A generalization of A ∘ A−1 ⩾I

A matrix inequality is obtained, in an elementary way, for the Schur product of two positive definite matrices. The inequality generalizes a well-known result due to Fiedler which asserts that if A is a positive definite matrix, then A ∘ A^-1-I is a positive semidefinite matrix, where ∘ denotes the Schur multiplication.     

Some remarks concerning matrices of the form A–A′, A−1A′

Matrices and operators of the formA ^–1 A ^* have received a certain amount of attention in recent years. Here some of the literature is surveyed and the caseA ^–1 A is studied for complex matrices withA denoting the transpose ofA. A generalization ofA ^–1 A is introduced.

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Zusammenfassung Matrizen und Operatoren, die in der FormA ^–1 A ^* ausgedrückt werden können, sind in den letzten Jahren häufig studiert worden. Hier wird ein Ueberblick über Teil der relevanten Literatur gegeben und auch der FallA ^–1 A für komplexe Matrizen studiert, wobeiA die Transponierte vonA ist. Es wird auch eine Verallgemeinerung vonA ^–1 A eingeführt.

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Dedicated to Eduard Stiefel by a long time admirer     

A 1Ψ1 Summation Theorem for Macdonald Polynomials

In this note we extend the Ramanujan's 11 summation formula to the case of a Laurent series extension of multiple q-hypergeometric series of Macdonald polynomial argument [7]. The proof relies on the elegant argument of Ismail [5] and the q-binomial theorem for Macdonald polinomials. This result implies a q-integration formula of Selberg type [3, Conjecture 3] which was proved by Aomoto [2], see also [7, Appendix 2] for another proof. We also obtain, as a limiting case, the triple product identity for Macdonald polynomials [8].     

满足Cond_l(A)=1(l=1,∞)的矩阵A的充要条件

本文导出了长方矩阵A的条件数Cond(A)=||A||l||A~ ||l=1(l=1,∞)的充要条件;而且给出了Cond(A)=1时A的矩阵结构和元素结构形式;同时指出文[6]中两个主要定理的错误.     

载脂蛋白 A1基因多态性的研究进展

载脂蛋白A1（Apo A1）是HDL的重要组成部分,参与激活软磷脂胆固醇酰基转移酶、识别HDL受体,参与胆固醇的逆向转运,是脂质代谢的重要调节者。其对脂质分子有高亲和力,因而具有抗动脉粥样硬化、抗内毒素血症、抗炎和抗氧化作用。其能与ATP结合转运体A1结合,使游离的胆固醇从表皮组织或血管内皮的巨噬细胞中流出[1];能与脂多糖结合蛋白-脂多糖（LBP-LPS）复合物结合,抑制炎症反应[2]。Apo A1在临床上显示了重要的作用,因此,国内外针对Apo A1基因的研究较多,在多态性方面的研究主要集中在包括肥胖及冠心病、2型糖尿病、阿尔茨海默病（AD）、胆囊结石在内的多种疾病上的研究,现将Apo A1基因的多态性研究进展综述如下。     

矩阵方程组A1XB1=C1,A2XB2=C2的相容性与通解

本文推广了R.Penrose关于矩阵方程组AX=C,XB=D的工作,给出了矩阵方程组 A_1XB_1=C_1,A_2XB_2=C_2的相容性条件,给出了通解表达式以及唯一解的充要条件。     

加强关系式sin A+cos A=1应用的训练

众所周知,关系式sin~2A+cos~2A=1是三角学中最基本的关系式之一。统编教材对其应用作了一定的安排,无疑对提高学生的三角式变形能力是有益的,但限于篇幅,无论深度或广度都很不够。据多年实践,我认为对关系式之应用,有必要进一步加强,在教学过程中不失时机地适当予以补充,从顺用、逆用、顺逆交替用等方面加强训练,充分发挥关系式之妙用,以利于培养学生恒等变形及综合运用知识的能力。一、顺用公式以达熟能生巧     

On the size of the set A(A + 1)

Let F _p be the field of a prime order p. For a subset A ì F_p{A \subset F_p} we consider the product set A(A + 1). This set is an image of A ×  A under the polynomial mapping f(x, y) = xy +  x : F _p ×  F _p → F _p . In the present note we show that if |A| <  p ^1/2, then

满足Condl(A)=1(l=1,∞)的矩阵A的充要条件

本文导出了长方矩阵A的条件数Cond_l(A)=‖A‖_l‖A⁺‖_l=1(l=1,∞)的充要条件;而且给出了Cond_l(A)=1时A的矩阵结构和元素结构形式;同时指出文[6]中两个主要定理的错误.     

When Min(A)−1 is Hausdorff

For a commutative ring with identity, say A, its collection of minimal prime ideals is denoted by Min(A). The hull-kernel topology on Min(A) is a well-studied structure. For example, it is known that the hull-kernel topology on Min(A) has a base of clopen subsets, and classifications of when Min(A) is compact abound. Recently, a program of studying the inverse topology on Min(A) has begun. This article adds to the growing literature. In particular, we characterize when Min(A)^?1 is Hausdorff. In the final section, we consider rings of continuous functions and supply examples.     

A Short Note on Essentially Σ1 Sentences

Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ₁ (i.e., it is Σ₁ under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ₁ formulas of languages including witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ ₀  + Supexp and we address a similar problem for IΔ ₀  + Exp.     

A Projection Proximal-Point Algorithm for ℓ1 Minimization

The problem of the minimization of least squares functionals with ?¹ penalties is considered in an infinite dimensional Hilbert space setting. Though there are several algorithms available in the finite dimensional setting there are only a few of them that come with a proper convergence analysis in the infinite dimensional setting.

In this work we provide an algorithm from a class that has not been considered for ?¹ minimization before, namely, a proximal-point method in combination with a projection step. We show that this idea gives a simple and easy-to-implement algorithm. We present experiments that indicate that the algorithm may perform better than other algorithms if we employ them without any special tricks. Hence, we may conclude that the projection proximal-point idea is a promising idea in the context of ?¹ minimization.     

A NOTE ON C~1-CURVED FINITE ELEMENT

This paper shows that the C1-curved finite element developed by Bernadon in general can not satisfy the essential boundary conditions on approximate boundary. Furthermore, a modified C1-curved finite element is given, which is compatible with the element of Argyris triangle and can satisfy the homogeneous Dirichlet boundary conditions onapproximate boundary.     

A note for tightening 0–1 models

We describe two approaches for 0–1 program model tightening that are based on the coefficient increasing and reduction methods proposed in Dietrich, Escudero and Chance (1993). We present some characterizations for the new formulations to be tighter than the original model. It can be shown that tighter models can be obtained even when applying any of both approaches to a redundant constraint; see Escudero and Muñoz (1998). We also present some situations where these approaches cannot be applied.     

A U(n) generalization of Ramanujan's 1Ψ1 summation

In this paper we derive a U(n) generalization of Ramanujan's ₁Ψ₁ summation directly from a recent U(n) multiple series refinement of the q-binomial theorem. We also obtain a new generalization of the Macdonald identities for A⁽¹⁾_l.     

LIMIT CYCLES FOR A CLASS OF E_3~1 SYSTEMWITH A FINE FOCUS AND A FINE SADDLE

1IntroductionManypracticalproblems,suchasnon--linearvibration,reactionofbiochemistrygandself-oscillatingsystem,etc.canbereducedtotheproblemsonthelimitcyclesofEjsystem.Therefore,thestudyoflimitcyclesforEjsystemhaspracticalsignificances.ThesocalledEdsystemi…     

A Tauberian theorem for ℓ-adic sheaves on \mathbb{A} 1

Let K ∈ L ¹(?) and let f ∈ L ^∞(?) be two functions on ?. The convolution $$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$ can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if $$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$ for some constant A, then $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$ We prove the following ?-adic analogue of this theorem: Suppose K, F, G are perverse ?-adic sheaves on the affine line $ \mathbb{A} $ over an algebraically closed field of characteristic p (p ≠ ?). Under suitable conditions, if $ \left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty } $ , then $ F|_{\eta _\infty } \cong G|_{\eta _\infty } $ , where η _∞ is the spectrum of the local field of $ \mathbb{A} $ at ∞.     

HARMONIC ANALYSIS ON THE QUOTIENT κ× ∝ κ \ A1 ∝ A

Let G be the semidirect product A¹ ∝ A of the adeles and the norm 1 ideles of a global field κ. Let Γ be the discrete subgroup κ^×∝ κ. In this paper the trace formula for this setting is established and used to give the complete decomposition of the G-representation on L²(Γ\G). It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.     

The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T

The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = [A1, A2] and B = [B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.