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

critical exponents for semilinear equations of mixed elliptic hyperbolic and degenerate types for semilinear gellerstedt equations with tricomi goursat or dirichlet boundary conditions we prove poho aev type identities and derive nonexistence results that exploit an invariance of the linear part with respect to certain nonhomogeneous dilations a critical exponent phenomenon of power type in the nonlinearity is exhibited in these mixed elliptic hyperbolic or degenerate settings where the power is 1 less than the critical exponent in a relevant sobolev embedding c 2003 wiley periodicals inc

modeling long memory in stock market volatility inspired by the idea that regime switching may give rise to persistence that is observationally equivalent to a unit root we derive a regime switching process that exhibits long memory the feature of the process that generates long memory is a heavy tailed duration distribution using this process for volatility we obtain a regime switching stochastic volatility rssv model that we fit to daily s p returns from 1928 through 1995 by means of the efficient method of moments estimation emm method forecasts of rssv volatility given past returns can be generated by reprojection as we illustrate the rssv model is accepted according to the emm chi squared statistic using this statistic we also evaluate several other models that have been proposed in the literature and some modifications to them we find that models that exhibit long memory in volatility and heavy tails conditionally as does the rssv model fit the data whereas models without these characteristics do not we also find weak evidence that suggests the presence of an additional short memory component of volatility over and above the long memory component

tangent cones to discriminant loci for families of hypersurfaces a deformation of a variety with nonisolated hypersurface singularities such as a projective hypersurface or a theta divisor of an abelian variety determines a rational map of the singular locus to projective space and the resulting projective geometry of the singular locus describes how the singularities propagate in the deformation the basic principle is that the projective model of the singular locus is dual to the tangent cone to the discriminant of the deformation a detailed study of the method which emerged from interpreting andreotti mayer s work on theta divisors in terms of schlessinger s deformation theory of singularities is given along with examples applications and multiplicity formulas introduction this paper is about the geometry associated to deformations of nonisolated hypersurface singularities the basic principle is that a deformation of a hypersurface determines a projective model of the singular locus whose dual variety is the projectivized tangent cone to the discriminant this includes an infinitesimal form of bertini s theorem for a linear system of divisors and a converse to andreotti mayer s condition for double points of theta divisors now we describe the main results more precisely working in the category of complex analytic spaces let x be a space which can locally be written as a hypersurface of a nonsingular space and let x s be a deformation of x ir 1 0 then t1 the sheaf of first order deformations of x is a line bundle on sg x the singular locus of x and ir determines a kodaira spencer linear map to s k h0 t1 i e a linear system on the singular locus if x and s are nonsingular this linear system has no base locus and the corresponding morphism sg x pto s satisfies 0 p ir p tpx for p e sg x hence is called the gauss map of ir for a projective hypersurface of degree d x is the d fold veronese map restricted to the singular locus and for the theta divisor of an abelian variety 0 is the rational map assigning to a double point its quadric tangent cone assuming the restriction of ir to the critical locus c c x is proper consider the discriminant locus d ir c with its reduced analytic structure in addition to the above suppose finally that c is nonsingular of the expected codimension in x duality theorem there is a finite collection of 8ubvarietie8 zz of sg x which is intrinsic to the analytic space sg x and includes the irreducible components received by the editors march 31 1987 the contents of this paper were presented october 18 1986 by the first author in the special session on singularities and algebraic geometry at the charlotte north carolina meeting of the ams 1980 mathematics subject classification 1985 revision primary 14d15 14n05 secondary 32g11 53c57 d1988 american mathematical society 0002 9947 88 1 00 25 per page

positive solutions of quasilinear elliptic equations with exponential nonlinearity combined with convection term abstract we establish the existence of positive solutions for a nonlinear elliptic dirichlet problem in dimension n involving the n laplacian the nonlinearity considered depends on the gradient of the unknown function and an exponential term in such case variational methods cannot be applied our approach is based on approximation scheme where we consider a new class of normed spaces of finite dimension as a particular case we extended the result achieved by de araujo and montenegro 2016 for any n 2

multivalued elliptic equation with exponential critical growth in r2 abstract in this work we study the existence of nontrivial solutions for the following class of multivalued elliptic problems p u v x u h x t f x u in r 2 where 0 v is a continuous function verifying some conditions h h 1 r 2 and t f x u is the generalized gradient of f x t with respect to t and f x t 0 t f x s d s assuming that f is a discontinuous function with exponential critical growth we have applied variational methods for locally lipschitz functional to get two solutions for p when is small enough

adams type inequalities and related elliptic partial differential equations in dimension four abstract motivated by ruf sani s recent work we prove an adams type inequality and a singular adams type inequality in the whole four dimensional euclidean space as applications of those inequalities a class of elliptic partial differential equations are considered existence of nontrivial weak solutions and multiplicity results are obtained via the mountain pass theorem and the ekeland s variational principle this is a continuation of our previous work about singular trudinger moser type inequality

simulated moments estimation of markov models of asset prices this paper provides a simulated moments estimator sme of the parameters of dynamic models in which the state vector follows a time homogeneous markov process conditions are provided for both weak and strong consistency as well as asymptotic normality various tradeoffs among the regularity conditions underlying the large sample properties of the sme are discussed in the context of an asset pricing model this paper provides conditions for the consistency and asymptotic normality of a simulated moments estimator sme of the parameters of asset pricing models with time homogeneous markov representations of the stochastic forc ing process sme s for economic models have been proposed by mcfadden 1989 and pakes and pollard 1989 for i i d environments and by lee and ingram 1991 for a time series environment the sme for time series models examined in this paper is as follows the state vector yt that determines asset prices is assumed to follow a time homogeneous markov process whose transi tion function depends on an unknown parameter vector 3 0 asset prices and possibly other relevant data are observed as f yt 0 for some given function f of the underlying state and parameter vector in parallel a simulated state process y is generated analytically or numerically from the economic model and corresponding simulated observations f yj3 13 are taken for a given parameter choice f3 the parameter is chosen so as to match moments that is to minimize the distance between sample moments of the data f y 8030 and those of the simulated series f yt f3 in a sense to be made precise the proposed sme extends the generalized method of moments gmm estimator hansen 1982 to a large class of asset pricing models for which the moment restrictions of interest do not have analytic representations in terms of observable variables and the unknown parameter vector we provide conditions on the transition function of yt and the observation function f under which the sme of 030 is consistent and characterize the normalized asymptotic distribu tion of the estimator for two reasons neither the regularity conditions underly ing hansen s 1982 analysis of gmm estimators for time series models without

a reexamination of diffusion estimators with applications to financial model validation time homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variables recently stanton proposed drift and diffusion estimators based on a higher order approximation scheme and kernel regression method he claimed that higher order approximations must outperform lower order approximations and concluded nonlinearity in the instantaneous return function of short term interest rates to examine the impact of higher order approximations we develop general and explicit formulas for the asymptotic behavior of both drift and diffusion estimators we show that these estimators will reduce the numerical approximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximation simulation studies also confirm our asymptotic results this variance inflation problem arises not only from nonparametric fitting but also from parametric fitting stanton s work also postulates the interesting qu

subvarieties of suc 2 and 2theta divisors in the jacobian let suc 2 l denote the projective moduli variety of semistable rank 2 vector bundles with determinant l pic c on a smooth curve c of genus g 2 and suppose that deg l is even it is well known that on the one hand the singular locus of suc 2 l is isomorphic to the kummer variety of the jacobian and on the other hand that when c is nonhyperelliptic suc 2 o has an injective morphism into the linear series 2 on the jacobian jg 1 c which restricts to the kummer embedding a 7 a a on the singular locus dually suc 2 k injects into the linear series l on j c where l o 2 for any theta characteristic and again this map restricts to the kummer map jg 1 c 2 l on the singular locus this map to projective space the two cases are of course isomorphic comes from the complete series on the ample generator of the picard group and at least for a generic curve is an embedding of the moduli space moreover its image contains much of the geometry studied in connection with the schottky problem notably the configuration of prym kummer varieties in this paper we explore a little of the interplay via this embedding between the geometry of vector bundles and the geometry of 2 divisors on the vector bundle side we are principally concerned with the brill noether loci wr suc 2 k defined by the condition h e r on stable bundes e these are analogous to the very classical varieties w r g 1 jg 1 c unlike the line bundle theory however general results connectedness dimension smoothness and so on are not known for the varieties wr see 6 on the 2 side we shall consider the fay trisecants of the 2 embedded kummer variety and the subseries p 00 l consisting of divisors having multiplicity 4 at the origin this subseries is known to be important in the study of principally polarised abelian varieties 10 in the jacobian of a curve its base locus is the surface c c j c plus a pair of isolated points

maximization problem on trudinger moser inequality involving lebesgue norm abstract we study a maximization problem on the trudinger moser inequality and clarify the effect of lower order perturbation on the attainability of the best constant in particular we show the explicit forms of threshold nonlinearity involving lower order perturbation our results are based on the estimates of the dirichlet energy of maximizers

efficient estimation of general dynamic models with a continuum of moment conditions there are two difficulties with the implementation of the characteristic function based estimators first the optimal instrument yielding the ml efficiency depends on the unknown probability density function second the need to use a large set of moment conditions leads to the singularity of the covariance matrix we resolve the two problems in the framework of gmm with a continuum of moment conditions a new optimal instrument relies on the double indexing and as a result has a simple exponential form the singularity problem is addressed via a penalization term we introduce hac type estimators for non markov models a simulated method of moments is proposed for non analytical cases

improved moser trudinger type inequalities in the hyperbolic space hn we establish an improved version of the moser trudinger inequality in the hyperbolic space hnhn n 2n 2 namely we prove the following result for any 0 n 1nn then we have supu c0 hn hn gu gndvolg hn u ndvolg 1 hn n n u nn 1 dvolg where n n n 11n 1 n 1 n 1 denotes the surface area of the unit sphere in rnrn and n t et j 0n 2tjj this improves the moser trudinger inequality in hyperbolic spaces obtained recently by mancini and sandeep 2010 by mancini et al 2013 and by adimurthi and tintarev 2010 for 0 0 in the limiting case n 1n n we prove a moser trudinger inequality with exact growth in hnhn supu c0 hn hn gu gndvolg n 1n n hn u ndvolg 11 hn u ndvolg hn n n u nn 1 1 u nn 1dvolg this improves the moser trudinger inequality with exact growth in hnhn established by lu and tang 2016 these inequalities are achieved from the comparison of the symmetric non increasing rearrangement of a function both in the hyperbolic and the euclidean spaces and the same inequalities in the euclidean space this approach seems to be new comparing with the previous ones

purebred or hybrid reproducing the volatility in term structure dynamics abstract this paper investigates the ability of mixtures of affine quadratic and non linear models to track the volatility in the term structure of interest rates term structure dynamics appear to exhibit pronounced time varying or stochastic volatility ahn et al rev financial stud xx 2001 xxx provide evidence suggesting that term structure models incorporating a set of quadratic factors are better able to reproduce term structure dynamics than affine models although neither class of models is able to fully capture term structure volatility in this study we combine affine quadratic and non linear factors in order to maximize the ability of a term structure model to generate heteroskedastic volatility we show that this combination entails a tradeoff between specification of heteroskedastic volatility and correlations among the factors by combining factors we are able to gauge the cost of this tradeoff using efficient method of moments gallant and tauchen econometric theory 12 1996 657 we find that augmenting a quadratic model with a non linear factor results in improvement in fit over a model comprised solely of quadratic factors when the model only has to confront first and second moment dynamics when the full dynamics are confronted this result reverses since the non linear factor is characterized by stronger dependence of volatility on the level of the factor we conclude that flexibility in the specification of both level dependence and correlation structure of the factors are important for describing term structure dynamics

existence of solutions for a higher order kirchhoff type problem with exponential critical growth abstract a higher order kirchhoff type equation r 2 m m u 2 0 m 1 a x u 2 d x m u 0 m 1 1 a x u f x u x h x in r 2 m is considered in this paper we assume that the nonlinearity of the equation has exponential critical growth and prove that for a positive which is small enough there are two distinct nontrivial solutions to the equation when 0 we also prove that the equation has a nontrivial mountain pass type solution

estimation of affine asset pricing models using the empirical characteristic function abstract the known functional form of the conditional characteristic function ccf of discretely sampled observations from an affine diffusion is used to develop computationally tractable and asymptotically efficient estimators of the parameters of affine diffusions and of asset pricing models in which the state vectors follow affine diffusions both time domain estimators based on fourier inversion of the ccf and frequency domain estimators based directly on the ccf are constructed a method of moments estimator based on the ccf is shown to approximate the efficiency of maximum likelihood for affine diffusion and asset pricing models

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

singularities of the prym theta divisor for the jacobian of a curve the riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve this paper proves an analogous theorem for prym varieties applications of this theorem to cubic threefolds and prym varieties of dimension five are also considered

a statistical method of estimation and simulation for systems of stochastic differential equations summary this paper proposes a new statistical method of estimation and simulation for systems of stochastic differential equations by using this method we can easily derive a discretised process from a system of stochastic differential equations the discretised process is tractable enough to simulate the system and to construct the likelihood which is used to estimate the coefficients of the stochastic differential equations from discrete observation

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

estimating continuous time stochastic volatility models of the short term interest rate abstract we obtain consistent parameter estimates of continuous time stochastic volatility diffusions for the u s risk free short term interest rate sampled weekly over 1954 1995 using the efficient method of moments procedure of gallant and tauchen the preferred model displays mean reversion and incorporates level effects and stochastic volatility in the diffusion function extensive diagnostics indicate that the cox ingersoll ross model with an added stochastic volatility factor provides a good characterization of the short rate process further they suggest that recently proposed garch models fail to approximate the discrete time short rate dynamics while level egarch models perform reasonably well

linear relations among holomorphic quadratic differentials and induced siegel s metric on mg we find the explicit form of the volume form on the moduli space of non hyperelliptic riemann surfaces induced by the siegel metric a long standing question in string theory this question is related to the explicit form of the g 2 g 3 2 linearly independent relations among the 2 fold products of holomorphic abelian differentials that are provided in the case of canonical curves of genus g 4 such relations can be completely expressed in terms of determinants of the standard normalized holomorphic abelian differentials remarkably it turns out that the induced volume form is the kodaira spencer map of the square of the bergman reproducing kernel

recovering of curves with involution by extended prym data the classical torelli theorem states that every smooth projective algebraic curve x is determined uniquely up to isomorphism by its principally polarized jacobian j x 0 in this paper we consider curves c with involution without fixed points a c c we let c c tr and denote by n c c the factor map one associates the principally polarized prym variety p where p p c a 1 tr d t and e is algebraically equivalent to 23 the natural question whether the prym variety determines uniquely the pair c tr up to isomorphism has negative answer in general if the genus g of c is 7 for some special loci of curves e g for hyperelliptic c the problem of spotting the pairs c rr which are not determined uniquely by the prym variety is still open some partial results have been obtained in f s k dol do2 del de2 n v in this paper we propose an extension of the prym data p 3 and prove that it determines uniquely up to isomorphism any pair c tr for g 2 our extension originates from the following observation consider the case of c of genus 1 here the involution is a translation t by some point p of order 2 in j c notice that 0 ker nm j c j c the prym variety is equal to 0 there is a classically known data which determines uniquely the pair c tj up to isomorphism see e g m3 c namely one can always represent j c as c z z where z belongs to the upper half plane so that

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

the relative efficiency of method of moments estimators abstract the asymptotic relative efficiency of efficient method of moments when implemented with a seminonparametric auxiliary model is compared to that of conventional method of moments when implemented with polynomial moment functions because the expectations required by these estimators can be computed by simulation these two methods are commonly used to estimate the parameters of nonlinear latent variables models the comparison is for the models in the marron wand test suite a scale mixture of normals and the second largest order statistic of the lognormal distribution the latter models are representative of financial market data and auction data respectively which are the two most common applications of simulation estimators efficient method of moments dominates conventional method of moments over these models

scrollar syzygies of general canonical curves with genus le 8 we prove that for a general canonical curve c c z 9 1 of genus g the space of g 5 2 th last scrollar syzygies is isomorphic to the brill noether locus c 1 g 2 2 schreyer has conjectured that these scrollar syzygies span the space of all g 5 2 th last syzygies of c using mukai varieties we prove this conjecture for genus 6 7 and 8

on a singular and nonhomogeneous n laplacian equation involving critical growth abstract in this paper we apply minimax methods to obtain existence and multiplicity of weak solutions for singular and nonhomogeneous elliptic equation of the form n u f x u x a h x in where u w 0 1 n n u div u n 2 u is the n laplacian a 0 n is a smooth bounded domain in r n n 2 containing the origin and h w 0 1 n w 1 n is a small perturbation h 0 motivated by a singular trudinger moser inequality we study the case when f x s has the maximal growth on s which allows to treat this problem variationally in the sobolev space w 0 1 n

notes on financial econometrics the first part of the discussion reviews recent successes in modeling of discrete time financial data and argues that a direct approach is better suited than stochastic volatility the second part reviews recent work on estimating continuous time models with emphasis on simulation based techniques and joint estimation of the risk neutral and objective probability distributions

simulation estimation of time series models abstract a formal econometric treatment of the estimation of the parameters of a fully specified stochastic equilibrium model is proposed the method estimation by simulation yields an estimator which is shown to have an asymptotic normal distribution a goodness of fit test based on a chi square statistic is also derived

inference for stochastic volatility models using time change transformations we address the problem of parameter estimation for diffusion driven stochastic volatility models through markov chain monte carlo mcmc to avoid degeneracy issues we introduce an innovative reparametrisation defined through transformations that operate on the time scale of the diffusion a novel mcmc scheme which overcomes the inherent difficulties of time change transformations is also presented the algorithm is fast to implement and applies to models with stochastic volatility the methodology is tested through simulation based experiments and illustrated on data consisting of us treasury bill rates

the surface c c on jacobi varieties and 2nd order theta functions preprint enviat per a la seva publicacio en una revista cientifica acta mathematica volume 157 1986 1 22 https projecteuclid org euclid acta 1485890436

existence of positive solutions to quasi linear elliptic equations with exponential growth in the whole euclidean space abstract in this paper a quasi linear elliptic equation in the whole euclidean space is considered the nonlinearity of the equation is assumed to have exponential growth or have critical growth in view of trudinger moser type inequality under some assumptions on the potential and the nonlinearity it is proved that there is a nontrivial positive weak solution to this equation also it is shown that there are two distinct positive weak solutions to a perturbation of the equation the method of proving these results is combining trudinger moser type inequality mountain pass theorem and ekeland s variational principle

extremal for a k hessian inequality of trudinger moser type we consider k hessian operators s k u in bounded domains varomega in mathbb r n such that partial varomega is k 1 convex for so called k admissible functions u in varphi 0 k one has sobolev type inequalities of the form begin aligned vert u vert l p varomega le c vert u vert varphi 0 k end aligned where vert u vert varphi 0 k k 1 int varomega u s k u dx and 1 le p le k frac n k 1 n 2k the case n 2k is a borderline case of trudinger moser type and it was recently shown by tian wang that a corresponding inequality of exponential type holds begin aligned sup vert u vert varphi 0 k le 1 int varomega left mathrm e alpha u frac n 2 n sum j 0 k 1 frac alpha j u j frac n 2 n j right mathrm d x le c end aligned for alpha le alpha n n left frac omega n 1 k n 1 atopwithdelims k 1 right 2 n in this article we prove an analogue to the famous result of carleson chang namely that for varomega b r 0 the above supremum is attained also in the limiting case alpha alpha n

sharp threshold nonlinearity for maximizing the trudinger moser inequalities abstract we study existence of maximizer for the trudinger moser inequality with general nonlinearity of the critical growth on r 2 as well as on the disk we derive a very sharp threshold nonlinearity between the existence and the non existence in each case in asymptotic expansions with respect to growth and decay of the function the expansions are explicit using apery s constant we also obtain an asymptotic expansion for the exponential radial sobolev inequality on r 2

sharp adams type inequalities in sobolev spaces wm nm rn for arbitrary integer m the main purpose of our paper is to prove sharp adams type inequalities in unbounded domains of rn for the sobolev space wm nm rn for any positive integer m less than n our results complement those of ruf and sani in press 35 where such inequalities have been established for even integer m we extend the main techniques of ruf and sani in press 35 which are the combinations of the comparison principle of talenti 1976 36 and trombetti and vazquez 1985 38 for polyharmonic operators and a symmetrization argument together with constructions of radial auxiliary functions to the case when m is odd moreover we offer a completely different but much simpler approach to prove the comparison principle using the power of bessel potentials and the riesz rearrangement see remarks 3 2 and 3 3 this approach is of independent interest and works for any differential operators with appropriate radial kernels as corollaries of our main theorems we will derive the adams type inequalities in the case when n 2m for all positive integer m by using different sobolev norms

estimating stochastic differential equations efficiently by minimum chi squared summary we propose a minimum chi squared estimator for the parameters of an ergodic system of stochastic differential equations with partially observed state we prove that the efficiency of the estimator approaches that of maximum likelihood as the number of moment functions entering the chi squared criterion increases and as the number of past observations entering each moment function increases the minimised criterion is asymptotically chi squared and can be used to test system adequacy when a fitted system is rejected inspecting studentised moments suggests how the fitted system might be modified to improve the fit the method and diagnostic tests are applied to daily observations on the u s dollar to deutschmark exchange rate from 1977 to 1992

existence and non existence of radial solutions for elliptic equations with critical exponent in ℝ2 where 2 2 n n 2 is the critical sobolev exponent for the embedding h 52 c l2 0 criticality and subcriticality play important rbles concerning the solvability of equation 1 1 using by now classical variational methods one sees that equation 1 1 is solvable i f f has subcritical growth and satisfies some additional hypotheses while for f u iuip u the well known pohoiaev identity shows that the solvability of equation 1 1 is lost if p reaches 2 and r is starshaped in their important paper 6 brezis nirenberg showed that for n e 4 solvability is regained if one adds a lower order linear term to the critical term more precisely they showed that for f r ar lr12 2r one has a denotes the 1st eigenvalue of a on ha r

high rank linear syzygies on low rank quadrics we study the linear syzygies of a homogeneous ideal i subseteq s rm sym bf k v focussing on the graded betti numbers b i i 1 rm dim bf k rm tor i s i bf k i 1 for a variety x and divisor d with v h 0 d what conditions on d ensure that b i i 1 ne 0 eisenbud has shown that a decomposition d sim a b such that a and b have at least two sections gives rise to determinantal equations and corresponding syzygies in i x and conjectured that if i 2 is generated by quadrics of rank le 4 then the last nonvanishing b i i 1 is a consequence of such equations we describe obstructions to this conjecture and prove a variant the obstructions arise from toric specializations of the rees algebra of koszul cycles and we give an explicit construction of toric varieties with minimal linear syzygies of arbitrarily high rank this gives one answer to a question posed by eisenbud and koh

ground state solutions of hamiltonian elliptic systems in dimension two the aim of this paper is to study hamiltonian elliptic system of the form 0 1 where 2 is a bounded domain in the second place we present existence results for the following stationary schrodinger systems defined in the whole plane 0 2 we assume that the nonlinearities f g have critical growth in the sense of trudinger moser by using a suitable variational framework based on the generalized nehari manifold method we obtain the existence of ground state solutions of both systems 0 1 and 0 2

a hardy moser trudinger inequality abstract in this paper we obtain an inequality on the unit disk b in r 2 which improves the classical moser trudinger inequality and the classical hardy inequality at the same time namely there exists a constant c 0 0 such that b e 4 u 2 h u d x c 0 u c 0 b 0 where h u b u 2 d x b u 2 1 x 2 2 d x this inequality is a two dimensional analog of the hardy sobolev maz ya inequality in higher dimensions which has been intensively studied recently we also prove that the supremum is achieved in a suitable function space which is an analog of the celebrated result of carleson chang for the moser trudinger inequality

semi nonparametric maximum likelihood estimation on developpe une approche basee sur une innovation due a phillips 1983 sur les approximations des fonctions de densite

alternative models for stock price dynamics abstract this paper evaluates the role of various volatility specifications such as multiple stochastic volatility sv factors and jump components in appropriate modeling of equity return distributions we use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns we consider two broad families of models 1 the multifactor loglinear family and 2 the affine jump family both classes of models have attracted much attention in the derivatives and econometrics literatures there are various tradeoffs in considering such diverse specifications if pure diffusion sv models are chosen over jump diffusions it has important implications for hedging strategies if logarithmic models are chosen over affine ones it may seriously complicate option pricing comparing many different specifications of pure diffusion multifactor models and jump diffusion models we find that 1 log linear models have to be extended to two factors with feedback in the mean reverting factor 2 affine models have to have a jump in returns stochastic volatility or probably both models 1 and 2 are observationally equivalent on the data set in hand in either 1 or 2 the key is that the volatility can move violently as we obtain models with comparable empirical fit one must make a choice based on arguments other than statistical goodness of fit criteria the considerations include facility to price options to hedge and parsimony the affine specification with jumps in volatility might therefore be preferred because of the closed form derivatives prices

estimation of continuous time markov processes sampled at random time intervals we introduce a family of generalized method of moments estimators of the parameters of a continuous time markov process observed at random time intervals the results include strong consistency asymptotic normality and a characterization of standard errors sampling is at an arrival intensity that is allowed to depend on the underlying markov process and on the parameter vector to be estimated we focus on financial applications including tick based sampling allowing for jump diffusions regime switching diffusions and reflected diffusions

existence and nonexistence of maximizers for variational problems associated with trudinger moser type inequalities in mathbb r n we discuss the existence of a maximizer for a maximizing problem associated with the trudinger moser type inequality in mathbb r n n geq 2 different from the bounded domain case we obtain both of the existence and the nonexistence results the proof requires a careful estimate of the maximizing level with the aid of normalized vanishing sequences

min max level estimate for a singular quasilinear polyharmonic equation in r2m abstract using the framework first presented by ruf and sani we give a proof of an adams type inequality which can be applied to the functional j u 1 2 r 2 m m u 2 0 m 1 a x u 2 d x r 2 m f x u x d x r 2 m h u d x under two kinds of assumptions on the nonlinearity we estimate the min max level of the functional as an application a multiplicity result for the related singular quasilinear elliptic equation is proved

parametric estimation from approximate data non gaussian diffusions we study the problem of parameters estimation in indirect observability contexts where x t in r r is an unobservable stationary process parametrized by a vector of unknown parameters and all observable data are generated by an approximating process y varepsilon t which is close to x t in l 4 norm we construct consistent parameter estimators which are smooth functions of the sub sampled empirical mean and empirical lagged covariance matrices computed from the observable data we derive explicit optimal sub sampling schemes specifying the best paired choices of sub sampling time step and number of observations we show that these choices ensure that our parameter estimators reach optimized asymptotic l 2 convergence rates which are constant multiples of the l 4 norm y varepsilon t x t

on nonlinear perturbations of a periodic elliptic problem in r2 involving critical growth we consider the equation u v x u f x u for x r2 where v r2 r is a positive potential bounded away from zero and the nonlinearity f r2 r r behaves like exp u 2 as u we also assume that the potential v x and the nonlinearity f x u are asymptotically periodic at infinity we prove the existence of at least one weak positive solution u h1 r2 by combining the mountain pass theorem with trudinger moser inequality and a version of a result due to lions for critical growth in r2

siegel metric and curvature of the moduli space of curves we study the curvature of the moduli space mg of curves of genus g with the siegel metric induced by the period map j mg ag we give an explicit formula for the holomorphic sectional curvature of mg along a schiffer variationp for p a point on the curve x in terms of the holomorphic sectional curvature of ag and the second gaussian map finally we extend the kahler form of the siegel metric as a closed current on m g and we determine its cohomology class as a multiple of