fractal reinforcement of elastic membranes we describe a homogenization model of an elastic membrane reinforced by the inclusion of a fractal string we follow a variational approach consisting in proving the convergence of certain energy functionals this leads to the spectral convergence of a sequence of weighted second order elliptic partial differential operators to a singular elliptic operator with a fractal term

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

variational convergence over metric spaces we introduce a natural definition of l p convergence of maps p 1 in the case where the domain is a convergent sequence of measured metric space with respect to the measured gromov hausdorff topology and the target is a gromov hausdorff convergent sequence with the l p convergence we establish a theory of variational convergences we prove that the poincare inequality with some additional condition implies the asymptotic compactness the asymptotic compactness is equivalent to the gromov hausdorff compactness of the energy sublevel sets supposing that the targets are cat 0 spaces we study convergence of resolvents as applications we investigate the approximating energy functional over a measured metric space and convergence of energy functionals with a lower bound of ricci curvature

γ convergence of equi coercive nonlinear energies defined on vector valued functions with non uniformly bounded coefficients abstract the present paper deals with the asymptotic behavior of equi coercive sequences n of nonlinear functionals defined over vector valued functions in w 0 1 p m where p 1 m 1 and is a bounded open set of r n n 2 the strongly local energy density f n d u of the functional n satisfies a lipschitz condition with respect to the second variable which is controlled by a positive sequence a n which is only bounded in some suitable space l r we prove that the sequence n converges for the strong topology of l p m to a functional which has a strongly local density f d u for sufficiently regular functions u this compactness result extends former results on the topic which are based either on maximum principle arguments in the nonlinear scalar case or adapted div curl lemmas in the linear case here the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional n the relevance of the conditions which are imposed to the energy density f n d u is illustrated by several examples including some classical hyperelastic energies

fundamental solutions for second order subelliptic operators estimation de la solution fondamentale d un operateur aux derivees partielles lineaire du second ordre sur une variete compacte avec une mesure reguliere

weak convergence of symmetric diffusion processes in this paper we show the convergence of forms in the sense of mosco associated with the part form on relatively compact open set of dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states under regular dirichlet space setting we also get the same assertion under dirichlet space in infinite dimensional setting as a result of this we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls defined by modified pseudo metric used in k th sturm

homogenization of non uniformly bounded operators critical barrier for nonlocal effects this paper deals with the homogenization of a class of highly oscillating monotonic operators which are uniformly elliptic but non uniformly bounded we determine an asymptotic barrier below which we obtain a classical behaviour and above which nonlocal effects appear we also prove that this condition is optimal in the case of a fibre reinforced medium

space time spectral method for the cattaneo equation with time fractional derivative abstract this paper introduces a high order accurate numerical method for solving the cattaneo equation with time fractional derivative it is based on the galerkin legendre spectral method in space and the chebyshev collocation method in time arbitrarily high order accurate can be made in both space and time optimal priori error bound of the semi discrete method and the stability and convergence of the full discrete method are strictly given extensive experimental results confirm the theoretical claims of this method in both space and time

regularity for quasilinear second order subelliptic equations in this paper we study the regularity of solutions of the quasilinear equation where x x1 xm is a system of real smooth vector fields aij b c rm 1 assume that x satisfies the hormander condition and aij x z is positive definite we prove that if u s2 see section 2 is a solution of the above equation then u c

mosco convergence of dirichlet forms in infinite dimensions with changing reference measures let e be an infinite dimensional locally convex space let n be a weakly convergent sequence of probability measures on e and let en be a sequence of dirichlet forms on e such that en is defined on l2 n general sufficient conditions for mosco convergence of the gradient dirichlet forms are obtained applications to gibbs states on a lattice and to the gaussian case are given weak convergence of the associated processes is discussed

grusin operators riesz transforms and nilpotent lie groups we establish that the riesz transforms of all orders corresponding to the grusin operator h n nabla x 2 x 2n nabla y 2 and the first order operators nabla x x nu nabla y where x in mathbf r n y in mathbf r m n in mathbf n and nu in 1 ldots n n are bounded on l p mathbf r n m for all p in langle 1 infty rangle and are also weak type 1 1 moreover the transforms of order less than or equal to n 1 corresponding to h n and the operators nabla x x n nabla y are bounded on l p mathbf r n m for all p in langle 1 infty rangle but if n is odd all transforms of order n 2 are bounded if and only if p in langle 1 n rangle the proofs are based on the observation that the nabla x x nu nabla y generate a finite dimensional nilpotent lie algebra the corresponding connected simply connected nilpotent lie group is isometrically represented on the spaces l p mathbf r n m and h n is the corresponding sublaplacian

pin 2 equivariant seiberg witten floer homology and the triangulation conjecture we define pin 2 equivariant seiberg witten floer homology for rational homology 3 spheres equipped with a spin structure the analogue of froyshov s correction term in this setting is an integer valued invariant of homology cobordism whose mod 2 reduction is the rokhlin invariant as an application we show that there are no homology 3 spheres y of rokhlin invariant one such that y y bounds an acyclic smooth 4 manifold by previous work of galewski stern and matumoto this implies the existence of non triangulable high dimensional manifolds

an endomorphism of the khovanov invariant abstract we construct an endomorphism of the khovanov invariant to prove h thinness and pairing phenomena of the invariants for alternating links as a consequence it follows that the khovanov invariant of an oriented nonsplit alternating link is determined by its jones polynomial signature and the linking numbers of its components

monopoles and lens space surgeries monopole floer homology is used to prove that real projective three space cannot be obtained from dehn surgery on a nontrivial knot in the three sphere to obtain this result we use a surgery long exact sequence for monopole floer homology together with a nonvanishing theorem which shows that monopole floer homology detects the unknot in addition we apply these techniques to give information about knots which admit lens space surgeries and to exhibit families of three manifolds which do not admit taut foliations

analysis of degenerate elliptic operators of grusin type we analyse degenerate second order elliptic operators h in divergence form on l 2 r n r m we assume the coefficients are real symmetric and a 1 h h a 2 h for some a 1 a 2 0 where h delta nabla x 1 cdot c delta 1 delta 1 x 1 nabla x 1 c delta 2 delta 2 x 1 nabla x 2 2 here x 1 r n x 2 r m and c delta i delta i are positive measurable functions such that c delta i delta i x behaves like x delta i as x 0 and x delta i as x to infty with delta 1 delta 1 in 0 1 rangle and delta 2 delta 2 geq 0 our principal results state that the submarkovian semigroup s t e th is conservative and its kernel k t satisfies bounds 0 leq k t x y leq a b x t 1 2 b y t 1 2 1 2 where b x r denotes the volume of the ball b x r centred at x with radius r measured with respect to the riemannian distance associated with h the proofs depend on detailed subelliptic estimations on h a precise characterization of the riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation we discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive nor continuous

small time gaussian behavior of symmetric diffusion semi groups this work is involved with the short time asymptotics of diffusion semigroups in a general setting a generalization of fang s version of varadhan s formula is proven for general dirichlet spaces that are local and conservative the intrinsic metric appearing in the formula is characterized by pointwise distance for canonical dirichlet spaces on loop groups

variational composition of a monotone operator and a linear mapping with applications to elliptic pdes with singular coefficients abstract this paper proposes a regularized notion of a composition of a monotone operator with a linear mapping this new concept called variational composition can be shown to be maximal monotone in many cases where the usual composition is not the two notions coincide however whenever the latter is maximal monotone the utility of the variational composition is demonstrated by applications to subdifferential calculus theory of measurable multifunctions and elliptic pdes with singular coefficients

weighted sobolev poincare inequalities and pointwise estimates for a class of degenerate elliptic equations in this paper we prove a sobolev poincare inequality for a class of function spaces associated with some degenerate elliptic equations these estimates provide us with the basic tool to prove an invariant harnack inequality for weak positive solutions in addition holder regularity of the weak solutions follows in a standard way let y z 7 ai aij9 a be a second order degenerate elliptic operator in divergence form with measurable coefficients in this paper we shall obtain pointwise estimates for the weak solutions of su 0 h61der continuity of the weak solutions and harnack inequality for nonnegative solutions let us recall that the original results for elliptic operators were obtained by de giorgi nash and moser an extensive bibliography about the degenerate case can be found in fli fl2 fs to introduce the results of the present paper let us recall some recent results in fl1 fl2 a suitable metric d is associated with the differential operator y in such a way that we obtain a new geometry which is natural for the degenerate operator as the euclidean geometry is natural for the laplace operator or more precisely as a suitable riemannian geometry is natural for a secondorder elliptic operator in the smooth case this idea is contained in many papers we refer to fp nsw the basic results in fl 1 fl2 are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the hormander hypoellipticity condition for sum of squares operators we note that the same idea is used in nsw s j v to obtain pointwise estimates for sum of squares operators on the other hand a different class of degenerate elliptic operators is considered in fks instead of a geometrical degeneracy a measure degeneracy is allowed a typical example of this class is given by yu div wo x vu where cl is a weight function belonging to the a2 class of muckenhoupt unified results for a class containing both the operators in fl 1 and in fks have received by the editors august 1 1989 1980 mathematics subject classification 1985 revision primary 46e35 35j70 partially supported by g n a f a of c n r and m u r s t italy 1991 american mathematical society 0002 9947 91 1 00 25 per page

mosco convergence of nonlocal to local quadratic forms abstract we study sequences of nonlocal quadratic forms and function spaces that are related to markov jump processes in bounded domains with a lipschitz boundary our aim is to show the convergence of these forms to local quadratic forms of gradient type under suitable conditions we establish the convergence in the sense of mosco our framework allows bounded and unbounded nonlocal operators to be studied at the same time moreover we prove that smooth functions with compact support are dense in the nonlocal function spaces under consideration

energy inequalities for cutoff functions and some applications let x d m be a metric measure space with a local regular dirichlet form we establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space time exponent to hold this characterization is stable under rough isometries that is it is preserved under bounded perturbations of the dirichlet form further we give a criterion for stochastic completeness in terms of a sobolev inequality for cutoff functions as an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space where the well established criterion in terms of volume growth fails

diffusion processes and heat kernels on metric spaces processes x p on any given locally compact metric space x d tx equipped with a radon measure m these processes are associated with local regular dirichlet forms which are obtained as limits of approxi mating nonlocal dirichlet forms this general method works without any restrictions on x d m and yields processes which are well defined for quasi every starting point the second main topic of this paper is to formulate and exploit the so called measure contraction property this is a condition on the original data x d m which can be regarded as a generalization of curvature bounds on the metric space x d it is a bound for distortions of the measure m under contractions of the state space x along suitable geodesics or quasi geodesics w r t the metric d in the case of rieman nian manifolds this condition is always satisfied several other examples will be discussed including uniformly elliptic operators operators with weights certain subelliptic operators manifolds with boundaries or cor ners and glueing together of manifolds the measure contraction property implies upper and lower gaussian estimates for the heat kernel and a harnack inequality for the associated harmonic functions therefore the above mentioned diffusion processes are strong feller processes and are well defined for every starting point

semilinear evolution transmission problems across fractal layers abstract a semilinear parabolic transmission problem across either a fractal layer s or the corresponding prefractal layer s h is studied local existence uniqueness and regularity results for the mild solution in both cases are established as well as a sufficient condition on the initial datum for global existence is given the asymptotic behaviour of the solutions of the approximating problems is studied

second order operators with degenerate coefficients we consider properties of second order operatorsh d i j 1 i cij j on r with bounded real symmetric measurable coefficients we assume that c cij 0 almost everywhere but allow for the possibility that c is singular we associate with h a canonical self adjoint viscosity operator h0 and examine properties of the viscosity semigroup s generated by h0 the semigroup extends to a positive contraction semigroup on the lp spaces with p 1 we establish that it conserves probability satisfies l2 off diagonal bounds and that the wave equation associated with h0 has finite speed of propagation nevertheless s is not always strictly positive because separation of the system can occur even for subelliptic operators this demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor holder continuous in particular one can construct examples for which both upper and lower gaussian bounds fail even with coefficients in c r with e 0

fractal snowflake domain diffusion with boundary and interior drifts abstract we study a parabolic ventsell problem for a second order differential operator in divergence form and with interior and boundary drift terms on the snowflake domain we prove that under standard conditions a related cauchy problem possesses a unique classical solution and explain in which sense it solves a rigorous formulation of the initial ventsell problem as a second result we prove that functions that are intrinsically lipschitz on the snowflake boundary admit euclidean lipschitz extensions to the closure of the entire domain our methods combine the fractal membrane analysis the vector analysis for local dirichlet forms and pde on fractals coercive closed forms and the analysis of lipschitz functions

variations on a strange semi continuity result abstract this paper is devoted to the study of some lower semi continuity results whose model example is lim inf n u n v n u v for any sequence u n v n which weakly converges to u v in h 0 1 2 and satisfies the positivity assumptions u n 0 and v n 0 extensions of this semi continuity result are obtained in the following cases two fixed linear operators two monotone operators two varying linear operators varying domains and non uniformly bounded operators in these various cases the operators under consideration are second order elliptic operators in divergence form and truncation arguments play an essential role finally the case of higher order operators is treated by a quite different approach

weak convergence of diffusion processes on wiener space let be a gaussian measure on a suslin space x h be the corresponding cameron martin space and e i h be an orthonormal basis of h suppose that n n is a sequence of probability measures which converges weakly to a probability measure consider a sequence of dirichlet forms varepsilon n where varepsilon n f int x nabla h f 2 h rho n rm d gamma and sqrt rho n in w 2 1 gamma we prove some sufficient conditions for mosco convergence varepsilon n to varepsilon where varepsilon f int x nabla h f 2 h rho rm d gamma in particular if x is a hilbert space sup n sqrt rho n w 2 1 gamma infty and partial e i rho n rho n can be uniformly approximated by finite dimensional conditional expectations rm ie mathcal f n mu n big partial e i rho n rho n big for every fixed e i then under broad assumptions varepsilon n to varepsilon mosco and the distributions of the associated stochastic processes converge weakly

realizable response matrices of multi terminal electrical acoustic and elastodynamic networks at a given frequency we give a complete characterization of the possible response matrices at a fixed frequency of n terminal electrical networks of inductors capacitors resistors and grounds and of n terminal discrete linear elastodynamic networks of springs and point masses both in three dimensional and two dimensional cases specifically we construct networks that realize any response matrix that is compatible with the known symmetry properties and thermodynamic constraints of response matrices owing to a mathematical equivalence we also obtain a characterization of the response matrices of discrete acoustic networks

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

a new div curl result applications to the homogenization of elliptic systems and to the weak continuity of the jacobian in this paper a new div curl result is established in an open set omega of mathbb r n n geq 2 for the product of two sequences of vector valued functions which are bounded respectively in l p omega n and l q omega n with 1 p 1 q 1 1 n 1 and whose respectively divergence and curl are compact in suitable spaces we also assume that the product converges weakly in w 1 1 omega the key ingredient of the proof is a compactness result for bounded sequences in w 1 q omega based on the imbedding of w 1 q s n 1 into l p s n 1 s n 1 the unit sphere of mathbb r n through a suitable selection of annuli on which the gradients are not too high in the spirit of de giorgi and manfredi the div curl result is applied to the homogenization of equi coercive systems whose coefficients are equi bounded in l rho omega for some rho n 1 over 2 if n 2 or in l 1 omega if n 2 it also allows us to prove a weak continuity result for the jacobian for bounded sequences in w 1 n 1 omega satisfying an alternative assumption to the l infty strong estimate of brezis and nguyen two examples show the sharpness of the results

fractional newton mechanics with conformable fractional derivative in this paper we use the conformable fractional derivative and integral to discuss the fractional newtonian mechanics the fractional version of the calculus of variations is introduced and the fractional euler lagrange equation is constructed some mechanical problems such as fractional harmonic oscillator problem the fractional damped oscillator problem and the forced oscillator problem are discussed in the one dimensional fractional dynamics

particle approximation of the wasserstein diffusion abstract we construct a system of interacting two sided bessel processes on the unit interval and show that the associated empirical measure process converges to the wasserstein diffusion von renesse and sturm 2009 25 assuming that markov uniqueness holds for the generating wasserstein dirichlet form the proof is based on the variational convergence of an associated sequence of dirichlet forms in the generalized mosco sense of kuwae and shioya 2003 19

intrinsic metrics for non local symmetric dirichlet forms and applications to spectral theory we present a study of what may be called an intrinsic metric for a general regular dirichlet form for such forms we then prove a rademacher type theorem for strongly local forms we show existence of a maximal intrinsic metric under a weak continuity condition and for dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals we then turn to applications on spectral theory and provide for measure perturbation of general regular dirichlet forms an allegretto piepenbrinck type theorem which is based on a ground state transform and a shnol type theorem our setting includes laplacian on manifolds on graphs and stable processes

riesz transform gaussian bounds and the method of wave equation for an abstract self adjoint operator l and a local operator a we study the boundedness of the riesz transform al on lp for some 0 a very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation we also discuss the relation between the gaussian bounds and the finite speed propagation property using the wave equation methods we obtain a new natural form of the gaussian bounds for the heat kernels for a large class of the generating operators we describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature

large deformations of planar extensible beams and pantographic lattices heuristic homogenization experimental and numerical examples of equilibrium the aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call pantographic structures the interest in these materials was increased by the possibilities opened by the diffusion of technology of three dimensional printing they can be regarded once choosing a suitable length scale as families of beams also called fibres interconnected to each other by pivots and undergoing large displacements and large deformations there are however relatively few ready to use results in the literature of nonlinear beam theory in this paper we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by piola to formulate a continuum fully nonlinear beam model the homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems furthermore we consider pantographic structures find the corresponding homogenized second gradient deformation energies and study some planar problems numerical solutions for these two dimensional problems are obtained via minimization of energy and are compared with some experimental measurements in which elongation phenomena cannot be neglected

nonlocal effects in two dimensional conductivity the paper deals with the asymptotic behaviour as e 0 of a two dimensional conduction problem whose matrix valued conductivity ae is e periodic and not uniformly bounded with respect to e we prove that only under the assumptions of equi coerciveness and l1 boundedness of the sequence ae the limit problem is a conduction problem of same nature this new result points out a fundamental difference between the two dimensional conductivity and the three dimensional one indeed under the same assumptions of periodicity equi coerciveness and l1 boundedness it is known that the high conductivity regions can induce nonlocal effects in three or greater dimensions

from refined estimates for spherical harmonics to a sharp multiplier theorem on the grushin sphere abstract we prove a sharp multiplier theorem of mihlin hormander type for the grushin operator on the unit sphere in r 3 and a corresponding boundedness result for the associated bochner riesz means the proof hinges on precise pointwise bounds for spherical harmonics

on singularity of energy measures on self similar sets we provide general criteria for energy measures of regular dirichlet forms on self similar sets to be singular to bernoulli type measures in particular every energy measure is proved to be singular to the hausdorff measure for canonical dirichlet forms on 2 dimensional sierpinski carpets

insulating layers and robin problems on koch mixtures abstract this paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random mixture of self similar koch curves we construct an e thin polygonal 2 dimensional fiber e n n n 0 e 1 around pre fractal approximating domains n and related suitable energy functionals the aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while simultaneously the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as n

n v limit for stochastic dynamics in continuous particle systems we provide an n v limit for the infinite particle infinite volume stochastic dynamics associated with gibbs states in continuous particle systems on d d 1 starting point is an n particle stochastic dynamic with singular interaction and reflecting boundary condition in a subset d with finite volume lebesgue measure v the aim is to approximate the infinite particle infinite volume stochastic dynamic by the above n particle dynamic in as n and v such that n v where is the particle density first we derive an improved ruelle bound for the canonical correlation functions under an appropriate relation between n and v then tightness is shown by using the lyons zheng decomposition the equilibrium measures of the accumulation points are identified as infinite volume canonical gibbs measures by an integration by parts formula and the accumulation points themselves are identified as infinite particle infinite volume stochastic dynamics via the associated martingale problem assuming a property closely related to markov uniqueness and weaker than essential self adjointness via mosco convergence techniques we can identify the accumulation points as markov processes and show uniqueness i e all accumulation corresponding to one invariant canonical gibbs measure coincide the proofs work for general repulsive interaction potentials of ruelle type and all temperatures densities and dimensions d 1 respectively may have a nontrivial negative part and infinite range as e g the lennard jones potential additionally our result provides as a by product an approximation of grand canonical gibbs measures by finite volume canonical gibbs measures with empty boundary condition

homogenization of stiff plates and two dimensional high viscosity stokes equations the paper deals with the homogenization of stiff heterogeneous plates assuming that the coefficients are equi bounded in l 1 we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients this compactness result is based on a div curl lemma for fourth order equations on the other hand using an intermediate stream function we deduce from the plates case a similar result for high viscosity stokes equations in dimension two so that the nature of the stokes equation is preserved in the homogenization process finally we show that the l 1 boundedness assumption cannot be relaxed indeed in the case of the stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem which violates the compactness result obtained under the l 1 boundedness condition

poincare inequality for weighted first order sobolev spaces on loop spaces abstract let e be the loop space over a compact connected riemannian manifold with a torsion skew symmetric connection let l d be the ornstein uhlenbeck operator on a nonempty connected component d of the loop space e and let v d r be the restriction on d of the potential in the logarithmic sobolev inequality found by l gross on the loop group by s aida and by f z gong and z m ma on the loop space respectively we prove that the schrodinger operator l v l d v always has a spectral gap at the bottom 0 v of its spectrum and thus has its ground state transformed operator 1 l v 0 v where is the unique ground state of l v in particular our result proves l gross s conjecture about the existence of a spectral gap for the ground state transform of the schrodinger operator studied by him on the loop group in addition in all the above cases we identify the domain of the dirichlet forms associated with the ground state transforms as weighted first order sobolev spaces with weight given by 2 thus establishing a poincare inequality for them all these results are consequences from some new results in this paper on dirichlet forms characterizing certain classes with spectral gaps and from results by s aida and m hino

determination of the closure of the set of elasticity functionals we determine the closure for mosco convergence in l2 3 of the set of elasticity functionals we prove that this closure coincides with the set of all non negative lower semicontinuous quadratic functionals which are objective i e which vanish for rigid motions the result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed poisson coefficient this shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients

solitary wave solutions for time fractional third order modified kdv equation using two reliable techniques g g expansion method and improved g g expansion method abstract in the present paper we construct the analytical exact solutions of a nonlinear evolution equation in mathematical physics namely time fractional modified kdv equation by using g g expansion method and improved g g expansion method as a result new types of exact analytical solutions are obtained

continuum and discrete models for structures including quasi inextensible elasticae with a view to the design and modeling of composite reinforcements abstract inspired by some composite fiber reinforcements used in aeronautical engineering and by the need of conceiving new metamaterials some discrete models including quasi inextensible elasticae are considered a class of continuum models approximately describing the macroscopic mechanical behavior of introduced structures is then heuristically proposed some of these continuum models can be regarded as a special kind of second gradient elastic media in which the higher gradient elasticity is conferred by the flexural stiffnesses of elasticae constituting the microscopic lattice the discrete models are studied by means of suitably tailored numerical codes designed to avoid numerical instabilities and locking and a comparison of discrete versus continuum models is attempted the obtained results show that the theory of generalized continua may be useful in some engineering applications and it plays a relevant role in the mechanics of woven composites the introduced discrete and continuum models are used to describe the so called bias extension test on woven fabrics and it is shown that a good choice to correctly reproduce the targeted phenomenology is to use a second gradient continuum theory however as discussed throughout the paper in the context of rigorous micro macro identification procedures there still remain many open problems to be solved especially when dealing with systems subjected to particular constraints such as inextensibility

nonlocal limits in the study of linear elliptic systems arising in periodic homogenization in the present paper we obtain the two scale limit system of a sequence of linear elliptic periodic problems with varying coefficients we show that this system has not the same structure than the classical one obtained when the coefficients are fixed this is due to the apparition of nonlocal effects our results give an example showing that the homogenization of elliptic problems with varying coefficients depending on one parameter gives in general a nonlocal limit problem

auxiliary equation method for time fractional differential equations with conformable derivative abstract in this paper the auxiliary equation method is applied to obtain analytical solutions of 2 1 dimensional time fractional zoomeron equation and the time fractional third order modified kdv equation in the sense of the conformable fractional derivative given equations are converted to the nonlinear ordinary differential equations of integer order and then the resulting equations are solved using a novel analytical method called the auxiliary equation method as a result some exact solutions for them are successfully established the exact solutions obtained by the proposed method indicate that the approach is easy to implement and effective

homogenization on nested fractals we study the homogenization problem on nested fractals letxt be the continuous time markov chain on the pre nested fractal given by puttingi i d random resistors on each cell it is proved that under some conditions alpha n x t e n t converges in law to a constant time change of the brownian motion on the fractal asn where is the contraction rate andte is a time scale constant as the brownian motion on fractals is not a semi martingale we need a different approach from the well developed martingale method

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

semilinear fractal problems approximation and regularity results abstract we prove existence and uniqueness of the weak solution for a second order semilinear transmission problem in a bounded domain in r 3 in the case of a koch type fractal layer s or the corresponding prefractal layer approximating s we study the asymptotic behavior of the solutions of the corresponding prefractal problem and we prove regularity results for the weak solution via a generalized green formula for domains with fractal boundaries

weighted restriction type estimates for grushin operators and application to spectral multipliers and bochner riesz summability we prove weighted restriction type estimates for grushin operators these estimates are then used to prove sharp spectral multiplier theorems as well as bochner riesz summability results with sharp exponent

homogenization and optimal bounds in linear elasticity in their celebrated paper of 1963 hashin shtrikman 1 addressed the problem of determining optimal bounds for the bulk and shear moduli of a statistically isotropic elastic composite material with arbitrary isotropic phase geometry they derived a set of bounds for these moduli in the physically meaningful case of three dimensional elasticity in the case of a two phase composite let k 1 1 and k 2 2 respectively denote the bulk and shear moduli for the first and the second phase let k denote their analogues for the composite and let 0 stand for the volume fraction of the first phase in the composite under the ordering restriction that both

optimal bounds and microgeometries for elastic two phase composites consider two isotropic linearly elastic materials with elasticity tensors c 1 c 2 and assume c 1 c 2 we prove that the class of effective tensors of composites built from these materials with given volume fractions that are invariant under a given symmetry group is such that each of its elements is bounded above and below by tensors in the same class corresponding to finite rank laminates this implies that for any imposed uniform strain or stress field optimal bounds on the effective strain or stress energy per unit volume are attained by finite rank laminates explicit bounds on the strain energy with no symmetry assumptions are given in dimensions 2 and 3 these bounds are more stringent than the classical voigt reuss bounds finally explicit bounds and microgeometries are given for the effective moduli of composites with cubic symmetry

a link surgery spectral sequence in monopole floer homology abstract to a link l s 3 we associate a spectral sequence whose e 2 page is the reduced khovanov homology of l and which converges to a version of the monopole floer homology of the branched double cover the pages e k for k 2 depend only on the mutation equivalence class of l we define a mod 2 grading on the spectral sequence which interpolates between the grading on khovanov homology and the mod 2 grading on floer homology we also derive a new formula for link signature that is well adapted to khovanov homology more generally we construct new bigraded invariants of a framed link in a 3 manifold as the pages of a spectral sequence modeled on the surgery exact triangle the differentials count monopoles over families of metrics parameterized by permutohedra we utilize a connection between the topology of link surgeries and the combinatorics of graph associahedra this also yields simple realizations of permutohedra and associahedra as refinements of hypercubes

holomorphic disks and topological invariants for closed three manifolds the aim of this article is to introduce certain topological invariants for closed oriented three manifolds y equipped with a spiny structure given a heegaard splitting of y u 0o u u 1 these theories are variants of the lagrangian floer homology for the g fold symmetric product of relative to certain totally real subspaces associated to u 0 and u 1

singular perturbations to semilinear stochastic heat equations we consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof the interesting feature of these perturbations is that as the small parameter e tends to zero their solutions converge to the wrong limit i e they do not converge to the solution obtained by simply setting e 0 a similar effect is also observed for some formally small stochastic perturbations of a deterministic semilinear parabolic pde our proofs are based on a detailed analysis of the spatially rough component of the equations combined with a judicious use of gaussian concentration inequalities

continuity of eigenvalues of subordinate processes in domains let x x t t 0 be a symmetric markov process in a state space e and d an open set of e let s n s n t t 0 be a subordinator with laplace exponent n and s s t t 0 a subordinator with laplace exponent suppose that x is independent of s and s n in this paper we consider the subordinate processes open image in new window and open image in new window and their subprocesses open image in new window and x d killed upon leaving d suppose that the spectra of the semigroups of open image in new window and x d are all discrete with open image in new window being the eigenvalues of the generator of open image in new window and open image in new window being the eigenvalues of the generator of x d we show that if lim n n for every 0 then

on the construction and convergence of traces of forms abstract we elaborate a new method for constructing traces of quadratic forms in the framework of hilbert and dirichlet spaces our method relies on monotone convergence of quadratic forms and the canonical decomposition into regular and singular part we give various situations where the trace can be described more explicitly and compute it for some illustrating examples we then show that mosco convergence of dirichlet forms implies mosco convergence of a subsequence of their approximating traces

a categorification of the jones polynomial author s khovanov mikhail abstract we construct a bigraded cohomology theory of links whose euler characteristic is the jones polynomial

extensions of levy khintchine formula and beurling deny formula in semi dirichlet forms setting abstract the levy khintchine formula or more generally courrege s theorem characterizes the infinitesimal generator of a levy process or a feller process on r d for more general markov processes the formula that comes closest to such a characterization is the beurling deny formula for symmetric dirichlet forms in this paper we extend these celebrated structure results to include a general right process on a metrizable lusin space which is supposed to be associated with a semi dirichlet form we start with decomposing a regular semi dirichlet form into the diffusion jumping and killing parts then we develop a local compactification and an integral representation for quasi regular semi dirichlet forms finally we extend the formulae of levy khintchine and beurling deny in semi dirichlet forms setting through introducing a quasi compatible metric

energy measures and indices of dirichlet forms with applications to derivatives on some fractals we introduce the concept of index for regular dirichlet forms by means of energy measures and discuss its properties in particular it is proved that the index of strong local regular dirichlet forms is identical with the martingale dimension of the associated diffusion processes as an application a class of self similar fractals is taken up as an underlying space we prove that first order derivatives can be defined for functions in the domain of the dirichlet forms and their total energies are represented as the square integrals of the derivatives

bar natan s deformation of khovanov homology and involutive monopole floer homology we study the conjugation involution in seiberg witten theory in the context of ozsvath szabo and bloom s spectral sequence for the branched double cover of a link l in s 3 we prove that there exists a spectral sequence of mathbb f q q 2 modules where q has degree 1 which converges to widetilde hmi sigma l an involutive version of the monopole floer homology of the branched double cover and whose e 2 page is a version of bar natan s deformation of khovanov homology in characteristic two of the mirror of l

the first integral method for some time fractional differential equations abstract in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the first integral method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations the power of this manageable method is presented by applying it to several examples this approach can also be applied to other nonlinear fractional differential equations

mosco convergence and wiener measures for conductive thin boundaries abstract the mosco convergence of energy functionals and the convergence of associated wiener measures are proved for a domain with highly conductive thin boundary we obtain those results for matrix valued conductivities and a family of speed measures measures of the underlying domain in particular this family includes the lebesgue measure and the one which makes the energy functional superposition the expectation of the displacement of the associated processes close to the boundary goes to due to the explosion of the conductivity at the limit

quenched invariance principles for random walks and elliptic diffusions in random media with boundary via a dirichlet form extension theorem and making full use of two sided heat kernel estimates we establish quenched invariance principles for random walks in random environments with a boundary in particular we prove that the random walk on a supercritical percolation cluster or amongst random conductances bounded uniformly from below in a half space quarter space etc converges when rescaled diffusively to a reflecting brownian motion which has been one of the important open problems in this area we establish a similar result for the random conductance model in a box which allows us to improve existing asymptotic estimates for the relevant mixing time furthermore in the uniformly elliptic case we present quenched invariance principles for domains with more general boundaries

a new definition of fractional derivative we give a new definition of fractional derivative and fractional integral the form of the definition shows that it is the most natural definition and the most fruitful one the definition for email protected email protected 1 coincides with the classical definitions on polynomials up to a constant further if a 1 the definition coincides with the classical definition of first derivative we give some applications to fractional differential equations

non lipschitz uniform domain shape optimization in linear acoustics we introduce new parametrized classes of shape admissible domains in r n n 2 and prove that they are compact with respect to the convergence in the sense of characteristic functions the hausdorff sense the sense of compacts and the weak convergence of their boundary volumes the domains in these classes are bounded e domains with possibly fractal boundaries that can have parts of any non uniform hausdorff dimension greater or equal to n 1 and less than n we prove the existence of optimal shapes in such classes for maximum energy dissipation in the framework of linear acous tics a by product of our proof is the result that the class of bounded e domains with fixed e is stable under hausdorff convergence an additional and related result is the mosco convergence of robin type energy functionals on converging domains

on the heegaard floer homology of branched double covers abstract let l s 3 be a link we study the heegaard floer homology of the branched double cover l of s 3 branched along l when l is an alternating link hf of its branched double cover has a particularly simple form determined entirely by the determinant of the link for the general case we derive a spectral sequence whose e 2 term is a suitable variant of khovanov s homology for the link l converging to the heegaard floer homology of l

uniform convergence of sequences of solutions of two dimensional linear elliptic equations with unbounded coefficients abstract this paper deals with the behavior of two dimensional linear elliptic equations with unbounded and possibly infinite coefficients we prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two dimensional capacitary estimate we give two applications of this result the continuity of the solutions of two dimensional linear elliptic equations by a constructive approach and the density of the continuous functions in the domain of the limit of equicoercive diffusion energies in dimension two we also build two counter examples which show that the previous results cannot be extended to dimension three

some diffusion processes associated with two parameter poisson dirichlet distribution and dirichlet process the two parameter poisson dirichlet distribution pd is the distribution of an infinite dimensional random discrete probability it is a generalization of kingman s poisson dirichlet distribution the two parameter dirichlet process pi alpha theta nu 0 is the law of a pure atomic random measure with masses following the two parameter poisson dirichlet distribution in this article we focus on the construction and the properties of the infinite dimensional symmetric diffusion processes with respective symmetric measures pd and pi alpha theta nu 0 the methods used come from the theory of dirichlet forms

homogenization for conductive thin layers of pre fractal type the aim of this work is to obtain convergence results for the solutions of transmission problems across highly conductive layers of pre fractal type from the point of view of homogenization we prove the m convergence of the energy functionals to an energy functional which incorporates a singular term supported within the layer from the convergence of the energy functionals we deduce the convergence of the approximating solutions to the limit solution in a suitable sense

materials with prescribed constitutive parameters an inverse homogenization problem abstract this paper deals with the construction of materials with arbitrary prescribed positive semi definite constitutive tensors the construction problem can be called an inverse problem of finding a material with given homogenized coefficients the inverse problem is formulated as a topology optimization problem i e finding the interior topology of a base cell such that cost is minimized and the constraints are defined by the prescribed constitutive parameters numerical values of the constitutive parameters of a given material are found using a numerical homogenization method expressed in terms of element mutual energies numerical results show that arbitrary materials including materials with poisson s ratio 1 0 and other extreme materials can be obtained by modelling the base cell as a truss structure furthermore a wide spectrum of materials can be constructed from base cells modelled as continuous discs of varying thickness only the two dimensional case is considered in this paper but formulation and numerical procedures can easily be extended to the three dimensional case

discrete approximation of symmetric jump processes on metric measure spaces in this paper we give general criteria on tightness and weak convergence of discrete markov chains to symmetric jump processes on metric measure spaces under mild conditions as an application we investigate discrete approximation for a large class of symmetric jump processes we also discuss some application of our results to the scaling limit of random walk in random conductance