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This paper introduces cutting planes that involve minimal structural assumptions, enabling the generation of strong polyhedral relaxations for a broad class of problems. We consider valid inequalities for the set S ∩ P , where S is a closed set, and P is a polyhedron. Given an oracle that provides the distance from a point to S, we construct a pure cutting plane algorithm which is shown to converge if the initial relaxation is a polyhedron. These cuts are generated from convex forbidden zones, or S-free sets, derived from the oracle. We also consider the special case of polynomial optimization. Accordingly we develop a theory of outer-product-free sets, where S is the set of real, symmetric matrices of the form x x T . All maximal outer-product-free sets of full dimension are shown to be convex cones and we identify several families of such sets. These families are used to generate strengthened intersection cuts that can separate any infeasible extreme point of a linear programming relaxation efficiently. Computational experiments demonstrate the promise of our approach. [ABSTRACT FROM AUTHOR]

The aim of this paper is to extend the domain of the Grace's theorem, Schur-Szegö composition theorem and Cohn-Egerváry theorem from the set of complex numbers to the set of bicomplex numbers. [ABSTRACT FROM AUTHOR]

In this paper, we give a complete description of the generalized hypergeometric functions, introduced by Faraut and Kor´anyi on the Cartan domain. We establish some Gauss type contiguous relations between these functions on the two Cartan domains of type I2 and type IV4 analogous to the classical relations in the one variable case. [ABSTRACT FROM AUTHOR]

Let G be a graph. A dissociation set of G is a subset of vertices that induces a subgraph with vertex degree at most 1. The dissociation polynomial of G is D G (λ) = ∑ D ∈ D (G) λ | D | , where D (G) is the set of all dissociation sets of G . In this paper, we prove that for any cubic graph G and any λ ∈ (0 , 1 ] , 1 | V (G) | ln ⁡ D G (λ) ≤ 1 4 ln ⁡ D K 4 (λ) with equality if and only if G is a disjoint union of copies of the complete graph K 4 . When λ = 1 , the value of D G (λ) is exactly the number of dissociation sets of G . Hence, for any cubic graph G on n vertices, | D (G) | ≤ | D (K 4) | n / 4 = 11 n / 4. Let be a graph. A dissociation set of is a subset of vertices that induces a subgraph with vertex degree at most 1. The dissociation polynomial of is , where is the set of all dissociation sets of . In this paper, we prove that for any cubic graph and any , with equality if and only if is a disjoint union of copies of the complete graph . When , the value of is exactly the number of dissociation sets of . Hence, for any cubic graph on vertices, [ABSTRACT FROM AUTHOR]

In this paper, on the basis of zero set of reverse indexed polynomial, first introduced by us, we have investigated the uniqueness of meromorphic functions under weighted sharing in wider sense criteria [4], which in turn extend some earlier results in different directions. We have succeeded to identify a subclass of meromorphic functions for which uniqueness property exists for higher reverse indexed polynomial in literature. In the last section, we have presented the application of our results in case of derivatives of the concerned functions accompanied by series of examples. [ABSTRACT FROM AUTHOR]

Let S be a ring of polynomials in finitely many variables over a field. In this paper, we give lower bounds for depth and Stanley depth of modules of the type S/It for t = 1, where I is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness of transcendental entire functions whose certain non-linear differential-difference polynomials share a small function with some weight in a relaxed manner. Our results generalize the earlier results of P. Sahoo and H. Karmakar [22]. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for polynomials of special forms to be permutation polynomials over finite field. More specifically, we construct several explicit classes of permutation polynomials from these forms over finite fields. We also present from these polynomials a new family of complete permutation polynomials. [ABSTRACT FROM AUTHOR]

The edge neighborhood graph Ne(G) of a simple graph G is the graph with the vertex set E ∪ S where S is the set of all open edge neighborhood sets of G and two vertices u,v ∈ V (Ne(G)) adjacent if u ∈ E and v is an open edge neighborhood set containing u. In this paper, we determine the domination number, the total domination number, the independent domination number and the 2-domination number in the edge neighborhood graph. We also obtain a 2-domination polynomial of the edge neighborhood graph for some certain graphs. [ABSTRACT FROM AUTHOR]

One central theme in quantum error-correction is to construct quantum codes that have a relatively large minimum distance. In this paper, we first present a construction of classical linear codes based on certain classes of polynomials. Through these classical linear codes, we are able to obtain some new quantum codes. It turns out that some of the quantum codes exhibited here have better parameters than the ones available in the literature. [ABSTRACT FROM PUBLISHER]

In this paper we show that if p is a polynomial which bifurcates then the product map (z,w)↦(p(z),q(w)) can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types: repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets: the bifurcation locus of Hd(P²) and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of Hénon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets. [ABSTRACT FROM AUTHOR]

If by s is denoted the number of independent sets of cardinality k in a graph G, then $${I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}$$ is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97-106, ), where α = α( G) is the size of a maximum independent set. The inequality | I ( G; −1)| ≤ 2, where ν( G) is the cyclomatic number of G, is due to (Engström in Eur. J. Comb. 30:429-438, ) and (Levit and Mandrescu in Discret. Math. 311:1204-1206, ). For ν( G) ≤ 1 it means that $${I(G;-1)\in\{-2,-1,0,1,2\}.}$$ In this paper we prove that if G is a unicyclic well-covered graph different from C, then $${I(G;-1)\in\{-1,0,1\},}$$ while if G is a connected well-covered graph of girth ≥ 6, non-isomorphic to C or K (e.g., every well-covered tree ≠ K), then I ( G; −1) = 0. Further, we demonstrate that the bounds {−2, 2} are sharp for I ( G; −1), and investigate other values of I ( G; −1) belonging to the interval [−2, 2]. [ABSTRACT FROM AUTHOR]

In this paper, we study some fundamental geometrical properties related to the S-procedure. Given a pair of quadratic functions (g, f), it asks when "g(x)=0⟹f(x)≥0" can imply "(∃λ∈R) (∀x∈Rn) f(x)+λg(x)≥0." Although the question has been answered by Xia et al. (Math Program 156:513-547, 2016), we propose a neat geometric proof for it (see Theorem 2): the S-procedure holds when, and only when, the level set {g=0} cannot separate the sublevel set {f<0}. With such a separation property, we proceed to prove that, for two polynomials (g, f) both of degree 2, the image set of g over {f<0},g({f<0}), is always connected (see Theorem 4). It implies that the S-procedure is a kind of the intermediate value theorem. As a consequence, we know not only the infimum of g over {f≤0}, but the extended results when g over {f≤0} is unbounded from below or bounded but unattainable. The robustness and the sensitivity analysis of an optimization problem involving the pair (g, f) automatically follows. All the results in this paper are novel and fundamental in control theory and optimization. [ABSTRACT FROM AUTHOR]

The present paper is a continuation of the previous papers of the author dealing with this subject. We study the cascade search of a given subset A, i.e., the construction on the metric space X of a multicascade with given limit subset A in X. A multicascade is a multivalued dynamical system with translation semigroup equal to the additive semigroup of nonnegative integers. We propose a finer (than in the author’s previous papers) version of cascade search for cases in which (1) A is the complete preimage of a closed subspace under the multivalued mapping of metric spaces; (2) A is the set of coincidence points n, n > 1, of the multivalued mappings. An estimate of the distance from the initial to any corresponding limit point is given. In particular, in case (2), a new generalization of a recent theorem due to Arutyunov is obtained for n = 2. [ABSTRACT FROM AUTHOR]

The main seismic design codes propose simplified formulations to evaluate the fundamental period of regular structures based on the total height. Indeed, the fundamental period depends on several parameters directly connected to the mass and stiffness of the structure and on its geometrical characteristics, including also irregularities. This paper proposes a set of mathematical formulations to evaluate the longitudinal and transversal fundamental period of vibration of 3D Reinforced Concrete (RC) frames, which have various vertical and plan irregularities and for different mechanical and geometrical design parameters. Several types of Reinforced Concrete Bare Moment Resisting Frame (RC‐BMRF) buildings have been designed according to the different versions of the Italian codes starting from 1916 to nowadays and then used as case studies. Modal analysis is performed on the entire building dataset to assess the fundamental periods in both longitudinal and transversal directions. Then, cluster analysis is carried out to classify the buildings based on similar design characteristics and construction details. Finally, a robust Evolutionary Polynomial Regression (EPR) technique is used to find the optimal polynomial forms of the natural period. Numerical results show a better performance of the proposed formulation compared with the existing methodologies available in the literature. [ABSTRACT FROM AUTHOR]

In the present paper, we consider 2 integral transforms involving the Appell function F3 in the kernels. They generalize the fractional integral operators given by Saigo (1978). Formulas for compositions of such generalized fractional integrals with the product of the Ħ-function and a general class of polynomials are proved. The results are established in terms of Ħ-function due to Inayat-Ħussain (1987(a), 1987(b)). The obtained results of this paper provide an extension of the results given by the literature. [ABSTRACT FROM AUTHOR]

We prove new bounds on the Betti numbers of real varieties and semialgebraic sets that have a more refined dependence on the degrees of the polynomials defining them than results known before. Our method also unifies several different types of results under a single framework, such as bounds depending on the total degrees, on multi-degrees, as well as in the case of quadratic and partially quadratic polynomials. The bounds we present in the case of partially quadratic polynomials offer a significant improvement over what was previously known. Finally, we extend a result of Barone and Basu on bounding the number of connected components of real varieties defined by two polynomials of differing degrees to the sum of all Betti numbers, thus making progress on an open problem posed in that paper. [ABSTRACT FROM AUTHOR]

In this paper we consider n-poised planar node sets, as well as more special ones, called G C sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ℓ is called k-node line for a node set $\mathcal X$ if it passes through exactly k nodes. An ( n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of $\mathcal X$ is used by each node in $\mathcal X\setminus M, $ meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C set $\mathcal {X}$ is used either by exactly $\binom {n}{2}$ nodes or by exactly $\binom {n-1}{2}$ nodes. We prove also similar statements concerning n-node or ( n − 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C sets. At the end we present a conjecture concerning any k-node line. [ABSTRACT FROM AUTHOR]

Solution to the solvability problem for a class of product-type systems of difference equations is given by presenting explicit formulas for its solutions. In the main/most complicated cases the problem is solved by using two different methods. This is the last system, out of the three non-equivalent ones, for which an associated polynomial, which essentially determines the structure of the solutions, is of the third order, so that the paper finishes the study of this kind of two-dimensional systems. [ABSTRACT FROM AUTHOR]

It is a well known fact that, in this modern era, the data visualization has become very important in almost all the areas of human life including science and technology. In this paper, we have made an attempt to visualize the behaviour of interpolation search by measuring its time in worst case for a varying size of equi - interval sets of data in a personal computer (desktop) using polynomial curve fitting technique. It has been observed that in the worst case this search technique behaviourally does not fit to any particular polynomial model i.e. polynomial model of a particular degree for the varying size of equi - interval sets of data. In this paper, the researchers have also shown the smooth spline curves passing through the predicted values obtained by using the best fit polynomial models for the varying size of equi - interval sets of data. [ABSTRACT FROM AUTHOR]

Three classes of binary sequences of period 4N with optimal autocorrelation value/magnitude have been constructed by Tang and Gong based on interleaving certain kinds of sequences of period N, i.e., the Legendre sequence, twin-prime sequence and generalized GMW sequence. In this paper, by means of sequence polynomials of the underlying sequences, the properties of roots of the corresponding sequence polynomials of the interleaved sequences with period 4N and optimal autocorrelation value/magnitude are discussed in the splitting field of x^N-1. As a consequence, both the minimal polynomials and linear complexities of these three classes of sequences are completely determined except for the case of the sequences obtained from the generalized GMW sequences. For the latter, the minimal polynomial and linear complexity can be specially obtained if the sequence is constructed based on m-sequences instead of generalized GMW sequences. [ABSTRACT FROM AUTHOR]

homomorphism from a graph G to a graph H (in this paper, both simple, undirected graphs) is a mapping f: V( G)→ V( H) such that if uv∈ E( G) then f( u) f( v)∈ E( H). The problem Hom ( G, H) of deciding whether there is a homomorphism is NP-complete, and in fact the fastest known algorithm for the general case has a running time of O( n( H)) (the notation O(⋅) signifies that polynomial factors have been ignored) for a constant 0< c<1. In this paper, we consider restrictions on the graphs G and H such that the problem can be solved in plain-exponential time, i.e. in time O( c) for some constant c. Previous research has identified two such restrictions. If H= K or contains K as a core (i.e. a homomorphically equivalent subgraph), then Hom ( G, H) is the k-coloring problem, which can be solved in time O(2) (Björklund, Husfeldt, Koivisto); and if H has treewidth at most k, then Hom ( G, H) can be solved in time O(( k+3)) (Fomin, Heggernes, Kratsch). We extend these results to cases of bounded cliquewidth: if H has cliquewidth at most k, then we can count the number of homomorphisms from G to H in time O((2 k+1)), including the time for finding a k-expression for H. The result extends to deciding Hom ( G, H) when H has a core with a k-expression, in this case with a somewhat worse running time. If G has cliquewidth at most k, then a similar result holds, with a worse dependency on k: We are able to count Hom ( G, H) in time O((2 k+1)+2), and this also extends to when G has a core of cliquewidth at most k with a similar running time. [ABSTRACT FROM AUTHOR]

In this paper, a description of the small-h limit of loci of zeros of subdominant solutions for polynomial potentials is given called as fundamental solutions in our earlier papers. The considered potentials are those which provide us with the simple turning points only. Three types of Stokes graphs (SG's) associated with the potentials are investigated - the general non-critical ones, the general critical ones but with only single internal Stokes line (SL), and the Stokes graphs corresponding to arbitrary multiple-well real even degree polynomial potentials with internal Stokes lines distributed on the real axis only. All these cases are considered in their both versions of the quantized and not quantized h. In particular due to the fact that the small-h limit is semiclassical it is shown that loci of roots of subdominant solutions in the cases considered are collected along Stokes lines. There are infinitely many roots of subdominant solutions on such lines escaping to infinity and a finite number of them on internal Stokes lines. [ABSTRACT FROM AUTHOR]

Let $${S =\{x\in \mathbb{R}^n: g_1(x)\geq 0, \ldots, g_m(x)\geq 0\}}$$ be a semialgebraic set defined by multivariate polynomials g i( x). Assume S is convex, compact and has nonempty interior. Let $${S_i =\{x\in \mathbb{R}^n:\, g_i(x)\geq 0\}}$$ , and ∂ S (resp. ∂ S i) be the boundary of S (resp. S i). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of $${\mathbb{R}^n}$$ , does there exist an LMI representable set Ŝ in some higher dimensional space $${ \mathbb{R}^{n+N}}$$ whose projection down onto $${\mathbb{R}^n}$$ equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g i( x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function ℓ T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g i( x) is either sos-concave ( − ∇2 g i( x) = W( x) T W( x) for some possibly nonsquare matrix polynomial W( x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S i is either sos-convex or poscurv-convex ( S i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇ g i( x) ≠ 0 on ∂ S i ∩ S), then S is SDP representable. This also holds for S i for which ∂ S i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmüdgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii). [ABSTRACT FROM AUTHOR]

In this paper,we study the growth of solutions of a k-th order linear differential equation and that of a k+1-th order linear differential equation. From this we affirmatively answer a uniqueness question concerning a conjecture given by Brück in 1996 under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its derivative sharing a finite nonzero complex number CM. The results in this paper also improve some known results. Some examples are provided to show that the results in this paper are best possible. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of computing the real dimension of a given semi-algebraic subset of R k, where R is a real closed field. We prove that the dimension k′ of a semi-algebraic set described by s polynomials of degree d in k variables can be computed in time . This result slightly improves the result by Vorobjov, who described an algorithm with complexity bound (sd)O(k′(k−k′)) for the same problem. The complexity bound of the algorithm described in this paper has a better dependence on the number s of polynomials in the input. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

The splitting method is one of the most powerful and well-studied approaches to solving various NP-hard problems. The main idea of this method is to split the input instance of a problem into several simpler instances (further simplified by certain simplification rules) such that when the solution for each of them is found, one can construct the solution for the initial instance in polynomial time. There exists a huge number of papers describing algorithms of this type, and usually a considerable part of such a paper is devoted to case analysis. In this paper, we present a program that, given a set of simplification rules, automatically generates a proof of an upper bound on the running time of a splitting algorithm using these rules. As an example, we report the results of experiments with such a program for the SAT, MAXSAT, and (n, 3)- MAXSAT (the MAXSAT problem for the case where every variable in the formula appears at most three times) problems. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the lower bound on the number of gates in a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining three approaches: the restriction applied to constant depth circuits due to Håstad, the approximation method applied to monotone circuits due to Razborov, and the boundary covering developed in the present paper. We prove that if a circuit $C$ with at most $\floor{(1/6) \log \log m}$ negation gates detects cliques of size $(\log m)^{3(\log m)^{1/2}}$ in a graph with $m$ vertices, then $C$ contains at least $2^{(1/5)(\log m)^{(\log m)^{1/2}}}$ gates. No nontrivial lower bounds on the size of such circuits were previously known, even if we restrict the number of negation gates to be a constant. Moreover, it follows from a result of Fischer [{\it Lect. Notes Comput. Sci.,} 33 (1974), pp. 71--82] that if one can improve the number of negation gates from $\floor{(1/6)\log\log m}$ to $\floor{2\log m}$ in the statement, then we have P $\neq$ NP. We also show that the problem of lower bounding the negation-limited circuit complexity can be reduced to the one of lower bounding the maximum of the monotone circuit complexity of the functions in a certain class of monotone functions. [ABSTRACT FROM AUTHOR]

Equivalence of aggregate queries is investigated for the class of conjunctive queries with comparisons and the aggregate operators count, count-distinct, min, max, and sum. Essentially, this class contains unnested SQL queries with the above aggregate operators, with a where clause consisting of a conjunction of comparisons, and without a having clause. The comparisons are either interpreted over a domain with a dense order (like the rationals) or with a discrete order (like the integers). Characterizations of equivalence differ for the two cases. For queries with either max or min, equivalence is characterized in terms of dominance mappings, which can be viewed as a generalization of containment mappings. For queries with the count-distinct operator, a sufficient condition for equivalence is given in terms of equivalence of conjunctive queries under set semantics. For some special cases, it is shown that this condition is also necessary. For conjunctive queries with comparisons but without aggregation, equivalence under bag-set semantics is characterized in terms of isomorphism. This characterization essentially remains the same also for queries with the count operator. Moreover, this characterization also applies to queries with the sum operator if the queries have either constants or comparisons, but not both. In the general case (i.e., both comparisons and constants), the characterization of the equivalence of queries with the sum operator is more elaborate. All the characterizations given in the paper are decidable in polynomial space. [ABSTRACT FROM AUTHOR]

A polarized polynomial form (PPF) (modulo k) is a modulo k sum of products of variables x1, . . . , xn or their Post negations, where the number of negations of each variable is determined by the polarization vector of the PPF. The length of a PPF is the number of its pairwise distinct summands. The length of a function f( x1, . . . , xn)of k-valued logic in the class of PPFs is the minimum length among all PPFs realizing the function. The paper presents a sequence of symmetric functions fn( x1, . . . , xn)of three-valued logic such that the length of each function fn in the class of PPFs is not less than ⌊3 n+1/4⌋, where ⌊a⌋ denotes the greatest integer less or equal to the number a. The complexity of a system of PPFs sharing the same polarization vector is the number of pairwise distinct summands entering into all of these PPFs. The complexity of a system F ={ f1,..., fm} of functions of k-valued logic depending on variables x1,..., xn in the class of PPFs is the minimum complexity among all systems of PPFs { p1,..., pm}such that all PPFs p1,..., pm share the same polarization vector and the PPF pj realizes the function fj, j = 1,..., m. Let , where F runs through all systems consisting of m functions of k-valued logic depending on variables x1,..., xn. For prime values of k it is easy to derive the estimate . In this paper it is shown that and for all m ≥ 2, n= 1, 2, . . . Moreover, it is demonstrated that the estimates remain valid when consideration is restricted to systems of symmetric functions only. [ABSTRACT FROM AUTHOR]

We show that all knots up to six crossings can be represented by polynomial knots of degree at most , among which except for and all are in their minimal degree representation. We provide concrete polynomial representation of all these knots. Durfee and O'Shea had asked a question: Is there any -crossing knot in degree ? In this paper we try to partially answer this question. For an integer , we define a set to be the set of all polynomial knots given by such that and . This set can be identified with a subset of and thus it is equipped with the natural topology which comes from the usual topology . In this paper we determine a lower bound on the number of path components of for . We define a path equivalence for polynomial knots in the space and show that it is stronger than the topological equivalence. [ABSTRACT FROM AUTHOR]

Hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials. A well-understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral. In this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone, that is, a spectrahedral shadow. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory. [ABSTRACT FROM AUTHOR]

We study the following problem. Given a multiset M of non-negative integers, decide whether there exist and, in the positive case, compute two non-trivial multisets whose Minkowski sum is equal to M. The Minkowski sum of two multisets A and B is a multiset containing all possible sums of any element of A and any element of B. This problem was proved to be NP-complete when multisets are replaced by sets. This version of the problem is strictly related to the factorization of boolean polynomials that turns out to be NP-complete as well. When multisets are considered, the problem is equivalent to the factorization of polynomials with non-negative integer coefficients. The computational complexity of both these problems is still unknown. The main contribution of this paper is a heuristic technique for decomposing multisets of non-negative integers. Experimental results show that our heuristic decomposes multisets of hundreds of elements within seconds, independently of the magnitude of numbers belonging to the multisets. Our heuristic can also be used for factoring polynomials in N[x]. We show that, when the degree of the polynomials gets larger, our technique is much faster than the state-of-the-art algorithms implemented in commercial software like Mathematica and MatLab. [ABSTRACT FROM AUTHOR]

The paper is concerned with representations of predicates over a finite set in the form of generalized conjunctive normal forms (GCNF). Properties of predicates GCNF are found which are preserved by some majority function. Such predicates are called generalized bijunctive predicates. These properties are used to construct new faster polynomial algorithms for the generalized satisfiability problem in the case when some majority function preserves all the original predicates. [ABSTRACT FROM AUTHOR]

An n-poised node set X in the plane is called GCn set if the (bivariate) fundamental polynomial of each node is a product of n linear factors. A line is called k-node line if it passes through exactly k-nodes of X. An (n + 1)-node line is called a maximal line. In 1982, Gasca and Maeztu conjectured that every GCn set has a maximal line. Until now the conjecture has been proved only for n ≤ 5. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. It is a simple fact that any maximal line λ is used by all n+12 nodes in X∖λ. We consider the main result of the paper—Bayramyan and Hakopian (Adv. Comput. Math. 43, 607-626, 2017) stating that any n-node line of GCn set is used either by exactly n2 nodes or by exactly n−12 nodes, provided that the Gasca-Maeztu conjecture is true. Here, we show that this result is not correct in the case n = 3. Namely, we bring an example of a GC3 set and a 3-node line there which is not used at all. Fortunately, then we were able to establish that this is the only possible counterexample, i.e., the abovementioned result is true for all n ≥ 4. We also characterize the exclusive case n = 3 and present some new results on the maximal lines and the usage of n-node lines in GCn sets. [ABSTRACT FROM AUTHOR]

Let x0,x1, ⋯, xn, be a set of n + 1 distinct real numbers (i.e., xm ≠ xj, for m ≠ j) and let ym,k, for m = 0, 1, ⋯, n, and k = 0, 1, ⋯, rm, with rm ∈ IN, be given real numbers. It is known that there exists a unique polynomial pN− 1 of degree N − 1 with N=∑m=0n(rm+1), such that pN−1(k)(xm)=ym,k, for m = 0, 1, ⋯, n and k = 0, ⋯, rm. pN− 1 is the Hermite interpolation polynomial for the set {(xm, ym,k), m = 0, 1, ⋯, n, k = 0, 1, ⋯, rm}. The polynomial pN− 1 can be computed by using the Lagrange generalized polynomials. Recently, Messaoudi et al. (2017) presented a new algorithm for computing the Hermite interpolation polynomial called the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA), for a particular case where rm = μ = 1, for m = 0, 1, ⋯, n. In this paper, we will give a new formulation of the Hermite polynomial interpolation problem and derive a new algorithm, called the Generalized Recursive Polynomial Interpolation Algorithm (GRPIA), for computing the Hermite polynomial interpolation in the general case. A new result of the existence of the polynomial pN− 1 will also be established, cost and storage of this algorithm will also be studied, and some examples will be given. [ABSTRACT FROM AUTHOR]

The notion of coupled fixed point was introduced in 2006 by Bhaskar and Lakshmikantham. On the other hand, Filipovićet al. [M. Filipovićet al., 'Remarks on 'Cone metric spaces and fixed-point theorems of T-Kannan and T-Chatterjea contractive mappings',' Math. Comput. Modelling 54, 1467-1472 (2011)] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper we prove some coupled fixed-point theorems for certain T-contractions and study the existence of solutions of a system of nonlinear integral equations using the results of our work. The results of this paper extend and generalize well-known comparable results in the literature. [ABSTRACT FROM AUTHOR]

Permutation polynomials with low differential uniformity are important candidate functions to design substitution boxes of block ciphers. In this paper, we investigate several classes of differential 4-uniform binomial and trinomial permutation polynomials over the finite field of elements. [ABSTRACT FROM AUTHOR]

The concept of energy of a graph was first defined in 1978 by Gutman as the sum of the absolute values of the eigenvalues of its adjacency matrix. Let λ1, λ2, ..., λn be eigenvalues of graph Γ, then the Estrada index of Γ is defined as . The aim of this paper is to estimate the energy and Estrada index of Cayley graphs (G, S) where G ≅ D2n, U6n and S is a normal symmetric generating subset of G. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the number of limit cycles for two classes of discontinuous Liénard polynomial perturbed differential systems. By the second-order averaging theorem of discontinuous differential equations, we provide several criteria on the lower upper bounds for the maximum number of limit cycles. The results show that the second-order averaging theorem of discontinuous differential equations can predict more limit cycles than the first-order one. [ABSTRACT FROM AUTHOR]

By a famous result, the subgroup generated by the n-cycle σ=(1,2,…,n) and the transposition τ=(a,b) is the full symmetric group Sn if and only if gcd(n,b-a)=1. In this paper, we first generalize the above result for one n-cycle and k arbitrary transpositions, and then provide similar necessary and sufficient conditions for the subgroups of Sn in the following three cases: first, the subgroup generated by the n-cycle σ and a 3-cycle δ=(a,b,c), second, the subgroup generated by the n-cycle σ and a set of transpositions and 3-cycles, and third, by the n-cycle σ and an involution (a, b)(c, d). In the first case, we also determine the structure of the subgroup generated by (1,2,…,n) and a 3-cycle δ=(a,b,c) in general. Finally, an application to unsolvability of a certain infinite family of polynomials by radicals is given. [ABSTRACT FROM AUTHOR]

In this paper, we present and study the concepts of fuzzifying pre-θ-neighborhood system of a point, fuzzifying pre-θ-closure of a set, fuzzifying pre-θ-interior of a set, fuzzifying pre-θ-open sets and fuzzifying pre-θ-closed sets in fuzzifying topological spaces. The basic properties of these concepts are investigated. Two types of functions in a fuzzifying topological spaces called fuzzifying strongly pre-irresolute and fuzzifying weakly pre-irresolute functions are introduced. Then the interrelations of these functions with the parallel existing allied concepts are established. Finally, several characterizations of these functions along with different conditions for their existence are obtained. [ABSTRACT FROM AUTHOR]

We consider the Multilinear set S defined as the set of binary points (x, y) satisfying a collection of multilinear equations of the form yI=∏i∈Ixi, I∈I, where I denotes a family of subsets of {1,…,n} of cardinality at least two. Such sets appear in factorable reformulations of many types of nonconvex optimization problems, including binary polynomial optimization. A great simplification in studying the facial structure of the convex hull of the Multilinear set is possible when S is decomposable into simpler Multilinear sets Sj, j∈J; namely, the convex hull of S can be obtained by convexifying each Sj, separately. In this paper, we study the decomposability properties of Multilinear sets. Utilizing an equivalent hypergraph representation for Multilinear sets, we derive necessary and sufficient conditions under which S is decomposable into Sj, j∈J, based on the structure of pair-wise intersection hypergraphs. Our characterizations unify and extend the existing decomposability results for the Boolean quadric polytope. Finally, we propose a polynomial-time algorithm to optimally decompose a Multilinear set into simpler subsets. Our proposed algorithm can be easily incorporated in branch-and-cut based global solvers as a preprocessing step for cut generation. [ABSTRACT FROM AUTHOR]

The aim of this research paper is to establish a quite general transformation involving the generalized hypergeometric function. Extensions of Kummer's first transformation, Gauss and Kummer summation, and its contiguous results are then applied to obtain a new class of summation formulas involving the generalized hypergeometric 2F2 polynomial, which have not previously appeared in the literature. A few well-known results obtained earlier by Kim et al. (Int. J. Math. Math. Sci. 2010:309503, 2010; Integral Transforms Spec. Funct. 23(6):435-444, 2012) and Exton have been obtained as special cases of our main findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of optimal solutions to a constrained polynomial optimization problem. More precisely, let $$f_0$$ and $$f_1, \ldots , f_p :{\mathbb {R}}^n \rightarrow {\mathbb {R}}$$ be convenient polynomial functions, and let $$S := \{x \in {\mathbb {R}}^n \ : \ f_i(x) \le 0, i = 1, \ldots , p\} \ne \emptyset .$$ Under the assumption that the map $$(f_0, f_{1}, \ldots , f_{p}) :{\mathbb {R}}^n \rightarrow {\mathbb {R}}^{p + 1}$$ is non-degenerate at infinity, we show that if $$f_0$$ is bounded from below on $$S,$$ then $$f_0$$ attains its infimum on $$S.$$ [ABSTRACT FROM AUTHOR]

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