on center singularity for compressible spherically symmetric nematic liquid crystal flows abstract the paper is concerned with a simplified system proposed by ericksen 6 and leslie 20 modeling the flow of nematic liquid crystals in the first part we give a new serrin s continuation principle for strong solutions of general compressible liquid crystal flows based on new observations we establish a localized serrin s regularity criterion for the 3d compressible spherically symmetric flows it is proved that the classical solution loses its regularity in finite time if and only if either the concentration or vanishing of mass forms or the norm inflammation of gradient of orientation field occurs around the center

incompressible limit of the compressible nematic liquid crystal flow abstract this paper is concerned with the incompressible limit of the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions in r n n 2 3 the local and global existence of strong solutions for the incompressible system with small initial data is rigorously proved via the incompressible limit furthermore the convergence rates are obtained in some sense

a blowup criterion for the compressible nematic liquid crystal flows in dimension two abstract in this paper we establish a blowup criterion for the two dimensional compressible nematic liquid crystal flows the criterion is given in terms of the density and the gradient of direction field where the later satisfies the serrin type blowup criterion for this result we do not need the initial density to be positive

the low mach number limit for the compressible flow of liquid crystals abstract in this paper we are concerned with the compressible flow of liquid crystals based on the convergence stability principle it is shown that for the mach number sufficiently small the cauchy problem of compressible liquid crystal flow has a unique smooth solution on the finite time interval where the incompressible liquid crystal flow exists furthermore it is justified that as the mach number tends to zero the smooth solutions converge rigorously to those of the incompressible equations and the sharp convergence orders are also obtained

a blow up criterion of strong solutions to a viscous liquid gas two phase flow model with vacuum in 3d abstract in this paper we get a blow up criterion for the strong solutions to the 3d viscous liquid gas two phase flow model with vacuum more precisely we prove that the bound of l t 1 l x norm of the deformation tensor of velocity gradient controls the possible breakdown of the strong solutions

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

strong solutions of the navier stokes equations for isentropic compressible fluids we study strong solutions of the isentropic compressible navier stokes equations in a domain r3 we first prove the local existence of unique strong solutions provided that the initial data 0 and u0 satisfy a natural compatibility condition the important point in this paper is that we allow the initial vacuum the initial density may vanish in an open subset of we then prove a new uniqueness result and stability result our results are valid for unbounded domains as well as bounded ones

global classical solutions to the one dimensional free boundary problem for compressible non isothermal liquid crystal flow with large data abstract in this paper we study the free boundary problem of compressible non isothermal liquid crystal flow in dimension one for the vacuum interface condition that is the density is assumed to vanish and connect to vacuum continuously we prove global existence and uniqueness of classical solutions to the liquid crystal system with large initial data

a free boundary problem for compressible hydrodynamic flow of liquid crystals in one dimension abstract in this paper we consider the free boundary problem for a simplified version of ericksen leslie equations modeling the compressible hydrodynamic flow of nematic liquid crystals in dimension one we obtain both existence and uniqueness of global classical solutions provided that the initial density is away from vacuum

existence and partial regularity of static liquid crystal configurations we establish the existence and partial regularity for solutions of some boundary value problems for the static theory of liquid crystals some related problems involving magnetic or electric fields are also discussed

rapid multiplication modulo the sum and difference of highly composite numbers we extend the work of richard crandall et al to demonstrate how the discrete weighted transform dwt can be applied to speed up multiplication modulo any number of the form a b where p abp is small in particular this allows rapid computation modulo numbers of the form k 2n 1 in addition we prove tight bounds on the rounding errors which naturally occur in floating point implementations of fft and dwt multiplications this makes it possible for fft multiplications to be used in situations where correctness is essential for example in computer algebra packages

the twentieth fermat number is composite the twentieth fermat number f20 2220 1 has been proven composite by machine computation the fermat numbers are the numbers fn 224 1 orginally conjectured by fermat to be prime for all n in fact only for n equal to 0 through 4 are they known to be prime and small factors of f9 fl1 fl2 fl5 fl6 have been known for some time as part of a long term test of the hardware reliability of the cray 2 supercomputer at the supercomputing research center the authors proved that f20 2220 1 which had been the smallest fermat number of unknown character is composite the test for compositeness was the standard technique of pepin 5 for n 1 fn is prime if and only if 3 f 2 1 mod fn this test for compositeness does not of course produce factors of the number but was the test used for proving the compositeness of f7 f8 flo f13 f14 2 3 4 6 7 the result of the computation on the cray 2 has been verified by performing the same computation on a cray x mp belonging to cray research the total computation time on the cray 2 was about 10 cpu days the time on the cray xmp was 82 hours both programs ran as single processor programs on any available cpu of the respective machines the ability of either computer to run in parallel on multiple cpu s was not used the time needed to test fn for n in the range 10 through 20 is just slightly more than four times the time needed to test fn the number of multiplications doubles in incrementing n and the time required for each multiplication doubles being dependent almost entirely on the length of the operands our programs would thus determine the character of f22 which is now the smallest fermat number of unknown character in a little more than 16 times the time needed for our computation on f20 the table below summarizes what is now known about the fermat numbers for n less than or equal to 22 a status list for larger n appears in 1 this computation roughly one million squarings modulo a one million bit number would be impossible to do even on supercomputers without fast fourier transform techniques for integer multiplication since one reason for performing this computation was to verify hardware reliablity and not to minimize the execution time the program was written entirely in cray fortran and called cray library functions for the fft s the program itself was very simple and only about 200 lines long much of which was used for checkpointing and restarting the program the program was called into execution every time the cray 2 was restarted and so received april 6 1987 revised june 5 1987 1980 mathematics subject classification 1985 revnsion primary ilyii lla51 11 04 d1988 american mathematical society 0025 5718 88 1 00 25 per page

dynamics of the ericksen leslie equations with general leslie stress ii the compressible isotropic case in this article the non isothermal compressible ericksen leslie system for nematic liquid crystals subject to general leslie stress is considered it is shown that this system is locally well posed within the l q setting and that for initial data close to equilibria points which are identical with the ones for the incompressible situation the solution exists globally moreover any global solution which does not develop singularities converges to an equilibrium in the topology of the natural state manifold note that no structural assumptions on the leslie coefficients are imposed and in particular parodi s relation is not being assumed the results can be viewed as an extension of the studies in hieber and pruss math ann 369 977 996 2017 for the incompressible case to the compressible situation

uniform global solutions of the 3d compressible mhd system in a bounded domain abstract in this paper we consider the 3d compressible mhd system in a bounded domain we first prove a regularity criterion and then use it and the bootstrap argument to show the uniform in global small solutions here is the resistivity coefficient

blowup criterion for viscous baratropic flows with vacuum states we prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth strong solutions for the 3 dimensional 3d barotropic compressible navier stokes equations more precisely if a solution of the 3d barotropic compressible navier stokes equations is initially regular and loses its regularity at some later time then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches our result is the same as ponce s criterion for 3 dimensional incompressible euler equations ponce in commun math phys 98 349 353 1985 in addition initial vacuum states are allowed in our cases

global weak solutions to the equations of compressible flow of nematic liquid crystals in two dimensions we consider weak solutions to a simplified ericksen leslie system of two dimensional compressible flow of nematic liquid crystals an initial boundary value problem is first studied in a bounded domain by developing new techniques and estimates to overcome the difficulties induced by the supercritical nonlinearity nabla mathbf d 2 mathbf d in the equations of angular momentum on the direction field and adapting the standard three level approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions under a restriction imposed on the initial energy including the case of small initial energy then the cauchy problem with large initial data is investigated and we prove the global existence of large weak solutions by using the domain expansion technique and the rigidity theorem provided that the second component of initial data of the direction field satisfies some geometric angle condition

existence of global solutions for 2 dimensional viscous compressible flows abstract we prove the existence of global weak solutions for 2 dimensional unsteady viscous compressible flows of an ideal isothermal gas whatever the size of the data may be this is achieved by a suitable coupling of the method of hopf 17 with standard techniques in orlicz spaces

incompressible limit of the compressible nematic liquid crystal flows in a bounded domain with perfectly conducting boundary abstract in this paper we study the asymptotic behavior of the regular solution to a simplified ericksen leslie model for the compressible nematic liquid crystal flow in a bounded smooth domain in r 2 as the mach number tends to zero the evolution system consists of the compressible navier stokes equations coupled with the transported heat flow for the averaged molecular orientation we suppose that the navier stokes equations are characterized by a navier s slip boundary condition while the transported heat flow is subject to neumann boundary condition by deriving a differential inequality with certain decay property the low mach limit of the solutions is verified for all time provided that the initial data are well prepared

global well posedness and decay of smooth solutions to the non isothermal model for compressible nematic liquid crystals abstract the cauchy problem for the three dimensional non isothermal model for compressible nematic liquid crystals is considered existence of global in time smooth solutions is established provided that the initial datum is close to a steady state 0 d by using the l q l p estimates and the fourier splitting method if the initial perturbation is small in h 3 norm and bounded in l q q 1 6 5 norm we obtain the optimal decay rates for the first and second order spatial derivatives of solutions in addition the third and fourth order spatial derivatives of director field d in l 2 norm are achieved

the twenty second fermat number is composite we have shown by machine proof that f 22 2 222 1 is composite in addition we reenacted young and buell s 1988 resolution of f 20 as composite finding agreement with their final selfridge hurwitz residues we also resolved the character of all extant cofactors of f n n 22 finding no new primes and ruling out prime powers

singularity formation to the two dimensional compressible non isothermal nematic liquid crystal flows in a bounded domain abstract we study the singularity formation of strong solutions to the two dimensional 2d compressible non isothermal nematic liquid crystal flows in a bounded domain under a geometric condition of the initial orientation field we show that the strong solution exists globally if the temporal integral of the maximum norm of the divergence of the velocity is bounded our method relies on critical sobolev inequalities of logarithmic type and delicate energy estimates

on multi dimensional compressible flows of nematic liquid crystals with large initial energy in a bounded domain abstract we study the global existence of weak solutions to a multi dimensional simplified ericksen leslie system for compressible flows of nematic liquid crystals with large initial energy in a bounded domain r n where n 2 or 3 by exploiting a maximum principle nirenberg s interpolation inequality and a smallness condition imposed on the n th component of initial direction field d 0 to overcome the difficulties induced by the supercritical nonlinearity d 2 d in the equations of angular momentum and then adapting a modified three dimensional approximation scheme and the weak convergence arguments for the compressible navier stokes equations we establish the global existence of weak solutions to the initial boundary problem with large initial energy and without any smallness condition on the initial density and velocity

liquid crystal flows in two dimensions this paper is concerned with a simplified hydrodynamic equation proposed by ericksen and leslie modeling the flow of nematic liquid crystals in dimension two we establish both interior and boundary regularity theorems for such a flow under smallness conditions as a consequence we establish the existence of global in time weak solutions on a bounded smooth domain in mathbb r 2 which are smooth everywhere with possible exceptions of finitely many singular times

classical solutions for the compressible liquid crystal flows with nonnegative initial densities abstract in this paper we study the cauchy problem of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in r 3 we first prove the local existence of the strong solutions provided that the data satisfies a natural compatibility condition then by deriving the smoothing effect of the solution in t 0 we conclude that it is indeed classical on some time interval 0 t 0 here we do not need the condition that initial density is bounded below away from zero

weak strong uniqueness property for the compressible flow of liquid crystals abstract weak strong uniqueness property in the class of finite energy weak solutions is established for two different compressible liquid crystal systems by the method of relative entropy to overcome the difficulties caused by the molecular direction with inhomogeneous dirichlet boundary condition new techniques are introduced to build up the relative entropy inequalities

percival variational principle for invariant measures and commensurate incommensurate phase transitions in one dimensional chains we prove for a one dimensional system of classical particles with potential energy u alpha gamma sum limits n left alpha v x n f x n 1 x n gamma right the existence of such a smooth function 0 0 that the system with potential energyu has the equilibrium state at the temperaturet 0 this is the incommensurate phase with the ratio of periods equal to the prescribed irrational number badly approximated by rational ones a simple geometric condition for the invariant curve of the corresponding dynamical system is established under which it is the support of the invariant measure minimizing percival s energy functional

remarks on a paper by j t beale t kato and a majda we prove that the maximum norm of the deformation tensor controls the possible breakdown of smooth solutions for the 3 dimensional euler equations more precisely the loss of regularity in a local smooth solution of the euler equations implies the growth without bound of the deformation tensor as the critical time approaches equivalently if the deformation tensor remains bounded the existence of a smooth solution is guaranteed

global weak solution and blow up criterion of the general ericksen leslie system for nematic liquid crystal flows abstract in this paper we investigate the three dimensional general ericksen leslie e l system with ginzburg landau type approximation modeling nematic liquid crystal flows first by overcoming the difficulties from lack of maximum principle for the director equation and high order nonlinearities for the stress tensor we prove existence of global in time weak solutions under physically meaningful boundary conditions and suitable assumptions on the leslie coefficients which ensures that the total energy of the e l system is dissipated moreover for the e l system with periodic boundary conditions we prove the local well posedness of classical solutions under the so called parodi s relation and establish a blow up criterion in terms of the temporal integral of both the maximum norm of the curl of the velocity field and the maximum norm of the gradient of the liquid crystal director field

the complexity of membership problems for circuits over sets of integers we investigate the complexity of membership problems for circuits computing sets of integers these problems are a natural modification of the membership problems for circuits computing sets of natural numbers studied by mckenzie and wagner the complexity of membership problems for circuits over sets of natural numbers lecture notes in computer science vol 2607 2003 pp 571 582 we show that there are several membership problems for which the complexity in the case of integers differs significantly from the case of the natural numbers testing membership in the subset of integers produced at the output of a circuit is nexptime complete whereas it is pspace complete for the natural numbers as another result evaluating circuits is shown to be p complete for the integers and pspace complete for the natural numbers the latter result extends mckenzie and wagner s work in nontrivial ways furthermore evaluating circuits is shown to be nl l complete and several other cases are resolved

blow up criterion for 3d viscous liquid gas two phase flow model abstract this work deals with the blow up criterion for the strong solution to the three dimensional viscous liquid gas two phase flow model in terms of the l 1 0 t l norm of the gradient of the velocity with two types of boundary conditions the dirichlet boundary condition and the navier slip boundary condition there is no extra restriction on viscosity coefficients the result also applies to the whole space case

the twenty fourth fermat number is composite we have shown by machine proof that f24 2224 1 is composite the rigorous pepin primality test was performed using independently developed programs running simultaneously on two different physically separated processors each program employed a floating point fft based discrete weighted transform dwt to effect multiplication modulo f24 the final respective pepin residues obtained by these two machines were in complete agreement using intermediate residues stored periodically during one of the floating point runs a separate algorithm for pure integer negacyclic convolution verified the result in a wavefront paradigm by running simultaneously on numerous additional machines to effect piecewise verification of a saturating set of deterministic links for the pepin chain we deposited a final pepin residue for possible use by future investigators in the event that a proper factor of f24 should be discovered herein we report the more compact traditional selfridge hurwitz residues for the sake of completeness we also generated a pepin residue for f23 and via the suyama test determined that the known cofactor of this number is composite

global low energy weak solution and large time behavior for the compressible flow of liquid crystals abstract in this paper we consider the weak solution of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in r 3 when the initial data are of small energy and initial density is positive and essentially bounded we prove the existence of a global weak solution in r 3 the large time behavior of a global weak solution is also established

long time behavior of solution for the compressible nematic liquid crystal flows in r 3 abstract in this paper we investigate the global existence and long time behavior of classical solution for the compressible nematic liquid crystal flows in three dimensional whole space first of all the global existence of classical solution is established under the condition that the initial data are close to the constant equilibrium state in h n r 3 n 3 framework then one establishes algebraic time decay for the classical solution by weighted energy method finally the algebraic decay rate of classical solution in l p r 3 norm with 2 p and optimal decay rate of their spatial derivative in l 2 r 3 norm are obtained if the initial perturbation belong to l 1 r 3 additionally

crack initiation in brittle materials in this paper we study the crack initiation in a hyper elastic body governed by a griffith type energy we prove that during a load process through a time dependent boundary datum of the type t tg x and in the absence of strong singularities e g this is the case of homogeneous isotropic materials the crack initiation is brutal that is a big crack appears after a positive time ti 0 conversely in the presence of a point x of strong singularity a crack will depart from x at the initial time of loading and with zero velocity we prove these facts for admissible cracks belonging to the large class of closed one dimensional sets with a finite number of connected components the main tool we employ to address the problem is a local minimality result for the functional epsilon nu gamma int omega f x nabla v rm d x k mathcal h 1 gamma where omega subseteq mathbb r 2 k 0 and f is a suitable caratheodory function we prove that if the uncracked configuration u of relative to a boundary displacement has at most uniformly weak singularities then configurations u with mathcal h 1 gamma small enough are such that epsilon u emptyset epsilon u gamma gamma

uniform well posedness and low mach number limit to the compressible nematic liquid crystal flows in a bounded domain abstract we prove global in time and uniform in of the strong solutions to the 3 d compressible nematic liquid crystal flows in a bounded domain where is the mach number consequently we obtain the strong solution of compressible nematic liquid crystal model that converges to that of incompressible nematic liquid crystal model this is the first result on the low mach number limit for compressible nematic liquid crystal flows in 3 d bounded domain our proof relies on the dedicated estimates on the solutions and the subtle use of the boundary conditions

heat flow of harmonic maps whose gradients belong to l n _ x l infty _ t for any compact n dimensional riemannian manifold m g without boundary a compact riemannian manifold n subset mathbb r k without boundary and 0 t we prove that for n 4 if u m 0 t n is a weak solution to the heat flow of harmonic maps such that nabla u in l n x l infty t m times 0 t then u c m 0 t n as a consequence we show that for n 3 if 0 t is the maximal time interval for the unique smooth solution u c m 0 t n of 1 1 then nabla u t l n m blows up as t t

irregular primes to one million using fast algorithms for power series inversion based on the fast fourier transform and multisectioning of power series we have calculated all irregular primes up to one million including their indices of irregularity and associated irregular pairs using this data we verified that fermat s last theorem and vandiver s conjecture are true for these primes two primes with index of irregularity five were already known we find that there are nine other primes less than one million with index five and that the prime 527377 is the unique prime less than one million with index six a pair of integers p k is said to be an irregular pair if p is a prime k is an even integer satisfying 2 p are known see 8 our calculations show that vandiver s conjecture is true for all p 106 the table of irregular pairs could also be used to calculate iwasawa invariants for the corresponding primes we did not do this as part of our calculations see 9 or 4 for a discussion of this problem previous computations of irregular pairs have used algorithms that take 0 p2 arithmetic operations for each prime p the bernoulli numbers are defined in received by the editor april 30 1991 and in revised form september 10 1991 1991 mathematics subject classification primary 11b68 secondary 1 1d41 65t20 1 1y99 the first author was partially supported by national science foundation grant dms 9012989 d 1992 american mathematical society 0025 5718 92 1 00 25 per page

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

global regularity and uniqueness of weak solution for the 2 d liquid crystal flows abstract we prove the global existence and regularity of weak solution for the 2 d liquid crystal flows with the large initial velocity the uniqueness of weak solution is also proved by using the littlewood paley analysis

strong solutions of the compressible nematic liquid crystal flow abstract we study strong solutions of the simplified ericksen leslie system modeling compressible nematic liquid crystal flows in a domain r 3 we first prove the local existence of a unique strong solution provided that the initial data 0 u 0 d 0 are sufficiently regular and satisfy a natural compatibility condition the initial density function 0 may vanish on an open subset i e an initial vacuum may exist we then prove a criterion for possible breakdown of such a local strong solution at finite time in terms of blow up of the quantities l t l x and d l t 3 l x