Moments and Projections of Semistable Probability Measures on p-adic Vector Spaces

In this paper, two topics on semistable probability measures on p-adic vector spaces are studied. One is the existence of absolute moments of operator-semistable probability measures and another is an answer to the question whether one can get semistability of a probability measure from that of all its projections. All results obtained here are extensions of known results for real vector spaces to p-adic vector spaces.     

Contractibility of the Efficient Set in Strictly Quasiconcave Vector Maximization

In this paper, we investigate the contractibility of the efficient frontier in a vector maximization problem defined by a continuous vector-valued strictly quasiconcave function and a convex compact set D in ^p . It is shown that the efficient frontier is contractible if one of the components of g is strongly quasiconcave on X. This work extends a result by Sun (see Ref. 1), which confirms the connectedness of the efficient frontier.     

Rank 4 vector bundles on the quintic threefold

By the results of the author and Chiantini in [3], on a general quintic threefold X⊂P ⁴ the minimum integer p for which there exists a positive dimensional family of irreducible rank p vector bundles on X without intermediate cohomology is at least three. In this paper we show that p≤4, by constructing series of positive dimensional families of rank 4 vector bundles on X without intermediate cohomology. The general member of such family is an indecomposable bundle from the extension class Ext ¹ (E, F), for a suitable choice of the rank 2 ACM bundles E and F on X. The existence of such bundles of rank p=3 remains under question.     

Connectedness of the Efficient Set for Strictly Quasiconcave Sets

Given a closed subset X in , we show the connectedness of its efficient points or nondominated points when X is sequentially strictly quasiconcave. In the particular case of a maximization problem with n continuous and strictly quasiconcave objective functions on a compact convex feasible region of , we deduce the connectedness of the efficient frontier of the problem. This work solves the open problem of the efficient frontier for strictly quasiconcave vector maximization problems.     

An efficient Fréchet differentiable high breakdown multivariate location and dispersion estimator

A good robust functional should, if possible, be efficient at the model, smooth, and have a high breakdown point. M-estimators can be made efficient and Fréchet differentiable by choosing appropriate ψ-functions but they have a breakdown point of at most 1/(p + 1) in p dimensions. On the other hand, the local smoothness of known high breakdown functionals has not been investigated. It is known that Rousseeuw's minimum volume ellipsoid estimator is not differentiable and that S-estimators based on smooth functions force a trade-off between efficiency and breakdown point. However, by using a two-step M-estimator based on the minimum volume ellipsoid we show that it is possible to obtain a highly efficient, Fréchet differentiable estimator whilst still retaining the breakdown point. This result is extended to smooth S-estimators.     

<Emphasis Type="Italic">I</Emphasis>-projection onto isotonic cones and its applications to maximum likelihood estimation for log-linear models

A frequently occurring problem is to find a probability vector,p∈D, which minimizes theI-divergence between it and a given probability vector π. This is referred to as theI-projection of π ontoD. Darroch and Ratcliff (1972,Ann. Math. Statist.,43, 1470–1480) gave an algorithm whenD is defined by some linear equalities and in this paper, for simplicity of exposition, we propose an iterative procedure whenD is defined by some linear inequalities. We also discuss the relationship betweenI-projection and the maximum likelihood estimation for multinomial distribution. All of the results can be applied to isotonic cone.     

On Threshold Probability for the Stability of Independent Sets in Distance Graphs

This paper considers the so-called distance graph G(n, r, s);its vertices can be identified with the r-element subsets of the set {1, 2,…,n}, and two vertices are joined by an edge if the size of the intersection of the corresponding subsets equals s. Note that, in the case s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s-Ko-Rado problem; they also play an important role in combinatorial geometry and coding theory.

We study properties of random subgraphs of the graph G(n, r, s) in the Erd?s-Rényi model, in which each edge is included in the subgraph with a certain fixed probability p independently of the other edges. It is known that if r > 2s + 1, then, for p = 1/2, the size of an independent set is asymptotically stable in the sense that the independence number of a random subgraph is asymptotically equal to that of the initial graph G(n, r, s). This gives rise to the question of how small p must be for asymptotic stability to cease. The main result of this paper is the answer to this question.

     

Vector lattice covers of ideals and bands in Pre-Riesz spaces

Abstract

Pre-Riesz spaces are ordered vector spaces which can be order densely embedded into vector lattices, their so-called vector lattice covers. Given a vector lattice cover Y for a pre-Riesz space X, we address the question how to find vector lattice covers for subspaces of X, such as ideals and bands. We provide conditions such that for a directed ideal I in X its smallest extension ideal in Y is a vector lattice cover. We show a criterion for bands in X and their extension bands in Y as well. Moreover, we state properties of ideals and bands in X which are generated by sets, and of their extensions in Y.     

Generalized Proximal Method for Efficient Solutions in Vector Optimization

This article is devoted to developing the generalized proximal algorithm of finding efficient solutions to the vector optimization problem for a mapping from a uniformly convex and uniformly smooth Banach space to a real Banach space with respect to the partial order induced by a pointed closed convex cone. In contrast to most published literature on this subject, our algorithm does not depend on the nonemptiness of ordering cone of the space under consideration and deals with finding efficient solutions of the vector optimization problem in question. We prove that under some suitable conditions the sequence generated by our method weakly converges to an efficient solution of this problem.     

On the connectedness of the set of weakly efficient points of a vector optimization problem in locally convex spaces

In vector optimization, topological properties of the set of efficient and weakly efficient points are of interest. In this paper, we study the connectedness of the setE ^w of all weakly efficient points of a subsetZ of a locally convex spaceX with respect to a continuous mappingp:X Y,Y locally convex and partially ordered by a closed, convex cone with nonempty interior. Under the general assumptions thatZ is convex and closed and thatp is a pointwise quasiconvex mapping (i.e., a generalized quasiconvex concept), the setE ^w is connected, if the lower level sets ofp are compact. Furthermore, we show some connectedness results on the efficient points and the efficient and weakly efficient outcomes. The considerations of this paper extend the previous results of Refs. 1–3. Moreover, some examples in vector approximation are given.The author is grateful to Dr. D. T. Luc and to a referee for pointing out an error in an earlier version of this paper.     

Robust Parametric Classification and Variable Selection by a Minimum Distance Criterion

We investigate a robust penalized logistic regression algorithm based on a minimum distance criterion. Influential outliers are often associated with the explosion of parameter vector estimates, but in the context of standard logistic regression, the bias due to outliers always causes the parameter vector to implode, that is, shrink toward the zero vector. Thus, using LASSO-like penalties to perform variable selection in the presence of outliers can result in missed detections of relevant covariates. We show that by choosing a minimum distance criterion together with an elastic net penalty, we can simultaneously find a parsimonious model and avoid estimation implosion even in the presence of many outliers in the important small n large p situation. Minimizing the penalized minimum distance criterion is a challenging problem due to its nonconvexity. To meet the challenge, we develop a simple and efficient MM (majorization–minimization) algorithm that can be adapted gracefully to the small n large p context. Performance of our algorithm is evaluated on simulated and real datasets. This article has supplementary materials available online.     

A GRASP algorithm for the multi-criteria minimum spanning tree problem

This paper proposes a GRASP (Greedy Randomized Adaptive Search Procedure) algorithm for the multi-criteria minimum spanning tree problem, which is NP-hard. In this problem a vector of costs is defined for each edge of the graph and the problem is to find all Pareto optimal or efficient spanning trees (solutions). The algorithm is based on the optimization of different weighted utility functions. In each iteration, a weight vector is defined and a solution is built using a greedy randomized constructive procedure. The found solution is submitted to a local search trying to improve the value of the weighted utility function. We use a drop-and-add neighborhood where the spanning trees are represented by Prufer numbers. In order to find a variety of efficient solutions, we use different weight vectors, which are distributed uniformly on the Pareto frontier. The proposed algorithm is tested on problems with r=2 and 3 criteria. For non-complete graphs with n=10, 20 and 30 nodes, the performance of the algorithm is tested against a complete enumeration. For complete graphs with n=20, 30 and 50 nodes the performance of the algorithm is tested using two types of weighted utility functions. The algorithm is also compared with the multi-criteria version of the Kruskal’s algorithm, which generates supported efficient solutions. This work was funded by the Municipal Town Hall of Campos dos Goytacazes city. The used computer was acquired with resource of CNPq.     

On the number of conjugacy classes of finite nilpotent groups

We establish the first super-logarithmic lower bound for the number of conjugacy classes of a finite nilpotent group. In particular, we obtain that for any constant c there are only finitely many finite p-groups of order pm with at most c⋅m conjugacy classes. This answers a question of L. Pyber.     

Input–output substitutability and strongly monotonic p-norm least distance DEA measures

In DEA, there are two frameworks for efficiency assessment and targeting: the greatest and the least distance framework. The greatest distance framework provides us with the efficient targets that are determined by the farthest projections to the assessed decision making unit via maximization of the p-norm relative to either the strongly efficient frontier or the weakly efficient frontier. Non-radial measures belonging to the class of greatest distance measures are the slacks-based measure (SBM) and the range-adjusted measure (RAM). Whereas these greatest distance measures have traditionally been utilized because of their computational ease, least distance projections are quite often more appropriate than greatest distance projections from the perspective of managers of decision-making units because closer efficient targets may be attained with less effort. In spite of this desirable feature of the least distance framework, the least distance (in) efficiency versions of the additive measure, SBM and RAM do not even satisfy weak monotonicity. In this study, therefore, we introduce and investigate least distance p-norm inefficiency measures that satisfy strong monotonicity over the strongly efficient frontier. In order to develop these measures, we extend a free disposable set and introduce a tradeoff set that implements input–output substitutability.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

Fuzzy subgroups of nilpotent groups

In 1991, Asaad has suggested the following question: What can be said about the structure of a group G when information is known about the structure of a fuzzy subgroup A of G and vice-versa? This question has been partially investigated in Fuzzy Sets and Systems39 (1991) 323–328. The purpose of this paper is to continue more investigations for the above mentioned question.     

The permutation action of finite symplectic groups of odd characteristic on their standard modules

We study the space of functions on a finite-dimensional vector space over a field of odd order as a module for a symplectic group. We construct a basis of this module with the following special properties. Each submodule generated by a single basis element under the symplectic group action is spanned as a vector space by a subset of the basis and has a unique maximal submodule. From these properties, the dimension and composition factors of the submodule generated by any subset of the basis can be determined. These results apply to incidence geometry of the symplectic polar space, yielding the symplectic analogue of Hamada's additive formula for the p-ranks of the incidence matrices between points and flats. A special case leads to a closed formula for the p-rank of the incidence matrix between the points and lines of the symplectic generalized quadrangle over a field of odd order. Together with earlier results on the 2-ranks, this result completes the determination of the p-ranks for these quadrangles.     

A note on <Emphasis Type="Italic">p</Emphasis>-groups of order ≤ <Emphasis Type="Italic">p</Emphasis><Superscript>4</Superscript>

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p ⁴, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.     

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASUREgS

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.