Boolean designs and self-dual matroids

A proper splitting of a rectangular matrix A is one of the form A = M ? N, where A and M have the same range and null spaces. This concept was introduced by R. Plemmons as a means of generalizing to rectangular and singular matrices the concept of a regular splitting of a nonsingular matrix as introduced by R. Varga. In consideration of the linear system Ax=b, A. Berman and R. Plemmons used a proper splitting of A into M ? N and showed that the iteration x⁽ⁱ⁺¹⁾=M⁺Nx⁽ⁱ⁾+M⁺b converges to A⁺b, the best least-squares solution to the system, if and only if the spectral radius of M⁺N is less than one. The purpose of this paper is to further develop the characteristics of proper splittings and to extend these previous results by replacing the Moore-Penrose generalized inverse with a least-squares g-inverse, a minimum-norm g-inverse, or a g-inverse. Also, some criteria are given for comparing convergence rates of M_i^?N_i, where A = M₁?N₁ = M₂?N₂, and a method is developed for constructing proper splittings of special types of matrices.     

Invariant Constructions of Simple and Maximal Sets

The main results of the present paper are the following theorems: 1. There is no e ∈ ω such that for any A, B ? ω, S^A = W is simple in A, and if A′ ?_T B′, then S^A =* S^B. 2 There is an e ∈ ω such that for any A, B ? ω, M^A = W_e is incomplete maximal in A, and if A =* B, then M^A ?_T M^B.     

A generalization of N-matrices

Ky Fan defines an N-matrix to be a matrix of the form A = tI ? B, B ? 0, λ < t < ?(B), where ?(B) is the spectral radius of B and λ is the maximum of the spectral radii of all principal submatrices of B of order n ? 1. In this paper, we define the closure (N₀-matrices) of N-matrices by letting λ ? t. It is shown that if A ∈ Z and A^-1 < 0, then A ∈ N. Certain inequalities of N-matrices are shown to hold for N₀-matrices, and a method for constructing an N-matrix from an M-matrix is given.     

Δ-Monotone arrangements of real numbers

Given a set of M × N real numbers, can these always be labeled as x_i,j; i = 1,…, M; j = 1,…, N; such that x_i+1,j+1 ? x_i+1,j ? x_i,j+1 + x_ij ≥ 0, for every (i, j) where 1 ≤ i ≤ M ? 1, 1 ≤ j ≤ N ? 1? For M = N = 3, or smaller values of M, N it is shown that there is a “uniform” rule. However, for max(M, N) > 3 and min(M, N) ≥ 3, it is proved that no uniform rule can be given. For M = 3, N = 4 a way of labeling is demonstrated. For general M, N the problem is still open although, for a special case where all the numbers are 0's and 1's, a solution is given.     

Minimisation of functions of a positive semidefinite matrix A subject to AX = 0

A common problem in multivariate analysis is that of minimising or maximising a function f of a positive semidefinite matrix A subject possibly to AX = 0. Typically A is a variance-covariance matrix. Using the theory of nearest point projections in Hilbert spaces, it is shown that the solution satisfies the equation f′(A) + M ? A = 0, where A = P₀(M) and P₀ is a certain projection operator.     

Extension de quelques theoremes sur les densites de series d'elements de n a des series de sous-ensembles finis de n

For a sequence A = {A_k} of finite subsets of N we introduce: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{n}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$, where A(m) is the number of subsets A_k ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation $\text{a ∪ b, (a ∩ b), (a 1 b = a ∪ b ? a ∩ b)}$ constitutes a finite semi-group N^∪ (semi-group N^∩) (group $\text{N}{}^1$). For N^∪, N^∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)^?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {A_kr}, where A_kr | A_kr+1 for r = 1, 2, …. In Section 6 we consider for $\text{N∪, N∩, N1}$ analogues of Rohrbach inequality: $\text{2n}\text{? g(n) ? 2}\text{n}$, where g(n) = min k over the subsets {a₁ < … < a_k} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = a_i + a_j.Pour une série A = {A_k} de sous-ensembles finis de N on introduit les densités: $\text{δ(A) =}\text{inf}{}_{\mathrm{m?n}}\text{A(m)}\text{2}{}^{\mathrm{m}}$, $\text{d(A) =}\text{lim inf}{}_{\mathrm{n\to \infty }}\text{A(n)}\text{2}{}^{\mathrm{n}}$ où A(m) est le nombre d'ensembles A_k ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations $\text{a ∪ b, a ∩ b, a 1 b = a ∪ b ? a ∩ b}$, un semi-groupe fini N^∪, N^∩ ou un groupe N¹ respectivement. Pour N^∪, N^∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)^?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour $\text{N}{}^{\mathrm{\cup }}\text{, N}{}^{\mathrm{\cap }}\text{, N}{}^1$ l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {A_kr}, où A_kr|A_kr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N^∪, $\text{N}{}^{\mathrm{\cap }}\text{, N}{}^1$ les analogues de l'inégalité de Rohrbach: $\text{2n}\text{? g(n) ? 2}\text{n}$, où g(n) = min k pour les ensembles {a₁ < … < a_k} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = a_i + a_j.     

Extensions of the Ostrowski-Reich theorem for SOR iterations

The Ostrowski-Reich theorem states that for a system Ax =b of linear equations with A nonsingular, if A is hermitian and if the diagonal of A is positive, then the SOR method converges for each relaxation parameter in (0,2) if and only if A is positive definite. This is actually a special case of the Householder-John theorem, which states that for A=M?N with A,M nonsingular, if A is hermitian and $\text{M}{}^1\text{+N}$ is positive definite, then M^?1N is a convergent matrix if and only if A is positive definite. Our purposes here are to generalize the Householder-John theorem and to provide an insight into how and why the SOR method can converge. As a result the Ostrowski-Reich theorem is extended in two directions; one is when A is hermitian but the diagonal of A is not necessarily positive, so that A is not necessarily positive definite, and the other is when $\text{A+A}{}^1$ is positive definite but A is not necessarily hermitian. In the process, several other convergence results are obtained for general splittings of A. However, no claims are made concerning the case in which the convergence results obtained here can be applied to practical situations.     

Omega- and Beta-Models of Alternative Set Theory

We present the axioms of Alternative Set Theory (AST) in the language of second-order arithmetic and study its ω- and β-models. These are expansions of the form (M, M), M ? P(M), of nonstandard models M of Peano arithmetic (PA) such that (M, M) ? AST and ω ? M. Our main results are: (1) A countable M ? PA is β-expandable iff there is a regular well-ordering for M. (2) Every countable β-model can be elementarily extended to an ω-model which is not a β-model. (3) The Ω-orderings of an ω-model (M, M) are absolute well-orderings iff the standard system SS(M) of M is a β-model of A^?_2. (4) There are ω-expandable models M such that no ω-expansion of M contains absolute Ω-orderings. (5) There are s-expandable models (i. e., their ω-expansions contain only absolute Ω-orderings) which are not β-expandable. (6) For every countable β-expansion M of M, there is a generic extension M[G] which is also a β-expansion of M. (7) If M is countable and β-expandable, then there are regular orderings <₁, <₂ such that neither <₁ belongs to the ramified analytical hierarchy of the structure (M, <₂), nor <₂ to that of (M, <₁). (8) The result (1) can be improved as follows: A countable M ? PA is β-expandable iff there is a semi-regular well-ordering for M.     

On simultaneous diagonalization of one Hermitian and one symmetric form

It is remarked that if A, B ? M_n(C), A = A^t, and B? = B^t, B positive definite, there exists a nonsingular matrix U such that (1) ū^tBU = I and (2) U^tAU is a diagonal matrix with nonnegative entries. Some related actions of the real orthogonal group and equations involving the unitary group are studied.     

Involutory functions and Moore-Penrose inverses of matrices in an arbitrary field

Let F be a field, and M be the set of all matrices over F. A function ? from M into M, which we write ?(A) = A^s for A∈M, is involutory if (1) (AB)^s = B^sA^s for all A, B in M whenever the product AB is defined, and (2) (A^s)^s = A for all A∈M. If ? is an involutory function on M, then A^s is n×m if A is m×n; furthermore, Rank A = Rank A^s, the restriction of ? to F is an involutory automorphism of F, and (aA + bB)^s = a^sA^s + b^sB^s for all m×n matrices A and B and all scalars a and b. For an A∈M, an Ã∈M is called a Moore-Penrose inverse of A relative to ? if (i) AÃA = A, ÃAÃ = Ã and (ii) (AÃ)^s = AÃ, (ÃA)^s = ÃA. A necessary and sufficient condition for A to have a Moore-Penrose inverse relative to ? is that Rank A = Rank AA^s = Rank A^sA. Furthermore, if an involutory function ? preserves circulant matrices, then the Moore-Penrose inverse of any circulant matrix relative to ? is also circulant, if it exists.     

On J. C. C. Nitsche type inequality for annuli on Riemann surfaces

Assume that (N, ?) and (M, S) are two Riemann surfaces with conformal metrics ? and S. We prove that if there is a harmonic homeomorphism between an annulus A ? N with a conformal modulus Mod(A) and a geodesic annulus A _S (p, ρ₁, ρ₂)?M, then we have ρ₂/ρ₁ ≥ Ψ _S Mod(A)²+ 1, where Ψ _S is a certain positive constant depending on the upper bound of Gaussian curvature of the metric S. An application for the minimal surfaces is given.     

A conditional dichotomy theorem for stochastic processes with independent increments

Let X(t) and Y(t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M_t and N_t, respectively, and let the “time-jump” measures M and N be defined over [0, T] × $\text{R}$?{0} by M((t₁, t₂] × A) = M_t2(A) ? M_t1(A) and N((T₁, t₂] × A) = N_t2(A) ? N_t1(A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X(t) and Y(t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X(t) if and only if it is an admissible translate for the Gaussian component of X(t). In particular, if X(t) has no Gaussian component, then every nontrivial translate of X(t) is orthogonal to it.     

Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems

In this paper we study linear differential systems (1) $\text{x′ =}\text{A}\text{?}\text{(θ + ωt)x,}\text{where}\text{A}\text{?}\text{(θ)}\text{is an}\text{(n × n)}$ matrix-valued function defined on the k-torus T^k and (θ, t) → θ + ωt is a given irrational twist flow on T^k. First, we show that if A ? C^N(T^k), where N ? {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class C^N on T^k. Next we assume that à is sufficiently smooth on T^k and ω satisfies a suitable “small divisors” inequality. We show that if (1) satisfies the “full spectrum” assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix $\text{A}\text{?}\text{(θ + ωt)}$ in (1) is the Jacobian matrix of a nonlinear vector field $\text{?(x)}$ evaluated along a quasi-periodic solution x = φ(t) of (2) $\text{x′ = ?(x)}$. We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, ?) defined in the vicinity of $\text{Ω = H(φ)}$, the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, ?′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be “close to” a constant coefficient system.     

Theorems on M-splittings of a singular M-matrix which depend on graph structure

Let $\text{A=M?Nε}\text{R}{}^{\mathrm{n\; n}}$ be a splitting. We investigate the spectral properties of the iteration matrix M^-1N by considering the relationships of the graphs of A, M, N, and M^-1N. We call a splitting an M-splitting if M is a nonsingular M-matrix and N?0. For an M-splitting of an irreducible Z-matrix A we prove that the circuit index of M^-1N is the greatest common divisor of certain sets of integers associated with the circuits of A. For M-splittings of a reducible singular M-matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting. These results hold under somewhat weaker assumptions.     

Explicit solution family for the equation of the resistively shunted Josephson junction model

We obtain and study a family of solutions of the equation $\dot \varphi $ + sin ? = B + A cos ωt, which is applicable to several problems in physics, mechanics, and geometry. We use polynomial solutions of double confluent Heun equations associated with this equation to construct the family. We describe the manifold M _P of parameters (A,B, ω) of these solutions and obtain explicit formulas for the rotation number and Poincaré map of the dynamical system on a torus corresponding to this equation with parameters (A,B, ω) ∈ M _P .     

L 2 existence theorems for the\bar \partial _b - Neumann problem on strongly pseudoconvex CR manifoldsproblem on strongly pseudoconvex CR manifolds

LetM be the boundary of a strongly pseudoconvex domain in \(\mathbb{C}^n \) ,n≥4 and ω be an open subset inM such that ?ω is the intersection ofM with a flat hypersurface. We establish theL ² existence theorems of the \(\bar \partial _b - Neumann\) problem on ω. In particular, we prove that the \(\bar \partial _b - Laplacian\) \(\square _b = \bar \partial _b \bar \partial _b^* + \bar \partial _b^* \bar \partial _b \) equipped with a pair of natural boundary conditions, the so-called \(\bar \partial _b - Neumann\) boundary conditions, has closed range when it acts on (0,q) forms, 1≤q≤n?3. Thus there exists a bounded inverse operator for \(\square _b \) , the \(\bar \partial _b - Neumann\) operatorN _b, and we have the following Hodge decomposition theorem on ω for \(\bar \partial _b \bar \partial _b^* N_b \alpha + \bar \partial _b^* \bar \partial _b N_b \alpha \) , for any (0,q) form α withL ²(ω) coefficients. The proof depends on theL ^p regularity of the tangential Cauchy-Riemann operators \(\bar \partial _b u = \alpha \) on ω?M under the compatibility condition \(\bar \partial _b \alpha = 0\) , where α is a (p, q) form on ω, where 1≤q≤n?2. The interior regularity ofN _b follows from the fact that \(\square _b \) is subelliptic in the interior of ω. The operatorN _b induces natural questions on the regularity up to the boundary ?ω. Near the characteristic point of the boundary, certain compatibility conditions will be present. In fact, one can show thatN _b is not a compact operator onL ²(ω).     

On the bilinear and cubic forms of some symplectic connected sums

Let ( M1 , M2 , N ) be three symplectic manifolds and suppose that we can do the symplectic connected sum of M1 and M2 along their submanifold N to obtain M1#NM2 . In this paper, we consider the bilinear and cubic forms of H* (M1#NM2 , Z) when dim M1#NM2 = 4, 6. Under some conditions, we get some relations of the bilinear and the cubic forms between M1#NM2 and M1■M2 .     

The weight distribution of irreducible cyclic codes with block lengths n1((q1−1)N)

We study the weight distribution of irreducible cyclic (n, k) codeswith block lengths n = n₁((q¹ ? 1)/N), where N|q ? 1, gcd(n₁,N) = 1, and gcd(l,N) = 1. We present the weight enumerator polynomial, A(z), when k = n₁l, k = (n₁ ? 1)l, and k = 2l. We also show how to find A(z) in general by studying the generator matrix of an (n₁, m) linear code, $\text{V}{}^1{}_{\mathrm{d}}$ over GF(q^d) where d = gcd (ord_n1(q), l). Specifically we study A(z) when $\text{V}{}^1{}_{\mathrm{d}}$ is a maximum distance separable code, a maximal shiftregister code, and a semiprimitive code. We tabulate some numbers A_μ which completely determine the weight distributionof any irreducible cyclic (n₁(2¹ ? 1), k) code over GF(2) for all n₁ ? 17.