Asymptotic properties of locally extensible designs

An (n+1, 1)-design D is locally extensible at a block B if D can be embedded in an (n+1, 1)-design having a block B ^* of cardinality n+1 and such that BB ^*. If D is embeddable in a finite projective plane of order n, then D is called globally extensible. In this paper, we investigate the asymptotic behaviour of locally extensible designs and Euclidean designs. We study the relationship between locally extensible and extensible designs and the uniqueness of such embeddings. It is shown that, for n, l and t sufficiently large, any (n+1, 1)-design which has minimum block length l and which is locally extensible at t of its blocks is globally extensible.     

A Characterization of Locally Semicomplete CKI-digraphs

A digraph is locally-in semicomplete if for every vertex of D its in-neighborhood induces a semicomplete digraph and it is locally semicomplete if for every vertex of D the in-neighborhood and the out-neighborhood induces a semicomplete digraph. The locally semicomplete digraphs where characterized in 1997 by Bang-Jensen et al. and in 1998 Bang-Jensen and Gutin posed the problem if finding a kernel in a locally-in semicomplete digraph is polynomial or not. A kernel of a digraph is a set of vertices, which is independent and absorbent. A digraph D such that every proper induced subdigraph of D has a kernel is said to be critical kernel imperfect digraph (CKI-digraph) if the digraph D does not have a kernel. A digraph without an induced CKI-digraph as a subdigraph does have a kernel. We characterize the locally semicomplete digraphs, which are CKI. As a consequence of this characterization we conclude that determinate whether a locally semicomplete digraph is a CKI-digraph or not, is polynomial.     

Group inverse for two classes of 2×2 anti-triangular block matrices over right Ore domains

Suppose ? is a right Ore domain with identity 1. Let ? ^m×n denote the set of all m×n matrices over ?. In this paper, we give the necessary and sufficient conditions for the existence and explicit representations of the group inverses of the block matrices in the following two cases, respectively:

1. A ²=A and CA=C;
2. r(C)≤r(B) and A=DC,

where A∈? ^n×n , B∈? ^n×m , C∈? ^m×n , D∈? ^n×m . The paper’s results generalize some relative results of Liu and Yang (Appl. Math. Comput. 218:8978–8986, 2012).     

Circle grids and bipartite graphs of distances

Fort fixed,n+t pointsA ₁,A ₂,...,A _n andB ₁,B ₂,...,B _t are constructed in the plane withO(n) distinct distancesd(A _i B _j ) As a by-product we show that the graph of thek largest distances can contain a complete subgraphK _{t, n} withn=(k ²), which settles a problem of Erds, Lovász and Vesztergombi.Research partially supported by the Hungarian National Science Fund (OTKA) # 2117.     

A ratio ergodic theorem for increasing additive functionals

Summary Let B be a 1-dimensional Brownian motion. In this paper ratios of the form A ⁺ (t)/A ^- (t), where A ⁺ is the (0, )-occupation time functional of B and A ^-is a local time integral of an infinite (but locally finite) measure m with support in (-, 0), are studied. Conditions on m are given which ensure that such a ratio will be unbounded a.s. (or go to zero a.s.) as t».The work was supported in part by a grant from the National Science Foundation.     

Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t ^k+1 and letG be a reductive complex Lie algebra with exponentsm ₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA ₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA ₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m _s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m _s +1, one has weight 0 and the others have weights (k+1)m _s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A ₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA _n ,B _n ,C _n , orD _n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship     

Digraphs containing every possible pair of dicycles

Let D = (V, A) be a directed graph of order n ≥ 4. Suppose that the minimum degree of D is at least (3n − 3)/2. Then for any two integers s and t with s ≥ 2, t ≥ 2 and s + t ≤ n, D contains two vertex‐disjoint directed cycles of lengths s and t, respectively. Moreover, the condition on the minimum degree is sharp. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 154–162, 2000     

Matrices with different Gondran–Minoux and determinantal ranks over max-algebras

Let GMr(A) be the row Gondran–Minoux rank of a matrix, GMc(A) be the column Gondran–Minoux rank, and d(A) be the determinantal rank, respectively. The following problem was posed by M. Akian, S. Gaubert, and A. Guterman: Find the minimal numbers m and n such that there exists an (m × n)-matrix B with different row and column Gondran–Minoux ranks. We prove that in the case GMr(B) > GMc(B) the minimal m and n are equal to 5 and 6, respectively, and in the case GMc(B) > GMr(B) the numbers m = 6 and n = 5 are minimal. An example of a matrix $ A \in {\mathcal{M}_{5 \times 6}}\left( {{\mathbb{R}_{\max }}} \right) $ such that GMr(A) = GMc(A ^t) = 5 and GMc(A) = GMr(A ^t) = 4 is provided. It is proved that p = 5 and q = 6 are the minimal numbers such that there exists an (p×q)-matrix with different row Gondran–Minoux and determinantal ranks.     

Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matrices

The method developed in [A.J. Durán, F.A. Grünbaum, Orthogonal matrix polynomials satisfying second order differential equations, Int. Math. Res. Not. 10 (2004) 461–484] led us to consider matrix polynomials that are orthogonal with respect to weight matrices W(t) of the form , , and (1−t)^α(1+t)^βT(t)T^*(t), with T satisfying T^′=(2Bt+A)T, T(0)=I, T^′=(A+B/t)T, T(1)=I, and T^′(t)=(−A/(1−t)+B/(1+t))T, T(0)=I, respectively. Here A and B are in general two non-commuting matrices. We are interested in sequences of orthogonal polynomials (P_n)_n which also satisfy a second order differential equation with differential coefficients that are matrix polynomials F₂, F₁ and F₀ (independent of n) of degrees not bigger than 2, 1 and 0 respectively. To proceed further and find situations where these second order differential equations hold, we only dealt with the case when one of the matrices A or B vanishes.The purpose of this paper is to show a method which allows us to deal with the case when A, B and F₀ are simultaneously triangularizable (but without making any commutativity assumption).     

Disjoint systems

A disjoint system of type (?, ?, k, n) is a collection ?? = {??₁,…, ??_m} of pairwise disjoint families of k-subsets of an n-element set satisfying the following condition. For every ordered pair ??_i and ??_j of distinct members of ?? and for every A ? ??_i there exists a B ? ??_j that does not intersect A. Let D_n (?, ?, k) denote the maximum possible cardinality of a disjoint system of type (?, ?, k, n). It is shown that for every fixed k ? 2,. This settles a problem of Ahlswede, Cai, and Zhang. Several related problems are considered as well.     

Geometry and inequalities of geometric mean

We study some geometric properties associated with the t-geometric means A ?_tB:= A^1/2(A^?1/2BA^?1/2)^tA^1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

Generalization of a theorem of Gonchar

Let X and Y be two complex manifolds, let D⊂X and G⊂Y be two nonempty open sets, let A (resp. B) be an open subset of ∂D (resp. ∂G), and let W be the 2-fold cross ((D∪A)×B)∪(A×(B∪G)). Under a geometric condition on the boundary sets A and B, we show that every function locally bounded, separately continuous on W, continuous on A×B, and separately holomorphic on (A×G)∪(D×B) “extends” to a function continuous on a “domain of holomorphy” and holomorphic on the interior of .     

Kings in locally semicomplete digraphs

A k‐king in a digraph D is a vertex which can reach every other vertex by a directed path of length at most k. We consider k‐kings in locally semicomplete digraphs and mainly prove that all strong locally semicomplete digraphs which are not round decomposable contain a 2‐king. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 279–287, 2010     

A Bipartite Analogue of Dilworth’s Theorem

Let m(n) be the maximum integer such that every partially ordered set P with n elements contains two disjoint subsets A and B, each with cardinality m(n), such that either every element of A is greater than every element of B or every element of A is incomparable with every element of B. We prove that . Moreover, for fixed ε ∈ (0,1) and n sufficiently large, we construct a partially ordered set P with n elements such that no element of P is comparable with other elements of P and for every two disjoint subsets A and B of P each with cardinality at least , there is an element of A that is comparable with an element of B.     

Deadbeat control: A special inverse eigenvalue problem

In this paper we give a numerical method to construct a rankm correctionBF (where then ×m matrixB is known and them ×n matrixF is to be found) to an ×n matrixA, in order to put all the eigenvalues ofA +BF at zero. This problem is known in the control literature as deadbeat control. Our method constructs, in a recursive manner, a unitary transformation yielding a coordinate system in which the matrixF is computed by merely solving a set of linear equations. Moreover, in this coordinate system one easily constructs the minimum norm solution to the problem. The coordinate system is related to the Krylov sequenceA ^–1 B,A ^–2 B,A ^–3 B, .... Partial results of numerical stability are also obtained.Dedicated to Professor Germund Dahlquist: on the occasion of his 60th birthday     

On the vector space of 0-configurations

Let α be a rational-valued set-function on then-element sexX i.e. α(B) εQ for everyB ⫅X. We say that α defines a 0-configuration with respect toA⫅2 ^x if for everyA εA we have α(B)=0. The 0-configurations form a vector space of dimension 2 ⁿ − |A| (Theorem 1). Let 0 ≦t<k ≦n and letA={A ⫅X: |A| ≦t}. We show that in this case the 0-configurations satisfying α(B)=0 for |B|>k form a vector space of dimension , we exhibit a basis for this space (Theorem 4). Also a result of Frankl, Wilson [3] is strengthened (Theorem 6).     

Endomorphism algebras and Igusa–Todorov algebras

Let A be an Artin algebra. If $V\in \operatorname{mod} A$ such that the global dimension of  $\operatorname{End}_{A}V$ is at most 3, then for any ${M\in \operatorname{add}_{A}V}$ , both B and B ^op are 2-Igusa–Todorov algebras, where ${B=\operatorname{End}_{A}M}$ . Let ${P\in \operatorname{mod} A}$ be projective and ${B=\operatorname{End}_{A}P}$ such that the projective dimension of P as a right B-module is at most n(<∞). If A is an m-syzygy-finite algebra (resp. an m-Igusa–Todorov algebra), then B is an (m+n)-syzygy-finite algebra (resp. an (m+n)-Igusa–Todorov algebra); in particular, the finitistic dimension of B is finite in both cases. Some applications of these results are given.     

On Short Time Asymptotic Behavior of Some Symmetric Diffusions on General State Spaces

For conservative symmetric diffusions on a general state space (X,m), the short time asymptotic behavior of tlog _X 1 _A T _t 1 _B dm is investigated, where T _t is the associated semigroup and A and B are measurable subsets of X. It is proved that the superior limit is dominated by the inferior limit up to some absolute constant. When ₂ of the associated Dirichlet form is lower bounded, it is shown that the limit exists for any A and B, and is described by the intrinsic metric between them. Applications to infinite-dimensional spaces and fractals are given.