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

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

concentration of haar measures with an application to random matrices abstract we show that the mixing times of random walks on compact groups can be used to obtain concentration inequalities for the respective haar measures as an application we derive a concentration inequality for the empirical distribution of eigenvalues of sums of random hermitian matrices with possible applications in free probability the advantage over existing techniques is that the new method can deal with functions that are non lipschitz or even discontinuous with respect to the usual metrics

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

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

two semi online scheduling problems on two uniform machines this paper considers two semi online scheduling problems one with known optimal value and the other with known total sum on two uniform machines with a machine speed ratio of s 1 for the first problem we provide an optimal algorithm for s 1 3 2 1 21 4 and improved algorithms or and lower bounds for s 1 21 4 3 over which the optimal algorithm is unknown as a result the largest gap between the competitive ratio and the lower bound decreases to 0 02192 for the second problem we also present algorithms and lower bounds for s 1 the largest gap between the competitive ratio and the lower bound is 0 01762 and the length of the interval over which the optimal algorithm is unknown is 0 47382 our algorithms and lower bounds for these two problems provide insights into their differences which are unusual from the viewpoint of the known results on these two semi online scheduling problems in the literature

spectral rigidity for addition of random matrices at the regular edge abstract we consider the sum of two large hermitian matrices a and b with a haar unitary conjugation bringing them into a general relative position we prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of a and b as the dimension of the matrix increases this implies optimal rigidity of the eigenvalues and optimal rate of convergence in voiculescu s theorem our previous works 4 5 established these results in the bulk spectrum the current paper completely settles the problem at the spectral edges provided they have the typical square root behavior the key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge our results also hold if the haar unitary matrix is replaced by the haar orthogonal matrix

the kinetic boundary layer for the linearized boltzmann equation around an absorbing sphere we construct approximate solutions of the linearized boltzmann equation for a gas outside of a completely absorbing sphere a simple model for a liquid droplet growing in a supersaturated vapor the solutions are linear combinations of two chapman enskog type solutions which carry heat and particle currents and boundary layer eigenfunctions that decay with increasing distance from the sphere on a distance of the order of a mean free path to construct the boundary layer eigenfunctions and the linear combination that satisfies the boundary condition at the sphere we expand the solution in burnett functions and truncate the resulting system of equations for the expansion coefficients for one particular truncation prescription which generalizes grad s 13 moment scheme good initial convergence with increasing order of truncation is obtained for both moderately small and large radii of the sphere the results for small radii extrapolate smoothly toward the known limit of zero radius we present results for the reaction rate the particle current arriving at the sphere divided by the density at infinity and for the density and temperature profiles in the boundary layer the explicit calculations are carried out for maxwell molecules but the method appears to be suitable for more general intermolecular potentials

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

concentration and convergence rates for spectral measures of random matrices the topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest including in particular matrices drawn from haar measure on the classical lie groups random compressions of random hermitian matrices and the so called random sum of two independent random matrices in each case we estimate the expected wasserstein distance from the empirical spectral measure to a deterministic reference measure and prove a concentration result for that distance as a consequence we obtain almost sure convergence of the empirical spectral measures in all cases

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

online bin stretching with bunch techniques we are given a sequence of items that can be packed into m unit size bins and the goal is to assign these items online to m bins while minimizing the stretching factor bins have infinite capacities and the stretching factor is the size of the largest bin we present an algorithm with stretching factor 26 17 1 5294 improving the best known algorithm by kellerer and kotov 2013 1 with a stretching factor 11 7 1 5714 our algorithm has 2 stages and uses bunch techniques we aggregate bins into batches sharing a common purpose

on the law of addition of random matrices normalized eigenvalue counting measure of the sum of two hermitian or real symmetric matrices a n and b n rotated independently with respect to each other by the random unitary or orthogonal haar distributed matrix u n i e a n u n b n u n is studied in the limit of large matrix order n convergence in probability to a limiting nonrandom measure is established a functional equation for the stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of a n and b n is obtained and studied

on line bin stretching we are given a sequence of items that can be packed into m unit size bins in the classical bin packing problem we fix the size of the bins and try to pack the items in the minimum number of such bins in contrast in the bin stretching problem we fix the number of bins and try to pack the items while stretching the size of the bins as least as possible we present two on line algorithms for the bin stretching problem that guarantee a stretching factor of 5 3 for any number m of bins we then combine the two algorithms and design an algorithm whose stretching factor is 1 625 for any m the analysis for the performance of this algorithm is tight the best lower bound for any algorithm is 4 3 for any m 2 we note that the bin stretching problem is also equivalent to the classical scheduling load balancing problem in which the value of the makespan maximum load is known in advance copyright 2001 elsevier science b v

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

local stability of the free additive convolution abstract we prove that the system of subordination equations defining the free additive convolution of two probability measures is stable away from the edges of the support and blow up singularities by showing that the recent smoothness condition of kargin is always satisfied as an application we consider the local spectral statistics of the random matrix ensemble a u b u where u is a haar distributed random unitary or orthogonal matrix and a and b are deterministic matrices in the bulk regime we prove that the empirical spectral distribution of a u b u concentrates around the free additive convolution of the spectral distributions of a and b on scales down to n 2 3

convergence rate for spectral distribution of addition of random matrices abstract let a and b be two n by n deterministic hermitian matrices and let u be an n by n haar distributed unitary matrix it is well known that the spectral distribution of the sum h a u b u converges weakly to the free additive convolution of the spectral distributions of a and b as n tends to infinity we establish the optimal convergence rate 1 n in the bulk of the spectrum

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

local law of addition of random matrices on optimal scale the eigenvalue distribution of the sum of two large hermitian matrices when one of them is conjugated by a haar distributed unitary matrix is asymptotically given by the free convolution of their spectral distributions we prove that this convergence also holds locally in the bulk of the spectrum down to the optimal scales larger than the eigenvalue spacing the corresponding eigenvectors are fully delocalized similar results hold for the sum of two real symmetric matrices when one is conjugated by haar orthogonal matrix

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

the lebesgue decomposition of the free additive convolution of two probability distributions we prove that the free additive convolution of two borel probability measures supported on the real line can have a component that is singular continuous with respect to the lebesgue measure on mathbb r only if one of the two measures is a point mass the density of the absolutely continuous part with respect to the lebesgue measure is shown to be analytic wherever positive and finite the atoms of the free additive convolution of borel probability measures on the real line have been described by bercovici and voiculescu in a previous paper

local spectral statistics of the addition of random matrices we consider the local statistics of h v x v u y u where v and u are independent haar distributed unitary matrices and x and y are deterministic real diagonal matrices in the bulk we prove that the gap statistics and correlation functions coincide with the gue in the limit when the matrix size n rightarrow infty under mild assumptions on x and certain rigidity assumptions on y the latter being an assumption on the convergence of the eigenvalues of y to the quantiles of its limiting spectral measure which we assume to have a density our method relies on running a carefully chosen diffusion on the unitary group and comparing the resulting eigenvalue process to dyson brownian motion our method also applies to the case when v and u are drawn from the orthogonal group our proof relies on the local law for h proved in bao et al commun math phys 349 3 947 990 2017 j funct anal 271 3 672 719 2016 adv math 319 251 291 2017 as well as the dbm convergence results of landon and yau commun math phys 355 3 949 1000 2017 and landon et al adv math 346 1137 1332 2019

a concentration inequality and a local law for the sum of two random matrices let h n a n u n b n u n ast where an and bn are two n by n hermitian matrices and un is a haar distributed random unitary matrix and let mu h n mu a n mu b n be empirical measures of eigenvalues of matrices hn an and bn respectively then it is known see pastur and vasilchuk in commun math phys 214 249 286 2000 that for large n the measure mu h n is close to the free convolution of measures mu a n and mu b n where the free convolution is a non linear operation on probability measures the large deviations of the cumulative distribution function of mu h n from its expectation have been studied by chatterjee j funct anal 245 379 389 2007 in this paper we improve chatterjee s concentration inequality and show that it holds with the rate which is quadratic in n in addition we prove a local law for eigenvalues of h n n by showing that the normalized number of eigenvalues in an interval approaches the density of the free convolution of a and b provided that the interval has width log n 1 2