on the measure of totally real algebraic integers ii for a certain measure s2 a defined for totally real algebraic integers a 0 1 we find the four smallest values of q a the methods used involve linear programming and the results are verified using interval arithmetic

a generalization of the schur siegel smyth trace problem abstract let be a totally positive algebraic integer and define its absolute trace to be tr deg the trace of divided by the degree of elementary considerations show that the absolute trace is always at least one while it is plausible that for any 0 the absolute trace is at least 2 with only finitely many exceptions this is known as the schur siegel smyth trace problem our aim in this paper is to show that the schur siegel smyth trace problem can be considered as a special case of a more general problem

distribution of algebraic numbers schur studied limits of the arithmetic means an of zeros for polynomials of degree n with integer coefficients and simple zeros in the closed unit disk if the leading coefficients are bounded schur proved that lim supn an 1 e 2 we show that an 0 and estimate the rate of convergence by generalizing the erd s turan theorem on the distribution of zeros as an application we show that integer polynomials have some unexpected restrictions of growth on the unit disk schur also studied problems on means of algebraic numbers on the real line when all conjugate algebraic numbers are positive the problem of finding the sharp lower bound for lim infn an was developed further by siegel and others we provide a solution of this problem for algebraic numbers equidistributed in subsets of the real line potential theoretic methods allow us to consider distribution of algebraic numbers in or near general sets in the complex plane we introduce the generalized mahler measure and use it to characterize asymptotic equidistribution of algebraic numbers in arbitrary compact sets of capacity one the quantitative aspects of this equidistribution are also analyzed in terms of the generalized mahler measure 1 schur s problems on means of algebraic numbers let e be a subset of the complex plane c consider the set of polynomials zn e of the exact degree n with integer coefficients and all zeros in e we denote the subset of zn e with simple zeros by zn e given m 0 we write pn anz zn e m if an m and pn zn e respectively pn zn e m if an m and pn zn e schur 45 4 8 studied the limit behavior of the arithmetic means of zeros for polynomials from zn e m as n where m 0 is an arbitrary fixed number his results may be summarized in the following statements let r 0 where r is the real line theorem a schur 45 satz ix given a polynomial pn z an n k 1 z k n define the arithmetic mean of squares of its zeros by sn n k 1 2 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n sn e 1 6487 1 1 theorem b schur 45 satz xi for a polynomial pn z an n k 1 z k n define the arithmetic mean of its zeros by an n k 1 k n n if pn zn r m is any sequence of polynomials with degrees n then lim inf n an e 1 6487 1 2 1991 mathematics subject classification primary 11c08 secondary 11r04 26c10 30c15

comparison and sub supersolution principles for the fractional p x laplacian abstract in this paper we study the new fractional sobolev space w s q x p x y where q and p are variable exponents and s 0 1 and the related nonlocal operator which is a fractional version of the nonhomogeneous p x laplace operator we first give some further qualitative properties of w s q x p x y we also show the strong comparison principle for the fractional p x laplace operator a sub super solution for the nonlocal equations involving the fractional p x laplacian is established

on the absolute mahler measure of polynomials having all zeros in a sector let be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z we compute the greatest lower bound c of the absolute mahler measure i 1 d max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in particular we show that c 2 1 12933793 from which it follows that any integer 1 and e i 3 all of whose conjugates have positive real part has absolute mahler measure at least c 2 this value is achieved for satisfying 1 0 2 where 0 1 3247 is the smallest pisot number the real root of 0 3 0 1

partitioned time stepping for a parabolic two domain problem there have been many numerical simulations but few analytical results of stability and accuracy of algorithms for computational modeling of fluid fluid and fluid structure interaction problems where two domains corresponding to different fluids ocean atmosphere or a fluid and deformable solid blood flow are separated by an interface as a simplified model of the first examples this report considers two heat equations in omega 1 omega 2 subset mathbb r 2 adjoined by an interface i omega 1 cap omega 2 subset mathbb r the heat equations are coupled by a condition that allows energy to pass back and forth across the interface i while preserving the total global energy of the monolithic coupled problem to compute approximate solutions to the above problem only using subdomain solvers two first order in time fully discrete methods are presented the methods consist of an implicit explicit approach in which the action across i is lagged and a partitioned method based on passing interface values back and forth across i stability and convergence results are derived for both schemes numerical experiments that support the theoretical results are presented

defect deferred correction method for the two domain convection dominated convection diffusion problem abstract we present a method for solving a fluid fluid interaction problem two convection dominated convection diffusion problems adjoined by an interface which is a simplified version of the atmosphere ocean coupling problem the method resolves some of the issues that can be crucial to the fluid fluid interaction problems it is a partitioned time stepping method yet it is of high order accuracy in both space and time the two step algorithm considered in this report provides second order accuracy it allows for the usage of the legacy codes which is a common requirement when resolving flows in complex geometries yet it can be applied to the problems with very small viscosity diffusion coefficients this is achieved by combining the defect correction technique for increased spatial accuracy and for resolving the issue of high convection to diffusion ratio with the deferred correction in time which allows for the usage of the computationally attractive partitioned scheme yet the time accuracy is increased beyond the usual result of partitioned methods being only first order accurate into the defect deferred correction method ddc the results are readily extendable to the higher order accuracy cases by adding more correction steps both the theoretical results and the numerical tests provided demonstrate that the computed solution is unconditionally stable and the accuracy in both space and time is improved after the correction step

on integer chebyshev polynomials we are concerned with the problem of minimizing the supremum norm on o 1 of a nonzero polynomial of degree at most n with integer coefficients we use the structure of such polynomials to derive an efficient algorithm for computing them we give a table of these polynomials for degree up to 75 and use a value from this table to answer an open problem due to p borwein and t erdelyi and improve a lower bound due to flammang et al

on the linear independence measure of logarithms of rational numbers in this paper we give a general theorem on the linear independence measure of logarithms of rational numbers and in particular the linear independence measure of 1 log 2 log 3 log 5 and of 1 log 2 log 3 log 5 log 7 we also give a method to search for polynomials of smallest norm on a real interval a b which may be suitable for computing or improving the linear independence measure of logarithms of rational numbers

phase segregation dynamics in particle systems with long range interactions ii interface motion we study properties of the solutions of a family of second order integrodifferential equations which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics we first establish existence and uniqueness as well as some properties of the instantonic solutions then we concentrate on formal asymptotic sharp interface limits we argue that the obtained interface evolution laws a stefan like problem and the mullins sekerka solidification model coincide with the ones which can be obtained in the analogous limits from the cahn hilliard equation the fourth order pde which is the standard macroscopic model for phase segregation with one conservation law

stabilization of explicit coupling in fluid structure interaction involving fluid incompressibility abstract in this work we propose a stabilized explicit coupling scheme for fluid structure interaction problems involving a viscous incompressible fluid the coupled discrete formulation is based on nitsche s method with a time penalty term giving l 2 control on the fluid pressure variations at the interface for a linear model problem we prove that the scheme is stable in the energy norm irrespectively of the so called added mass effect namely the fluid structure density ratio and the geometry of the domain numerical experiments in the linear and non linear case show that optimal time accuracy can be obtained by performing one defect correction iteration

a new approach to front propagation problems theory and applications in this paper we present a new definition for the global in time propagation motion of fronts hypersurfaces boundaries with a prescribed normal velocity past the first time they develop singularities we show that if this propagation satisfies a geometric maximum principle inclusion avoidance type property then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures this new approach which is more geometric and as it turns out equivalent to the level set method is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction diffusion equations we finally present a number of new asymptotic results among them are the asymptotics of i reaction diffusion equations with rapidly oscillating coefficients ii fully nonlinear nonlocal integral differential equations and iii stochastic ising models with long range anisotropic interactions and general spin flip dynamics

the smallest perron numbers a perron number is a real algebraic integer of degree d 2 whose conjugates are i such that max 2 i d i in this paper we compute the smallest perron numbers of degree d 24 and verify that they all satisfy the lind boyd conjecture moreover the smallest perron numbers of degree 17 and 23 give the smallest house for these degrees the computations use a family of explicit auxiliary functions these functions depend on generalizations of the integer transfinite diameter of some compact sets in c

periodic patterns for a model involving short range and long range interactions abstract we consider a physical model where the total energy is governed by surface tension and repulsive screened coulomb potential on the 3 dimensional space we obtain different periodic equilibrium patterns i e stationary sets for this energy under some volume constraints the patterns bifurcate smoothly from straight lamellae lattices of round solid cylinders and lattices of round balls

the integer chebyshev problem the main problem of this chapter is to find a polynomial in z n of minimal supremum norm on an interval this is p1 and it is of a slightly different flavour than most of the other problems in this book in that there is no restriction on the size of the coefficients we now state p1 with greater precision

fractional laplacian in conformal geometry abstract in this note we study the connection between the fractional laplacian operator that appeared in the recent work of caffarelli and silvestre and a class of conformally covariant operators in conformal geometry

the trace of totally positive algebraic integers for all totally positive algebraic numbers except a finite number of explicit exceptions the following inequality holds 1 d 1 d max 1 780022 1 66 1 where d is the degree of a and 0 1 d its conjugates this improves previous results of smyth flammang and rhin

algebraic integers whose conjugates all lie in an ellipse we find all 15909 algebraic integers whose conjugates all lie in an ellipse with two of them nonreal while the others lie in the real interval 1 2 this problem has applications to finding certain subgroups of sl 2 c we use explicit auxiliary functions related to the generalized integer transfinite diameter of compact subsets of c this gives good bounds for the coefficients of the minimal polynomial of a

convergence of nonlocal threshold dynamics approximations to front propagation in this note we prove that appropriately scaled threshold dynamics type algorithms corresponding to the fractional laplacian of order 0 2 converge to moving fronts when 1 the resulting interface moves by weighted mean curvature while for 1 the normal velocity is nonlocal of fractional type the results easily extend to general nonlocal anisotropic threshold dynamics schemes

front propagation and phase field theory the connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction diffusion equations is rigorously established two different scalings are studied in the first the limiting geometric equation is a first order equation in the second it is a generalization of the mean curvature equation intrinsic definitions for the geometric equations are obtained and uniqueness under a geometric condition on the initial surface is proved in particular in the case of the mean curvature equation this condition is satisfied by surfaces that are strictly starshaped that have positive mean curvature or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions

decoupled time stepping methods for fluid fluid interaction a model of two incompressible newtonian fluids coupled across a common interface is studied the nonlinearity of the coupling condition exacerbates the problem of decoupling the fluid calculations in each subdomain a natural parallelization strategy employed in current climate models a specialized partitioned time stepping method is studied which decouples the discrete fluid equations without sacrificing stability and maintaining convergence this is accomplished through explicit updating of the size of the jump in tangential velocities across the fluid fluid interface by a geometric averaging of this data over the previous two time levels a full numerical analysis is presented and computational tests are performed demonstrating the robustness of this method

minimal polynomials of integral symmetric matrices abstract we investigate what the possible minimal polynomials are for integral symmetric matrices we show that the obvious necessary conditions are sufficient for polynomials of degree at most 4 we show that necessary conditions are sufficient for minimal polynomials of self adjoint operators on positive definite unimodular lattices we also give a relatively elementary proof of the result of estes that any total real algebraic integer is the eigenvalue of an integral symmetric matrix this question was asked by alan hoffman who also showed that this result implies that any totally real algebraic integer is the eigenvalue of the adjacency matrix of some graph

bivariate factorizations connecting dickson polynomials and galois theory in his ph d thesis of 1897 dickson introduced certain permutation polynomials whose galois groups are essentially the dihedral groups these are now called dickson polynomials of the first kind to distinguish them from their variations introduced by schur in 1923 which are now called dickson polynomials of the second kind in the last few decades there have been extensive investigations of both of these types which are related to the classical chebyshev polynomials we give new bivariate factorizations involving both types of dickson polynomials these factorizations demonstrate certain isomorphisms between dihedral groups and orthogonal groups and lead to the construction of explicit equations with orthogonal groups as galois groups

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

threshold growth dynamics we study the asymptotic shape of the occupied region for monotone deterministic dynamics in d dimensional euclidean space parametrized by a threshold 0 and a borel set n r d with positive and finite lebesgue measure if a n denotes the oocupied set of the dynamics at integer time n then a n 1 is obtained by adjoining any point x for which the volume of overlap between x n and a n exceeds except in some degenerate cases we prove that n 1 a n converges to a unique limiting shape l starting from any bounded initial region a 0 that is suitably large moreover l is computed as the polar transform for 1 w where w is an explicit width function that depends on n and

trace of totally positive algebraic integers and integer transfinite diameter the diameter of a totally real algebraic integer of degree d with conjugates 1 2 d is diam d 1 for all positive integers k and n diam 2 cos 2k n is less than 4 r m robinson has computed modulo integer translations all the other totally real algebraic integers with diam 4 for d 8 we have done the computations for all d 15 we use a large family of explicit auxiliary functions related to generalized integer transfinite diameter of real intervals they give good bounds for the coefficients of the minimal polynomial of for d 15 we prove a recent conjecture of capparelli et al mathematics subject classification 2000 primary 11c08 11y40

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

polynomials with small mahler measure we describe several searches for polynomials with integer coefficients and small mahler measure we describe the algorithm used to test mahler measures we determine all polynomials with degree at most 24 and mahler measure less than 1 3 test all reciprocal and antireciprocal polynomials with height 1 and degree at most 40 and check certain sparse polynomials with height 1 and degree as large as 181 we find a new limit point of mahler measures near 1 309 four new salem numbers less than 1 3 and many new polynomials with small mahler measure none has measure smaller than that of lehmer s degree 10 polynomial

integer transfinite diameter and polynomials with small mahler measure in this work we show how suitable generalizations of the integer transfinite diameter of some compact sets in c give very good bounds for coefficients of polynomials with small mahler measure by this way we give the list of all monic irreducible primitive polynomials of z x of degree at most 36 with mahler measure less than 1 324 and of degree 38 and 40 with mahler measure less than 1 31

a trace bound for positive definite connected integer symmetric matrices abstract let a be a connected integer symmetric matrix i e a a i j m n z for some n a a t and the underlying graph vertices corresponding to rows with vertex i joined to vertex j if a i j 0 is connected we show that if all the eigenvalues of a are strictly positive then tr a 2 n 1 there are two striking corollaries first the analogue of the schur siegel smyth trace problem is solved for characteristic polynomials of connected integer symmetric matrices second we find new examples of totally real separable irreducible monic integer polynomials that are not minimal polynomials of integer symmetric matrices

a second order decoupled penalty projection method based on deferred correction for mhd in elsasser variable abstract we study the deferred correction method for the magnetohydrodynamics mhd system written in elsasser variables the proposed algorithm is based on the penalty projection with grad div stabilized taylor hood solutions of the elsasser formulation in this way second order accuracy of the method in time is obtained through the deferred correction method with excellent mass conservation properties for the proposed method stability is rigorously proven and numerical experiments are presented to verify the proposed scheme and theory

gevrey regularity for integro differential operators abstract we prove for some singular kernels k x y that viscosity solutions of the integro differential equation r n u x y u x y 2 u x k x y d y f x locally belong to some gevrey class if so does f the fractional laplacian equation is included in this framework as a special case

a variational model for dislocations in the line tension limit we study the interaction of a singularly perturbed multiwell energy with an anisotropic nonlocal regularizing term of h1 2 type and a pinning condition this functional arises in a phase field model for dislocations which was recently proposed by koslowski cuitino and ortiz but it is also of broader mathematical interest in the context of the dislocation model we identify the limit of the energy in all scaling regimes for the number n of obstacles the most interesting regime is n ln where is a nondimensional length scale related to the size of the crystal lattice in this case the limiting model is of line tension type one important feature of our model is that the set of energy wells is periodic and hence not compact thus a key ingredient in the proof is a compactness estimate up to a single translation for finite energy sequences which generalizes earlier results from alberti bouchitte and seppecher for the two well problem with a h1 2 regularization

generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and glauber dynamics we study themacroscopic limit of an appropriately rescaledstochastic ising model withlong range interactions evolving withglauber dynamics as well as the correspondingmean field equation which is nonlinear and nonlocal in the limit we obtain an interface evolving with normal velocity k wherek isthe mean curvature and thetransport coefficient is identified by aneffective green kubo type formula the above assertions are valid for all positive times the motion of the interface being interpreted in theviscosity sense after the onset of the geometric singularities

a defect correction method for the incompressible navier stokes equations a defect correction method for the incompressible navier stokes equation with a high reynolds number is considered in the defect step the artificial viscosity parameter s is added to the reynolds number as a stability factor and the residual is taken care of in the correction step h 1 and l 2 error estimations are derived for the one step defect correction method and the results of some numerical experiments are presented these results show that for the driven cavity two defect correction steps antidiffuse the artificial viscosity approximation nearly optimally this combination gives on a very coarse mesh results indistinguishable from a benchmark very fine mesh calculation

on the absolute mahler measure of polynomials having all zeros in a sector ii kklet be an algebraic integer of degree d not 0 or a root of unity all of whose conjugates i are confined to a sector arg z in the paper on the absolute mahler measure of polynomials having all zeros in a sector g rhin and c smyth compute the greatest lower bound c of the absolute mahler measure d i 1 max 1 i 1 d of for belonging to nine subintervals of 0 2 3 in this paper we improve the result to thirteen subintervals of 0 and extend some existing subintervals

a simple proof of convergence for an approximation scheme for computing motions by mean curvature we prove the convergence of an approximation scheme recently proposed by bence merriman and osher for computing motions of hypersurfaces by mean curvature our proof is based on viscosity solutions methods

nonlocal curvature flows this paper aims at building a unified framework to deal with a wide class of local and nonlocal translation invariant geometric flows we introduce a class of nonlocal generalized mean curvatures and prove the existence and uniqueness for the level set formulation of the corresponding geometric flows we then introduce a class of generalized perimeters whose first variation is an admissible generalized curvature within this class we implement a minimizing movements scheme and we prove that it approximates the viscosity solution of the corresponding level set pde we also describe several examples and applications besides recovering and presenting in a unified way existence uniqueness and approximation results for several geometric motions already studied and scattered in the literature the theory developed in this paper also allows us to establish new results

on the bourgain brezis and mironescu theorem concerning limiting embeddings of fractional sobolev spaces the article is concerned with the bourgain brezis and mironescu theorem on the asymptotic behaviour of the norm of the sobolev type embedding operator ws p lpn n sp as s 1 and s n p thei

boundary behavior of nonlocal minimal surfaces abstract we consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary by a series of detailed examples we show that nonlocal minimal surfaces may stick at the boundary of the domain even when the domain is smooth and convex this is a purely nonlocal phenomenon and it is in sharp contrast with the boundary properties of the classical minimal surfaces in particular we show stickiness phenomena to half balls when the datum outside the ball is a small half ring and to the side of a two dimensional box when the oscillation between the datum on the right and on the left is large enough when the fractional parameter is small the sticking effects may become more and more evident moreover we show that lines in the plane are unstable at the boundary namely small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary by a quantity that is proportional to a power of the perturbation in all the examples we present concrete estimates on the stickiness phenomena also we construct a family of compactly supported barriers which can have independent interest

regularity properties of nonlocal minimal surfaces via limiting arguments abstract we prove an improvement of flatness result for nonlocal minimal surfaces which is independent of the fractional parameter s when s 1 as a consequence we obtain that all the nonlocal minimal cones are flat and that all the nonlocal minimal surfaces are smooth when the dimension of the ambient space is less or equal than 7 and s is close to 1

on the smallest value of the maximal modulus of an algebraic integer the house of an algebraic integer of degree d is the largest modulus of its conjugates for d 1 of degree d say m d as a consequence we improve matveev s theorem on the lower bound of m d we show that in this range the conjecture of schinzelzassenhaus is satisfied the minimal polynomial of any algebraic integer whose house is equal to m d is a factor of a bi tri or quadrinomial the computations use a family of explicit auxiliary functions these functions depend on generalizations of the integer transfinite diameter of some compact sets in c they give better bounds than the classical ones for the coefficients of the minimal polynomial of an algebraic integer whose house is small

computing totally positive algebraic integers of small trace we construct minimal polynomials of totally positive algebraic integers of small absolute trace by consideration of their reductions modulo auxiliary polynomials many new examples of such polynomials of minimal absolute trace for given degree are found the computations are pushed to degrees that previously were unattainable and one consequence is that the new examples form the majority of all those known as an application we produce a new bound for the schur siegel smyth trace problem

finite element discretizations of nonlocal minimal graphs convergence abstract in this paper we propose and analyze a finite element discretization for the computation of fractional minimal graphs of order s 0 1 2 on a bounded domain such a plateau problem of order s can be reinterpreted as a dirichlet problem for a nonlocal nonlinear degenerate operator of order s 1 2 we prove that our numerical scheme converges in w 1 2 r for all r s where w 1 2 s is closely related to the natural energy space moreover we introduce a geometric notion of error that for any pair of h 1 functions in the limit s 1 2 recovers a weighted l 2 discrepancy between the normal vectors to their graphs we derive error bounds with respect to this novel geometric quantity as well in spite of performing approximations with continuous piecewise linear lagrangian finite elements the so called stickiness phenomenon becomes apparent in the numerical experiments we present

nonlocal minimal graphs in the plane are generically sticky we prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities more precisely we show that if a nonlocal minimal graph in a slab is continuous up to the boundary then arbitrarily small perturbations of the far away data produce boundary discontinuities hence either a nonlocal minimal graph is discontinuous at the boundary or a small perturbation of the prescribed conditions produces boundary discontinuities the proof relies on a sliding method combined with a fine boundary regularity analysis based on a discontinuity smoothness alternative namely we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is holder continuous up to the boundary in this spirit we prove that the boundary regularity of nonlocal minimal graphs in the plane jumps from discontinuous to c 1 gamma with no intermediate possibilities allowed in particular we deduce that the nonlocal curvature equation is always satisfied up to the boundary as an interesting byproduct of our analysis one obtains a detailed understanding of the switch between the regime of continuous and hence differentiable nonlocal minimal graphs to that of discontinuous and hence with differentiable inverse ones

approximation schemes for propagation of fronts with nonlocal velocities and neumann boundary conditions the convergence of schemes for propagation of fronts in a bounded domain moving with normal velocities is studied the velocities considered depend on the principal curvatures the normal direction the location as well as some nonlocal properties of the front most of the schemes considered are in essence threshold dynamics type approximation schemes modified for neumann boundary conditions and nonlocal terms the existence and uniqueness of appropriately defined viscosity solutions of the level set equations describing the nonlocal motions is also shown

high order multi implicit spectral deferred correction methods for problems of reactive flow models for reacting flow are typically based on advection diffusion reaction a d r partial differential equations many practical cases correspond to situations where the relevant time scales associated with each of the three sub processes can be widely different leading to disparate time step requirements for robust and accurate time integration in particular interesting regimes in combustion correspond to systems in which diffusion and reaction are much faster processes than advection the numerical strategy introduced in this paper is a general procedure to account for this time scale disparity the proposed methods are high order multi implicit generalizations of spectral deferred correction methods misdc methods constructed for the temporal integration of a d r equations spectral deferred correction methods compute a high order approximation to the solution of a differential equation by using a simple low order numerical method to solve a series of correction equations each of which increases the order of accuracy of the approximation the key feature of misdc methods is their flexibility in handling several sub processes implicitly but independently while avoiding the splitting errors present in traditional operator splitting methods and also allowing for different time steps for each process the stability accuracy and efficiency of misdc methods are first analyzed using a linear model problem and the results are compared to semi implicit spectral deferred correction methods furthermore numerical tests on simplified reacting flows demonstrate the expected convergence rates for misdc methods of orders three four and five the gain in efficiency by independently controlling the sub process time steps is illustrated for nonlinear problems where reaction and diffusion are much stiffer than advection although the paper focuses on this specific time scales ordering the generalization to any ordering combination is straightforward

on higher dimensional singularities for the fractional yamabe problem a nonlocal mazzeo pacard program we consider the problem of constructing solutions to the fractional yamabe problem which are singular at a given smooth submanifold for which we establish the classical gluing method of mazzeo and pacard j differential geom 1996 for the scalar curvature in the fractional setting this proof is based on the analysis of the model linearized operator which amounts to the study of a fractional order ordinary differential equation ode thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of nonlocal odes note however that no traditional phase plane analysis is available here instead we first provide a rigorous construction of radial fast decaying solutions by a blowup argument and a bifurcation method then second we use conformal geometry to rewrite this nonlocal ode giving a hint of what a nonlocal phase plane analysis should be third for the linear theory we use complex analysis and some non euclidean harmonic analysis to examine a fractional schrodinger equation with a hardy type critical potential we construct its green s function deduce fredholm properties and analyze its asymptotics at the singular points in the spirit of frobenius method surprisingly enough a fractional linear ode may still have a 2 dimensional kernel as in the second order case