On convection with nearly insulating boundaries in a low Prandtl number fluid

Summary For the Prandtl numberP in the rangeP ^1/3 1 ( is the ratio between the thermal conductivities of the boundary and of the fluid) two-dimensional rolls are preferred in contrast to square-pattern convection that represents the preferred stable convection in the rangeP ^1/3.

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Zussammenfasung Für PrandtlzahlenP, die der RelationP ^1/3 1 genügen (\ ist das Verhältnis zwischen den thermischen Leitfähigkeiten der Wand und des Fluid), sind zweidimensionale Rollen bevorzugt im Gegensatz zur quadratartigen Konvektion, welche die bevorzugte stabile Konvektionsform im BereichP ^1/3 darstellt.

     

The self-similar solutions to a fast diffusion equation

A quasilinear equation u -x·u/2+f(u)=0 is studied, wheref(u)=–u+u , > 0, 0<. <1, >1 andx R ⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation _t =+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.     

On a generalized Eulerian distribution

The distribution with probability function p _k(n, , ) = A _{n, k}(, )/(+ )^[p], k = 0, 1, 2, ..., n, where the parameters and are positive real numbers, A _{n, k} (, ) is the generalized Eulerian number and ( + )^[n] = ( + )( + +1) ... ( + +n – 1), introduced and discussed by Janardan (1988, Ann. Inst. Statist. Math., 40, 439–450), is further studied. The probability generating function of the generalized Eulerian distribution is expressed by a generalized Eulerian polynomial which, when expanded suitably, provides the factorial moments in closed form in terms of non-central Stirling numbers. Further, it is shown that the generalized Eulerian distribution is unimodal and asymptotically normal.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

Singular perturbations of variational problems arising from a two-phase transition model

Given that, are two Lipschitz continuous functions of to ₊ and thatf(x, u, p) is a continuous function of × × ^N to [0, + [ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two points(x) and(x), we prove the convergence ofF (u) = (1/) f (x, u, Du) dx when the perturbation parameter goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs.     

EpGs with minimal μ: II

There are two known lower bounds for (P, Q) in an EpG, called ₁ and ₂, see for example [3]. In [4], =₁ was studied for the case of triangular EGQs and, in [3], =₂ was considered for EpGs in general. Here we extend this to the case =₁ for EpGs in general, including non-triangular EGQs, and we give a number of characterizations. For instance a triangular EpG with =₁ locally is an EGQ, an extended dual net or a semibiplane; if t>2–1, then an EpG(s, t) with =₁ locally is an EGQ. In general we have only partial results for t2–1.     

On asymptotically efficient estimation for a semiparametric regression model

1.IntroductionConsiderthemodelY=X"0 g(T) E,(1'1)whereX"~(xl,',xo)areexplanatoryvariablesthatenterlinearly,Pisakx1vectorofunknownparameters,Tisanotherexplanatoryvariablesthatentersinanonlinearfashion,g')isanunknownsmoothfunctionofTinR',(X,T)andeareindependent,andeistheerrorwithmean0andvariancea2.Trangesoveranondegeneratecompact1-dimensionalilltervalC*;withoutlossofgenerality,C*=[0,1].Chenl2]discussedasymptoticnormalityofestimatorsP.of0byusingpiecewisepolynthacaltoapproximateg.Speckmanls…     

The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model

Summary We prove a variational inequality linking the values of the free energy per site at different temperatures. This inequality is based on the Legendre transform of the free energy of two replicas of the system. We prove that equality holds when1/ and fails when 1/ <1. We deduce from this that the mean entropy per site of the uniform distribution with respect to the distribution of the coupling _i ¹ _i ² = _i between two replicas is null when 01/ and strictly positive when 1/ <1. We exhibit thus a new secondary critical phenomenon within the high temperature region 01. We given an interpretation of this phenomenon showing that the fluctuations of the law of the coupling with the interactions remains strong in the thermodynamic limit when>1/ . We also use our inequality numerically within the low temperature region to improve (slightly) the best previously known lower bounds for the free energy and the ground state energy per site.     

A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL _{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T _t;t0 of class (C ₀) on a family C _,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL _{n, x} and a characterization of convex functions inC _, through an inequality involving the action ofL _{n, x}.     

A high-indices theorem for strong Cesàro summability of double series

[4] , [C,(, )]_;- (, 0 1) s, [C,(+,+)]- s , 0 0< , (i) , ; > > 1 , (ii) ,>0; >=1 ,>1-1/, [C,(,)- [C,(+,+)]- .

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This work was done while the author was a visiting researcher at the Steklov Mathematical Institute, Moscow, U.S.S.R.     

Strictly Cyclic Algebra of Operators Acting on Banach Spaces H p(β)

Let be a sequence of positive numbers and 1 p< . We consider the space H ^p() of all power series such that . We investigate strict cyclicity of the weakly closed algebra generated by the operator of multiplication by zacting on H ^p(), and determine the maximal ideal space, the dual space and the reflexivity of the algebra . We also give a necessary condition for a composition operator to be bounded on H ^p() when is strictly cyclic.     

Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Let (0,1) be an irrational, and [a ₁,a ₂,...] be the continued fraction expansion of . Let H be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum (H ) is strictly great than zero for any irrational , and is strictly less than 1 if and only if liminf _k(a ₁ a ₂a _k ))^1/k <.     

A note on a set-mapping problem of Hajnal and Máté

It is consistent that there exists a set mappingF with <F(, )< for + 2 >w ₂ with no uncountable free sets.Research supported by Hungarian National Research Fund No. 1805 and 1908.     

On the accuracy of empirical likelihood confidence regions for linear regression model

The coverage errors of the empirical likelihood confidence regions for in a linear regression model,Y _i=x _i+ _i , 1in, are of ordern ^–1. Bartlett corrections may be employed to reduce the order of magnitude of the coverage errors ton ^–2. For practical implementation of Bartlett correction, an empirical Bartlett correction is given.     

On sequences of Fourier coefficients of functions of Hölder classes

The following theorem is proved. Let { _k(t)} be an arbitrary complete orthonormal system on [0, 1] and let 1/2<<1. Then anf(t) C exists for all< such that _k=1 · |c_k(f)|^p=, p=2/(l+2), where .Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 567–572, November, 1969.The authors wish to thank P. P. Zabreiko and P. L. Ul'yanov for helpful discussions and remarks.     

A space of vector-valued measures and a strict topology

Let X be a completely regular Hausdorff space and let E be a real locally convex Hausdorff space. Katsaras [2] has studied the topologies ₀, , and ₁, for the vector-valued case on C_rc(X,E), the space of all continuous E-valued functions on X with relatively compact range. The corresponding dual spaces are the spaces M_t (B,E'), M (B,E'), and M (B,E') of all t-additive, all -additive, and all -additive members of M(B,E'), the dual space of C_rc (X,E') under the uniform topology. In this paper we study the subspace M_e(B,E') of M(B,E'). A locally convex topology _e is defined on C_rc(X,E) that yields M_e (B,E') as a dual space. It is proved that if E is strongly Mackey then (C (X,E),_e) is strongly Mackey.The author is grateful to Professor J. Schmets for useful suggestions.     

Asymptotics of the partial sums of a set of integral transforms

In this paper, we explore the asymptotic distribution of the zeros of the partial sums of the family of entire functions of order 1 and type 1, defined by G(,,z)=₀ ¹(t)t ^–1×(1–t)^–1e ^zt dt, where Re,Re>0, is Riemann-integrable on [0,1], continuous at t=0, 1 and satisfies (0)(1)0.     

On the circles of Apollonius associated with a triangle

Given a ratio , >>0, and a triangle ABC, on the sides and , using ratios , and , three circles of Apollonius are denned. In this paper, we will show that the three centers are collinear, the circles are coaxal and develop a necessary and sufficient condition that these circles intersect. J. A. Hoskins, W. D. Hoskins and R. G. Stanton obtained these results in a recent paper using algebraic computation. Our aim is to establish all these results using only results from elementary Euclidean geometry and thereby uncovering more geometric insights and avoid lengthy calculations.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.