optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions if stiff inclusions are closely located then the stress which is the gradient of the solution may become arbitrarily large as the distance between two inclusions tends to zero in this paper we investigate the asymptotic behavior of the stress concentration factor which is the normalized magnitude of the stress concentration as the distance between two inclusions tends to zero for that purpose we show that the gradient of the solution to the case when two inclusions are touching decays exponentially fast near the touching point we also prove a similar result when two inclusions are closely located and there is no potential difference on boundaries of two inclusions we then use these facts to show that the stress concentration factor converges to a certain integral of the solution to the touching case as the distance between two inclusions tends to zero we then present an efficient way to compute this integral

courbes algebriques et equations multiplicatives we study the intersection of an algebraic curve c lying in a multiplicative torus over bar mathbb q with the union of all algebraic subgroups of codimension 2 finiteness of this set has already been proved by bombieri masser and zannier under the assumption that c is not contained in a translate of a proper subtorus following this result the question of the minimal hypothesis implying finiteness has been raised by these authors giving rise to the conjecture finiteness holds precisely when c is not contained in a proper subgroup we prove here this statement which is also a special case of more general conjectures stated independently by zilber and pink our proof takes its inspiration from an article by remond and viada concerning the zilber pink conjecture for curves lying in a power of an elliptic curve hence it relies on a uniform version of the vojta inequality proven via the generalized vojta inequality of remond the main task is to establish a lower bound for some intersection numbers here on a whole family of surfaces obtained by blowing up a compactification of c c

stability of a mach configuration we prove the stability of a mach configuration which occurs in shock reflection off an obstacle or shock interaction in compressible flow the compressible flow is described by a full steady euler system of gas dynamics the unperturbed mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity among four regions divided by these four lines in the neighborhood of the intersection two are supersonic regions and other two are subsonic regions we prove that if the constant states in the supersonic regions are slightly perturbed then the structure of the whole configuration holds while the other two shock fronts and the slip line as well as the flow field in the subsonic regions are also slightly perturbed such a conclusion asserts the existence and stability of the general mach configuration in shock theory in order to prove the result we reduce the problem to a free boundary value problem where two unknown shock fronts are free boundaries while the slip line is transformed to a fixed line by a lagrange transformation in the region where the solution is to be determined we have to deal with an elliptic hyperbolic composed system by decoupling this system and combining the technique for both hyperbolic equations and elliptic equations we establish the required estimates which are crucial in the proof of the existence of a solution to the free boundary value problem 2005 wiley periodicals inc

torsion points on families of squares of elliptic curves in a recent paper we proved that there are at most finitely many complex numbers 0 1 such that the points 2 sqrt 2 2 lambda and 3 sqrt 6 3 lambda are both torsion on the elliptic curve defined by y 2 x x 1 x here we give a generalization to any two points with coordinates algebraic over the field q and even over c this implies a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes

gradient estimates for solutions to the conductivity problem in this paper we derive very precise gradient estimates for solutions to the conductivity problem in the case where two circular conductivity inclusions are very close but not touching the novelty of these estimates is that they give very specific information about the blow up of the gradient as the conductivities of the inclusions degenerate

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

lp gradient estimate for elliptic equations with high contrast conductivities in rn abstract uniform estimate for the solutions of elliptic equations with high contrast conductivities in r n is concerned the space domain consists of a periodic connected sub region and a periodic disconnected matrix block subset the elliptic equations have fast diffusion in the connected sub region and slow diffusion in the disconnected subset suppose 0 1 is the diameter of each matrix block and 2 0 1 is the conductivity ratio of the disconnected matrix block subset to the connected sub region it is proved that the w 1 p norm of the elliptic solutions in the connected sub region is bounded uniformly in when the l p norm of the elliptic solutions in the whole space is bounded uniformly in the w 1 p norm of the elliptic solutions in perforated domains is bounded uniformly in however the l p norm of the second order derivatives of the solutions in the connected sub region may not be bounded uniformly in

an optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions abstract we consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions when inclusions with an extreme conductivity insulators or perfect conductors are closely located the gradient can be concentrated in between inclusions and then becomes arbitrarily large as the distance between inclusions approaches zero the gradient estimate in between insulators in three dimensions has been regarded as a challenging problem while the optimal blow up rates in terms of the distance were successfully obtained for the other extreme conductivity problems in two and three dimensions and are attained on the shortest line segment between inclusions in this paper we establish upper and lower bounds of gradients on the shortest line segment between two insulating unit spheres in three dimensions these bounds present the optimal blow up rate of gradient on the line segment which is substantially different from the rates in the other problems

stable determination of an inhomogeneous inclusion by local boundary measurements we deal with the problem of determining an inclusion within an electrostatic conductor from boundary measurements under mild a priori assumptions we establish optimal stability estimates measurements are performed on a portion of the boundary of the conductor

holder estimate for non uniform parabolic equations in highly heterogeneous media abstract uniform bound for the solutions of non uniform parabolic equations in highly heterogeneous media is concerned the media considered are periodic and they consist of a connected high permeability sub region and a disconnected matrix block subset with low permeability parabolic equations with diffusion depending on the permeability of the media have fast diffusion in the high permeability sub region and slow diffusion in the low permeability subset and they form non uniform parabolic equations each medium is associated with a positive number denoting the size ratio of matrix blocks to the whole domain of the medium let the permeability ratio of the matrix block subset to the connected high permeability sub region be of the order 2 for 0 1 it is proved that the holder norm of the solutions of the above non uniform parabolic equations in the connected high permeability sub region are bounded uniformly in one example also shows that the holder norm of the solutions in the disconnected subset may not be bounded uniformly in

height uniformity for integral points on elliptic curves we recall the result of d abramovich and its generalization by p pacelli on the uniformity for stably integral points on elliptic curves it says that the lang vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves in this paper we will investigate its analogue for their heights under the assumption of the vojta conjecture basically we will show that the vojta conjecture gives a naturally expected simple uniformity for their heights

averaging of dilute random media a rigorous proof of the clausius mossotti formula we consider a large number of randomly dispersed spherical identical inclusions in a bounded domain with conductivity different than that of the host medium in the dilute limit with some mild assumptions on the first few marginal probability densities no periodicity or stationarity is assumed we prove convergence in the h1 norm of the expectation of the solution of the steady state heat equation to the solution of an effective medium problem where the conductivity is given by the clausius mossotti formula error estimates are provided as well

elliptic and parabolic equations in fractured media abstract the elliptic and the parabolic equations with dirichlet boundary conditions in fractured media are considered the fractured media consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected subset to the connected sub region it is proved that the w 1 p norm of the elliptic and the parabolic solutions in the high permeability sub region are bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in for the elliptic and the parabolic equations in periodic perforated domains it is also shown that the w 1 p norm of their solutions are bounded uniformly in

non uniform elliptic equations in convex lipschitz domains abstract non uniform elliptic equations in convex lipschitz domains are concerned the non smooth domains consist of a periodic connected high permeability sub region and a periodic disconnected matrix block subset with low permeability let 0 1 denote the size ratio of the matrix blocks to the whole domain and let 2 0 1 denote the permeability ratio of the disconnected matrix block subset to the connected sub region the w 1 p norm for p 1 of the elliptic solutions in the high permeability sub region is shown to be bounded uniformly in however the w 1 p norm of the solutions in the low permeability subset may not be bounded uniformly in roughly speaking if the sources in the low permeability subset are small enough the solutions in that subset are bounded uniformly in otherwise the solutions cannot be bounded uniformly in relations between the sources and the variation of the solutions in the low permeability subset are also presented in this work

interior regularity estimates in high conductivity homogenization and application in this paper uniform pointwise regularity estimates for the solutions of conductivity equations are obtained in a unit conductivity medium reinforced by an e periodic lattice of highly conducting thin rods the estimates are derived only at a distance e1 for some 0 away from the fibres this distance constraint is rather sharp since the gradients of the solutions are shown to be unbounded locally in lp as soon as p 2 one key ingredient is the derivation in dimension two of regularity estimates to the solutions of the equations deduced from a fourier series expansion with respect to the fibres direction and weighted by the high contrast conductivity the dependence on powers of e of these two dimensional estimates is shown to be sharp the initial motivation for this work comes from imaging and enhanced resolution phenomena observed experimentally in the presence of micro structures lerosey et al science 315 1120 1124 2007 we use these regularity estimates to characterize the signature of low volume fraction heterogeneities in the fibred reinforced medium assuming that the heterogeneities stay at a distance e1 away from the fibres

on the spectrum of the poincare variational problem for two close to touching inclusions in 2d we study the spectrum of the poincare variational problem for two close to touching inclusions in r2 we derive the asymptotics of its eigenvalues as the distance between the inclusions tends to zero

an infinite family of elliptic curves over q with large rank via neron s method in his famous article n1 n6ron has given a method for constructing an infinite family of elliptic curves over q with rank at least 11 by studying this from the viewpoint of mordell weil lattices is 1 s 2 s 3 we find that n6ron s method gives a complete algorithm for such a construction in particular we can write down an explicit numerical example n6ron s idea is a very ingeneous beautiful combination of some deep results in algebraic geometry and number theory the former is based on the theory of del pezzo surfaces cf m dp and the latter is the specialization argument originated by n6ron which has been strengthened by silverman and tate cf sil t another ingredient implicit in in1 seems to be the idea leading to the kodaira n6ron model an elliptic surface of an elliptic curve over a function field which was to be developed later in n2 our contribution if any will be

root numbers and ranks in positive characteristic abstract for a global field k and an elliptic curve e over k t silverman s specialization theorem implies rank e k t rank e t k for all but finitely many t p 1 k if this inequality is strict for all but finitely many t the elliptic curve e is said to have elevated rank all known examples of elevated rank for k q rest on the parity conjecture for elliptic curves over q and the examples are all isotrivial some additional standard conjectures over q imply that there does not exist a non isotrivial elliptic curve over q t with elevated rank in positive characteristic an analogue of one of these additional conjectures is false inspired by this for the rational function field k u over any finite field with characteristic 2 we construct an explicit 2 parameter family e c d of non isotrivial elliptic curves over k t depending on arbitrary c d such that under the parity conjecture each e c d has elevated rank

the difference between the weil height and the canonical height on elliptic curves estimates for the difference of the weil height and the canonical height of points on elliptic curves are used for many purposes both theoretical and computational in this note we give an explicit estimate for this difference in terms of the j invariant and discriminant of the elliptic curve the method of proof suggested by serge lang is to use the decomposition of the canonical height into a sum of local heights we illustrate one use for our estimate by computing generators for the mordell weil group in three examples let e be an elliptic curve defined over a number field k say given by a weierstrass equation 1 y2 x ax b with a and b in the ring of integers of k the canonical height on e is a quadratic form h e k r for the definition and basic properties of h see 10 chapter viii 9 or 6 chapter vi the canonical height is determined by this property together with the fact that the difference

hilbert s tenth problem for function fields of varieties over number fields and p adic fields abstract let k be a subfield of a p adic field of odd residue characteristic and let l be the function field of a variety of dimension n 1 over k then hilbert s tenth problem for l is undecidable in particular hilbert s tenth problem for function fields of varieties over number fields of dimension 1 is undecidable

three region inequalities for the second order elliptic equation with discontinuous coefficients and size estimate abstract in this paper we would like to derive a quantitative uniqueness estimate the three region inequality for the second order elliptic equation with jump discontinuous coefficients the derivation of the inequality relies on the carleman estimate proved in our previous work 5 we then apply the three region inequality to study the size estimate problem with one boundary measurement

elliptic equations with singular bmo coefficients in reifenberg domains abstract w 1 p estimate for the solutions of elliptic equations whose coefficient matrix can have large jump along the boundary of subdomains is obtained the principal coefficients are supposed to be in the john nirenberg space with small bmo seminorms and the domain and subdomains are reifenberg flat domains moreover it has been shown that the estimates are uniform with respect to the distance between the subdomains

characterization of electric fields between two spherical perfect conductors with general radii in 3d abstract in composite materials the inclusions are frequently spaced very closely the electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large in this paper we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions an explicit blowup factor relying on radii is obtained which also involves the digamma function and euler mascheroni constant and so the role of inclusions radii played in such blowup analysis is identified

strong influence of a small fiber on shear stress in fiber reinforced composites abstract in stiff fiber reinforced composites it has been known that the shear stress increases at the rate of 1 as the distance between adjacent fibers approaches 0 this paper reveals a strong influence of a combination of a triple fiber as well as the distance between a pair of fibers on the blow up so that the stress concentration can be significantly accelerated by adding a small fiber in between fibers specifically if a fiber f 2 with a small diameter is located in between fibers f 1 and f 3 1 dist f 1 f 2 and 2 dist f 2 f 3 then the stress blows up at the exact rates of 1 1 and 1 2 between f 1 and f 2 and between f 2 and f 3 respectively this estimate still holds even when a part of f 2 overlaps with f 3 the magnification factor 1 yields the enormous increase in the stress that greatly surpasses the expectancy by previous methods

optimization algorithm for reconstructing interface changes of a conductivity inclusion from modal measurements we propose an original optimization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenvectors associated with the transmission problem for the laplacian let be a smooth domain and d be an inclusion contained in whose boundary is also assumed to be smooth shape deformation of d causes a perturbation of modal parameters the aim of this work is to show how this information can be used to reconstruct the unknown deformation for doing so we rigorously derive an asymptotic formula for the perturbations in the eigenvalues of the transmission problem for the laplacian that are due to small deformations of the interface of an inclusion based on this formula we design an efficient reconstruction algorithm from modal measurements our algorithm consists on minimizing a functional whose minimizer yields certain geometric properties of the unknown inclusion it naturally follows from a key identity that is in some sense dual to the asymptotic formula numerical experiments showing the viability of our algorithm are presented our results in this paper extend those established in the context of small volume inclusions as well as those for the conductivity interface problem in fact on one hand in a series of recent papers 5 3 1 2 we have derived high order asymptotic expansions of the eigenvalue perturbations due to the presence of small inclusions and used them for locating the inclusions and identifying some of their geometric features on the other hand in 4 we have derived highorder terms in the asymptotic expansions of the boundary perturbations of steady state voltage potentials resulting from small perturbations of the shape of a conductivity inclusion based on these derivations we have designed an effective algorithm to determine some geometric features of the shape perturbation of the inclusion based on boundary measurements

asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities abstract we consider the conductivity problem in the presence of adjacent circular inclusions with constant conductivities when two inclusions get closer and their conductivities degenerate to zero or infinity the gradient of the solution can be arbitrary large in this paper we derive an asymptotic formula of the solution which characterizes the gradient blow up of the solution in terms of conductivities of inclusions as well as the distance between inclusions the asymptotic formula is expressed in bipolar coordinates in terms of the lerch transcendent function and it is valid for inclusions with arbitrary constant conductivities we illustrate our results with numerical calculations

fair petri nets and structural induction for rings of processes abstract we present a structural induction theorem for rings consisting of an arbitrary number of identical components the components of a ring are modeled using a fair petri net in which the firing of a prespecified set of transitions is assumed to occur fairly i e any of these transitions that becomes firable infinitely often must fire infinitely often specifically we introduce the concept of similarity between rings of different sizes and give a condition under which the similarity between the rings of sizes two and three guarantees the similarity among the rings of all sizes so if the given condition is satisfied then the correctness of a ring of any large size can be inferred from the correctness of a ring having only a few components the usefulness of the theorem is demonstrated using the examples of token passing mutual exclusion and a simple producer consumer system

smoothness of the beurling transform in lipschitz domains let c be a lipschitz domain and consider the beurling transform of b z lime 0 1 w z w e1 z w 2dm w let 1 1 in this paper we show that if the outward unit normal n on belongs to the besov space bp p 1 p then b is in the sobolev space w p this result is sharp further together with recent results by cruz mateu and orobitg this implies that the beurling transform is bounded in w p if n belongs to bp p 1 p assuming that p 2

an asymptotic formula for the voltage potential in a perturbed e periodic composite medium containing misplaced inclusions of size e we consider composite media made of homogeneous inclusions with c1 boundaries our goal is to compare the potential ue in a perfectly periodic composite with the potential ue d of a perturbed periodic medium where the periodicity defects consist of misplaced inclusions we give an asymptotic expansion of the difference ue d ue away from the defects and show that to first order a misplaced inclusion manifests itself via a polarization tensor which is characterized

special points on fibered powers of elliptic surfaces consider a fibered power of an elliptic surface we characterize its subvarieties that contain a zariski dense set of points that are torsion points in fibers with complex multiplication this result can be viewed as a mix of the manin mumford and andre oort conjecture and is related to a conjecture of pink the main technical tool is a new height inequality we also use it to give another proof of a case of gubler s result on the bogomolov conjecture over function fields

multiscale homogenization with bounded ratios and anomalous slow diffusion we show that the effective diffusivity matrix d v n for the heat operator t 1 2 rv n r in a periodic potential v n d p nd0 uk x rk obtained as a superposition of hlder continuous periodic potentials uk of period t d vd r d z d d 2 n uk 0 d 0 decays exponentially fast with the number of scales when the scale ratios rkc1 rk are bounded above and below from this we de duce the anomalous slow behavior for a brownian motion in a potential obtained as a superposition of an innite number of scales dyt d d t rv 1 yt dt

gradient estimates for parabolic and elliptic systems from linear laminates we establish several gradient estimates for second order divergence type parabolic and elliptic systems the coefficients and data are assumed to be holder or dini continuous in the time variable and all but one spatial variable this type of system arises from the problems of linearly elastic laminates and composite materials for the proof we use campanato s approach in a novel way non divergence type equations under similar conditions are also discussed

gradient estimates for solutions of the lame system with partially infinite coefficients we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

boundary contact problems for domains with conical singularities abstract we study boundary contact problems for elliptic equations and systems with interfaces that have conical singularities such problems represent continuous operators between weighted sobolev spaces and subspaces with asymptotics ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities we show regularity and asymptotics of solutions in weighted spaces and we construct parametrices the result will be illustrated by a number of explicit examples

x automata on ω words abstract for any storage type x the languages accepted by x automata are investigated six accepting conditions including those introduced by landweber are compared for x automata the inclusions between the corresponding six families of languages are essentially the same as for finite state automata apart from unrestricted automata also real time and deterministic automata are considered the main tools for this investigation are 1 a characterization of the languages accepted by x automata in terms of inverse x transductions of finite state languages and 2 the existence of topological upper bounds on some of the families of accepted languages independent of the storage type x

gradient estimates for the perfect conductivity problem we establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect stiff conductors are closely spaced inside an open bounded domain and away from the boundary these results give the optimal blow up rates of the stress for conductors with arbitrary shape and in all dimensions

multiscale asymptotic method for steklov eigenvalue equations in composite media in this paper we consider the multiscale analysis of a steklov eigenvalue equation with rapidly oscillating coefficients arising from the modeling of a composite media with a periodic microstructure there are mainly two new results in the present paper first we obtain the convergence rate with varepsilon 1 2 for the multiscale asymptotic expansions of the eigenvalues and the eigenfunctions of the steklov eigenvalue problem second the boundary layer solution is defined numerical simulations are then carried out to validate the above theoretical results

existence of simple jacobian varieties of genus g with rank at least 4g 5 we prove the existence of infinitely many curves of genus g over the rational number field which have absolutely simple jacobian varieties with mordell weil rank at least 4 g 5

decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions abstract when two inclusions get closer and their conductivities degenerate to zero or infinity the gradient of the solution to the conductivity equation blows up in general in this paper we show that the solution to the conductivity equation can be decomposed into two parts in an explicit form one of them has a bounded gradient and the gradient of the other part blows up using the decomposition we derive the best possible estimates for the blow up of the gradient we then consider the case when the inclusions have positive permittivities we show quantitatively that in this case the size of the blow up is reduced

a priori estimate for non uniform elliptic equations abstract a priori estimate for non uniform elliptic equations with periodic boundary conditions is concerned the domain considered consists of two sub regions a connected high permeability region and a disconnected matrix block region with low permeability let denote the size ratio of one matrix block to the whole domain it is shown that in the connected high permeability sub region the holder and the lipschitz estimates of the non uniform elliptic solutions are bounded uniformly in but holder gradient estimate and l p estimate of the second order derivatives of the solutions in general are not bounded uniformly in

effective boundary conditions resulting from anisotropic and optimally aligned coatings the two dimensional case we discuss a thermally conducting body insulated by a thin anisotropically conducting coating the coating is optimally aligned in the sense that the normal vector inside the coating is always an eigenvector of the thermal tensor we study the effects of the coating by investigating the limiting behavior of solutions u of the heat equation with either dirichlet or neumann boundary conditions imposed on the outer boundary of the coating as the thickness of the coating shrinks to zero in the two dimensional case we find the complete list of effective boundary conditions satisfied by the limit of u on the boundary of the uncoated body this list contains not only the usual dirichlet neumann and robin boundary conditions but also some new and even nonlocal ones involving the dirichlet to neumann mapping and the hilbert transform on the circle we also prove that u converges to its limit in various norms that include the l 2 the sobolev and the holder ones during the course of this study we establish a schauder theory for the regularity of weak solutions of general second order parabolic equations near an interface where the transmission condition is satisfied

torsion anomalous points and families of elliptic curves we prove that there are at most finitely many complex lambda neq 0 1 such that two points on the legendre elliptic curve y 2 x x 1 x lambda with coordinates x 2 3 both have finite order this is a very special case of some conjectures on unlikely intersections in semiabelian schemes

gradient estimates for solutions of the lame system with partially infinite coefficients in dimensions greater than two abstract we establish upper bounds on the blow up rate of the gradients of solutions of the lame system with partially infinite coefficients in dimensions greater than two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero

gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients in this paper we derive global w1 and piecewise c1 estimates for solutions to divergence form elliptic equations with piecewise holder continuous coefficients the novelty of these estimates is that even though they depend on the shape and on the size of the surfaces of discontinuity of the coefficients they are independent of the distance between these surfaces

spectral analysis of the neumann poincare operator and characterization of the stress concentration in anti plane elasticity when holes or hard elastic inclusions are closely located stress which is the gradient of the solution to the anti plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero it is important to precisely characterize the blow up of the gradient of such an equation in this paper we show that the blow up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1 2 of a neumann poincare type operator defined on the boundaries of the inclusions by comparing the singular function with the one corresponding to two disks osculating to the inclusions we quantitatively characterize the blow up of the gradient in terms of explicit functions in electrostatics our results apply to the electric field which is the gradient of the solution to the conductivity equation in the case where perfectly conducting or insulating inclusions are closely located

optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross sections abstract we consider high stresses in stiff fiber reinforced materials which increase rapidly as fibers approximate to one another this paper presents the optimal blow up rate of the stresses with respect to the distance between a pair of stiff fibers in r 3 the blow up result plays an important role in our understanding of low strengths of fiber reinforced composites referring to a problem of anti plane shear the stresses can be interpreted as the electric fields outside closely spaced perfect conductors in r 2 under the action of applied electric field h it has been shown by ammari kang et al that in the particular case of circular inclusions the electric field blows up at the optimal rate 1 2 as 0 where is the distance between conductors recently yun has extended the blow up result to pairs of conductors associated with a large class of shapes whose complements can be transformed conformally to the outside of a circle with c 2 mapping however it presented a suboptimal result that only for a special uniform field h 1 0 the electric fields blow up at the exact rate 1 2 in this paper an upper bound with the rate 1 2 of electric field for any harmonic function h is established this yields the optimal blow up rate 1 2 for the inclusions in the same class of shapes as yun

elliptic equations with measurable nonlinearities in nonsmooth domains abstract we study elliptic equations with measurable nonlinearities in nonsmooth domains we establish an optimal global w 1 p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small bmo semi norm in the other variables while the underlying domain is sufficiently flat in the reifenberg sense that the boundary of the domain is locally trapped between two narrow strips

reconstruction of a penetrable obstacle by complex spherical waves we consider an inverse acoustic scattering problem for identifying a non convex penetrable obstacle in three dimensions in a homogeneous medium we apply the complex geometrical optics solutions with logarithmic phase which is called complex spherical waves to reconstruction problem the reconstruction schemes will be demonstrated in the last section

mach configuration in pseudo stationary compressible flow when a shock front in unsteady compressible flow is moving along an inclined ramp the incident shock will be reflected by the ramp as it moves forward in the process of reflection various reflected wave pattern may occur depending on the slope of the ramp and the strength of the incident shock 14 23 this problem is one of the most important multidimensional prototype problems in gas dynamics generally if the angle of inclination is larger than a critical value then the reflected shock starting from the intersection of the incident shock with the ramp forms a smooth bubble which expands when time goes on such a reflection is called regular reflection on the other hand if the angle of inclination is small then the intersection of the incident shock and the reflected shock will not touch the ramp and an additional shock front called mach stem appears which connects the ramp with the intersection of the incident shock and the reflected shock in this case three shock fronts meet at one point and form a triple shock configuration an important fact is that the structure solely containing three shock fronts separating the neighborhood of the triple intersection into three zones with different continuous states does not exist 14 27 many physical experiments indicate that there is a slip line issuing from the triple intersection such a local wave pattern is called mach configuration or mach structure correspondingly the whole process of reflection is called mach reflection because of the complexity of the whole picture of mach reflection it is natural and necessary to start work on getting a clear understanding of the local mach configuration mach reflection also occurs in various other cases for compressible flow for instance in the study on oblique shock reflection in stationary compressible flow von neumann indicated that see 14 25 if the incident shock hit a wall with an angle larger than a critical value then the reflection must be mach reflection many physical experiments approximate analysis and numerical computations verified the appearance of mach configuration and a tremendous amount of literature is devoted to study such problems see 3 4 11 19 20 24 and the references therein however for the rigorous proof on the stability of mach configuration we can only mention a related result on stationary flow in 13

infinite behaviour of petri nets abstract the infinite behaviour of a petri net is the set of infinite labelled or unlabelled firing sequences for the well known notion of i acceptance for i 1 1 2 2 3 two hierarchies are established and their interrelation is studied

torsion points on families of products of elliptic curves abstract in a recent paper we proved a special case of a variant of pink s conjecture for a variety inside a semiabelian scheme namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non simple surfaces this involves some entirely new and crucial issues

torsion points pell s equation and integration in elementary terms the main results of this paper involve general algebraic differentials omega on a general pencil of algebraic curves we show how to determine if omega is integrable in elementary terms for infinitely many members of the pencil in particular this corrects an assertion of james davenport from 1981 and provides the first proof even in rather strengthened form we also indicate analogies with work of andre and hrushovski and with the grothendieck katz conjecture to reach this goal we first provide proofs of independent results which extend conclusions of relative manin mumford type allied to the zilber pink conjectures we characterize torsion points lying on a general curve in a general abelian scheme of arbitrary relative dimension at least 2 in turn we present yet another application of the latter results to a rather general pencil of pell equations a 2 db 2 1 over a polynomial ring we determine whether the pell equation with squarefree d is solvable for infinitely many members of the pencil

boundary contact problems for domains with edge singularities abstract we study boundary contact problems for elliptic equations and systems with interfaces that have edge singularities such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics ellipticity is formulated in terms of a principal symbolic hierarchy containing interior transmission and edge symbols we construct parametrices show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary contact problems for the laplacian with jumping coefficients

electric field concentration in the presence of an inclusion with eccentric core shell geometry in this paper we analyze the gradient blow up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core shell geometry assuming that the core and shell have circular boundaries that are nearly touching we derive an asymptotic formula for the solution in terms of the single and double layer potentials with image line charges we also deduce an integral formula with image line charges for the problem relating to two nearly touching separated conductors

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

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

a note on divisible points of curves let c be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3 we partially answer a question posed by levin on points on c for which a non trivial power lies again on c our results have connections to zilber s conjecture on intersections with tori and yield to methods arising in transcendence theory and the theory of o minimal structures

stress concentration for closely located inclusions in nonlinear perfect conductivity problems abstract we study the stress concentration which is the gradient of the solution when two smooth inclusions are closely located in a possibly anisotropic medium r n n 2 the governing equation may be degenerate of p laplace type with 1 p n we prove optimal l estimates for the blow up of the gradient of the solution as the distance between the inclusions tends to zero

convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients we develop the convergence analysis of a numerical scheme for approximating the solution of the elliptic problem l epsilon u epsilon frac partial partial x i a ij x epsilon frac partial partial x j u epsilon f mbox in omega u epsilon 0 mbox on partial omega where a y a ij y is a periodic symmetric positive definite matrix and omega 0 1 2 the major goal of the numerical scheme is to capture the epsilon scale of the oscillations of the solution u epsilon on a mesh size h epsilon mbox or h epsilon the numerical scheme is based on asymptotic expansions constructive boundary corrector and finite element approximations new a priori error estimates are established for the asymptotic expansions and for the constructive boundary correctors under weak assumptions on the regularity of the problem these estimates permit to establish sharp finite element error estimates and to consider composite materials applications depending on the regularity of the problem we establish for the numerical scheme a priori error estimates of o h 2 epsilon 3 2 epsilon h on the l 2 norm and o h epsilon 1 hat delta for the broken h 1 norm where hat delta in frac 1 4 0

a multiscale algorithm for radiative heat transfer equation with rapidly oscillating coefficients this paper as a continued work of huang and cao 2014 discusses the multiscale computation of the radiative heat transfer in composite materials or porous media a novel multiscale asymptotic expansion is presented and an explicit rate of convergence is derived we develop a multiscale algorithm for solving this kind of problem a fully implicit scheme is carefully studied and an iterative algorithm is given the convergence of the iterative algorithm is proved by the fixed point method numerical results confirm the efficiency and accuracy of this approach and show that the novel multiscale asymptotic expansion is essential for the radiative dominated cases

neron models of jacobians over base schemes of dimension greater than 1 we investigate to what extent the theory of neron models of jacobians and of abel jacobi maps extends to relative curves over base schemes of dimension greater than 1 we give a necessary and sufficient criterion for the existence of a neron model we use this to show that in general neron models do not exist even after making a modification or even alteration of the base on the other hand we show that neron models do exist outside some codimension 2 locus

optimal estimates for the conductivity problem by green s function method we study a class of second order elliptic equations of divergence form with discontinuous coefficients and data which models the conductivity problem in composite materials we establish optimal gradient estimates by showing the explicit dependence of the elliptic coefficients and the distance between interfacial boundaries of inclusions these extend the known results in the literature and answer open problem b proposed by li and vogelius 2000 for the isotropic conductivity problem we also obtain more interesting higher order derivative estimates which answers open problem c of li and vogelius 2000 it is worth pointing out that the equations under consideration in this paper are non homogeneous

free transmission problems we study transmission problems with free interfaces from one random medium to another solutions are required to solve distinct partial differential equations rm l and rm l within their positive and negative sets respectively a corresponding flux balance from one phase to another is also imposed we establish existence and l infty bounds of solutions we also prove that variational solutions are non degenerate and develop the regularity theory for solutions of such free boundary problems

bounded height in pencils of finitely generated subgroups in this paper we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric mordell lang theorem by replacing finiteness by emptyness for the intersection of varieties and subgroups all moving in a pencil except for bounded height values of the parameters and excluding identical relations more precisely an instance of the results is as follows consider the torus scheme g r m c over a curve c defined over q and let be a subgroup scheme generated by finitely many sections satisfying some necessary conditions further let v be any subscheme then there is a bound for the height of the points p c q such that for some which does not generically lie in v p lies in the fiber vp we further offer some direct diophantine applications to illustrate once again that the results implicitly contain information absent from the previous bounds in this context

convergence for elliptic equations in periodic perforated domains abstract convergence for the solutions of elliptic equations in periodic perforated domains is concerned let denote the size ratio of the holes of a periodic perforated domain to the whole domain it is known that by energy method the gradient of the solutions of elliptic equations is bounded uniformly in in l 2 space also when approaches 0 the elliptic solutions converge to a solution of some simple homogenized elliptic equation in this work above results are extended to general w 1 p space for p 1 more precisely a uniform w 1 p estimate in for p 1 and a w 1 p convergence result for p n n 2 for the elliptic solutions in periodic perforated domains are derived here n is the dimension of the space domain one also notes that the l p norm of the second order derivatives of the elliptic solutions in general cannot be bounded uniformly in

specializations of finitely generated subgroups of abelian varieties given a generic mordell weil group over a function field we can specialize it down to a number field it has been known for some time that the resulting homomorphism of groups is injective infinitely often we prove that this is in fact true almost always in a sense that is quantitatively nearly best possible

optimal boundary gradient estimates for lame systems with partially infinite coefficients abstract in this paper we derive the pointwise upper bounds and lower bounds on the gradients of solutions to the lame systems with partially infinite coefficients as the surface of discontinuity of the coefficients of the system is located very close to the boundary when the distance tends to zero the optimal blow up rates of the gradients are established for inclusions with arbitrary shapes and in all dimensions

intersections of polynomial orbits and a dynamical mordell lang conjecture we prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection then the polynomials have a common iterate we also prove a special case of a conjectured dynamical analogue of the mordell lang conjecture

regularity in oscillatory nonlinear elliptic systems we give uniform bounds in morrey sobolev spaces h1 for weak solutions ue to oscillatory nonlinear elliptic systems the main novelty is in the precise qualitative description of the e uniform morrey exponent in terms of the ellipticity ratio of the elliptic systems this is achieved by showing that weak solutions of general nonlinear elliptic systems with lipschitz coefficients are of class w2 p locally or globally depending on assumptions for an exponent p 2 that is independent of dimensions and for which p 2 is inversely proportional to the ellipticity ratio

global gradient estimates for nonlinear equations of p laplace type from composite materials abstract we study global gradient estimates for the weak solution to divergence form nonlinear elliptic equations with p growth nonlinearities from composite materials we assume that the domain is bounded and is composed of disjoint reifenberg flat domains and the nonlinearities have small bmo seminorms in each subdomain under these assumptions based on our new geometric observation for disjoint reifenberg domains and gradient estimates for nonlinear equations with measurable p growth nonlinearities we establish global w 1 q estimates for p q