Showing papers in "Open Mathematics in 2017"
TL;DR: The concept of learnheuristics is introduced, a novel type of hybrid algorithms used to solve combinatorial optimization problems with dynamic inputs (COPDIs) that require a coordination between the learning mechanism and the metaheuristic algorithm.
Abstract: Abstract This paper reviews the existing literature on the combination of metaheuristics with machine learning methods and then introduces the concept of learnheuristics, a novel type of hybrid algorithms. Learnheuristics can be used to solve combinatorial optimization problems with dynamic inputs (COPDIs). In these COPDIs, the problem inputs (elements either located in the objective function or in the constraints set) are not fixed in advance as usual. On the contrary, they might vary in a predictable (non-random) way as the solution is partially built according to some heuristic-based iterative process. For instance, a consumer’s willingness to spend on a specific product might change as the availability of this product decreases and its price rises. Thus, these inputs might take different values depending on the current solution configuration. These variations in the inputs might require from a coordination between the learning mechanism and the metaheuristic algorithm: at each iteration, the learning method updates the inputs model used by the metaheuristic.
110 citations
TL;DR: In this paper, the authors presented several new and generalized Hermite-Hadamard type inequalities for s-convex functions via classical and Riemann-Liouville fractional integrals.
Abstract: Abstract In this paper, we present several new and generalized Hermite-Hadamard type inequalities for s-convex as well as s-concave functions via classical and Riemann-Liouville fractional integrals. As applications, we provide new error estimations for the trapezoidal formula.
110 citations
TL;DR: In this article, Nieto et al. have partially supported by the Ministerio de Economia and Competitividad of Spain under grants MTM2016-75140 P, MTM2013-43014 P, Xunta de Galicia, Grants GRC2015-004 and R2016-022, and co-financed by the European Community fund FEDER
Abstract: The research of J.J. Nieto has been partially supported by the Ministerio de Economia
y Competitividad of Spain under grants MTM2016–75140–P, MTM2013–43014–P, Xunta de Galicia, Grants GRC2015-004 and R2016-022, and co- nanced by the European Community fund FEDER
38 citations
TL;DR: In this article, the authors used the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk (ψ), and gave an interesting fourth-order linear recurrence formula for it.
Abstract: Abstract Let p be an odd prime with p ≡ 1 mod 4, k be any positive integer, ψ be any fourth-order character mod p. In this paper, we use the analytic method and the properties of character sums mod p to study the computational problem of G(k, p) = τk(ψ)+τk(ψ), and give an interesting fourth-order linear recurrence formula for it, where τ(ψ) denotes the classical Gauss sums.
34 citations
TL;DR: It is shown that the strong edge geodetic problem is NP-complete and derive lower and upper bounds for the strong edges of this problem and demonstrate that these bounds are sharp.
Abstract: Geodesic covering problems form a widely researched topic in graph theory. One such problem is geodetic problem introduced by Harary et al. Here we introduce a variation of the geodetic problem and call it strong edge geodetic problem. We illustrate how this problem is evolved from social transport networks. It is shown that the strong edge geodetic problem is NP-complete. We derive lower and upper bounds for the strong edge geodetic number and demonstrate that these bounds are sharp. We produce exact solutions for trees, block graphs, silicate networks and glued binary trees without randomization.
32 citations
TL;DR: In this paper, the description of the Leibniz algebras whose subalgebra is an ideal is given, which is the same as the description given in this paper.
Abstract: Abstract In this paper we obtain the description of the Leibniz algebras whose subalgebras are ideals.
25 citations
TL;DR: A novel calibration procedure is provided that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods and improves the QE scheme by adapting the antithetic variates technique for variance reduction.
Abstract: Abstract We calibrate Heston stochastic volatility model to real market data using several optimization techniques. We compare both global and local optimizers for different weights showing remarkable differences even for data (DAX options) from two consecutive days. We provide a novel calibration procedure that incorporates the usage of approximation formula and outperforms significantly other existing calibration methods. We test and compare several simulation schemes using the parameters obtained by calibration to real market data. Next to the known schemes (log-Euler, Milstein, QE, Exact scheme, IJK) we introduce also a new method combining the Exact approach and Milstein (E+M) scheme. Test is carried out by pricing European call options by Monte Carlo method. Presented comparisons give an empirical evidence and recommendations what methods should and should not be used and why. We further improve the QE scheme by adapting the antithetic variates technique for variance reduction.
22 citations
TL;DR: In this article, the existence of positive solutions for Hadamard type fractional differential systems with coupled nonlocal fractional integral boundary conditions on an infinite domain was investigated, based on Guo-Krasnoselskii's and Leggett-Williams fixed point theorem.
Abstract: Abstract In this paper, we investigate the existence of positive solutions for Hadamard type fractional differential system with coupled nonlocal fractional integral boundary conditions on an infinite domain. Our analysis relies on Guo-Krasnoselskii’s and Leggett-Williams fixed point theorems. The obtained results are well illustrated with the aid of examples.
21 citations
TL;DR: The simulation results conclude that the proposed nature–inspired metaheuristic optimization algorithms are superior to the existing conventional and nature-inspired algorithms to find near–OGRs in terms of ruler length, total optical channel bandwidth, computation time, and computational complexity.
Abstract: Abstract Nowadays, nature–inspired metaheuristic algorithms are most powerful optimizing algorithms for solving the NP–complete problems. This paper proposes three approaches to find near–optimal Golomb ruler sequences based on nature–inspired algorithms in a reasonable time. The optimal Golomb ruler (OGR) sequences found their application in channel–allocation method that allows suppression of the crosstalk due to four–wave mixing in optical wavelength division multiplexing systems. The simulation results conclude that the proposed nature–inspired metaheuristic optimization algorithms are superior to the existing conventional and nature–inspired algorithms to find near–OGRs in terms of ruler length, total optical channel bandwidth, computation time, and computational complexity. Based on the simulation results, the performance of proposed different nature–inspired metaheuristic algorithms are being compared by using statistical tests. The statistical test results conclude the superiority of the proposed nature–inspired optimization algorithms.
21 citations
TL;DR: In this paper, it was shown that the homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable has only trivial solution in the case when the characteristic determinant of the problem is nonzero.
Abstract: Abstract We prove that homogeneous problem for PDE of second order in time variable, and generally infinite order in spatial variables with local two-point conditions with respect to time variable, has only trivial solution in the case when the characteristic determinant of the problem is nonzero. In another, opposite case, we prove the existence of nontrivial solutions of the problem, and we propose a differential-symbol method of constructing them.
21 citations
TL;DR: In this article, the authors used the analytic method and properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss and Kloosterman sums, and gave an exact computational formula for it.
Abstract: Abstract The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind fourth hybrid power mean of the quartic Gauss sums and Kloosterman sums, and give an exact computational formula for it.
TL;DR: In this paper, the rank and inertia formulas for covariance matrices related to covariance matrix of predictors and estimators of parameter spaces in general linear models (GLMs) were derived and used in statistical analysis of GLMs.
Abstract: Abstract Matrix mathematics provides a powerful tool set for addressing statistical problems, in particular, the theory of matrix ranks and inertias has been developed as effective methodology of simplifying various complicated matrix expressions, and establishing equalities and inequalities occurred in statistical analysis. This paper describes how to establish exact formulas for calculating ranks and inertias of covariances of predictors and estimators of parameter spaces in general linear models (GLMs), and how to use the formulas in statistical analysis of GLMs. We first derive analytical expressions of best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) of all unknown parameters in the model by solving a constrained quadratic matrix-valued function optimization problem, and present some well-known results on ordinary least-squares predictors/ordinary least-squares estimators (OLSPs/OLSEs). We then establish some fundamental rank and inertia formulas for covariance matrices related to BLUPs/BLUEs and OLSPs/OLSEs, and use the formulas to characterize a variety of equalities and inequalities for covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. As applications, we use these equalities and inequalities in the comparison of the covariance matrices of BLUPs/BLUEs and OLSPs/OLSEs. The work on the formulations of BLUPs/BLUEs and OLSPs/OLSEs, and their covariance matrices under GLMs provides direct access, as a standard example, to a very simple algebraic treatment of predictors and estimators in linear regression analysis, which leads a deep insight into the linear nature of GLMs and gives an efficient way of summarizing the results.
TL;DR: In this paper, the smallest strongly regular relation defined on a general hyperring R is defined, such that the quotient R/ εm∗ $\\varepsilon^{*}_{m} $ is a ring.
Abstract: Abstract The γ*-relation defined on a general hyperring R is the smallest strongly regular relation such that the quotient R/γ* is a ring. In this note we consider a particular class of hyperrings, where we define a new equivalence, called εm∗ $\\varepsilon^{*}_{m} $, smaller than γ* and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R/ εm∗ $\\varepsilon^{*}_{m} $ is a ring. Moreover, we introduce the concept of m-idempotent hyperrings, show that they are a characterization for Krasner hyperfields, and that εm∗ $\\varepsilon^{*}_{m} $ is a new exhibition for γ* on the above mentioned subclass of m-idempotent hyperrings.
TL;DR: In this article, it was shown that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsusu structure.
Abstract: Abstract Let (M3, g) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either ℍ3(−1) or ℍ2(−4) × ℝ.
TL;DR: In this paper, the (1-4)-problem for monogenic ǫ-valued functions is reduced to a system of integral equations on the real axes, and sufficient conditions under which this system has the Fredholm property and the unique solution are established.
Abstract: Abstract We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions (e12+e22)2=0,e12+e22≠0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\
eq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = {xe1 + ye2 : (x, y) ∈ D}: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when values of two component-functions U1, U4 are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions.
TL;DR: In this article, a translation surface in G3 with a log-linear density and a surface with vanishing weighted mean curvature was studied and a classification of the surface was made based on the isotropic and non-isotropic plane curves.
Abstract: Abstract Translation surfaces in the Galilean 3-space G3 have two types according to the isotropic and non-isotropic plane curves. In this paper, we study a translation surface in G3 with a log-linear density and classify such a surface with vanishing weighted mean curvature.
TL;DR: In this paper, the first, third and fourth authors were supported by MINECO and FEDER, grant MTM2016-75963-P. The second author has been partially supported by CONICYT under FONDECYT grant number 1140258.
Abstract: The first, third and fourth authors were supported by MINECO and FEDER, grant MTM2016-75963-P. The second author has been partially supported by CONICYT under FONDECYT grant number 1140258 and CONICYT-PIA-Anillo ACT1416.
TL;DR: In this paper, the authors prove uniqueness theorems of meromorphic functions, which show how two meromorphomorphic functions are uniquely determined by their two finite shared sets, and some examples are provided to demonstrate that all the conditions are necessary.
Abstract: Abstract We prove uniqueness theorems of meromorphic functions, which show how two meromorphic functions are uniquely determined by their two finite shared sets. This answers a question posed by Gross. Moreover, some examples are provided to demonstrate that all the conditions are necessary.
TL;DR: Prodinger and Prodinger as mentioned in this paper represented derivatives of Chebyshev polynomials by Chebyshaev polynomorphisms and related questions, including related questions.
Abstract: CITATION: Prodinger, H. 2017. Representing derivatives of Chebyshev polynomials by Chebyshev polynomials and related questions. Open Mathematics, 15(1):1156-1160, doi:10.1515/math-2017-0096.
TL;DR: In this paper, the authors used modular forms to evaluate the number of representations of a positive integer by the octonary quadratic forms for all natural numbers n. Since the modular space of level 22 is contained in that of level 44, they almost completely use the basis elements of the modular spaces of the level 44 to carry out the evaluation of the convolution sums for αβ = 22.
Abstract: Abstract The convolution sum, ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $ \\begin{array}{} \\sum\\limits_{{(l\\, ,m)\\in \\mathbb{N}_{0}^{2}}\\atop{\\alpha \\,l+\\beta\\, m=n}} \\sigma(l)\\sigma(m), \\end{array} $ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).
TL;DR: In this article, the authors consider the problem of establishing additive matrix decompositions of estimators in the context of a general linear model with partial parameter restrictions, and demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear models (CGLM) as the sums of the estimators under submodels with parameter restrictions.
Abstract: Abstract A general linear model can be given in certain multiple partitioned forms, and there exist submodels associated with the given full model. In this situation, we can make statistical inferences from the full model and submodels, respectively. It has been realized that there do exist links between inference results obtained from the full model and its submodels, and thus it would be of interest to establish certain links among estimators of parameter spaces under these models. In this approach the methodology of additive matrix decompositions plays an important role to obtain satisfactory conclusions. In this paper, we consider the problem of establishing additive decompositions of estimators in the context of a general linear model with partial parameter restrictions. We will demonstrate how to decompose best linear unbiased estimators (BLUEs) under the constrained general linear model (CGLM) as the sums of estimators under submodels with parameter restrictions by using a variety of effective tools in matrix analysis. The derivation of our main results is based on heavy algebraic operations of the given matrices and their generalized inverses in the CGLM, while the whole contributions illustrate various skillful uses of state-of-the-art matrix analysis techniques in the statistical inference of linear regression models.
TL;DR: In this paper, the graded comultiplication modules over a commutative graded ring were investigated and the authors obtained some results concerning the graded COMPLIATION modules over the graded ring.
Abstract: Abstract Let G be a group with identity e. Let R be a G-graded commutative ring and M a graded R-module. In this paper we will obtain some results concerning the graded comultiplication modules over a commutative graded ring.
TL;DR: In this paper, the authors give a complete description of finite groups with enhanced power graphs admitting a perfect code, and describe all groups in the following two classes of infinite groups: the class of groups with power graphs with a total perfect code (i.e., two distinct elements being adjacent if one is a power of the other).
Abstract: Abstract The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.
TL;DR: In this paper, the quadratic matrix equation AXA = XAX was solved for any given matrix A of rank-two, where A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular.
Abstract: Abstract Let A = PQT, where P and Q are two n × 2 complex matrices of full column rank such that QTP is singular. We solve the quadratic matrix equation AXA = XAX. Together with a previous paper devoted to the case that QTP is nonsingular, we have completely solved the matrix equation with any given matrix A of rank-two.
TL;DR: In this article, it was shown that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number.
Abstract: Abstract In this paper, we prove that a bounded double sequence of fuzzy numbers which is statistically convergent is also statistically (C, 1, 1) summable to the same number. We construct an example that the converse of this statement is not true in general. We obtain that the statistically (C, 1, 1) summable double sequence of fuzzy numbers is convergent and statistically convergent to the same number under the slowly oscillating and statistically slowly oscillating conditions in certain senses, respectively.
TL;DR: In this paper, the existence of solution of a quasilinear generalized Kirchhoff equation with initial boundary conditions of Dirichlet type was studied using the Leray- Schauder principle.
Abstract: Abstract In this paper the existence of solution of a quasilinear generalized Kirchhoff equation with initial – boundary conditions of Dirichlet type will be studied using the Leray – Schauder principle.
TL;DR: In this article, the second-order nonlinear discrete Robin boundary value problem with parameter dependence was studied and sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions were established.
Abstract: Abstract In this paper, we study second-order nonlinear discrete Robin boundary value problem with parameter dependence. Applying invariant sets of descending flow and variational methods, we establish some new sufficient conditions on the existence of sign-changing solutions, positive solutions and negative solutions of the system when the parameter belongs to appropriate intervals. In addition, an example is given to illustrate our results.
TL;DR: In this paper, the oscillation constant for half-linear Euler type differential equations with α-periodic positive coefficients is computed explicitly, and an example and corollaries which illustrate cases that are solved with their result are given.
Abstract: Abstract In this paper, we compute explicitly the oscillation constant for certain half-linear second-order differential equations having different periodic coefficients. Our result covers known result concerning half-linear Euler type differential equations with α—periodic positive coefficients. Additionally, our result is new and original in case that the least common multiple of these periods is not defined. We give an example and corollaries which illustrate cases that are solved with our result.
TL;DR: In this paper, the notions of Jordan homomorphism and Jordan derivation of inverse semirings are introduced, and a few results of Herstein and Brešar on Jordan homomorphic properties and derivations of rings are generalized in the setting of inverse semiirings.
Abstract: Abstract In this paper, the notions of Jordan homomorphism and Jordan derivation of inverse semirings are introduced. A few results of Herstein and Brešar on Jordan homomorphisms and Jordan derivations of rings are generalized in the setting of inverse semirings.