Comparison theorems for eigenvalues

Summary Comparison theorems are proved relating the smallest positive eigenvalues λ and λ* of two operator equations of type Au=λBu and A*v=λ*B*v, respectively. Sufficient conditions on the operators are given which guarantee that λ⩽λ*, with special reference to the case that A and A* are elliptic differential operators. One novelty of the theory is that B is not required to be positive. Two general techniques are described: 1) A generalization of the classical minimum principle for eigenvalues, which is appropriate for selfadjoint elliptic operators A of arbitrary even order; and 2) A differential identity related to Picone's identity, appropriate for nonselfadjoint second order elliptic operators and strongly elliptic quasilinear systems. Entrata in Redazione il 24 maggio 1972.     

Theory of Completeness for Logical Spaces

A logical space is a pair (A, B){(A, {\mathcal{B}})} of a non-empty set A and a subset B{{\mathcal{B}}} of P A{{\mathcal{P}} A} . Since P A{{\mathcal{P}} A} is identified with {0, 1}^A and {0, 1} is a typical lattice, a pair (A, F){(A, {\mathcal{F}})} of a non-empty set A and a subset F{{\mathcal{F}}} of \mathbbB^A{{\mathbb{B}}^A} for a certain lattice \mathbbB{{\mathbb{B}}} is also called a \mathbbB{{\mathbb{B}}} -valued functional logical space. A deduction system on A is a pair (R, D) of a subset D of A and a relation R between A* and A. In terms of these simplest concepts, a general framework for studying the logical completeness is constructed.     

The dimension conjecture for polydisc algebras

It is shown that form≠n the polydisc algebrasA(D ^m) andA(D ⁿ) are not isomorphic as Banach spaces. More precisely, there is no linear embedding of the dual spaceA(D ⁿ)* intoA(D ^m)* form<n. The invariant is infinite dimensional and is based on certain, multi-indexed martingales related to those considered by Davis et al. [10]. In the one-dimensional case, i.e. for the spaceA(D)*, a finite inequality is proved, implying thatA(D ²)* is not finitely representable inA(D)*. Extensions to algebras on products of strictly pseudoconvex domains are outlined. They imply in particular the non-isomorphism of certain algebras in the same number of variables, for instance A (D⁴) ≠ A (B₂xB₂).     

Property T for Actions

We study property T for an action α of a discrete group Γ on a unital C*-algebra $$\mathscr{A}$$ . Our main results improve some well-known results about property T for groups. Moreover, we introduce Hilbert $$\mathscr{A}$$ -module property T and show that the action α has property T if and only if the reduced crossed product $$\mathscr{A}\;{\rtimes_{\alpha, r}}$$ Γ has Hilbert $$\mathscr{A}$$ -module property T.     

Formulas for functions of ordered operators

In an algebra with a lattice of functions of ordered elements (e.g., in an algebra of operators), we investigate the expansions of functions of the type f(A+B) and ? (A ¹,B ² in powers of the commutators A, B. In particular, we obtain all the terms of the expansion $$f(A + B) = f(\mathop A\limits^1 + \mathop B\limits^2 ) + \frac{1}{2}[\mathop {\overline {A,B} }\limits^2 ] f^{(2)} (\mathop A\limits^1 + \mathop B\limits^3 ) + ....$$ A diagram method for a similar type of calculation is developed. Our discussion is based on Maslov's technique of ordered operators.     

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

Our first result is a ‘sum-product’ theorem forsubsets A of the finite field F_p, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums     

Some sieves for partition theory

A Combinatorial lemma due to Zolnowsky is applied to partition theory in a new way. Several even-odd formulas are derived. A combinatorial proof of the Jacobi triple-product identity is given. Some sieve formulas are proved which relate to the Euler pentagonal number theorem and the Rogers-Ramanujan identities and their generalizations. A formula is given for the number of partitions of n into parts not congruent to 0, ±x (mod y).     

A transfer morphism for Witt cogroups

Let A be a ring with involution in which 2 is invertible, ε be 1 or −1, and s(∈ A) be a central regular element such that s* = s. A transfer homomorphism _εW′₀(A/s) → _ε W′₁(A) for Witt cogroups is constructed and a projection formula is proved. Bibliography: 11 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 356, 2008, pp. 149–158.     

Hall Polynomials for Nakayama Algebras

Let \(\mathcal{A}\) be a representation finite algebra over finite field k. In this note we first show that the existence of Hall polynomials for \(\mathcal{A}\) equivalent to the existence of the Hall polynomial \(\varphi^{M}_{N L}\) for each \(M, L \in mod\mathcal{A}\) and \(N\in ind\mathcal{A}\). Then we show that for a basic connected Nakayama algebra \(\mathcal{A}\), \(\mathcal{H}(\mathcal{A})=\mathcal{L}(\mathcal{A})\) and Hall polynomials exist for this algebra. We also provide another proof of the existence of Hall polynomials for the representation directed split algebras.     

On three problems for sequences

If ξ∈ (0,1) and A=a_n, n?? is a sequence of real numbers define S_n(ξ,A)∶=Σ{a_k∶:k=[nξ]+1 to n}, n??, where [x] is the greatest integer less than or equal to x. In the theory of regularly varying sequences the problem arose to conclude from the convergence of the sequence S_n (ξ,A), n??, for all ξ in an appropriate set K of real numbers, that the sequence a_n, n??, converges to zero. It was shown that such a conclusion is possible if K={ξ,1?ξ} with ξ∈ (0,1) irrational. Then the following three questions were posed and will be answered in this paper:

1. does the convergence of S_n (ξ,A), n??, for a single irrational number ξ imply a_n→0.
2. does the convergence of S_n(ξ,A), n??, for finitely many rational numbers ξ∈ (0, 1) imply a_n→0.
3. does the convergence of S_n (ξ,A), n??, for all rational numbers ξ∈ (0,1) imply a_n→0?

     

The Kantor-Koecher-Tits construction for Jordan coalgebras

The relationship between Jordan and Lie coalgebras is established. We prove that from any Jordan coalgebra 〈L(A), Δ〉, it is possible to construct a Lie coalgebra 〈L(A), Δ_L〉. Moreover, any dual algebra of the coalgebra 〈L(A), Δ_L〉 corresponds to a Lie algebra that can be determined from the dual algebra for (A, Δ), following the Kantor-Koecher-Tits process. The structure of subcoalgebras and coideals of the coalgebra 〈L(A), Δ_L〉 is characterized. Supported by ISF grant No. RB 6000. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 173–189, March–April, 1996.     

A Commutative Algebra for Oriented Matroids

Let V be a vector space of dimension d over a field K and let A be a central arrangement of hyperplanes in V. To answer a question posed by K. Aomoto, P. Orlik and H. Terao construct a commutative K -algebra U(A) in terms of the equations for the hyperplanes of A. In the course of their work the following question naturally occurred: \circ Is U(A) determined by the intersection lattice L(A) of the hyperplanes of A? We give a negative answer to this question. The theory of oriented matroids gives rise to a combinatorial analogue of the algebra of Orlik—Terao, which is the main tool of our proofs. Received November 7, 2000, and in revised form May 18, 2001. Online publication November 2, 2001.     

行块矩阵M-P逆的充要条件

通过使用矩阵秩方法,证明了如下结果:[A,B]+=[αA+βB+]βAA*αBB*.[A,B][A,B]+=αAA++βBB+R(A)=R(B).这里,α+β=1,α>0,β>0.这两个结果是2007年田永革在国际线性代数学会会刊中获得的相应结果的推广.     

正则局部环的判别法（英）

本文给出了非Artin的Noether局部环(A, m)为正则环的充要条件.设n为正整数，则(A, m)为正则环的充要条件为A/mⁿ的投射维数或mⁿ的内射维数是有限的.     

Centralizing and normalizing constants for extrema and for records

A series of relations are given for the domains of attraction of limiting distributions and the corresponding centralizing and normalizing constants for extremal order statistics, record values, and some generalizations of record values.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 103–111, 1988.     

交换代数的lazy上循环

该文对于Hopf代数H和交换代数A, 给出了lazy左2 -上循环$\sigma$的定义, 并且证得lazy 2 -上循环和lazy H-cleft对象之间存在着对应关系.     

Sensitivity of Bayes decisions for the success probability for samples from a nonbinomial population

Summary A family of generalised negative binomial distributions is employed to investigate inference robustness of the Bayes estimator of the unknown parameter of the binomial distribution. A zone of sensitivity for the test of significance is constructed to forewarn the pro-Jeffreys Bayesians against indiscriminate choice of the probability in favour of the null hypothesis. A few selected tables are presented to illustrate the effect of relaxation of the ‘binomiality’ assumption.     

A construction of multiscale bases for Petrov–Galerkin methods for integral equations

In this paper, a construction of multiscale bases for Petrov–Galerkin methods for Fredholm integral equations of the second kind is proposed. The properties of multiscale bases are presented including additional order of vanishing moments, compact supports and stability. Communicated by A. Zhou Dedicated to Professor Charles A. Micchelli on the occasion of his sixtieth birthday with friendship and esteem Mathematics subject classifications (2000) 41A10, 65R20, 65D15. Min Huang: Supported in part by Professor Yuesheng Xu's support under the program of “One Hundred Distinguished Young Scientists” of the Chinese Academy of Sciences and by the Graduate Innovation Foundation of the Chinese Academy of Sciences.     

Rank formulae for factorized groups

The following inequalities for the torsion-free rank r₀(G) of the group G=AB and for the p^∞-rank r_p(G) of the soluble-by-finite group G=AB are stated: $$\begin{gathered} r_0 (G) \leqslant r_0 (A) + r_0 (B) - r_0 (A \cap B), \hfill \\ r_p (G) \leqslant r_p (A) + r_p (B) - r_p (A \cap B). \hfill \\ \end{gathered} $$