexpansive mappings is equivalent to the existence of best proximity pairs for noncyclic nonexpansive mappings in the setting of strictly convex Banach spaces by using the projection operator. In this way, we conclude that the main result of the paper [Proximal normal structure and nonexpansive mappings, Studia Math. 171 (2005), 283–293] immediately follows. We then discuss the convergence of best proximity pairs for noncyclic contractions by applying the convergence of iterative sequences for cyclic contractions and show that the convergence method of a recent paper [Convergence of Picard's iteration using projection algorithm for noncyclic contractions, Indag. Math. 30 (2019), no. 1, 227–239] is obtained exactly from Picard’s iteration sequence.

All groups in the family of Baumslag-Solitar groups (i.e., groups of the form G(m,n) = 〈a, b; a−1bma = bn〉, where m and n are nonzero integers) for which the residual nilpotence condition holds if and only if the residual p-finiteness condition holds for some prime number p are described. It has turned out, in particular, that the group G(pr, −pr), where p is an odd prime and r ≥ 1, is residually nilpotent, but it is residually q-finite for no prime q. Thus, an answer to the existence problem for noncyclic one-relator groups possessing such a property (formulated by McCarron in his 1996 paper) is obtained. A simple proof of the statement that an arbitrary residually nilpotent noncyclic one-relator group which has elements of finite order is residual p-finite for some prime p, which was announced in the same paper of McCarron, is also given.

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages

Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.

In this paper, the structure of relative disposition of 3-vertex cutsets in a 3-connected graph is studied. All such cutsets are divided into structural units – complexes of flowers, of cuts, of single cutsets, and trivial complexes. The decomposition of the graph by a complex of each type is described in detail. It is proved that for any two complexes C1 and C2 of a 3-connected graph G there is a unique part of the decomposition of G by C1 that contains C2. The relative disposition of complexes is described with the help of a hypertree T (G) – a hypergraph any cycle of which is a subset of a certain hyperedge. It is also proved that each nonempty part of the decomposition of G by the set of all of its 3-vertex cutsets is either a part of the decomposition of G by one of the complexes or corresponds to a hyperedge of T (G). This paper can be considered as a continuation of studies begun in the joint paper by D. V. Karpov and A. V. Pastor “On the structure of a 3-connected graph,” published in 2011. Bibliography: 10 titles.

In the paper, the elementary net closure problem is considered. An elementary net (net without a diagonal) σ = (σij)i ≠ j of additive subgroups σij of field k is called “closed” if elementary net group E(σ) does not contain new elementary transvections. Elementary net σ = (σij) is called “supplemented” if table (with a diagonal) σ = (σij), 1 ≤ i, j ≤ n, is a (full) net for some additive subgroups σii of field k. The supplemented elementary nets are closed. The necessary and sufficient condition for the supplementarity of elementary net σ = (σij) is the implementation of inclusions σijσjiσij ⊆ σij (for any i ≠ j). The question (Kourovka Notebook, Problem 19.63) is investigated of whether it true that, for closure of elementary net σ = (σij) it suffices to implement inclusions $$\sigma _{{ij}}^{2}{{\sigma }_{{ji}}}$$ ⊆ σji for any i ≠ j (here, ($$\sigma _{{ij}}^{2}$$ denotes the additive subgroup of field k generated by the squares from σij). The elementary nets for which the latter inclusions are satisfied are called “weakly supplemented elementary nets.” The concepts of supplemented and weakly supplemented elementary nets coincide for fields of odd characteristic. Thus, the aforementioned question of the sufficiency of weak supplementarity for the closure of an elementary net is relevant for the fields of characteristics 0 and 2. In this paper, examples of weakly supplemented but not supplemented elementary nets are constructed for the fields of characteristics 0 and 2. An example of a closed elementary net that is not weakly supplemented is constructed.

In this paper we consider a link $$B_{n,\rho }$$Bn,ρ between Baskakov type operators $$B_{n,\infty }$$Bn,∞ and genuine Baskakov–Durrmeyer type operators $$ B_{n,1}$$Bn,1 depending on a positive real parameter $$\rho $$ρ. The topic of the present paper is the pointwise limit relation $$\left( B_{n,\rho }f\right) \left( x\right) \rightarrow \left( B_{n,\infty }f\right) \left( x\right) $$Bn,ρfx→Bn,∞fx as $$\rho \rightarrow \infty $$ρ→∞ for $$x\ge 0.$$x≥0. As a main result we derive uniform convergence on each compact subinterval of the positive real axis for all continuous functions f of polynomial growth.

The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

The theory of mixed multiplicities of filtrations by $m$-primary ideals in a ring is introduced in a recent paper by Cutkosky, Sarkar and Srinivasan. In this paper, we consider the positivity of mixed multiplicities of filtrations. We show that the mixed multiplicities of filtrations must be nonnegative real numbers and give examples to show that they could be zero or even irrational. When $R$ is analytically irreducible, and $\mathcal I(1),\ldots,\mathcal I(r)$ are filtrations of $R$ by $m_R$-primary ideals, we show that all of the mixed multiplicities $e_R(\mathcal I(1)^{[d_1]},\ldots,\mathcal I(r)^{[d_r]};R)$ are positive if and only if the ordinary multiplicities $e_R(\mathcal I(i);R)$ for $1\le i\le r$ are positive. We extend this to modules and prove a simple characterization of when the mixed multiplicities are positive or zero on a finitely generated module., 15 pages

The current paper is devoted to the Cauchy problem for the stochastic generalized Benjamin-Ono equation. By establishing the bilinear estimate, trilinear estimates in some Bourgain spaces, we prove that the Cauchy problem for the stochastic generalized Benjamin-Ono equation is locally well-posed for the initial data u0(x, ω) ∈ L2 (Ω; Hs (ℝ)) which is $$\mathscr{F}_{0}$$ measurable with $$s \geqslant \frac{1}{2}-\frac{\alpha}{4}$$ and $$\Phi \in L_{2}^{0, s}$$ . In particular, when α = 1, we prove that it is globally well-posed for the initial data u0(x, ω) ∈ L2(Ω; H1(ℝ)) which is $$\mathscr{F}_{0}$$ measurable and $${\rm{\Phi }} \in L_2^{0,1}$$ . The key ingredients that we use in this paper are trilinear estimates, the Ito formula and the Burkholder-Davis-Gundy (BDG) inequality as well as the stopping time technique.

In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| < a, can be extended to a positive definite one on R if and only if 0 < a ≤ 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| ≤ a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a ∈ [1/2,1] and Re ϕ(x) = 1 - |x|, |x| ≤ a, for some ϕ ∈ T, then ϕ(x) = (1 - |x|) +, x G R. In addition, for any a G (0,1/2), there exists a function ϕ ∈ T such that ϕ(x) = 1 - |x|, |x| ≤ a, but ϕ(x) ≠ (1 - |x|)+. Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space $$\Pi _n^d$$ of spherical polynomials of degree at most n on the unit sphere $$\mathbb{S}{^d} \subset {^{d + 1}}$$ as n → ∞. It is shown that for 0 < p < ∞, $$\mathop {\lim }\limits_{x \to \infty } \sup \left\{ {\frac{{{{\left\| P \right\|}_{{L^\infty }({\mathbb{S}^d})}}}}{{{n^{\tfrac{d}{p}}}{{\left\| P \right\|}_{{L^p}({\mathbb{S}^d})}}}}:P \in \Pi _n^d} \right\} = \sup \left\{ {\frac{{{{\left\| f \right\|}_{{L^\infty }({\mathbb{R}^d})}}}}{{{{\left\| f \right\|}_{{L^p}({\mathbb{R}^d})}}}}:f \in \varepsilon _p^d} \right\},$$ where $$\varepsilon _p^d$$ denotes the space of all entire functions of spherical exponential type at most 1 whose restrictions to ℝd belong to the space Lp(ℝd), and it is agreed that 0/0 = 0. It is also proved that for 0 < p < q < ∞, $$\liminf _{n \rightarrow \infty} \sup \left\{\frac{\|P\|_{L^{q}\left(\mathbb{S}^{d}\right)}}{n^{d(1 / p-1 / q)}\|P\|_{L^{p}\left(\mathbb{S}^{d}\right)}}: P \in \Pi_{n}^{d}\right\} \geq \sup \left\{\frac{\|f\|_{L^{q}\left(\mathbb{R}^{d}\right)}}{\|f\|_{L^{p}\left(\mathbb{R}^{d}\right)}}: f \in \mathcal{E}_{p}^{d}\right\}.$$ These results extend the recent results of Levin and Lubinsky for trigonometric polynomials on the unit circle. The paper also determines the exact value of the Nikolskii constant for nonnegative functions with p = 1 and q = ∞: $$\lim _{n \rightarrow \infty} \sup _{0 \leq P \in \Pi_{n}^{d}} \frac{\|P\|_{L^{\infty}\left(\mathbb{S}^{d}\right)}}{\|P\|_{L^{1}\left(\mathbb{S}^{d}\right)}}=\sup _{0 \leq f \in \mathcal{E}_{1}^{d}} \frac{\|f\|_{L^{\infty}\left(\mathbb{R}^{d}\right)}}{\|f\|_{L^{1} \mathbb{R}^{d}}}=\frac{1}{4^{d} \pi^{d / 2} \Gamma(d / 2+1)}.$$

A graph Γ is called a bi-circulant if it admits a cyclic group as a group of automorphisms acting semiregularly on the vertices of Γ with two orbits. The characterization of tetravalent edgetransitive bi-circulants was given in several recent papers. In this paper, a classification is given of connected tetravalent vertex-transitive bi-circulants of order twice an odd integer.

In this paper, we consider weak, directional and strong matrix majorizations. Namely, for square matrices A and B of the same size we say that A is weakly majorized by B if there is a row stochastic matrix X such that A = XB. Further, A is strongly majorized by B if there is a doubly stochastic matrix X such that A = XB. Finally, A is directionally majorized by B if Ax is majorized by Bx for any vector x where the usual vector majorization is used. We introduce the notion of majorization of matrix tuples which is defined as a natural generalization of matrix majorizations: for a chosen type of majorization we say that one tuple of matrices is majorized by another tuple of the same size if every matrix of the “smaller” tuple is majorized by a matrix in the same position in the “bigger” tuple. We say that a linear operator preserves majorization if it maps ordered pairs to ordered pairs and the image of the smaller element does not exceed the image of the bigger one. This paper contains a full characterization of linear operators that preserve weak, strong or directional majorization of tuples of matrices and linear operators that map tuples that are ordered with respect to strong majorization to tuples that are ordered with respect to directional majorization. We have shown that every such operator preserves respective majorization of each component. For all types of majorization we provide counterexamples that demonstrate that the inverse statement does not hold, that is if majorization of each component is preserved, majorization of tuples may not.

Bidimensionality Theory was introduced by [E.D. Demaine, F.V. Fomin, M.Hajiaghayi, and D.M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J. ACM, 52 (2005), pp.866--893] as a tool to obtain sub-exponential time parameterized algorithms on H-minor-free graphs. In [E.D. Demaine and M.Hajiaghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (respectively contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO), admits a linear kernel for classes of graphs that exclude a fixed graph (respectively an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems g graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work., Comment: An an earlier version of this paper appeared in SODA 2010. That paper contained preliminary versions of some of the results of this paper

In this paper, we apply the Faber polynomial expansions to find upper bounds for the general coefficients |amk+1| (k >= 3) of functions in the subclass N?m (?,?;?). The results presented in this paper would generalize and improve some recent works.

The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. More precisely, Eells and Sampson suggested the investigation of the so-called $ES-r$-energy functionals $ E_r^{ES}(\varphi)=(1/2)\int_{M}\,|(d^*+d)^r (\varphi)|^2\,dV$, where $r \geq 2 $ and $ \varphi:M \to N$ is a map between two Riemannian manifolds. The initial part of this paper is a short overview on basic definitions, properties, recent developments and open problems concerning the functionals $ E_r^{ES}(\varphi)$ and other, equally interesting, higher order energy functionals $E_r(\varphi)$ which were introduced and studied in various papers by Maeta and other authors. If a critical point $\varphi$ of $E_r^{ES}(\varphi)$ (respectively, $E_r(\varphi)$) is an \textit{isometric immersion}, then we say that its image is an $ES-r$-harmonic (respectively, $r$-harmonic) submanifold of $N$. We observe that \textit{minimal} submanifolds are trivially both $ES-r$-harmonic and $r$-harmonic. Therefore, it is natural to say that an $ES-r$-harmonic ($r$-harmonic) submanifold is \textit{proper} if it is not minimal. In the special case that the ambient space $N$ is the Euclidean space $\mathbb{R}^n$ the notions of $ES-r$-harmonic and $r$-harmonic submanifolds coincide. The Chen-Maeta conjecture is still open: it states that, for all $r \geq2$, any proper, $r$-harmonic submanifold of $\mathbb{R}^n$ is minimal. In the second part of this paper we shall focus on the study of $G={\rm SO}(p+1) \times {\rm SO}(q+1)$-invariant submanifolds of $\mathbb{R}^n$, $n=p+q+2$. In particular, we shall obtain an explicit description of the relevant Euler-Lagrange equations in the case that $r=3$ and we shall discuss difficulties and possible developments towards the proof of the Chen-Maeta conjecture for $3$-harmonic $G$-invariant hypersurfaces.

If $G$ is a connected graph, the $distance$ $d(u, v)$ between two vertices $u, v \in V(G)$ is the length of a shortest path between them. Let $W = \{w_1,w_2, \dots ,w_k\}$ be an ordered set of vertices of $G$ and let $v$ be a vertex of $G$. The $representation$ $r(v|W)$ of $v$ with respect to $W$ is the k-tuple $(d(v,w_1), d(v,w_2), \dots , d(v,w_k))$. $W$ is called a $resolving set$ or a $locating set$ if every vertex of $G$ is uniquely identified by its distances from the vertices of $W$, or equivalently if distinct vertices of $G$ have distinct representations with respect to $W$. A resolving set of minimum cardinality is called a $metric basis$ for $G$ and this cardinality is the $metric dimension$ of $G$, denoted by $\beta(G)$. The metric dimension of some wheel related graphs is studied recently by Siddiqui and Imran. In this paper, we study the metric dimension of wheels with $k$ consecutive missing spokes denoted by $W(n,k)$. We compute the exact value of the metric dimension of $W(n,k)$ which shows that wheels with consecutive missing spokes have unbounded metric dimensions. It is natural to ask for the characterization of graphs with an unbounded metric dimension. The exchange property for resolving a set of $W(n,k)$ has also been studied in this paper and it is shown that exchange property of the bases in a vector space does not hold for minimal resolving sets of wheels with $k$-consecutive missing spokes denoted by $W(n,k)$.

The paper proves that an odd composite integer N can be factorized in O((log2N)4) bit operations if N = pq, the divisor q is of the form 2αu +1 or 2αu-1 with u being an odd integer and α being a positive integer and the other divisor p satisfies 1 < p ≤ 2α+1 or 2α +1 < p ≤ 2α+1-1. Theorems and corollaries are proved with detail mathematical reasoning. Algorithm to factorize the odd composite integers is designed and tested in Maple. The results in the paper demonstrate that fast factorization of odd integers is possible with the help of valuated binary tree.

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

In this paper we introduce a direct family of simple polytopes $P^{0}\subset P^{1}\subset\ldots$ such that for any $k$, $2\leq k\leq n$ there are non-trivial strictly defined Massey products of order $k$ in the cohomology rings of their moment-angle manifolds $\mathcal Z_{P^n}$. We prove that the direct sequence of manifolds $\ast\subset S^{3}\hookrightarrow\ldots\hookrightarrow\mathcal Z_{P^n}\hookrightarrow\mathcal Z_{P^{n+1}}\hookrightarrow\ldots$ has the following properties: every manifold $\mathcal Z_{P^n}$ is a retract of $\mathcal Z_{P^{n+1}}$, and one has inverse sequences in cohomology (over $n$ and $k$, where $k\to\infty$ as $n\to\infty$) of the Massey products constructed. As an application we get that there are non-trivial differentials $d_k$, for arbitrarily large $k$ as $n\to\infty$ in the Eilenberg--Moore spectral sequence connecting the rings $H^*(\Omega X)$ and $H^*(X)$ with coefficients in a field, where $X=\mathcal Z_{P^n}$., Comment: 53 pages, 5 figures; extended version of a paper to appear in Izv. Math

If $G$ is a group, a virtual retract of $G$ is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns., 30 pages, 1 figure. v3: added Lemma 5.8 and made minor corrections following referee's comments. This version of the paper has been accepted for publication

In this paper we study the b b -ary expansions of the square roots of the function defined by the recurrence f b ( n ) = b f b ( n − 1 ) + n f_b(n)=b f_b(n-1)+n with initial value f ( 0 ) = 0 f(0)=0 taken at odd positive integers n n , of which the special case b = 10 b=10 is often referred to as the “schizophrenic” or “mock-rational” numbers. Defined by Darling in 2004 2004 and studied in more detail by Brown in 2009 2009 , these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 b\geq 2 by formally defining the schizophrenic pattern present in the b b -ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.

In our previous paper we have elaborated a certain signed count of real lines on real projective n-dimensional hypersurfaces of degree 2n-1. Contrary to the honest "cardinal" count, it is independent of the choice of a hypersurface, and by this reason provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic threefolds., Comment: 20 pages, typos are corrected (most essential, in Proposition 4.3.3)

Let $f\in\mathbb{Z}[T]$ be any polynomial of degree $d>1$ and $F\in\mathbb{Z}[X_{0},...,X_{n}]$ an irreducible homogeneous polynomial of degree $e>1$ such that the projective hypersurface $V(F)$ is smooth. In this paper we give a bound for \[ N(f,F,B):=|\{\textbf{x}\in\mathbb{Z}^{n+1}:\max_{0\leq i\leq n}|x_{i}|\leq B,\exists t\in\mathbb{Z}\text{ such that }f(t)=F(\textbf{x})\}|, \] To do this, we introduce a generalization of the Heath-Brown and Munshi's power sieve and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables., Theorem 1 has been improved. The paper has been reorganized to improve the exposition

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the action of inertia groups on the l-torsion of the Jacobian, including at primes where the Jacobian has non-semistable reduction. We also give a framework for systematically dealing with primitivity of symplectic mod l Galois representations. The main result of the paper is the following. Suppose n=2g+2 is an even integer that can be written as a sum of two primes in two different ways, with none of the primes being the largest primes less than n (this hypothesis appears to hold for all g different from 0,1,2,3,4,5,7 and 13). Then there is an explicit integer N and an explicit monic polynomial $f_0(x)\in \mathbb{Z}[x]$ of degree n, such that the Jacobian $J$ of every curve of the form $y^2=f(x)$ has $Gal(\mathbb{Q}(J[l])/\mathbb{Q})\cong GSp_{2g}(\mathbb{F}_l)$ for all odd primes l and $Gal(\mathbb{Q}(J[2])/\mathbb{Q})\cong S_{2g+2}$, whenever $f(x)\in\mathbb{Z}[x]$ is monic with $f(x)\equiv f_0(x) \bmod{N}$ and with no roots of multiplicity greater than $2$ in $\overline{\mathbb{F}}_p$ for any p not dividing N., Comment: 24 pages, minor corrections

Michael Somos discovered several theta-function identities of various levels by computer and offered no proof for them. These identities highly resemble some of Ramanujan’s identities. The main focus of this paper is to prove some of these theta-function identities, in particular those of level 6 that have been discovered using computational searches. Some of the the Somos identities that we are discussing in this paper cannot be expressed in the form of P −−Q type. Furthermore, we establish certain colored partition identities for them.

An endofunction on an Abelian group A is a fonction f: An → A such that φf (x1, …, xn) = f (φ(x1), …, φ(xn)) for all endomorphisms φ of group A and all n from ℕ. If each endofunction has the form $$f\left(x_{1}, \ldots, x_{n}\right)=\sum\nolimits_{i=1}^{n} \lambda_{i} x_{i}$$ for some central endomorphisms λ1, …, λn of group A, then such a group is called generalized endoprimal (GE) group. In this paper, we find GE-groups in the class of nonreduced Abelian groups. In addition, results paper, we find GE-group in the class of nonreduced Abelian groups. In addition, results concerning connections of GE-groups with Abelian groups whose endomorphism rings are unique addition rings have been obtained.

The word maps $$ \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) $$ and $$ \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } $$ for a word $$ w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], $$ where D is a division algebra over a field K, are considered. It is proved that if $$ \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], $$ then $$ \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), $$ where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then $$ \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). $$ The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.

Let G be a graph with vertex set V(G), edge set E(G) and maximum degree Δ respectively. G is called degree-magic if it admits a labelling of the edges by integers {1, 2, …, |E(G)|} such that for any vertex v the sum of the labels of the edges incident with v is equal to $${{1 + \left| {E(G)} \right|} \over 2} \cdot d(v)$$ , where d(v) is the degree of v. Let f be a proper edge coloring of G such that for each vertex v ∈ V(G), |{e : e ∈ Ev, f(e) ≤ Δ/2}| = |{e : e ∈ Ev, f(e) > Δ/2}|, and such an f is called a balanced edge coloring of G. In this paper, we show that if G is a supermagic even graph with a balanced edge coloring and m ≥ 1, then (2m + 1)G is a supermagic graph. If G is a d-magic even graph with a balanced edge coloring and n ≥ 2, then nG is a d-magic graph. Results in this paper generalise some known results.

Associated to a presentable $\infty$-category $\mathcal{C}$ and an object $X \in \mathcal{C}$ is the tangent $\infty$-category $\mathcal{T}_X\mathcal{C}$, consisting of parameterized spectrum objects over $X$. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $\mathcal{T}_X\mathcal{C}$. When $\mathcal{C}$ consists of algebras over a nice $\infty$-operad in a stable $\infty$-category, $\mathcal{T}_X\mathcal{C}$ is equivalent to the $\infty$-category of operadic modules, by work of Basterra--Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper., Comment: The section concerning stabilization of model categories was separated into an independent paper, appearing now as arXiv:1802.08031

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area

In this paper, we introduce and study a new kind of graph related to a unitary module [Formula: see text] on a commutative ring [Formula: see text] with identity, namely the annihilators comaximal graph of submodules of [Formula: see text], denoted by [Formula: see text]. The (undirected) graph [Formula: see text] is with vertices of all non-trivial submodules of [Formula: see text] and two vertices [Formula: see text] of [Formula: see text] are adjacent if and only if their annihilators are comaximal ideals of [Formula: see text], i.e. [Formula: see text]. The main purpose of this paper is to investigate the interplay between the graph-theoretic properties of [Formula: see text] and the module-theoretic properties of [Formula: see text]. We study the annihilators comaximal graph [Formula: see text] in terms of the powers of the decomposition of [Formula: see text] to product distinct prime numbers in some special cases.

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

Let G be a finite 2-group. In our recent papers, we proved that in a finite 2-group of almost maximal class, the set of all commuting automorphisms, $$\mathcal {A}(G)=\lbrace \alpha \in Aut(G) :x\alpha (x)=\alpha (x)x~~for~ all~ x\in G\rbrace $$ is equal to the group of all central automorphisms, $$Aut_{c}(G)$$ , except only for five ones. Also, we determined the structure of $$Aut_{c}(G)$$ and $$\mathcal {A}(G)$$ for these five groups. Using these results, in this paper, we find the structure of $$\mathcal {A}(G)=Aut_{c}(G)$$ for the remaining 2-groups of almost maximal class. Also, we prove the following results

In this paper, we define D-magic labelings for oriented graphs where D is a distance set. In particular, we label the vertices of the graph with distinct integers {1,2,…,|V(G)|} in such a way that the sum of all the vertex labels that are a distance in D away from a given vertex is the same across all vertices. We give some results related to the magic constant, construct a few infinite families of D-magic graphs, and examine trees, cycles, and multipartite graphs. This definition grew out of the definition of D-magic (undirected) graphs. This paper explores some of the symmetries we see between the undirected and directed version of D-magic labelings.

In the first part of the paper, we clarify the connections between several algebraic objects appearing in matroid theory: both partial fields and hyperfields are fuzzy rings, fuzzy rings are tracts, and these relations are compatible with the respective matroid theories. Moreover, fuzzy rings are ordered blueprints and lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection pastures. In the second part, we construct moduli spaces for matroids over pastures. We show that, for any non-empty finite set $E$, the functor taking a pasture $F$ to the set of isomorphism classes of rank-$r$ $F$-matroids on $E$ is representable by an ordered blue scheme $Mat(r,E)$, the moduli space of rank-$r$ matroids on $E$. In the third part, we draw conclusions on matroid theory. A classical rank-$r$ matroid $M$ on $E$ corresponds to a $\mathbb{K}$-valued point of $Mat(r,E)$ where $\mathbb{K}$ is the Krasner hyperfield. Such a point defines a residue pasture $k_M$, which we call the universal pasture of $M$. We show that for every pasture $F$, morphisms $k_M\to F$ are canonically in bijection with $F$-matroid structures on $M$. An analogous weak universal pasture $k_M^w$ classifies weak $F$-matroid structures on $M$. The unit group of $k_M^w$ can be canonically identified with the Tutte group of $M$. We call the sub-pasture $k_M^f$ of $k_M^w$ generated by ``cross-ratios' the foundation of $M$,. It parametrizes rescaling classes of weak $F$-matroid structures on $M$, and its unit group is coincides with the inner Tutte group of $M$. We show that a matroid $M$ is regular if and only if its foundation is the regular partial field, and a non-regular matroid $M$ is binary if and only if its foundation is the field with two elements. This yields a new proof of the fact that a matroid is regular if and only if it is both binary and orientable., 85 pages; some additional material, e.g. a new section 5.6; the terminology has been adapted to the usage in follow-up papers

We are interested in the McKay quiver $\Gamma(G)$ and skew group rings $A*G$, where $G$ is a finite subgroup of $\mathrm{GL}(V)$, where $V$ is a finite dimensional vector space over a field $K$, and $A$ is a $K-G$-algebra. These skew group rings appear in Auslander's version of the McKay correspondence. In the first part of this paper we consider complex reflection groups $G \subseteq \mathrm{GL}(V)$ and find a combinatorial method, making use of Young diagrams, to construct the McKay quivers for the groups $G(r,p,n)$. We first look at the case $G(1,1,n)$, which is isomorphic to the symmetric group $S_n$, followed by $G(r,1,n)$ for $r >1$. Then, using Clifford theory, we can determine the McKay quiver for any $G(r,p,n)$ and thus for all finite irreducible complex reflection groups up to finitely many exceptions. In the second part of the paper we consider a more conceptual approach to McKay quivers of arbitrary finite groups: we define the Lusztig algebra $\widetilde A(G)$ of a finite group $G \subseteq \mathrm{GL}(V)$, which is Morita equivalent to the skew group ring $A*G$. This description gives us an embedding of the basic algebra Morita equivalent to $A*G$ into a matrix algebra over $A$., Comment: v2: minor revision, final version to appear in Algebr. Represent. Theory

Consider an elliptic quadric $$Q^-(3,q)$$ in $$\mathrm{PG}(3,q)$$ . Let $$\mathcal {E}$$ and $$\mathcal {T}$$ denote the set of all lines of $$\mathrm{PG}(3,q)$$ which meet $$Q^-(3,q)$$ in 0 and 1 point, respectively. In this paper, we characterize the minimum size $$(\mathcal {T}\cup \mathcal {E})$$ -blocking sets and give a different proof for the characterization of minimum size $$\mathcal {E}$$ -blocking sets in $$\mathrm{PG}(3,q)$$ which works for all q. We also discuss whether the main results of this paper (Theorems 1.6 and 1.7) can be extended to ovoids in $$\mathrm{PG}(3,q)$$ .

We revisit Rozansky’s construction of Khovanov homology for links in S 2 × S 1 , extending it to define the Khovanov homology Kh ( L ) for links L in M r = # r ( S 2 × S 1 ) for any r . The graded Euler characteristic of Kh ( L ) can be used to recover WRT invariants at certain roots of unity and also recovers the evaluation of L in the skein module S ( M r ) of Hoste and Przytycki when L is null-homologous in M r . The construction also allows for a clear path toward defining a Lee’s homology Kh ' ( L ) and associated s -invariant for such L , which we will explore in an upcoming paper. We also give an equivalent construction for the Khovanov homology of the knotification of a link in S 3 and show directly that this is invariant under handle-slides, in the hope of lifting this version to give a stable homotopy type for such knotifications in a future paper.

[EN] Let G be a finite group and let K=xG be the conjugacy class of an element x of G. In this paper, it is proved that if N is a normal subgroup of G such that the coset xN is the union of K and K-1 (the conjugacy class of the inverse of x), then N and the subgroup ¿K¿ are solvable. As an application, we prove that if there exists a natural number n >= 2 such that Kn=K?K-1, then ¿K¿ is solvable., The authors are grateful to the referee for careful reading and many helpful comments and improvements on the paper. This research is partially supported by the Spanish Government, Proyecto PGC2018-096872-B-I00 and by Generalitat Valenciana, Proyecto AICO-2020-298. The first named author is also supported by Proyecto UJI-B2019-03.

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.

Let E∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to E∗ G-isomorphism, there exists at most one Hughes-free division E∗G-ring. However, the existence of a Hughes-free division E∗ G-ring DE∗G for an arbitrary locally indicable group G is still an open question. Nevertheless, DE∗G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether DE∗G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists DE[G] and it is universal. In Appendix we give a description of DE[G] when G is a RFRS group, This paper is partially supported by the Spanish Ministry of Science and Innovation through the grant MTM2017-82690-P and the “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S4). I would like to thank Dawid Kielak and an anonymous referee for useful suggestions and comments

A multiobjective optimization problem is $C^r$ simplicial if the Pareto set and the Pareto front are $C^r$ diffeomorphic to a simplex and, under the $C^r$ diffeomorphisms, each face of the simplex corresponds to the Pareto set and the Pareto front of a subproblem, where $0\leq r\leq \infty$. In the paper titled "Topology of Pareto sets of strongly convex problems," it has been shown that a strongly convex $C^r$ problem is $C^{r-1}$ simplicial under a mild assumption on the ranks of the differentials of the mapping for $2\leq r \leq \infty$. On the other hand, in this paper, we show that a strongly convex $C^1$ problem is $C^0$ simplicial under the same assumption. Moreover, we establish a specialized transversality theorem on generic linear perturbations of a strongly convex $C^r$ mapping $(r\geq 2)$. By the transversality theorem, we also give an application of singularity theory to a strongly convex $C^r$ problem for $2\leq r \leq \infty$., Comment: 17 pages, to appear in Journal of the Mathematical Society of Japan

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)&lt, k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

In this paper a second order elliptic equation with nonsmooth coefficients is considered in grand-Sobolev classes $$W_{q)}^{2} \left( \varOmega \right) $$ on a bounded n-dimensional domain $$\varOmega \subset R^{n} $$ with a sufficiently smooth boundary $$\partial \varOmega $$ , generated by the norm of the grand-Lebesgue space $$L_{q)}\left( \varOmega \right) $$ . These spaces are non-separable and therefore the definition of a reasonable solution in them faces certain difficulties. For this purpose, a subspace $$N_{q)}^{2} \left( \varOmega \right) $$ is distinguished in which infinitely differentiable and finite functions are dense. The strict inclusion $$W_{q}^{2} \left( \varOmega \right) \subset N_{q)}^{2} \left( \varOmega \right) $$ holds, where $$W_{q}^{2} \left( \varOmega \right) $$ is the classical Sobolev space. This raises specific questions dictated by the theory of spaces $$W_{q}^{2} \left( \varOmega \right) $$ , for example, the characterization of the space of traces of functions from $$N_{q)}^{1} \left( \varOmega \right) $$ cannot be characterized following the classical case. In this paper, the corresponding theorems concerning traces, extensions, and compactness of a family of functions from $$N_{q)}^{k} \left( \varOmega \right) $$ are proved. These results are applied to obtain a Schauder-type estimate up to the boundary. Schauder-type estimates make it possible to establish the fredholmness of the Dirichlet problem for the considered equation in spaces $$N_{q)}^{2} \left( \varOmega \right) $$ with data from grand-Lebesgue type spaces that are different from Lebesgue spaces. Therefore, the results of this work cannot be directly obtained from the results of the $$L_{p}$$ -theory. This work is a continuation of the research carried out by the authors in articles (Bilalov and Sadigova in Complex Var Elliptic Equ, 2020. https://doi.org/10.1080/17476933.2020.1807965 ; Bilalov and Sadigova in Sahand Commun Math Anal, 2021. https://doi.org/10.22130/scma.2021.521544.893 .

In this paper, we will establish a formula that relates the product $$\prod _{\omega ^n=1} {\omega x-1 \atopwithdelims ()p-1}$$ to generalized and homogeneous multiple harmonic sums, this would allow us to derive new identities and congruences. The congruences considered in this paper are congruences in $$\mathbb {C}_p$$ which are also valid in $$\mathbb {Z}_p$$ and $$\mathbb {Z}$$ . In order to prove these congruences, we employ well-known theorems for symmetric functions and harmonic numbers.