D'Alembert's functional equations on metabelian groups

Summary. We extend d'Alembert's classical functional equation by replacing the domain of definition Bbb R {Bbb R} of the solutions by a metabelian group G and simultaneously replacing the group involution by an arbitrary involution of G. We find all complex valued solutions. In particular we show that the continuous solutions have the same form as in the abelian case if G is connected.     

A note on purely inseparable morphisms from an elliptic ruled surface

In this note, we shall study compositions of finite purely inseparable morphisms of degree p from an elliptic ruled surface in characteristic p > 0 and in particular, we shall give sufficient conditions for nonsingular minimal models of their images to be of general type.     

On Strong Reality of Finite Simple Groups

In this paper we shall consider the following problem. Which finite simple groups have the property: any element is inverted by an involution?     

Boundedness and convergence of martingales in rearrangement-invariant function spaces

Let (W, F, P)(\Omega, \cal F, P) be a complete nonatomic probability space. We shall give a characterization of rearrangement-invariant spaces X over W\Omega with the property that every martingale f = (f_n)_{n \geqq 0}f = (f_n)_{n \geqq 0} bounded in X converges with respect to the norm topology of X. Using the results, we shall consider the summability of martingales by Toeplitz matrices.     

Absolute valued algebras satisfying (x,x,x2) = 0

Let A be an absolute valued algebra satisfying the identity (x,x,x²) = 0. We give some conditions which imply that A is isomorphic to R, \mathbbC \mathbb{C} , H or D. These results enable us to show that if A is an algebra with involution then A is one of those classical algebras. We construct an example of A having dimension two and is not isomorphic to \mathbbC \mathbb{C} .     

A classification theorem for complete minimal surfaces in S6 with constant Kähler angles

In the present paper, we shall determine all complete minimal surfaces in the nearly Kähler sphere S⁶ with constant Kähler angles q ? (0, p)\theta \in (0,\,\pi ) and nonnegative curvatures.     

Group-invariant Percolation on Graphs

Let G be a closed group of automorphisms of a graph X. We relate geometric properties of G and X, such as amenability and unimodularity, to properties of G-invariant percolation processes on X, such as the number of infinite components, the expected degree, and the topology of the components. Our fundamental tool is a new masstransport technique that has been occasionally used elsewhere and is developed further here.¶ Perhaps surprisingly, these investigations of group-invariant percolation produce results that are new in the Bernoulli setting. Most notably, we prove that critical Bernoulli percolation on any nonamenable Cayley graph has no infinite clusters. More generally, the same is true for any nonamenable graph with a unimodular transitive automorphism group.¶ We show that G is amenable if for all $ \alpha < 1 $ \alpha < 1 , there is a G-invariant site percolation process w \omega on X with $ {\bf P} [x \in \omega] > \alpha $ {\bf P} [x \in \omega] > \alpha for all vertices x and with no infinite components. When G is not amenable, a threshold $ \alpha < 1 $ \alpha < 1 appears. An inequality for the threshold in terms of the isoperimetric constant is obtained, extending an inequality of Häggström for regular trees.¶ If G acts transitively on X, we show that G is unimodular if the expected degree is at least 2 in any G-invariant bond percolation on X with all components infinite.¶ The investigation of dependent percolation also yields some results on automorphism groups of graphs that do not involve percolation.     

完全分配格上的全有界一致结构与邻近结构

本文的目的是在具有逆序对合对应的完全分配格上研究点式（拟）一致结构与（拟）邻近中构的联系,证明了在全有界点式一致结构与邻近结构间存在一个一一对应关系。     

The Cartan matrix of the Schur algebra S(2, r)

Let k be an infinite field of prime characteristic and let r be a positive integer. Using admissible decompositions, we determine explicitly the entries of the decomposition matrix of the Schur algebra S(2, r) over k and prove that any two blocks with the same number of simple modules have the same decomposition matrix and hence the same Cartan matrix.     

Sur les tours localement cyclotomiques

For a given number field K and any prime l\ell we construct an increasing sequence of l\ell -extensions K_n of K which are locally cyclotomic over K. We give various criterious of finiteness or non-finiteness of this l\ell -tower and we characterise the number fields which have such a finite tower in terms of Iwasawa theory.     

The 2-Pebbling Property and a Conjecture of Graham's

The pebbling number of a graph G, f(G), is the least m such that, however m pebbles are placed on the vertices of G, we can move a pebble to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. It is conjectured that for all graphs G and H, f(G 2H)hf(G)f(H).¶Let C_m and C_n be cycles. We prove that f(C_m 2C_n)hf(C_m) f(C_n) for all but a finite number of possible cases. We also prove that f(G2T)hf(G) f(T) when G has the 2-pebbling property and T is any tree.     

The Wave Function of the Lyman-Alpha Photon Part I. The Wave Function in the Angular and Linear Momentum Representations

To the best of the writer‘s knowledge no one has given the wave function of a photon emitted in an atomic, molecular, or nuclear transition. In the present paper we derive the wave function in the angular momentum and linear momentum representations for the photon emitted by a non-relativistic hydrogen atom, when the electron of the atom falls from the first excited state to the ground state. This is the simplest transition which produces a photon. A two level model for the atom is used, in which the lower level (the ground state energy) is associated with a non-degenerate wave function and the upper level (the energy of the first excited state) is associated with wave functions corresponding to the four-fold degeneracy of that state. We use a generalization of Dirac‘s method for finding the eigenfunctions in resonance scattering. We find the exact solution of the two-level problem using the exact matrix elements of the interaction. The calculations are finite without renormalization. In the next paper we shall introduce the [(x)\vec] \vec{x} -representation and thereby obtain the "position", "shape", and "trajectory" of the photon.     

Variational analysis of non-Lipschitz spectral functions

We consider spectral functions f : 5 where f is any permutation-invariant mapping from Cⁿ to R, and 5 is the eigenvalue map from the set of n 2 n complex matrices to Cⁿ, ordering the eigenvalues lexicographically. For example, if f is the function "maximum real part", then f : 5 is the spectral abscissa, while if f is "maximum modulus", then f : 5 is the spectral radius. Both these spectral functions are continuous, but they are neither convex nor Lipschitz. For our analysis, we use the notion of subgradient extensively analyzed in Variational Analysis, R.T. Rockafellar and R. J.-B. Wets (Springer, 1998). We show that a necessary condition for Y to be a subgradient of an eigenvalue function f : 5 at X is that Y^* commutes with X. We also give a number of other necessary conditions for Y based on the Schur form and the Jordan form of X. In the case of the spectral abscissa, we refine these conditions, and we precisely identify the case where subdifferential regularity holds. We conclude by introducing the notion of a semistable program: maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non-positive. Semistable programming is a nonconvex generalization of semidefinite programming. Using our analysis, we derive a necessary condition for a local maximizer of a semistable program, and we give a generalization of the complementarity condition familiar from semidefinite programming.     

On p-chief factors and extensions of $ \mathbb{K} $ G-modules

The subject of this paper is the relationship between the set of chief factors of a finite group G and extensions of an irreducible \mathbbK \mathbb{K} G-module U ( \mathbbK \mathbb{K} a field). Let H / L be a p-chief factor of G. We prove that, if H / L is complemented in a vertex of U, then there is a short exact sequence of Ext-functors for the module U and any \mathbbK \mathbb{K} G-module V. In some special cases, we prove the converse, which is false in general. We also consider the intersection of the centralizers of all the extensions of U by an irreducible module and provide new bounds for this group.     

Finding Convex Sets in Convex Position

Let F{\cal{F}} denote a family of pairwise disjoint convex sets in the plane. F{\cal{F}} is said to be in convex position, if none of its members is contained in the convex hull of the union of the others. For any fixed k 3 5k\ge5, we give a linear upper bound on P_k(n)P_k(n), the maximum size of a family F{\cal{F}} with the property that any k members of F{\cal{F}} are in convex position, but no n are.     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

How Tight Is the Bollobás-Komlós Conjecture?

The bipartite case of the Bollobás and Komlós conjecture states that for every j₀, %>0 there is an !=!(j₀, %) >0 such that the following statement holds: If G is any graph with minimum degree at least n$\displaystyle {n \over 2}+%n then G contains as subgraphs all n vertex bipartite graphs, H, satisfying¶H)hj₀ \quad {\rm and} \quad b(H)h!n.$j (H)hj₀ \quad {\rm and} \quad b(H)h!n.¶Here b(H), the bandwidth of H, is the smallest b such that the vertices of H can be ordered as v₁, …, v_n such that v_i~_Hv_j implies |imj|hb.¶ This conjecture has been proved in [1]. Answering a question of E. Szemerédi [6] we show that this conjecture is tight in the sense that as %̂ then !̂. More precisely, we show that for any 0 such that that !(j₀, %)Д %.     

Remarks on varieties of involution bands

In the present paper, we study varieties consisting of bands (idempotent semigroups) endowed with an involutorial antiautomorphism * as a fundamental operation. Our principal aim is here to provide an insight to some classes of these varieties from the structural point of view, especially in terms of semilattice decompositions, subdirect products and ideal extensions. In the course of such considerations, we shall extend the result of C. L. Adair [1], who described the lattice of all varieties of bands with a regular involution (i.e. with the identity x = xx ^* x) We depict a broader lattice of involution band varieties, which incorporates Adair’s lattice.     

The concept of hyperellipticity on surfaces with nodes

A surface with nodes X is hyperelliptic if there exists an involution such that the genus of X/〈h〉 is 0. We prove that this definition is equivalent, as in the category of surfaces without nodes, to the existence of a degree 2 morphism satisfying an additional condition where the genus of Y is 0. Other question is if the hyperelliptic involution is unique or not. We shall prove that the hyperelliptic involution is unique in the case of stable Riemann surfaces but is not unique in the case of Klein surfaces with nodes. Finally, we shall prove that a complex double of a hyperelliptic Klein surface with nodes could not be hyperelliptic.     

Topological algebras with ascending or descending chain condition

We examine some topological algebras with ascending or descending chain condition. We prove that a commutative noetherian F-algebra is necessarily a Q-algebra. We characterize noetherian F-algebras which are Q-algebras among those whose left ideals are closed. We show that any commutative artinian m-convex algebra is finite dimensional.