Best approximation of a class of functions by another class

Let B be the space C=C(I) or L=L₁(I), where I=(–, ). By ⁽ⁿ⁾(x)_c we will mean the upper bound of the modulus of the values of the derivative of the function 相似文献   

Relation between the vectors Э and S for a plastic material with a sequence of ray-like loading paths

Two different complex loading paths are investigated in stress space — longitudinal tension with subsequent total unloading and reloading in transverse tension or compression. For these loading paths the local strains theory [4] is used to determine the values of the components of the strain vector {₁, ₂} in five-dimensional space for nonlinearities n=3 and n=5 together with the components of the stress vector S{S₁, S₂}. A relation between the vectors and S is established in terms of the given loading parameter k.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 241–245, March–April, 1970.     

Maximum monoreflections

It is shown that, in a category with a specified class of monics and under some mild hypothesis,there is a monoreflection maximum among those whose reflection maps lie in . Thus, for example, any variety, and most SP-classes in a variety, have both amaximum monoreflection and amaximum essential reflection (which might be the same, but frequently aren't, and which might be the identity functor, but frequently aren't). And, for example, under some mild hypotheses, beneath each completion lies a maximum monoreflection, so that, for example, any category of rings has amaximum functorial ring of quotients.     

Variational Inequalities and the Elastic-Plastic Torsion Problem

We show that, under suitable conditions, the variational inequality that expresses the elastic-plastic torsion problem is equivalent to a variational inequality on a convex set which depends on (x)=d(x, ). Such an equivalence allows us to find the related Lagrange multipliers and to exhibit a computational procedure based on the subgradient method.     

Use of the augmented penalty function in mathematical programming problems,part 2

In this paper, the problem of minimizing a functionf(x) subject to a constraint (x)=0 is considered. Here,f is a scalar,x ann-vector, and aq-vector, withq<n. The use of the augmented penalty function is explored in connection with theconjugate gradient-restoration algorithm. The augmented penalty functionW(x, ,k) is defined to be the linear combination of the augmented functionF(x, ) and the constraint errorP(x), where theq-vector is the Lagrange multiplier and the scalark is the penalty constant.The conjugate gradient-restoration algorithm includes a conjugate-gradient phase involvingn-q iterations and a restoration phase involving one iteration. In the conjugate-gradient phase, one tries to improve the value of the function, while avoiding excessive constraint violation. In the restoration phase, one reduces the constraint error, while avoiding excessive change in the value of the function.Concerning the conjugate-gradient phase, two classes of algorithms are considered: for algorithms of Class I, the Lagrange multiplier is determined so that the error in the optimum condition is minimized for givenx; for algorithms of Class II, the Lagrange multiplier is determined so that the constraint is satisfied to first order. For each class, two versions are studied. In version (), the penalty constant is held unchanged throughout the entire algorithm. In version (), the penalty constant is updated at the beginning of each conjugate-gradient phase so as to achieve certain desirable properties.Concerning the restoration phase, the minimum distance algorithm is employed. Since the use of the augmented penalty function automatically prevents excessive constraint violation, single-step restoration is considered.If the functionf(x) is quadratic and the constraint (x) is linear, all the previous algorithms are identical, that is, they produce the same sequence of points and converge to the solution in the same number of iterations. This number of iterations is at mostN*=n–q if the starting pointx _s is such that (x _s)=0 and at mostN*=1+n–q if the starting pointx _s is such that (x _s)0.In order to illustrate the theory, seven numerical examples are developed. The first example refers to a quadratic function and a linear constraint. The remaining examples refer to nonquadratic functions and nonlinear constraints. For the linear-quadratic example, all the algorithms behave identically, as predicted by the theory. For the nonlinear-nonquadratic examples, algorithms of Class II generally exhibit faster convergence than algorithms of Class I, and algorithms of type () generally exhibit faster convergence than algorithms of type ().This research was supported by the National Science Foundation, Grant No. GP-27271.     

On a Type of Mixed Norm Spaces

In this paper the mixed norm spaces L(B,p,q) and their duals are investigated. In the case p,q < it is proved that the dual of L(B,p,q) is L(B,p,q), where p-1 + p-1 = 1 and q-1 + q-1 = 1. For p = 2 and q = an isometric isomorphism is discussed between the mixed norm space L(B,2,) and L(B,2), the L-space of 2-valued functions. Here a measurability theorem is proved for 2-valued functions. The dual of an important subspace of L(B,2,) is characterized as a space of vector measures. Finally, as an application we show that if B is finitely generated then the dual of L(B,2,) is L(B,2,1).     

Maximum principle and second-order conditions for minimax problems of optimal control

In this paper, we study the optimal control problem of minimizing the functionalJ(x, u)=max_t1tt2(x(t),t). We formulate and prove necessary optimality conditions for this problem. We establish the equivalence between the initial minimax problem and a problem involving a terminal functional and phase constraints.     

Saturation theorems for discretized linear operators

, , , - .

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Part of this research was completed at a conference of Hungarian and American mathematicians held in Madison, Wisconsin, August 1974, and sponsored by the National Science Foundation (USA) and the Institute for Cultural Relations (Hungary).

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The first author gratefully acknowledges NSF support in Grant GP 19620.     

Spectral synthesis on a system of unbounded domains starlike in a common direction

, . . - ₁, ..., ₄, — ; =(₁,)×...×H(₄), — H(₁, ..., H(₄), r H(₁) — , ₁ ; D: HH- . , D. , ₁..., ₄ , (.. z₁ z+te^ia ₁ t>0), W H .     

Finite rank,relatively bounded perturbations of semigroups generators

Summary This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator A_F=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b H. While Part I studied the question of generation of a s.c. semigroup on H by A_F and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_F, given A and a H, via a suitable vector b H; alternatively, given A, via a suitable pair of vectors a, b H; (ii) spectrality of A_F—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {_n} of A, the given vector a H and a given suitable sequence {_n} of nonzero complex numbers, which guarantee the existence of a suitable vector b H such that A_F possesses the following two desirable properties: (i) the eigenvalues of A_F are precisely equal to _n+_n; (ii) the corresponding eigenvectors of A_F form a Riesz basis (a fortiori, A_F is spectral). While finitely many _ns can be preassigned arbitrarily, it must be however that _n 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Direct and converse imbedding theorems for seminormed spaces

. n=2 . ., 103 498      

Exact penalty functions in proximal bundle methods for constrained convex nondifferentiable minimization

This paper presents new versions of proximal bundle methods for solving convex constrained nondifferentiable minimization problems. The methods employ ₁ or exact penalty functions with new penalty updates that limit unnecessary penalty growth. In contrast to other methods, some of them are insensitive to problem function scaling. Global convergence of the methods is established, as well as finite termination for polyhedral problems. Some encouraging numerical experience is reported. The ideas presented may also be used in variable metric methods for smooth nonlinear programming.This research was supported by the Polish Academy of Sciences.     

On the absolute summability of trigonometric series

{_n} a _{n n} , s _n n- , _n -n- (, 1)- . . . -, . , «|s₂n+1–s ₂ n¦< . .» «¦ _n – _n–1¦< . .» . s. , {a _n } 0, a _n/n<, {a _n } , {it|_n¦<. , , -.     

Trigonometric Cesàro bases in the spaces of functions integrable with power weight

(2), k1, >0, L _p (0,), 1p L =C. , , p, k, (C, )- L _p (0,), , , {sinnx} _n ^=k/ (C, )- L _p (0,) |x|^p . , 1p, {x sinnx} _n=k , k2 2k–2–1/p<2k–1/p, (C, )- L ^p [0,] , >–(p–1)/.     

A-stable spline-collocation methods of multivalue type

In this paper the general classV of spline-collocation methods presented by Mülthei is investigated. The methods ofV approximate solutions of first order initial value problems. ClassV contains as subclass the methods of so-called multivalue type, and in particular contains the generalized singly-implicit methods treated by Butcher.Any multivalue type representativeU V yields a matrix valued function corresponding toU, which characterizes the region of absolute stability ofU. If a sequence (U()) of multivalue type representatives ofV tending to some singlevalue type representative V is considered, it can easily be seen by the structure of , that the sequence of the greatest eigenvalues of the (.,) tends to the stability function corresponding to . This fact allows one to construct one-parameter families of A-stable methods of multivalue type.     

Integrability of double trigonometric series with special coefficients

- , , . , L ¹=L ¹([0, ]×]0, ]). , ; , L ¹ , - . . . . 1976 ., ; 1989 .

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The basic part of this research was done while the author was a visiting professor at the Syracuse University, U.S.A., during the academic year 1986/87.     

Asymptotic analysis of the exponential penalty trajectory in linear programming

We consider the linear program min{cx: Axb} and the associated exponential penalty functionf _r(x) = cx + rexp[(A _ix – b_i)/r]. Forr close to 0, the unconstrained minimizerx(r) off _r admits an asymptotic expansion of the formx(r) = x ^* + rd^* + (r) wherex ^* is a particular optimal solution of the linear program and the error term(r) has an exponentially fast decay. Using duality theory we exhibit an associated dual trajectory(r) which converges exponentially fast to a particular dual optimal solution. These results are completed by an asymptotic analysis whenr tends to : the primal trajectory has an asymptotic ray and the dual trajectory converges to an interior dual feasible solution.Corresponding author. Both authors partially supported by FONDECYT.     

A globally convergent primal—dual interior point algorithm for convex programming

In this paper, we study the global convergence of a large class of primal—dual interior point algorithms for solving the linearly constrained convex programming problem. The algorithms in this class decrease the value of a primal—dual potential function and hence belong to the class of so-called potential reduction algorithms. An inexact line search based on Armijo stepsize rule is used to compute the stepsize. The directions used by the algorithms are the same as the ones used in primal—dual path following and potential reduction algorithms and a very mild condition on the choice of the centering parameter is assumed. The algorithms always keep primal and dual feasibility and, in contrast to the polynomial potential reduction algorithms, they do not need to drive the value of the potential function towards — in order to converge. One of the techniques used in the convergence analysis of these algorithms has its root in nonlinear unconstrained optimization theory.Research supported in part by NSF Grant DDM-9109404.     

Spektrale Invarianz bei verallgemeinerten Eigenfunktionsentwicklungen

Let T be a selfadjoint operator in a Gelfand triplet H. Examples show that there can occur generalized eigenvalues of T which are not in the Hilbert space spectrum (T) of T. Moreover, the fuction d, which assigns to each real number s the dimension of the generalized eigenspace corresponding to s, can be essentially greater then the von Neumann multiplicity function of T. We therefore construct a new triplet H, closely related to the given Gelfand triplet, according to which the set of generalized eigenvalues of T is contained in (T), and the function d essentially equals the von Neumann multiplicity function of T. Then, in particular, the closure of the set of generalized eigenvalues equals (T). The expansion theorems in H are transferred to H.

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Grundlage dieser Arbeit ist ein Teil meiner Dissertation. Ich danke Herrn Prof. Dr. H.G. Tillmann für die Anregung hierzu und für viele wertvolle Hinweise.     

On Riesz means with respect to a cylindric distance function

(1–) ₊ , — R ⁿ =R ^j ×R ^k , ()=max{¦ ₁¦, ¦ ₁¦},=( ₁, ₂), ₁R ^J , ₂R ^k ,j,k1,n=j+k. n=3 , (1–) ₊ [L ¹(R ⁿ )]¹, >1/2; j=4, (1–) ₊ R L ^p (R ⁿ ). .

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The author would like to thank Professor W. Trebels for encouragement and valuable advice.