Thermodynamics,real and complex periods of the Volterra model

The Lotka-Volterra model is converted into a Hamiltonian system. For this system the Gibbs state and the thermodynamic functions as free energy, entropy, expectation and variances of the Hamiltonian and its summands are calculated together with expansions for low and high temperature ^–1.Since the canonical partition functionZ (, , ) is the Laplace transform of the energy-period functionT(E, , ), we obtain expansions ofT (E, , ) for small and large energyE=H (p, q) and arbitrary parameters, by inverse Laplace transformation of the partition functionZ (, , ). Expansions of the partial orbit times T_±±, i.e. the part of the periodT with predator and prey above or below equilibrium values, are available, too. By expressingT(E, , ) as a convolution integral, we derive global inequalities, e. g. the function (log (E/), log (E/)) log (ET) (E, , ) is convex on the whole domainR ² and (E/T) (T/E) (0, 1) globally. Finally the analytic continuation of the functionT=T (E, , ) to the left- as well as right halfplane ReE<0 and ReE>0 can be expressed by Laplace transforms.

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Zusammenfassung Das Lotka-Volterra Modell wird in ein Hamiltonsches System umgeformt. Für dieses System kann man den Gibbszustand und die thermodynamischen Funktionen wie freie Energie, Entropie sowie Erwartungswerte und Varianzen der Hamiltonfunktion und ihrer Summanden berechnen und Entwicklungen für niedrige als auch hohe Temperatur ^–1 angeben.Da die kanonische ZustandssummeZ (, , ) die Laplacetransformierte der Energie-Perioden FunktionT=T(E, , ) ist, erhält man durch Umkehrung der Laplacetransformation Entwicklungen vonT (E, , ) für kleine sowie große EnergieE=H(p, q). Durch spezielle Manipulationen bekommt man sogar Entwicklungen der PeriodenteileT _±±, das sind die Intervalle während denen Räuber bzw. Beute ober- bzw. unterhalb der Gleichgewichte liegen. Mit Hilfe einer Darstellung vonT(E, , ) als Faltungsintegral beweist man globale Ungleichungen, z. B. ist die Funktion (log (E/), log (E/)) log (ET) (E, , ) auf dem ganzen DefinitionsbereichR ² konvex und (E/T) (T/E) (0,1).Schlußendlich läßt sich die analytische Fortsetzung der FunktionT(E, , ) in die linke sowie rechte Halbebene ReE< 0 bzw. ReE> 0 durch ein Laplaceintegral darstellen.

     

ε-Expansions in Dyson's Hierarchical Model

We consider Dyson's hierarchical model on a d-dimensional hierarchical lattice and define a renormalization group (RG) transformation for complex values of d as a map in the space of sequences of coupling constants determining the model Hamiltonian. We show that d=4 is a bifurcation value of this transformation for the RG transformation parameter equal to 1+2/d, and we construct a non-Gaussian RG-invariant Hamiltonian in terms of the (4–d)-expansion. We establish that the (–3/2)- and (4–d)-expansion coefficients for a non-Gaussian fixed point in the dimension d=3 have the same asymptotic representation as the size of the elementary cell tends to infinity, thus confirming that both the expansions describe the same nontrivial fixed point in the dimension three.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

On local oscillatory integrals with variable Calderón-Zygmund kernels

Letp(1, ). In this paper, the authors investigate the uniformL ^p ( ⁿ ) in of the oscillatory singular integral operatorT defined by

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Formal Flows on the Non-Archimedean Open Unit Disk

This paper deals with group actions of one-dimensional formal groups defined over the ring of integers in a finite extension of the p-adic field, where the space acted upon is the maximal ideal in the ring of integers of an algebraic closure of the p-adic field. Given a formal group F as above, a formal flow is a series (t,x) satisfying the conditions (0,x)=x and (F(s,t),x)=(s,(t,x)). With this definition, any formal group will act on the disk by left translation, but this paper constructs flows with any specified divisor of fixed points, where a point of the open unit disk is a fixed point of order n if (x–) ⁿ |((t,x)–x). Furthermore, if is an analytic automorphism of the open unit disk with only finitely many periodic points, then there is a flow , an element of the maximal ideal of the ring of constants, and an integer m such that the m-fold iteration of (x) is equal to (,x). All the formal flows constructed here are actions of the additive formal group on the unit disk. Indeed, if the divisor of fixed points of a formal flow is of degree at least two, then the formal group involved must become isomorphic to the additive group when the base is extended to the residue field of the constant ring.     

On an optimum test of the equality of two covariance matrices

Let X: p × 1, Y: p × 1 be independently and normally distributed p-vectors with unknown means ₁, ₂ and unknown covariance matrices ₁, ₂ (>0) respectively. We shall show that Pillai's test, which is locally best invariant, is locally minimax for testing H ₀: ₁=₂ against the alternative H ₁: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaeiDaiaabkhacaqGOaWaaabmaeaadaaeqaqaaiabgkHiTiaadMea% caGGPaGaaiiiaiabg2da9iaacccacqaHdpWCcaGGGaGaeyOpa4Jaai% iiaiaaicdaaSqaaiaaigdaaeqaniabggHiLdaaleaacaqGYaaabaGa% aeylaiaabgdaa0GaeyyeIuoaaaa!4E3F!\[{\rm{tr(}}\sum\nolimits_{\rm{2}}^{{\rm{ - 1}}} {\sum\nolimits_1 { - I) = \sigma > 0} }\]as 0. However this test is not of type D among G-invariant tests.Research supported by the Canadian N.S.E.R.C. Grant.     

Hecke Algebras of Classical Groups over p-adic Fields II

In the previous part of this paper, we constructed a large family of Hecke algebras on some classical groups G defined over p-adic fields in order to understand their admissible representations. Each Hecke algebra is associated to a pair (J , ) of an open compact subgroup J and its irreducible representation which is constructed from given data = (, P₀, ). Here, is a semisimple element in the Lie algebra of G, P₀ is a parahoric subgroup in the centralizer of in G, and is a cuspidal representation on the finite reductive quotient of P₀. In this paper, we explicitly describe those Hecke algebras when P₀ is a minimal parahoric subgroup, is trivial and is a character.     

Estimates of Green Function for Relativistic α-Stable Process

We call an R ^d -valued stochastic process X _t with characteristic function exp{–t{(m ^2/+²)^/2–m}},R ^d ,m>0, the relativistic -stable process. In the paper we derive sharp estimates for the Green function of the relativistic -stable process on C ^1,1 domains. Using these estimates we provide lower and upper bounds for the Poisson kernel. As another application we derive 3G Theorem and Boundary Harnack Principle for C ^1,1 domains.     

The Eta Invariant and the Real Connective K-Theory of the Classifying Space for Quaternion Groups

We express the real connective K-theory groups o_4k–1(B Q ) ofthe quaternion group Q of order = 2 ^j 8 in terms of therepresentation theory of Q by showing o_4k–1(B Q ) = Sp(S ^4k+3/Q )where is any fixed point free representation of Q in U(2k + 2).     

Linear structures on the collections of minimal surfaces inℝ 3 andℝ 4

The collection of minimal herissons in ³ is endowed with a vector space structure. The existence of this structure is related to the fact that null curves inC ³ are described by a single map from the étalé space of the sheaf of germs of holomorphic sections of the line bundle of degree 2 over ₁ to C³, which islinear on stalks. There is an analogous construction for null curves inC ⁴. This gives a similar class of minimal surfaces in ⁴.     

Some <Emphasis Type="Italic">L</Emphasis>-Function of a Regular Covering of a Graph

Let ₁:KH, ₂:HG and ₂₁:KG be three finite regular coverings of graphs, and let be a representation of the covering transformation group of ₁. We show that the (Bartholdi type) L-function of G associated to the representation of the covering transformation group of ₂₁ induced from is equal to that of H associated to by means of ordinary voltage assignments.Acknowledgments. We would like to thank the referee for many valuable comments and suggestions. This is partially supported by Grant-in-Aid for Science Research (C).Final version received: February 16, 2004     

Differential geometry of tensor product immersions II

In the first part of this series, we prove that the tensor product immersionf ₁ f _2k of2k isometric spherical immersions of a Riemannian manifoldM in Euclidean space is of-type with k and classify tensor product immersionsf ₁ f _2k which are ofk-type. In this article we investigate the tensor product immersionsf ₁ f _2k which are of (k+1)-type. Several classification theorems are obtained.     

Formal Dimension for Semisimple Symmetric Spaces

If G is a semisimple Lie group and (, ) an irreducible unitary representation of G with square integrable matrix coefficients, then there exists a number d() such that

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

Random fixed point theorems for a certain class of mappings in banach spaces

Let (, ) be a measurable space and C a nonempty bounded closed convex separable subset of p-uniformly convex Banach space E for some p > 1. We prove random fixed point theorems for a class of mappings T: × C C satisfying: for each x, y C, and integer n 1,

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(), } is said to be parametrically stochastically increasing and convex (concave) in if X() is stochastically increasing in , and if for any increasing convex (concave) function , E(X()) is increasing and convex (concave) in whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {X _n(), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {X _n(), }, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205. Reproduction in whole or in part is permitted for any purpose by the United States Government.     

On the Broadwell model of the Boltzmann equation for a simple discrete velocity gas

The Broadwell model of the Boltzmann equation for a simple discrete velocity gas is investigated on two asymptotic problems. (a) The decay of solutions inxR ast+. (b) The hydrodynamical limit in the compressible Euler level as the mean free path0.     

Fuchsian groups,finite simple groups and representation varieties

Let be a Fuchsian group of genus at least 2 (at least 3 if is non-oriented). We study the spaces of homomorphisms from to finite simple groups G, and derive a number of applications concerning random generation and representation varieties. Precise asymptotic estimates for |Hom(,G)| are given, implying in particular that as the rank of G tends to infinity, this is of the form |G|^()+1+o(1), where () is the measure of . We then prove that a randomly chosen homomorphism from to G is surjective with probability tending to 1 as |G|. Combining our results with Lang-Weil estimates from algebraic geometry, we obtain the dimensions of the representation varieties , where is GL_n(K) or a simple algebraic group over K, an algebraically closed field of arbitrary characteristic. A key ingredient of our approach is character theory, involving the study of the zeta function ^G(s)=(1)^-s, where the sum is over all irreducible complex characters of G.