mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

winding of planar brownian curves the authors compute the joint probability for a closed brownian curve to wind n times around a prescribed point and to enclose a given algebraic area an estimate from below of the arithmetic area is obtained

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

algebraic areas enclosed by brownian curves on bounded domains using constrained path integrals and perturbation theory the authors study the algebraic area a enclosed by closed brownian curves on various domains such as rectangles or strips in the limit of infinitely long curves the probability distribution p a is shown to be gaussian the standard deviation a2 1 2 is simply expressed in terms of the length scales of the problem

plane polymer configurations enclosing a fixed area in an electric field generating functional and statistical mechanical properties the statistical mechanical properties of plane polymer configurations which enclose a fixed area and are subject to an external electric field are investigated for this purpose an exact expression for the generating functional is obtained and subsequently used to derive a the distribution function for the enclosed area b the mean square distance of a given repeating unit from the origin c the entropic force on a repeating unit

oscillatory integrals with quadratic phase function on a real abstract wiener space abstract stochastic oscillatory integrals with quadratic phase function are studied in the case where amplitude functions are multiple wiener integrals we show that a principle of stationary phase holds when kernels of multiple wiener integrals are of trace class on the way we establish a new criterion for a wiener functional to be analytic

semiflexible polymers in straining flows we introduce a model for semiflexible polymer chains based on the integral of an appropriate gaussian process the stiffness is characterized physically by adding a bending energy the degree of stiffness in the polymer chain is quantified by means of a parameter and as this parameter tends to infinity the limiting case reduces to the brownian model of completely flexible chains studied in earlier work the calculation of the partition function for the configuration statistical mechanics i e the distribution of shapes of such polymers in elongational flow or quadratic potentials is equivalent to the probabilistic problem of finding the law of a quadratic functional of the associated gaussian process an exact formula for the partition function is presented however in practice this formula is too complicated for most computations we therefore develop an asymptotic expansion for the partition function in terms of the stiffness parameter and obtain the first order term which gives the first order deviation from the completely flexible case in addition to the partition function the method presented here can also deal with other quadratic functionals such as the stochastic area associated with two polymer chains

injectivity almost everywhere for weak limits of sobolev homeomorphisms abstract let r n be an open set and let f w 1 p r n be a weak sequential limit of sobolev homeomorphisms then f is injective almost everywhere for p n 1 both in the image and in the domain for p n 1 we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and the topological image of a point is a continuum for every point in a set of positive measure in the domain

the atiyah singer theorems a probabilistic approach ii the lefschetz fixed point formulas abstract in this paper the lefschetz fixed point formulas of m f atiyah and i m singer ann of math 87 1968 484 530 546 604 for classical elliptic complexes are proved by using probabilistic methods

cyclically stationary brownian local time processes local time processes parameterized by a circle defined by the occupation density up to time t of brownian motion with constant drift on the circle are studied for various random times t while such processes are typically non markovian their laplace functionals are expressed by series formulae related to similar formulae for the markovian local time processes subject to the ray knight theorems for bm on the line and for squares of bessel processes and their bridges for t the time that bm on the circle first returns to its starting point after a complete loop around the circle the local time process is cyclically stationary with same two dimensional distributions but not the same three dimensional distributions as the sum of squares of two i i d cyclically stationary gaussian processes this local time process is the infinitely divisible sum of a poisson point process of local time processes derived from brownian excursions the corresponding intensity measure on path space and similar levy measures derived from squares of bessel processes are described in terms of a 4 dimensional bessel bridge by williams decomposition of ito s law of brownian excursions

asymptotic laws of planar brownian motion on presente des resultats recents sur la distribution asymptotique des nombres de passages et d enroulements

the transient dynamics of a cable mass system due to the motion of an attached accelerating mass abstract the objective of this paper is an analytical and numerical study of the transient vibrations of a taut inclined cable with a riding accelerating mass that is suspended in space the moving mass is accelerated by a thrust down the inclined cable and is able to be aerodynamically brought to a halt the thrust is a follow force that is tangential to the deformed configuration of the cable the cable is suspended from two points of different elevation the higher point of the cable is anchored to the ground and the lower point of the cable is controlled so that its tension at that point is held at a constant level hence the length of the cable is not fixed and is a parameter to be determined during any run the restriction in which an inextensible cannot vibrate in the limit of vanishing sag then is removed the mechanics of the problem is newtonian methods of analysis will consist of the dynamics of small deformations superimposed on the static catenary state the problem is nonlinear due to the presence of friction and the convective acceleration interaction of the moving mass and the cable galerkin s procedure for removal of spatial dependence and numerical integration are used to obtain convergent solutions

monotone hopf harmonics we introduce the concept of monotone hopf harmonics in 2d as an alternative to harmonic homeomorphisms much of the foregoing is motivated by the principle of non interpenetration of matter in the mathematical theory of nonlinear elasticity ne the question we are concerned with is whether or not a dirichlet energy minimal mapping between jordan domains with a prescribed boundary homeomorphism remains injective in the domain the classical theorem of rado kneser choquet asserts that this is the case when the target domain is convex an alternative way to deal with arbitrary target domains is to minimize the dirichlet energy subject to only homeomorphisms and their limits this leads to the so called hopf laplace equation among its solutions some rather surreal are continuous monotone mappings of sobolev class mathscr w 1 2 text loc called monotone hopf harmonics it is at the heart of the present paper to show that such solutions are correct generalizations of harmonic homeomorphisms and in particular are legitimate deformations of hyperelastic materials in the modern theory of ne we make this clear by means of several examples

etude de processus generalisant l aire de levy given ann dimensional brownian motionb t b 1 t b n t we consider the stochastic process n dimensional levy s stochastic area l n t sum limits sigma varepsilon sigma int limits 0 t ldots int limits 0 t 2 b sigma 1 t 1 db sigma 2 t 1 ldots db sigma n t n 1 were e is the signature of the permutation we show that this process can be explicitly expressed as a functionalf of 2 dimensional ordinary levy s stochastic areas f is calculated then evaluation of the characteristic function ofl 3 t follows then an iterated logarithm theorem is proved for l n t

analytic functions on abstract wiener spaces abstract let x h be a real abstract wiener space a new definition of analytic functions on x is introduced and it is shown that stochastic line integrals of real analytic 1 forms along brownian motion and solutions to stochastic differential equations with real analytic coefficients are analytic wiener functionals an l p theoretical sufficient condition for wiener functionals to be analytic and an associated cauchy formula are also established

strict limits of planar bv homeomorphisms abstract we introduce a new class of planar mappings that allows for cavitations and fractures the class is the set of strict limits of planar bv homeomorphisms each mapping from this class has a proper pointwise representative which is a multifunction we show that it maps disjoint sets to essentially disjoint sets and that they have an inverse as a proper multifunction we also characterize and study cavities and fractures of these mappings

triangulation of diffeomorphisms is it possible to approximate a diffeomorphism of euclidean domains with piecewise affine homeomorphisms locally uniformly up to the first order derivatives the answer is yes however any effort to provide a rigorous and clear proof reveals the complexity of this question especially in higher dimensions it is the objective of the present paper to formulate this question in its greatest generality as well as to provide all details for the affirmative answer theorem 1 1 a novelty which has broader applications is the construction of selfsimilar isotropic triangulation of the euclidean domains theorem 1 2

on the exponential decay of oscillatory integrals on an abstract wiener space abstract let x h be a real abstract wiener space the saddle point method is carried out on x and an exact formula for deformation of contour integration that is a formula transferring a fourier type integral to a fourier laplace one will be concluded as an application an exponentially fast decay of stochastic oscillatory integrals on x is studied

central limit theorem for the edwards model the edwards model in one dimension is a transformed path measure for standard brownian motion discouraging self intersections we prove a central limit theorem for the endpoint of the path extending a law of large numbers proved by westwater the scaled variance is characterized in terms of the largest eigenvalue of a one parameter family of differential operators introduced and analyzed by van der hofstad and den hollander interestingly the scaled variance turns out to be independent of the strength of self repellence and to be strictly smaller than one the value for free brownian motion

levy s stochastic area and the principle of stationary phase abstract let w 0 h 0 0 be the 2 dimensional classical pinned wiener space over 0 1 a localization phenomenon will be shown for stochastic oscillatory integrals with levy s stochastic area as phase function

dynamic analysis of suspended cables carrying moving oscillators abstract an efficient procedure for analyzing in plane vibrations of flat sag suspended cables carrying an array of moving oscillators with arbitrarily varying velocities is presented the cable is modelled as a mono dimensional elastic continuum fully accounting for geometrical nonlinearities by eliminating the horizontal displacement component through a standard condensation procedure the nonlinear integro differential equation governing vertical cable vibrations is derived due to the dynamic interaction at the contact points with the moving oscillators such equation is coupled to the set of ordinary differential equations ruling the response of the travelling sub systems an improved series representation of vertical cable displacement is proposed which allows to overcome the inability of the traditional galerkin method to reproduce the kinks and abrupt changes of cable configuration at the interface with the moving sub systems following the philosophy of the well known mode acceleration method the convergence of the series expansion of cable response in terms of appropriate basis functions is improved through the introduction of the so called quasi static solution numerical results demonstrate that despite the basis functions are continuous the improved series enables to capture with very few terms the abrupt changes of cable profile at the contact points between the cable and the moving oscillators

some brownian functionals and their laws we develop some topics about brownian motion with a particular emphasis on the study of principal values of brownian local times we show some links between principal values and doob s h transforms of brownian motion for nonpositive harmonic functions h we also give a survey and complement some martingale approaches to ray knight theorems for local times

brownian motion constrained to enclose a given area the probability of a planar brownian closed curve enclosing a fixed area is rederived using a simple method of functional integration the mean square distance between two points of the ring is calculated when the constraint on the enclosed area increases one passes from a diffusion regime of the brownian ring to a normal regime where the curve approaches a circle

magnetic fields and brownian motion on the 2 sphere using constrained path integrals the authors study some statistical properties of brownian paths on the two dimensional sphere a generalized levy law for the probability p a that a closed brownian path encloses an algebraic area a is obtained distributions of scaled variables related to the winding of paths around some fixed point are recovered in the asymptotic regime t to infinity

on wiener functionals of order 2 associated with soliton solutions of the kdv equation abstract the eigenvalues and eigenvectors of the hilbert schmidt operators corresponding to the wiener functionals of order 2 which give a rise of soliton solutions of the kdv equation are determined two explicit expressions of the stochastic oscillatory integral with such wiener functional as phase function are given one is of infinite product type and the other is of levy s formula type as an application the asymptotic behavior of the stochastic oscillatory integral will be discussed

wavelet galerkin method for the free vibrations of an elastic cable carrying an attached mass a multilevel representation of daubechies compactly supported wavelet has been used to study the free vibrations of elastic catenary cables carrying an attached mass anti derivatives of wavelets are used to guarantee satisfaction of boundary conditions natural frequencies mode shapes and dynamic tensions are obtained and compared with the classical fourier series representation the localization feature of wavelets has been implemented to enter the singularity region that is produced by the attached mass more wavelets are used near the mass location and the spurious oscillations in the solution are minimized with few number of terms in the series however the fourier solution shows many oscillations along the cable length and gibbs phenomenon at the mass location in both methods reverting and swapping modes are discovered in which higher modes revert to lower modes and that the horizontal displacement components become greater than the vertical ones even for cables with small sag to span ratios

the dynamic analysis of a suspended cable due to a moving load the dynamic behaviour of a saggy suspension cable under a moving load was investigated first of all the updated lagrangian formulation and the finite element method were used to derive the property matrices of a saggy suspension cable in order to define the discretized equations of motion then the jacobi method was applied to the determination of the natural frequencies and the mode shapes of the suspended cable the moving load induced dynamic responses of the saggy suspended cable were obtained by using the newmark direct integration method incorporated with the newton raphson iteration technique the influence of some pertinent factors such as speed of moving load ratio of axial rigidity to total cable weight ae 0mg0l and ratio of moving load mass to total cable mass is the key point of the dynamic analysis

analytic functions cauchy formula and stationary phase on a real abstract wiener space abstract a new complexification of a real abstract wiener space will be introduced and some analogs of the algebra of analytic functions on finite dimensional euclidean space will be considered analytic functions on the original space their holomorphic prolongation to the complexified space and holomorphic functions and a cauchy formula on the complexified space the cauchy formula is a key tool to study probabilistic quantities via deformation of the contour integration namely it will be applied to establish i an explicit representation of stochastic oscillatory integrals with quadratic phase function and ii a stationary phase estimation of the integrals further the later estimation is applicable to study gevrey type smoothness of density functions an integration by parts formula on a totally real submanifold in the complexified space is also studied

a topological problem in polymer physics configurational and mechanical properties of a random walk enclosing a constant are the authors show how a random walk in a plane constrained to enclose a given area can be used to approximately represent the properties of an entangled polymer molecule the statistical mechanical properties of the loop are calculated exactly and the distribution function for the enclosed areas is found for the case of a random walk with free ends joined by a straight line segment the distribution function is given by the cauchy distribution this implies that the area has statistical fractal properties but does not have a mean for a genuinely closed random walk a mean exists but the distribution of areas is not fractal the spatial and mechanical properties of the constrained configurations have also been calculated analytically if the unrestricted coil can be regarded as an entropic spring of zero natural length then the area constrained configurations behave qualitatively like springs with a finite natural length the deformation behaviour also shows both softening and hardening dependent on the area imposed

topology of sobolev mappings ii this paper addresses some topological and analytical issues concern ing sobolev mappings between compact riemannian manifolds among the results we obtained are unified proofs of various generalizations of results obtained in a recent work of brezis and li in particular we solved two conjectures in bl we also give a topological obstruction for the weak sequential density of smooth maps in a given sobolev mapping space finally we show a necessary and suf ficient topological condition under which the smooth maps are strongly dense in the sobolev spaces the earlier result theorem 1 of b2 was shown to be not correct

area versus length distribution for closed random walks using a connection between the q oscillator algebra and the coefficients of the high temperature expansion of the frustrated gaussian spin model we derive an exact formula for the number of closed random walks of given length and area on a hypercubic lattice in the limit of infinite number of dimensions the formula is investigated in detail and asymptotic behaviour is evaluated the area distribution in the limit of long loops is computed as a byproduct we also obtain an infinite set of new nontrivial identities

travelling load response of an axially moving string abstract the classical moving load problem in which a mass traverses an elastic structure is extended to systems in which the guideway also has prescribed axial motion a monocable ropeway such as a tramway or ski lift is an example of such a traveling structure traveling load system and it is modeled herein as an axially moving string that transports an attached discrete mass between two supports the green s function methods for the two constituent systems the particle and the uniform traveling string are combined to derive a volterra integral equation with delay that governs the interaction force in the coupled system the calculated response of the traveling string particle system to arbitrary initial conditions and external excitation is compared to that predicted by the derived asymptotic solution for vanishing discrete mass repeated scattering by the particle of transverse waves in the string dominates the motion transient vibration as the particle is transported between the supports is maximized at a transport speed that can be substantially lower than the critical speed of divergence instability for the uniform traveling string agreement between responses measured in the laboratory and those predicted by the traveling structure traveling load theory supports the analysis

a parametrix for step two hypoelliptic diffusion equations in this paper i construct a parametrix for the hypoelliptic diffusion equations 0 ot l u 0 where l ea 1 9a2 and where the ga are vector fields which satisfy the property that they together with all of the commutators ga gbi for a b are at each point linearly independent and span the tangent space

deformations of annuli with smallest mean distortion we determine the extremal mappings with smallest mean distortion for mappings of annuli as a corollary we find that the nitsche harmonic maps are dirichlet energy minimizers among all homeomorphisms h mathbb a r r to mathbb a r r however outside the nitsche range of the modulus of the annuli within the class of homeomorphisms no such energy minimizers exist in this case we identify the bv limits of minimizers

asymptotic expansion of the one loop approximation of the chern simons integral in an abstract wiener space setting abstract in an abstract wiener space setting we construct a rigorous mathematical model of the one loop approximation of the perturbative chern simons integral and derive its explicit asymptotic expansion for stochastic wilson lines

on stochastic areas and averages of planar brownian motion duplantier 1989 has computed the characteristic function of the stochastic area of planar brownian motion taken with respect to its centre of gravity in this paper it is shown that this result can be deduced from levy s stochastic area formula moreover duplantier s stochastic area is shown to have the same law as the stochastic area of the restriction to the time interval 0 1 of the brownian ring defined on the time interval 0 4 3 the same method also applies to the stochastic area formulae obtained by biane and yor 1986 which are relative to certain planar processes associated with the orthogonal decomposition of brownian motion along the l2 0 1 basis of legendre polynomials

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

stochastic areas winding numbers and hopf fibrations we define and study stochastic areas processes associated with brownian motions on the complex symmetric spaces mathbb cp n and mathbb ch n the characteristic functions of those processes are computed and limit theorems are obtained in the case n 1 we also study windings of the brownian motion on those spaces and compute the limit distributions for mathbb cp n the geometry of the hopf fibration plays a central role whereas for mathbb ch n it is the anti de sitter fibration

levy s stochastic area formula in higher dimensions let wt w t 1 w t 2 w t d d 2 be a d dimensional standand brownian motion and let a t be a bounded measurable function from into the space of dxd skew symmetric matrices and x t into d we are concerned with a class of stochastic processes l t a x a particular example of which is p l vy s stochastic area open image in new window

sobolev homeomorphism that cannot be approximated by diffeomorphisms in w 1 1 we construct a sobolev homeomorphism in dimension n geqq 4 f in w 1 1 0 1 n mathbb r n such that j f rm det df 0 on a set of positive measure and jf 0 on a set of positive measure it follows that there are no diffeomorphisms or piecewise affine homeomorphisms fk such that f k to f in w 1 1 rm loc

convex hull of n planar brownian paths an exact formula for the average number of edges we establish an exact formula for the average number of edges appearing on the boundary of the global convex hull of n independent brownian paths in the plane this requires the introduction of a counting criterion which amounts to cutting off edges that are in a specific sense small the main argument consists in a mapping between planar brownian convex hulls and configurations of constrained independent linear brownian motions this new formula is confirmed by retrieving an existing exact result on the average perimeter of the boundary of brownian convex hulls in the plane

areas of planar brownian curves addresses the problem of the algebraic area enclosed by a brownian curve in two dimensions recently reconsidered by khandekar and wiegel 1988 the author recalls the principal results actually first obtained by paul levy in 1950 another derivation by functional integrals is given for several probability distributions that of the area of a brownian ring the area between an open brownian arc and its chord and finally the area swept with respect to the mean position of the brownian path as a result which seems to be new

la filtration de b l let btbe a one dimensional brownian motion and let ltbe its local time at zero we answer a question of j pitman by showing that the canonical filtrations of b and b cl are equal for all real constants c it is also shown that the filtrations of b and b cl are equal if and only if c 1

a stochastic flow arising in the study of local times a stochastic flow of homeomorphisms of open image in new window previously studied by bass and burdzy 2 and hu and warren 4 is shown to arise in the study of the local times of brownian motion this leads to a new proof of the ray knight theorems for the flow via the classical ray knight theorems for brownian motion

algebraic areas distributions for two dimensional levy flights the author studies curves generated by planar levy flights and calculate the probability distribution of the algebraic area between the curve and the chord he compares his analytical result with computer simulations he also considers closed curves and give the result for the distribution of areas enclosed by those curves

an approximation of small time probability density functions in a general jump diffusion model we propose a method for approximating probability density functions related to multidimensional jump diffusion processes for small time horizons a closed form approximation of the characteristic function is derived based on the ito taylor expansion the probability density function is then approximated numerically by inverting the characteristic function using fast fourier transform as application we consider a general stochastic volatility model which involves time state dependent drift and diffusion functions as well as jump components we test our approach under the heston model and the bates model and show that our method provides accurate approximations

lipschitz regularity for inner variational equations we obtain lipschitz regularity results for a fairly general class of nonlinear first order pdes these equations arise from the inner variation of certain energy integrals even in the simplest model case of the dirichlet energy the inner stationary solutions need not be differ entiable everywhere the lipschitz continuity is the best possible but the proofs even in the dirichlet case turn out to relay on topological arguments the appeal to the inner stationary solutions in this context is motivated by the classical problems of existence and regularity of the energy minimal deformations in the theory of harmonic mappings and certain mathematical models of nonlinear elasticity specifically neo hookian type problems

martingale transforms and lp norm estimates of riesz transforms on complete riemannian manifolds under the condition that the bakry emery ricci curvature is bounded from below we prove a probabilistic representation formula of the riesz transforms associated with a symmetric diffusion operator on a complete riemannian manifold using the burkholder sharp l p inequality for martingale transforms we obtain an explicit and dimension free upper bound of the l p norm of the riesz transforms on such complete riemannian manifolds for all 1 p in the euclidean and the gaussian cases our upper bound is asymptotically sharp when p 1 and when p

existence of energy minimal diffeomorphisms between doubly connected domains the paper establishes the existence of homeomorphisms between two planar domains that minimize the dirichlet energy

monotone sobolev mappings of planar domains and surfaces an approximation theorem of youngs duke math j 15 87 94 1948 asserts that a continuous map between compact oriented topological 2 manifolds surfaces is monotone if and only if it is a uniform limit of homeomorphisms analogous approximation of sobolev mappings is at the very heart of geometric function theory gft and nonlinear elasticity ne in both theories the mappings in question arise naturally as weak limits of energy minimizing sequences of homeomorphisms as a result of this the energy minimal mappings turn out to be monotone in the present paper we show that conversely monotone mappings in the sobolev space fancyscript w 1 p 1 p infty are none other than fancyscript w 1 p weak also strong limits of homeomorphisms in fact these are limits of diffeomorphisms by way of illustration we establish the existence of traction free energy minimal deformations for p harmonic type energy integrals

on polymer conformations in elongational flows we consider various models of polymer conformations using paths of gaussian processes such as brownian motion in each case the calculation of the law of the moment of inertia of a random polymer structure which is equivalent to the calculation of the partition function is reduced to the problem of finding the law of a certain quadratic functional of a gaussian process we present a new method for computing the laplace transforms of these quadratic functionals which exploit their special form via the ray knight theorem and which does not involve the classical method of eigenvalue expansions we apply the method to several simple examples then show how the same techniques can be applied to more complicated cases with the aid of a little excursion theory

global invertibility for orientation preserving sobolev maps via invertibility on or near the boundary by a result of ball proc r soc edinb sect a math 88 315 328 1981 https doi org 10 1017 s030821050002014x a locally orientation preserving sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension as shown here for any dimension the conclusions of ball s theorem and related results can be reached while completely avoiding the problem of homeomorphic extension for suitable domains it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps an application in nonlinear elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed as a tool in the proofs strictly orientation preserving maps and their global invertibility properties are studied from a purely topological point of view

distribution of the area enclosed by a plane random walk the authors derive an exact expression for the distribution function for the algebraic area enclosed by a plane closed random walk using a continuous model from the start their result has a different analytical form but is numerically close to an expression for the same quantity of a discrete model derived by brereton and butler 1987

algebraic areas enclosed by 2d brownian curves in random media we study closed brownian trajectories when the brownian particle is subjected to a random potential for a poissonian repulsive potential the enclosed algebraic area a is a gaussian variable that scales like t3 4 when t the length of the curve goes to infinity this is intermediate between the situations where the particle is allowed to wander i everywhere on the plane a t and ii only on a bounded domain a t1 2 for the lloyd model we show that the probability distribution p a is the same as in the absence of disorder this surprising result is related to some peculiarities of the cauchy law

vibrations of slack cables with discrete masses this paper describes a new method for obtaining the natural frequencies and mode shapes of arbitrarily slack cables which may contain concentrated masses as well as distributed mass and tensile stiffness the method of imaginary reactions is used to perform the static calculation and stodola s method is applied to do the dynamics integral formulas for the required influence functions appropriate to both fixed fixed and fixed forced boundary conditions are included solutions to sample problems obtained with a program implemented on naval research laboratory s advanced scientific computer are presented the efficiency of this method is believed to exceed that of available finite element programs when the required number of integration intervals is large that is when numerous masses are present and or high order modes are required