The intersection of complete geodesics on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript> with positive curvature

Let M be a Riemannian manifold. A complete geodesic on M means that :(-,+)M is a normalized geodesic. In this paper, we prove that on (S²,g) with positive curvature, any two complete geodesics must intersect an infinite number of times, and a complete geodesic must self-intersect an infinite number of times. Mathematics Subject Classification (2000) 53C40 (53C22)     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.     

Min-max game theory and algebraic Riccati equations for boundary control problems with continuous input-solution map. Part II: The general case

We consider the abstract dynamical framework of [LT3, class (H.2)] which models a variety of mixed partial differential equation (PDE) problems in a smooth bounded domain ⁿ , arbitraryn, with boundaryL ₂-control functions. We then set and solve a min-max game theory problem in terms of an algebraic Riccati operator, to express the optimal quantities in pointwise feedback form. The theory obtained is sharp. It requires the usual Finite Cost Condition and Detectability Condition, the first for existence of the Riccati operator, the second for its uniqueness and for exponential decay of the optimal trajectory. It produces an intrinsically defined sharp value of the parameter, here called _c (critical), _c0, such that a complete theory is available for > _c, while the maximization problem does not have a finite solution if 0 < < _c. Mixed PDE problems, all on arbitrary dimensions, except where noted, where all the assumptions are satisfied, and to which, therefore, the theory is automatically applicable include: second-order hyperbolic equations with Dirichlet control, as well as with Neumann control, the latter in the one-dimensional case; Euler-Bernoulli and Kirchhoff equations under a variety of boundary controls involving boundary operators of order zero, one, and two; Schroedinger equations with Dirichlet control; first-order hyperbolic systems, etc., all on explicitly defined (optimal) spaces [LT3, Section 7]. Solution of the min-max problem implies solution of theH -robust stabilization problem with partial observation.The research of C. McMillan was partially supported by an IBM Graduate Student Fellowship and that of R. Triggiani was partially supported by the National Science Foundation under Grant NSF-DMS-8902811-01 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321.     

A priori estimates of the solution of the dirichlet problem for the Monge-ampére equation in weight spaces

One proves that a priori boundedness of the norm of the solution of the problem det(U_xx)=f(x,u,u_x)>>0,u¦=0. The magnitudes of the exponents,() depends on whether the arguments u p occur or not in f (x,u,p).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 74–90, 1983.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

The compact support property for solutions to the heat equation with noise

Summary We consider all solutions of a martingale problem associated with the stochastic pde and show thatu(t,·) has compact support for allt0 ifu(0,·) does and if <1. This extends a result of T. Shiga who derived this compact support property for 1/2 and complements a result of C. Mueller who proved this property fails if 1.The author's research was supported by an NSF grant and an NSERC operating grantThe author's research was supported by an NSERC operating grant     

Compact Embedded Minimal Surfaces of Positive Genus Without Area Bounds

Let M ³ be a three-manifold (possibly with boundary). We will show that, for any positive integer , there exists an open nonempty set of metrics on M (in the C ²-topology on the space of metrics on M) for each of which there are compact embedded stable minimal surfaces of genus with arbitrarily large area. This extends a result of Colding and Minicozzi, who proved the case =1.     

k-regular factors and semi-k-regular factors in bipartite graphs

LetG be a graph, andk1 an integer. LetU be a subset ofV(G), and letF be a spanning subgraph ofG such that deg _F (x)=k for allx V(G)–U. If deg _F (x)k for allxU, thenF is called an upper semi-k-regular factor with defect setU, and if deg _F (x)k for allxU, thenF is called a lower semi-k-regular factor with defect setU. Now letG=(X, Y;E(G)) be a bipartite graph with bipartition (X,Y) such that X=Yk+2. We prove the following two results.(1) Suppose that for each subsetU ₁X such that U ₁=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=max{k+1, X+1/2},G has an upper semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.(2) Suppose that for each subsetU ₁X such that U ₁=X–1/k+1,G has a lower semi-k-regular factor with defect setU ₁Y, and for each subsetU ₂Y such that U ₂=X–1/k+1,G has a lower semi-k-regular factor with defect setXU ₂. ThenG has ak-factor.     

Codes from Affine Permutation Groups

It is shown that A(22,10) 50, A(23,10) 76, A(25,10) 166, A(26,10) 270, A(29,10) 1460, and A(28,12) 178, where A(n,d) denotes the maximum cardinality of a binary code of length n and minimum Hamming distance d. The constructed codes are invariant under permutations of some affine (or closely related) permutation group and have been found using computer search.     

On the isotopy of Legendrian knots

Let ₀ and ₁ be Legendrian knots which are isotopic as usual knots, and which have the same obvious invariants rot and link. It seems to be an open question whether ₀ and ₁ are isotopic as Legendrian knots. In the paper we give a positive answer to this question for the (rather restricted) class of Legendrian knots with nonintersecting fronts.     

Linearization of boundary eigenvalue problems

It is shown that certain eigenvalue problems for ordinary differential operators with boundary conditions depending holomorphically on the eigenvalue parameter can be linearized by making use of the theory of operator colligations. As examples, first order systems with boundary conditions depending polynomially on and Sturm-Liouville problems with -holomorphic boundary conditions are considered.     

The closure of a similarity orbit is always arcwise connected

If T, A are Hilbert space operators such that A is the norm-limit of operators similar to T, then there exists a continuous function from [0,1] into the algebra of operators such that (t) is a limit of similarities of T for all t in the interval [0,1], (0)=T and (1)=A.This research was partially supported by a Grant of the National Science Foundation.     

Maximal inequalities,convexity inequality,and their duality. II

, , . . . [1], , . , , ., , L logL. , , . . . . [5]. , .     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

A Theorem of Liouville Type for Harmonic Morphisms

Let : M N be a harmonic morphism from a complete noncompact Riemannian manifold M with nonnegative Ricci curvature to a complete Riemannian manifold N with nonpositive scalar curvature. We show if the energy of is finite, then is constant. This can be compared with a similar result for harmonic maps when N has nonpositive sectional curvature due to Schoen and Yau.     

Angular limits of holomorphic functions which satisfy an integrability condition

Let (–1,1), let 2/(1–)p<, letp denote the Hölder conjugate ofp, and let be an open arc of the unit circle. It is shown that, iff is a holomorphic function on the unit disc such that: (i) (1–|z|)log⁺|f(z)| isL ^p -integrable on the sector {r:0f has an infinite asymptotic value has -finite (2–(1+)p)-dimensional Hausdorff, measure, thenf has finite angular limits on a subset of of positive linear measure. In fact, a stronger conclusion will be established.     

Integrability of multiple Walsh series and Parseval's formula

f(x,y) _jk . , {c _jk} , f(x, )(, ) [0,1)&#x0445;[0,1) , - (0,0). , , f, - f. , , , [1] . . - [5] [6].

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This research is supported by National Science Council, Taipei, R.O.C. under Grant #NSC 84-2121-M-007-026.     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

Explicitly solvable model of Mössbauer scattering

An explicitly solvable model of Mössbauer scattering of rays by a nucleus bound in a harmonic-oscillator potential is constructed. The probability of elastic scattering, which is proportional to the Debye—Waller factor, is calculated in the framework of the explicitly solvable scattering problem. It is assumed that the rms deviation x of the nucleus and the photon wave numberk satisfykxE /E , whereE andE are typical energy levels of the photon and the oscillator states.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 439–430, June, 1993.     

An elementary proof of the existence of uncountably many nonclosed isometry-invariant geodesics

An isometry A on a Riemannian manifold M will be called of compact type if the subgroup generated by A is relatively compact in the isometry group of M. A geodesic : RM is called invariant under the isometry A if there exists a nonzero constant b such that A(t)=(t+b) for any real t. We prove that if A is an isometry of compact type on a connected Riemannian manifold, and if A has a nonclosed invariant geodesic, then A has uncountably many invariant geodesics.