Recognition of finite groups by a set of orders of their elements

For G a finite group, ω(G) denotes the set of orders of elements in G. If ω is a subset of the set of natural numbers, h(ω) stands for the number of nonisomorphic groups G such that ω(G)=ω. We say that G is recognizable (by ω(G)) if h(ω(G))=1. G is almost recognizable (resp., nonrecognizable) if h(ω(G)) is finite (resp., infinite). It is shown that almost simple groups PGL_n(q) are nonrecognizable for infinitely many pairs (n, q). It is also proved that a simple group S₄(7) is recognizable, whereas A₁₀, U₃(5), U₃(7), U₄(2), and U₅(2) are not. From this, the following theorem is derived. Let G be a finite simple group such that every prime divisor of its order is at most 11. Then one of the following holds: (i) G is isomorphic to A₅, A₇, A₈, A₉, A₁₁, A₁₂, L₂(q), q=7, 8, 11, 49, L₃(4), S₄(7), U₄(3), U₆(2), M₁₁, M₁₂, M₂₂, HS, or M^cL, and G is recognizable by the set ω(G); (ii) G is isomorphic to A₆, A₁₀, U₃(3), U₄(2), U₅(2), U₃(5), or J₂, and G is nonrecognizable; (iii) G is isomorphic to S₆(2) or O ₈ ⁺ (2), and h(ω(G))=2. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 651–666, November–December, 1998.     

On a class of matrices which arise in the numerical solution of Euler equations

Summary We study block matricesA=[A_ij], where every blockA _ij ^k,k is Hermitian andA _ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld     

Rank One Perturbations with Infinitesimal Coupling

We consider a positive self-adjoint operator A and formal rank one perturbations B = A + α(φ, ·)φ, where φ ₋₂(A) but φ ₋₁ (A), with _s(A) the usual scale of spaces. We show that B can be defined for such φ and what are essentially negative infinitesimal values of α. In a sense we will make precise, every rank one perturbation is one of three forms: (i) φ ₋₁(A), α ; (ii) φ ₋₁, α = ∞; or (iii) the new type we consider here.     

Cramér-type conditions and quadratic mean differentiability

Summary Let (Ω,A) be a measurable space, let Θ be an open set inR ^k , and let {P _θ; θ∈Θ} be a family of probability measures defined onA. Let μ be a σ-finite measure onA, and assume thatP _θ≪μ for each θ∈Θ. Let us denote a specified version ofdP _θ /d _μ byf(ω; θ). In many large sample problems in statistics, where a study of the log-likelihood is important, it has been convenient to impose conditions onf(ω; θ) similar to those used by Cramér [2] to establish the consistency and asymptotic normality of maximum likelihood estimates. These are of a purely analytical nature, involving two or three pointwise derivatives of lnf(ω; θ) with respect to θ. Assumptions of this nature do not have any clear probabilistic or statistical interpretation. In [10], LeCam introduced the concept of differentially asymptotically normal (DAN) families of distributions. One of the basic properties of such a family is the form of the asymptotic expansion, in the probability sense, of the log-likelihoods. Roussas [14] and LeCam [11] give conditions under which certain Markov Processes, and sequences of independent identically distributed random variables, respectively, form DAN families of distributions. In both of these papers one of the basic assumptions is the differentiability in quadratic mean of a certain random function. This seems to be a more appealing type of assumption because of its probabilistic nature. In this paper, we shall prove a theorem involving differentiability in quadratic mean of random functions. This is done in Section 2. Then, by confining attention to the special case when the random function is that considered by LeCam and Roussas, we will be able to show that the standard conditions of Cramér type are actually stronger than the conditions of LeCam and Roussas in that they imply the existence of the necessary quadratic mean derivative. The relevant discussion is found in Section 3. This research was supported by the National Science Foundation, Grant GP-20036.     

Optimality relationships forp-cyclic SOR

Summary The optimality question for blockp-cyclic SOR iterations discussed in Young and Varga is answered under natural conditions on the spectrum of the block Jacobi matrix. In particular, it is shown that repartitioning a blockp-cyclic matrix into a blockq-cyclic form,q, results in asymptotically faster SOR convergence for the same amount of work per iteration. As a consequence block 2-cyclic SOR is optimal under these conditions.Research supported in part by the US Air Force under Grant no. AFOSR-88-0285 and the National Science Foundation under grant no. DMS-85-21154 Present address: Boeing Computer Services, P.O. Box 24346, MS 7L-21, Seattle, WA 98124-0346, USA     

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.     

The inverse problem of bisymmetric matrices with a submatrix constraint

An n×n real matrix A is called a bisymmetric matrix if A=A^T and A=S_nAS_n, where S_n is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A₀, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A^*, find an n×n matrix  in S_E such that where ∥·∥ is Frobenius norm, and S_E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Extremal problems for convex polygons

Consider a convex polygon V _n with n sides, perimeter P _n , diameter D _n , area A _n , sum of distances between vertices S _n and width W _n . Minimizing or maximizing any of these quantities while fixing another defines 10 pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). We survey research on these problems, which uses geometrical reasoning increasingly complemented by global optimization methods. Numerous open problems are mentioned, as well as series of test problems for global optimization and non-linear programming codes.     

Finding blocks and other patterns in a random coloring of ℤ

Let ξ = (ξ_k)_k∈? be i.i.d. with P(ξ_k = 0) = P(ξ_k = 1) = 1/2, and let S: = (S_k) be a symmetric random walk with holding on ?, independent of ξ. We consider the scenery ξ observed along the random walk path S, namely, the process (χ_k := ξ). With high probability, we reconstruct the color and the length of blockⁿ, a block in ξ of length ≥ n close to the origin, given only the observations (χ_k). We find stopping times that stop the random walker with high probability at particular places of the scenery, namely on blockⁿ and in the interval [?3ⁿ,3ⁿ]. Moreover, we reconstruct with high probability a piece of ξ of length of the order 3 around blockⁿ, given only 3 observations collected by the random walker starting on the boundary of blockⁿ. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

Scattering techniques for a one dimensional inverse problem in geophysics

A one dimensional problem for SH waves in an elastic medium is treated which can be written as v_tt = A^?1 (Av_y)_y, A = (?μ)^1/2, ? = density, and μ = shear modulus. Assume A ? C¹ and A′/A ? L¹; from an input v_y(t, 0) = ?(t) let the response v(t, 0) = g(t) be measured (v(t, y) = 0 for t < 0). Inverse scattering techniques are generalized to recover A(y) for y > 0 in terms of the solution K of a Gelfand-Levitan type equation, .     

Characterizations of finite groups by sets of orders of their elements

For a finite group G, ω(G) denotes the set of orders of its elements. If ω is a subset of the set of natural numbers, h(ω) stands for the number of pairwise nonisomorphic finite groups G for which ω(G)=ɛ. We prove that h(ω(G))=1, if G is isomorphic to S₉, S₁₁, S₁₂, S₁₃, or A₁₂, and h(ω(G))=2 if G is isomorphic to S₂(6) or to O ₈ ⁺ (2). 01 Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 36, No. 1, pp. 37–53, January–February, 1997.     

On Singular Phenomena in Certain Time-Optimal Problem

The purpose of this paper is to present the solution of time-optimal problem ofthe controlled object the dynamics of which is given by: , , where and motion resistance function if ,f(x)=-A if x > 0 where . That model describes dynamicsof a very important class of industrial installations. As the time-optimalproblem will be understood a transfer of the initial state to the target state in a minimumtime . There has been shown that in the formula defining resistancefunction f(x)there exists a value that plays an essentialrole in time-optimal structure formation. Namely, if then thetime-optimal control process is typical, analogous as in classical case , i.e. there exists a switching curve formed by thetrajectories of time-optimal solutions reaching the target state and thetime-optimal process is formed by at most one switching operation. For the caseA>A_bwe will examine two following singular phenomena.(a) If the target state z₁=(0, 0) then there exists theswitching curve, dividing the state plane into two sets, however only one itsbranch is formed by the time-optimal solution reaching the target z₁=(0, 0) and generated by the control u=-1. None of solution formsthe second branch of switching curve. It is formed by a state-locus dependingon the value of Aonly. In dependency of the starting state z₀ thetime-optimal control process is generated by bang-bang control with none,one or two switching operations. This is the first singular phenomenon,because any small decrease of the value Aover A _b requires to change thestructure which would be able to generate the time-optimal process.(b) The paper shows, that if the target state z ₁(x_1, 0), x₁>0then there exists a set of the starting states from which there start twotrajectories reaching the target in the same minimum time. This is thesecond phenomenon.Finally, some suggestions as to practical applications have been given too.     

Stability of generalized solutions to equations of one-dimensional motion of viscous heat-conducting gases

Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spacesL () orL ₂() and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved, i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 835–846, June, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00621 and No. 97-01-00214, and by the INTAS Foundation under grant No. 93-27-16.     

Inverse subspace problems with applications

Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large‐scale linear systems of equations and eigenvalue problems as well as to the construction of blurring matrices. Extensions that allow the matrix A to be rectangular and applications to Lanczos bidiagonalization, as well as to the recently proposed subspace‐restricted SVD method for the solution of linear discrete ill‐posed problems, also are considered.Copyright © 2013 John Wiley & Sons, Ltd.     

A modified truncated singular value decomposition method for discrete ill‐posed problems

Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank‐k approximant, which we denote by A_k. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A_k. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A_k.Copyright © 2014 John Wiley & Sons, Ltd.     

Skew codes of prescribed distance or rank

In this paper we generalize the notion of cyclic code and construct codes as ideals in finite quotients of non-commutative polynomial rings, so called skew polynomial rings of automorphism type. We propose a method to construct block codes of prescribed rank and a method to construct block codes of prescribed distance. Since there is no unique factorization in skew polynomial rings, there are much more ideals and therefore much more codes than in the commutative case. In particular we obtain a [40, 23, 10]₄ code by imposing a distance and a [42,14,21]₈ code by imposing a rank, which both improve by one the minimum distance of the previously best known linear codes of equal length and dimension over those fields. There is a strong connection with linear difference operators and with linearized polynomials (or q-polynomials) reviewed in the first section.      

The best approximation to a class of functions of several variables by another class and related extremum problems

We study the relationship between several extremum problems for unbounded linear operators of convolution type in the spaces , m ≥ 1, 1 ≤ γ ≤ ∞. For the problem of calculating the modulus of continuity of the convolution operatorA on the function classQ defined by a similar operator and for the Stechkin problem on the best approximation of the operatorA on the classQ by bounded linear operators, we construct dual problems in dual spaces, which are the problems on, respectively, the best and the worst approximation to a class of functions by another class. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 323–340, September, 1998. This research was supported by INTAS under grant No. 94-4070.     

On the [28, 7, 12] binary self-complementary codes and their residuals

Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary selfcomplementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distributionA ₀=A ₂₈=1,A ₁₂=A ₁₆=63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified.This work has been done during the visit of the author as a guest researcher to the Department of Electrical Engineering at Linköping University, Sweden. The research was partially supported by the Bulgarian National Science Foundation under contracts No. 37/1987 and No. 876/1988.     

Procrustes problems and inverse eigenproblems for multilevel block α‐circulants

Let n = (n₁,n₂,…,n_k) and α = (α₁,α₂,…,α_k) be integer k‐tuples with α_i∈{1,2,…,n_i?1} and for all i = 1,2,…,k. Multilevel block α ‐circulants are (k + 1)‐level block matrices, where the first k levels have the block α_i‐circulant structure with orders and the entries in the (k + 1)‐st level are unstructured rectangular matrices with the same size . When k = 1, Trench discussed on his paper "Inverse problems for unilevel block α‐circulants" the Procrustes problems and inverse problems of unilevel block α‐circulants and their approximations. But the results are not perfect for the case gcd( α , n ) > 1 (i.e., gcd(α₁,n₁) > 1). In this paper, we also discuss Procrustes problems for multilevel block α ‐circulants. Our results can further make up for the deficiency when k = 1. When , inverse eigenproblems for this kind of matrices are also solved. By using the related results, we can design an artificial Hopfield neural network system that possesses the prescribed equilibria, where the Jacobian matrix of this system has the constrained multilevel α ‐circulative structure. Finally, some examples are employed to illustrate the effectiveness of the proposed results. Copyright © 2016 John Wiley & Sons, Ltd.