A note on sign symmetry for a subclass of efficient,symmetric, and linear values

Symmetry requires that equally productive players should get the same payoff. Sign symmetry is a considerable weakening of symmetry, which requires equally productive players' payoffs to have the same sign. In this note, we apply sign symmetry, as a substitute of symmetry to characterize a subclass of efficient, symmetric, and linear values.     

A note on extrema of linear combinations of elementary symmetric functions

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377–385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. Math. Anal. Appl. 15 (1966), pp. 269–272] and Eberlein [P.J. Eberlein, Remarks on the van der Waerden conjecture, II, Linear Algebra Appl. 2 (1969), pp. 311–320]. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof in [Foregger, 1987] is flawed.     

A note on simultaneous preconditioning and symmetrization of non‐symmetric linear systems

Motivated by the theory of self‐duality that provides a variational formulation and resolution for non‐self‐adjoint partial differential equations (Ann. Inst. Henri Poincaré (C) Anal Non Linéaire 2007; 24 :171–205; Selfdual Partial Differential Systems and Their Variational Principles. Springer: New York, 2008), we propose new templates for solving large non‐symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well‐known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill‐conditioned, and highly non‐symmetric systems. The numerical and theoretical results are provided to show the efficiency of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Note On linear consistency of anonymous values for TU-games

In the framework of values for TU-games, it is shown that a particular type of consistency, called linear consistency, together with some kind of standardness for two-person games, imply efficiency, anonymity, linearity, as well as uniqueness of the value. Among others, this uniform treatment generalizes Sobolev's axiomatization of the Shapley value. Revised version: December 2001     

A note on pseudo-invexity and symmetric duality

A pair of symmetric dual nonlinear programming problems is presented and duality theorems are established under pseudoinvexity type assumptions on the kernel function. This formulation removes certain inconsistencies in a recently introduced primal-dual pair and gives the correct proof of various duality theorems.     

A note on labeled symmetric graphs

By using the known results of groups and graphs, we determine the number of labeled symmetric graphs with a prime number of vertices.     

A note on symmetric doubly-stochastic matrices

Each extreme point in the convex set Δ¹_n of all n×n symmetric doubly-stochastic matrices is shown to have the form $\text{1}\text{2}\text{(P + P}{}^{\mathrm{t}}\text{)}$, for some n×n permutation matrix P. The convex hull Σ_n of the integral points in Δ¹_n (i.e., the symmetric permutation matrices) is shown to consist of those matrices, X = (x_ij) in Δ¹_n satisfying Σ_i∈SΣ_j∈S?{i}x_ij ? 2k, for each subset S of {1, 2, …, n} having cardinality 2k + 1, for some k > 0.     

A note on the eigenvalue consistency index

In this paper we show the relation between the eigenvalue consistency index and cycles of length k in a complete directed graph. Cardinal consistency, which requires the introduction of a scale to represent the degree of preferences among different alternatives, is discussed, and a new index to measure this concept is derived.     

A note on multiobjective second-order symmetric duality

In this paper, we establish a strong duality theorem for a pair of multiobjective second-order symmetric dual programs. This removes an omission in an earlier result by Yang et al. [X.M. Yang, X.Q. Yang, K.L. Teo, S.H. Hou, Multiobjective second-order symmetric duality with F-convexity, Euro. J. Oper. Res. 165 (2005) 585–591].     

A note on homogeneous locally symmetric spaces

Let denote an orthogonal symmetric Lie algbra and let (G, K) be an associated pair, i.e., Lie(G = and Lie(K°) = . In this paper we prove that the homogeneous spaceG/K has a structure of a globally symmetric space for every choice ofG andK, especially forG being compact.     

A note on a consistency property for permutations

In cooperative game theory allocation of earnings to players may take place on the basis of selectors or—more restrictively—consistent selectors, or on the basis of a permutation representing the queueing of the players. This note gives a graph theoretic characterization of those situations in which the latter allocation method results in allocation with consistent selectors.     

A note on multinomial probabilistic values

Multinomial values were previously introduced by one of the authors in reliability and extended later to all cooperative games. Here, we present for this subfamily of probabilistic values three new results, previously stated only for binomial semivalues in the literature. They concern the dimension of the subspace spanned by the multinomial values and two characterizations: one, individual, for each multinomial value; another, collective, for the whole subfamily they form. Finally, an application to simple games is provided.     

A note on Cartan matrices for symmetric groups

Using generating functions, a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.Received: 5 September 2001     

A note on skew characters of symmetric groups

In this paper we establish the following estimate:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$

where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log⁺(t)). This inequality relies upon the following sharp L ^p estimate:

$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$

where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]:

$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$

We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.

     

A note on the pivoted symmetric LR iteration

Summary. We prove that the diagonally pivoted symmetric LR algorithm on a positive definite matrix is globally convergent. Received December 23, 1997 / Revised version received August 3, 1998 / Published online August 19, 1999     

A note on refining eigenelements of symmetric matrices

By means of a Fixed Slope Inexact Newton Method we define an Iterative Refinement Process for approximating eigenvalues and eigenvectors of real symmetric matrices.     

A note on linear permutation polynomials

Kai Zhou (2008) [8] gave an explicit representation of the class of linear permutation polynomials and computed the number of them. In this paper, we give a simple proof of the above results.