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In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

In his seminal papers, Koblitz proposed curves for cryptographic use. For fast operations on these curves, these papers also initiated a study of the radix-τ expansion of integers in the number fields and . The (window) nonadjacent form of τ -expansion of integers in was first investigated by Solinas. For integers in , the nonadjacent form and the window nonadjacent form of the τ -expansion were studied. These are used for efficient point multiplications on Koblitz curves. In this paper, we complete the picture by producing the (window) nonadjacent radix-τ expansions for integers in all Euclidean imaginary quadratic number fields.

Let (X, O) be a germ of a normal surface singularity, π: → X be the minimal resolution of singularities and let A = (ai,j) be the n × n symmetrical intersection matrix of the exceptional set of In an old preprint Nash proves that the set of arcs on a surface singularity is a scheme , and defines a map from the set of irreducible components of to the set of exceptional components of the minimal resolution of singularities of (X,O). He proved that this map is injective and ask if it is surjective. In this paper we consider the canonical decomposition •For any couple (Ei,Ej) of distinct exceptional components, we define Numerical Nash condition (NN(i,j)). We have that (NN(i,j)) implies In this paper we prove that (NN(i,j)) is always true for at least the half of couples (i,j).•The condition (NN(i,j)) is true for all couples (i,j) with i ≠ j, characterizes a certain class of negative definite matrices, that we call Nash matrices. If A is a Nash matrix then the Nash map N is bijective. In particular our results depend only on A and not on the topological type of the exceptional set.•We recover and improve considerably almost all results known on this topic and our proofs are new and elementary.•We give infinitely many other classes of singularities where Nash Conjecture is true.The proofs are based on my old work [8] and in Plenat [10].

Suppose [Formula: see text] is a finite field extension of [Formula: see text] containing a primitive [Formula: see text]th root of unity. Let [Formula: see text] be a maximal [Formula: see text]-extension of [Formula: see text] with the Galois group of period [Formula: see text] and nilpotent class [Formula: see text]. In this paper, we develop formalism which allows us to study the structure of [Formula: see text] via methods of Lie theory. In particular, we introduce an explicit construction of a Lie [Formula: see text]-algebra [Formula: see text] and an identification [Formula: see text], where [Formula: see text] is a [Formula: see text]-group obtained from the elements of [Formula: see text] via the Campbell–Hausdorff composition law. In the next paper, we apply this formalism to describe the ramification filtration [Formula: see text] and an explicit form of the Demushkin relation for [Formula: see text].

The question of a non-[Formula: see text]-type [Formula: see text] sequence wherein the fourth term shares the property [Formula: see text] with the first term has not been investigated so far. The present paper seeks to fill up the gap in this unexplored area. Let [Formula: see text] denote the set of all natural numbers and [Formula: see text] the sequence of Fibonacci numbers. Choose two integers [Formula: see text] and [Formula: see text] with [Formula: see text] such that their product increased by [Formula: see text] is a square [Formula: see text]. Certain properties of the sequence [Formula: see text] defined by the relation [Formula: see text] are established in this paper and polynomial expressions for Diophantine quadruples from the [Formula: see text] sequence [Formula: see text] are derived. The concept of a near-Diophantine quintuple is introduced and it is proved that there exist an infinite number of near-Diophantine quintuples.

WOS: 000389311800021, In [Finite presentability of Bruck-Reilly extensions of groups, J. Algebra 242 (2001) 2030], Araujo and RuAtic studied finite generation and finite presentability of Bruck Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck Reilly s -extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck Reilly s-extension of a group, G R. (G; beta, gamma; u), to be finitely generated and finitely presented. Let G be a group, (3,-y: be morphisms and u is an element of H-1 (H-1(*) and H-1 are the ?-G- and ?-t-classes, respectively, contains the identity element kr of T). We prove that GE R* (G; 3, 7; u) is finitely generated if and only if there exists a finite subset Xo C G such that G is generated by ((boolean OR(i >= 0) X-0,beta(i))U(Uj >= 0X0 gamma(j)). We also prove that C superset of R.* (C; beta, gamma; u) is finitely presented if and only if G is presented by (X; R), where X is a finite set and R = (U i >= 0 R-0 beta(i)) boolean OR (U-j >= 0 R0 gamma(j)) = {w(1)beta(i) = upsilon(1)beta(i) : i >= 0, w(1) = v(1) is an element of H-1* for some finite set of relations R0 subset of X* x X*.

Doty and Giaquinto constructed a certain set denoted by [Formula: see text] in this paper, and conjectured that the set [Formula: see text] forms a [Formula: see text]-basis for the integral [Formula: see text]-Schur algebra [Formula: see text], where [Formula: see text]. In this paper, we show that the set [Formula: see text] is a [Formula: see text]-basis of [Formula: see text]-Schur algebra [Formula: see text] over [Formula: see text]. The set [Formula: see text] is a truncated form of PBW basis for quantum [Formula: see text]. Moreover, we give an example to show that the set [Formula: see text] is not [Formula: see text]-basis of [Formula: see text] in general.

In this paper, we establish some nontrivial and effective upper bounds for the least common multiple of consecutive terms of a finite arithmetic progression. Precisely, we prove that for any two coprime positive integers $a$ and $b$, with $b\geq 2$, we have \[\mathrm{lcm}\left(a,a+b,\dots,a+nb\right) \leq \left(c_1\cdot b\log b\right)^{n+\left\lfloor \frac{a}{b}\right\rfloor}~~~~(\forall n\geq b+1),\] where $c_1=41.30142$. If in addition $b$ is a prime number and $a, Comment: 8 pages

LetKbe an algebraically closed field and$\mathbb{A}$n≅Knaffinen-space. It is known that a finite group$\frak{G}$can only act freely on$\mathbb{A}$nifKhas characteristicp> 0 and$\frak{G}$is ap-group. In that case the group action is “non-linear” and the ring of regular functionsK[$\mathbb{A}$n] must be atrace-surjectiveK−$\frak{G}$-algebra.Now letkbe an arbitrary field of characteristicp> 0 and letGbe a finitep-group. In this paper we study the category$\mathfrak{Ts}$of all finitely generated trace-surjectivek−Galgebras. It has been shown in [13] that the objects in$\mathfrak{Ts}$are precisely those finitely generatedk−GalgebrasAsuch thatAG≤Ais a Galois-extension in the sense of [7], with faithful action ofGonA. Although$\mathfrak{Ts}$is not an abelian category it has “s-projective objects”, which are analogues of projective modules, and it has (s-projective) categorical generators, which we will describe explicitly. We will show thats-projective objects and their rings of invariants are retracts of polynomial rings and therefore regular UFDs. The category$\mathfrak{Ts}$also has “weakly initial objects”, which are closely related to the essential dimension ofGoverk. Our results yield a geometric structure theorem for free actions of finitep-groups on affinek-varieties. There are also close connections to open questions on retracts of polynomial rings, to embedding problems in standard modular Galois-theory ofp-groups and, potentially, to a new constructive approach to homogeneous invariant theory.

In this paper, we present various forms of the inverse of a special Schur complement C ⁢ D - 1 ⁢ B {CD^{-1}B} in the case where B, C, D are operators on Hilbert spaces. We show that when C ⁢ D - 1 ⁢ B {CD^{-1}B} is invertible, there always exist operators X and Y which belong to some special classes of generalized inverses of B and C, for which ( C ⁢ D - 1 ⁢ B ) - 1 = X ⁢ D ⁢ Y {(CD^{-1}B)^{-1}=XDY} . Some explicit expressions for such X and Y are given.

In this paper, we generalize monotone operators, their resolvents and the proximal point algorithm to complete CAT(0) spaces. We study some properties of monotone operators and their resolvents. We show that the sequence generated by the inexact proximal point algorithm $\unicode[STIX]{x1D6E5}$-converges to a zero of the monotone operator in complete CAT(0) spaces. A strong convergence (convergence in metric) result is also presented. Finally, we consider two important special cases of monotone operators and we prove that they satisfy the range condition (see Section 4 for the definition), which guarantees the existence of the sequence generated by the proximal point algorithm.

In this paper, we establish the following sharp Adams inequality with exact growth condition in the entire space ℝn (n ≥ 3): There exists a constant C(n) > 0 such that for all with , , where . This extends the main result in [27] when n = 4 to all dimensions n ≥ 3. A crucial technical lemma we need is Lemma 4.2 for all p > 1 (corresponding to the Adams inequality for all n ≥ 3) whose proof is quite involved. As an application, we obtain the best constant for Ozawa’s inequality of Adams type in the Sobolev space : For any α < βn, there exists a constant C(α, n) > 0 such that for all satisfying , we have . Moreover, if α ≥ βn then the inequality cannot hold with a uniform constant C(α, n).

Given a positive integer [Formula: see text], in this paper, we investigate some more properties of the [Formula: see text]-capability of groups. For instance, the relationship between [Formula: see text]-capability and the varietal capability is determined. Moreover, we introduce the notion of [Formula: see text]-epicenter for a group and then we obtain some criteria for the [Formula: see text]-capability of groups. Finally, as an application, we characterize all [Formula: see text]-capable extra-special [Formula: see text]-groups when [Formula: see text] is a power of [Formula: see text]

M(x) = max[kj Mk _ x]. M(x) + 1 is then the first passage time out of the interval ( oo, x] for the random walk process S,. In this paper we shall concern ourselves with just those cases in which M(x) is a proper random variable with EM(x) < co and, when suitably normed, possesses a limit distribution as x -) co. It will be shown that M(x) can possess such a limit if and only if the random variables Xi belong to the domain of attraction of one of a certain group of stable laws and the limit law will be obtained under these circumstances. This result (Theorem 2) constitutes a generalization from the case of non-negative summands of various limit results of classical renewal theory (see for example Feller [6], 359-360). The significance of this type of generalization has previously been explored in Heyde [8]; Theorem 2 of this paper is in fact an extension of Theorem 4 of [8]. In order to obtain the limit distribution mentioned above we need a version of

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that [Formula: see text] is the finite field of order [Formula: see text] where [Formula: see text] is a power of the prime [Formula: see text] and [Formula: see text] are distinct odd primes, and [Formula: see text] are positive integers. We determine generator polynomials of all constacyclic codes of length [Formula: see text] over the finite field [Formula: see text] We also determine their dual codes.

Let [Formula: see text] be a finite simple graph. The [Formula: see text]-distance graph [Formula: see text] of [Formula: see text] is the graph with the same vertex set as [Formula: see text] and two vertices are adjacent if and only if their distance in [Formula: see text] is exactly [Formula: see text]. A graph [Formula: see text] is a [Formula: see text]-distance graph if there exists a graph [Formula: see text] such that [Formula: see text]. In this paper, we give three characterizations of [Formula: see text]-distance graphs, and find all graphs [Formula: see text] such that [Formula: see text] or [Formula: see text], where [Formula: see text] is an integer, [Formula: see text] is the path of order [Formula: see text], and [Formula: see text] is the complete graph of order [Formula: see text].

Let [Formula: see text] be the modular group algebra of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text]. The classification of the group algebras [Formula: see text] with upper Lie nilpotency index [Formula: see text] greater than or equal to [Formula: see text] has already been done. In this paper, we determine the group algebras [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text].

Let [Formula: see text] be an associative ring with non-zero identity and [Formula: see text] a non-zero unital left [Formula: see text]-module. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of [Formula: see text] and algebraic-theoretic properties of [Formula: see text] and [Formula: see text]. Also, we study girth, independence number, clique number and planarity of this graph.

A ring [Formula: see text] is called [Formula: see text]-coherent if every [Formula: see text]-presented ideal of [Formula: see text] is [Formula: see text]-presented. In this paper, the [Formula: see text]-coherence of the perfect closure [Formula: see text] is investigated. We show that if [Formula: see text] is [Formula: see text]-coherent, then it is normal. Also, it is shown that the perfect closure of a Noetherian reduced ring is [Formula: see text]-coherent if and only if the perfect closure of its Henselization is [Formula: see text]-coherent. Among other results, we study the [Formula: see text]-coherence of [Formula: see text] when [Formula: see text] is a retract algebra.

Let [Formula: see text] be a Lie algebra and [Formula: see text] and [Formula: see text] be the set of all derivations and inner derivations of [Formula: see text], respectively. A derivation [Formula: see text] of a Lie algebra [Formula: see text] is pointwise inner if [Formula: see text] for all [Formula: see text]. The set of all pointwise inner derivations of Lie algebra [Formula: see text] denoted by [Formula: see text] form a subalgebra of [Formula: see text] containing [Formula: see text]. In this paper, we prove that, if [Formula: see text] is nilpotent of class [Formula: see text] (solvable of length [Formula: see text]), then [Formula: see text] is nilpotent of class [Formula: see text] (solvable of length [Formula: see text] or [Formula: see text]). We also prove that if [Formula: see text] is nilpotent of class [Formula: see text], then [Formula: see text] is nilpotent of class at most [Formula: see text], in which [Formula: see text] and [Formula: see text] is the [Formula: see text]th term in the upper central series of [Formula: see text].

The distance signless Laplacian matrix [Formula: see text] of a connected graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the distance matrix of [Formula: see text] and [Formula: see text] is the diagonal matrix whose main entries are the vertex transmissions of [Formula: see text], and the spectral radius of a connected graph [Formula: see text] is the largest eigenvalue of [Formula: see text]. In this paper, first we obtain the [Formula: see text]-eigenvalues of the join of certain regular graphs. Next, we give some new bounds on the distance signless Laplacian spectral radius of a graph [Formula: see text] in terms of graph parameters and characterize the extremal graphs. Utilizing these results we present some upper and lower bounds on the distance signless Laplacian energy of a graph [Formula: see text].

Consider a selfadjoint unbounded operator D on a Hilbert space H and a one parameter norm continuous family of selfadjoint bounded operators {A(t)} parametrized by the real line. Then under certain conditions \cite{RS95} that include the assumption that the operators {D(t)= D+A(t)} all have discrete spectrum then the spectral flow along the path { D(t)} can be shown to be equal to the index of d/dt+D(t) when the latter is an unbounded Fredholm operator on L^2(R, H). In \cite{GLMST11} an investigation of the index=spectral flow question when the operators in the path may have some essential spectrum was started but under restrictive assumptions that rule out differential operators in general. In \cite{CGPST14a} the question of what happens when the Fredholm condition is dropped altogether was investigated. In these circumstances the Fredholm index is replaced by the Witten index. In this paper we take the investigation begun in \cite{CGPST14a} much further. We show how to generalise a formula known from the setting of the L^2 index theorem to the non-Fredholm setting. Our main theorem gives a trace formula relating the homological index of \cite{CaKa:TIH} to an integral formula that is known, for a path of selfadjoint Fredholms with compact resolvent and with unitarily equivalent endpoints, to compute spectral flow. Our formula however, applies to paths of selfadjoint non-Fredholm operators. We interpret this as indicating there is a generalisation of spectral flow to the non-Fredholm setting., Comment: 40 pages, the abstract has been shortened in order to accommodate the arXiv regulations

In this paper, we define two [Formula: see text] matrices [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text] respectively, where [Formula: see text] are a generalized hyperharmonic numbers of order [Formula: see text]. We give some new factorizations and determinants of the matrices [Formula: see text] and [Formula: see text].

In this paper, we explore the existence and nonlinearity of affine [Formula: see text]-cycle permutations on [Formula: see text] for different values of [Formula: see text] and [Formula: see text]. It is proved that a transposition [Formula: see text] on [Formula: see text] has nonlinearity [Formula: see text] for [Formula: see text], a [Formula: see text]-cycle permutation on [Formula: see text] has nonlinearity [Formula: see text] for [Formula: see text], [Formula: see text] a prime. The number of affine [Formula: see text]-cycle permutations on [Formula: see text] and a bound for their nonlinearity have been obtained. Some results on the cyclic decomposition of a permutation on [Formula: see text], [Formula: see text] a prime, have also been proved.

In this paper, the notions of derivations of types [Formula: see text] and [Formula: see text] on a BL-algebra [Formula: see text] are introduced. We extend the notions of these derivations by introducing the notions of [Formula: see text]-derivations of types [Formula: see text] and [Formula: see text], and discuss some related properties. The conditions for [Formula: see text]-derivations of types [Formula: see text] and [Formula: see text] to be isotone are provided. The set [Formula: see text] is defined and conditions for [Formula: see text] to be a down closed set and an ideal on [Formula: see text] are given where [Formula: see text] is a Boolean algebra and [Formula: see text] is a [Formula: see text]-derivation of type [Formula: see text] on [Formula: see text]. Finally, the relationship between [Formula: see text]-derivations of types [Formula: see text] and [Formula: see text] for a Boolean algebra [Formula: see text] is determined.

Let [Formula: see text] be a unital prime Banach algebra over complex field [Formula: see text] with unity and [Formula: see text] be a nonzero continuous linear generalized derivation associated with a nonzero continuous linear derivation [Formula: see text]. In this paper, we investigate the commutativity of [Formula: see text]. In particular, we prove that a unital prime Banach algebra [Formula: see text] is commutative if one of the following holds; (i) either [Formula: see text] or [Formula: see text], (ii) either [Formula: see text] or [Formula: see text], for sufficiently many [Formula: see text], for any complex numbers [Formula: see text] and an integer [Formula: see text].

Let [Formula: see text] be a non-abelian nilpotent Lie algebra of dimension [Formula: see text] and put [Formula: see text], where [Formula: see text] denotes the Schur multiplier of [Formula: see text]. Niroomand and Russo in 2011 proved that [Formula: see text] and that [Formula: see text] if and only if [Formula: see text], in which [Formula: see text] is the Heisenberg algebra of dimension [Formula: see text] and [Formula: see text] is the abelian [Formula: see text]-dimensional Lie algebra. In the same year, they also classified all nilpotent Lie algebras [Formula: see text] satisfying [Formula: see text] or [Formula: see text]. In this paper, we obtain all nilpotent Lie algebras [Formula: see text] provided that [Formula: see text].

In this paper we are concerned with transversely orientable codimension one foliations. Let be a Cr-foliation as above in a smooth manifold M, r ≧ 1, and let F0 be a closed leaf of . A neighborhood U of F0 is called a bicollar of F0 in this paper if there is a normal line bundle : U → F0 with respect to a fixed Riemannian metric on M such that each fibre of is transverse to .

Let T be a sectorial operator. It is known that the existence of a bounded (suitably scaled) H∞ calculus for T, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra . Such an algebra includes functions defined byMikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for T. In this paper, we use fractional derivation to analyse in detail the relationship between and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

Let R be a commutative Noetherian ring, α an ideal of R and M a finitely generated R-module. Let t be a non-negative integer. It is known that if the local cohomology module is finitely generated for all i < t, then is finitely generated. In this paper it is shown that if is Artinian for all i < t, then need not be Artinian, but it has a finitely generated submodule N such that /N is Artinian.

Let v ≥ k ≥ 1 and λ ≥ 0 be integers. A block design BD(v, k, λ) is a collection of k-subsets of a v-set X in which every unordered pair of elements from X is contained in exactly λ elements of . More generally, for a fixed simple graph G, a graph design GD(v, G, λ) is a collection of graphs isomorphic to G with vertices in X such that every unordered pair of elements from X is an edge of exactly λ elements of . A famous result of Wilson says that for a fixed G and λ, there exists a GD(v, G, λ) for all sufficiently large v satisfying certain necessary conditions. A block (graph) design as above is resolvable if can be partitioned into partitions of (graphs whose vertex sets partition) X. Lu has shown asymptotic existence in v of resolvable BD(v, k, λ), yet for over twenty years the analogous problem for resolvable GD(v, G, λ) has remained open. In this paper, we settle asymptotic existence of resolvable graph designs.

This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γg of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

An involuted semilattice [Formula: see text] is a semilattice [Formula: see text] with an identity element [Formula: see text] and with an involution [Formula: see text] satisfying [Formula: see text] and [Formula: see text]. We consider involuted semilattices [Formula: see text] with an identity [Formula: see text] such that there is a subsemilattice [Formula: see text] without [Formula: see text] with the property that any [Formula: see text] belongs to exactly one of the following four sets : [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. In this paper, we introduce an associative binary operation [Formula: see text] on [Formula: see text] in the following quite natural way: [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text] and characterize all endomorphisms of the orthodox semigroup [Formula: see text].

Let [Formula: see text] be a surjective map between some operator algebras such that [Formula: see text] for all [Formula: see text], where [Formula: see text] defined by [Formula: see text] and [Formula: see text] is Jordan product, i.e. [Formula: see text]. In this paper, we determine the concrete form of map [Formula: see text] on some operator algebras. Such operator algebras include standard operator algebras, properly infinite von Neumann algebras and nest algebras. Particularly, if [Formula: see text] is a factor von Neumann algebra that satisfies [Formula: see text] for all [Formula: see text] and idempotents [Formula: see text] then there exists nonzero scalar [Formula: see text] with [Formula: see text] such that [Formula: see text] for all [Formula: see text]

Let [Formula: see text] be a finite and simple digraph. A [Formula: see text]-rainbow dominating function ([Formula: see text]RDF) of a digraph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the set of in-neighbors of [Formula: see text]. The weight of a [Formula: see text]RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a digraph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a [Formula: see text]RDF of [Formula: see text]. The [Formula: see text]-rainbow reinforcement number [Formula: see text] of a digraph [Formula: see text] is the minimum number of arcs that must be added to [Formula: see text] in order to decrease the [Formula: see text]-rainbow domination number. In this paper, we initiate the study of [Formula: see text]-rainbow reinforcement number in digraphs and we present some sharp bounds for [Formula: see text]. In particular, we determine the [Formula: see text]-rainbow reinforcement number of some classes of digraphs.

The reciprocal complementary distance (RCD) matrix of a graph [Formula: see text] is defined as [Formula: see text], in which [Formula: see text] if [Formula: see text] and [Formula: see text] if [Formula: see text], where [Formula: see text] is the diameter of [Formula: see text] and [Formula: see text] is the distance between the [Formula: see text]th and [Formula: see text]th vertex of [Formula: see text]. The [Formula: see text]-energy [[Formula: see text]] of [Formula: see text] is defined as the sum of the absolute values of the eigenvalues of RCD-matrix of [Formula: see text]. Two graphs [Formula: see text] and [Formula: see text] are said to be RCD-equienergetic if [Formula: see text]. In this paper, we obtain the RCD-eigenvalues and RCD-energy of the join of certain regular graphs and thus construct the non-RCD-cospectral, RCD-equienergetic graphs on [Formula: see text] vertices, for all [Formula: see text].

Let [Formula: see text] be a group and let [Formula: see text] be a field of characteristic [Formula: see text]. In this paper, we have classified the group algebras [Formula: see text] which are strongly Lie nilpotent of index [Formula: see text], [Formula: see text] or [Formula: see text].

We study the existence of solutions of the semilinear equations (1) in which the non-linearity g may grow superlinearly in u in one of directions u → ∞ and u → −∞, and (2) −Δu + g(x, u) = h, in which the nonlinear term g may grow superlinearly in u as |u| → ∞. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that h may satisfy are arbitrarily nonnegative constants, . The proofs are based upon degree theoretic arguments.

In this paper the integrated three-valued telegraph process is examined. In particular, the third-order equations governing the distributions , (where N(t) denotes the number of changes of the telegraph process up to time t) are derived and recurrence relationships for them are obtained by solving suitable initial-value problems. These recurrence formulas are related to the Fourier transform of the conditional distributions and are used to obtain explicit results for small values of k. The conditional mean values (where V(0) denotes the initial velocity of motions) are obtained and discussed.

Let p be an odd prime and d be a positive integer prime to p such that d ≢ 2 mod 4. For technical reasons, we also assume that . For each integer n ≥ 1, we choose a primitive nth root ζn of 1 so that whenever n | m. Let be its cyclotomic Zp-extension, where is the nth layer of this extension. For n ≤ 1, we denote the Galois group Ga\(Kn/K0) by Gn, the unit group of the ring of integers of Kn by En, and the group of cyclotomic units of Kn by Cn. For the definition and basic properties of cyclotomic units such as the index theorem, we refer [6] and [7]. In this paper we examine the injectivity of the homomorphism between the first cohomology groups induced by the inclusion Cn → En.

In a graph [Formula: see text], a non-empty set [Formula: see text] is said to be an open packing set if no two vertices of [Formula: see text] have a common neighbor in [Formula: see text] Let [Formula: see text] and let [Formula: see text] denote the maximum cardinality of an open packing set in [Formula: see text] which contains [Formula: see text]. Then [Formula: see text] is called the open packing saturation number of [Formula: see text]. In this paper, we initiate a study on this parameter.

Let [Formula: see text] be a ring, [Formula: see text] a strictly totally ordered monoid which is also artinian and [Formula: see text] a monoid homomorphism. Given a right [Formula: see text]-module [Formula: see text], denote by [Formula: see text] the generalized inverse polynomial module over the skew generalized power series ring [Formula: see text]. It is shown in this paper that if [Formula: see text] is a completely [Formula: see text]-compatible module and [Formula: see text] an attached prime ideal of [Formula: see text], then [Formula: see text] is an attached prime ideal of [Formula: see text], and that if [Formula: see text] is a completely [Formula: see text]-compatible Bass module, then every attached prime ideal of [Formula: see text] can be written as the form of [Formula: see text] where [Formula: see text] is an attached prime ideal of [Formula: see text].

Let [Formula: see text] be a family of connected graphs. A spanning subgraph [Formula: see text] of [Formula: see text] is called an [Formula: see text]-factor (component factor) of [Formula: see text] if each component of [Formula: see text] is in [Formula: see text]. In this paper, we study the component factors of the Cartesian product of graphs. Here, we take [Formula: see text] and show that every connected graph [Formula: see text] has a [Formula: see text]-factor where [Formula: see text] and [Formula: see text] is the maximum degree of an induced subgraph [Formula: see text] in [Formula: see text] or [Formula: see text]. Also, we characterize graphs [Formula: see text] having a [Formula: see text]-factor.

It is well-known that the transformation semigroup [Formula: see text] is regular, but their subsemigroups need not be. A fence is an ordered set that the order forms a path with alternating orientation. Consider [Formula: see text] as the base set of a fence [Formula: see text]. Two subsemigroups of [Formula: see text] are studied. Namely, the semigroup [Formula: see text] of all order-decreasing self-mappings of [Formula: see text] and the semigroup [Formula: see text] of all order-preserving self-mappings of [Formula: see text]. In this paper, we obtain that [Formula: see text] is a coregular subsemigroup of [Formula: see text]. A characterization of regular subsemigroups [Formula: see text] of [Formula: see text] is given, that is, [Formula: see text] is regular if and only if [Formula: see text]. Finally, we discuss the regularity of elements in [Formula: see text].

A [Formula: see text]-rainbow dominating function (2RDF) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] such that for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the open neighborhood of [Formula: see text]. The weight of a 2RDF [Formula: see text] is the value [Formula: see text]. The [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum weight of a 2RDF of G. The [Formula: see text]-rainbow domination subdivision number [Formula: see text] is the minimum number of edges that must be subdivided (each edge in [Formula: see text] can be subdivided at most once) in order to increase the 2-rainbow domination number. It is conjectured that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. In this paper, we first prove this conjecture for some classes of graphs and then we prove that for any connected graph [Formula: see text] of order [Formula: see text], [Formula: see text].

Let [Formula: see text] be the loop algebra of a loop [Formula: see text] over a field [Formula: see text]. In this paper, we characterize the structure of the unit loop of [Formula: see text] modulo its Jacobson radical when [Formula: see text] is an [Formula: see text] loop obtained from the dihedral group of order [Formula: see text], [Formula: see text] is an odd number and [Formula: see text] is a finite field of characteristic [Formula: see text]. The structure of [Formula: see text] is also determined.

Our setting for this paper is projective 3-spaceover an algebraically closed fieldK. By a curveC⊂is meant a 1-dimensional, equidimensional projective algebraic set, which is locally Cohen-Macaulay. Letbe the Hartshorne-Rao module of finite length (cf. [R]). HereZis the set of integers andℐcthe ideal sheaf ofC. In [GMV] it is shown that, whereis the homogeneous ideal ofC,is the first local cohomology module of theR-moduleMwith respect to. Thus there exists a smallest nonnegative integerk∊Nsuch that, (see also the discussion on the 1-st local cohomology module in [GW]). Also in [GMV] it is shown thatk= 0 if and only ifCis arithmetically Cohen-Macaulay andCis arithmetically Buchsbaum if and only ifk≤ 1. We therefore have the following natural definition.

In this paper, we use the Lagrange neighbour and our equivalence theorem for primitive ideals to obtain lower bounds which are sharper than those given in the literature for class numbers of real quadratic fields in general, but applied to greatest advantage when d is of ERD type.

Let $A$ be a unital commutative associative algebra over a field of characteristic zero, $\k$ be a Lie algebra, and $\z$ a vector space, considered as a trivial module of the Lie algebra $\g := A \otimes \k$. In this paper we give a description of the cohomology space $H^2(\g,\z)$ in terms of well accessible data associated to $A$ and $\k$. We also discuss the topological situation, where $A$ and $\k$ are locally convex algebras., Some minor corrections have been made in this version