The smooth Banach submanifold B*(E, F) in B(E, F)

Let E,F be two Banach spaces,B(E,F),B+(E,F),Φ(E,F),SΦ(E,F) and R(E,F) be bounded linear,double splitting,Fredholm,semi-Frdholm and finite rank operators from E into F,respectively. Let Σ be any one of the following sets:{T ∈Φ(E,F):Index T=constant and dim N(T)=constant},{T ∈ SΦ(E,F):either dim N(T)=constant< ∞ or codim R(T)=constant< ∞} and {T ∈ R(E,F):Rank T=constant< ∞}. Then it is known that Σ is a smooth submanifold of B(E,F) with the tangent space TAΣ={B ∈ B(E,F):BN(A)-R(A) } for any A ∈Σ. However,for ...     

The number of connected sparsely edged graphs. II. Smooth graphs and blocks

A smooth graph is a connected graph without endpoints; f(n, q) is the number of connected graphs, v(n, q) is the number of smooth graphs, and u(n, q) is the number of blocks on n labeled points and q edges: W_k, V_k, and U_k are the exponential generating functions of f(n, n + k), v(n, n + k), and u(n, n + k), respectively. For any k ? 1, our reduction method shows that V_k can be deduced at once from W_k, which was found for successive k by the computer method described in our previous paper. Again the reduction method shows that U_k must be a sum of powers (mostly negative) of 1 - X and, given this information, we develop a recurrence method well suited to calculate U_k for successive k. Exact formulas for v(n, n + k) and u(n, n + k) for general n follow at once.     

Upper and lower bounds of the valence-functional

For the non-negative integerg let (M, g) denote the closed orientable 2-dimensional manifold of genusg. K-realizationsP of (M, g) are geometric cell-complexes inP with convex facets such that set (P) is homeomorphic toM. ForK-realizationsP of (M, g) and verticesv ofP, val (v,P) denotes the number of edges ofP incident withv and the weighted vertex-number Σ(val(v, P)-3) taken over all vertices ofP is called valence-valuev (P) ofP. The valence-functionalV, which is important for the determination of all possiblef-vectors ofK-realisations of (M, g), in connection with Eberhard's problem etc., is defined byV(g):=min[v(P)|P is aK-realization of (M,g)]. The aim of the note is to prove the inequality 2g+1≦V(g)≦3g+3 for every positive integerg.     

Power cocentralizing generalized derivations on prime rings

Let R be a prime ring, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, H and G non-zero generalized derivations of R. Suppose that there exists an integer n ≥ 1 such that (H(u)u − uG(u)) ⁿ = 0, for all u ∈ L, then one of the following holds: (1) there exists c ∈ U such that H(x) = xc, G(x) = cx; (2) R satisfies the standard identity s ₄ and char (R) = 2; (3) R satisfies s ₄ and there exist a, b, c ∈ U, such that H(x) = ax+xc, G(x) = cx+xb and (a − b) ⁿ = 0.     

A new infinite class of S(3, {4, 5, 7}, v)

A t‐wise balanced design ( at BD) of order v and block sizes from K , denoted by S ( t , K , v ), is a pair ( X , ??), where X is a v ‐element set and ?? is a set of subsets of X , called blocks , with the property that | B |∈ K for any B ∈?? and every t ‐element subset of X is contained in a unique block. In this article, we shall show that there is an S ( 3 , { 4 , 5 , 7 }, v ) for any positive integer v ≡ 7 ( mod12 ) with v ≠ 19 . Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:68–80, 2012     

On the differences between Szeged and Wiener indices of graphs

Let G be a connected graph and η(G)=Sz(G)−W(G), where W(G) and Sz(G) are the Wiener and Szeged indices of G, respectively. A well-known result of Klav?ar, Rajapakse, and Gutman states that η(G)≥0, and by a result of Dobrynin and Gutman η(G)=0 if and only if each block of G is complete. In this paper, a path-edge matrix for the graph G is presented by which it is possible to classify the graphs in which η(G)=2. It is also proved that there is no graph G with the property that η(G)=1 or η(G)=3. Finally, it is proved that, for a given positive integer k,k≠1,3, there exists a graph G with η(G)=k.     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

[a,b]-factorizations of graphs

Let a and b be integers with b ? a ? 0. A graph G is called an [a,b]-graph if a ? d_G(v) ? b for each vertex v ∈ V(G), and an [a,b]-factor of a graph G is a spanning [a,b]-subgraph of G. A graph is [a,b]-factorable if its edges can be decomposed into [a,b]-factors. The purpose of this paper is to prove the following three theorems: (i) if 1 ? b ? 2a, every [(12a + 2)m + 2an,(12b + 4)m + 2bn]-graph is [2a, 2b + 1]-factorable; (ii) if b ? 2a ?1, every [(12a ?4)m + 2an, (12b ?2)m + 2bn]-graph is [2a ?1,2b]-factorable; and (iii) if b ? 2a ?1, every [(6a ?2)m + 2an, (6b + 2)m + 2bn]-graph is [2a ?1,2b + 1]-factorable, where m and n are nonnegative integers. They generalize some [a,b]-factorization results of Akiyama and Kano [3], Kano [6], and Era [5].     

New Results on EX Graphs

By the extremal number ex(n; t) = ex(n; {C ₃, C ₄, . . . , C _t }) we denote the maximum size (that is, number of edges) in a graph of order n > t and girth at least g ≥ t + 1. The set of all the graphs of order n, containing no cycles of length ≥ t, and of size ex(n; t), is denoted by EX(n; t) = EX(n; {C ₃, C ₄, . . . , C _t }), these graphs are called EX graphs. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n; 4) of a graph of order n and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5. In particular, we prove that ex(29; 6) = 45, also we improve some lower bounds and upper bounds of ex _u (n; t), for some particular values of n and t.     

Reconstructible and Half-Reconstructible Tournaments: Application to Their Groups of Hemimorphisms

Let T and T¹ be tournaments with n elements, E a basis for T, E′ a basis for T′, and k ≥ 3 an integer. The dual of T is the tournament T” of basis E defined by T(x, y) = T(y, x) for all x, y ε E. A hemimorphism from T onto T′ is an isomorphism from T onto T” or onto T. A k-hemimorphism from T onto T′ is a bijection f from E to E′ such that for any subset X of E of order k the restrictions T/X and T¹/f(X) are hemimorphic. The set of hemimorphisms of T onto itself has group structure, this group is called the group of hemimorphisms of T. In this work, we study the restrictions to n – 2 elements of a tournament with n elements. In particular, we prove: Let k ≥ 3 be an integer, T a tournament with n elements, where n ≥ k + 5. Then the following statements are equivalent: (i) All restrictions of T to subsets with n – 2 elements are k-hemimorphic. (ii) All restrictions of T to subsets with n – 2 elements are 3-hemimorphic. (iii) All restrictions of T to subsets with n – 2 elements are hemimorphic. (iv) All restrictions of T to subsets with n – 2 elements are isomorphic, (v) Either T is a strict total order, or the group of hemimorphisms of T is 2-homogeneous.     

Gaussian Property of the Rings R(X) and R〈X〉

The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. A commutative ring R is said to be Gaussian if c(fg) = c(f)c(g) for every polynomials f and g in R[X]. A number of authors have formulated necessary and sufficient conditions for R(X) (respectively, R?X?) to be semihereditary, have weak global dimension at most one, be arithmetical, or be Prüfer. An open question raised by Glaz is to formulate necessary and sufficient conditions that R(X) (respectively, R?X?) have the Gaussian property. We give a necessary and sufficient condition for the rings R(X) and R?X? in terms of the ring R in case the square of the nilradical of R is zero.     

Circular consecutive choosability of k‐choosable graphs

Let S(r) denote a circle of circumference r. The circular consecutive choosability ch_cc(G) of a graph G is the least real number t such that for any r≥χ_c(G), if each vertex v is assigned a closed interval L(v) of length t on S(r), then there is a circular r‐coloring f of G such that f(v)∈L(v). We investigate, for a graph, the relations between its circular consecutive choosability and choosability. It is proved that for any positive integer k, if a graph G is k‐choosable, then ch_cc(G)?k + 1 ? 1/k; moreover, the bound is sharp for k≥3. For k = 2, it is proved that if G is 2‐choosable then ch_cc(G)?2, while the equality holds if and only if G contains a cycle. In addition, we prove that there exist circular consecutive 2‐choosable graphs which are not 2‐choosable. In particular, it is shown that ch_cc(G) = 2 holds for all cycles and for K_{2, n} with n≥2. On the other hand, we prove that ch_cc(G)>2 holds for many generalized theta graphs. © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 178‐197, 2011     

The group inverse of the combinations of two idempotent matrices

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈ ? and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ.     

Plane Sections of Convex Bodies of Maximal Volume

Let K = {K ₀ ,... ,K _k } be a family of convex bodies in R ⁿ , 1≤ k≤ n-1 . We prove, generalizing results from [9], [10], [13], and [14], that there always exists an affine k -dimensional plane A _k (subset, dbl equals) R ⁿ , called a common maximal k-transversal of K , such that, for each i∈ {0,... ,k} and each x∈ R ⁿ , where V _k is the k -dimensional Lebesgue measure in A _k and A _k +x . Given a family K = {K _i } _i=0 ^l of convex bodies in R ⁿ , l < k , the set C _k ( K ) of all common maximal k -transversals of K is not only nonempty but has to be ``large' both from the measure theoretic and the topological point of view. It is shown that C <sub>k</sub> ( K ) cannot be included in a ν -dimensional C <sup>1</sup> submanifold (or more generally in an ( H <sup>ν</sup> , ν) -rectifiable, H <sup>ν</sup> -measurable subset) of the affine Grassmannian AGr <sub>n,k</sub> of all affine k -dimensional planes of R <sup>n</sup> , of O(n+1) -invariant ν -dimensional (Hausdorff) measure less than some positive constant c <sub>n,k,l</sub> , where ν = (k-l)(n-k) . As usual, the ``affine' Grassmannian AGr _n,k is viewed as a subspace of the Grassmannian Gr _n+1,k+1 of all linear (k+1) -dimensional subspaces of R ⁿ⁺¹ . On the topological side we show that there exists a nonzero cohomology class θ∈ H ^n-k (G _n+1,k+1 ;Z ₂ ) such that the class θ ^l+1 is concentrated in an arbitrarily small neighborhood of C _k ( K ) . As an immediate consequence we deduce that the Lyusternik—Shnirel'man category of the space C _k ( K ) relative to Gr _n+1,k+1 is ≥ k-l . Finally, we show that there exists a link between these two results by showing that a cohomologically ``big' subspace of Gr _n+1,k+1 has to be large also in a measure theoretic sense. Received May 22, 1998, and in revised form March 27, 2000. Online publication September 22, 2000.     

Hodge algebra structures on certain rings of invariants and applications

Let B be a commutative ring with identity, m, n, and r be positive integers such that r ≤ min{m, n}, a ₁, …, a _r (resp. b ₁, … b _r ) be integers such that 1 ≤ a ₁< … < a _r ≤ m (resp. 1 ≤ b ₁ < … < b _r < n) and U (resp. V) be the most general m × r (resp. r × n) matrix such that s-minors of first a _s ? 1 rows (resp. b _s ? 1 columns) of U (resp. V) are all zero for s = 1, …, r. We investigate the B-algebra C generated by all the entries of UV and all the r-minors of U and V. We introduce a Hodge algebra structure, to which the discrete Hodge algebra associate is Cohen Macaulay, on C and prove that C is Cohen-Macaulay if so is B. Using this Hodge algebra structure, we show that C is the ring of absolute invariants of a certain group action, compute the divisor class group and the canonical class of C, and give a criterion of Gorenstein property of C in terms of a ₁ ,…, a _r and b ₁…, b _r .     

A spectrum result on maximal partial ovoids of the generalized quadrangle , even

This article presents a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q even. We prove that for every integer k in an interval of, roughly, size [q²/10,9q²/10], there exists a maximal partial ovoid of size k on Q(4,q), q even. Since the generalized quadrangle W(q), q even, defined by a symplectic polarity of PG(3,q) is isomorphic to the generalized quadrangle Q(4,q), q even, the same result is obtained for maximal partial ovoids of W(q), q even. As equivalent results, the same spectrum result is obtained for minimal blocking sets with respect to planes of PG(3,q), q even, and for maximal partial 1-systems of lines on the Klein quadric Q⁺(5,q), q even.     

On an extension of a theorem on conjugacy class sizes

Let G be a finite group. We extend Alan Camina’s theorem on conjugacy classes sizes which asserts that if the conjugacy classes sizes of G are {1, p ^a , q ^b , p ^a q ^b }, where p and q are two distinct primes and a and b are integers, then G is nilpotent. We show that let G be a group and assume that the conjugacy classes sizes of elements of primary and biprimary orders of G are exactly {1, p ^a , n,p ^a n} with (p, n) = 1, where p is a prime and a and n are positive integers. If there is a p-element in G whose index is precisely p ^a , then G is nilpotent and n = q ^b for some prime q ≠ p.     

Vanishing and cocentralizing generalized derivations on Lie ideals

Let R be a prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid and L a not central Lie ideal of R. Suppose that F, G and H are generalized derivations of R, with F≠0, such that F(G(x)x?xH(x)) = 0, for any x∈L. In this paper we describe all possible forms of F, G and H.