Locally forced critical surface waves in channels of arbitrary cross section

Critical long surface waves forced by locally distributed external pressure applied on the free surface in channels of arbitrary cross section are studied in this paper. The fluid under consideration is inviscid and has constant density. The upstream flow is uniform and the upstream velocity is assumed to be near critical, i.e.,u ₀=u _c ++0(²), where 0<1 andu _c is the critical velocity determined by the geometry of the channel. The external pressure applied on the free surface as the forcing is ² P(x). Then the first order perturbation of the free surface elevation satisfies a forced Korteweg-de Vries equation (fK-dV). It is shown in this paper that: (i) If (supercritical), the stationaryfK-dV has two cusped solitary wave solutions; (ii) if (subcritical), the stationaryfK-dV has a downstream cnoidal wave solution; (iii) when= _L , the unique stationary solution of thefK-dV is a wave free hydraulic fall; (iv) if= _d =– _L , thefK-dV has a jump solution; and (v) if _L << _c , thefK-dV does not have stationary solutions. Some free surface profiles and bifurcation diagrams are presented.     

Estimates of Entire Functions of Exponential Type Less than π in Terms of Logarithmic Sums over Real Duffin and Schaeffer Sequences

We give uniform estimates of entire functions of exponential type less than having sufficiently small logarithmic sums over real sequences { _n } satisfying | _n –n|L and _n+1– _n for fixed positive constants L and . We thereby generalize results about logarithmic sums over the set of integers and so-called relatively h-dense sequences.     

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

Let m , 0 m₊ in Kato's class. We investigate the spectral function s( + m) where s( + m) denotes the upper bound of the spectrum of the Schrödinger operator + m. In particular, we determine its derivative at 0. If m_- is sufficiently large, we show that there exists a unique ₁ > 0 such that s( + ₁m) = 0. Under suitable conditions on m⁺ it follows that 0 is an eigenvalue of + ₁m with positive eigenfunction.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

Completeness of the plans of sequential estimation for Wiener spaces with a drift and some uniqueness theorems

One considers a family of n-dimensional Wiener processess x(t) =(t) +t, depending on a drift parameter , where is the standard Wiener process. Let be a closed subset of the space of trajectories ⁿ×₊ (plan) and assume that the measure , defined by the first occurrence of the trajectory of the process x in the set , is a probability measure. One gives conditions which the plan has to satisfy in order that from the equality for any there should follow that f 0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 69–72, 1983.The authors are grateful to V. P.Khavin for discussions on certain questions of potential theory.     

A Proof of the Jungnickel-Tonchev Conjecture on Quasi-Multiple Quasi-Symmetric Designs

A proof of the following conjecture of Jungnickel and Tonchev on quasi-multiple quasi-symmetric designs is given: Let D be a design whose parameter set (v,b,r,k,) equals (v,sv,sk,k, s) for some positive integer s and for some integers v,k, that satisfy (v-1) = k(k-1) (that is, these integers satisfy the parametric feasibility conditions for a symmetric (v,k,)-design). Further assume that D is a quasi-symmetric design, that is D has at most two block intersection numbers. If (k, (s-1)) = 1, then the only way D can be constructed is by taking multiple copies of a symmetric (v,k, )-design.     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

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Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Subdivision algorithms with nonnegative masks generally converge

In this paper, a subdivision scheme consists of an operator froml () tol () determined by a doubly infinite sequence, called the mask. This operator convolutes, in a certain sense, sequences l () with the mask, thus producing a new sequence inl (). Moreover, this new sequence is placed on a finer grid. If we iterate this process with a positive mask infinitely many times, it is known that this process will produce a continuous function, which we callf . In this paper, we consider the extent to which non-negative masks yield similar results. An important application of subdivision schemes in computer graphics is the generation of curves and surfaces from an initial sequence.     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

Quasi-conformal mapping theorem and bifurcations

LetH be a germ of holomorphic diffeomorphism at 0 . Using the existence theorem for quasi-conformal mappings, it is possible to prove that there exists a multivalued germS at 0, such thatS(ze ²ⁱ )=HS(z) (1). IfH is an unfolding of diffeomorphisms depending on (,0), withH ₀=Id, one introduces its ideal . It is the ideal generated by the germs of coefficients (a _i (), 0) at 0 ^k , whereH (z)–z=a _i ()z ⁱ . Then one can find a parameter solutionS (z) of (1) which has at each pointz ₀ belonging to the domain of definition ofS ₀, an expansion in seriesS (z)=z+b _i ()(z–z ₀) ⁱ with , for alli.This result may be applied to the bifurcation theory of vector fields of the plane. LetX be an unfolding of analytic vector fields at 0 ² such that this point is a hyperbolic saddle point for each . LetH (z) be the holonomy map ofX at the saddle point and its associated ideal of coefficients. A consequence of the above result is that one can find analytic intervals , , transversal to the separatrices of the saddle point, such that the difference between the transition mapD (z) and the identity is divisible in the ideal . Finally, suppose thatX is an unfolding of a saddle connection for a vector fieldX ₀, with a return map equal to identity. It follows from the above result that the Bautin ideal of the unfolding, defined as the ideal of coefficients of the difference between the return map and the identity at any regular pointz, can also be computed at the singular pointz=0. From this last observation it follows easily that the cyclicity of the unfoldingX , is finite and can be computed explicity in terms of the Bautin ideal.Dedicated to the memory of R. Mañé     

Limit theorems for multitype continuous time Markov branching processes

Summary Let X(t)=(X ₁ (t), X ₂ (t), , X _t (t)) be a k-type (2k<) continuous time, supercritical, nonsingular, positively regular Markov branching process. Let M(t)=((m _ij (t))) be the mean matrix where m _ij (t)=E(X _j (t)¦X _r (0)= _ir for r=1, 2, , k) and write M(t)=exp(At). Let be an eigenvector of A corresponding to an eigenvalue . Assuming second moments this paper studies the limit behavior as t of the stochastic process . It is shown that i) if 2 Re >₁, then · X(t)e{–t¦ converges a.s. and in mean square to a random variable. ii) if 2 Re ₁ then [ · X(t)] f(v · X(t)) converges in law to a normal distribution where f(x)=(x) ^–1 if 2 Re <₁ and f(x)=(x log x)^–1 if 2 Re =₁, ₁ the largest real eigenvalue of A and v the corresponding right eigenvector.Research supported in part under contracts N0014-67-A-0112-0015 and NIH USPHS 10452 at Stanford University.     

On eigenfunctions of nonlinear heat generation

Summary We consider the equation u+ expu=0, >0,u(boundary)0 in the formv= exp (K,v), whereK ^–1=–. We give bounds on for the latter equation to be solvable by the contraction mapping principle, and estimate theL ² norm of the solution so obtained. We also give a bound on for the topological index of the solution to be non-zero and apply Krasnoselskii's results to the least squares method of approximating the solution.

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Sommario Consideriamo l'equazione u+ expu=0, >0,u(frontiera)=0 nella formav= exp (Kv), doveK ^–1=–. In questo lavoro diamo limitazioni per per cui la seconda equazione e risolubile col metodo delle contrazioni, e diamo una stima della norma inL ² della soluzione cosi ottenuta. Diamo anche una limitazione per per cui l'indice topologico della soluzione diventa non zero, e applichiamo i risultati di Krasnoselskii al metodo dei minimi quadrati per approssimare la soluzione.

     

The Euler characteristic of local systems on the moduli of genus 3 hyperelliptic curves

For a partition ={₁₂₃0} of non-negative integers, we calculate the Euler characteristic of the local system on the moduli space of genus 3 hyperelliptic curves using a suitable stratification. For some of low degree, we make a guess for the motivic Euler characteristic of using counting curves over finite fields.Mathematics Subject Classification (1991): 14J15, 20B25     

A note on a nonlinear eigenvalue problem

Summary In this paper a necessary and sufficient condition for the existence of negative eigenvalues for the problem-u – u=(x)¦u¦^p–2u in u¦=0 is given. Here Rⁿ is supposed a smooth bounded domain, 0 a bounded nonnegative function, (₁, ₂), ₁ and ₁ being the first and the second eigenvalue of - in with zero Dirichlet boundary data, p2 and, if n 3, p < 2n¦(n–2). Moreover in the linear case (p=2) a uniqueness result is proved.Work supported by G.N.A.F.A. and by M.P.I, of Italy Fondi 40% Equazioni Differenziali e Calcolo delle Variazioni and Fondi 60% Analisi matematica.     

A Result on Ext Over Kac–Moody Algebras

We prove the following result for a not necessarily symmetrizable Kac–Moody algebra: Let x,y W with x y, and let P⁺. If n=l(x)-l(y), then Ext _C() ⁿ (M(x·),L(y·))=1.     

Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire

Summary A finite element method (P₁) with numerical integration for approximating the boundary value problem –u=e ^u is considered. It is shown that the discrete problem has a solution branch (with turning point) which converges uniformely to a solution branch of the continuous problem. Error estimates are given; for example it is found that , >0, where ₀ and _h ⁰ are critical values of the parameter for continuous and discrete problems.

     

An Extension of the Roots Separation Theorem

Let T _n be an n×n unreduced symmetric tridiagonal matrix with eigenvalues ₁<₂<< _n and W _k is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T _n , k=1,2,...,n. Let _{12 n–1} be the eigenvalues of W _k . It is proved that if W _k has no multiple eigenvalue, then ₁<₁<₂<₂<< _n–1< _n–1< _n ; otherwise if _i = _i+1 is a multiple eigenvalue of W _k , then the above relationship still holds except that the inequality _i < _i+1< _i+1 is replaced by _i = _i+1= _i+1.     

Sur les Mesures Simultanement Invariantes Pour les Transformations x » {λx} ET x » {βx}

We show that for reasonable couples of Pisot number and , there is no measure simultaneously invariant by the two transformations of [0, 1], x {x} and x {x}, and Bernoulli (or weak Bernoulli) for one of the transformations.     

A shift-invariant algebra of singular integral operators with oscillating coefficients

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL ^p (T, ) with an arbitrary Muckenhoupt weight on the unit circleT, and the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse _h,S _T e _h, ^–1 I (hR, T) whereS _T is the Cauchy singular integral operator ande _h,(t)=exp(h(t+)/(t–)),tT. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra and its matrix analogue . These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.Partially supported by CONACYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.