Partitioned matrices and Seidel convergence

Summary Without using spectral resolution, an elementary proof of convergence of Seidel iteration. The proof is based on the lemma (generalizing a lemma of P. Stein): If (A+A ^*)–B ^*(A+A ^*)B>0, whereB=–(P+L) ^–1 R,A=P+L (Lower)+R (upper), then Seidel iteration ofAX=Y ₀ converges if and only ifA+A ^*>0. This lemma has as corollaries not only the well-known results of E. Reich and Stein, but also applications to a matrix that can be far from symmetric, e.g.M=[A _ij ] ₁ ² , whereA ₂₁=–A ₁₂ ^* ,A ₁₁,A ₂₂ are invertible;A ₁₁ +A ₁₁ ^* =A₂₂+A ₂₂ ^* ; and the proper values ofA ₁₂ ^–1 A ₁₁,A ₁₂ ^*–1 A ₂₂ are in the interior of the unit disk.Supported under NSF GP 32527.Supported under NSF GP 8758.     

Optimal quadrature problem on classes defined by kernels satisfying certain oscillation properties

We consider some classes of 2π-periodic functions defined by a class of operators having certain oscillation properties, which include the classical Sobolev class and a class of analytic functions which can not be represented as a convolution class as its special cases. Let be the largest integer not bigger than x. We prove that on these classes of functions the rectangular formula

Multiadditive functions satisfying certain functional equations

Aequationes mathematicae - As a generalization of a result proved independently by Kurepa and Jurkat on additive functions on $${mathbb R}$$ satisfying certain functional equations, we determine...     

On generalized derivations satisfying certain identities

Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R. We study the generalized derivation d satisfying any of the following identities:

Elastic properties of certain polymer matrices at low temperatures

A study is made of the viscoelastic behavior of the polyimide DFO and polybenzoxasol (PBZ) at low temperatures. Pulsed ultrasound is used to measure ultrasonic velocity in polymers at the frequency f=5 MHz within the temperature interval 4.2–240 K. Data obtained from acoustic measurements is used to calculate the dynamic elastic modulus, shear modulus, compressive bulk modulus, Poisson's ratio, and certain thermophysical characteristics (Debye temperature, specific heat, coefficient of linear expansion, Grünheisen parameter). It is established that the investigated polymers have low dynamic moduli at 4.2 K. Their values change by 18–20% within the temperature range 4.2–240 K. The polymers also have a large free volume at the temperature of liquid helium. The results show that polyimide DFO and PBZ have good service properties at low temperatures.Paper presented at the IX International Conference on the Mechanics of Composite Materials, Riga, October, 1995.Moscow State Academy of Automotive and Tractor Engineering, Russia. Translated from Mekhanika Kompozitmykh Materialov, No. 4, pp. 454–459, July–August, 1996.     

Bernstein n-widths for classes of convolution functions with kernels satisfying certain oscillation properties

In this paper, we consider some classes of 2π-periodic convolution functions Bp, and Kp with kernels having certain oscillation properties, which include the classical Sobolev class as special case. With the help of the spectral of nonlinear integral equations, we determine the exact values of Bernstein n-width of the classes Bp, Kp in the space Lp for 1 〈 p 〈 ∞.     

Some properties of matrices from certain binary mathematical programming problems

It is shown that if a feasible solution to a binary mathematical programming problem is represented as an N × N matrix consisting of zeroes and ones, where there exists a single one in each row, then this matrix is periodic. From this result, it is proved that the eigenvalues of the constraint matrix are the N roots of unity.     

Eigenvectors of certain matrices

An algorithm is given for calculation of eigenvalues and eigenvectors of centrosymmetric and some related matrices, and some desirable properties of the algorithm are proved. Centrosymmetric matrices are characterized by a symmetry property of their eigenvectors and this result is used to establish a property of certain methods for the numerical solution of differential equations.     

Conformal changes of riemannian metrics satisfying certain conditions

Summary The authors try to find necessary and sufficient conditions for the existence of conformal changes of metrics of Riemannian manifolds which implies that the Riemannian manifolds are isometric to spheres. They first obtain a series of integral inequalities in Riemannian manifolds relative to conformal changes of metrics and then find necessary and sufficient conditions under which the Riemannian manifolds are isometric to spheres. Entrata in Redazione il 14 giugno 1976.     

Zeros for accretive operators satisfying certain boundary conditions

Let X be a Banach space with the dual space $\text{X}{}^1$ to be uniformly convex, let D ? X be open, and let $\text{T:}\text{D}\text{?}\text{→ X}$ be strongly accretive (i.e., for some k < 1: (λ ? k)∥ u ? v∥ ? ∥(λ ? 1)(u ? v)+ T(u) ? T(v)∥ for all $\text{u, v ?}\text{D}\text{?}$ and λ > k). Suppose T is demicontinuous and strongly accretive and suppose there exists z?D satisfying: T(x) t(x ? z) for all x??D and t < 0. Then it is shown that T has a unique zero in $\text{D}\text{?}$. This result is then applied to the study of existence of zeros of accretive mappings under apparently different types of boundary conditions on T.     

Maps between Banach function algebras satisfying certain norm conditions

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y, respectively, and let $\bar A$ and $\bar B$ be their uniform closures. Let I, I′ be arbitrary non-empty sets, α ∈ ?\{0}, ρ: I → A, τ: l′ → a and S: I → B T: l′ → B be maps such that ρ(I, τ(I′) and S(I), T(I′) are closed under multiplications and contain exp A and expB, respectively. We show that if ‖S(p)T(p′)?α‖Y=‖ρ(p)τ(p′) ? α‖ _x for all p ∈ I and p′ ∈ I′, then there exist a real algebra isomorphism S: A → B, a clopen subset K of M _B and a homeomorphism ?: M _B → M _A between the maximal ideal spaces of B and A such that for all f ∈ A, where $\hat \cdot$ denotes the Gelfand transformation. Moreover, S can be extended to a real algebra isomorphism from $\bar A$ onto $\bar B$ inducing a homeomorphism between $M_{\bar B}$ and $M_{\bar A}$ . We also show that under an additional assumption related to the peripheral range, S is complex linear, that is A and B are algebraically isomorphic. We also consider the case where α = 0 and X and Y are locally compact.     

Partitions of a set satisfying certain set of conditions

In this paper we present a new combinatorial class enumerated by Catalan numbers. More precisely, we establish a bijection between the set of partitions π₁π₂?π_n of [n] such that π_i+1−π_i≤1 for all i=,1,2…,n−1, and the set of Dyck paths of semilength n. Moreover, we find an explicit formula for the generating function for the number of partitions π₁π₂?π_n of [n] such that either π_i+?−π_i≤1 for all i=1,2,…,n−?, or π_i+1−π_i≤m for all i=1,2,…,n−1.     

On groups satisfying the minimal condition on certain subgroups

Periodica Mathematica Hungarica -     

Identities among certain triangular matrices

The object of this paper is to develop the ideas introduced in the author's paper [1] on matrices which generate families of polynomials and associated infinite series. A family of infinite one-subdiagonal non-commuting matrices Q_m is defined, and a number of identities among its members are given. The matrix Q₁ is applied to solve a problem concerning the derivative of a family of polynomials, and it is shown that the solution is remarkably similar to a conventional solution employing a scalar generating function. Two sets of infinite triangular matrices are then defined. The elements of one set are related to the terms of Laguerre, Hermite, Bernoulli, Euler, and Bessel polynomials, while the elements of the other set consist of Stirling numbers of both kinds, the two-parameter Eulerian numbers, and numbers introduced in a note on inverse scalar relations by Touchard. It is then shown that these matrices are related by a number of identities, several of which are in the form of similarity transformations. Some well-known and less well-known pairs of inverse scalar relations arising in combinatorial analysis are shown to be derivable from simple and obviously inverse pairs of matrix relations. This work is an explicit matrix version of the umbral calculus as presented by Rota et al. [24-26].