Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

An isolation theorem for decomposable forms of purely real algebraic fields of degree n⩾3

Let M be a complete module of a purely algebraic field of degree n3, let be the lattice of this module and let F(X) be its form. By we denote any lattice for which we have = , where is a nondiagonal matrix satisfying the condition ¦-I¦ , I being the identity matrix. The complete collection of such lattices will be denoted by {}. To each lattice we associate in a natural manner the decomposable form F(X). The complete collection of forms, corresponding to the set {}, will be denoted by {F} It is shown that for any given arbitrarily small interval (N–, N+), one can select an such that for each F(X) from {F} there exists an integral vector X₀ such that N– < F(X₀) < N+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 112, pp. 167–171, 1981.     

Regularity and Unit-regularity of Generalized Semigroups of Linear Transformations

Let V and W be vector spaces over a division ring D and L_D (V, W) the set of all linear transformations from V into W. For L_D(W, V), let (L_D (V, W), ) denote the semigroup L_D (V, W) with the operation * defined by * = for all , L_D(V, W). By a unit-regular semigroup we mean a semigroup S with identity having the property that for each a S, a = aua for some unit u S. The main purpose of this paper is to prove the following statements. The semigroup (L_D(V, W), ) is regular if and only if V = {0}, W = {0} or is an isomorphism from W onto V. The semigroup (L_D (V, W), ) is unit-regular if and only if (i) V = {0}, (ii) W = {0} or (iii) is an isomorphism from W onto V and dim_D V .     

Level set representation for the Gibbs states of the ferromagnetic ising model

Summary We prove the existence of a real valued random field with parameters in thed-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter (,h) in the uniqueness region except on the critical curve _c ={(,h _c ())}, whereh _c() is the critical level of the external field above which percolation takes place.Supported in part by JSPS, BiBoS, Grant in Aid for Cooperative research no. 62303006 and Grant in Aid for Scientific Research no. 63540168     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

Application of Topological Degree to the Study of Bifurcation in von Kármán Equations

The aim of this paper is to illustrate the use of topological degree for the study of bifurcation in von Kármán equations with two real positive parameters and for a thin elastic disk lying on the elastic base under the action of a compressing force, which may be written in the form of an operator equation F(x, , ) = 0 in some real Banach spaces X and Y. The bifurcation problem that we study is a mathematical model for a certain physical phenomenon and it is very important in the mechanics of elastic constructions. We reduce the bifurcation problem in the solution set of equation F(x, , ) = 0 at a point (0, ₀, ₀) X × IR ₊ ² to the bifurcation problem in the solution set of a certain equation in IR ⁿ at a point (0, ₀, ₀) IR ⁿ × IR ₊ ², where n = dim Ker F _x (0, ₀, ₀) and F _x (0, ₀, ₀): X Y is a Fréchet derivative of F with respect to x at (0, ₀, ₀). To solve the bifurcation problem obtained as a result of reduction, we apply homotopy and degree theory.     

Equivalent Sets of Solutions of the Klein–Gordon Equation with a Constant Electric Field

We argue extensively in favor of our earlier choice of the in and out states (among the solutions of a wave equation with one-dimensional potential). In this connection, we study the nonstationary and stationary families of complete sets of solutions of the Klein–Gordon equation with a constant electric field. A nonstationary set _Pv consists of the solutions with the quantum number p _v=p ⁰ v–p₃. It can be obtained from the nonstationary set _P3 with the quantum number p ₃ by a boost along the x ₃ axis (in the direction of the electric field) with the velocity –v. By changing the gauge, we can bring the solutions in all sets to the same potential without changing quantum numbers. Then the transformations of solutions in one set (with the quantum number p _v) to the solutions in another set (with the quantum number p _v) have group properties. The stationary solutions and sets have the same properties as the nonstationary ones and are obtainable from stationary solutions with the quantum number p ⁰ by the same boost. It turns out that each set can be obtained from any other by gauge manipulations. All sets are therefore equivalent, and the classification (i.e., assigning the frequency sign and the in and out indices) in any set is determined by the classification in the set _P3, where it is obvious.     

On the construction and calculation of optimal nonlifting critical airfoils

Numerical calculations are carried out in the hodograph plane to construct optimal critical airfoil shapes and the flow about them. These optimal airfoil shapes give the highest free-stream Mach numberM for a given thickness ratio and tail angle _t (nonlifting) for which the flow is nowhere supersonic. A relationship betweenM and for various _t is given. Analytical and numerical solutions to the same problem are found on the basis of transonic small-disturbance theory. These results provide a limiting case asM 1, 0 and agree well with the calculations of the full problem. Using a numerical method to calculate the flow about general (subsonic) airfoils, a comparison is made between the critical free-stream Mach numbers for some standard airfoil shapes and the optimal free stream Mach number of the corresponding and _t . A significant increase in the critical free-stream Mach number is found for the optimal airfoils.     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

Über Automorphismenbahnen von Elementen u≠O mit (ad u)3=αad u in endlichdimensionalen einfachen komplexen Lie-Algebren

In this paper we study certain semisimple elements in simple complex Koecher-Tits-constructions from Jordan-triplesystems. Let L be a finite dimensional simple complex Lie-Algebra and u O an element in L with (ad u)³=-ad u. Then there is a compact real form L of L, which contains u. The involutorial automorphism id_L+2 (ad_Lu)² of L induces a Cartan-decomposition of a real form L (u) of L and this gives us a criterion of conjugacy under Aut L for two such elements u₁, u₂L.Using this result, we show that the number of conjugacy classes of elements uL (u O) with (ad u)³=ad u (\{O}, under Aut L is equal to the number of similarity classes of Jordantriplesystems, the Koecher-Tits-construction of which is isomorphic to L. The corresponding data are finally listed for all possible types of L.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

Quasiconvex uniform-convergence factors for Fourier series of functions with a given modulus of continuity

It is proved that a quasiconvex sequence _v of convergence factors transforms Fourier series of functions whose moduli of continuity do not exceed a given modulus of continuity(gd) into uniformly convergent series if and only if _n (1/n) log n 0 for n . The sufficiency of this condition is already known.Translated from Matematicheskie Zametki, Vol. 8, No. 5,pp. 619–623, November, 1970.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Transformation of Wiener measure under anticipative flows

Summary LetT()=+F() be a transformation from the Wiener space to itself with the range ofF() assumed to be in the Cameron-Martin space. The absolute continuity and the density function associated withT is considered;T is assumed to be embedded in or defined through a parameterizationT _t =+F _t () andF _t is assumed to be differentiable int. The paper deals first with the case where the range of thet-derivative ofF _t () is also in the Cameron-Martin space and new representations for the Radon-Nikodym derivative and the Carleman-Fredholm determinant are derived. The case where thet-derivative ofF _t is not in the Cameron-Martin space is considered next and results on the absolute continuity and the density function, under conditions which are considerably weaker than previously known conditions, are presented.The work of the second author was supported by the fund for promotion of research at the Technion     

Computing the Hilbert transform of a Jacobi weight function

We discuss the evaluation of the Hilbert transformf _–1 ¹ (t-)^–1 w(, )(t)dt,–1<<1, of the Jacobi weight functionw(, )(t)=(1–t))(1+t) by analytic and numerical means and also comment on the recursive computation of the quantitiesf _–1 ¹ )(t–)^–1 _n (t;w ^{(, ))} w ^{(, )}(t)dt,n=0, 1, 2, ..., where _n (·;w ^{(, )}) is the Jacobi polynomial of degreen.The work of the first author was supported in part by the National Science Foundation under grant DCR-8320561. The work of the second author was supported by the National Science Foundation under grant DMS-8419086.     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates

We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0, ). We assume that u(x, 0), u_t(x, 0), u(0, t) are given, and that they satisfy u(x, 0)2q, u_t(x, 0)0, for large x, u(0, t)2p for large t, where q, p are integers. We also assume that u_x(x, 0), u_t(x, 0), u_t(0, t), u(0, t)-2p, u(x, 0)-2q L₂. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large t shows how the boundary conditions can generate solitons.The authors dedicate this paper to the memory of M. C. PolivanovDepartment of Mathematics and Computer Science; Institute for Nonlinear Studies, Clarkson University, Postdam, New York. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 387–403, September, 1992.