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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 this paper, we are interested in obtaining fixed point theorems by keeping the orthogonal completeness of the orthogonal metric space and replacing the [Formula: see text]-contraction condition in theorems by another slightly modified conditions. The paper contains an example illustrating our results.

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.

The "similarity degree" of a unital operator algebra A was defined and studied in two recent papers of ours, where in particular we showed that it coincides with the "length" of an operator algebra. This paper brings several complements: we give direct proofs (with slight improvements) of several known facts on the length which were only known via the degree, and we show that the length of a type II1 factor with property Γ is at most 5, improving on a previous bound (≤ 44) due to E. Christensen.

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.

L systems generalise context-free grammars by incorporating parallel rewriting, and generate languages such as EDT0L and ET0L that are strictly contained in the class of indexed languages. In this paper we show that many of the languages naturally appearing in group theory, and that were known to be indexed or context-sensitive, are in fact ET0L and in many cases EDT0L. For instance, the language of primitives in the free group on two generators, the Bridson-Gilman normal forms for the fundamental groups of 3-manifolds or orbifolds, and the co-word problem of Grigorchuk's group can be generated by L systems. To complement the result on primitives in free groups, we show that the language of primitives, and primitive sets, in free groups of rank higher than two is context-sensitive. We also show the existence of EDT0L and ET0L languages of intermediate growth., Revised following referees suggestions. 21 pages, 2 figures

In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

We study some properties of the generalized local cohomology modules [Formula: see text] of [Formula: see text] with respect to a pair of ideals [Formula: see text] in Serre subcategories. Some results concerning to [Formula: see text]-cofinite modules are also given in this paper.

We show that, given an equation over a finitely generated free group, the set of all solutions in reduced words forms an effectively constructible EDT0L language. In particular, the set of all solutions in reduced words is an indexed language in the sense of Aho. The language characterization we give, as well as further questions about the existence or finiteness of solutions, follow from our explicit construction of a finite directed graph which encodes all the solutions. Our result incorporates the recently invented recompression technique of Je\.z, and a new way to integrate solutions of linear Diophantine equations into the process. As a byproduct of our techniques, we improve the complexity from quadratic nondeterministic space in previous works to $\mathsf{NSPACE}(n\log n)$ here., Comment: 38 pages, 3 figures. A conference version of this paper was presented at ICALP 2015, Kyoto (Japan), July 4-10, 2015, see http://arxiv.org/abs/1502.03426

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]

In this paper it is shown that a classical continuation principle due to Granas for contractions holds under weaker contractive assumptions. This leads to a Leray–Schauder principle for such contractions in hyperbolic spaces. Some applications to nonexpansive mappings in hyperbolic geodesic spaces are also discussed.

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

In this paper, we present a fixed point theorem for multivalued mappings satisfying new inequalities without using the Pompeiu–Hausdorff distance. The results generalize the famous Feng and Liu’s result [Feng and Liu, J. Math. Anal. Appl. 317 (2006) 103–112] and some known results in the literature. Some examples are given to illustrate our results.

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.

This paper studies the global bifurcation phenomenon for the following homogeneous nonlocal eigenvalue problem [Formula: see text] Under some natural hypotheses on [Formula: see text] and [Formula: see text], we show that [Formula: see text] is a bifurcation point of the nontrivial solution set of the above problem. As application of the above result, we determine the interval of [Formula: see text], in which there exist positive solutions for the following Kirchhoff type problem [Formula: see text] where [Formula: see text] is asymptotically 3-linear at zero and infinity. Our results provide a positive answer to an open problem. Moreover, we also study the spectral structure for a homogeneous nonlocal eigenvalue problem.

The global determinism of a ternary semigroup [Formula: see text] is the set of all nonempty subsets of [Formula: see text], denoted by [Formula: see text] equipped with the naturally defined multiplication. A class [Formula: see text] of ternary semigroups is said to be globally determined if any two members [Formula: see text] and [Formula: see text] of [Formula: see text] with isomorphic globals are themselves isomorphic i.e. [Formula: see text] implies that [Formula: see text] for any two ternary semigroups [Formula: see text] and [Formula: see text] in the class [Formula: see text]. In this paper, we mainly discuss that the class of all ternary semilattices are globally determined.

The frame set of a function [Formula: see text] is the subset of all parameters [Formula: see text] for which the time-frequency shifts of [Formula: see text] along [Formula: see text] form a Gabor frame for [Formula: see text] In this paper, we investigate the frame set of a class of compactly supported continuous functions which includes the [Formula: see text]-splines. In particular, we add some new points to the frame sets of these functions. In the process, we generalize and unify some recent results on the frame sets for this class of functions.

In the present paper, we classify varieties of algebraic systems of the type [Formula: see text], for natural numbers [Formula: see text] and [Formula: see text], which are closed under particular derived algebraic systems. If we replace in an algebraic system the [Formula: see text]-ary operation by an [Formula: see text]-ary term operation and the [Formula: see text]-ary relation by the [Formula: see text]-ary relation generated by an [Formula: see text]-ary formula, we obtain a new algebraic system of the same type, which we call derived algebraic system. We shall restrict the replacement to so-called “linear” terms and atomic “linear” formulas, respectively.

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].

In this paper, we obtain a complete rewriting system for monoid presentation of Schützenberger product of [Formula: see text] groups, which is firstly defined in [G. M. S. Gomes, H. Sezinando and J. E. Pin, Presentations of the Schützenberger product of [Formula: see text] groups, Comm. Algebra 34(4) (2006) 1213–1235]. It gives an algorithm for getting normal form of elements and hence solving the word problem in this group.

Our purpose in this paper is to investigate commutativity of a ring with involution [Formula: see text] which admits a generalized derivation satisfying certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Moreover, we provide examples to show that the assumed restrictions cannot be relaxed.

In this paper, we prove a multivariate central limit theorem for [Formula: see text]-norms of high-dimensional random vectors that are chosen uniformly at random in an [Formula: see text]-ball. As a consequence, we provide several applications on the intersections of [Formula: see text]-balls in the flavor of Schechtman and Schmuckenschläger and obtain a central limit theorem for the length of a projection of an [Formula: see text]-ball onto a line spanned by a random direction [Formula: see text]. The latter generalizes results obtained for the cube by Paouris, Pivovarov and Zinn and by Kabluchko, Litvak and Zaporozhets. Moreover, we complement our central limit theorems by providing a complete description of the large deviation behavior, which covers fluctuations far beyond the Gaussian scale. In the regime [Formula: see text] this displays in speed and rate function deviations of the [Formula: see text]-norm on an [Formula: see text]-ball obtained by Schechtman and Zinn, but we obtain explicit constants.

Given a collection [Formula: see text] of k2 commutative polynomials in 2k2 variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative (nc) polynomial p(X, Y) in two k × k matrix variables X and Y. In this paper, we develop algorithms that will generically determine whether the given family [Formula: see text] has a nc representation and will produce such a representation if it exists. In particular, we determine an open, dense subset of the space of nc polynomials in two variables that satisfies the following property: if a family [Formula: see text] of polynomials admits a nc representation in this subset, then our algorithms will determine this representation.

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.

In this paper, we gives lower and upper bounds on the covering radius of codes over [Formula: see text], where [Formula: see text] is a prime integer with respect to Lee distance. We also determine the covering radius of various Repetition codes over [Formula: see text], where [Formula: see text] is a prime integer.

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].

In this paper, we introduce the concept of algebra cone generalized [Formula: see text]-metric space over Banach algebra by replacing the constant [Formula: see text] with [Formula: see text] where [Formula: see text] is the spectral radius of [Formula: see text], with this modification we shall prove Banach and Kannan fixed point results for contractive generalized Lipschitz mappings in such a space. Moreover, we present one example in support of our results.

Let [Formula: see text] be a Lie algebra and [Formula: see text] be a proper ideal of [Formula: see text]. Then [Formula: see text] is called a Camina pair if [Formula: see text] for all [Formula: see text]. Also, [Formula: see text] is called a Camina Lie algebra if [Formula: see text] is a Camina pair. In this paper, we give some properties of Camina Lie algebras. Moreover, we show that a nilpotent Camina Lie algebra of finite dimension over an algebraically closed field is nilpotent with nilindex at most [Formula: see text].

Let [Formula: see text] be a simple graph. A hop dominating set [Formula: see text] is called a connected hop dominating set of [Formula: see text] if the induced subgraph [Formula: see text] of [Formula: see text] is connected. The smallest cardinality of a connected hop dominating set of [Formula: see text], denoted by [Formula: see text], is called the connected hop domination number of [Formula: see text]. In this paper, we characterize the connected hop dominating sets in the join, corona and lexicographic product of graphs and determine the corresponding connected hop domination number of these graphs. The study of these concepts is motivated with a social network application.

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].

The tensor product and the Cartesian product of two graphs are very well-known graph products and studied in detail. Many graph parameters, particularly independence number, have been studied for these graph products. These two graph products have been generalized by [Formula: see text]-tensor product and [Formula: see text]-Cartesian product, respectively, and studied in detail. In this paper, we discuss the independence number for [Formula: see text]-tensor product [Formula: see text] and [Formula: see text]-Cartesian product [Formula: see text]. In general, we obtain lower bound and upper bound for the independence number.

In this paper, we show some Second Main Theorems for zero-order meromorphic mappings intersecting slowly moving targets in [Formula: see text] by considering their [Formula: see text]-Casorati determinant. Our results are [Formula: see text]-difference analogues of Cartan’s Second Main Theorem for moving targets. As an application, we give an unicity theorem for meromorphic mappings of [Formula: see text] into [Formula: see text] under the growth condition “order [Formula: see text]”.

The main purpose of this paper is to improve the bound of complexity of the well-known algorithms on polynomial ideals having complexities polynomial in dn, where d is the maximal degree of input polynomials and n is the number of variables. Instead of this bound, we present the more accurate bound max {S, Dn} where S is the size of the input polynomials in dense representation, and D is the arithmetic mean value of the degrees of input polynomials.

Let K be a complete ultrametric algebraically closed field. This paper is aimed at studying problems of interpolation for entire functions on K. We prove an analogue of M . Lazard's divisors theorem for entire functions and then we show that sequences of interpolation are the injective sequences (ai)i∈ N in K such that lim i→∞ |ai| = +∞. This result does not assume K to be spherically complete. In order to construct concrete entire interpolation functions, we introduce Lagrange entire functions and define L-sequences. We show that a sequence (ai)i∈ N in K defines a Lagrange entire function if and only if (ai)i∈ N is an L-sequence.

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

Let [Formula: see text] be a commutative ring and [Formula: see text] an [Formula: see text]-algebra. An [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is an [Formula: see text]-linear map of the form [Formula: see text] for some [Formula: see text]-algebra endomorphism [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the identity map of [Formula: see text]. In this paper, we discuss some open problems on whether or not the image of a locally finite (LF) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] is a Mathieu subspace [W. Zhao, Generalizations of the image conjecture and the Mathieu conjecture, J. Pure Appl. Algebra 214 (2010) 1200–1216; Mathieu subspaces of associative algebras, J. Algebra 350(2) (2012) 245–272] of [Formula: see text], and whether or not a locally nilpotent (LN) [Formula: see text]-derivation or [Formula: see text]-[Formula: see text]-derivation of [Formula: see text] maps every ideal of [Formula: see text] to a Mathieu subspace of [Formula: see text]. We propose and discuss two conjectures which state that both questions above have positive answers if the base ring [Formula: see text] is a field of characteristic zero. We give some examples to show the necessity of the conditions of the two conjectures, and discuss some positive cases known in the literature. We also show some cases of the two conjectures. In particular, both the conjectures are proved for LF or LN algebraic derivations and [Formula: see text]-[Formula: see text]-derivations of integral domains of characteristic zero.

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].

For a group algebra [Formula: see text] of a group [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text], it is well known that [Formula: see text] is the minimal upper as well as the minimal lower Lie nilpotency index. Group algebras of upper Lie nilpotency index upto [Formula: see text] have already been characterized completely. In this paper, we classify the modular group algebra [Formula: see text] having upper Lie nilpotency index [Formula: see text] which is the possible next higher Lie nilpotency index.

In this paper, the description of hereditary saturated lattice formations and lattice subgroup functors in the class of all finite metanilpotent groups was found.

This paper provides the several homological characterization of perfect rings and semi-simple rings in terms of semi-projective modules. We investigate whether Hopkins–Levitzki Theorem extend to semi-projective module i.e. whether there exists an artinian semi-projective module which are noetherian. Unfortunately, the answer we have is negative; counter example is given. However, it is shown that the answer is positive for certain large classes of semi-projective modules in Proposition 2.26. We have discussed the summand intersection property, summand sum property for semi-projective modules. Apart from this, we have introduced the idea of [Formula: see text]-hollow modules, also several necessary and sufficient conditions are established when the endomorphism rings of a semi-projective modules is a local ring.

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, we introduce the concept of generalized [Formula: see text]-Suzuki-contractions in the context of quasi [Formula: see text]-metric-like spaces and we establish some related fixed point theorems. As consequences, we obtain some known fixed point results in the literature. Some concrete examples are also provided illustrating the obtained results.

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.

By the use of the celebrated Kato’s inequality, we obtain in this paper some new inequalities for trace class operators on a complex Hilbert space [Formula: see text] Natural applications for functions defined by power series of normal operators are given as well.

Let [Formula: see text] be an associative ring with identity. A ring [Formula: see text] is called reversible if [Formula: see text], then [Formula: see text] for [Formula: see text]. The total graph of unfaithful submodules of a module [Formula: see text] over a reversible ring [Formula: see text], denoted by [Formula: see text], is a graph whose vertices are all nonzero unfaithful submodules of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] is unfaithful. In this paper, we determine the diameter and girth of [Formula: see text]. Also, we study some combinatorial properties of [Formula: see text] such as independence number and clique number. Moreover, we study the case that the degree of a vertex of [Formula: see text] is finite.

The fractional derivative operator is an extension of the usual derivative operator [Formula: see text] to an arbitrary (integer, rational, irrational or complex) values of [Formula: see text]. In this paper, weighted fractional differentiation composition operators from mixed-norm spaces to Zygmund spaces are characterized in terms of boundedness and compactness.