Some minimax problems of vector-valued functions

The concepts of cone extreme points, cone saddle points, and cone saddle values are introduced. The relation of inclusion among the sets mini _xX max _yY f(x, y), maxi _yY min _xX f(x, y), and the set of all weak cone saddle values is investigated in the case where the image space ⁿ off is ordered by an acute convex cone.The author is grateful for the useful suggestions and comments given by Prof. K. Tanaka, Niigata University, Niigata, Japan.The author would like to thank the referees for their valuable suggestions on the original draft.     

On the controllability of a class of differential inclusions depending on a parameter

In this paper, we deal with the following stability problem: given a differential inclusion of the formx'F(t,x,), where is a parameter varying in a topological space , find conditions under which the set of all , such that the differential inclusion is controllable, is open in . Applying Theorem 3.1 of Ref. 3, we get a result in this direction, assuming, as leader hypotheses, thatF(t,·,) is a convex process, from into itself, and thatF(t,x,·) is lower semicontinuous.     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

On the Projections of Multivalued Maps

This paper considers analogues of the Helmholtz projections of the set of selections of a piecewise smooth multivalued map , n2. It is shown that, for mn–1 (m=1), the closure of the projection of on the subspace of gradient fields (solenoidal vector fields) is a convex set. For the general case, there are given point-wise conditions on the values of the map which ensure that the closure of the projection of contains the zero element. Possible applications to optimal control problems are discussed.     

On the convergence of projected gradient processes to singular critical points

The projected gradient methods treated here generate iterates by the rulex _k+1=P (x _k –s _k F(x _k )),x ₁ , where is a closed convex set in a real Hilbert spaceX,s _k is a positive real number determined by a Goldstein-Bertsekas condition,P projectsX into ,F is a differentiable function whose minimum is sought in , and F is locally Lipschitz continuous. Asymptotic stability and convergence rate theorems are proved for singular local minimizers in the interior of , or more generally, in some open facet in . The stability theorem requires that: (i) is a proper local minimizer andF grows uniformly in near ; (ii) –F() lies in the relative interior of the coneK of outer normals to at ; and (iii) is an isolated critical point and the defect P (x – F(x)) –x grows uniformly within the facet containing . The convergence rate theorem imposes (i) and (ii), and also requires that: (iv)F isC ⁴ near and grows no slower than x–⁴ within the facet; and (v) the projected Hessian operatorP _{F 2} F()F is positive definite on its range in the subspaceF orthogonal toK . Under these conditions, {x _k } converges to from nearby starting pointsx ₁, withF(x _k ) –F() =O(k ^–2) and x _k – =O(k ^–1/2). No explicit or implied local pseudoconvexity or level set compactness demands are imposed onF in this analysis. Furthermore, condition (v) and the uniform growth stipulations in (i) and (iii) are redundant in ⁿ .     

Arithmetic of the Modular Function j1,8

Since the genus of the modular curve X_1 (8) = _1 (8) * is zero, we find a field generator j _1,8(z) = ₃(2z)/₃(4z) (₃(z) := _n ein ^2z ) such that the function field over X ₁(8) is (j _1,8). We apply this modular function j _1,8 to the construction of some class fields over an imaginary quadratic field K, and compute the minimal polynomial of the singular value of the Hauptmodul N(j _1,8) of (j _1,8).     

p-Banach Spaces and p-Totally Convex Spaces. II

We show that the results about the set S : ={ [0, 1] ^{1 / p} x + (1 – )^{1 / p} z ^{1 / p} y + (1 – )^{1 / p} z}, where x, y, z elements of a p-absolutely convex space D and `' is a congruence relation on D are the best possible. Finally, we give an explicit construction of the left adjoint of the comparison functor Ô _p : B an _p T C _p (resp. Ô _{p, fin} : V ec _p A C _p ).     

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Maximal Points of Convex Sets in Locally Convex Topological Vector Spaces: Generalization of the Arrow–Barankin–Blackwell Theorem

In 1953, Arrow, Barankin, and Blackwell proved that, if C is a nonempty compact convex set in Rⁿ with its standard ordering, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. In this paper, we present a generalization of this result. We show that that, if C is a compact convex set in a locally convex topological space X and if K is an ordering cone on X such that the quasi-interiors of K and the dual cone K* are nonempty, then the set of points in C maximizing strictly positive linear functionals is dense in the set of maximal points of C. For example, our work shows that, under the appropriate conditions, the density results hold in the spaces Rⁿ, L^p(, ), 1p, l^p, 1p, and C (), a compact Hausdorff space, when they are partially ordered with their natural ordering cones.     

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk _t on the grid =^m X ⁿ is considered. Let _t ⁰ be a random walk homogeneous in time and space, and let _t be obtained from it by changing transition probabilities on the set A= X ⁿ, || < , so that the walk remains homogeneous only with respect to the subgroup ⁿ of the group . It is shown that if >m 2 or the drift is distinct from zero, then the central limit theorem holds for _t.     

A class of nonlinear complementarity problems for multifunctions

Given a continuous mapF:R ⁿ R ⁿ and a lower semicontinuous positively homogeneous convex functionh:R ⁿ R, the nonlinear complementarity problem considered here is to findxR ₊ ⁿ andyh(x), the subdifferential ofh atx, such thatF(x)+y0 andx ^T (F(x)+y)=0. Some existence theorems for the above problem are given under certain conditions on the mapF. An application to quasidifferentiable convex programming is also shown.The authors are grateful to Professor O. L. Mangasarian and the referee for their substantive suggestions.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Generalized convex functions and vector variational inequalities

In this paper, (, ,Q)-invexity is introduced, where :X ×X intR _m ⁺ , :X ×X X,X is a Banach space,Q is a convex cone ofR ^m . This unifies the properties of many classes of functions, such asQ-convexity, pseudo-linearity, representation condition, null space condition, andV-invexity. A generalized vector variational inequality is considered, and its equivalence with a multi-objective programming problem is discussed using (, ,Q)-invexity. An existence theorem for the solution of a generalized vector variational inequality is proved. Some applications of (, ,Q)-invexity to multi-objective programming problems and to a special kind of generalized vector variational inequality are given.The author is indebted to Dr. V. Jeyakumar for his constant encouragement and useful discussion and to Professor P. L. Yu for encouragement and valuable comments about this paper.     

The Exterior Cube L-Function for GL(6)

We construct a Rankin Selberg integral to represent the exterior cube L function L(,³,s) of an automorphic cuspidal module of GL₆( _F ) (where F is a number field). We determine the poles of this L function and find period conditions for the special value L(,³,1/2). We use the Siegal Weil formula. We also state an analogue of the Gross–Prasad conjecture concerning a criterion for the nonvanishing of L(,³,1/2).     

Théorème de division avec des conditions au bord

Let be an open subset of ⁿ and be a subalgebra of the algebra of analytic functions on . We suppose that satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of . We also suppose that satifies global £ojasiewicz's inequalities. We prove the following: Let andf C on flat on ; if for eacha the Taylor's serie off ata, T _a f, is in the ideal generated byT _a f ₁,...,T _a f _p in the ring of formal power series, then there exist ₁,..., _p ,C on flat on such that . This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.Reherches menées dans le cadre du Programme d'Appui à la Recherche Scientifique (PARS MI 33)     

On the growth order of partial sums of non-orthogonal series

a _k f _k , f _k L ₂, w-, (2), w(n) — . a _k f _k N {a _k }l ₂, {a _k }l ₂ ( 1, 2, 1a, 2a). ( 2) [8]. , {a _k } w-.     

On contiguity of probability measures corresponding to semimartingales

, (ⁿ), - (P ⁿ ), P ⁿ (A ⁿ )>0P ⁿ (A ⁿ )0,n. [15] - , . , P ⁿ P ⁿ T ⁿ T ⁿ .     

An Inequality for Tail Probabilities of Martingales with Bounded Differences

Let M _n =X₁+...+X_n be a martingale with bounded differences X_m=M_m-M_m-1 such that {|X_m| _m}=1 with some nonnegative _m. Write ²= ₁ ² + ... + _n ² . We prove the inequalities {M _nx}c(1-(x/)), {M _n x} 1- c(1- (-x/)) with a constant . The result yields sharp inequalities in some models related to the measure concentration phenomena.     

Stability of Completeness for Systems of Exponentials on Compact Convex Sets in ℂ

A sufficient condition for the stability of completeness of systems of exponentials with exponent shifts _n in spaces of functions continuous onb and holomorphic inside a compact convex set with nonempty interior is obtained. This condition is a generalization of an important special case of the corresponding result for the closed interval due to Redheffer and Alexander.