Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

Non-linear Grassmannians as coadjoint orbits

For a given manifold M we consider the non-linear Grassmann manifold Gr _n (M) of n–dimensional submanifolds in M. A closed (n+2)–form on M gives rise to a closed 2–form on Gr _n (M). If the original form was integral, the 2–form will be the curvature of a principal S ¹ –bundle over Gr _n (M). Using this S ¹ –bundle one obtains central extensions for certain groups of diffeomorphisms of M. We can realize Gr _m–2 (M) as coadjoint orbits of the extended group of exact volume preserving diffeomorphisms and the symplectic Grassmannians SGr _2k (M) as coadjoint orbits in the group of Hamiltonian diffeomorphisms. Mathematics Subject Classification (2000):58B20Both authors are supported by the Fonds zur Förderung der wissenschaftlichen Forschung (Austrian Science Fund), project number P14195-MAT     

Prime non-Lie modules for Mal'tsev superalgebras

A module V for a superalgebra A is called prime if any two of its nonzero submodules have a nonzero intersection, and no nonzero submodule is annihilated by a nonzero ideal of A. We prove that if V is a prime module for a Mal'tsev superalgebra M = M₀+M₁, one of the following cases is realized:

Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).     

Criteria for Convergence of the Number of Near Maxima for Long Tails

Let M_n denote the maximum of a random sample of size n and K_n(a) be the number of near maxima, i.e. the number of sample observations in the fixed-width window (M_n – a, M_n]. There is a known integral criterion for almost sure convergence (to unity) of K_n(a), and we establish a similar criterion for complete convergence. We obtain simple but quite general sufficient conditions on the survivor function for satisfying the integral criteria. Further insight is obtained by seeking the rate at which P(K_n(a > 1)) tends to zero.AMS 2000 Subject Classification. 62G30, 60F15     

Regular J -classes of Subspace Semigroups

Subspace semigroup S(M _n (K)) of the matrix algebra over a field K is studied. Idempotents and regular J -classes are characterized and some symmetries of S(M _n (K)) are established.     

Distribution of integral points on some hyperboloids of one sheet

Results of B. F. Skubenko (Izv. Akad. Nauk SSSR, Ser. Mat.,26, 721–752 (1962)) are generalized to indefinite ternary quadratic formsf(x)=f ₀(Cx), which are contained in the simplest formf ₀(x)=x ₁ x ₃–x _r ^e We prove that the integral points on the hyperboloid of one sheetf(x)=m,m<0, are uniformly distributed over area (in the sense of hyperbolic metric) and over residue classes for given modulus.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 121, pp. 83–93, 1983.     

Dirac index classes and the noncommutative spectral flow

We present a detailed proof of the existence-theorem for noncommutative spectral sections (see the noncommutative spectral flow, unpublished preprint, 1997). We apply this result to various index-theoretic situations, extending to the noncommutative context results of Booss–Wojciechowski, Melrose–Piazza and Dai–Zhang. In particular, we prove a variational formula, in K_*(C_r^*(Γ)), for the index classes associated to 1-parameter family of Dirac operators on a Γ-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K_*(C_r^*(Γ)), for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r:M→BΓ when we assume that M is the union along a hypersurface F of two manifolds with boundary M=M_{+ F} M₋. Finally, we prove a defect formula for the signature-index classes of two cut-and-paste equivalent pairs (M₁,r₁:M₁→BΓ) and (M₂,r₂:M₂→BΓ), where

M₁=M_{+ (F,φ1)} M₋, M₂=M_{+ (F,φ2)} M₋

and φ_jDiff(F). The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F. We give applications to the problem of cut-and-paste invariance of Novikov's higher signatures on closed oriented manifolds.     

The Semigroup Generated by the Similarity Class of a Singular Matrix

Let A be a singular matrix of M _n (𝕂), where 𝕂 is an arbitrary field. Using canonical forms, we give a new proof that the sub-semigroup of ( _n (𝕂), ×) generated by the similarity class of A is the set of matrices of M _n (𝕂) with a rank lesser than or equal to that of A.     

Finite Topological Type and Volume Growth

We show that a complete noncompact n-dimensional Riemannian manifold Mwith Ricci curvature Ric _M –(n – 1) and conjugateradius conj _M c > 0 has finite topological type, provided that the volume growth of geodesic balls in M is not very far from that of the balls in an n-dimensional hyperbolic space H ⁿ (–1)of sectional curvature –1. We also show that a complete open Riemannian manifold M with nonnegative intermediate Ricci curvature and quadratic curvature decay has finite topological typeif the volume of geodesic balls of M around the base point grows slowly.     

由本性随机阵导致的陈代数

A matrix of order n whose row sums are all equal to 1 is called an essentially stochastic matrix (see Johnsen [4]). We extend this notion as the following. Let F be a field of characteristic 0 or a prime greater than n. M_n(F) denotes the set of all n×n matrices over F. Let t be an elernent of F. A matrix A=(a_ij) in M_n(F) is called essentially t-stochastic' provided its row sums are each equal to t. We denote by R_n(t) the set of all essentially t-stochastic matrices over F. We shall mainly study R_n(0) and R_n(F)=(?)R_n(t). Our main references are Johnson [2,4] and Kim [5].     

A construction of maximal linearly independent array

Let M be a subset of r-dimensional vector space Vτ （F2） over a finite field F2, consisting of n nonzero vectors, such that every t vectors of M are linearly independent over F2. Then M is called （n, t）-linearly independent array of length n over Vτ（F2）. The （n, t）-linearly independent array M that has the maximal number of elements is called the maximal （r, t）-linearly independent array, and the maximal number is denoted by M（r, t）. It is an interesting combinatorial structure, which has many applications in cryptography and coding theory. It can be used to construct orthogonal arrays, strong partial balanced designs. It can also be used to design good linear codes, In this paper, we construct a class of maximal （r, t）-linearly independent arrays of length r ＋ 2, and provide some enumerator theorems.     

Quantum -modules and generalized mirror transformations

In the previous paper [Hiroshi Iritani, Quantum D-modules and equivariant Floer theory for free loop spaces, Math. Z. 252 (3) (2006) 577–622], the author defined equivariant Floer cohomology for a complete intersection in a toric variety and showed that it is isomorphic to the small quantum D-module after a mirror transformation when the first Chern class c₁(M) of the tangent bundle is nef. In this paper, even when c₁(M) is not nef, we show that the equivariant Floer cohomology reconstructs the big quantum D-module under certain conditions on the ambient toric variety. The proof is based on a mirror theorem of Coates and Givental [T. Coates, A.B. Givental, Quantum Riemann — Roch, Lefschetz and Serre, Ann. of Math. (2) 165 (1) (2007) 15–53]. The reconstruction procedure here gives a generalized mirror transformation first observed by Jinzenji in low degrees [Masao Jinzenji, On the quantum cohomology rings of general type projective hypersurfaces and generalized mirror transformation, Internat. J. Modern Phys. A 15 (11) (2000) 1557–1595; Masao Jinzenji, Co-ordinate change of Gauss–Manin system and generalized mirror transformation, Internat. J. Modern Phys. A 20 (10) (2005) 2131–2156].     

A rigidity result for highest order local cohomology modules

Let M be a finitely generated faithful module over a noetherian ring R of dimension d < ￥ \infty and let \mathfrak a \subseteqq R {\mathfrak a} \subseteqq R be an ideal. We describe the (finite) set Supp_R(H_{\mathfrak a}^d (M)) = Ass_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) = \textrm{Ass}_R(H_{\mathfrak a}^d (M)) of primes associated to the highest local cohomology module H_{\mathfrak a}^d (M) H_{\mathfrak a}^d (M) in terms of the local formal behaviour of \mathfrak a {\mathfrak a} . If R is integral and of finite type over a field, Supp_R(H_{\mathfrak a}^d (M)) \textrm{Supp}_R(H_{\mathfrak a}^d (M)) is the set of those closed points of X = Spec(R) whose fibre under the normalization morphism n: X￠? X \nu : X' \rightarrow X contains points which are isolated in n^-1(Spec(R/\mathfrak a)) \nu^{-1}(\textrm{Spec}(R/{\mathfrak a})) .     

The curvature of the Sasaki metric of tangent sphere bundles

We study the sectional curvaturesK of the Sasaki metric of tangent sphere bundles over spaces of constant curvatureK(T ₁(M ⁿ, K)). We give precise bounds on the variation of the Ricci curvature and a bound on the scalar curvature ofT ₁ (M ⁿ, K) that is uniform onK. In an appendix we calculate and give lower bounds for the lengths of closed geodesics onT ₁ S ⁿ. titles.Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 132–145.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

Limits on the real line of symmetric spaces on segments

In the same way as the known spacesM _p ,M _p , andI _p are constructed on the basis of the spaceL _p (–1, 1), we construct the corresponding limit spacesM _E ,M _E , andI _E on the real line on the basis of a symmetric function spaceE on a segment and study some of their Banach properties.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 46–55, January, 1995.     

Asymptotic distribution of sequences with random indices

A general form is determined for the limit distribution function of a sequence of random vectors with random indices (S₁(N _n ⁽¹⁾ ) ..., Sr(N_n ^(r)) in the case when sequences (S₁(n), ..., S_r(n)) and (N _n ⁽¹⁾ , ..., N _n ^(r) ) for appropriate normalization have a nonsingular joint limit distribution.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 705–712, December, 1969.     

Products of Ratios of Consecutive Integers

Take the product of the numbers (n/(n+1))^∊_n for 1≤ n < N, where each ∊_n is ± 1. Express the product as a/b in lowest terms. Evidently the minimal possible value for a over all choices for ∊_n is 1; just take each ∊_n = 1, or each ∊_n = 0. Denote the maximal possible value of a by A(N). It is known from work of Nicolas and Langevin that (log 4+o(1))N≤ log A(N)≤(2/3+o(1))Nlog N. Using the Rosse–Iwaniec sieve, we improve the lower bound to the same order of magnitude as the upper bound.For Jean-Louis Nicolas, on his sixtieth birthday2000 Mathematics Subject Classification: Primary—11N56; Secondary—11N36     

Metrics on products of surfaces with non-positive sectional curvature

Let (S _i, g_i),i=1, 2 be two compact riemannian surfaces isometrically embedded in euclidean spaces. In this paper we show that ifM=S ₁×S₂,then for any functionF: M→R, the graph ofF, i.e. the manifold {(x, F(x)): x∈M}, does not have positive sectional curvature.