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In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

The concept of graph coloring has become a very active field of research that enhances many practical applications and theoretical challenges. Various methods have been applied in carrying out this study. Let G be a finite group. In this paper, we introduce a new graph of groups, which is a commuting order product prime graph of finite groups as a graph having the elements of G as its vertices and two vertices are adjacent if and only if they commute and the product of their order is a prime power. This is an extension of the study for order product prime graph of finite groups. The graph's general presentations on dihedral groups, generalized quaternion groups, quasi-dihedral groups, and cyclic groups have been obtained in this paper. Moreover, the commuting order product prime graph on these groups has been classified as connected, complete, regular, or planar. These results are used in studying various and recently introduced chromatic numbers of graphs.

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.

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.

In this paper, the compositional structure of a finite group G is investigated, which has the Sylow 2-subgroup that is permutable with some non p-nilpotent biprimary subgroups, which contain the Sylow р-subgroup of G for all odd simple divisors of the р order of the group G, and such biprimary subgroups are taken one by one for each odd р, and mark the set SB(G). In this work, the existence of the subset SB(G)* in SB(G) is proved, which consists of р-closed subgroups. The main result of this paper is as follows: if the Sylow 2-subgroup of the group G is permutable with all subgroups SB(G)*, then G may have simple non-abelian compositional factors only of L2(7) type, if p > 3, and additionally of L2(3f) type, f = 3a, a ≥ 1, if p = 3.

We present a theorem in the present paper on an approximation to functions of a Holder class on a countable union of segments lying on a positive ray by entire functions of order 1/2 bounded on this ray. Problems related to the approximation of entire functions on subsets of the semiaxis by using entire functions of order 1/2 are closely related to problems of approximating functions on subsets of the whole axis using entire functions of exponential type but have their own specifics. We consider segments In in this paper with lengths of order n such that the distance between In and In + 1 is also of order n. Cases of the whole semiaxis or the union of finitely many segments and a ray were considered in previous papers. As for the problem of approximating functions of the Holder class on the union of a countable set of segments on the whole axis, it turns out that the approximation rate at neighborhoods of the segment endpoints as the type of the functions increases is higher than that in a neighborhood of their midpoints.

In 1998, R. Downey in his review paper outlined a research agenda for the study and description of sufficient conditions for computable representability of linear orderings, namely, the problem of describing order properties P such that, for any low linear ordering Limplies that, the feasibility of the condition P(L) implies that L has a computable presentation. This paper is a part of the research program initiated by R. Downey. It is shown in the paper that any low linear ordering whose factor order (in other words, condensation) is η (the order type of naturally ordered natural numbers) has a computable presentation via a 0—computable isomorphism if this ordering does not contain a strongly η-like infinite interval. A countable linear ordering is said to be strongly η-like if there exists some natural number k such that each maximal block of the ordering is of cardinality no more than k. It is also proved that the above result does not hold for a 0—computable isomorphism instead of the 0—computable one. Namely, we construct a low linear ordering L with condensation η and without strongly η-like infinite interval so that L has no computable presentation via a 0— computable isomorphism.

In this paper, we study Carlitz&rsquo, s type higher-order (p,q)-Genocchi polynomials. To be specific, we define Carlitz&rsquo, s type higher-order (p,q)-Genocchi polynomials and Carlitz&rsquo, s type higher-order (h,p,q)-Genocchi polynomials. This paper also explores properties including distribution relation and symmetric identities. In addition, we find alternating (p,q)-power sums. We identify symmetric identities using Carlitz&rsquo, s type higher-order (h,p,q)-Genocchi polynomials and alternating (p,q)-power sums.

In the paper, we extend known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G+Dn, where the graph G consists of two leaves incident with two opposite vertices of one 4-cycle, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for a calculating the distances between all (k − 1)! vertices of the graph. Finally, by adding new edges to the graph G, we are able to obtain the crossing number of the join product with the discrete graph Dn for two other graphs. The methods used in the paper are new, and they are based on combinatorial properties of cyclic permutations.

The Farey sequence of order n consists of all reduced fractions a / b between 0 and 1 with positive denominator b less or equal to n. In a series of former papers we obtained a limit function which describes a scaling property of the Farey sequence of order n for $$n \rightarrow \infty $$ in the vicinity of any fixed fraction a / b and which is independent of a / b. In this paper we derive a representation formula for a sequence of functions used in the Franel–Landau theorem to determine the number of Farey-fractions in prescribed intervals, and we establish a corresponding limit function for this sequence as well. This is combined with a former result to derive a remarkable representation formula for arithmetic means of related functions.

In a paper in 1962, Guido Zappa asked whether a non-trivial coset of a Sylow p-subgroup of a finite group could contain only elements whose orders are powers of p, and also in that case, at least one element of order p. The first question was raised again recently in a 2014 paper by Daniel Goldstein and Robert Guralnick, when generalising an answer by John Thompson in 1967 to a similar question by L.J. Paige. In this paper we give a positive answer to both questions of Zappa, showing somewhat surprisingly that in a number of non-abelian finite simple groups (including PSL ( 3 , 4 ) , PSU ( 5 , 2 ) and the Janko group J 3 ), some non-trivial coset of a Sylow 5-subgroup (of order 5) contains only elements of order 5. Also Zappa's first question is studied in more detail. Various possibilities for the group and its Sylow p-subgroup P are eliminated, and it then follows that | P | ≥ 5 and | P | ≠ 8 . It is an open question as to whether the order of the Sylow p-subgroup can be 7 or 9 or more.

Recently a lot of papers have been devoted to partial infinite reciprocal sums of a higher-order linear recursive sequence. In this paper, we continue this program by finding a sequence which is asymptotically equivalent to partial infinite sums, including a reciprocal of polynomial applied to linear higher order recurrences.

Chen et al. (2004) strongly conjectured that R(Tn,Wm)=2n-1 if the maximum degree of Tn is small and m is even. Related to the conjecture, it is interesting to know for which tree Tn, we have R(Tn,Wm) > 2n-1 for even m. In this paper, we find the Ramsey number R(Tn,W8) for tree Tn with the maximum degree of Tn is at least n-3, namely R(Tn,W8) > 2n-1 for almost all such tree Tn. We also prove that if the maximum degree of Tn is large, then R(Tn,Wm) is a function of both m and n. In the end, we refine the conjecture of Chen et al. by giving the condition of small maximum degree. For a tree Tn with large maximum degree and even m, the R(Tn,Wm) is also unknown in general. In this paper, we shall determine the Ramsey number R(Tn,W8) for all trees Tn of order n with the maximum degree of Tn is at least n-3., 7 pages

Given a digital image (or digital object) (X,k),X⊂Zn, this paper initially establishes a group structure of the set of self-k-isomorphisms of (X,k) with the function composition, denoted by Isok(X) or Autk(X). In particular, let Ckn,l be a simple closed k-curve with l elements in Zn. Then, the group Isok(Ckn,l) is proved to be isomorphic to the standard dihedral group Dl with order l. The calculation of this quantity Isok(Ckn,l) is a key step for obtaining many new results. Indeed, it is essential for exploring many features of Isok(X). Furthermore, this quantity is proved to be a digital topological invariant. After proceeding with an Isok(X)-action on (X,k), we investigate some properties of fixed point sets by this action. Finally, we explore various features of fixed point sets by this action from the viewpoint of digital k-curve theory. This paper only deals with k-connected digital images (X,k) whose cardinality is equal to or greater than 2. Besides, we discuss some errors that have appeared in the lilterature.

In this paper, we introduce a new operator in order to derive some new symmetric properties of [Formula: see text]-Fibonacci and [Formula: see text]-Lucas numbers and Fibonacci polynomials. By making use of the new operator defined in this paper, we give some new generating functions for [Formula: see text]-Fibonacci and Pell numbers and Fibonacci polynomials.

The purpose of the paper is to study new algebraic operations and fundamental properties of totally dependent-neutrosophic sets and totally dependent-neutrosophic soft sets. First, the in-coordination relationships among the original inclusion relations of totally dependent-neutrosophic sets (called type-1 and typ-2 inclusion relations in this paper) and union (intersection) operations are analyzed, and then type-3 inclusion relation of totally dependent-neutrosophic sets and corresponding type-3 union, type-3 intersection, and complement operations are introduced. Second, the following theorem is proved: all totally dependent-neutrosophic sets (based on a certain universe) determined a generalized De Morgan algebra with respect to type-3 union, type-3 intersection, and complement operations. Third, the relationships among the type-3 order relation, score function, and accuracy function of totally dependent-neutrosophic sets are discussed. Finally, some new operations and properties of totally dependent-neutrosophic soft sets are investigated, and another generalized De Morgan algebra induced by totally dependent-neutrosophic soft sets is obtained.

In this paper we give a combinatorial characterization of tight fusion frame (TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our characterization does not have this limitation. We also develop some methods for generating TFF sequences. The basic technique is a majorization principle for TFF sequences combined with spatial and Naimark dualities. We use these methods and our characterization to give necessary and sufficient conditions which are satisfied by the first three highest ranks. We also give a combinatorial interpretation of spatial and Naimark dualities in terms of Littlewood-Richardson coefficients. We exhibit four classes of TFF sequences which have unique maximal elements with respect to majorization partial order. Finally, we give several examples illustrating our techniques including an example of tight fusion frame which can not be constructed by the existing spectral tetris techniques. We end the paper by giving a complete list of maximal TFF sequences in dimensions less than ten., 31 pages

Let f be a transcendental meromorphic function of nite order in the plane such that f (m) has nitely many zeros for some positive integer m ≥ 2: Suppose that f (k) and f share a CM, where k ≥ 1 is a positive integer, a 0 is a nite complex value. Then f is an entire function such that f (k) − a = c(f − a); where c 0 is a nonzero constant. The results in this paper are concerning a conjecture of Bruck (5). An example is provided to show that the results in this paper, in a sense, are the best possible.

In the paper [4, Theorem 1] Gat, Goginava and the author proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer means of Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the space Lp for p > 2/3 . Moreover, Goginava and the author showed that σκ,∗ is not bounded from the Hardy space H2/3 to the space L2/3 [6, Theorem 1]. The main aim of this paper is to show that the maximal operator σκ,∗ f := supn∈P |σκ n f | log3/2(n+1) , is bounded from the Hardy space H2/3 into the space L2/3. Moreover, we prove that the order of deviant behavior of the n th Walsh-Kacmarz-Marcinkiewicz-Fejer mean is exactly log3/2(n+1) in the endpoint p = 2/3 . Mathematics subject classification (2010): 42C10.

In this paper, we establish some new Ostrowski type inequalities for s logarithmically convex functions. Some applications of our results to P.D.F.’s and in numerical integration are given. 1. INTRODUCTION Let f : I [0;1] ! R be a di¤erentiable mapping on I , the interior of the interval I, such that f 0 2 L [a; b] where a; b 2 I with a < b. If jf 0 (x)j M , then the following inequality holds (see [2]): (1.1) f(x) 1 b a Z b a f(u)du M b a " (x a) + (b x) 2 # : This inequality is well known in the literature as the Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([2]-[6]) and the references therein. Let us recall some known de…nitions and results which we will use in this paper. The function f : I R ! [0;1) is said to be log convex or multiplicatively convex if log t is convex, or, equivalenly, if for all x; y 2 I and t 2 [0; 1] ; one has the inequality (See [7], p.7): f (tx+ (1 t) y) [f (x)] + [f (y)] t) : In [1], Akdemir and Tunc were introduced the class of s logarithmically convex functions in the …rst sense as the following: De…nition 1. A function f : I R0 ! R+ is said to be s logarithmically convex in the …rst sense if (1.2) f ( x+ y) [f (x)] s [f (y)] s for some s 2 (0; 1], where x; y 2 I and s + s = 1: In [8], authors introduced the class of s logarithmically convex functions in the second sense as the following: De…nition 2. A function f : I R0 ! R+ is said to be s logarithmically convex in the second sense if (1.3) f (tx+ (1 t) y) [f (x)] s [f (y)] (1 t) 2000 Mathematics Subject Classi…cation. 26D10, 26A15, 26A16, 26A51. Key words and phrases. Hadamard’s inequality, s geometrically convex functions . 1 2 MEVLUT TUNC AND AHMET OCAK AKDEMIR for some s 2 (0; 1], where x; y 2 I and t 2 [0; 1]. Clearly, when taking s = 1 in De…nition 1.2 or De…nition 1.3, then f becomes the standard logarithmically convex function on I. The main purpose of this paper is to establish some new Ostrowski’s type inequalities for s logarithmically convex functions. We also give some applications to P.D.F.’s and to midpoint formula. 2. THE NEW RESULTS In order to prove our main results, we will use following Lemma which was used by Alomari and Darus (see [3]): Lemma 1. Let f : I R ! R, be a di¤erentiable mapping on I where a; b 2 I, with a < b. Let f 0 2 L[a; b]; then the following equality holds

In this paper, we determine the bound of the valency of Cayley graphs of Frobenius groups with respect to normal Cayley subsets which guarantees to be Ramanujan. We see that if the ratio between the orders of the Frobenius kernel and complement is not so small, then this bound coincides with the trivial one coming from the trivial estimate of the largest non-trivial eigenvalue of the graphs. Moreover, in the cases of the dihedral groups of order twice odd primes, which are special cases of the Frobenius groups, we determine the same bound for the Cayley graphs of the groups with respect to not only normal but also all Cayley subsets. As is the case of abelian groups which we have treated in the previous papers, such a bound is equal to the trivial one in the above sense or, as exceptional cases, exceeds one from it. We then clarify that the latter occurs if and only if the corresponding prime is represented by a quadratic polynomial in a finite family., Comment: 12 pages

By the classification theorem by F. Oort and J. Tate [6], any group scheme of prime order is isomorphic to a group scheme Ga,b under the suitable choice of a and b. We computed the torsors for some kinds of group schemes Ga,b in [8], which is a joint work with T. Sekiguchi, as in the following way: denote by p a prime number and by m = φ(p−1) the value of the Euler function φ. Suppose p is a prime ideal lying over p (which splits completely in Z[ζ]), where ζ is a primitive (p − 1)-st root of the unity. In case p is principal, the sequence 0 → μp,B → G m m,B p −→ Gm,B → 0 is exact, and the Galois descent of μp,B is isomorphic to Ga,b under the suitable choice of a and b, thus one can compute the torsors for this kinds of group schemes. The non-principal case is solved by Y. Koide [3] by using our method. The aim of this paper is to study some group schemes of order a power of a prime number. In section from 1 to 3, we would like to review the main result of the papers [6] by F. Oort and J. Tate, [4] by Y. Koide and T. Sekiguchi, and [8] by T. Sekiguchi and Y. Toda. In section 4, we give our main result, namely, the torsor for the Galois descent of μpn,B.

Let G = ((A,B),E) be an instance of the stable marriage problem where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if M does not lose a head-to-head election against any matching. Popular matchings are a well-studied generalization of stable matchings, introduced with the goal of enlarging the set of admissible solutions, while maintaining a certain level of fairness. Every stable matching is a min-size popular matching. Unfortunately, when there are edge costs, it is NP-hard to find a popular matching of minimum cost -- even worse, the min-cost popular matching problem is hard to approximate up to any factor. Let opt be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most opt by paying the price of mildly relaxing popularity. Our main positive results are two bi-criteria algorithms that find in polynomial time a near-popular or quasi-popular matching of cost at most opt. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching, which could be much smaller than opt. Key to the other algorithm is a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost, and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity. This version of the paper goes beyond the conference version [12] in the following two points: (i) the algorithm for finding a quasi-popular matching of cost at most that of a min-cost popular fractional matching is new; (ii) the proofs from Section 6.1 and Section 7.3 are now self-contained (the conference version used constructions from [10] to show these lower bounds).

In the present paper we study $$G_2$$ -periodic $$p$$ -adic quasi Gibbs measures for $$p$$ -adic Potts model on a Cayley tree of order two. In the case $$q=3$$ , we prove the occurrence of a phase transition and construct ART quasi Gibbs measures for $$p$$ -adic Potts model on a Cayley tree of order $$k\geq3$$ .

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W ⊆ V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .

Let Z2 denote a cyclic group of 2 order and Z 2 2 = Z2 × Z2 the direct product of groups. Suppose that (M, Φ) is a closed and smooth manifold M with a smooth Z 2 2 -action whose fixed point set is the disjoint union of two real projective spaces with the same dimension. In this paper, the authors give a sufficient condition on the fixed data of the action for (M, Φ) bounding equivariantly.

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

In this article we consider representing properties of subspaces generated by the Szego kernel. We examine under which conditions on the sequence of points of the unit disk the order-preserving weak greedy algorithm for appropriate subspaces generated by the Szego kernel converges. Previously, we constructed a representing system based on discretized Szego kernels. The aim of this paper is to find an effective algorithm to get such representation, and we draw on the work of Silnichenko that introduced the notion of the order-preserving weak greedy algorithm. By selecting a special sequence of discretization points we refine one of Totik's results on the approximation of functions in the Hardy space using Szego kernels. As the main result we prove the convergence criteria of the order-preserving weak greedy algorithm for subspaces generated by the Szego kernel in the Hardy space.

Denote by $G$ a finite group and by $\psi(G)$ the sum of element orders in $G$. If $t$ is a positive integer, denote by $C_t$ the cyclic group of order $t$ and write $\psi(t)=\psi(C_t)$. In this paper we proved the following Theorem A: Let $G$ be a non-cyclic group of odd order $n=qm$, where $q$ is the smallest prime divisor of $n$ and $(m,q)=1$. Then the following statements hold. (1) If $q=3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {85}{301}$, and equality holds if and only if $n=3\cdot 7\cdot m_1$ with $(m_1,42)=1$ and $G=(C_7\rtimes C_3)\times C_{m_1}$, with $C_7\rtimes C_3$ non-abelian. (2) If $q>3$, then $\frac {\psi(G)}{\psi(|G|)}\leq \frac {p^4+p^3-p^2+1}{p^5+1}$, where $p$ is the smallest prime bigger than $q$ and equality holds if and only if $n=qp^2m_1$ with $(m_1,p!)=1$ and $G=C_q\times C_p\times C_p \times C_{m_1}$.

A graph H of order n is said to be $$k-placeable$$ into a graph G, having the same order n, if G contains k edge-disjoint copies of H. Kaneko et al. [9] proved that any non-star tree T is $$2-placeable$$ into its third power $$T^3$$ . In this paper, we give a particular interest on the $$3-placement$$ of a tree T into its sixth power $$T^6$$ .

In this paper, we try to give a classification of spacelike hypersurfaces of the Lorentz-Minkowski space-time E1n+1, whose mean curvature vector field of order (k+ 1) is an eigenvector of the kth linearized operator Lk, for a non-negative integer k less than n. The operator Lk is defined as the linear part of the first variation of the (k + 1)th mean curvature of a hypersurface arising from its normal variations. We show that any spacelike hypersurface of E1n+1 satisfying the condition LkHk+1 = λHk+1 (where 0 ≤ k ≤ n − 1) belongs to the class of Lk-biharmonic, Lk-1-type or Lk-null-2-type hypersurface. Furthermore, we study the above condition on a well-known family of spacelike hypersurfaces of Lorentz-Minkowski spaces, named the weakly convex hypersurfaces (i.e. on which all of principle curvatures are nonnegative). We prove that, on any weakly convex spacelike hypersurface satisfying the above condition for an integer k (where, 0 ≤ r ≤ n−1), the (k + 1)th mean curvature will be constant. As an interesting result, any weakly convex spacelike hypersurfaces, having assumed to be Lk-biharmonic, has to be k-maximal.

The purpose of this paper is the study of simple graphs that are generalized Cayley graphs over LA-polygroups GCLAP − graphs . In this regard, we construct two new extensions for building LA-polygroups. Then, we define Cayley graph over LA-group and GCLAP-graph. Further, we investigate a few properties of them to show that each simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAP-graph) is a GCLAP-graph and then we prove this result.

Let $ F $ be a finite field of characteristic $ p $ having $ q = p^n $ elements and $ G $ be an abelian group. In this paper, we determine the structure of the group of units of the group algebra $ FG $, where $ G $ is an abelian group of order $ 17\leq |G|\leq 20 $.

We present a sufficient condition for the $kG$-Scott module with vertex $P$ to remain indecomposable under the Brauer construction for any subgroup $Q$ of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$, and $P$ is a semidihedral $2$-subgroup of a finite group $G$. This generalizes results for the cases where $P$ is abelian or dihedral. The Brauer indecomposability is defined \linebreak by R.~Kessar, N.~Kunugi and N.~Mitsuhashi. The motivation of \linebreak this paper is a fact that the Brauer indecomposability of a $p$-permutation bimodule ($p$ is a prime) is one of the key steps in order to obtain a splendid stable equivalence of Morita type by making use of the gluing method due to Brou\'e, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence., Comment: 10 pages

Let G be a group. If for every proper normal subgroup N and element x of G with N〈x〉≠G, N〈x〉 is an FC-group, but G is not an FC-group, then we call G an NFC-group. In the present paper we consider the NFC-groups. We prove that every non-perfect NFC-group with non-trivial finite images is a minimal non-FC-group. Also we show that if G is a non-perfect NFC-group having no nontrivial proper subgroup of finite index, then G is a minimal non-FC-group under the condition “every Sylow p-subgroup is an FC-group for all primes p”. In the perfect case, we show that there exist locally nilpotent perfect NFC-p-groups which are not minimal non-FC-groups and also that McLain groups $M(\mathbb {Q},GF(p))$ for any prime p contain such groups. We give a characterization for torsion-free case. We also consider the p-groups such that the normalizer of every element of order p is an FC-subgroup.

Let G be a simple graph of order n with vertex set V= {v1, v2, ..., vn} and edge set E = {e1, e2, ..., em}. A subset of E is called an edge dominating set of G if every edge of E - is adjacent to some edge in .Any edge dominating set with minimum cardinality is called a minimum edge dominating set [2]. Let be a minimum edge dominating set of a graph G. The minimum edge dominating matrix of G is the m x m matrix defined by G)= , where = The characteristic polynomial of is denoted by fn (G, ρ) = det (ρI - (G) ). The minimum edge dominating eigen values of a graph G are the eigen values of (G). Minimum edge dominating energy of G is defined as (G) = [12] In this paper we have computed the Minimum Edge Dominating Energy of a graph. Its properties and bounds are discussed. All graphs considered here are simple, finite and undirected.

Let G be a graph with edge set E G and e = u v ∈ E G . Define n u e , G and m u e , G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v , respectively. The numbers n v e , G and m v e , G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as M o G = ∑ u v ∈ E G n u u v , G − n v u v , G and M o e G = ∑ u v ∈ E G m u u v , G − m v u v , G , respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.

A well-known conjecture of Frankl and F\"{u}redi from 1989 states that an initial segment of colex of has the largest Lagrangian of any $r$-uniform hypergraph with $m$ hyperedges. We show that this is true when $r=3$. We also give a new proof of a related conjecture of Nikiforov and a counterexample to an old conjecture of Ahlswede and Katona., Comment: We split our original paper (arXiv:1807.00793v2) into two parts. The first part can be found in arXiv:1807.00793. This is the second part, which consists of 18 pages, including a two-page appendix

The Italian reinforcement number of a digraph is the minimum number of arcs that have to be added to the digraph in order to decrease the Italian domination number. In this paper, we present some new sharp upper bounds on the Italian reinforcement number of a digraph. We also determine the exact values of the Italian reinforcement number of the Cartesian products of directed paths and directed cycles: $ P_2\square P_n $, $ P_3\square P_n $, $ P_3\square C_n $, $ C_3\square P_n $ and $ C_3\square C_n $.

The vulnerability in a communication network is the measurement of the strength of the network against damage that occurs in nodes or communication links. It is important that a communication network is still effective even when it loses some of its nodes or links. In other words, since a network can be modelled by a graph, it is desired to know whether the graph is still connected when some of the vertices or edges are removed from a connected graph. The vulnerability parameters aim to find the nature of the network when a subset of the nodes or links is removed. One of these parameters is domination. Domination is a measure of the connection of a subset of vertices with its complement. In this paper, we study porous exponential domination as a vulnerability parameter and obtain certain results on the Cartesian product and lexicographic product graphs. We determine the porous exponential domination number, denoted by $$\gamma_e^*$$ , of the Cartesian product of $$P_2$$ with $$P_n$$ and $$C_n$$ , separately. We also determine the porous exponential domination number of the Cartesian product of $$ P_n $$ with complete bipartite graphs and any graph $$ G $$ which has a vertex of degree $$ |V(G)|-1 $$ . Moreover, we obtain the porous exponential domination number of the lexicographic product of $$ P_n $$ and $$ G_m $$ , denoted by $$ P_n[G_m] $$ , for the case where $$ G_m $$ is a graph of order $$ m $$ with a vertex of degree $$ m-1 $$ and for the opposite case where $$ G_m $$ is a graph of order $$ m $$ which has no vertex of degree $$ m-1 $$ . We further show that $$\gamma_e^*(P_n[G_m])=\gamma_e^*(G_m[P_n])=\gamma_e^*(P_n)$$ by proving $$\gamma_e^*(G_m[G_n])=\gamma_e^*(G_n)$$ , where $$ G_m $$ is a graph of order $$ m $$ with a vertex of degree $$ m-1 $$ and $$ G_n $$ is any graph of order $$ n $$ .

In this paper we characterize groups according to the number of end vertices in the associated coprime graphs. An upper bound on the order of the group that depends on the number of end vertices is obtained. We also prove that $$2-$$ groups are the only groups whose coprime graphs have odd number of end vertices. Classifications of groups with small number of end vertices in the coprime graphs are given. We give a complete answer to [4, Question 3.7], where we show that $$\mathbb {Z}_4$$ and $$\mathbb {Z}_2\times \mathbb {Z}_2$$ are the only groups whose coprime graph has exactly three end vertices.

In this paper the classification is given for finite groups in which the normalizer of every non-normal cyclic subgroup of order divided by the minimal prime of |G| is a maximal subgroup.

Three edges $ e_1 $, $ e_2 $ and $ e_3 $ in a graph $ G $ are $ consecutive $ if they form a cycle of length $ 3 $ or a path in this order. A $ k $-$ injective\; edge\; coloring $ of a graph $ G $ is an edge coloring of $ G $, (not necessarily proper), such that if edges $ e_1 $, $ e_2 $, $ e_3 $ are consecutive, then $ e_1 $ and $ e_3 $ receive distinct colors. The minimum $ k $ for which $ G $ has a $ k $-injective edge coloring is called the $ injective\; edge\; coloring\; number $, denoted by $ \chi_i^{\prime}(G) $. In this paper, we consider the injective edge coloring numbers of generalized Petersen graphs $ P(n, 1) $ and $ P(n, 2) $. We determine the exact values of injective edge coloring numbers for $ P(n, 1) $ with $ n\geq 3 $, and for $ P(n, 2) $ with $ 4\leq n\leq 7 $. For $ n\geq 8 $, we show that $ 4\leq \chi_{i}^{'}(P(n, 2))\leq 5. $

The paper continues the authors ’ research on the evaluation of trigonometric sums of an algebraic grid with weights. The case of an arbitrary weight function of infinite order is considered. For the parameter 𝑚 of the trigonometric sum 𝑆𝑀(𝑡),𝜌∞(𝑚), three cases are highlighted. If 𝑚 belongs to the algebraic lattice Λ(𝑡·𝑇(𝑎)), then for any natural 𝑟 the asymptotic formula is valid 𝑆𝑀(𝑡),𝜌∞(𝑡(𝑚, . . . ,𝑚)) = 1 + 𝑂 ( ln𝑠−1 det Λ(𝑡) (detΛ(𝑡))𝑟+1). If 𝑚 does not belong to the algebraic lattice Λ(𝑡·𝑇(𝑎)), then two vectors are defined 𝑛Λ(𝑚) =(𝑛1, . . . , 𝑛𝑠) and 𝑘Λ(𝑚) from the conditions 𝑘Λ(𝑚) ∈ Λ, 𝑚 = 𝑛Λ( 𝑀)+ 𝐾𝜆(𝑚) and the product𝑞(𝑛𝜆(𝑚)) = 𝑛1 · . . . · 𝑛𝑠 is minimal. Asymptotic estimation is proved|𝑆𝑀(𝑡),𝜌∞(𝑚)| 6 𝐵(𝑟,∞)(1 − 𝛿(𝑘Λ(𝑚))(𝑞(𝑛Λ(𝑚)))𝑟+1 + 𝑂(𝑞(𝑛Λ(𝑚))𝑟+1 ln𝑠−1 det Λ(𝑡)(det Λ(𝑡))𝑟+1)).

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. Let p be a prime. In this paper, we proved that there is only one connected heptavalent symmetric graphs of order 40p, it is a vertex primitive graph of order 40.3 = 120 admitting S7 as its full automorphism group.

A finite group $G$ is called $\psi$-divisible if $\psi(H)|\psi(G)$ for any subgroup $H$ of $G$, where $\psi(H)$ and $\psi(G)$ are the sum of element orders of $H$ and $G$, respectively. In this paper, we extend a result provided in [10], by classifying the finite groups whose all subgroups are $\psi$-divisible. Since the existence of $\psi$-divisible groups is related to the class of square-free order groups, we also study the sum of element orders and the $\psi$-divisibility property of ZM-groups. In the end, we introduce the concept of $\psi$-normal divisible group, i.e. a group for which the $\psi$-divisibility property is satisfied by all its normal subgroups. Using simple and quasisimple groups, we are able to construct infinitely many $\psi$-normal divisible groups which are neither simple nor nilpotent., Comment: 10 pages

The paper is devoted to such sets of $$E'$$ , where E is a Banach lattice, on which every order bounded disjoint sequence $$(x_{n}) \subset E$$ converges uniformly to zero. We characterize this class of sets [almost order (L) sets] which is largest than the class of almost (L) sets and we study some of its properties. Consequently, we obtain some new important characterizations of order weakly compact operators. Moreover, we derive the following main result: a subset $$A \subset E'$$ is almost (L) set if, and only if, every positive weakly null sequence converges uniformly to zero on the subset A.

Recently, Brietzke, Silva and Sellers [‘Congruences related to an eighth order mock theta function of Gordon and McIntosh’, J. Math. Anal. Appl.479 (2019), 62–89] studied the number $v_{0}(n)$ of overpartitions of $n$ into odd parts without gaps between the nonoverlined parts, whose generating function is related to the mock theta function $V_{0}(q)$ of order 8. In this paper we first present a short proof of the 3-dissection for the generating function of $v_{0}(2n)$. Then we establish three congruences for $v_{0}(n)$ along certain progressions which are subsequences of the integers $4n+3$.

The purpose of this paper is to determine L(G), the absolute center of the group G, when G is a p-group of maximal class. Particularly we find L(G) for metabelian p-groups of maximal class, all p-groups of maximal class of order less than p6 and p-groups of maximal class for p = 2,3.

In this paper, we solve Diophantine equation in the tittle in nonnegative integers m,n, and a. In order to prove our result, we use lower bounds for linear forms in logarithms and and a version of the Baker-Davenport reduction method in diophantine approximation.