The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Quasisymmetric geometry of the Julia sets of McMullen maps

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z^m + λ/z^?, where λ ∈ ? {0} and ? and m are positive integers satisfying 1/?+1/m < 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet (which is also standard in some sense). If the free critical points are not escaped, we give a suffcient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

Inclusion results on statistical cluster points

We study the concepts of statistical cluster points and statistical core of a sequence for A _λ methods defined by deleting some rows from a nonnegative regular matrix A. We also relate A _λ-statistical convergence to A _μ-statistical convergence. Finally we give a consistency theorem for A-statistical convergence and deduce a core equality result.     

Lebesgue decomposition in action via semidefinite relaxations

Given all (finite) moments of two measures μ and λ on \(\mathbb {R}^{n}\), we provide a numerical scheme to obtain the Lebesgue decomposition μ = ν + ψ with ν?λ and ψ ⊥ λ. When ν has a density in \(L_{\infty }(\lambda )\) then we obtain two sequences of finite moments vectors of increasing size (the number of moments) which converge to the moments of ν and ψ respectively, as the number of moments increases. Importantly, no à priori knowledge on the supports of μ,ν and ψ is required.     

A computational study of hybrid approaches of metaheuristic algorithms for the cell formation problem

In this paper we solve the 0–1 cell formation problem where the number of cells is fixed a priori and where the objective is to maximize the overall efficiency of a production system by grouping together machines providing service to similar parts into a subsystem (denoted cell). Three different methods are introduced and compared numerically. The first local search method is an implementation of simulated annealing (SA) where the definition of the neighbourhood is specific to the application and requires using a diversification and intensification strategies. The second local search method is an adaptive simulated annealing method where the neighbourhood is selected randomly at each iteration. The procedure is adaptive in the sense that the probability of selecting a neighbourhood is updated during the process. The third method is a hybrid method (HM) of a population-based method and a local search method. To improve the solution obtained with HM, we apply a SA method afterward. The best variants are very efficient to solve the 35 benchmark problems commonly used in the literature.     

Transmission of Waves Through a Small Aperture in the Cross-Wall in an Acoustic Waveguide

We study wave diffraction at near-threshold frequencies in an acoustic waveguide with a cross-wall that has a small aperture of diameter ε > 0. We describe the effects of almost complete reflection or transmission of waves related to the classical Vainstein anomaly and the presence of almost standing waves for the threshold value Λ _k of the spectral parameter λ in continuous spectrum. The greatest attention is paid to analyzing the range λ ^ε = Λ _k + ε²μ² of the spectral parameter with μ ≤ μ₀, which generates scattering coefficients depending on μ > 0 and presents the greatest difficulties in constructing and justifying the asymptotics. Almost complete reflection and transmission correspond to the cases of going away from the threshold (as μ → +∞) and approaching it (as μ → +0) characterized by simpler asymptotics.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).     

On Defining Sets of Full Designs with Block Size Three

A defining set of a t-(v, k, λ) design is a subcollection of its blocks which is contained in no other t-design with the given parameters, on the same point set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set {|M| | M is a minimal defining set of D}. We show that if a t-(v, k, λ) design D is contained in a design F, then for every minimal defining set d _D of D there exists a minimal defining set d _F of F such that \({d_D = d_F\cap D}\). The unique simple design with parameters \({{\left(v,k, {v-2\choose k-2}\right)}}\) is said to be the full design on v elements; it comprises all possible k-tuples on a v set. Every simple t-(v, k, λ) design is contained in a full design, so studying minimal defining sets of full designs gives valuable information about the minimal defining sets of all t-(v, k, λ) designs. This paper studies the minimal defining sets of full designs when t = 2 and k = 3. Several families of non-isomorphic minimal defining sets of these designs are found. For given v, a lower bound on the size of the smallest and an upper bound on the size of the largest minimal defining set are given. The existence of a continuous section of the spectrum comprising approximately v values is shown, where just two values were known previously.     

On the <Emphasis Type="Italic">σ</Emphasis>-countable compactness of spaces of continuous functions with the set-open topology

For a Tychonoff space X, we obtain a criterion of the σ-countable compactness of the space of continuous functions C(X) with the set-open topology. In particular, for the class of extremally disconnected spaces X, we prove that the space C _λ(X) is σ-countably compact if and only if X is a pseudocompact space, the set X(P) of all P-points of the space X is dense in X, and the family λ consists of finite subsets of the set X(P).     

Strongly regular locally <Emphasis Type="Italic">GQ</Emphasis>(4,<Emphasis Type="Italic">t</Emphasis>)-graphs

Amply regular with parameters (v, k, λ, μ) we call an undirected graph with v vertices in which the degrees of all vertices are equal to k, every edge belongs to λ triangles, and the intersection of the neighborhoods of every pair of vertices at distance 2 contains exactly μ vertices. An amply regular diameter 2 graph is called strongly regular. We prove the nonexistence of amply regular locally GQ(4,t)-graphs with (t,μ) = (4, 10) and (8, 30). This reduces the classification problem for strongly regular locally GQ(4,t)-graphs to studying locally GQ(4, 6)-graphs with parameters (726, 125, 28, 20).     

Subgraph ejection chains and tabu search for the crew scheduling problem

The tabu search algorithms for the Crew Scheduling Problem (CSP) reported in this paper are part of a decision support system for crew scheduling management of the Lisbon Underground. The CPS is formulated as the minimum number of duties necessary to cover a pre-defined timetable under a set of contractual rules. An initial solution is constructed following a traditional run-cutting approach. Two alternative improvement algorithms are subsequently used to reduce the number of duties in the initial solution. Both algorithms are embedded in a tabu search framework: Tabu-crew takes advantage of a form of strategic oscillation for the neighbourhood search while the run-ejection algorithm considers compound moves based on a subgraph ejection chain method. Computational results are reported for a set of real problems.     

Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues λ _mn (b, c) and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters b and c. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain b _s and c _s are second-order branch points for λ _mn (b, c) with different indices n ₁ and n ₂, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite-Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points b _s and c _s and the double eigenvalues to high accuracy. A large number of these singular points are calculated.     

Bose–Einstein distribution as a problem of analytic number theory: The case of less than two degrees of freedom

The problem of finding the number and the most likely shape of solutions of the system \(\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N\), where λ_j,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λ_j is of the order of λ^d/2, where d, the number of degrees of freedom, is less than two.     

Permutation resolutions for Specht modules

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M ^μ for the symmetric group \(\mathfrak{S}_{r}\) with Young subgroup stabilizers \(\mathfrak{S}_{\mu}\). The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S ^λ . We prove the exactness in special cases.     

Ultrafilter semigroups generated by direct sums

Let λ be an infinite cardinal and for every ordinal α<λ, let A _α be a set with a distinguished element 0 _α ∈A _α . The direct sum of sets A _α , α<λ, is the subset \(X=\bigoplus_{\alpha<\lambda}A_{\alpha}\) of the Cartesian product ∏_α<λ A _α consisting of all x with finite supp?(x)={α<λ:x(α)≠0 _α }. Endow X with a topology by taking as a neighborhood base at x∈X the subsets of the form {y∈X:y(α)=x(α) for all α<γ} where γ<λ. Let Ult?(X) denote the set of all nonprincipal ultrafilters on X converging to 0∈X. There is a natural partial semigroup operation on X which induces a semigroup operation on Ult?(X). We show that if direct sums X and Y are homeomorphic, then the semigroups Ult?(X) and Ult?(Y) are isomorphic.     

The size of the set of <Emphasis Type="Italic">μ</Emphasis>-irregular points of a measure <Emphasis Type="Italic">μ</Emphasis>

Let μ be a compactly suppported positive measure on the real line. A point x∈supp?[μ] is said to be μ-regular, if, as n→∞,

$\sup_{\deg\, (P) \le n}\left(\frac{|P(x)|}{{\|P\|}_{L_{2}(d\mu)}}\right )^{1/n}\to1.$

Otherwise it is a μ-irregular point. We show that for any such measure, the set of μ-irregular points in {μ′>0} (with a suitable definition of this set) has Hausdorff \(m_{h_{\beta}}\) measure 0, for \(h_{\beta}(t) =\left(\log \frac{1}{t}\right)^{-\beta}\), any β>1.

     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

Greedy random adaptive memory programming search for the capacitated clustering problem

In the capacitated clustering problem (CCP), a given set of n weighted points is to be partitioned into p clusters such that, the total weight of the points in each cluster does not exceed a given cluster capacity. The objective is to find a set of p centers that minimises the total scatter of points allocated to these centers. In this paper, we propose a merger of Greedy Random Adaptive Search Procedure (GRASP) and Adaptive Memory Programming (AMP) into a new GRAMPS framework for the CCP. A learning process is kept in charge of tracking information on the best components in an elite set of GRAMPS solutions. The information are strategically combined with problem-domain data to restart the construction search phase. At early stage of constructions, priorities are given to problem-domain data and progressively shifted towards generated information as the learning increases. GRAMPS is implemented with an efficient local search descent based on a restricted λ-interchange neighbourhood. Extensive experiments are reported on on a standard set of bench-marks from the literature and on a new set of large instances. The results show that GRAMPS has an efficient learning mechanism and is competitive with the existing methods in the literature.