Positive definite and Gram tensor complementarity problems

Given a continuous function Open image in new window and Open image in new window , the non-linear complementarity problem \(\text{ NCP }(g,q)\) is to find a vector Open image in new window such that

$$\begin{aligned} x \ge 0,~~y:=g(x) +q\ge 0~~\text{ and }~~x^Ty=0. \end{aligned}$$

We say that g has the Globally Uniquely Solvable (\(\text{ GUS }\))-property if \(\text{ NCP }(g,q)\) has a unique solution for all Open image in new window and C-property if \(\mathrm{NCP}(g,q)\) has a convex solution set for all Open image in new window . In this paper, we find a class of non-linear functions that have the \(\text{ GUS }\)-property and C-property. These functions are constructed by some special tensors which are positive semidefinite. We call these tensors as Gram tensors.

     

Preserving preference rankings under non-financial background risk

We investigate the impact of a non-financial background risk ??? on thepreference rankings between two independent financial risks Open image in new window ₁ and Open image in new window ₂ for anexpected-utility maximizer. More precisely, we provide necessary and sufficientconditions for the alternative (x₀+ Open image in new window ₁,y₀+ ???) to be preferred to(x₀+ Open image in new window ₂,y₀+ ???)whenever (x₀+ Open image in new window ₁,y₀) ispreferred to (x₀+ Open image in new window ₂,y₀). Utilityfunctions that preserve the preference rankings are fully characterized. Theirpractical relevance is discussed in light of recent results on the constraintsfor the modelling of the preference for the disaggregation of harms.     

Monitoring the process mean based on attribute inspection when a small sample is available

This paper proposes a new control chart, denoted by Open image in new window to evaluate the stability of a process mean when a small sample is available. This chart is based on attribute inspection rather than the physical measurements (taken with an instrument, such as a caliper or precision balance) of the quality characteristics of interest of the sampled items. The main goal is to recover measurements on a continuous scale by generating random measurements using the frequencies observed for the sample as inputs. The average sample obtained using these recovery measures ( Open image in new window ) is calculated and used to draw the standard Open image in new window chart. The average sample Open image in new window can be shown to be a mixture of normal distributions. The values of the lower control limit (LCL) and the upper control limit (UCL) are chosen to minimize the average run length (ARL).     

Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into

Let Open image in new window denote a weight in Open image in new window which belongs to the Muckenhoupt class Open image in new window and let Open image in new window denote the uncentered Hardy–Littlewood maximal operator defined with respect to the measure Open image in new window . The sharp Tauberian constant of Open image in new window with respect to Open image in new window , denoted by Open image in new window , is defined by

Open image in new window

In this paper, we show that the Solyanik estimate

$$\begin{aligned} \lim _{\alpha \rightarrow 1^-}\mathsf{C}_{w}(\alpha ) = 1 \end{aligned}$$

holds. Following the classical theme of weighted norm inequalities we also consider the sharp Tauberian constants defined with respect to the usual uncentered Hardy–Littlewood maximal operator Open image in new window and a weight Open image in new window :

Open image in new window

We show that we have Open image in new window if and only if Open image in new window . As a corollary of our methods we obtain a quantitative embedding of Open image in new window into Open image in new window .

     

Symmetric quiver Hecke algebras and <Emphasis Type="Italic">R</Emphasis>-matrices of quantum affine algebras IV

Let \(U'_q(\mathfrak {g})\) be a twisted affine quantum group of type \(A_{N}^{(2)}\) or \(D_{N}^{(2)}\) and let \(\mathfrak {g}_{0}\) be the finite-dimensional simple Lie algebra of type \(A_{N}\) or \(D_{N}\). For a Dynkin quiver of type \(\mathfrak {g}_{0}\), we define a full subcategory \({\mathcal C}_{Q}^{(2)}\) of the category of finite-dimensional integrable \(U'_q(\mathfrak {g})\)-modules, a twisted version of the category \({\mathcal C}^{(1)}_{Q}\) introduced by Hernandez and Leclerc. Applying the general scheme of affine Schur–Weyl duality, we construct an exact faithful KLR-type duality functor \({\mathcal F}_{Q}^{(2)}:\mathrm{Rep}(R) \rightarrow {\mathcal C}_{Q}^{(2)}\), where \(\mathrm{Rep}(R)\) is the category of finite-dimensional modules over the quiver Hecke algebra R of type \(\mathfrak {g}_{0}\) with nilpotent actions of the generators \(x_k\). We show that \({\mathcal F}_{Q}^{(2)}\) sends any simple object to a simple object and induces a ring isomorphism Open image in new window .     

Power boundedness in the Fourier and Fourier–Stieltjes algebras on homogeneous spaces

We study power boundedness in the Fourier and Fourier–Stieltjes algebras, Open image in new window and Open image in new window of a homogeneous space Open image in new window The main results characterizes when all elements with spectral radius at most one, in any of these algebras, are power bounded.     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

A weird relation between two cardinals

For a set M, let \({\text {seq}}(M)\) denote the set of all finite sequences which can be formed with elements of M, and let \([M]^2\) denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \(\textsf {ZF}\): There exists a set M such that Open image in new window and no function Open image in new window is finite-to-one.     

A duality theorem for Tate–Shafarevich groups of curves over algebraically closed fields

In this note, we prove a duality theorem for the Tate–Shafarevich group of a finite discrete Galois module over the function field K of a curve over an algebraically closed field: there is a perfect duality of finite groups Open image in new window for F a finite étale Galois module on K of order invertible in K and with \(F' = {{\mathrm{Hom}}}(F,\mathbf{Q}/\mathbf {Z}(1))\). Furthermore, we prove that \(\mathrm {H}^1(K,G) = 0\) for G a simply connected, quasisplit semisimple group over K not of type \(E_8\).     

Invariant and stationary measures for the action on Moduli space

We prove some ergodic-theoretic rigidity properties of the action of Open image in new window on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of Open image in new window is supported on an invariant affine submanifold.The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.     

Nonseparable closed vector subspaces of separable topological vector spaces

In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space Open image in new window have a nonseparable closed vector subspace, where \(\hbox {c}\) is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space Open image in new window has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line \(\mathbb M\) the space of all continuous real-valued functions on \(\mathbb M\) endowed with the pointwise convergence topology, \(C_p(\mathbb M)\) contains a nonseparable closed vector subspace while \(C_p(\mathbb M)\) is separable.     

A WIN-WIN approach to multiple objective linear programming problems

Interactive decision making arose as a means to overcome the uncertainty concerning the decision maker's (DM) value function. So far the search is confined to nondominated alternatives, which assumes that a win–lose strategy is adopted. The purpose of this paper is to suggest a complementary interactive algorithm that uses an interior point method to solve multiple objective linear programming problems. As the algorithm proceeds, the DM has access to intermediate solutions. The sequence of intermediate solutions has a very interesting characteristic: all of the criteria are improved, that is, a solution Open image in new window , that follows another solution Open image in new window , has the values of all objectives greater than those of Open image in new window . This WIN-WIN feature allows the DM to reach a nondominated solution without making any trade-off among the objective functions. However, there is no impediment in proceeding with traditional multiobjective methods.     

<Emphasis Type="Italic">p</Emphasis>-Saturations of Welter’s game and the irreducible representations of symmetric groups

We establish a relation between the Sprague–Grundy function Open image in new window of p-saturations of Welter’s game and the degrees of the ordinary irreducible representations of symmetric groups. In these games, a position can be regarded as a partition \(\lambda \). Let \(\rho ^\lambda \) be the irreducible representation of the symmetric group \(\mathrm{Sym}(\left| \lambda \right| )\) corresponding to \(\lambda \). For every prime p, we show the following results: (1) \(\mathrm{sg}(\lambda ) \le \left| \lambda \right| \) with equality if and only if the degree of \(\rho ^\lambda \) is prime to p; (2) the restriction of \(\rho ^\lambda \) to \(\mathrm{Sym}(\mathrm{sg}(\lambda ))\) has an irreducible component with degree prime to p. Further, for every integer p greater than 1, we obtain an explicit formula for \(\mathrm{sg}(\lambda )\).     

Unbounded norm convergence in Banach lattices

A net \((x_\alpha )\) in a vector lattice X is unbounded order convergent to \(x \in X\) if \(|x_\alpha - x| \wedge u\) converges to 0 in order for all \(u\in X_+\). This convergence has been investigated and applied in several recent papers by Gao et al. It may be viewed as a generalization of almost everywhere convergence to general vector lattices. In this paper, we study a variation of this convergence for Banach lattices. A net \((x_\alpha )\) in a Banach lattice X is unbounded norm convergent to x if Open image in new window for all \(u\in X_+\). We show that this convergence may be viewed as a generalization of convergence in measure. We also investigate its relationship with other convergences.     

On Shalika germs

Let G be (the group of F-points of) a reductive group over a local field F satisfying the assumptions of Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002), sections 2.2, 3.2, 4.3. Let \(G_{{\text {reg}}}\subset G\) be the subset of regular elements. Let \(T\subset G\) be a maximal torus. We write \(T_{{\text {reg}}}=T\cap G_{{\text {reg}}}\). Let dg, dt be Haar measures on G and T. They define an invariant measure Open image in new window on Open image in new window . Let \(\mathcal {H}\) be the space of complex valued locally constant functions on G with compact support. For any \(f\in \mathcal {H}\), \(t\in T_{{\text {reg}}}\), we put \(I_t(f)=\int _{G/T}f(\dot{g}t\dot{g}^{-1})dg/dt\). Let \(\mathcal U\) be the set of conjugacy classes of unipotent elements in G. For any \(\Omega \in \mathcal U\) we fix an invariant measure \(\omega \) on \(\Omega \). It is well known—see, e.g., Rao (Ann Math 96:505-510, 1972)—that for any \(f\in \mathcal {H}\) the integral

$$\begin{aligned} I_\Omega (f)=\int _\Omega f\omega \end{aligned}$$

is absolutely convergent. Shalika (Ann Math 95:226–242, 1972) showed that there exist functions \(j_\Omega (t)\), \(\Omega \in \mathcal U\), on \(T\cap G_{{\text {reg}}}\), such that

$$\begin{aligned} I_t(f)=\sum _{\Omega \in \mathcal U}j_\Omega (t)I_\Omega (f) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad ({\star }) \end{aligned}$$

for any \(f\in \mathcal {H}\), \(t\in T\) near to e, where the notion of near depends on f. For any \(r\ge 0\) we define an open \({\text {Ad}}(G)\)-invariant subset \(G_r\) of G, and a subspace \(\mathcal {H}_r\) of \(\mathcal {H}\), as in Debacker (Ann Sci Ecole Norm Sup 35(4):391–422, 2002). Here I show that for any \(f\in \mathcal {H}_r\) the equality \((\star )\) holds for all \(t\in T_{{\text {reg}}}\cap G_r\).

     

Some remarks on the ring

Some properties like factoriality, seminormality and being a Krull domain, … are studied on power series rings , and over a commutative ring A. If \(\mathbb{X}\) is an uncountable set, there is an other sub-ring of that stands strictly between and , we denote it by . In this paper, we study properties mentioned before on the ring .     

A comparison of several enumerative algorithms for Sudoku

Sudoku is a puzzle played of an n × n grid Open image in new window where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, …, n} in each row and each column of Open image in new window as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02?s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm.     

The Nevo–Zimmer intermediate factor theorem over local fields

The Nevo–Zimmer theorem classifies the possible intermediate G-factors Y in Open image in new window , where G is a higher rank semisimple Lie group, P a minimal parabolic and X an irreducible G-space with an invariant probability measure. An important corollary is the Stuck–Zimmer theorem, which states that a faithful irreducible action of a higher rank Kazhdan semisimple Lie group with an invariant probability measure is either transitive or free, up to a null set. We present a different proof of the first theorem, that allows us to extend these two well-known theorems to linear groups over arbitrary local fields.     

Global existence of small solutions for the fourth-order nonlinear Schrödinger equation

We consider the Cauchy problem for the fourth-order nonlinear Schrödinger equation

Open image in new window

where \(n=1,2\). We prove global existence of small solutions under the growth condition of \(f\left( u\right) \) satisfying \(\left| \partial _{u}^{j}f\left( u\right) \right| \le C\left| u\right| ^{p-j},\) where \(p>1+\frac{4}{n},0\le j\le 3\).

     

MIP models for minimizing total tardiness in a two-machine flow shop

We propose compact mixed-integer programming models for the Open image in new window -hard problem of minimizing tardiness in a two-machine flow shop. Also, we propose valid inequalities that aim at tightening the models’ representations. We provide empirical evidence that the linear programming relaxation of an enhanced formulation yields an excellent lower bound that consistently outperforms the best bound from the literature. We further provide the results of extensive computational experiments that attest to the efficacy of the proposed MIP models. In particular, our computational study reveals that most of the 30-job hard instances are optimally solved using the proposed MIP models. Furthermore, we found that even much larger instances, with up to 70 jobs, can be solved for several problem classes.