Local boundedness of the number of solutions of Plateau’s problem for polygonal boundary curves

In the present article, the author proves two generalizations of his “finiteness-result” (I.H.P. Anal. Non-lineaire, 2006, accepted) which states for any extreme simple closed polygon that every immersed, stable disc-type minimal surface spanning Γ is an isolated point of the set of all disc-type minimal surfaces spanning Γ w.r.t. the C ⁰-topology. First, it is proved that this statement holds true for any simple closed polygon in , provided it bounds only minimal surfaces without boundary branch points. Also requiring that the interior angles at the vertices of such a polygon Γ have to be different from the author proves the existence of some neighborhood O of Γ in and of some integer , depending only on Γ, such that the number of immersed, stable disc-type minimal surfaces spanning any simple closed polygon contained in O, with the same number of vertices as Γ, is bounded by .      

拟圆周的两个几何性质

§1　IntroductionLetΓbe a Jordan curve of R2 and f∶R2→R2 be a k-quasiconformal mapping,where1≤k<+∞.Γis called a quasicirlce ifΓis the image of the unit circle B2 under f.It is well-known that quasicircles play a very important role in quasiconformalmapping theory,complex dynamics,Fuchsian groups,Teichmuller space theory and lowdimensional topology,( see[1—5] etc.)In1 963 ,Ahlfors obtained the three-point property of quasidisks[6] .Later,Gehring[7] ,Osgood[8] ,Krzyz[9] ,Ch…     

Finiteness of the number of solutions of Plateau’s problem for polygonal boundary curves II

In this article, the author proves that a simple closed polygon can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles at its N + 3 vertices, has to be smaller than 6π.      

Modeling teletraffic arrivals by a Poisson cluster process

In this paper we consider a Poisson cluster process N as a generating process for the arrivals of packets to a server. This process generalizes in a more realistic way the infinite source Poisson model which has been used for modeling teletraffic for a long time. At each Poisson point Γ _j , a flow of packets is initiated which is modeled as a partial iid sum process , with a random limit K _j which is independent of (X _ji ) and the underlying Poisson points (Γ _j ). We study the covariance structure of the increment process of N. In particular, the covariance function of the increment process is not summable if the right tail P(K _j > x) is regularly varying with index α∊ (1, 2), the distribution of the X _ji ’s being irrelevant. This means that the increment process exhibits long-range dependence. If var(K _j ) < ∞ long-range dependence is excluded. We study the asymptotic behavior of the process (N(t)) _{t≥ 0} and give conditions on the distribution of K _j and X _ji under which the random sums have a regularly varying tail. Using the form of the distribution of the interarrival times of the process N under the Palm distribution, we also conduct an exploratory statistical analysis of simulated data and of Internet packet arrivals to a server. We illustrate how the theoretical results can be used to detect distribution al characteristics of K _j , X _ji , and of the Poisson process. AMS Subject Classifications Primary—60K30; Secondary—60K25 A large part of this research was done with support of Institut Mittag-Leffler of the Royal Swedish Academy of Sciences when the authors participated in the Fall 2004 program on Queuing Theory and Teletraffic Theory. Mikosch’s research is also partially supported by MaPhySto, the Danish research network for mathematical physics and stochastics and the Danish Research Council (SNF) Grant No 21-04-0400. Samorodnitsky’s research is also partially supported by NSF grant DMS-0303493 and NSA grant MSPF-02G-183 at Cornell University. González-Arévalo’s research is partially supported by BoRSF grant LEQSF(2004-2007)-RD-A-31 at the University of Louisiana at Lafayette.     

An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in . As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary Γ_∞ is a Jordan curve homologous to zero in such that Γ_∞ is contained in a slab between two horizontal circles of with width equal to π. We construct vertical minimal graphs in over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains Ω in are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition. The first author wish to thank Laboratoire Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu for the kind hospitality and support. The authors would like to thank CNPq, PRONEX of Brazil and Accord Brasil-France, for partial financial support.     

Explicit Bounded-Degree Unique-Neighbor Concentrators

Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite family of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈ satisfy the following properties. The output-set Y has cardinality times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of is simple to specify, and each has fewer than edges. We then modify to obtain explicit unique-neighbor concentrators of maximum degree 3. * Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

On one extremal problem for numerical series

Let Γ be the set of all permutations of the natural series and let α = {α _j} _j∈ℕ, ν = {ν_j} _j∈ℕ, and η = {η_j} _j∈ℕ be nonnegative number sequences for which

On the branch point index of minimal surfaces

Let Γ be a closed, sufficiently smooth Jordan curve in and denote by the class of disk-type surfaces which map ∂B continuously and monotonically onto Γ. Then any minimal surface possesses only finitely many branch points in , and the order of any such point is well-defined, and also the index of an interior branch point is defined in a natural way if X is nonplanar. We show that also the index of boundary branch points can be defined if the curvature κ and the torsion τ of Γ are strictly nonzero. Secondly we derive upper bounds for the index of any branch point in terms of the total curvature of Γ or of its cut number. Dedicated to Professor Heinz K?nig on the occasion of his 80^th birthday     

On the rate of convergence of <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> norms in the CLT for poisson and gamma random variables

In the paper, we present upper bounds of L _p norms of order ( X)^-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X− X)/ √ X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter α > 0. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09.     

Spectra of ergodic group actions

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ^∞-spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim sup_g→∞Re (g)<1, where is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim sup_g→∞Re (g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.     

A generalization of the rogosinski-rogosinski theorem

We establish necessary and sufficient conditions for numerical functions αj(x), j ∈ N, x ∈ X, under which the conditions K(f _j ⊂ K(f ₁) ∀j≥2 and yield The functions fj(x) are uniformly bounded on the set X and take values in a boundedly compact space L, and K(fj) is the kernel of the function fj. The well-known Rogosinski-Rogosinski theorem follows from the proved statements in the case where X = N, α _j (x) ≡ α_j, and the space L is the m-dimensional Euclidean space.     

On the Similarity of Analytic Matrix Functions

Consider a domain , and two analytic matrix-valued functions functions . Consider also points and positive integers n ₁, n ₂, . . . , n _N . We are interested in the existence of an analytic function such that X(ω) is invertible, and G(ω) coincides with X(ω)F(ω)X(ω)⁻¹ up to order n _j at the point ω _j . We will see that such a function exists provided that F(ω _j ),G(ω _j ) have cyclic vectors, and the characteristic polynomials of F,G coincide up to order n _j at ω _j . This allows one to give a short proof to a result of Huang, Marcantognini and Young concerning spectral interpolation in the unit disk. The author was partially supported by a grant from the National Science Foundation. Received: September 8, 2006. Accepted: January 11, 2007.     

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel ∧↑μ(t) =∑v=1^∞exp(-it μv^1/2), where i= √-1 and {μv}v=1^∞ are the eigenvalues of the negative Laplacian -△=-∑β=1^2(δ/δx^β)^2 in the (x^1, x^2)-plane, is studied for a multi-connected vibrating membrane Ω in R^2 surrounded by simply connected bounded domains Ωj with smooth boundaries δΩj(j=1,...,n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Гi(i=1 κj-1,...,κj) of the boundaries δΩj are considered, such that δΩj=∪i=1 κj-1^κj Гi and κ0=0. The basic problem is to extract information on the geometry of Ω using the wave equation approach. Some geometric quantities of Ω (e.g. the area of Ω, the total lengths of its boundary, the curvature of its boundary, the number of the holes of Ω, etc.) are determined from the asymptotic expansion of the trace of the wave kernel ∧↑μ(t) for small |t|.     

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel θ(t)=∑^∞v=1^exp(-tλv) for small positive t,where {λv} are the eigenvalues of the negative Laplacian -△n=-∑^ni=1(D/Dx^1)^2 in R^2(n=2 or 3),are studied for a general annular bounded domain Ω with a smooth inner boundary DΩ1 and a smooth outer boundary DΩ2,where a finite number of piecewise smooth Robin boundary conditions(D/Dnj γh)Ф=0 on the components Гj(j= 1,...,m) of (DΩ1 and on the components Гj (j=k 1,…,m) of of DΩ2 are considered such that DΩl=U^kj=lГj and DΩ2= U^m=k 1Гj and where the coefficients γj(j=1,...,m) are piecewise smooth positive functions. Some applications of θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given. Further results are also obtained.     

A free boundary problem of type-I superconductivity

1.IntroductionSuperconductorsofTypeIarematerialswhicharecapableofchangingfromthephaseofbeingnormalconductorstoaphasewherethereisnoresistancetothemotionoffreeelections.InnormalconductorphasethenormalizedMaxwellequations(neglectingdisplacementcurrents)aretogetherwithOhm'slawj~acEwhereuistheelectricconductivity.InasuperconductingphaseOhm'slawisnolongervalidandMaxwell'sequationsaresupplemelltedbytheGinzburg-Landaufieldequationsll].Underisothermalconditions,thechangeofphasefromsuperconductingto…     

The zeta functions of complexes from PGL(3): A representation-theoretic approach

The zeta function attached to a finite complex X _Γ arising from the Bruhat-Tits building for PGL₃(F) was studied in [KL], where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of X _Γ. In this paper we re-establish the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.     

Marcinkiewicz multiplier theorem for the Weyl functional calculus

Let {X _j , Y _j , T : 1 ≤ j ≤ n} be a basis satisfying the commutation relation for the Heisenberg Lie algebra . Then we obtain a multi-parameter Marcinkiewicz multiplier theorem for the operator defined by m(X ₁,..., X _n , Y ₁,..., Y _n , T).     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Finiteness properties of chevalley groups overFq[t]

LetG be a simple Chevalley group of rankn and Γ=G( ). Then the finiteness length of Γ shall be determined by studying the action of Γ on the Bruhat-Tits buildingX ofG . This is always possible provided that certain subcomplexes of the links of simplices inX are spherical. As a consequence, one obtains that Γ is of typeF _n−1 but not of typeFP _n ifG is of typeA _n, B_n, C_n orD _n andq≥2²ⁿ⁻¹.