Rigidity and Separation Indices of Graphs in Surfaces

Let Σ be a surface. We prove that rigidity indices of graphs which admit a polyhedral embedding in Σ and 5-connected graphs admitting an embedding in Σ are bounded by a constant depending on Σ. Moreover if the Euler characteristic of Σ is negative, then the separation index of graphs admitting a polyhedral embedding in Σ is also bounded. As a side result we show that distinguishing number of both Σ-polyhedral and 5-connected graphs which admit and embedding in Σ is also bounded.     

Determination of solvable algebraic groups by dense integral subgroups

Let G and G′ be triangularizable algebraic groups defined over the field Q of rational numbers, and let Γ ? G_Q and Γ′ ? G′_Q be dense subgroups of them containing integral subgroups of finite index. A study is made of the conditions under which a birational isomorphism of G and G′ follows from an abstract isomorphism of Γ and Γ′.     

The girth of a thin distance-regular graph

Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE _i ^* (x)W≤1 for 0≤i≤d, whereE _i ^* (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:

1. Γis a regular generalized quadrangle.
2. Γis thin and c ₃=1.

Corollary. Let Γ=(X,R) be a thin distance-regular graph with diameter d≥3 and valency k≥3. Then Γ has girth 3, 4, 6, or 8. Then girth of Γ is 8 exactly when Γ is a regular generalized quadrangle.     

Vertex-transitive graphs: Symmetric graphs of prime valency

Let G be a group acting symmetrically on a graph Σ, let G₁ be a subgroup of G minimal among those that act symmetrically on Σ, and let G₂ be a subgroup of G₁ maximal among those normal subgroups of G₁ which contain no member except 1 which fixes a vertex of Σ. The most precise result of this paper is that if Σ has prime valency p, then either Σ is a bipartite graph or G₂ acts regularly on Σ or G₁ | G₂ is a simple group which acts symmetrically on a graph of valency p which can be constructed from Σ and does not have more vertices than Σ. The results on vertex-transitive groups necessary to establish results like this are also included.     

On some minimal graphs of the torus

Let Γ be the set of finite, simple and nondirected graphs being not embeddable into the torus. Furthermore let >⁴ be a partial order-relation and M⁴ (Γ) the minimal basis of Γ. In this paper we determine three graphs of M⁴ (Γ) being embeddable into the projective plane and containing the subgraph W.     

Equations implying congruence n-permutability and semidistributivity

T. Dent, K. Kearnes and Á. Szendrei have defined the derivative, Σ′, of a set of equations Σ and shown, for idempotent Σ, that Σ implies congruence modularity if Σ′ is inconsistent ${(\Sigma^\prime \models x \approx y)}$ . In this paper we investigate other types of derivatives that give similar results for congruence n-permutability for some n, and for congruence semidistributivity.     

On the problems of Hartshorne and Serre for some -analytic surfaces

Let M be a compact -analytic surface, let Γ ⊂ M be a compact analytic subvariety, and let X := Mx00393;. We are interested in the following two problems: Assume that X does not contain any compact curve and that Γ is an irreducible compact curve such that Γ² ≥ 0 (resp. assume that the analytic cohomology groups H¹ (X, Ω^p) = 0, for all 0 ≤ p ≤ 2). Is X always Stein? It is our main purpose here to provide an affirmative answer to those two problems, provided M is either a (minimal) ruled surface or a non-algebraic compact surface. Also, the affine structure of such Stein surfaces will be discussed.     

一个代数的预投射分支和τ-预投射分支

设A是一个Artin代数,Γ_A是A的Auslander-Reiten箭图。我们得到:如果Γ是Γ_A的一个不包含有向循环的预投射分支,那么Γ是Γ_A的一个τ-预投射分支,且对一个拟倾斜代数A,Γ是Γ_A的一个预投射分支当且仅当Γ是Γ_A的一个,τ-预投射分支。     

Hamiltonian κ-factors in graphs

Let κ ≥ 2 be an integer. A k-factor F of a graph G is called a hamiltonian k-factor if F contains a Hamiltonian cycle. In this paper, we shall prove that if G is a graph of order n with κ ≥ 2,n ≥ 8κ - 4, κn even and δ(G) ≥ n/2, then G has a hamiltonian k-factor. © 1997 Wiley & Sons, Inc. J Graph Theory 25: 217–227, 1997     

Martin boundaries of random walks on Fuchsian groups

Let Γ be a finitely generated non-elementary Fuchsian group, and let μ be a probability measure with finite support on Γ such that supp μ generates Γ as a semigroup. If Γ contains no parabolic elements we show that for all but a small number of co-compact Γ, the Martin boundaryM of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ. If Γ has cusps, we prove that Γ can be deformed into a group Γ', abstractly isomorphic to Γ, such thatM can be identified with Λ', the limit set of Γ'. Our method uses the identification of Λ with a certain set of infinite reduced words in the generators of Γ described in [15]. The harmonic measure ν (ν is the hitting distribution of random paths in Γ on Λ) is a Gibbs measure on this space of infinite words, and the Poisson boundary of Γ, μ can be identified with Λ, ν.     

On the boundedness of topological categories

Let K be a complete and cocomplete category with a given proper (E,M)-factorization. K is called well-bounded if K is moreover bounded with a generator and cowellpowered with respect to the given factorization. Freyd-Kelly proved the following theorem about well-bounded categories: Let K be a well-bounded category and let Γ be a class of cylinders in the small category C¹, and let all but a set of these cylinders be cones. Then Γ(C,K) is a reflective subcategory of [C,K]. The main results of this paper are: (I) If F: K→L is a Top-functor and L is well-bounded, then K is well-bounded. (II) If U is an E-reflective subcategory of a well-bounded category,then U is again wellbounded. As a corollary one obtains for instance that all coreflective and all epireflective subcategories of the category of topological spaces are well-bounded.     

La monotonia di rappresentazioni del gruppo libero

Let Γ be a free nonabelian group on finitely many generators. Let Ω be the boundary of Γ, letC(Ω) be theC ^*-algebra of continuous functions on Ω, and let λ be the natural action of Γ onC(Ω). Aboundary representation is a representation of the crossed productC ^*-algebra Γ×_λ C(Ω). Given a unitary representation π of Γ onH, aboundary realization of π is an isometric Γ-inclusion ofH into the space of a boundary representation whose image is cyclic for that boundary representation. If the Γ-inclusion is bijective, we call, the realizationperfect. We prove below that if π admits an imperfect boundary realization, then there exists a nonzero vectorv ₀∈H satisfying $$\sum\limits_{|x| = n} {|\left\langle {v,\pi (x)v_0 } \right\rangle |^2 \leqslant |v|^2 } for each v \in {\mathcal{H}} (GVB)$$ If π is irreducible and weakly contained in the regular representation, and if no suchv ₀ exists, it follows that π satisfiesmonotony: up to equivalence, there exists exactly one realization of π, and that realization is perfect.     

关于解析函数的一个唯一性定理及其应用

<正> §1.在近代解析函数论中边界值的唯一性定理有许多的研究,其中有我们所习知著名的(?)氏唯一性定理,即:若 D 是某一可求长约当曲线Γ所范围的内域,而 f(z)是 D 内的半纯函数,如在Γ上存在某一测度大于零的集 E_z,对 E_z 任一点 z_0 上,f(z)的角形边界值为零.则必致f(z)≡0于 D 内.同时,卢洵(?)与普里瓦洛夫还指出:存在有单位圆域内非常数的解析函数,而在一个正测度的集(?)上具有等于零的射形边界值.除此之外还有一个很有用的 Koebe 氏定理.即:     

Varieties of Algebras without the Amalgamation Property

Let α be an ordinal and κ be a cardinal, both infinite, such that κ ≤ |α|. For τ ∈^αα, let sup(τ) = {i ∈ α: τ(i) ≠ i}. Let G _κ = {τ ∈^αα: |sup(τ)| < κ}. We consider variants of polyadic equality algebras by taking cylindrifications on Γ ? α, |Γ| < κ and substitutions restricted to G _κ. Such algebras are also enriched with generalized diagonal elements. We show that for any variety V containing the class of representable algebas and satisfying a finite schema of equations, V fails to have the amalgamation property. In particular, many varieties of Halmos’ quasi-polyadic equality algebras and Lucas’ extended cylindric algebras (including that of the representable algebras) fail to have the amalgamation property.     

Local boundary regularity of holomorphic mappings

Let f be a holomorphic mapping between two bounded domains D and D' in complex space ?ⁿ. Suppose that D and D' contain smooth real hypersurfaces Γ and Γ′ as open subsets of their respective boundaries, which correspond under a continuous extension of f. We shall show that this extension is smooth, given certain restrictions on Γ, Γ, and f.     

The Szegö function on a non-rectifiable arc

Let Γ be a simple Jordan arc in the complex plane. The Szegö function, by definition, is a holomorphic in ? \ Γ function with a prescribed product of its boundary values on Γ. The problem of finding the Segö function in the case of piecewise smooth Γ was solved earlier. In this paper we study this problem for non-rectifiable arcs. The solution relies on properties of the Cauchy transform of certain distributions with the support on Γ.     

Integral Closure of Graded Integral Domains

Let Γ be a torsion-free cancellative commutative monoid and let R = ?_α∈ΓR_α be a commutative Γ-graded ring. We show that if R is a graded Noetherian domain, then its integral closure is a graded Krull domain. This is a graded analog of the Mori–Nagata theorem. We also show that for a graded Strong Mori domain, its complete integral closure is a graded Krull domain but its integral closure is not necessarily a graded Krull domain.     

The deformation theory of representations of fundamental groups of compact Kähler manifolds

Let Γ be the fundamental group of a compact Kähler manifold M and let G be a real algebraic Lie group. Let ?(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈ ?(Γ, G) it is shown that there exists a neighborhood of ρ in ?(Γ, G) which is analytically equivalent to a cone defined by homogeneous quadratic equations. Furthermore this cone may be identified with the quadratic cone in the space \(Z^1 (\Gamma ,g_{Ad\rho } )\) of Lie algebra-valued l-cocycles on Γ comprising cocyclesu such that the cohomology class of the cup/Lie product square [u, u] is zero in \(H^2 (\Gamma ,g_{Ad\rho } )\) . We prove that ?(Γ, G) is quadratic at ρ if either (i) G is compact, (ii) ρ is the monodromy of a variation of Hodge structure over M, or (iii) G is the group of automorphisms of a Hermitian symmetric space X and the associated flat X-bundle over M possesses a holomorphic section. Examples are given where singularities of ?(Γ, G) are not quadratic, and are quadratic but not reduced. These results can be applied to construct deformations of discrete subgroups of Lie groups.     

Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDEs

Suppose L is a second-order elliptic differential operator in ℝ^d and D is a bounded, smooth domain in ℝ^d. Let 1 < α ≤ 2 and let Γ be a closed subset of ∂D. It is known [13] that the following three properties are equivalent: (α) Γ is ∂-polar; that is, Γ is not hit by the range of the corresponding (L, α)-superdiffusion in D; (β) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where ρ(x) is the distance to the boundary and k(x, y) is the corresponding Poisson kernel; and (γ) Γ is a removable boundary singularity for the equation Lu = u^α in D; that is, if u ≥ 0 and Lu = u^α in D and if u = 0 on ∂D \ Γ, then u = 0. We investigate a similar problem for a parabolic operator in a smooth cylinder 𝒬 = ℝ₊ × D. Let Γ be a compact set on the lateral boundary of 𝒬. We show that the following three properties are equivalent: (a) Γ is 𝒢-polar; that is, Γ is not hit by the graph of the corresponding (L, α)-superdiffusion in 𝒬; (b) the Poisson capacity of Γ is equal to 0; that is, the integral is equal to 0 or ∞ for every measure ν, where k(r, x; t, y) is the corresponding (parabolic) Poisson kernel; and (c) Γ is a removable lateral singularity for the equation u˙ + Lu = u^α in 𝒬; that is, if u ≥ 0 and u˙ + Lu = u^α in 𝒬 and if u = 0 on ∂𝒬 \ Γ and on {∞} × D, then u = 0. © 1998 John Wiley & Sons, Inc.     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .