on an inequality by n trudinger and j moser and related elliptic equations it has been shown by trudinger and moser that for normalized functions u of the sobolev space w 1 n whereis a bounded domain in r n one has r exp njuj n n 1 dx cn where n is an explicit constant depending only on n and cn is a constant depending only on n and carleson and chang proved that there exists a corresponding extremal function in the case that is the unit ball in r n in this paper we give a new proof a generalization and a new interpretation of this result in particular we give an explicit sequence that is maximizing for the above integral among all normalized concentrating sequences as an application the existence of a nontrivial solution for a related elliptic equation with trudinger moser growth is proved c 2002 john wiley sons inc

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

an improved geometric inequality via vanishing moments with applications to singular liouville equations we consider a class of singular liouville equations on compact surfaces motivated by the study of electroweak and self dual chern simons theories the gaussian curvature prescription with conical singularities and onsager s description of turbulence we analyse the problem of existence variationally and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the moser trudinger inequality and in turn to lower bounds on the euler lagrange functional we then discuss existence and non existence results

on the lieb thirring constantslγ 1 for γ 1 2 letei h denote the negative eigenvalues of the one dimensional schrodinger operatorhu u vu v 0 onl2 we prove the inequality mathop sum limits i e i h gamma leqq l gamma 1 mathop smallint limits mathbb r v gamma 1 2 x dx 1 for the limit case 1 2 this will imply improved estimates for the best constantsl 1 in 1 as 1 2 3 2

mean field flow difference model with consideration of on ramp and off ramp abstract in this paper an extended lattice hydrodynamic model is proposed by incorporating the effects of mean field flow difference on ramp and off ramp the linear stability condition is obtained through linear stability theory and the mkdv equation is derived by using nonlinear analysis method therefore the propagation behavior of traffic jam can be described by the kink antikink soliton solution of the mkdv equation furthermore the theoretical findings are verified with the use of numerical simulation which confirms that the stability of traffic flow can be efficiently improved with the consideration of the effects of mean field flow difference on ramp and off ramp

self duality of the sl 2 hitchin integrable system at genus 2 we revisit the hitchin integrable system 11 21 whose phase space is the bundle cotangent to the moduli space of holomorphic sl 2 bundles over a smooth complex curve of genus 2 as shown in 18 may be identified with the 3 dimensional projective space of theta functions of the 2nd order i e we prove that the hitchin system on possesses a remarkable symmetry it is invariant under the interchange of positions and momenta this property allows to complete the work of van geemen previato 21 which basing on the classical results on geometry of the kummer quartic surfaces specified the explicit form of the hamiltonians of the hitchin system the resulting integrable system resembles the classic neumann systems which are also self dual its quantization produces a commuting family of differential operators of the 2nd order acting on homogeneous polynomials in four complex variables as recently shown by van geemen dejong 11 these operators realize the knizhnik zamolodchikov bernard hitchin connection for group su 2 and genus 2 curves

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

a new car following model for autonomous vehicles flow with mean expected velocity field abstract due to the development of the modern scientific technology autonomous vehicles may realize to connect with each other and share the information collected from each vehicle an improved forward considering car following model was proposed with mean expected velocity field to describe the autonomous vehicles flow behavior the new model has three key parameters adjustable sensitivity strength factor and mean expected velocity field size two lemmas and one theorem were proven as criteria for judging the stability of homogeneousautonomous vehicles flow theoretical results show that the greater parameters means larger stability regions a series of numerical simulations were carried out to check the stability and fundamental diagram of autonomous flow from the numerical simulation results the profiles hysteresis loop and density waves of the autonomous vehicles flow were exhibited the results show that with increased sensitivity strength factor or field size the traffic jam was suppressed effectively which are well in accordance with the theoretical results moreover the fundamental diagrams corresponding to three parameters respectively were obtained it demonstrates that these parameters play almost the same role on traffic flux i e before the critical density the bigger parameter is the greater flux is and after the critical density the opposite tendency is in general the three parameters have a great influence on the stability and jam state of the autonomous vehicles flow

moser trudinger trace inequalities sharp constants in exponential inequalities involving a general class of measures in domains are exhibited in the limiting case of the sobolev embedding theorem a comprehensive approach is presented yielding as special instances trace inequalities on o on smooth submanifolds of o of arbitrary dimension and also on fractal subsets of o and recovering in particular the classical moser trudinger inequality

sobolev inequalities for the symmetric gradient in arbitrary domains abstract a form of sobolev inequalities for the symmetric gradient of vector valued functions is proposed which allows for arbitrary ground domains in r n in the relevant inequalities boundary regularity of domains is replaced with information on boundary traces of trial functions the inequalities so obtained exhibit the same exponents as in classical inequalities for the full gradient of sobolev functions in regular domains furthermore they involve constants independent of the geometry of the domain and hence yield novel results yet for smooth domains our approach relies upon a pointwise estimate for the functions in question via a riesz potential of their symmetric gradient and an unconventional potential depending on their boundary trace

sharp moser trudinger inequalities for the laplacian without boundary conditions we derive a sharp moser trudinger inequality for the borderline sobolev imbedding of w2 n 2 bn into the exponential class where bn is the unit ball of rn the corresponding sharp results for the spaces w0d n d are well known for general domains and are due to moser and adams when the zero boundary condition is removed the only known results are for d 1 and are due to chang yang cianchi and leckband the proof of our result is based on a new integral representation formula for the canonical solution of the poisson equation on the ball that is the unique solution of the equation u f which is orthogonal to the harmonic functions on the ball the main technical difficulty of the paper is to establish an asymptotically sharp growth estimate for the kernel of such representation expressed in terms of its distribution function

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

a thermodynamical formalism for monge ampere equations moser trudinger inequalities and kahler einstein metrics we develop a variational calculus for a certain free energy functional on the space of all probability measures on a kahler manifold x this functional can be seen as a generalization of mabuchi s k energy functional and its twisted versions to more singular situations applications to monge ampere equations of mean field type twisted kahler einstein metrics and moser trudinger type inequalities on miller manifolds are given tian s alpha invariant is generalized to singular measures allowing in particular a proof of the existence of kahler einstein metrics with positive ricci curvature that are singular along a given anti canonical divisor which combined with very recent developments concerning miller metrics with conical singularities confirms a recent conjecture of donaldson as another application we show that if the calabi flow in the anti canonical class exists for all times then it converges to a kahler einstein metric when a unique one exists which is in line with a well known conjecture c 2013 elsevier inc all rights reserved

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

on the discriminant locus of a lagrangian fibration let x rightarrow mathbb p n be an irreducible holomorphic symplectic manifold of dimension 2n fibred over mathbb p n matsushita proved that the generic fibre is a holomorphic lagrangian abelian variety in this article we study the discriminant locus delta subset mathbb p n parametrizing singular fibres our main result is a formula for the degree of leading to bounds on the degree when x is a fourfold

stabilization effect of traffic flow in an extended car following model based on an intelligent transportation system application an extended car following model is proposed by incorporating an intelligent transportation system in traffic the stability condition of this model is obtained by using the linear stability theory the results show that anticipating the behavior of more vehicles ahead leads to the stabilization of traffic systems the modified korteweg de vries equation the mkdv equation for short near the critical point is derived by applying the reductive perturbation method the traffic jam could be thus described by the kink antikink soliton solution for the mkdv equation from the simulation of space time evolution of the vehicle headway it is shown that the traffic jam is suppressed efficiently with taking into account the information about the motion of more vehicles in front and the analytical result is consonant with the simulation one

sharp exponential integrability for critical riesz potentials and fractional laplacians on rn abstract we derive sharp adams inequalities for the riesz and more general riesz like potentials on the whole of r n as a consequence we obtain sharp moser trudinger inequalities for the critical sobolev spaces w n r n 0 n these inequalities involve fractional laplacians higher order gradients general homogeneous elliptic operators with constant coefficients and general trace type borel measures

fundamental solution for the q laplacian and sharp moser trudinger inequality in carnot groups abstract for a general carnot group g with homogeneous dimension q we prove the existence of a fundamental solution of the q laplacian u q and a constant a q 0 such that exp a q u q is a homogeneous norm on g this implies a representation formula for smooth functions on g which is used to prove the sharp carnot group version of the celebrated moser trudinger inequality

adams inequalities on measure spaces abstract in 1988 adams obtained sharp moser trudinger inequalities on bounded domains of r n the main step was a sharp exponential integral inequality for convolutions with the riesz potential in this paper we extend and improve adams results to functions defined on arbitrary measure spaces with finite measure the riesz fractional integral is replaced by general integral operators whose kernels satisfy suitable and explicit growth conditions given in terms of their distribution functions natural conditions for sharpness are also given most of the known results about moser trudinger inequalities can be easily adapted to our unified scheme we give some new applications of our theorems including sharp higher order moser trudinger trace inequalities sharp adams moser trudinger inequalities for general elliptic differential operators scalar and vector valued for sums of weighted potentials and for operators in the cr setting

bellman functions and dimension free lp estimates for the riesz transforms in bessel settings abstract in this article we prove dimension free l p boundedness of riesz transforms associated with a bessel differential operator we obtain explicit estimates of the l p norms for the bessel riesz transforms in terms of p establishing a linear behavior with respect to p we use the bellman function technique to prove a bilinear dimension free inequality involving poisson semigroups defined through this bessel operator

on negative eigenvalues of two dimensional schrodinger operators with singular potentials we present upper estimates for the number of negative eigenvalues of two dimensional schrodinger operators with potentials generated by ahlfors regular measures of arbitrary fractional dimension 0 2 the estimates are given in terms of integrals of the potential with a logarithmic weight and of its l log l type orlicz norms in the case 1 our results are stronger than the known ones about schrodinger operators with potentials supported by lipschitz curves

a novel car following model considering average speed of preceding vehicles group abstract in this paper a new car following model is presented by considering the average speed effect of preceding vehicles group in cyber physical systems cps environment the effect of this new consideration upon the stability of traffic flow is examined through linear stability analysis a modified korteweg de vries mkdv equation was derived via nonlinear analysis to describe the propagating behavior of traffic density wave near the critical point good agreement between the simulation and the analytical results shows that average speed of preceding vehicles group leads to the stabilization of traffic systems and thus can efficiently suppress the emergence of traffic jamming

nonlinear analysis of traffic flow on a gradient highway abstract we investigate the slope effects upon traffic flow on a single lane gradient uphill downhill highway analytically and numerically the stability condition neutral stability condition and instability condition are obtained by the use of linear stability theory it is found that stability of traffic flow on the gradient varies with the slopes the burgers korteweg de vries kdv and modified korteweg de vries mkdv equations are derived to describe the triangular shock waves soliton waves and kink antikink waves in the stable meta stable and unstable region respectively a series of simulations are carried out to reproduce the triangular shock waves kink antikink waves and soliton waves results show that amplitudes of the triangular shock waves and kink antikink waves vary with the slopes the soliton wave appears in an upward form when the average headway is less than the safety distance and a downward form when the average headway is more than the safety distance moreover both the kink antikink waves and the solitary waves propagate backwards the numerical simulation shows a good agreement with the analytical result

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

critical loci for higgs bundles the paper studies the locus in the moduli space of rank 2 higgs bundles over a curve of genus g corresponding to points which are critical for d of the poisson commuting functions defining the integrable system these correspond to the higgs field vanishing on a divisor d of degree d the degree d critical locus is shown to have an induced integrable system related to k d twisted higgs bundles it is embedded in the singular part of the fibration and a description of these singular fibres using hecke curves is given the methods used are also applied to give information about the cohomology classes of components of the nilpotent cone and of the first critical locus it is shown that whereas in the extreme case d 2g 2 the locus is a hyperkahler submanifold this does not hold in general the example of genus 2 is studied concretely and the d 1 integrable system is seen to be described by a pencil of kummer surfaces

on the schrodinger equation and the eigenvalue problem if k is thekth eigenvalue for the dirichlet boundary problem on a bounded domain in n h weyl s asymptotic formula asserts that lambda k sim c n left frac k v d right 2 n hence sum limits i 1 k lambda i sim frac nc n n 2 k frac n 2 n v d 2 n we prove that for any domain and for all sum limits i 1 k lambda i geqq frac nc n n 2 k frac n 2 n v d 2 n a simple proof for the upper bound of the number of eigenvalues less than or equal to for the operator v x defined on n n 3 in terms of int limits mathbb r n v alpha n 2 is also provided

on the hitchin system 1 1 what is known as the hitchin system is a completely integrable hamiltonian system cihs involving vector bundles over algebraic curves identified by hitchin in h1 h2 it was recently generalized by faltings f in this paper we only consider the case of ranktwo vector bundles with trivial determinant in that case the hitchin system corresponding to a curve c of genus g is obtained as follows let

a car following model considering the effect of electronic throttle opening angle over the curved road abstract this paper proposes an extended car following model by taking the effect of electronic throttle dynamics into account on the curved road in which the electronic throttle opening angle difference from multiple preceding vehicles at the previous moment is considered as a delay feedback control signal then the stability condition of the proposed model is gained using the control theory method considering the effects of friction coefficient radius of curved road and control signal the numerical simulations are designed to study the change of traffic flow stability under different parameters results show that the radius and friction coefficient have a negative effect on stabilizing traffic flow while the stability increases with an increase in the delay feedback control coefficient in addition the numerical simulation also verifies that the control signal can reduce the fuel consumption of traffic flow

bound states in one and two spatial dimensions in this article we study the number of bound states for potentials in one and two spatial dimensions we first show that in addition to the well known fact that an arbitrarily weak attractive potential has a bound state it is easy to construct examples where weak potentials have an infinite number of bound states these examples have potentials which decrease at infinity faster than expected using somewhat stronger conditions we derive explicit bounds on the number of bound states in one dimension using known results for the three dimensional zero angular momentum a change of variables which allows us to go from the one dimensional case to that of two dimensions results in a bound for the zero angular momentum case finally we obtain a bound on the total number of bound states in two dimensions first for the radial case and then under stronger conditions for the noncentral case

a new car following model with the consideration of anticipation optimal velocity abstract in this paper a new anticipation optimal velocity model aovm is proposed by considering anticipation effect on the basis of the full velocity difference model fvdm for car following theory on single lane the linear stability condition is derived from linear stability analysis starting and braking process is investigated for the car motion under a traffic signal which shows that the results accord with empirical traffic values especially aovm can avoid the disadvantage of the unrealistically high deceleration appearing in fvdm furthermore numerical simulation shows that aovm might avoid the disadvantage of negative velocity and headway that occur at small sensitivity coefficients in the fvdm since the anticipation effect is taken into account in aovm which means that collision disappears with the consideration of an appropriate anticipation parameter

an extended continuum model considering optimal velocity change with memory and numerical tests in this paper an extended continuum model of traffic flow is proposed with the consideration of optimal velocity changes with memory the new model s stability condition and kdv burgers equation considering the optimal velocities change with memory are deduced through linear stability theory and nonlinear analysis respectively numerical simulation is carried out to study the extended continuum model which explores how optimal velocity changes with memory affected velocity density and energy consumption numerical results show that when considering the effects of optimal velocity changes with memory the traffic jams can be suppressed efficiently both the memory step and sensitivity parameters of optimal velocity changes with memory will enhance the stability of traffic flow efficiently furthermore numerical results demonstrates that the effect of optimal velocity changes with memory can avoid the disadvantage of historical information which increases the stability of traffic flow on road and so it improve the traffic flow stability and minimize cars energy consumptions

moser inequalities in gauss space the sharp constants in a family of exponential sobolev type inequalities in gauss space are exhibited they constitute the gaussian analogues of the moser inequality in the borderline case of the sobolev embedding in the euclidean space interestingly the gaussian results have features in common with the euclidean ones but also reveal marked diversities

spectral dimension and bohr s formula for schrodinger operators on unbounded fractal spaces we establish an asymptotic formula for the eigenvalue counting function of the schrodinger operator for some unbounded potentials v on several types of unbounded fractal spaces we give sufficient conditions for bohr s formula to hold on metric measure spaces which admit a cellular decomposition and then verify these conditions for fractafolds and fractal fields based on nested fractals in particular we partially answer a question of fan khandker and strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow up of a sierpinski gasket

resolvent expansion and time decay of the wave functions for two dimensional magnetic schrodinger operators we consider two dimensional schrodinger operators h b v given by eq 1 1 below we prove that under certain regularity and decay assumptions on b and v the character of the expansion for the resolvent h b v 1 as 0 is determined by the flux of the magnetic field b through mathbb r 2 subsequently we derive the leading term of the asymptotic expansion of the unitary group e ith b v as t and show how the magnetic field improves its decay in t with respect to the decay of the unitary group e ith 0 v

new classes of potentials for which the radial schrodinger equation can be solved at zero energy in a previous paper chadan and kobayashi 2006 j phys a math gen 39 3381 we presented an infinite class of potentials for which the radial schrodinger equation at zero energy can be solved explicitly for part of them the angular momentum must be zero but for the other part also infinite one can have any angular momentum in the present paper we study first a simple subclass also infinite of the whole class for which the solution of the schrodinger equation is simpler than in the general case this subclass is obtained by combining another approach together with the general approach of the previous paper once this is achieved one can then see that one can in fact combine the two approaches in full generality and obtain a much larger class of potentials than the class found in chadan and kobayashi 2006 we mention here that our results are explicit and when exhibited one can check in a straightforward manner their validity the main result of the present paper is given in theorem 2

on estimates for the number of negative eigenvalues of two dimensional schrodinger operators with potentials supported by lipschitz curves abstract in this paper we present quantitative upper estimates for the number of negative eigenvalues of a two dimensional schrodinger operator with potential supported by an unbounded lipschitz curve the estimates are given in terms of weighted l 1 and l log l type orlicz norms of the potential

relations in the cohomology ring of the moduli space of rank 2 higgs bundles paper is to characterize the rational cohomology ring of 7 n when the rank is 2 and the degree is odd in fact we have given a complete set of generators for this ring in another paper 14 so it is now a question of finding the relations between these generators even though this is a natural companion to the generation problem the ideas and methods with which it is studied have quite a different flavor in particular there is much more explicit calculation what makes the rank 2 case tractable is that the number of generators is man

full velocity difference model for a car following theory in this paper we present a full velocity difference model for a car following theory based on the previous models in the literature to our knowledge the model is an improvement over the previous ones theoretically because it considers more aspects in car following process than others this point is verified by numerical simulation then we investigate the property of the model using both analytic and numerical methods and find that the model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion

an estimate for the morse index of a stokes wave stokes waves are steady periodic water waves on the free surface of an infinitely deep irrotational two dimensional flow under gravity without surface tension they can be described in terms of solutions of the euler lagrange equation of a certain functional this allows one to define the morse index of a stokes wave it is well known that if the morse indices of the elements of a set of non singular stokes waves are bounded then none of them is close to a singular one the paper presents a quantitative variant of this result