bilinear restriction estimates for surfaces with curvatures of different signs recently the sharp l 2 bilinear adjoint restriction estimates for the cone and the paraboloid were established by wolff and tao respectively their results rely on the fact that for the cone and the paraboloid the nonzero principal curvatures have the same sign we generalize those bilinear restriction estimates to surfaces with curvatures of different signs

restriction theorems for a surface with negative curvature we prove a bilinear restriction theorem for a surface of negative curvature this is the analogue of the results of t wolff 19 and t tao 14 15 for cones and paraboloids as a consequence we obtain an almost sharp linear restriction theorem

mean exit times for particles driven by weakly colored noise consider a particle whose velocity is an ornstein uhlenbeck process with correlation time varepsilon 2 frac dx dt frac sigma z varepsilon qquad frac dz dt frac z varepsilon 2 frac sqrt 2 xi t varepsilon where xi t is gaussian white noise this represents brownian motion with memory i e a non markovian brownian motion singular perturbation techniques reduce the mean exit time problem to a boundary layer problem which turns out to be the half range expansion problem for the fokker planck operator this is solved via complex variable theory it is found that the spatial density has a milne extrapolation length varepsilon sigma zeta tfrac 1 2 varepsilon sigma 1 46035 cdots where zeta is the riemann zeta function it is also found that the average time at which x t first escapes an interval a 0 b ven that it starts at x 0 0 and z 0 z 0 is e t text ex frac 1 2 sigma 2 left a varepsilo

asymptotic solution of the wang uhlenbeck recurrence time problem a langevin particle is initiated at the origin with positive velocity its trajectory is terminated when it returns to the origin in 1945 wang and uhlenbeck posed the problem of finding the joint probability density function pdf of the recurrence time and velocity naming it the recurrence time problem we show that the short time asymptotics of the recurrence pdf is similar to that of the integrated brownian motion solved in 1963 by mckean we recover the long time t ensuremath 3 2 decay of the first arrival pdf of diffusion by solving asymptotically an appropriate variant of mckean s integral equation

improved restriction estimate for hyperbolic surfaces in r3 abstract recently l guth improved the restriction estimate for the surfaces with strictly positive gaussian curvature in r 3 in this paper we extend his restriction estimate to the quadratic surfaces with strictly negative gaussian curvature in r 3

the analytic solutions of some boundary layer problems in the theory of brownian motion the authors give analytic solutions of the generalised albedo and milne problems for the uhlenbeck ornstein process and discuss the relation between these and the original wang uhlenbeck 1945 problem of determining the distribution of first passage times

the density profile for the klein kramers equation near an absorbing wall we derive asymptotic series for the expansion coefficients of a function in terms of the pagani functions which occur in the boundary layer solutions of the klein kramers equation the results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall with an accuracy of about 2 we also obtain information about the analytic behavior of the density profile this profile increases near the wall with the square root of the distance to the wall finally the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz this is used to obtain a better variational estimate for the density at the wall

the analytic structure of the stationary kinetic boundary layer for brownian particles near an absorbing wall a method for solving kinetic boundary layer problems for the 1d klein kramers equation the kinetic equation for brownian particles was proposed a few years ago by marshall and watson 1987 for the special case of particles moving under the influence of a constant or zero external force in this paper the authors apply this method to a number of classical stationary boundary layer problems for completely or partially absorbing walls these are the milne problem in which a current flows towards the wall from infinity and the albedo problem in which particles are injected into the system at the wall with a prescribed velocity distribution the solutions are known to be non analytic at the wall for zero normal velocity the authors pay particular attention to the nature of this singularity for several special cases the results are compared with results obtained by approximate methods

the convex hull for a random acceleration process in two dimensions we compute exactly the mean perimeter l t and the mean area a t of the convex hull of a random acceleration process of duration t in two dimensions we use an exact mapping that relates via cauchy s formulae the computation of the perimeter and the area of the convex hull of an arbitrary two dimensional stochastic process x t y t to the computation of the extreme value statistics of the associated one dimensional component process x t the latter can be computed exactly for the one dimensional random acceleration process even though the process is non markovian physically our results are relevant in describing the average shape of a semi flexible ideal polymer chain in two dimensions

solution to the linear bgk equation via the principles of invariance the solution to the linearized bgk equation of kinetic theory is obtained via the principles of invariance as formulated by chandrasekhar the solution represents an alternative to the singular eigenfunction method featuring regular integral equations which are more easily evaluated laplace transforms are obtained for the net flow velocity and velocity distribution for both the albedo and milne problems with wall reflection the transforms are then numerically inverted to a high degree of accuracy results are compared to those of kainz and titulaer

exact solution of the reaction diffusion problem for a particle generating band on a surface by riemann hilbert matching the existence of a class of exact solutions for diffusion reaction problems is demonstrated for the example of a band the method may be applied to any particle source which can be analytically mapped to a band or a disc situated on an infinite plane a simple asymptotic expression for the experimentally measurable collector efficiency current current response function has been found

semiflexible polymer in the half plane and statistics of the integral of a brownian curve a continuum model of a polymer with non zero bending energy fluctuating without overhangs in the half plane is considered the exact partition function is obtained from the marshall watson solution of the klein kramers equation for brownian motion in the half space the partition function contains information on probabilities associated with the integral of a brownian curve and reproduces sinai s t 5 4 result for the asymptotic first passage time density the t 5 2 dependence of a different passage probability implies a first order polymer adsorption transition for short range pinning potentials

symmetry properties of higher numerical ranges abstract let a be a linear operator on a finite dimensional unitary space v of dimension n the k th higher numerical range of a denoted by w k a is the totality of complex numbers tr pap where p runs over all k dimensional orthogonal projections on v we show that w k a is a polygon with the real axis as a line of symmetry k 1 n if and only if a is normal with a real characteristic polynomial we also construct several nonnormal examples to investigate the extent to which the symmetry of all of the w k a is required

the growth of small droplets from a gas mixture results and estimates for very small radii abstract to test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture we develop a solution method especially suitable for radii smaller or not much larger than a mean free path the method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere for the simplest kinetic equation a variant of the linear bgk equation and for radii below two mean free paths this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent for radii around two mean free paths the new results agree with those of a variant of the moment method for lower radii discrepancies of up to a few percent are found our new method does not work for the klein kramers equation a kinetic equation for brownian particles for the latter equation we obtain an estimate for the current at very low radii from the free space solution by uhlenbeck and ornstein its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the bgk case

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

first passage times for the uhlenbeck ornstein process a numerical solution is obtained for the laplace transformed backward kramers equation from which the mean first passage time may be obtained the main difficulties are associated with a the parabolic nature of the time development operator and b the existence of a double structure in the solution near the absorbing barrier both of these difficulties are resolved by computational methods derived from boundary layer theory the reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field the authors also present the results for a harmonic force field for which no analytic solution is yet known

the kinetic boundary layer for the klein kramers equation a new numerical approach we explore a numerical technique for determining the structure of the kinetic boundary layer of the klein kramers equation for noninteracting brownian particles in a fluid near a wall that absorbs the brownian particles the equation is of interest in the theory of diffusion controlled reactions and of the coagulation of colloidal suspensions by numerical simulation of the langevin equation equivalent to the klein kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities the data can be used to construct the solutions of the standard problems at an absorbing wall the milne and the albedo problem we confirm and extend earlier results by burschka and titulaer obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion we briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field

half range completeness for the fokker planck equation with an external force this paper deals with a generalization of a classical result obtained by r beals and v protopopescu for the fokker planck equations to the case in which a constant external force is present

on the erdős distinct distances problem in the plane in this paper we prove that a set of n points in r 2 has at least c n log n distinct distances thus obtaining the sharp exponent in a problem of erd os we follow the setup of elekes and sharir which in the spirit of the erlangen program allows us to study the problem in the group of rigid motions of the plane this converts the problem to one of point line incidences in space we introduce two new ideas in our proof in order to control points where many lines are incident we create a cell decomposition using the polynomial ham sandwich theorem this creates a dichotomy either most of the points are in the interiors of the cells in which case we immediately get sharp results or alternatively the points lie on the walls of the cells in which case they are in the zero set of a polynomial of suprisingly low degree and we may apply the algebraic method in order to control points incident to only two lines we use the ecnode polynomial of the rev george salmon to conclude that most of the lines lie on a ruled surface then we use the geometry of ruled surfaces to complete the proof

the structure of the stationary kinetic boundary layer for the linear bgk equation recently very accurate numerical methods for solving kinetic boundary layer problems for linear kinetic equations have been developed to test such methods detailed information about exactly solvable models can be very helpful to this purpose the authors present results for the stationary 1d linear bgk equation obtained by the singular eigenfunction method that extend results already known in the literature both in scope and in precision special attention is paid to the nature of the singularity in the distribution function for small velocities near the boundary this singularity cannot be reproduced exactly by the numerical methods available explicit expressions are presented for the milne problem and for the albedo problem with input particles of a single velocity they compare the results with those of a recently developed variant of the two stream moment method they find excellent agreement except close to the singularity for many quantities of interest the accuracy obtainable by the novel two stream moment method cannot be reproduced with comparable numerical effort by evaluation of the exact solution

analytic solution for a boundary value problem in the theory of brownian motion a closed form formula for the solution of the first boundary value problem for the kolmogorov equation is found by constructing explicitly its green function

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

the endpoint case of the bennett carbery tao multilinear kakeya conjecture we prove the endpoint case of the multilinear kakeya conjecture of bennett carbery and tao the proof uses the polynomial method introduced by dvir

conditions for the generalized numerical range to be real abstract necessary and sufficient conditions are given for the c numerical range of a matrix a to be a subset of the real axis in particular it is shown that both a and c must be translates of hermitian matrices

birkhoff s theorem and multidimensional numerical range abstract we show that under certain conditions birkhoff s theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections using this result we study the relation between the spectrum of a self adjoint operator a and its multidimensional numerical range it turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator a every collection of eigenvalues which can be obtained by the rayleigh ritz formula generates an extreme point of the multidimensional numerical range however it may also have other extreme points

an accurate two stream moment method for kinetic boundary layer problems of linear kinetic equations the authors develop a method for solving 1d kinetic boundary layer problems for linear equations that uses separate hermite moment expansions in the velocity variable for particles moving towards and away from a plane wall this so called two stream method is tested for two especially simple kinetic equations the linear bgk equation and the klein kramers equation the kinetic equation for brownian particles for both of these equations extensive exact information is available for two simple boundary layer problems the milne and the albedo problem a comparison of these exact results with those obtained with this version of the two stream moment method shows that the accuracy obtainable by this method exceeds that of earlier methods by several orders of magnitude in particular the method allows them to obtain accurate results not only for the moments of the distribution function but also for the distribution function itself

on the wang uhlenbeck problem in discrete velocity space the arguably simplest model for dynamics in phase space is the one where the velocity can jump between only two discrete values v with rate constant k for this model which is the continuous space version of a persistent random walk analytic expressions are found for the first passage time distributions to the origin since the evolution equation of this model can be regarded as the two state finite difference approximation in velocity space of the kramers klein equation this work constitutes a solution of the simplest version of the wang uhlenbeck problem formal solution in laplace space of generalizations where the velocity can assume an arbitrary number of discrete states that mimic the maxwell distribution is also provided

a drop of ink falls from my pen it comes to earth i know not when the authors obtain for a brownian particle in a uniform force field the mean and asymptotic first passage times as functions of the particle s initial position and velocity with the recurrence times given as a special case they discuss the region of phase space for which the diffusion model of brownian motion provides an adequate approximation and conclude that there is no possibility of obtaining the recurrence times within that model they find that the nature of the boundary value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space because the time development operator is then parabolic rather than elliptic they argue that such a change in the treatment of brownian motion places it within the sphere of transport theory rather than diffusion theory and that consequently results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport

half range expansion analysis for langewin dynamics in the high friction limit with a singular absorbing boundary condition noncharacteristic case we consider the dynamics of a brownian particle given by the langevin equation in a strip under the effects of a deterministic force the trajectories of particles originate at a source whose spatial location in the phase space coincides with the location of adsorbing boundaries this leads to singular behavior of trajectories in the high friction limit we use the half range expansion technique and systematic asymptotics to solve a boundary value problem for the fokker planck operator and to calculate the steady state transition probability density the mean time to absorption and the distribution of exit points we do not make assumptions about other parameters in the problem except that they areo 1 relative to the friction coefficient we calculate explicitly the correct location of the milne type extrapolation for absorbing boundary conditions for the smoluchowski approximation to the langevin equation

conjugate dynamical systems classical analogue of the quantum energy translation an aspect of quantum mechanics that has not been fully understood is the energy shift generated by the time operator in this study we introduce the use of the eigensurfaces of dynamical variables and commutators in classical mechanics to study the classical analogue of the quantum translation of energy we determine that there is a conjugate dynamical system that is conjugate to hamilton s equations of motion and then we generate the analogue of the time operator and use it in the translation of points along the energy direction i e the classical analogue of the pauli theorem the theory is illustrated with a nonlinear oscillator model

a numerical approach to study the kramers equation for finite geometries boundary conditions and potential fields the kramers equation for the phase space function which models the dynamics of an underdamped brownian particle is the subject of our study numerical solutions of this equation for natural boundaries unconfined geometries have been well reported in the literature but not much has been done on the kramers equation for finite confining geometries which require a set of additional constraints imposed on the phase space function at physical boundaries in this paper we present numerical solutions for the kramers equation with a variety of potential fields namely constant linear harmonic and periodic in the presence of fully absorbing and fully reflecting boundary conditions bcs the choice of the numerical method and its implementation take into consideration the type of bcs in order to avoid the use of ghost points or artificial conditions we study and assess the conditions under which the numerical method converges various aspects of the solutions for the phase space function are presented with figures and discussed in detail