Spectral approximation in L(H)

We introduce the concept of numerical range approximant which is related to that of a spectral approximant. We obtain a variant of Bouldin result on spectral approximation of normal operators [3] and an analogous result for numerical range approximation . We also extend Maher's result on partially isometric approximation [9] and Halmos' on positive approximation [6].     

Learning-by-doing may not be enough to sustain competitiveness in a market

The economic order quantity model and its variants are the oldest reported scientific inventory models in the literature. They are mathematically simple to use and understand. This characteristic contributed to their popularity. However, they have limitations that restrict their application in practice. Researchers have extended those models by relaxing some of their restrictive assumptions. One of the main limitations of the EOQ model is the estimation of its cost parameters, such as setup and holding costs, which include some hidden (or difficult to estimate) components. Emerging research proposes using the second law of thermodynamics when to measure the hidden (entropy) costs of inventory systems. Another research stream focuses on learning-by-doing, which reduces the unit cost of a product, thus reducing its price and enhancing a firm's competitiveness. This paper assumes that a buyer receives a shipment from its vendor following the EOQ model. The buyer invests in the vendor's process to accelerate learning (mainly workers training, among other things). The buyer's unit purchase cost reduces because of learning at the vendor's side. The buyer's competitors also learn, increasing competitiveness. For this purpose, we develop an EOQ model where price-dependent demand is analogous to a heat transfer equation, which is used to determine the entropy cost. It accounts for improvement by learning as a result of the buyer's investment to compete in a market. The implementation and management of improvement programs are not usually smooth, resulting in disorder quantified by an entropy cost function. The developed EOQ model is investigated with and without entropy costs. Numerical results representing different scenarios are provided to illustrate the behavior of the models and identify the factors affecting the decision variables and, subsequently, the buyer's profitability. The results showed that in a competitive market, the buyer has to maintain a learning rate faster than those of its competitors to ensure a prolonged, but indefinite, profitability. The results also showed that investment in learning is meaningful when the competition is fierce.     

Spurious modes in finite-element discretizations of the wave equation may not be all that bad

If higher-order finite elements are used to discretize the wave equation, spurious modes may occur. These modes are classified as unphysical and supposedly make elements of high order useless for accurate computations. This is in conflict with numerical experiments which appear to provide good results. Here Fourier analysis is used to investigate the behaviour of the numerical error for a number of higher-order one-dimensional finite elements. It is shown that the spurious modes have a contribution to the numerical error that behaves in a reasonable manner, and that higher-order elements can be more accurate than lower-order elements. Lumped elements with Gauss–Lobatto nodes appear to be the best choice.     

A class of (4,2)-disk pairs which are trivial

Suppose Δ?S³ is a ribbon disk and let D (Δ) denote the cononical properly embedded 2-disk obtained by pushing the interior of Δ into B⁴. A well-known conjecture states that the disk pair (B⁴, D(Δ)) is trivial provided the sphere pair ?(B⁴, D(Δ)) is trivial. We show here that the conjecture is true for those D(Δ) with the property that there is an embedded 2-disk, D²?S³, whose boundary is ?D(Δ) and which intersects Δ in ‘transverse double points’.     

集合H上矩阵A的左(右)逆、伪左(右)逆

以集合 S与空集Φ的交、并运算为背景 ,定义集合 H ={ 0 ,1 }中的加法与乘法运算 0 ,并考虑 H上一个 s×n级矩阵的左逆矩阵、右逆矩阵以及伪左逆矩阵、伪右逆矩阵的定义 ,并且证明了矩阵 A有左、右逆矩阵 ,A有伪左、右逆矩阵的充分必要条件 .     

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let L _p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L _p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L _p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L _p ⁰ (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S _p (G, H) be the L ₁(G) Banach module of functions in L _p (G, H) having unconditionally convergent Fourier series in L _p -norm. We show that S _p (G, H) coincides with the derived space L _p ⁰ (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L _p ⁰ (G, H) = L ₂(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L _p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L _p -norm, then F ∈ L ₂(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L _p ⁰ (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l ²(Γ, H).     

算子子空间的渐近自反性和遗传性

研究了算子子空间的渐近自反性问题，渐近自反子空间的遗传斯近自反性以及某些单个算子的渐近自反性．我们也讨论了投影网类的浙近自反性。     

A characterizing property of commutative Banach algebras may not be sufficient only on the invertible elements

In this article, we construct a commutative unital Banach algebra, in which the property $6a^{2}6={6a6}^2$ is true for the invertible elements, but cannot be extended to the whole algebra.     

The interior of the Šilov boundary of M(G) is trivial

Let G be a non-discrete locally compact abelian group, and let M(G) be the convolution algebra of regular bounded Borel measures on G. Let Γ denote the dual group of G. Then the interior of the ?ilov boundary of M(G) is exactly Γ. The proof uses generalized Riesz products for the compact metrizable case and standard liftings from that case.     

Multigrid in H (div) and H (curl)

Summary. We consider the solution of systems of linear algebraic equations which arise from the finite element discretization of variational problems posed in the Hilbert spaces and in three dimensions. We show that if appropriate finite element spaces and appropriate additive or multiplicative Schwarz smoothers are used, then the multigrid V-cycle is an efficient solver and preconditioner for the discrete operator. All results are uniform with respect to the mesh size, the number of mesh levels, and weights on the two terms in the inner products. Received June 12, 1998 / Revised version received March 12, 1999 / Published online January 27, 2000     

The exact condition of the B-spline basis may be hard to determine

The author's conjecture concerning the knot sequence whose associated B-spline sequence has maximum max-norm condition number is disproved. Related condition numbers are explored and the corresponding conjecture concerning the “worst” knot sequence for them is further supported by numerical results.     

SL(2,R)上一类极大算子的(H∞,s1,L1)型

本文我们引入了函数类B_δ(G//K)={φ∈L¹(G//K)||φ(t)|≤Δ^-1(t)(1+t)^1-δ,δ>0),对f∈L^p(G//K),1≤p≤∞,和极大算子(?),证明了这类算子是(H_∞,s¹,L¹)型的.     

Extension of Operators Defined on Reflexive Subspaces of L 1 or L 1/H 1 0

Inperpolation theory is used to develop a general pattern for proving the extension theorems mentioned in the title. In the case where the range space G is a w ^*-closed subspace of L or H with reflexive annihilator F, a necessary and sufficient condition on G is found for such an extension to always be possible. Specifically, F must be Hilbertian and become complemented in L ^p (1 < p 2) after a suitable change of density. Bibliography: 17 titles.