Generalizations of the odd degree theorem and applications

LetV ⊂ ℙℝ ⁿ be an algebraic variety, such that its complexificationV _ℂ ⊂ ℙ ⁿ is irreducible of codimensionm ≥ 1. We use a sufficient condition on a linear spaceL ⊂ ℙℝ ⁿ of dimensionm + 2r to have a nonempty intersection withV, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3.     

Linear bounds for levels of stable rationality

Let G be one of the groups SL _n (ℂ), Sp_2n (ℂ), SO _m (ℂ), O _m (ℂ), or G ₂. For a generically free G-representation V, we say that N is a level of stable rationality for V/G if V/G × ℙ ^N is rational. In this paper we improve known bounds for the levels of stable rationality for the quotients V/G. In particular, their growth as functions of the rank of the group is linear for G being one of the classical groups.     

On the Equicontinuity Region of Discrete Subgroups of <Emphasis Type="Italic">PU</Emphasis>(1,<Emphasis Type="Italic">n</Emphasis>)

Let G be a discrete subgroup of PU(1,n). Then G acts on ℙ_ℂ ⁿ preserving the unit ball ℍ_ℂ ⁿ , where it acts by isometries with respect to the Bergman metric. In this work we look at its action on all of ℙ_ℂ ⁿ and determine its equicontinuity region Eq(G). This turns out to be the complement of the union of all complex projective hyperplanes in ℙ_ℂ ⁿ which are tangent to ∂ℍ_ℂ ⁿ at points in the Chen-Greenberg limit set Λ(G), a closed G-invariant subset of ∂ℍ_ℂ ⁿ which is minimal for non-elementary groups. We also prove that the action on Eq(G) is discontinuous. Also , if the limit set is “sufficiently general” (i.e. it is not contained in any proper k -chain), then each connected component of Eq(G) is a holomorphy domain and it is a complete Kobayashi hyperbolic space.     

The signature of generalized flag manifolds and applications

Let G be a linear algebraic group over C and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in CP^n are prime if n satisfies certain conditions.     

Orthogonal factorizations of digraphs

Let G be a digraph with vertex set V(G) and arc set E(G) and let g = (g ⁻, g ⁺) and ƒ = (ƒ ⁻, ƒ ⁺) be pairs of positive integer-valued functions defined on V(G) such that g ⁻(x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ ƒ ⁺(x) for each x ∈ V(G). A (g, ƒ)-factor of G is a spanning subdigraph H of G such that g ⁻(x) ⩽ id _H (x) ⩽ ƒ ⁻(x) and g ⁺(x) ⩽ od _H (x) ⩽ ƒ ⁺(x) for each x ∈ V(H); a (g, ƒ)-factorization of G is a partition of E(G) into arc-disjoint (g, ƒ)-factors. Let = {F ₁, F ₂,…, F _m} and H be a factorization and a subdigraph of G, respectively. is called k-orthogonal to H if each F _i , 1 ⩽ i ⩽ m, has exactly k arcs in common with H. In this paper it is proved that every (mg+m−1,mƒ−m+1)-digraph has a (g, f)-factorization k-orthogonal to any given subdigraph with km arcs if k ⩽ min{g ⁻(x), g ⁺(x)} for any x ∈ V(G) and that every (mg, mf)-digraph has a (g, f)-factorization orthogonal to any given directed m-star if 0 ⩽ g(x) ⩽ f(x) for any x ∈ V(G). The results in this paper are in some sense best possible.      

On Independent Cycles in a Bipartite Graph

 Let G=(V ₁,V ₂;E) be a bipartite graph with 2k≤m=|V ₁|≤|V ₂|=n, where k is a positive integer. We show that if the number of edges of G is at least (2k−1)(n−1)+m, then G contains k vertex-disjoint cycles, unless e(G)=(2k−1)(n−1)+m and G belongs to a known class of graphs. Received: December 9, 1998 Final version received: June 2, 1999     

Finiteness conditions on subgroups of profinite p-Poincaré duality groups

For a prime number p let G be a profinite p-PD _n group with a closed normal subgroup N such that G/N is a profinite p-PD _m group and that H _i (V, $ \mathbb{F} $ \mathbb{F} _p ) is finite for every open subgroup V of N and all i ≤ [n/2]. Generalising [12, Thm. 3.7.4] we show that m ≤ n and N is a profinite p-PD _{n − m} group. In case that G is a pro-p PD _n group of Euler characteristic 0 with a closed normal subgroup N of type FP _{[n−1 / 2]} such that G/N is soluble-by-finite pro-p group of finite rank, we show that N is a pro-p PD _{n − m} group, where m = vcd _p (G/N). As a corollary we obtain that a pro-p PD ₃ group with infinite abelianization is either soluble or contains a free nonprocyclic pro-p subgroup.     

On Hua-Tuan’s conjecture

Let G be a finite group and |G| = pn, p be a prime. For 0 m n, sm(G) denotes the number of subgroups of of order pm of G. Loo-Keng Hua and Hsio-Fu Tuan have ever conjectured: for an arbitrary finite p-group G, if p > 2, then sm(G) ≡ 1, 1 + p, 1 + p + p2 or 1 + p + 2p2 (mod p3). In this paper, we investigate the conjecture, and give some p-groups in which the conjecture holds and some examples in which the conjecture does not hold.     

Discrete subgroups of <Emphasis Type="Italic">PU</Emphasis> (2, 1) acting on P<Stack><Subscript>ℂ</Subscript><Superscript>2</Superscript></Stack> and the Kobayashi metric

In this paper, we prove following: If G ⊂ PU (2, 1) is an infinite, discrete group, acting on P_ℂ² without complex invariant lines, then the component containing ℍP_ℂ² of the domain of discontinuity Ω(G) = PP_ℂ²∖ Λ (G), according to Kulkarni, is G-invariant complete Kobayashi hyperbolic. The authors were supported by the Universidad Autónoma de Yucatán and the Universidad Nacional Autónoma de México.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

Engel Values in Residually Finite Groups

The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x _1,n y ₁] ⋯ [x _m,n y _m ])^q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.     

The hamiltonian chromatic number of a connected graph without large hamiltonian-connected subgraphs

If G is a connected graph of order n ⩾ 1, then by a hamiltonian coloring of G we mean a mapping c of V (G) into the set of all positive integers such that |c(x) − c(y)| ⩾ n − 1 − D _G (x, y) (where D _G (x, y) denotes the length of a longest x − y path in G) for all distinct x, y ∈ V (G). Let G be a connected graph. By the hamiltonian chromatic number of G we mean

On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

Additive summable processes and their stochastic integral

We define and study a class of summable processes, called additive summable processes, that is larger than the class used by Dinculeanu and Brooks [D-B]. We relax the definition of a summable processesX:Ω×ℝ₊→E⊂L(F, G) by asking for the associated measureI _X to have just an additive extension to the predictableσ-algebra ℘, such that each of the measures (I _X) _z , forz∈(L _G ^p )*, beingσ-additive, rather than having aσ-additive extension. We define a stochastic integral with respect to such a process and we prove several properties of the integral. After that we show that this class of summable processes contains all processesX:Ω×ℝ₊→E⊂L(F, G) with integrable semivariation ifc ₀ ∋G.     

On Enomoto’s problems in a bipartite graph

In this paper, we obtain the following result: Let k, n ₁ and n ₂ be three positive integers, and let G = (V ₁,V ₂;E) be a bipartite graph with |V1| = n ₁ and |V ₂| = n ₂ such that n ₁ ⩾ 2k + 1, n ₂ ⩾ 2k + 1 and |n ₁ − n ₂| ⩽ 1. If d(x) + d(y) ⩾ 2k + 2 for every x ∈ V ₁ and y ∈ V ₂ with xy $ \notin $ \notin E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.     

Harmonic Analysis on SO (n, ℂ)/SO (n−1, ℂ), n≥3

In this article we compute the Plancherel measure for SO(n, ℂ)/SO(n − 1, ℂ) following the approach of Van den Ban. This result is required in order to calculate the explicit decomposition of the oscillator representation wn for the dual pair G = SL(2, ℂ) × SO(n, ℂ) and to prove that every wn(G)-invariant Hilbert subspace of the space of tempered distributions decomposes multiplicity free.     

On degrees of maps between Grassmannians

Let G_n,k denote the oriented grassmann manifold of orientedk-planes in ℝⁿ. It is shown that for any continuous mapf: G_n,k → G_n,k, dim G_n,k = dim G_m,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂG_m,l → ℂG_n,k,h: ℍG_m,l → ℍG_n,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.     

Homogenous projective factors for actions of semi-simple Lie groups

We analyze the structure of a continuous (or Borel) action of a connected semi-simple Lie group G with finite center and real rank at least 2 on a compact metric (or Borel) space X, using the existence of a stationary measure as the basic tool. The main result has the following corollary: Let P be a minimal parabolic subgroup of G, and K a maximal compact subgroup. Let λ be a P-invariant probability measure on X, and assume the P-action on (X,λ) is mixing. Then either λ is invariant under G, or there exists a proper parabolic subgroup Q⊂G, and a measurable G-equivariant factor map ϕ:(X,ν)→(G/Q,m), where ν=∫ _K kλdk and m is the K-invariant measure on G/Q. Furthermore, The extension has relatively G-invariant measure, namely (X,ν) is induced from a (mixing) probability measure preserving action of Q. Oblatum 14-X-1997 & 18-XI-1998 / Published online: 20 August 1999