Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

Proof-theoretic analysis by iterated reflection

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity ₁⁰ and, similarly, for any class _n⁰. We provide a more general version of the fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including I_n, I_n^–, I_n^– and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform ₁-reflection principle for T is ₂-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl's theorem. Supported by Alexander von Humboldt Foundation, INTAS grant 96-753, and RFBR grant 8-01-00282.     

Almost Covers Of 2-Arc Transitive Graphs

Let be a G-symmetric graph whose vertex set admits a nontrivial G-invariant partition with block size v. Let be the quotient graph of relative to and [B,C] the bipartite subgraph of induced by adjacent blocks B,C of . In this paper we study such graphs for which is connected, (G, 2)-arc transitive and is almost covered by in the sense that [B,C] is a matching of v-1 2 edges. Such graphs arose as a natural extremal case in a previous study by the author with Li and Praeger. The case K _v+1 is covered by results of Gardiner and Praeger. We consider here the general case where K _v+1, and prove that, for some even integer n 4, is a near n-gonal graph with respect to a certain G-orbit on n-cycles of . Moreover, we prove that every (G, 2)-arc transitive near n-gonal graph with respect to a G-orbit on n-cycles arises as a quotient of a graph with these properties. (A near n-gonal graph is a connected graph of girth at least 4 together with a set of n-cycles of such that each 2-arc of is contained in a unique member of .)     

A dissipative transformation with a trivial tail-algebra

Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where . Here we shall show that for this transformation we actually have R ={ØX}[].     

A Characterization of n s as a Multiplicative Function

All those complex valued multiplicative functions f and g are characterized for which g(n + k) – f(n) 0 (n ) is satisfied (k is an arbitrary nonzero integer).     

Gray codes for reflection groups

LetG be a finite group generated by reflections. It is shown that the elements ofG can be arranged in a cycle (a Gray code) such that each element is obtained from the previous one by applying one of the generators. The case G =A ₁ ⁿ yields a conventional binary Gray code. These generalized Gray codes provide an efficient way to run through the elements of any finite reflection group.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

A sum-product estimate in finite fields,and applications

Let A be a subset of a finite field for some prime q. If for some > 0, then we prove the estimate for some = () > 0. This is a finite field analogue of a result of [ErS]. We then use this estimate to prove a Szemerédi-Trotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the three-dimensional Kakeya problem in finite fields.     

The Hilbert function of curves on certain smooth quartic surfaces

The Riemann-Roch problem for divisors on a smooth surface in ³ is studied. This problem is solved for some smooth quartic surfaces which are called Mori quartics; as a consequence the Hilbert function of any integral curve on a Mori quartic is determined.     

Model theory of the regularity and reflection schemes

Abstract  This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here is a language with a distinguished linear order <, and REF consists of formulas of the form

On strong forms of reflection in set theory

In this paper we review the most common forms of reflection and introduce a new form which we call sharp‐generated reflection. We argue that sharp‐generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp‐maximality with the corresponding hypothesis . The statement is an analogue of the (Inner Model Hypothesis, introduced in 3 ) which is compatible with the existence of large cardinals.     

Cooperation in an asymmetric Volunteer's dilemma game theory and experimental evidence

The symmetric Volunteer's dilemma game (VOD) models a situation in which each ofN actors faces the decision of either producing a step-level collective good (action C) or freeriding (D). One player's cooperative action suffices for producing the collective good. Unilateral cooperation yields a payoffU forD-players andU-K for the cooperative player(s). However, if all actors decide for freeriding, each player's payoff is zero (U>K>0). In this article, an essential modification is discussed. In an asymmetric VOD, the interest in the collective good and/or, the production costs (i.e. U or K) may vary between actors. The generalized asymmetric VOD is similar to market entry games. Alternative hypotheses about the behaviour of subjects are derived from a game-theoretical analysis. They are investigated in an experimental setting. The application of the mixed Nash-equilibrium concept yields a rather counter-intuitive prediction which apparently contradicts the empirical data. The predictions of the Harsanyi-Selten-theory and Schelling's focal point theory are in better accordance with the data. However, they do not account for the diffusion-of-responsibility-effect also observable in the context of an asymmetric VOD game.I am indebted to Wulf Albers, Norman Braun, Werner Güth, Norbert L. Kerr, Reinhard Selten, and the participants of the V^th International Social Dilemma Conference in Bielefeld for critical and helpful comments. I am grateful to Axel Franzen who organized the experiment at the University of Mannheim. This work was supported by a grant of the Deutsche Forschungsgemeinschaft (DFG).     

Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities

We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.Mathematics Subject Classification (2000): 14E15, 16Rxx, 16S38, 18E30The first author was partially supported by the Nuffield Foundation grant NAL/00625/G and by the University of Washingtons Milliman Fund. The second author was partially supported by an NSF grant DMS-0070560. Both authors are grateful to the Edinburgh Mathematical Society and the Leverhulme Trust for support.Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday     

Time decay for solutions of Schrödinger equations with rough and time-dependent potentials

In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions

A sphere theorem for non-reversible Finsler metrics

For a non-reversible Finsler metric F on a compact smooth manifold M we introduce the reversibility = max {F(–X)|F(X)=1}1. We prove the following generalization of the classical sphere theorem in Riemannian geometry: A simply-connected and compact Finsler manifold of dimension n3 with reversibility and with flag curvature is homotopy equivalent to the n-sphere.Mathematics Subject Classification (2000):53C60; 53C20, 53C22     

A Note on Regular Near Polygons

Let be a regular near polygon of order (s,t) with s>1 and t3. Let d be the diameter of , and let r:= max{i(c_i,a_i,b_i)=(c₁,a₁,b₁)}. In this note we prove several inequalities for . In particular, we show that s is bounded from above by function in t if We also consider regular near polygons of order (s,3).This work was partly supported by the Grant-in-Aid for Scientific Research (No 14740072), the Ministry of Education, Culture, Sports, Science and Technology, JapanThis work was partly done when the author was at the Com²MaC center at the Pohang University of Science and Technology. He would like to thank the Com²MaC-KOSEF for its support     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Interpolation and approximation in

Assume a standard Brownian motion W=(W_t)_t[0,1], a Borel function such that f(W₁)L₂, and the standard Gaussian measure γ on the real line. We characterize that f belongs to the Besov space , obtained via the real interpolation method, by the behavior of , where is a deterministic time net and the orthogonal projection onto a subspace of ‘discrete’ stochastic integrals with X being the Brownian motion or the geometric Brownian motion. By using Hermite polynomial expansions the problem is reduced to a deterministic one. The approximation numbers a_X(f(X₁);τ) can be used to describe the L₂-error in discrete time simulations of the martingale generated by f(W₁) and (in stochastic finance) to describe the minimal quadratic hedging error of certain discretely adjusted portfolios.     

A pinching theorem for extrinsically symmetric submanifolds of Euclidean space

We show that a compact connected manifold which can be immersed into ^m with almost parallel second fundamental form, admits an extrinsically symmetric immersion into ^m.Mathematics Subject Classification (2000): 53C20, 53C24, 53C30, 53C35, 53C40, 53C42Acknowledgement I am most grateful to J.–H. Eschenburg, P. Ghanaat and E. A. Ruh for valuable discussions and helpful remarks. This work was supported by the Swiss National Science Foundation Grants 20-67619.02 and PBFR-106367.     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.