Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Nonconvergence results for the application of least-squares estimation to Ill-posed problems

One standard approach to solvingf(x)=b is the minimization of f(x)–b² overx in , where corresponds to a parametric representation providing sufficiently good approximation to the true solutionx*. Call the minimizerx=A( ). Take = _N for a sequence { _N } of subspaces becoming dense, and so determine an approximating sequences {x _N A ( _N )}. It is shown, withf linear and one-to-one, that one need not havex _Nx* iff ^–1 is not continuous.This work was supported by the US Army Research Office under Grant No. DAAG-29-77-G-0061. The author is indebted to the late W. C. Chewning for suggesting the topic in connection with computing optimal boundary controls for the heat equation (Ref. 2).     

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

An Algorithm Constructing Solutions for a Family of Linear Semi-Infinite Problems

A family of linear semi-infinite problems depending on a parameter is considered. For fixed , a sensitivity analysis of the problem solution is carried out and rules constructing the solutions of this family for in a right-side neighborhood of ₀ are described. Results on the one-sided derivatives of the solution with respect to the parameter are presented. On the basis of the results, an active-set-strategy and a path-following algorithm are suggested.     

Regular and singular Hermitian operators

Let A be a closed Hermitian operator, let be the orthogonal complement of the domain of definition of A, and let be the defect subspace. An operator A is called regular if the orthogonal projection of on is closed. Criteria for regularity are established.Translated from Matematicheskie Zametki, Vol. 8, No. 2, pp. 197–203, August, 1970.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Approximation of $$\bar \omega $$ -integrals of continuous functions defined on the real axis by Fourier operators

We obtain asymptotic formulas for the deviations of Fourier operators on the classes of continuous functions and in the uniform metric. We also establish asymptotic laws of decrease of functionals characterizing the problem of the simultaneous approximation of -integrals of continuous functions by Fourier operators in the uniform metric.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 663–676, May, 2004.     

Inertial Gradient-Like Dynamical System Controlled by a Stabilizing Term

Let H be a real Hilbert space and let be a function that we wish to minimize. For any potential and any control function which tends to zero as t+, we study the asymptotic behavior of the trajectories of the following dissipative system:

On convergence of types and processes in Euclidean space

Let be an Euclidean space; Y _n , Z, U random vectors in ; h _n , g _n affine transformations and let þ be a subgroup of the group G of all the in vertible affine transformations, closed relative to G. Suppose that g_n and where Z is nonsingular. The behaviour of _n = h _n g _n ^–1 as n is discussed first. The results are used then to prove that if for all t(0, ), where h _n þ and Z ₁ is nonsingular and nonsymmetric with respect to þ then H, for all t(0,) and is a continuous homomorphism of the multiplicative group of (0, ) into þ. The explicit forms of the possible are shown.     

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

Optimizing a linear function over an efficient set

The problem (P) of optimizing a linear functiond ^T x over the efficient set for a multiple-objective linear program (M) is difficult because the efficient set is typically nonconvex. Given the objective function directiond and the set of domination directionsD, ifd ^T 0 for all nonzero D, then a technique for finding an optimal solution of (P) is presented in Section 2. Otherwise, given a current efficient point , if there is no adjacent efficient edge yielding an increase ind ^T x, then a cutting plane is used to obtain a multiple-objective linear program ( ) with a reduced feasible set and an efficient set . To find a better efficient point, we solve the problem (I_i) of maximizingc _i ^T x over the reduced feasible set in ( ) sequentially fori. If there is a that is an optimal solution of (I_i) for somei and , then we can choosex ⁱ as a current efficient point. Pivoting on the reduced feasible set allows us to find a better efficient point or to show that the current efficient point is optimal for (P). Two algorithms for solving (P) in a finite sequence of pivots are presented along with a numerical example.The authors would like to thank an anonymous referee, H. P. Benson, and P. L. Yu for numerous helpful comments on this paper.     

The Boundary Equivalence for Rings and Matrix Rings over Them

We study into the question of whether some rings and their associated matrix rings have equal decidability boundaries in the scheme and scheme-alternative hierarchies. Let be a decidability boundary for an algebraic system A; w.r.t. the hierarchy H. For a ring R, denote by an algebra with universe . On this algebra, define the operations + and in such a way as to extend, if necessary, the initial matrices by suitably many zero rows and columns added to the underside and to the right of each matrix, followed by ordinary addition and multiplication of the matrices obtained. The main results are collected in Theorems 1-3. Theorem 1 holds that if R is a division or an integral ring, and R has zero or odd characteristic, then the equalities hold for any n1. And if R is an arbitrary associative ring with identity then for any n 1 and i,j { 1,..., n}, where e _ij is a matrix identity. Theorem 2 maintains that if R is an associative ring with identity then . Theorem 3 proves that for any n 1.     

Convergence of series of translations

For a mean zero norm one sequence (f _n )L ₂[0, 1], the sequence (f _n {nx+y}) is an orthonormal sequence inL ₂([0, 1]²); so if , then converges for a.e. (x, y)[0, 1]² and has a maximal function inL ₂([0, 1]²). But for a mean zerofL ₂[0, 1], it is harder to give necessary and sufficient conditions for theL ₂-norm convergence or a.e. convergence of . Ifc _n 0 and , then this series will not converge inL ₂-norm on a denseG subset of the mean zero functions inL ₂[0, 1]. Also, there are mean zerofL[0, 1] such that never converges and there is a mean zero continuous functionf with a.e. However, iff is mean zero and of bounded variation or in some Lip() with 1/2<1, and if |c _n | = 0(n ^–) for >1/2, then converges a.e. and unconditionally inL ₂[0, 1]. In addition, for any mean zerof of bounded variation, the series has its maximal function in allL _p[0, 1] with 1p<. Finally, if (f _n )L [0, 1] is a uniformly bounded mean zero sequence, then is a necessary and sufficient condition for to converge for a.e.y and a.e. (x _n )[0, 1]. Moreover, iffL [0, 1] is mean zero and , then for a.e. (x _n )[0, 1], converges for a.e.y and in allL _p [0, 1] with 1p<. Some of these theorems can be generalized simply to other compact groups besides [0, 1] under addition modulo one.     

An Improvement of an Inequality of Fiedler Leading to a New Conjecture on Nonnegative Matrices

Suppose that A is an n × n nonnegative matrix whose eigenvalues are = (A), ₂, ..., _n. Fiedler and others have shown that \det( I -A) ⁿ - ⁿ, for all > with equality for any such if and only if A is the simple cycle matrix. Let a _i be the signed sum of the determinants of the principal submatrices of A of order i × i, i=1, ..., n - 1. We use similar techniques to Fiedler to show that Fiedler's inequality can be strengthened to: for all . We use this inequality to derive the inequality that: . In the spirit of a celebrated conjecture due to Boyle-Handelman, this inequality inspires us to conjecture the following inequality on the nonzero eigenvalues of A: If ₁ = (A), ₂,...,_k are (all) the nonzero eigenvalues of A, then . We prove this conjecture for the case when the spectrum of A is real.     

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form where Z, Z ₁, Z ₂, are arbitrary i.i.d. random variables and N is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate . The truncation level introduced depends only on and Z and not on the overall exceedance level a.     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

Self-bounded controlled invariant subspaces: A straightforward approach to constrained controllability

A lattice-type structure is shown to exist in a particular subset of the set of all (A, )-controlled invariants contained in and containing , whereA denotes a linear map inR ⁿ ; , are arbitrary subspaces ofR ⁿ ; andD is an arbitrary subspace ofJ, the maximum (A, )-controlled invariant contained in . In linear system theory, this property can be used for a more direct theoretical and algorithmic approach to constrained controllability and disturbance rejection problems.     

Scalar-quadratic stabilizability of the Petersen counterexample via a linear static controller

A new property called scalar-quadratic is presented for establishing the stabilizability of linear time-varyring uncertain systems. It is applied to a well-known linear time-varying system _OL which contains two uncertainties ₁(t) and ₂(t). Using the Lyapunov functionsV(x)=x ^T Px, whereP is a constant postitive-definite symmetric matrix, previous authors have shown that _OL is stabilizable by linear static controllers when the time-varying uncertainties are bounded by a normalized bound satisfying < 0.8. We extend the bound to < 1.0 by using the more general Lyapunov functions satisfying the scalar-quadratic propertyV(ax)=a ² V(x), aR, xR ₀ ² .Our proof uses a hexagon as a closed, convex hypersuface to construct a scalar-quadratic Lyapunov function, so that the Lyapunov time derivative satisfies the quadratic convergence condition , >0, for the closed-loop system _CL formed from _OL and a stabilizing linear static controller. The critical condition in the proof of the quaratic convergence ondition is the satisfaction of the inequality , where _max is a normalization bound for ₁(t) and ₂(t) and wheree ₁ ande ₂ are parameters for the controller. For the controller parametrized bye ₁=8 ande ₂=20, this inequality reduces to _max < 2.2096. This result, in particular, establishes that the Petersen counterexample is stabilitzable by the linear static controller withe ₁=8 ande ₂=20. Furthermore, it establishes the amazing result that _OL is stabilizable by a linear static controlle on any compact subset of the constant uncertainaty controllability space defined by ₁>0 and ₂>0.     

On sequences of Dirichlet polynomials uniformly bounded on half planes

Given and a sequence of Dirichlet polynomials estimates for the coefficientsa _n are proved if {_n} is uniformly bounded on a region containing a half plane. Thereby a result is obtained which is an analogue of a known result for polynomials, that is for theA-transforms of the geometric sequence; moreover a Jentzsch type theorem for {_n(z)} is derived.