Syzygy algebras for Segre embeddings

We describe the syzygy spaces for the Segre embedding ?(U) × ?(V) ? ?(U ? V) in terms of representations of GL(U) × GL(V) and construct the minimal resolutions of the sheaves (a, b) in D(?(U ? V)) for a ? ?dim(U) and b ? ?dim(V). We also prove a property of multiplication in syzygy spaces of the Segre embedding.     

Commuting and centralizing generalized derivations on lie ideals in prime rings

Let R be a noncommutative prime ring of characteristic different from 2, U the Utumi quotient ring of R, C the extended centroid of R, and L a noncentral Lie ideal of R. If F and G are generalized derivations of R and k ≥1 a fixed integer such that [F(x), x] _k x ? x[G(x), x] _k = 0 for any x ∈ L, then one of the following holds:

1. either there exists an a ∈ U and an α ∈ C such that F(x) = xa and G(x) = (a + α)x for all x ∈ R
2. or R satisfies the standard identity s ₄(x ₁, …, x ₄) and one of the following conclusions occurs

1. there exist a, b, c, q ∈ U, such that a ?b + c ?q ∈ C and F(x) = ax + xb, G(x) = cx + xq for all x ∈ R
2. there exist a, b, c ∈ U and a derivation d of U such that F(x) = ax+d(x) andG(x) = bx+xc?d(x) for all x ∈ R, with a + b ? c ∈ C.

     

On stability of the homogeneity condition

Let ? be a function defined on a cone S with the values in a sequentially complete locally convex linear topological Hausdorff space Y. If there exist a bounded subset V of Y and an open interval (a, b) ? (1,∞) such that for all x ∈ S and every A ∈ (a, b) the condition λ^?1 ?(λx) ? ?(x) ∈ V holds, then there exists a unique positively homogeneous mapping F: S → Y such that the difference F(x) ? ?(x) is uniformly bounded on S.     

On a generalization of Minkowski's convex body theorem

Given a lattice $\text{Λ ? R}{}^{\mathrm{n}}$ and a bounded function g(x), x ∈ Rⁿ, vanishing outside of a bounded set, the functions ?(x)$\text{g}\text{?}\text{(x)?}$max_u∈Λg(u +x), ?(x)?Σ_{u∈Λ g(u +x)}, and ?₊(x)?Σ_u∈Λ max_{v∈Λ min {g(v + x); g(u + v + x)}} are defined and periodic mod Λ on Rⁿ. In the paper we prove that ?(x) + ?₊(x) ? 2?(x) ≥ ?(x) + h?₊(x) ? 2?(x) holds for all x ∈ Rⁿ, where h(x) is any “truncation” of g by a constant c ≥ 0, i.e., any function of the form h(x)?g(x) if g(x) ≤ c and h(x)?c if g(x) > c. This inequality easily implies some known estimations in the geometry of numbers due to Rado [1] and Cassels [2]. Moreover, some sharper and more general results are also derived from it. In the paper another inequality of a similar type is also proved.     

Embedding and unsolvability theorems for modular lattices

LetR be a nontrivial ring with 1 and δ a cardinal. Let,L(R, δ) denote the lattice of submodules of a free unitaryR-module on δ generators. Let ? be the variety of modular lattices. A lattice isR-representable if embeddable in the lattice of submodules of someR-module; ?(R) denotes the quasivariety of allR-representable lattices. Let ω denote aleph-null, and let a (m, n) presentation havem generators andn relations,m, n≤ω. THEOREM. There exists a (5, 1) modular lattice presentation having a recursively unsolvable word problem for any quasivarietyV,V ? ?, such thatL(R, ω) is inV. THEOREM. IfL is a denumerable sublattice ofL(R, δ), then it is embeddable in some sublatticeK ofL(R,δ^*) having five generators, where δ^*=δ for infinite δ and δ^*=4δ(m+1) if δ is finite andL has a set ofm generators. THEOREM. The free ?(R)-lattice on ω generators is embeddable in the free ?(R)-lattice on five generators. THEOREM. IfL has an (m, n), ?(R)-presentation for denumerablem and finiten, thenL is embeddable in someK having a (5, 1) ?(R)-presentation.     

Distance-hereditary graphs and signpost systems

An ordered pair (U,R) is called a signpost system if U is a finite nonempty set, R⊆U×U×U, and the following axioms hold for all u,v,w∈U: (1) if (u,v,w)∈R, then (v,u,u)∈R; (2) if (u,v,w)∈R, then (v,u,w)∉R; (3) if u≠v, then there exists t∈U such that (u,t,v)∈R. (If F is a (finite) connected graph with vertex set U and distance function d, then U together with the set of all ordered triples (u,v,w) of vertices in F such that d(u,v)=1 and d(v,w)=d(u,w)−1 is an example of a signpost system). If (U,R) is a signpost system and G is a graph, then G is called the underlying graph of (U,R) if V(G)=U and xy∈E(G) if and only if (x,y,y)∈R (for all x,y∈U). It is possible to say that a signpost system shows a way how to travel in its underlying graph. The following result is proved: Let (U,R) be a signpost system and let G denote the underlying graph of (U,R). Then G is connected and every induced path in G is a geodesic in G if and only if (U,R) satisfies axioms (4)-(8) stated in this paper; note that axioms (4)-(8)-similarly as axioms (1)-(3)-can be formulated in the language of the first-order logic.     

Binding number and minimum degree for the existence of (g,f,n)-critical graphs

Let G be a graph of order p. The binding number of G is defined as $\mbox{bind}(G):=\min\{\frac{|N_{G}(X)|}{|X|}\mid\emptyset\neq X\subseteq V(G)\,\,\mbox{and}\,\,N_{G}(X)\neq V(G)\}$ . Let g(x) and f(x) be two nonnegative integer-valued functions defined on V(G) with g(x)≤f(x) for any x∈V(G). A graph G is said to be (g,f,n)-critical if G?N has a (g,f)-factor for each N?V(G) with |N|=n. If g(x)≡a and f(x)≡b for all x∈V(G), then a (g,f,n)-critical graph is an (a,b,n)-critical graph. In this paper, several sufficient conditions on binding number and minimum degree for graphs to be (a,b,n)-critical or (g,f,n)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.     

Unbounded derivations tangential to compact groups of automorphisms

We consider unbounded derivations in C¹-algebras commuting with compact groups of ¹-automorphisms. A closed ¹-derivation δ in a C¹-algebra $\text{U}$ is said to be a generator if there exists a strongly continuous one-parameter subgroup t∈$\text{R}$→τ(t)? Aut($\text{U}$) such that $\text{δ =}\text{d}\text{dt}\text{τ(t)¦}{}_{\mathrm{t\; =\; 0}}$. If δ is known to commute with a compact abelian action α:G→Aut($\text{U}$), and if δ(a) = 0 for all a in the fixed point algebra $\text{U}$^α of the action G, then we show that δ is necessarily a generator. Moreover, in any faithful G-covariant representation, there is a commutative operator field γ ∈ ? → v(γ) such that $\text{v(γ)}{}^1\text{= ?v(γ), v(γ)}$ is possibly unbounded but affiliated with the center of {$\text{U}$^α}″, and e^tδ(x) = xe^tv(γ) for all x in the Arveson spectral subspace $\text{U}$^α(γ). In particular, if $\text{U}$ is the CAR algebra over an infinite-dimensional Hilbert space and α is the gauge group, then any such derivation δ is a scalar multiple of the generator of the gauge group.     

Long Paths Through Specified Vertices In 3-Connected Graphs

Let h ≥ 6 be an integer, let G be a 3-connected graph with ∣V(G)∣ ≥ h − 1, and let x and z be distinct vertices of G. We show that if for any nonadjacent distinct vertices u and v in V(G) − {x, z}, the sum of the degrees of u and v in G is greater than or equal to h, then for any subset Y of V(G) − {x, z} with ∣Y∣ ≤ 2, G contains a path which has x and z as its endvertices, passes through all vertices in Y, and has length at least h − 2. We also show a similar result for cycles in 2-connected graphs.     

New bounds for the broadcast domination number of a graph

A dominating broadcast on a graph G = (V, E) is a function f: V → {0, 1, ..., diam G} such that f(v) ≤ e(v) (the eccentricity of v) for all v ∈ V and such that each vertex is within distance f(v) from a vertex v with f(v) > 0. The cost of a broadcast f is σ(f) = Σ _v∈V f(v), and the broadcast number λ _b (G) is the minimum cost of a dominating broadcast. A set X ? V(G) is said to be irredundant if each x ∈ X dominates a vertex y that is not dominated by any other vertex in X; possibly y = x. The irredundance number ir (G) is the cardinality of a smallest maximal irredundant set of G. We prove the bound λ_b(G) ≤ 3 ir(G)/2 for any graph G and show that equality is possible for all even values of ir (G). We also consider broadcast domination as an integer programming problem, the dual of which provides a lower bound for λ_b.     

On the extension of vertex maps to graph homomorphisms

A reflexive graph is a simple undirected graph where a loop has been added at each vertex. If G and H are reflexive graphs and U⊆V(H), then a vertex map f:U→V(G) is called nonexpansive if for every two vertices x,y∈U, the distance between f(x) and f(y) in G is at most that between x and y in H. A reflexive graph G is said to have the extension property (EP) if for every reflexive graph H, every U⊆V(H) and every nonexpansive vertex map f:U→V(G), there is a graph homomorphism φ_f:H→G that agrees with f on U. Characterizations of EP-graphs are well known in the mathematics and computer science literature. In this article we determine when exactly, for a given “sink”-vertex s∈V(G), we can obtain such an extension φ_f;s that maps each vertex of H closest to the vertex s among all such existing homomorphisms φ_f. A reflexive graph G satisfying this is then said to have the sink extension property (SEP). We then characterize the reflexive graphs with the unique sink extension property (USEP), where each such sink extensions φ_f;s is unique.     

The recognition of the class of indecomposable digraphs under low hemimorphy

Given a digraph G=(V,A), the subdigraph of G induced by a subset X of V is denoted by G[X]. With each digraph G=(V,A) is associated its dual G^?=(V,A^?) defined as follows: for any x,y∈V, (x,y)∈A^? if (y,x)∈A. Two digraphs G and H are hemimorphic if G is isomorphic to H or to H^?. Given k>0, the digraphs G=(V,A) and H=(V,B) are k-hemimorphic if for every X⊆V, with |X|≤k, G[X] and H[X] are hemimorphic. A class C of digraphs is k-recognizable if every digraph k-hemimorphic to a digraph of C belongs to C. In another vein, given a digraph G=(V,A), a subset X of V is an interval of G provided that for a,b∈X and x∈V−X, (a,x)∈A if and only if (b,x)∈A, and similarly for (x,a) and (x,b). For example, 0?, {x}, where x∈V, and V are intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable.     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Monotonic Solutions of a Functional Equation Arising from Simultaneous Utility Representations

The functional equation ?(h(x)) ? ?(h(y)) + ?(y) = ?(h(x ? y) + y) was solved by Aczél, Luce and Marley on the assumption that the functions are different iable. They posed the question of its strictly monotonic continuous solutions. The problem is solved using a uniqueness theorem. The continuity assumption on the functions is removed and the equation is also solved on a restricted domain.     

Max-min of polynomials and exponential diophantine equations

In this paper we study the maximum-minimum value of polynomials over the integer ring Z. In particular, we prove the following: Let F(x,y) be a polynomial over Z. Then, maxx∈Z(T)miny∈Z|F(x,y)|=o(T1/2) as T→∞ if and only if there is a positive integer B such that maxx∈Zminy∈Z|F(x,y)|?B. We then apply these results to exponential diophantine equations and obtain that: Let f(x,y), g(x,y) and G(x,y) be polynomials over Q, G(x,y)∈(Q[x,y]−Q[x])∪Q, and b a positive integer. For every α in Z, there is a y in Z such that f(α,y)+g(α,y)bG(α,y)=0 if and only if for every integer α there exists an h(x)∈Q[x] such that f(x,h(x))+g(x,h(x))bG(x,h(x))≡0, and h(α)∈Z.     

About noncommuting graphs

The noncommuting graph ?(G) of a nonabelian finite group G is defined as follows: The vertices of ?(G) are represented by the noncentral elements of G, and two distinct vertices x and y are joined by an edge if xy ≠ yx. In [1], the following was conjectured: Let G and H be two nonabelian finite groups such that ?(G) ? ?(H); then ¦G¦ = ¦H¦. Here we give some counterexamples to this conjecture.     

Convergence of Gaussian Quadrature Formulas for Power Orthogonal Polynomials

In classical theorems on the convergence of Gaussian quadrature formulas for power orthogonal polynomials with respect to a weight w on I =（a,b）,a function G ∈ S（w）:= { f:∫_I | f（x）| w（x）d x < ∞} satisfying the conditions G ^2j（x） ≥ 0,x ∈（a,b）,j = 0,1,...,and growing as fast as possible as x → a + and x → b,plays an important role.But to find such a function G is often difficult and complicated.This implies that to prove convergence of Gaussian quadrature formulas,it is enough to find a function G ∈ S（w） with G ≥ 0 satisfying sup n ∑λ0knG（xkn） k=1 n<∞ instead,where the xkn ’s are the zeros of the n th power orthogonal polynomial with respect to the weight w and λ0kn ’s are the corresponding Cotes numbers.Furthermore,some results of the convergence for Gaussian quadrature formulas involving the above condition are given.     

Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Let R be a ring. We recall that R is called a near pseudo-valuation ring if every minimal prime ideal of R is strongly prime. Let now σ be an automorphism of R and δ a σ-derivation of R. Then R is said to be an almost δ-divided ring if every minimal prime ideal of R is δ-divided. Let R be a Noetherian ring which is also an algebra over ? (? is the field of rational numbers). Let σ be an automorphism of R such that R is a σ(*)-ring and δ a σ-derivation of R such that σ(δ(a)) = δ(σ(a)) for all a ∈ R. Further, if for any strongly prime ideal U of R with σ(U) = U and δ(U) ? δ, U[x; σ, δ] is a strongly prime ideal of R[x; σ, δ], then we prove the following:

1. R is a near pseudo valuation ring if and only if the Ore extension R[x; σ, δ] is a near pseudo valuation ring.
2. R is an almost δ-divided ring if and only if R[x; σ, δ] is an almost δ-divided ring.

     

Balanced decomposition of a vertex-colored graph

A balanced vertex-coloring of a graph G is a function c from V(G) to {−1,0,1} such that ∑{c(v):v∈V(G)}=0. A subset U of V(G) is called a balanced set if U induces a connected subgraph and ∑{c(v):v∈U}=0. A decomposition V(G)=V₁∪?∪V_r is called a balanced decomposition if V_i is a balanced set for 1≤i≤r.In this paper, the balanced decomposition number f(G) of G is introduced; f(G) is the smallest integer s such that for any balanced vertex-coloring c of G, there exists a balanced decomposition V(G)=V₁∪?∪V_r with |V_i|≤s for 1≤i≤r. Balanced decomposition numbers of some basic families of graphs such as complete graphs, trees, complete bipartite graphs, cycles, 2-connected graphs are studied.