Diameters of uniform subset graphs

Let r,s be positive integers with r>s, k a nonnegative integer, and n=2r−s+k. A uniform subset graph G(n,r,s) is a graph with vertex set [n]^r and where two r-subsets A,B∈[n]^r are adjacent if and only if |A∩B|=s. Let denote the diameter of a graph G.In this paper, we prove the following results: (1) If k>0, then if r≥2s+k+2, 2 if k≥s and 2s≤r≤s+k, or k<s and s+k≤r≤2s, and 3 otherwise; (2) If k=0, then . This generalizes a result in [M. Valencia-Pabon, J.-C. Vera, On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

Graded Lie algebras with finite polydepth

If A is a graded connected algebra then we define a new invariant, polydepthA, which is finite if for some A-module M of at most polynomial growth. Theorem 1: If f:X→Y is a continuous map of finite category, and if the orbits of acting in the homology of the homotopy fibre grow at most polynomially, then has finite polydepth. Theorem 5: If L is a graded Lie algebra and polydepthUL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, ∑_i=k+1^k+ddimL_i?k^r, k? some k(r).     

Weighted-Hardy functions with Hadamard gaps on the unit ball

We prove that an analytic function f on the unit ball B with Hadamard gaps, that is, (the homogeneous polynomial expansion of f) satisfying n_k+1/n_k?λ>1 for all k∈N, belongs to the space if and only if . Moreover, we show that the following asymptotic relation holds . Also we prove that lim_r→1(1-r²)^α‖Rf_r‖_p=0 if and only if . These results confirm two conjectures from the following recent paper [S. Stevi?, On Bloch-type functions with Hadamard gaps, Abstr. Appl. Anal. 2007 (2007) 8 pages (Article ID 39176)].     

On the Brauer group of a semisimple algebraic group

Let k be a field with algebraic closure , G a semisimple algebraic k-group, and a maximal torus with character group X(T). Denote Λ the abstract weight lattice of the roots system of G, and by and the n-torsion subgroup of the Brauer group of k and G, respectively. We prove that if chark does not divide n and n is prime to the order of Λ/X(T) then the natural homomorphism is an isomorphism.     

Duo group rings

It is shown that the group algebra of a torsion group G over a field K is duo if and only if it is reversible.     

Extensions for finite Chevalley groups I

Let G be a connected semisimple algebraic group defined and split over the field with p elements, and k be the algebraic closure of . Assume further that G is almost simple and simply connected and let be the finite Chevalley group consisting of -rational points of G where q=p^r for a non-negative integer r. In this paper, formulas are found relating extensions between simple -modules and extensions over G (considered as an algebraic group over k). One of these formulas, which only holds for primes p?3(h−1) (where h is the Coxeter number of G), is then used to show the vanishing of self-extensions between simple -modules except for certain simple modules when r=1 and the underlying root system is of type A₁ or C_n.     

Quasi-co-local subgroups of abelian groups

Let {0}≠K be a subgroup of the abelian group G. In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 29-37], K was called a co-local (cl) subgroup of G if is naturally isomorphic to . We generalize this notion to the quasi-category of abelian groups and call the subgroup K≠{0} of G a quasi-co-local (qcl) subgroup of G if is naturally isomorphic to . We show that qcl subgroups behave quite differently from cl subgroups. For example, while cl subgroups K are pure in G, i.e. G/K is torsion-free if G is torsion-free, any reduced torsion group T can be the torsion subgroup t(G/K) of G/K where G is torsion-free and K is a qcl subgroup of G.     

Hypersurfaces with constant mean curvature in a hyperbolic space

Let Mⁿ be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space Hⁿ⁺¹（c） with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that （1） if the multiplicities of the two distinct principal curvatures are greater than 1,then Mⁿ is isometric to the Riemannian product S^k（r）×H^n-k（-1/（r² + ρ²））, where r > 0 and 1 < k < n - 1;（2）if H² > -c and one of the two distinct principal curvatures is simple, then Mⁿ is isometric to the Riemannian product S^n-1（r） × H¹（-1/（r² +ρ²）） or S¹（r） × H^n-1（-1/（r² +ρ²））,r > 0, if one of the following conditions is satisfied （i） S≤（n-1）t²2+c²t^-22 on Mⁿ or （ii）S≥ （n-1）t²1+c²t^-21 on Mⁿ or（iii）（n-1）t²2+c²t^-22≤ S≤（n-1）t²1+c²t^-21 on Mⁿ, where t₁ and t₂ are the positive real roots of （1.5）.     

Invariant manifolds of partial functional differential equations

This paper is concerned with the existence, smoothness and attractivity of invariant manifolds for evolutionary processes on general Banach spaces when the nonlinear perturbation has a small global Lipschitz constant and locally C^k-smooth near the trivial solution. Such a nonlinear perturbation arises in many applications through the usual cut-off procedure, but the requirement in the existing literature that the nonlinear perturbation is globally C^k-smooth and has a globally small Lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut-off function. Our general results are illustrated by and applied to partial functional differential equations for which the phase space (where r>0 and being a Banach space) has no smooth inner product structure and for which the validity of variation-of-constants formula is still an interesting open problem.     

Constructions of optimal quaternary constant weight codes via group divisible designs

Generalized Steiner systems were first introduced by Etzion and used to construct optimal constant weight codes over an alphabet of size g+1 with minimum Hamming distance 2k−3, in which each codeword has length v and weight k. As to the existence of a , a lot of work has been done for k=3, while not so much is known for k=4. The notion k-^∗GDD was first introduced by Chen et al. and used to construct . The necessary condition for the existence of a is v≥14. In this paper, it is proved that there exists a for any prime power and v≥19. By using this result, the known results on the existence of optimal quaternary constant weight codes are then extended.     

On complete subsets of the cyclic group

A subset X of an abelian G is said to be complete if every element of G can be expressed as a nonempty sum of distinct elements from X.Let A⊂Zn be such that all the elements of A are coprime with n. Solving a conjecture of Erd?s and Heilbronn, Olson proved that A is complete if n is a prime and if . Recently Vu proved that there is an absolute constant c, such that for an arbitrary large n, A is complete if , and conjectured that 2 is essentially the right value of c.We show that A is complete if , thus proving the last conjecture.     

Kummer subfields of tame division algebras over Henselian fields

By generalizing the method used by Tignol and Amitsur in [J.-P. Tignol, S.A. Amitsur, Kummer subfields of Malcev-Neumann division algebras, Israel Journal of Math. 50 (1985), 114-144], we determine necessary and sufficient conditions for an arbitrary central division algebra D over a Henselian valued field E to have Kummer subfields when the characteristic of the residue field of E does not divide the degree of D. We prove also that if D is a semiramified division algebra of degree n [resp., of prime power degree p^r] over E such that does not divide n and [resp., and p³ divides ], then D is non-cyclic [resp., D is not an elementary abelian crossed product].     

n-Torsion of Brauer groups as relative Brauer groups of abelian extensions

It is now known [H. Kisilevsky, J. Sonn, Abelian extensions of global fields with constant local degrees, Math. Res. Lett. 13 (4) (2006) 599-607; C.D. Popescu, Torsion subgroups of Brauer groups and extensions of constant local degree for global function fields, J. Number Theory 115 (2005) 27-44] that if F is a global field, then the n-torsion subgroup of its Brauer group Br(F) equals the relative Brauer group Br(Ln/F) of an abelian extension Ln/F, for all n∈Z?1. We conjecture that this property characterizes the global fields within the class of infinite fields which are finitely generated over their prime fields. In the first part of this paper, we make a first step towards proving this conjecture. Namely, we show that if F is a non-global infinite field, which is finitely generated over its prime field and ?≠char(F) is a prime number such that μ?²⊆F^×, then there does not exist an abelian extension L/F such that . The second and third parts of this paper are concerned with a close analysis of the link between the hypothesis μ?²⊆F^× and the existence of an abelian extension L/F such that , in the case where F is a Henselian valued field.     

Classification of generalized symmetries of the Yang-Mills fields with a semi-simple structure group

A complete classification of generalized (or local) symmetries of the Yang-Mills equations on four dimensional Minkowski space with a semi-simple structure group is carried out. It is shown that any generalized symmetry, up to a generalized gauge symmetry, agrees with a first order symmetry on solutions of the Yang-Mills equations. Let be the decomposition of the Lie algebra of the structure group into simple ideals. First order symmetries for -valued Yang-Mills fields are found to consist of gauge symmetries, conformal symmetries for -valued Yang-Mills fields, 1?m?n, and their images under a complex structure of .     

Minimal pseudocompact group topologies on free abelian groups

A Hausdorff topological group G is minimal if every continuous isomorphism f:G→H between G and a Hausdorff topological group H is open. Significantly strengthening a 1981 result of Stoyanov, we prove the following theorem: For every infinite minimal abelian group G there exists a sequence of cardinals such that

     

On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

A note on edge-graceful spectra of the square of paths

For a simple path P_r on r vertices, the square of P_r is the graph on the same set of vertices of P_r, and where every pair of vertices of distance two or less in P_r is connected by an edge. Given a (p,q)-graph G with p vertices and q edges, and a nonnegative integer k, G is said to be k-edge-graceful if the edges can be labeled bijectively by k,k+1,…,k+q−1, so that the induced vertex sums are pairwise distinct, where the vertex sum at a vertex is the sum of the labels of all edges incident to such a vertex, modulo the number of vertices p. We call the set of all such k the edge-graceful spectrum of G, and denote it by egI(G). In this article, the edge-graceful spectrum for the square of paths is completely determined for odd r.     

Relaxed very asymmetric coloring games

This paper investigates a competitive version of the coloring game on a finite graph G. An asymmetric variant of the (r,d)-relaxed coloring game is called the (r,d)-relaxed (a,b)-coloring game. In this game, two players, Alice and Bob, take turns coloring the vertices of a graph G, using colors from a set X, with |X|=r. On each turn Alice colors a vertices and Bob colors b vertices. A color α∈X is legal for an uncolored vertex u if by coloring u with color α, the subgraph induced by all the vertices colored with α has maximum degree at most d. Each player is required to color an uncolored vertex legally on each move. The game ends when there are no remaining uncolored vertices. Alice wins the game if all vertices of the graph are legally colored, Bob wins if at a certain stage there exists an uncolored vertex without a legal color. The d-relaxed (a,b)-game chromatic number, denoted by , of G is the least r for which Alice has a winning strategy in the (r,d)-relaxed (a,b)-coloring game.The (r,d)-relaxed (1,1)-coloring game has been well studied and there are many interesting results. For the (r,d)-relaxed (a,1)-coloring game, this paper proves that if a graph G has an orientation with maximum outdegree k and a≥k, then for all d≥k²+2k; If a≥k³, then (a,1)- for all d≥2k+1.