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.     

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.     

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.     

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.     

Über gerade unimodulare Gitter der Dimension 32, III

Let A_j, B_j be complex B-spaces, j = 0, 1, A_θ and B_θ–the complex-interpolation spaces generated by the couples (A₀, A₁) and (B₀, B₁), resp., by CALDERON'S/LIONS'S method. Let T: A₀ ∧ A₁ → B₀ → B₁ be an operator satisfying some conditions such as continuity, estimates etc. in terms of the norms of A_j, B_j (j = 0, 1). We consider the question which one of these properties is inherited to T when A₀ → A₁ and B₀ → B₁ are equipped with the norm of A_θ and B_θ.     

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.     

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

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.     

C n and D n Very-Well-Poised 10 φ 9 Transformations

In this paper we derive multivariable generalizations of Bailey's classical terminating balanced very-well-poised _{10 9} transformation. We work in the setting of multiple basic hypergeometric series very-well-poised on the root systems A _n , C _n , and D _n . Following the distillation of Bailey's ideas by Gasper and Rahman [11], we use a suitable interchange of multisums. We obtain C _n and D _{n 10 9} transformations combined with A _n , C _n , and D _n extensions of Jackson's _{8 7} summation. Milne and Newcomb have previously obtained an analogous formula for A _n series. Special cases of our _{10 9} transformations include several new multivariable generalizations of Watson's transformation of an _{8 7} into a multiple of a _{4 3} series. We also deduce multidimensional extensions of Sears' _{4 3} transformation formula, the second iterate of Heine's transformation, the q -Gauss summation theorem, and of the q -binomial theorem. August 28, 1996. Date revised: September 8, 1997.     

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.     

Line-graphical degree sequences

A degree sequence π = (d₁, d₂,…,d_p), with d₁ ≥ d₂ ≥…≥ d_p, is line graphical if it is realized by the line graph of some graph. Degree sequences with line-graphical realizations are characterized for the cases d₁ = p - 1, d₁ = p - 2, d₁ ≤ 3, and d₁ = d_p. It is also shown that if a degree sequence with d₁ = p-1 is line graphical, it is uniquely line graphical. It follows that with possibly one exception each line-graphical realization of an arbitrary degree sequence must have either C₅, 2K₁, + K₂, K₁ + 2K₂, or 3K₁, as an induced subgraph.     

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.     

Covering the vertices of a graph by cycles of prescribed length

The main theorem of that paper is the following: let G be a graph of order n, of size at least (n² - 3n + 6)/2. For any integers k, n₁, n₂,…,n_k such that n = n₁ + n₂ +. + n_k and n_i ? 3, there exists a covering of the vertices of G by disjoint cycles (C_i) =l…k with |C_i| = ni, except when n = 6, n₁ = 3, n₂ = 3, and G is isomorphic to G₁, the complement of G₁ consisting of a C₃ and a stable set of three vertices, or when n = 9, n₁ = n₂ = n₃ = 3, and G is isomorphic to G₂, the complement of G₂ consisting of a complete graph on four vertices and a stable set of five vertices. We prove an analogous theorem for bipartite graphs: let G be a bipartite balanced graph of order 2n, of size at least n² - n + 2. For any integers s, n₁, n₂,…,n_s with n_i ? 2 and n = n₁ + n₂ + ? + n_s, there exists a covering of the vertices of G by s disjoint cycles C_i, with |C_i| = 2n_i.     

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     

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.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

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.     

Elementary equivalence of the automorphism groups of Abelian <Emphasis Type="Italic">p</Emphasis>-groups

We consider Abelian p-groups (p ≥ 3) A ₁ = D ₁ ⊕ G ₁ and A ₂ = D ₂ ⊕ G ₂, where D ₁ and D ₂ are divisible and G ₁ and G ₂ are reduced subgroups. We prove that if the automorphism groups Aut A ₁ and Aut A ₂ are elementarily equivalent, then the groups D ₁, D ₂ and G ₁, G ₂ are equivalent, respectively, in the second-order logic.     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.