Reflecting the pascal matrix about its main antidiagonal

Let B denote either of two varieties of order n Pascal matrix, i.e., one whose entries are the binomial coefficients. Let B_R denote the reflection of B about its main antidiagonal. The matrix B is always invertible modulo n; our main result asserts that B^-1 ≡ B^R mod n if and only if n is prime. In the course of motivating this result we encounter and highlight some of the difficulties with the matrix exponential under modular arithmetic. We then use our main result to extend the "Fibonacci diagonal" property of Pascal matrices.     

Continuity properties of the Drazin pseudoinverse

Let B^D denote that Drazin inverse of the n×n complex matrix B. Define the core-rank of B as rank (B^i(B)) where i(B) is the index of B. Let j = 1,2,…, and A_j and A be square matrices such that A_i converges to A with respect to some norm. The main result of this paper is that A_j^D converges to A^D if and only if there exist a j₀ such that core-rank A_j=core-rankA for j ? j₀.     

On free semilattice-ordered semigroups satisfying x n =x

Let B(k,0,n) denote the group with k generators which is free in the group variety defined by the identity x ⁿ =1. Let B _slo (k,1,n) denote the semilattice-ordered semigroup with k generators which is free in the semilattice-ordered semigroup variety defined by the identity x ⁿ =x. We prove a generalization of the Green-Rees theorem: B _slo (k,1,n) is finite for all k≥1 if and only if B(k,0,n−1) is finite for all k≥1. We find a formula for card(B _slo (1,1,n)). We construct B _slo (k,1,n) for some concrete values of k and n.     

The singular values of the hadamard product of a positive semidefinite and a skew-symmetric matrix

Let σ₁(X)≤ · ≤ σ_N(X)≤0 denote the ordered singular values ofan n × n matrix X and let α₁ (X) ≤ α₂(X)≤ · ≤ α_n(X) denote its ordered main diagonal entries (assuming that they are real). Let B be any complex n × n skew-symmetric matrix and ||.|| any unitarily invariant norm. It is shown that for any rea positive semidefinite n × n matrix A.     

On the exponential map of borei subgroups in a semi-simple lie group

Let G be a connected, complex, semi-simple Lie group Let g be an element in G. Let B be a Borel subgroup of G and g in B. Let m and n be the least positive integers such that the element g^m lies on a one-parameter subgroup in G and the element gⁿ lies on a one-parameter subgroup in B. We denote these integers by ind _G (g) and ind _B (g). In this note we prove the conjecture ind _G (g) = ind _B (g), if g is regular.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

Study of Hopficity in Certain Classes of Rings

The main results proved in this paper are:

1. For any non-zero vector space V _Dover a division ring D, the ring R= End(V _D) is hopfian as a ring

2. Let Rbe a reduced π-regular ring &; B(R) the boolean ring of idempotents of R. If B(R) is hopfian so is R.The converse is not true even when Ris strongly regular.

3. Let Xbe a completely regular spaceC(X) (resp. C ^?(X)) the ring of real valued (resp. bounded real valued) continuous functions on X. Let Rbe any one of C(X) or C ^?(X). Then Ris an exchange ring if &; only if Xis zero dimensional in the sense of Katetov. for any infinite compact totally disconnected space X C(X) is an exchange ring which is not von Neumann regular.

4. Let Rbe a reduced commutative exchange ring. If Ris hopfian so is the polynomial ring R[T ₁,…,T _n] in ncommuting indeterminates over Rwhere nis any integer ≥ 1.

5. Let Rbe a reduced exchange ring. If Ris hopfian so is the polynomial ring R[T].     

Certain decompositions of matrices over Abelian rings

A ring R is (weakly) nil clean provided that every element in R is the sum of a (weak) idempotent and a nilpotent. We characterize nil and weakly nil matrix rings over abelian rings. Let R be abelian, and let n ∈ ?. We prove that M ⁿ (R) is nil clean if and only if R/J(R) is Boolean and M _n (J(R)) is nil. Furthermore, we prove that R is weakly nil clean if and only if R is periodic; R/J(R) is ?₃, B or ?₃ ⊕ B where B is a Boolean ring, and that M _n (R) is weakly nil clean if and only if M _n (R) is nil clean for all n ≥ 2.     

Main Q-eigenvalues and generalized Q-cospectrality of graphs

Let Q ^G denote the signless Laplacian matrix of a graph G. An eigenvalue μ of Q ^G is said to be a main Q-eigenvalue of G if μ has an eigenvector which is not orthogonal to an all-ones vector e. We give some basic properties of main Q-eigenvalues. For a graph G of order n, G is called Q-controllable if G has n distinct main Q-eigenvalues. We show that a graph H is generalized Q-cospectral with a Q-controllable G if and only if H is Q-controllable and there exists a unique rational orthogonal matrix R such that R _e = e, Q _H = R ^? Q _G R.     

On Polygons Enclosing Point Sets II

Let R and B be disjoint point sets such that R ∪ B is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of R ∪ B belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56 |Conv(B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices p ₁ , ... , p _n and S a set of m points contained in the interior of P, m ≤ n − 1. Then there is a convex decomposition {P ₁ , ... , P _n } of P such that all points from S lie on the boundaries of P ₁ , ... , P _n , and each P _i contains a whole edge of P on its boundary. F. Hurtado partially supported by projects MEC MTM2006-01267 and DURSI 2005SGR00692. C. Merino supported by CONACYT of Mexico, Proyecto 43098. J. Urrutia supported by CONACYT of Mexico, Proyecto SEP-2004-Co1-45876, and MCYT BFM2003-04062. I. Ventura partially supported by Project MCYT BFM2003-04062.     

Homological connectivity of random k‐dimensional complexes

Let Δ_n?1 denote the (n ? 1)‐dimensional simplex. Let Y be a random k‐dimensional subcomplex of Δ_n?1 obtained by starting with the full (k ? 1)‐dimensional skeleton of Δ_n?1 and then adding each k‐simplex independently with probability p. Let H_k?1(Y; R) denote the (k ? 1)‐dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

On equivalence of derived categories

Let A be an Abelian category and B be a thick subcategory of A. Let D ^b(B) denote the derived category of cohomologically bounded chain complexes of objects in A and D _B ^b (A) denote the derived category of cohomologically bounded chain complexes of objects in A with cohomology in B. We give two if and only if conditions for equivalence of D(B) and D _B ^b (A), and we give an example where D ^b (B) and D _B ^b (A) are not equivalent.     

Classical operators on BLOCH spaces

Let \mathbbD \mathbb{D} ⁿ denote the unit polydisk and let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥1. We study weighted composition operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } ( \mathbbD \mathbb{D} ⁿ ), α > 1. We also study Cesàro type operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } (B _n ), α > 0. Bibliography: 15 titles.     

由本性随机阵导致的陈代数

A matrix of order n whose row sums are all equal to 1 is called an essentially stochastic matrix (see Johnsen [4]). We extend this notion as the following. Let F be a field of characteristic 0 or a prime greater than n. M_n(F) denotes the set of all n×n matrices over F. Let t be an elernent of F. A matrix A=(a_ij) in M_n(F) is called essentially t-stochastic' provided its row sums are each equal to t. We denote by R_n(t) the set of all essentially t-stochastic matrices over F. We shall mainly study R_n(0) and R_n(F)=(?)R_n(t). Our main references are Johnson [2,4] and Kim [5].     

The characteristic polynomial of a principal subpencil of a Hermitian matrix pencil

Given n-square Hermitian matrices A,B, let A_i,B_i denote the principal (n?1)- square submatrices of A,B, respectively, obtained by deleting row i and column i. Let μ, λ be independent indeterminates. The first main result of this paper is the characterization (for fixed i) of the polynomials representable as det(μA_i?λB_i) in terms of the polynomial det(μA?λB) and the elementary divisors, minimal indices, and inertial signatures of the pencil μA?λB. This result contains, as a special case, the classical interlacing relationship governing the eigenvalues of a principal sub- matrix of a Hermitian matrix. The second main result is the determination of the number of different values of i to which the characterization just described can be simultaneously applied.     

Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

Constructing Maximal Commutative Subalgebras of Matrix Rings in Small Dimensions

Let k denote an algebraically closed field of arbitrary characteristic. Let C denote the set of all commutative, finite dimensional, local k-algebras of the form (B, m, k) with i(m) ?2. Here i(m) denotes the index of nilpotency of the maximal ideal m. A Akalgebra (R, J,k)∈L is called a (c₁-construction if there exists (B, m, k)∈ £ ? {(k, (0), k)} and a finitely generated, faithful B-module N such that R,?B?(the idealization of N). (R.J.k) is called a (c_2::-construction if there exist a (B,m k)∈ L, a positive integer p $ge;2 and a nonzero z £ S_B(the socle of B) such that R?B[x]/(mX, X^p- z). Let M_n×n(K) denote the set of all n x n matrices, over k with n≥2. Let .M_n(k) denote the set of all maximal, commutative A;-subalgebras of M_n×n(k). In this paper, we show any (R J, k) ∈£?M_n;(k) with n>5 is a C₁ or C₂ -construction except for one isomorphism class. The one exception occurs when n = 5.