(n,d)-Injective covers,n-coherent rings,and (n,d)-rings

It is known that a ring R is left Noetherian if and only if every left R-module has an injective (pre)cover. We show that (1) if R is a right n-coherent ring, then every right R-module has an (n, d)-injective (pre)cover; (2) if R is a ring such that every (n, 0)-injective right R-module is n-pure extending, and if every right R-module has an (n, 0)-injective cover, then R is right n-coherent. As applications of these results, we give some characterizations of (n, d)-rings, von Neumann regular rings and semisimple rings.     

Some panconnected and pancyclic properties of graphs with a local ore-type condition

Asratian and Khachatrian proved that a connected graphG of order at least 3 is hamiltonian ifd(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| for any pathuwv withuv ? E(G), whereN(x) is the neighborhood of a vertexx. We prove that a graphG with this condition, which is not complete bipartite, has the following properties:

1. For each pair of verticesx, y with distanced(x, y) ≥ 3 and for each integern, d(x, y) ≤ n ≤ |V(G)| ? 1, there is anx ? y path of lengthn.
2. For each edgee which does not lie on a triangle and for eachn, 4 ≤ n ≤ |V(G)|, there is a cycle of lengthn containinge.
3. Each vertex ofG lies on a cycle of every length from 4 to |V(G)|.

This implies thatG is vertex pancyclic if and only if each vertex ofG lies on a triangle.     

Estimation of parameters in ARUMA models

Let X(n) be a time series satisfying the following ARUMA(p, d, q) models:U (B) A (B)X (n)=C (B) W (n)where U(B)=1+u(1)B+…+u(d) B~d is a polynomial with all roots on the unit circle, A(B)=1+a(1)B+…+a(p)Bp is a polynomial with all roots outside the unit circle, C(B)=1+c(1) B+…+c(q)Bq is a polynomial which is relatively prime with the polynomial U(B)A(B), B is thebackshift operator such that BX(n)=X(n-1), and (W (n), F(n), n≥1) is a sequence of martingaledifferences satisfying the following conditions:lim E (W (n)~2|F(n-1))=σ~2 a.s.n→∞sup E |W(n)|γ<∞ for some γ>2.n≥1The purpose of this paper is to provide consistent estimates of the parameters p, d, q, u(j) (j=1,2,…,d), and a(k) (k=1, 2.…, p).     

On Commutativity of Rings With Derivations

Let R be a ring and d : R → R a derivation of R. In the present paper we investigate commutativity of R satisfying any one of the properties (i)d([x,y]) = [x,y], (ii)d(x o y) = xoy, (iii)d(x) o d(y) = 0, or (iv)d(x) o d(y) = x o y, for all x, y in some apropriate subset of R.     

Dense subgroups ofC(K)-Stone-Weierstrass-type theorems for groups

LetB be a subgroup ofC(K) which separates points and contains the constants. An elementh∈C(R) operates onB iff∈B implies thath°f∈B. An elementh∈C(R) is condensing if its operation onB implies the density ofB inC(K). Similar notation applies to subgroupsB ofC(K, G) whereG is a metrizable group. We study the setD(G) of condensing functions inC(G, G) whenG is the additive group of a real Banach space and in particular whenG=R ⁿ.     

On functions of companion matrices

Presented in this paper are some new properties of a function f(C) of a companion matrix C, including (1) a representation of any entry of f(C) as a divided difference of f(λ) times a polynomial, (2) a similarity decomposition of f(C) generalizing that based on the Jordan form of C, and (3) characterization (construction) of all matrices that transform f(C) (by similarity) to companion form. The connection between functions of general and companion matrices is also dealt with, and a pair of dual relations is established.     

Random problems

A problem (a Boolean function f: {0, 1}^N → {0, 1}) is characterized by its randomness (à la Kolmogorov) R(f) and its entropy (à la Shannon) H(f). Random problems have large values of R(f) and are a good model for many natural pattern recognition problems. R(f) and H(f) are shown to be lower and upper bounds, respectively, for a minimum-size circuit that computes f False entropy, namely the hidden structure of a problem, is related to the difference between H(f) and R(f).     

Handcuffed designs

Handcuffed designs are a particular case of block designs on graphs. A handcuffed design with parametersv, k, λ consists of a system of orderedk-subsets of av-set, called handcuffed blocks. In a block {A ₁,A ₂,?, A _k } each element is assumed to be handcuffed to its neighbours and the block containsk ? 1 handcuffed pairs (A ₁,A ₂), (A ₂,A ₃), ? (A _k?1,A _k ). These pairs are considered unordered. The collection of handcuffed blocks constitute a hundcuffed design if the following are satisfied: (1) each element of thev-set appears amongst the blocks the same number of times (and at most once in a block) and (2) each pair of distinct elements of thev-set are handcuffed in exactly λ of the blocks. If the total number of blocks isb and each element appears inr blocks the following conditions are necessary for the handcuffed design to exist:

1. λv(v?1) = (k?1) b,
2. rv = kb.

We denote byH(v, k, λ) the class of all handcuffed designs with parametersv, k, λ and sayH (v, k, λ) exists if there is a design with parametersv, k, λ. In this paper we prove that the necessary conditions forH (v, k, λ) exist are also sufficient in the following cases: (a)λ = 1 or 2; (b)k = 3; (c)k is evenk = 2h, and (λ, 2h ? 1) = 1; (d)k is odd,k = 2h + 1, and (λ, 4h)=2 or (λ, 4h)=1.     

Realizability of p-point graphs with prescribed minimum degree,maximum degree,and point-connectivity

In a recent paper, we gave a generalization of extremal problems involving certain graph-theoretic invariants. In that work, we defined a (p, Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree δ, and line-connectivity λ. An arbitrary quadruple of integers (a, b, c, d) was called (p, Δ, δ, λ) realizable if there is a (p, Δ, δ, λ) graph with p = a, Δ = b, δ = c, and λ = d. In this work, we consider the more difficult case of (p, Δ, δ, κ) realizability, where κ is the point-connectivity. Necessary and sufficient conditions for a quadruple to be (p, Δ, δ, κ) realizable are derived.     

On the structure of k-connected graphs without Kk-minor

Suppose G is a k-connected graph that does not contain K_k as a minor. What does G look like? This question is motivated by Hadwiger’s conjecture (Vierteljahrsschr. Naturforsch. Ges. Zürich 88 (1943) 133) and a deep result of Robertson and Seymour (J. Combin. Theory Ser. B. 89 (2003) 43).It is easy to see that such a graph cannot contain a (k−1)-clique, but could contain a (k−2)-clique, as K_k−5+G′, where G′ is a 5-connected planar graph, shows. In this paper, however, we will prove that such a graph cannot contain three “nearly” disjoint (k−2)-cliques. This theorem generalizes some early results by Robertson et al. (Combinatorica 13 (1993) 279) and Kawarabayashi and Toft (Combinatorica (in press)).     

Connected and disconnected fields

Examples are sought of Hausdorff ring topologies on a field that are (i) arcwise connected; (ii) connected but not arcwise connected; (iii) totally disconnected but not ultraregular; (iv) ultraregular but not basically disconnected; (v) basically disconnected but neither a P-space nor extremally disconnected; (vi) P-spaces; (vii) extremally disconnected. Examples of type (i), (ii), (iv) and (vi) are given. For a field with a ring topology, properties of F-zerosets are considered. In particular, it is shown that the intersection of each pair of F-zerosets is again an F-zeroset if and only if {(0, 0)} is an F-zeroset of F².     

On the orthogonal component of BSDEs in a Markovian setting

In this note we consider a quadratic growth backward stochastic differential equation (BSDE) driven by a continuous martingale M. We prove (in Theorem 3.2) that if M is a strong Markov process and if the BSDE has the form (2.2) with regular data then the unique solution (Y,Z,N) of the BSDE is reduced to (Y,Z), i.e. the orthogonal martingale N is equal to zero, showing that in a Markovian setting the “usual” solution (Y,Z) (of a BSDE with regular data) has not to be completed by a strongly orthogonal component even if M does not enjoy the martingale representation property.     

The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits

We first describe a mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a quantum differential system (that is a trivial bundle equipped with a suitable flat meromorphic connection and a flat bilinear form) and we give an explicit isomorphism between these two quantum differential systems. On the A-side (resp. on the B-side), the quantum differential system alluded to is naturally produced by the small quantum cohomology (resp. a solution of the Birkhoff problem for the Brieskorn lattice of a Landau–Ginzburg model). Then we study the degenerations of these quantum differential systems and we apply our results to the construction of (classical, limit, logarithmic) Frobenius manifolds.     

Minimum degree of a graph and the existence ofk-factors

Let G be a graph andk a positive integer such that (i)k|V(G)| is even; (ii) δ(G) ≥1/2[|V(G)|], and (iii) |V(G)|≥4k-t. ThenG has ak-factor.     

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball By admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed *-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)^**.     

On domination numbers of graph bundles

Letγ(G) be the domination number of a graphG. It is shown that for anyκ ≥ 0 there exists a Cartesian graph bundleB█φF such thatγ(B█φF) =γ(B)γ(F) — 2κ. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing’s conjecture on strong graph bundles is shown not to be true by proving the inequalityγ(B █ φF) ≤γ(B)γ(F) for strong graph bundles. Examples of graphsB andF withγ(B █ φF) γ(B)γ(F) are given.     

Field invariants under totally positive quadratic extensions

K = F(√d) is a formally real field and a totally positive quadratic extension of F. A decomposition theorem for quadratic forms in Fed (K) is given. The invariants r(q) and ud(KF) are defined and relations between the invariants β_F(i), β_K(i), ud(F), ud(K), l(F), l(K) are studied, using the theory of quadratic forms.     

The embedding of (0, α)-geometries in PG(n, q): part II

In Part I we obtained results about the embedding of (0, α)-geometries in PG(3, q). Here we determine all (0, α)-geometries with q+1 points on a line, which are embedded in PG(n, q), n>3 and q>2. As a particular case all semi partial geometries with parameters s=q,t,α(>1),μ, which are embeddable in PG(n, q), q≠2, are obtained. We also prove some theorems about the embedding of (0, 2)-geometries in PG(n, 2): we show that without loss of generality we may restrict ourselves to reduced (0, 2)-geometries, we determine all (0, 2)-geometries in PG(4, 2), and we describe an unusual embedding of U_2,3(9) in PG(5, 2).     

Quasi-implication algebras,Part I: Elementary theory

Quasi-implication algebras (QIA's) are intended to generalize orthomodular lattices (OML's) in the same way that implication algebras (J. C. Abbott) generalize Boolean lattices. A QIA is defined to be a setQ together with a binary operation → satisfying the following conditions (a→b is denotedab). (Q1) $$\left( {ab} \right)a = a$$ (Q2) $$\left( {ab} \right)\left( {ac} \right) = \left( {ba} \right)\left( {bc} \right)$$ (Q3) $$\left( {\left( {ab} \right)\left( {ba} \right)} \right)a = \left( {\left( {ba} \right)\left( {ab} \right)} \right)b$$ Every OML induces a QIA, wherea → b=a ^⊥?(a?b). On the other hand, every QIA induces a join semi-lattice with a greatest element 1, where 1=aa,a≤b iffab=1, anda?b=((ab)(ba))a. A bounded QIA is defined to be a QIA with a least element 0 (w.r.t.≤). The QIA associated with any OML is bounded, the zero elements being the same. Conversely, every bounded QIA induces an OML, wherea ^⊥=a0, anda?b=((ab)(a0))0. The relationC of compatibility is defined so thataCb iffa≤ba, and it is shown that every compatible sub-QIA of a QIA is an implication algebra.