Accelerated convergence for the Powell/Hestenes multiplier method

It is known that augmented Lagrangian or multiplier methods for solving constrained optimization problems can be interpreted as techniques for maximizing an augmented dual functionD _c(). For a constantc sufficiently large, by considering maximizing the augmented dual functionD _c() with respect to, it is shown that the Newton iteration for based on maximizingD _c() can be decomposed into taking a Powell/Hestenes iteration followed by a Newton-like correction. Superimposed on the original Powell/Hestenes method, a simple acceleration technique is devised to make use of information from the previous iteration. For problems with only one constraint, the acceleration technique is equivalent to replacing the second (Newton-like) part of the decomposition by a finite difference approximation. Numerical results are presented.     

Implicit elliptic differential equations

Let be a bounded domain in ⁿ (n 3) having a smooth boundary, letY be a closed, connected and locally connected subset of ^h , letf be a real-valued function defined on × ^h × ^nh ×Y, and letL be a linear, second-order elliptic operator. In this paper, the existence of strong solutionsu W ^2,p (, ^h ) W ₀ ^1,p (, ^h ) (n<p<+) to the implicit elliptic equationf(x, u, Du, Lu)=0, whereu=(u ₁,u ₂, ...,u _h ),Du=(Du ₁,Du ₂, ...,Du _h ) andLu=(Lu ₁,Lu ₂, ...,Lu _h ), is established. The abstract framework where the equation is studied is that of set-valued analysis.Dedicated to Professor G. Pulvirenti on the occasion of his sixtieth birthday     

On Modified Strong Dyadic Integral and Derivative

For functions f L(R ₊), we define a modified strong dyadic integral J(f) L(R ₊) and a modified strong dyadic derivative D(f) L(R ₊). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition f(x)dx = 0, we prove the equalities J(D(f)) = f and D(J(f)) = f. We find a countable set of eigenfunctions of the operators J and D. We prove that the linear span L of this set is dense in the dyadic Hardy space H(R ₊). For the functions f H(R ₊), we define a modified uniform dyadic integral J(f) L (R ₊).     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

Some mildly wild circles in Sn arising from algebraic K-Theory

We study wild embeddings of S ¹ in S ⁿ which are tame in a sense introduced by Quinn. We show that if is a finitely presented group with H ₁()=H ₂()=0, then any finiteness obstruction K ₀() can be realized on the complement of such an embedded S ¹. We also realize trivially symmetric K _–1() obstructions on the complements of such embeddings. For trivially symmetric , the embeddings constructed are shown to be isotopy homogeneous.     

Reduction of operators with almost periodic symbols to operators overC *-Algebras on sections of associated bundles over a torus

A decomposition of any pseudodifferential operator (D) onR ⁿ with almost periodic symbol as 113-1 1 is obtained in the paper, where _A (D) is a pseudodifferential operator over a certainC ^*-algebraA acting on sections of a vector bundle over a torusT ⁿ whose fibre isA. The coincidence of spectra sp (D) = sp _A (D) is proved for all (D) either bounded or elliptic.     

Koassoziierte Primideale in 2-Dimensionalen Regulären Ringen

Suppose that R is a 2-dimensional regular local ring, pa prime ideal of height 1, and u = r/s an element of the quotient field of R. We give criteria for pbeing coassociated to the R-module B = R[u], i.e., that there is an Artinian factor module B/X such that p= Ann _R (B/X). These criteria are formulated using the ideal a= (r, s) and the intersection numbers ( _i , r), ( _i , s), where ₁, ... _h are the branches of pin the completion R.     

Relative komplexe quotienten

Let XS be a holomorphic map, and let RX×SX be an equivalence relation. The restriction of R to the fibre ^–1(S) is denoted by R_s. The quotient X/R is called a relative complex quotient, if the quotient map XX/R is holomorphic over S. Two cases are studied: (C) All fibres of are locally R_s-separable (relative Cartan quotient); (R) All fibres of are holomorphically convex, and R_s is given by tke holomorphic functions on ^–1 (s) (relative Remmert quotient).     

Nonlocal problems in the theory of equations of motion of Kelvin-Voight fluids. 2

This article studies nonlocal problems for equations of motion of Kelvin-Voight fluids (2): 1) global solvability of initial-boundary-value problem (2)-(3) on halfaxisR ⁺ with free termf(x, t) S₂(⁺; L₂(0)) (see (4)); 2global solvability of system (2) on the entire axis R in the class of functions that are bounded as t±with free term f(x,t)S₂(; L₂());3) the existence of periodic solutions for system (2) that are periodic in t with period with free term,f(x,t)L₂((0,); L₂());4) the existence of solutions of system (2) that are almost periodic in t with free term f(x,t)S₂(, L₂()).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 111–124, 1990.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Two variable <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis> functions attached to eigenfamilies of positive slope

Consider a classical cusp eigenform f= _n=1 a _n (f)q ⁿ of weight k2 for ₀(N) with a Dirichlet character mod N, and let L _f (s,)= _n=1 (n)a _n (f)n ^-s denote the L-function of f twisted with an arbitrary Dirichlet character . For a prime number p5, consider a family of cusp eigenforms f _(k) of weight k , k {f _(k)= _n=1 a _n (f _(k))q ⁿ } containing f=f _(k), such that the Fourier coefficients a _n (f _(k)) are given by certain p-adic analytic functions k a _n (f _(k)). The purpose of this paper is to construct a two variable p-adic L function attached to Colemans family {f _(k)} of cusp eigenforms of a fixed positive slope =v _p ( _p )>0 where _p = _p (k ) is an eigenvalue (which depends on k ) of the Atkin operator U=U _p . Our p-adic L-function interpolates the special values L _f(k)(s,) at points (s,k ) with s=1,2,...,k -1. We give a construction using the Rankin-Selberg method and the theory of p-adic integration on a profinite group Y with values in an affinoid K-algebra A, where K is a fixed finite extension of Q _p . Our p-adic L-functions are p-adic Mellin transforms of certain A-valued measures. In their turn, such measures come from Eisenstein distributions with values in certain Banach A-modules M =M (N;A) of families of overconvergent forms over A. To Robert Alexander Rankin in memoriam     

On the Structure of Spaces of Polyanalytic Functions

Suppose that A_mL_p(D,) is the space of all m-analytic functions on the disk D={z:|z| < 1} which are pth power integrable over area with the weight (1-|z|²), > -1. In the paper, we introduce subspaces A_kL_p ⁰(D,), k=1,2,...,m, of the space A _mL_p(D,) and prove that A _mL_p(D,) is the direct sum of these subspaces. These results are used to obtain growth estimates of derivatives of polyanalytic functions near the boundary of arbitrary domains.     

On the extreme spectral properties of Toeplitz matrices generated byL 1 functions with several minima/maxima

In this paper we are concerned with the asymptotic behavior of the smallest eigenvalue ₁ ⁽ⁿ⁾ of symmetric (Hermitian)n ×n Toeplitz matricesT _n (f) generated by an integrable functionf defined in [–, ]. In [7, 8, 11] it is shown that ₁ ⁽ⁿ⁾ tends to essinff =m _f in the following way: ₁ ⁽ⁿ⁾ –m _f 1/n ^2k . These authors use three assumptions:A1)f –m _f has a zero inx =x ₀ of order 2k.A2)f is continuous and at leastC ^2k in a neighborhood ofx ₀.A3)x =x ₀ is the unique global minimum off in [–, ]. In [10] we have proved that the hypothesis of smoothnessA2 is not necessary and that the same result holds under the weaker assumption thatf L ¹[–, ]. In this paper we further extend this theory to the case of a functionf L ¹[–, ] having several global minima by suppressing the hypothesisA3 and by showing that the maximal order 2k of the zeros off –m _f is the only parameter which characterizes the rate of convergence of ₁ ⁽ⁿ⁾ tom _f .     

Geometric Properties of Universally Measurable Mappings

We study conditions under which universally measurable mappings from a separable topological space S into a metric space R belong to the class D of mappings f : SR: such that for any compact subset KS and number > 0 there exists an open (in the induced topology) set VK such that the oscillation (f;V) of an R-valued function f on V is less than . Bibliography: 7 titles.     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

Generalizations of Perfect, Semiperfect, and Semiregular Rings

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:

The uniform random tree in a Brownian excursion

Summary To any Brownian excursione with duration (e) and anyt ₁, ...,t _p [0,(e)], we associate a branching tree withp branches denoted byT _p (e, t ₁,...,t _p ), which is closely related to the structure of the minima ofe. Our main theorem states that, ife is chosen according to the Itô measure and (t ₁, ...,t _p ) according to Lebesgue measure on [0,(e)] ^p , the treeT _p (e, t ₁, ...,t _p ) is distributed according to the uniform measure on the set of trees withp branches. The proof of this result yields additional information about the subexcursions ofe corresponding to the different branches of the tree, thus generalizing a well-known representation theorem of Bismut. If we replace the Itô measure by the law of the normalized excursion, a simple conditioning argument leads to another remarkable result originally proved by Aldous with a very different method.     

Submodule lattice quasivarieties and exact embedding functors for rings with prime power characteristic

Given ringsR with prime power characteristicp ^k , quasivarieties (R) of lattices generated by lattices of submodules ofR-modules are studied. An algebra of expressionsd not dependent onR is developed, such that each suchd uniquely determines a two-sides ideald _R ofR. The main technical result is that (R) (S) makes all implications of the formd _s =S d_R=R true, for any such expressiond. The proof makes use of the known equivalence between (R) (S) and existence of an exact embedding functorR-Mod S -Mod. Fork 2, the ordered setW(p ^k ) of all lattice quasivarieties (R),R having characteristic p _K , is shown to be large and complicated, with ascending and descending chains and antichains having continuously many elements. More precisely,W(p ^k ) has a subset which is order isomorphic to the Boolean algebra of all subsets of a denumerably infinite set. Also, given any prime powerp ^k ,k 2, a ringR can be constructed so that (R) and (R ^op) for the opposite ringR ^op are distinct elements ofW(p ^k ).Presented by R. Freese.Research partially supported by Hungarian National Foundation for Scientific Research grant no. 1903.     

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

Vertex-reinforced random walk

Summary This paper considers a class of non-Markovian discrete-time random processes on a finite state space {1,...,d}. The transition probabilities at each time are influenced by the number of times each state has been visited and by a fixed a priori likelihood matrix,R, which is real, symmetric and nonnegative. LetS _i (n) keep track of the number of visits to statei up to timen, and form the fractional occupation vector,V(n), where . It is shown thatV(n) converges to to a set of critical points for the quadratic formH with matrixR, and that under nondegeneracy conditions onR, there is a finite set of points such that with probability one,V(n)p for somep in the set. There may be more than onep in this set for whichP(V(n)p)>0. On the other handP(V(n)p)=0 wheneverp fails in a strong enough sense to be maximum forH.This research was supported by an NSF graduate fellowship and by an NSF postdoctoral fellowship