Lagrange multipliers for set-valued optimization problems associated with coderivatives

In this paper we investigate a vector optimization problem (P) where objective and constraints are given by set-valued maps. We show that by mean of marginal functions and suitable scalarizing functions one can characterize certain solutions of (P) as solutions of a scalar optimization problem (SP) with single-valued objective and constraint functions. Then applying some classical or recent results in optimization theory to (SP) and using estimates of subdifferentials of marginal functions, we obtain optimality conditions for (P) expressed in terms of Lagrange or sequential Lagrange multipliers associated with various coderivatives of the set-valued data.     

Zero Divisor Graph of a Poset with Respect to an Ideal

In this paper, we introduce the zero divisor graph G _I (P) of a poset P (with 0) with respect to an ideal I. It is shown that G _I (P) is connected with its diameter ??3, and if G _I (P) contains a cycle, then the core K of G _I (P) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of G _I (P) are shown to be equal. This proves a form of Beck??s conjecture for posets with 0. The complete bipartite zero divisor graphs are characterized.     

Single-valued extension property at the points of the approximate point spectrum

A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point in the case that λ_oI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

Generalized Kato decomposition, single-valued extension property and approximate point spectrum

In this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ₀∈C in the case that λ₀I−T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

Remarks on ideal boundedness, convergence and variation of sequences

We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal I⊂P(N) the quotient space l^∞(I)/c₀(I) is complete. The second main result states that consistently there is an admissible ideal I⊂P(N) such that the sets W(I), of all real sequences with finite I-variation, and c^?(I), of all restrictively I-convergent sequences, are equal.     

The stochastic unit root model and fractional integration: An extension to the seasonal case

In a recent paper, Yoon (Working Paper, Department of Economics and Related Studies, University of York, 2003. Presented at the ESF-EMM Second Annual Meeting, Rome, Italy, 2003) asserts that the stochastic unit root (STUR) model is closely related to long memory processes, and, in particular, that it is a special case of an I(d) process, with d = 1.5. In this paper we question this claim by using parametric and semiparametric techniques for modelling long memory. We extend the analysis by considering both non-normality and seasonality, and shed light, theoretically and by means of Monte Carlo methods, on the relationship between the seasonal STUR and the seasonal I(d) models. The results show that methods that are specifically designed for testing I(d) statistical models are not appropriate for testing the STUR model. Moreover, they have in some cases very low power against STUR alternatives. Copyright © 2007 John Wiley & Sons, Ltd.     

The inverse laplace transform of some analytic functions with an application to the eternal solutions of the Boltzmann equation

When the Laplace transform F(p) of a function f(x) has no poles but is singular only on the real negative semiaxis because of a cut required to make it single-valued, the inverse transform f(x) can easily be computed by means of the integral of a real-valued function. This result is applied to the calculation of a class of exact eternal solutions of the Boltzmann equation, recently found by the authors. The new approach makes it easier to prove that these solutions are positive, as well as to study their asymptotics.     

Variations on Weyl's theorem

In this note we study the property (w), a variant of Weyl's theorem introduced by Rako?evi?, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T^*) coincide whenever T^* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).     

The set of irreducible graphs for the projective plane is finite

In 1930 Kuratowski proved that a graph does not embed in the real plane R² if and only if it contains a subgraph homeomorphic to one of two graphs, K₅ or K_{3, 3}. For positive integer n, let I_n (P) denote a smallest set of graphs whose maximal valency is n and such that any graph which does not embed in the real projective plane contains a subgraph homeomorphic to a graph in I_n (P) for some n. Glover and Huneke and Milgram proved that there are only 6 graphs in I₃ (P), and Glover and Huneke proved that I_n (P) is finite for all n. This note proves that I_n (P) is empty for all but a finite number of n. Hence there is a finite set of graphs for the projective plane analogous to Kuratowski's two graphs for the plane.     

Generalized bi-circular projections

Recall that a projection P on a complex Banach space X is a generalized bi-circular projection if P+λ(I−P) is a (surjective) isometry for some λ such that |λ|=1 and λ≠1. It is easy to see that every hermitian projection is generalized bi-circular. A generalized bi-circular projection is said to be nontrivial if it is not hermitian. Botelho and Jamison showed that a projection P on C([0,1]) is a nontrivial generalized bi-circular projection if and only if P−(I−P) is a surjective isometry. In this article, we prove that if P is a projection such that P+λ(I−P) is a (surjective) isometry for some λ, then either P is hermitian or λ is an nth unit root of unity. We also show that for any nth unit root λ of unity, there are a complex Banach space X and a nontrivial generalized bi-circular projection P on X such that P+λ(I−P) is an isometry.     

On the normal ideals of exchange rings

An ideal I of a ring R is called normal if all idempotent elements in I lie in the center of R. We prove that if I is a normal ideal of an exchange ring R then: (1) R and R/I have the same stable range; (2) V(I) is an order-ideal of the monoid C(Specc(R), N), where Specc(R) consists of all prime ideals P such that R/P is local.     

On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/λ-hypoconvex if its proximal mapping Pλf is single-valued. When the function f is bounded below, and Pλf is single-valued for every λ>0, the function must be convex. Similarly, we show that the function f is 1/μ-strongly convex if the farthest mapping Qμf is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in Rn. We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.     

On the field of values of oblique projections

We highlight some properties of the field of values (or numerical range) W(P) of an oblique projector P on a Hilbert space, i.e., of an operator satisfying P²=P. If P is neither null nor the identity, we present a direct proof showing that W(P)=W(I-P), i.e., the field of values of an oblique projection coincides with that of its complementary projection. We also show that W(P) is an elliptical disk (i.e., the set of points circumscribed by an ellipse) with foci at 0 and 1 and eccentricity 1/‖P‖. These two results combined provide a new proof of the identity ‖P‖=‖I-P‖. We discuss the influence of the minimal canonical angle between the range and the null space of P, on the shape of W(P). In the finite dimensional case, we show a relation between the eigenvalues of matrices related to these complementary projections and present a second proof to the fact that W(P) is an elliptical disk.     

Graphs with bounded valency that do not embed in the projective plane

In 1930 Kuratowski proved that a graph does not embed in the real plane R² if and only if it contains a subgraph homeomorphic to one of two graphs, K₅ or K₃₃. Let I_n(P) denote the minimal set of graphs whose vertices have miximal valency n such that any graph which does not embed in the real projective plane (or equivalently, does not embed in the Möbius band) contains a subgraph homeomorphic to a graph in I_n(P) for some positive integer n. Glover and Huneke and Milgram proved that there are only 6 graphs in I₃(P). This note proves that for each n, I_n(P) is finite.     

Unit Interval Orders of Open and Closed Intervals

A poset P = (X, ?) is a unit OC interval order if there exists a representation that assigns an open or closed real interval I(x) of unit length to each x ∈ P so that x ? y in P precisely when each point of I (x) is less than each point in I (y). In this paper we give a forbidden poset characterization of the class of unit OC interval orders and an efficient algorithm for recognizing the class. The algorithm takes a poset P as input and either produces a representation or returns a forbidden poset induced in P.     

A simplification of the Kovarik formula

Let P,Q be two idempotents on a Hilbert space. Z.V. Kovarik (Z.V. Kovarik, Similarity and interpolation between projectors, Acta Sci. Math. (Szeged) 39 (1977) 341-351) showed that when P+Q−I is invertible, the formula K(P,Q)=P⁻²(P+Q−I)Q gives the only idempotent such that R(K)=R(P), N(K)=N(Q), where N(T) and R(T) denote the nullspace and the range of a bounded linear operator T on a Hilbert space, respectively. This formula was later extended to the context of Banach algebras and used in 1983 by J. Esterle to show that two homotopic idempotents may always be connected by a polynomial idempotent valued path. In the present paper, we give a simplification of Kovarik's original formula and one natural generalization of it.     

Contracting towards subspaces when estimating the mean of a multivariate normal distribution

The problem of estimating, under unweighted quadratic loss, the mean of a multinormal random vector X with arbitrary covariance matrix V is considered. The results of James and Stein for the case V = I have since been extended by Bock to cover arbitrary V and also to allow for contracting X towards a subspace other than the origin; minimax estimators (other than X) exist if and only if the eigenvalues of V are not “too spread out.” In this paper a slight variation of Bock's estimator is considered. A necessary and sufficient condition for the minimaxity of the present estimator is (1): the eigenvalues of (I ? P) V should not be “too spread out,” where P denotes the projection matrix associated with the subspace towards which X is contracted. The validity of (1) is then examined for a number of patterned covariance matrices (e.g., intraclass covariance, tridiagonal and first order autocovariance) and conditions are given for (1) to hold when contraction is towards the origin or towards the common mean of the components of X. (1) is also examined when X is the usual estimate of the regression vector in multiple linear regression. In several of the cases considered the eigenvalues of V are “too spread out” while those of (I ? P) V are not, so that in these instances the present method can be used to produce a minimax estimate.     

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

The number of uncountable models of ω-stable theories

In this paper we give a complete answer to the question of determining all possible functions I(μ, T), where I(μ, T) denotes the number of non-isomorphic models of a fixed countable ω-stable theory T or cardinality μ (μ uncountable).     

Some estimates for the number of vertices of integer polyhedra

A polyhedron is called integer if its every vertex has integer coordinates. We consider integer polyhedra P _I = conv(P ∩ ? ^d ) defined implicitly; that is, no system of linear inequalities is known for P _I but some is known for P. Some estimates are given for the number of vertices of P _I .