The solvability of two linear matrix equations

The solvability conditions of the following two linear matrix equations (i)A₁X₁B₁ +A₂X₂B₂ +A₃X₃B₃ =C,(ii) A₁XB₁ =C₁ A₂XB₂ =C₂ are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations

(iii) A ₁ X ₁ B ₁+A ₂ X ₂ B ₂+A ₃ X ₃ B ₃+A ₄ X ₄ B ₄=C, (iv) A ₁ XB ₁=C ₁ A ₂ XB ₂=C ₂ A ₃ XB ₃=C ₃ A ₄ XB ₄=C ₄, (v) AXB+CXD=E are also considered.     

Rank properties of certain semigroups

In this paper we consider certain ranks of some semigroups. These ranks are r ₁(S),r ₂(S),r ₃(S),r ₄(S) and r ₅(S) as defined below. We have r ₁≤r ₂≤r ₃≤r ₄≤r ₅. The semigroups are CL _n ,CL _m ×CL _n ,Z _n and SL _n . Here CL _n is a chain with n elements, Z _n is the zero semigroup on n elements and SL _n is the free semilattice generated by n elements and having 2 ⁿ −1 elements. We find many of the ranks for these classes of semigroups.     

Two-Sample T 3 Plot: A Graphical Comparison of Two Distributions

Abstract

Consider two independent random variables x and y with means and standard deviations μ _x ,μ _y ,σ _x , and σ _y , respectively. Let F _x (t) = P[(x - μ, _x )/σ _x ≤ t] and F _y (t) = P[(y - μ _y )/σ _y ≤ t]. In this article we address the problem of testing the null hypothesis H ₀ : F _x ≡ F _y , against the alternative H ₁ : F _x ≡ F _y . A graphical tool called T ₃ plot for checking normality of independently and identically distributed univariate data was proposed in an earlier article by Ghosh. In the present article we develop a two-sample T ₃ plot where the basic statistic is the normalized difference between the T ₃ functions for the two samples. Significant departure of this difference function from the horizontal zero line is indicative of evidence against the null hypothesis. In contrast to the one-sample problem, the common distribution function under the null hypothesis is not specified in the two-sample case. Bootstrap is used to construct the acceptance region under H ₀, for the two-sample T ₃ plot.     

A Note on Relative Efficiency of Axiom Systems

We introduce a notion of relative efficiency for axiom systems. Given an axiom system A_β for a theory T consistent with S¹₂, we show that the problem of deciding whether an axiom system A_α for the same theory is more efficient than A_β is II₂-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems A_α, A_β, with A_α ? A_β, both in the case of A_α, A_β having the same language, and in the case of the language of A_α extending that of A_β: in the latter case, letting Pr_α, Pr_β denote the theories axiomatized by A_α, A_β, respectively, and assuming Pr_α to be a conservative extension of Pr_β, we show that if A_α — A_β contains no nonlogical axioms, then A_α can only be a linear speed-up of A_β; otherwise, given any recursive function g and any A_β, there exists a finite extension A_α of A_β such that A_α is a speed-up of A_β with respect to g. Mathematics Subject Classification: 03F20, 03F30.     

The Permutations 123p 4…p m and 321p 4…p m are Wilf-Equivalent

Write p ₁, p ₂…p _m for the permutation matrix δ _{pi, j} . Let S _n (M) be the set of n×n permutation matrices which do not contain the m×m permutation matrix M as a submatrix. In [7] Simion and Schmidt show bijectively that |S _n (123) |=|S _n (213) |. In [9] this was generalised to a bijection between S _n (12 p ₃…p _m ) and S _n (21 p ₃…p _m ). In the present paper we obtain a bijection between S _n (123 p ₄…p _m ) and S _n (321 p ₄…p _m ). Revised: March 24, 1999     

Zinbiel algebras under <Emphasis Type="Italic">q</Emphasis>-commutators

An algebra with the identity t ₁(t ₂ t ₃) = (t ₁ t ₂+t ₂ t ₁)t ₃ is called Zinbiel. For example, ℂ[x] under the multiplication is Zinbiel. Let a ○ _q b = a ○ b + q b ○ a be a q-commutator, where q ∈ ℂ. We prove that for any Zinbiel algebra A the corresponding algebra under the commutator A ⁽⁻¹⁾ = (A, ○₋₁) satisfies the identities t ₁ t ₂ = −t ₂ t ₁ and (t ₁ t ₂)(t ₃ t ₄) + (t ₁ t ₄)(t ₃ t ₂) = jac(t ₁, t ₂, t ₃)t ₄ + jac(t ₁, t ₄, t ₃)t ₂, where jac(t ₁, t ₂, t ₃) = (t ₁ t ₂)t ₃ + (t ₂ t ₃)t ₁ + (t ₃ t ₁)t ₂. We find basic identities for q-Zinbiel algebras and prove that they form varieties equivalent to the variety of Zinbiel algebras if q ² ≠ 1. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 3, pp. 57–78, 2005.     

On the graded identities ofM 1,1(E)

We determine theS _n ×S _m -cocharacterX _n,m of the algebraM _1,1(E) and prove that theT ₂-ideal of its graded identities is generated by the polynomialsy ₁ y ₂−y ₂ y ₁ andz ₁ z ₂ z ₃+z ₃ z ₂ z ₁.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.     

Maximal L p regularity for parabolic difference equations

In this paper, we consider a family of finite difference operators {Ah }_{h >0} on discrete L _q ‐spaces L _q (?^N _h ). We show that the solution u _h to u ′_h (t) – A _h u _h(t) = f _h (t), t > 0, u _h (0) = 0 satisfies the estimate ‖A _h u _h ‖equation/tex2gif-inf-15.gif ≤ C ‖f _h ‖equation/tex2gif-inf-21.gif, where C is independent of h and f _h . In this case, the family {A _h }_{h >0} is said to have discrete maximal L _p regularity on the discrete L _q ‐space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Selfsimilar Jordan Curves on the Plane

We study the attractors of a finite system of planar contraction similarities S _j (j=1,...,n) satisfying the coupling condition: for a set {x ₀,...,x _n} of points and a binary vector (s ₁,...,s _n ), called the signature, the mapping S _j takes the pair {x ₀,x _n} either into the pair {x _j-1,x _j } (if s _j =0) or into the pair {x _j , x _j-1} (if s _j =1). We describe the situations in which the Jordan property of such attractor implies that the attractor has bounded turning, i.e., is a quasiconformal image of an interval of the real axis.     

Reducing system of parameters and the Cohen-Macaulay property

Let R be a local ring and let (x ₁, …, x _r) be part of a system of parameters of a finitely generated R-module M, where r < dim_R M. We will show that if (y ₁, …, y _r) is part of a reducing system of parameters of M with (y ₁, …, y _r) M = (x ₁, …, x _r) M then (x ₁, …, x _r) is already reducing. Moreover, there is such a part of a reducing system of parameters of M iff for all primes P ε Supp M ∩ V _R(x ₁, …, x _r) with dim_R R/P = dim_R M − r the localization M _P of M at P is an r-dimensional Cohen-Macaulay module over R _P. Furthermore, we will show that M is a Cohen-Macaulay module iff y _d is a non zero divisor on M/(y ₁, …, y _d−1) M, where (y ₁, …, y _d) is a reducing system of parameters of M (d:= dim_R M).     

Bounded edge-connectivity and edge-persistence of Cartesian product of graphs

The bounded edge-connectivity λ_k(G) of a connected graph G with respect to is the minimum number of edges in G whose deletion from G results in a subgraph with diameter larger than k and the edge-persistence D⁺(G) is defined as λ_d(G)(G), where d(G) is the diameter of G. This paper considers the Cartesian product G₁×G₂, shows λ_k1+k2(G₁×G₂)≥λ_k1(G₁)+λ_k2(G₂) for k₁≥2 and k₂≥2, and determines the exact values of D⁺(G) for G=C_n×P_m, C_n×C_m, Q_n×P_m and Q_n×C_m.     

Strongly subadditive functions

Let f: ℝ⁺ → ℝ. The subject is the trace inequality Tr f(A) + Tr f(P ₂ AP ₂) ≦ Tr f(P ₁₂ AP ₁₂) + Tr f(P ₂₃ AP ₂₃), where A is a positive operator, P ₁; P ₂; P ₃ are orthogonal projections such that P ₁ + P ₂ + P ₃ = I, P ₁₂ = P ₁ + P ₂ and P ₂₃ = P ₂ + P ₃. There are several examples of functions f satisfying the inequality (called (SSA)) and the case of equality is described.     

The Poincaré series of the algebras of simultaneous invariants and covariants of two binary forms

Let ? _{d 1, d 2} and 𝒞 _{d 1, d 2} be the algebras of simultaneous invariants and simultaneous covariants of the two binary forms of degrees d ₁ and d ₂. Formulas for computation of the Poincaré series 𝒫? _{d 1, d 2} (z), 𝒫𝒞 _{d 1, d 2} (z) of the algebras are found. By using these formulas, we have computed the series for d ₁, d ₂ ≤ 20.     

Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

On trigonometric functional equations of rectangular type

Summary Letf, G₁ × G₂ C, where G _i (i = 1, 2) denote arbitrary groups and C denotes the set of complex numbers. The general solutions of the following functional equationsf(x ₁ y ₁ ,x ₂ y ₂ ) +f(x ₁ y ₁ ,x ₂ y ₂ ^-1 ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ^-1 ) =f(x ₁ ,x ₂ )F(y ₁ ,y ₂ ) +F(x ₁ ,x ₂ )f(y ₁ ,y ₂ ) (1) andf(x ₁ y ₁ ,x ₂ y ₂ ) +f(x ₁ y ₁ ,x ₂ y ₂ ^-1 ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ) +f(x ₁ y ₁ ^-1 ,x ₂ y ₂ ^-1 ) =f(x ₁ ,x ₂ )f(y ₁ ,y ₂ ) +F(x ₁ ,x ₂ )F(y ₁ ,y ₂ ) (2) are determined assuming thatf satisfies the conditionf(x ₁y₁z₁, x₂) = f(x₁z₁y₁, x₂), f(x₁, x₂y₂z₂) = f(x₁, x₂z₂y₂) (C) for allx _i, y_i, x_i G_i (i = 1, 2). The functional equations (1) and (2) are generalizations of the well known rectangular type functional equationf(x ₁ + y₁, x₂ + y₂) + f(x₁ + y₁, x₂ – y₂) + f(x₁ – y₁, x₂ + y₂) + f(x₁ – y₁, x₂ – y₂) = 4f(x₁, x₂) studied by J. Aczel, H. Haruki, M. A. McKiernan and G. N. Sakovic in 1968.     

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

We study the existence and shape preserving properties of a generalized Bernstein operator B _n fixing a strictly positive function f ₀, and a second function f ₁ such that f ₁/f ₀ is strictly increasing, within the framework of extended Chebyshev spaces U _n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B _n : C[a, b] → U _n with strictly increasing nodes, fixing f₀, f₁ ? U_n{f_{0}, f_{1} \in U_{n}} . If U_n ì U_{n + 1}{U_{n} \subset U_{n + 1}} and U _n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B _n+1 : C[a, b] → U _n+1 with strictly increasing nodes, fixing f ₀ and f ₁. In particular, if f ₀, f ₁, . . . , f _n is a basis of U _n such that the linear span of f ₀, . . . , f _k is an extended Chebyshev space over [a, b] for each k = 0, . . . , n, then there exists a Bernstein operator B _n with increasing nodes fixing f ₀ and f ₁. The second main result says that under the above assumptions the following inequalities hold

The Real Solutions to a System of Quaternion Matrix Equations with Applications

In this article we establish necessary and sufficient conditions for the existence and the expressions of the general real solutions to the classical system of quaternion matrix equations A ₁ XB ₁ = C ₁, A ₂ XB ₂ = C ₂. Moreover, formulas of the maximal and minimal ranks of four real matrices X ₁, X ₂, X ₃, and X ₄ in solution X = X ₁ + X ₂ i + X ₃ j + X ₄ k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A ₁ XB ₁ = C ₁, A ₂ XB ₂ = C ₂, A ₃ XB ₃ = C ₃ to have common real solutions. In addition, the maximal and minimal ranks of four real matrices E, F, G, and H in the common generalized inverse of A ₁ + B ₁ i + C ₁ j + D ₁ k and A ₂ + B ₂ i + C ₂ j + D ₂ k, which can be expressed as E + Fi + Gj + Hk are also presented.     

A Class of Sparse Invertible Matrices and Their Use for Nonlinear Prediction of Nearly Periodic Time Series with Fixed Period

We introduce a class of sparse matrices U _m (A _{p 1} ) of order m by m, where m is a composite natural number, p ₁ is a divisor of m, and A _{p 1} is a set of nonzero real numbers of length p ₁. The construction of U _m (A _{p 1} ) is achieved by iteration, involving repetitive dilation operations and block-matrix operations. We prove that the matrices U _m (A _{p 1} ) are invertible and we compute the inverse matrix (U _m (A _{p 1} ))^?1 explicitly. We prove that each row of the inverse matrix (U _m (A _{p 1} ))^?1 has only two nonzero entries with alternative signs, located at specific positions, related to the divisors of m. We use the structural properties of the matrix (U _m (A _{p 1} ))^?1 in order to build a nonlinear estimator for prediction of nearly periodic time series of length m with fixed period.     

The genus of the 2-amalgamations of graphs

A graph G is called the 2-amalgamation of subgraphs G₁ and G₂ if G = G₁ ∪ G₂ and G₁ ∩ G₂ = {x, y}, 2 distinct points. in this case we write G = G₁∪_{{x, y}} G₂. in this paper we show that the orientable genus, γ(G), satisfies the inequalities γ(G₁) + γ(G₂) ? 1 ≤ γ(G₁ ∪_{{x, y}} G₂) ≤ γ(G₁) + γ(G₂) + 1 and that this is the best possible result, i. e., the resulting three values for γ(G₁ ∪_{{x, y}} G₂) which are possible can actually be realized by appropriate choices for G₁ and G₂.