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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.

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*.

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, 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.

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.

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]”.

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 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.

The aim of this paper is to study the probability that the commutator of an arbitrarily chosen pair of elements, each from two different additive subgroups of a finite non-commutative ring equals a given element of that ring. We obtain several results on this probability including a computing formula, some bounds and characterizations.

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.

In this paper, we generalize Nadler’s result by establishing the existence and uniqueness of coincidence and common fixed points of Nadler’s type set-valued mappings in ultrametric spaces. Examples are given to illustrate the results. As an application, well-posedness of a common fixed point problem is proved. The presented results generalize many known results in literature in the framework of ultrametric spaces.

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].

In this paper, we introduce the concept of set-valued Prešić–Reich type contractive condition in ultrametric spaces and establish the existence and uniqueness of coincidence and common fixed point of a set-valued and a single-valued mapping besides furnishing illustrative examples to highlight the realized improvements in the context of ultrametric spaces. Our results generalize and extend some known results in the literature.

Given a group [Formula: see text], we study right and left zeros, idempotents, the minimal ideal, left cancelable and right cancelable elements of the semigroup [Formula: see text] of [Formula: see text]-linked upfamilies and characterize groups [Formula: see text] whose extensions [Formula: see text] are commutative. We finish the paper with the complete description of the structure of the semigroups [Formula: see text] for all groups [Formula: see text] of cardinality [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].

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].

The Wiener number [Formula: see text] of a graph [Formula: see text] was introduced by Harold Wiener in connection with the modeling of various physic-chemical, biological and pharmacological properties of organic molecules in chemistry. Milan Randić introduced a modification of the Wiener index for trees (acyclic graphs), and it is known as the hyper-Wiener index. Then Klein et al. generalized Randić’s definition for all connected (cyclic) graphs, as a generalization of the Wiener index, denoted by [Formula: see text] and defined as [Formula: see text]. In this paper, we establish some upper and lower bounds for [Formula: see text], in terms of other graph-theoretic parameters. Moreover, we compute hyper-Wiener number of some classes of graphs.

This paper deals with the multiple values and algebraic dependences problem of meromorphic mappings sharing moving hyperplanes in projective space. We give some algebraic dependences theorems for meromorphic mappings sharing moving hyperplanes without counting multiplicity, where all zeros with multiplicities more than a certain number are omitted. Basing on these results, some unicity theorems regardless of multiplicity for meromorphic mappings in several complex variables are given. These results are extensions and strong improvements of some recent results.

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 the set of all nonempty subsets of a ternary semigroup [Formula: see text]. Then [Formula: see text] is a ternary semigroup with respect to the ternary multiplication defined by [Formula: see text] for all [Formula: see text]. If [Formula: see text] and [Formula: see text] are isomorphic ternary semigroups, then the corresponding power ternary semigroups [Formula: see text] and [Formula: see text] are obviously isomorphic. It is quite natural to ask whether the converse is true, i.e. is it true that for any ternary semigroups [Formula: see text] and [Formula: see text], [Formula: see text] implies that [Formula: see text]? If the class [Formula: see text] of algebra has this property, we say that [Formula: see text] is a globally determined class. In this paper, we provide some class of globally determined ternary semigroups. We show that all ternary semigroups are not globally determined but some special classes of ternary semigroups are globally determined. We show that the class of finite left (right) zero ternary semigroups, ternary groups and principal ideal (P. I.) ternary semigroups are globally determined but the class of involution ternary semigroups is not globally determined.

In this paper, assuming an appropriate condition, we prove that [Formula: see text]-generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern–Ishikawa type and a nonlinear ergodic theorem for [Formula: see text]-generalized hybrid mappings have been proven in Hilbert spaces.

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.

In this paper, we study the zero-divisor graph of Rickart *-rings. We determine the condition on Rickart *-ring so that its zero-divisor graph contains a cut vertex. It is proved that the set of cut vertices form a complete subgraph. We characterize Rickart *-rings for which the complement of the zero-divisor graph is connected. The diameter and girth of these graphs are characterized. Further, for a *-ring [Formula: see text] with unity we associate a graph, [Formula: see text], having all nonzero elements of [Formula: see text] as vertices and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text] is a unit in [Formula: see text]. For a finite Rickart *-ring [Formula: see text] it is proved that [Formula: see text] is connected if and only if [Formula: see text] is not isomorphic to [Formula: see text] or [Formula: see text] (for any [Formula: see text].

In 2011, Düntsch and Orlowska obtained a discrete duality for regular double Stone algebras. On the other hand, it is well known that regular double Stone algebras are polinominally equivalent to 3-valued Lukasiewicz-Moisil algebras (or LM3-algebras). In [R. Cignoli, Injective De Morgan and Kleene algebra, Proc. Amer. Math. Soc. 47 (1975) 269-278], LM3-algebras are considered as a Kleene algebras (L,∨,∧,∼, 0, 1) endowed with a unary operation : L → L, satisfying the properties: a∨ ∼ a = 1, ∼ a ∧ a = a∧ ∼ a and a∨b ≤(a ∨ b). Motivated by this result, in this paper, we determine another discrete duality for LM3-algebras, extending the discrete duality to De Morgan algebras described in [W. Dzik, E. Orlowska and C. van Alten, Relational representation theorems for general lattices with negations, in Relations and Kleene Algebra in Computer Science, Lecture Notes in Computer Science, Vol. 4136 (Springer, Berlin, 2006), pp. 162-176]. Fil: Pelaitay, Gustavo Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Juan; Argentina. Universidad Nacional de San Juan. Facultad de Filosofía, Humanidades y Artes; Argentina

In this paper, we introduce the neighborhood [Formula: see text] of analytic functions [Formula: see text] and [Formula: see text] defined in the open unit disc. Furthermore, we derive some sufficient and necessary conditions to be in [Formula: see text]. In addition, we see some applications of Jack’s lemma.

In this paper, we study the decay of the smallest singular value of submatrices that consist of bounded column vectors. We find that the smallest singular value of submatrices is related to the minimal distance of points to the lines connecting other two points in a bounded point set. Using a technique from integral geometry and from the perspective of combinatorial geometry, we show the decay rate of the minimal distance for the sets of points if the number of the points that are on the boundary of the convex hull of any subset is not too large, relative to the cardinality of the set. In the numeral or computational aspect, we conduct some numerical experiments for many sets of points and analyze the smallest distance for some extremal configurations.