The general Stone-Weierstrass problem and extremal completely positive maps

Let A be a C*-algebra with identity e and let B be a C*-subalgebra of A that contains e. We show that if B separates the pure states of A, then, for each n, B also separates the set ECP(A,ⁿ;I) of extremal completely positive unital maps of A into ⁿ, thus giving another equivalent condition for the general Stone-Weierstrass conjecture for C*-algebras.     

Equivalent formulations and necessary optimality conditions for the Lennard–Jones problem

The minimization of molecular potential energy functions is one of the most challenging, unsolved nonconvex global optimization problems and plays an important role in the determination of stable states of certain classes of molecular clusters and proteins. In this paper, some equivalent formulations and necessary optimality conditions for the minimization of the Lennard–Jones potential energy function are presented. A new strategy, the code partition algorithm, which is based on a bilevel optimization formulation, is proposed for searching for an extremal Lennard–Jones code. The convergence of the code partition algorithm is proved and some computational results are reported.     

A boundary-value problem for parabolic equations with general nonlocal conditions

We study a boundary-value problem with general two-point conditions with respect to the time coordinate, and periodic conditions on the spatial coordinates for Shilov-parabolic equations with constant coefficients. We construct the solution in the form of a Fourier series. We establish conditions for existence and uniqueness of a classical solution of the problem. We prove quantitative theorems on a lower bound for the small denominators that arise in solving the problem. Translated fromMatematichni Methody i Fiziko-mekhanichni Polya, Vol. 38, 1995.     

Equivalent conditions for hyperbolic coordinates

The existence of hyperbolic coordinates is equivalent to the pseudo orbits tracing property and expansiveness. Further equivalent conditions are also considered.     

f-Rings and the Stone-Weierstrass Theorem

Using an appropriate notion of separating subring, it is shown that the classical Stone-Weierstrass Theorem for compact Hausdorff spaces is ultimately a result about f-rings. As an application the constructively valid Stone-Weierstrass Theorem for compact completely regular frames is obtained.     

Existence theorems and necessary conditions for a general formulation of the minimum effort problem

The basic problem considered may be described briefly as follows. LetX,Y, andZ be normed linear spaces,T:D(T)→Y,S:D(S)→Z linear operators withD(T) \( \subseteq\) X andD(S) \( \subseteq\) X,Ω \( \subseteq\) X a convex set containing the zero elementθ, andJ a real-valued convex function defined onX×Y such that

1. J(x,y)?-0 for (x,y)teX×Y,
2. J(θ,θ)=0,
3. J(x,y)→+∞, as (∥x∥²+∥y∥²)^1/2→+∞.

Givenζ∈Y andη∈S[core _T Ω∩;D(S)], find an elementx=x ₀ which minimizesJ(x,ζ?Tx) on the set {x∈[Ω∩;D(S)∩;D(T)]:Sx=η}. The abovementioned problem, together with certain special cases, is analyzed using the classical techniques of functional analysis. Existence problems are considered for a certain class of closed linear operators. In particular, existence of an optimal solution is determined by evaluating a generalized Minkowski functional at the point (ζ,η) inY×Z. A necessary condition is presented for special cases, and corresponding characterizations of optimal solutions are made in terms of the adjoint operators. These results are applicable to linear minimum effort problems, constrained variational problems, optimal control of distributive systems, and certain ill-posed variational problems.     

On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem

We present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined. Different sets of conditions allow this examination to take place when null-space, range-space or Lagrangian methods are used to find stationary points for the quadratic program.This research supported in part by the Natural Sciences and Engineering Research Council, Canada.     

A Stone-Weierstrass theorem for certain discontinuous functions

Let A be point separating unital subalgebra of C(T) where T is a compact metric space. For each bounded function f:T→R which is continuous on the complement of a meagre subset of T there exists a sequence (w_n) of elements of the algebra A such that the sequence (w_n) convergence uniformly to the function f on each compact subset of the interior of the continuity points of the function f.     

Greedy algorithm for the general multidimensional knapsack problem

In this paper, we propose a new greedy-like heuristic method, which is primarily intended for the general MDKP, but proves itself effective also for the 0-1 MDKP. Our heuristic differs from the existing greedy-like heuristics in two aspects. First, existing heuristics rely on each item’s aggregate consumption of resources to make item selection decisions, whereas our heuristic uses the effective capacity, defined as the maximum number of copies of an item that can be accepted if the entire knapsack were to be used for that item alone, as the criterion to make item selection decisions. Second, other methods increment the value of each decision variable only by one unit, whereas our heuristic adds decision variables to the solution in batches and consequently improves computational efficiency significantly for large-scale problems. We demonstrate that the new heuristic significantly improves computational efficiency of the existing methods and generates robust and near-optimal solutions. The new heuristic proves especially efficient for high dimensional knapsack problems with small-to-moderate numbers of decision variables, usually considered as “hard” MDKP and no computationally efficient heuristic is available to treat such problems. Supported in part by the NSF grant DMI 9812994.     

广义双对称矩阵反问题

利用矩阵的奇异值分解讨论了一类广义双对称矩阵反问题,得到了此类矩阵反问题有解的充要条件及通解的表达式.     

Equivalence of the Stone-Weierstrass conjectures for C* and JB*-algebras

We prove that the Stone-Weierstrass conjectures for C*-algebras and JB*-algebras are equivalent.     

Equivalent conditions for n-equivalences modulo a classc of groups

In this work we define the notion of a mapf:XY to be ann-equivalence modulo a classC of groups. Then we show an equivalent condition, which is more close to a homological condition, in order to a mapf:XY to be ann-equivalence modulo a classC of groups. Finally, at least for a complexK which is finite and is a suspension of a connected space, the notion above is also given in terms of the mapf _#: [K, X][K, Y].     

Equivalent conditions of complete convergence for independent weighted sums

For a very general weight function, the equivalent conditions of complete convergence for weighted sums of independent but not necessary identically distributed random variables are given. The previous situation of only sufficient results except for particular weight functions is changed. These results may help deduce many known ones and bring to light richer content. Project supported by the National Natural Science Foundation of China (Grant No. 19671078). the Natural Science Foundation of the Highter Education Department of Cuangdong Province and the National Science Foundation of the Education Commission of Jiangsu Province.     

Equivalent conditions for representing analytic functions by exponential series

Let L() be an entire function of exponential type with simple zeros ₁, ₂, ...; let ¯D be the smallest closed convex set which contains all of the singularities of the function which is associated with L() in the sense of Borel. In [1] there are necessary and sufficient conditions on L() under which a function f(z) which is analytic in ¯D can be represented in D by a Dirichlet series with exponents ₁, ₂, ... We obtain new equivalent conditions on L().Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 91–104, July, 1976.