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     

Subexponential distributions and characterizations of related classes

LetL(),0, 0, denote the class of distributionsF satisfying

Turán's extremal problem in random graphs: Forbidding odd cycles

For 0<1 and graphsG andH, we writeGH if any -proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC ^k for a cycle of lengthk. We show that, for every fixed integerl1 and real >0, there exists a real constantC=C(l, ), such that almost every random graphG _{n, p} withp=p(n)Cn ^–1+1/2l satisfiesG _n,p1/2+ C ^2l+1. In particular, for any fixedl1 and >0, this result implies the existence of very sparse graphsG withG _1/2+ C ^2l+1.The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNP_q (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.     

On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

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.     

Some asymptotic results related to the law of iterated logarithm for Brownian motion

For 0<<1, let . The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t U ((t))=1–exp{–4(^–1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU (t), t>0. Also, when =1,U (t)0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form as 0, whereD is a suitable discount function. These results also hold for symmetric random walks.     

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.     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

Convex independent sets and 7-holes in restricted planar point sets

For a finite setA of points in the plane, letq(A) denote the ratio of the maximum distance of any pair of points ofA to the minimum distance of any pair of points ofA. Fork>0 letc (k) denote the largest integerc such that any setA ofk points in general position in the plane, satisfying for fixed , contains at leastc convex independent points. We determine the exact asymptotic behavior ofc (k), proving that there are two positive constants=(), such thatk ^1/3c (k)k ^1/3. To establish the upper bound ofc (k) we construct a set, which also solves (affirmatively) the problem of Alonet al. [1] about the existence of a setA ofk points in general position without a 7-hole (i.e., vertices of a convex 7-gon containing no other points fromA), satisfying . The construction uses Horton sets, which generalize sets without 7-holes constructed by Horton and which have some interesting properties.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Self-intersections of curves on surfaces

Let M be a compact orientable surface with nonempty boundary (x(M)<0) and fundamental group . Let be a geodesic on M (with a fixed hyperbolic structure), and let W be a (cyclically reduced) word in a fixed set of generators of which represents . In this paper, we give an algorithm to count the number of self-intersections of in terms of W, generalizing a result of Birman and Series, where an algorithm was given to decide if was simple. Some applications of the algorithm to surfaces with one boundary and the Markoff spectrum are also given.     

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.     

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.     

Bonds for a functional on a class of functions with positive real parts

We establish sharp upper and lower bounds for the functionalJ (p)- Re{p (z)-zp'(z)/p(z), 1. where ¦z¦=r is fixed, 0 <r <1, p(z).P_n (A, B), –1 B < a 1, a certain class of regular functions in the disk with values in the right halfplane.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 686–689, May, 1990.     

Über die Anzahl der Gitterpunkte in verallgemeinerten Kreissektoren

In this paper we consider lattice points in domains bounded by algebraic curves of the formx ⁿ+yⁿ=Rⁿ fulfilling the additional condition where and are fixed positive real numbers. The number of these lattice points is estimated for largeR and it appears that for rational or badly approximable and the error term in the final result can be made smaller (at least forn3) than it is best possible when counting the lattice points without the additional condition indicated above.     

The length,the width and the cutset-number of finite ordered sets

The partially ordered set P is an (, , ) ordered set if the width of P, the length of any chain of P and the cut-set number . We will prove that if P is an (, , ) ordered set then P contains a simple (, , ) ordered set and use this result to prove that if P has the 3 cutset property, then width of P length of P+3.     

Torsion and closed geodesics on complex hyperbolic manifolds

Summary LetX be a compact complex manifold covered by complex hyperbolicn-space with the induced metric. Each stable horocycle has a cocomplex structure preserved by the geodesic flow. To a closed geodesic one can thus associate a piece of the Poincaré map with a holomorphic fixed point. The resulting Atiyah-Bott fixed point indices, together with the length and multiplicity of as a periodic orbit, determine the contribution of to certain zeta functionsR _p(z), 0pn. From the leading coefficient ofR _p atZ=0 and the Hodge numbersh ^ij (X) we calculate the Ray-Singer -torsionT _p (X). This indicates that the known connections between torsion and the dynamical features of closed orbits continue to hold in the holomorphic category.Corresponding results hold for the -torsion of a flat unitary bundle, extending certain formulas of Ray and Singer to the casen>1.Partially supported by the Sloan Foundation and the National Science Foundation     

Small Ball Estimates in <Emphasis Type="Italic">p</Emphasis>-Variation for Stable Processes

Let Z _t , t 0 be a strictly stable process on with index (0, 2]. We prove that for every p > , there exists = _{, p} and such that

Dehn twists and pseudo-Anosov diffeomorphisms

Summary We show that it is possible to obtain many pseudo-Anosov diffeomorphisms from Dehn twists. In particular, we generalize a theorem of Long and Morton to obtain that iff is a pseudo-Anosov diffeomorphism of an oriented surface andT is the Dehn twist around the simple closed curve , then the isotopy class ofT ⁿ f contains a pseudo-Anosov diffeomorphism except for at most 7 consecutive values ofn.     

Impossibility of a Certain Cyclotomic Equation with Applicationsto Difference Sets

Under certain conditions, we show the nonexistence ofan element in the p-th cyclotomicfield over , that satisfies . As applications, we establish the nonexistence ofsome difference sets and affine difference sets.