hamiltonian structures and lax equations generated by matrix differential operators with polynomial dependence on a parameter we investigate a general set of equations which can be studied by the inverse scattering method

the construction of rational iterative methods for solving nonlinear equations abstract in this paper based on the idea of constructing iterative methods by sharma j r sharma a family of third order methods to solve nonlinear equations appl math comput 184 2007 210 215 jiang and han dongdong jiang danfu han some one parameter families of third order methods for solving nonlinear equations appl math comput 195 2008 392 396 we propose some rational iterative methods by linearizing the quadratic curve equation by analysis of asymptotic errors we prove the new families are all cubically convergent as an example we present a new concrete iterative family with better convergent performance than classic third order methods through the adjustment of parameter n automatically

classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

nonlinear quantum fields in 4 dimensions and cohomology of the infinite heisenberg group aspects of the cohomology of the infinite dimensional heisenberg group as represented on the free boson field over a given hilbert space are treated the 1 cohomology is shown to be trivial in certain spaces of generalized vectors from this derives a canonical quantization mapping from classical unquantized forms to generalized operators on the boson field an example applied here to scalar relativistic fields is the quantization of a given classical interaction lagrangian or hamiltonian i e the establishment and characterization of corresponding boson field operators for example if q denotes the free massless scalar field in d dimensional minkowski space d 4 even and if q is an even integer greater than or equal to 4 then f smo x q dx exists as a nonvanishing poincare invariant hermitian selfadjointly extendable operator where o x q denotes the wick power applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory including certain massive field theories to a class of pseudo interacting fields obtained by substituting the free field into desingularized expressions for the total hamiltonian in the conformally invariant case d q 4 and to corresponding scattering theory

holomorphic dirichlet forms on complex manifolds a complete description of all holomorphic dirichlet forms on complex hyperbolic space is given they are classified in terms of the lie algebra of the automorphism group which we take to act in the unit ball of n we show that the measure defining the holomorphic dirichlet form is finite if and only if the real part of the associated holomorphic vector field points outward at the boundary of the unit ball a logarithmic sobolev inequality is proven to hold whenever this measure is finite all holomorphic dirichlet forms over complex hyperbolic space complex projective space and n are now known

on complex hypercontractivity abstract we give a new proof of a hypercontractivity theorem for the mehler transform with a complex parameter earlier proved by weissler 1979 j funct anal 32 102 121 and epperson 1989 j funct anal 87 1 30 the proof uses stochastic integrals and ito calculus the method also yields new proofs of some related results

markov semigroups with hypocoercive type generator in infinite dimensions ergodicity and smoothing we start by considering finite dimensional markovian dynamics in rm generated by operators of hypocoercive type and for such models we obtain short and long time pointwise estimates for all the derivatives of any order and in any direction along the semigroup we then look at infinite dimensional models in rm zd produced by the interaction of infinitely many finite dimensional dissipative dynamics of the type indicated above for these infinite dimensional models we study finite speed of propagation of information well posedness of the semigroup time behaviour of the derivatives and strong ergodicity problem

strong hypercontractivity and strong logarithmic sobolev inequalities for log subharmonic functions on stratified lie groups abstract on a stratified lie group g equipped with hypoelliptic heat kernel measure we study the behavior of the dilation semigroup on l p spaces of log subharmonic functions we consider a notion of strong hypercontractivity and a strong logarithmic sobolev inequality and show that these properties are equivalent for any group g moreover if g satisfies a classical logarithmic sobolev inequality then both properties hold this extends similar results obtained by graczyk kemp and loeb in the euclidean setting

estimation optimale du gradient du semi groupe de la chaleur sur le groupe de heisenberg abstract en utilisant l inegalite de poincare et la formule de representation on montre que sur le groupe de heisenberg de dimension reelle 3 h 1 il existe une constante c 0 telle que e t f g c e t f g g h 1 t 0 f c o h 1 ce resultat repond par l affirmation a la question ouverte de b k driver t melcher hypoelliptic heat kernel inequalities on the heisenberg group j funct anal 221 2005 340 365 aussi le resultat principal de b k driver t melcher hypoelliptic heat kernel inequalities on the heisenberg group j funct anal 221 2005 340 365 peut etre considere comme une consequence immediate de l inegalite precedente

strong hypercontractivity and relative subharmonicity abstract a hermitian metric g on a complex manifold m together with a smooth probability measure on m determine minimal and maximal dirichlet forms q d and q max given by q f m g grad f z grad f z d z q d is the form closure of q on c c m and q max is the form closure of q on c 1 b m the corresponding operators a d and a max generate semigroups having standard hypercontractivity properties in the scale of l p spaces p 1 when the corresponding form q satisfies a logarithmic sobolev inequality it was shown by the author 1999 acta math 182 159 206 that the semigroup e ta d has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of l p these results are extended here to a max when m g is not complete it is necessary that the elliptic differential operator a max degenerate on the boundary of m a second proof of these strong hypercontractive inequalities for both a d and a max is given which depends on an extension of the submean value property of subharmonic functions the riemann surface for z 1 n and the weighted bergman spaces in the unit disc are given as examples

the contractivity of the free hamiltonian semigroup in the lp space of entire functions abstract a semigroup of operators t t t 0 generated by the number operator in the complex wave representation is given by t t z e t z for any entire function on c n the problem is to find all possible r p 1 such that t t hlr cn dgn cn dgn is contractive or bounded it turns out that t t hlr cn dgn cn dgn is contractive if and only if p 1 re2t and is unbounded if and only if p re2t furthermore all maximizers are identified they are constant functions if p 1 re2t and exponential functions if p re2t t 0

some second derivative free variants of super halley method with fourth order convergence in this paper we present some new variants of super halley method for solving nonlinear equations these methods are free from second derivatives and require one function and two first derivative evaluations per iteration analysis of convergence shows that the methods are fourth order several numerical examples are given to illustrate the performance of the presented methods

simulated annealing via sobolev inequalities we use sobolev inequalities to study the simulated annealing algorithm this approach takes advantage of the local time reversibility of the process and yields the optimal freezing schedule as well as quantitative information about the rate at which the process is tending to its ground state

on gradient bounds for the heat kernel on the heisenberg group it is known that the couple formed by the two dimensional brownian motion and its levy area leads to the heat kernel on the heisenberg group which is one of the simplest sub riemannian space the associated diffusion operator is hypoelliptic but not elliptic which makes difficult the derivation of functional inequalities for the heat kernel however driver and melcher and more recently h q li have obtained useful gradient bounds for the heat kernel on the heisenberg group we provide in this paper simple proofs of these bounds and explore their consequences in terms of functional inequalities including cheeger and bobkov type isoperimetric inequalities for the heat kernel

ornstein uhlenbeck processes on lie groups we consider ornstein uhlenbeck processes ou processes associated to hypo elliptic diffusion processes on finite dirnensional lie groups let l be a hypo elliptic left invariant sum of the squares operator on a lie group g with associated markov process x then we construct ou processes by adding negative horizontal gradient drifts of functions u in the natural case u x log p l x where p 1 x is the density of the law of x starting at identity e at time t 1 with respect to the right invariant haar measure on g we show the poincare inequality by applying the driver melcher inequality for sum of the squares operators on lie groups the resulting markov process is called the natural ou process associated to the hypo elliptic diffusion on g we prove the global strong existence of these ou type processes on g under an integrability assumption on u the poincare inequality for a large class of potentials u is then shown by a perturbation technique these results are applied to obtain a hypo elliptic equivalent of standard results on cooling schedules for simulated annealing on compact homogeneous spaces m c 2008 elsevier inc all rights reserved

harmonic maps into lie groups classical solutions of the chiral model on considere deux aspects de la structure algebrique de 7b m l espace des applications harmoniques d un domaine a 2 dimensions simplement connexe dans un groupe de lie reel g r la forme reelle d un groupe complexe g on construit une representation du groupe boucle 7b a s 1 g r sur 7b m correspondant a l algebre de lie de kacmoody des deformations infinitesimales on decrit l action sur 7b m et l action d un sous groupe sur l espace des applications harmoniques dans des gaussmanniens dans une seconde partie on se restreint a une theorie qui s applique seulement quand est un domaine riemannien a 2 dimensions simplement connexe et le groupe est g r u n

levy gromov s isoperimetric inequality for an infinite dimensional diffusion generator we establish by simple semigroup arguments a levy gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the gaussian measure this produces in particular a new proof of the gaussian isoperimetric inequality this isoperimetric inequality strengthens the classical logarithmic sobolev inequality in this context a local version for the heat kernel measures is also proved which may then be extended into an isoperimetric inequality for the wiener measure on the paths of a riemannian manifold with bounded ricci curvature

hypoelliptic heat kernel inequalities on the heisenberg group we study the existence of l p type gradient estimates for the heat kernel of the natural hypoelliptic laplacian on the real three dimensional heisenberg lie group using malliavin calculus methods we verify that these estimates hold in the case

gundy varopoulos martingale transforms and their projection operators on manifolds and vector bundles this paper proves the l p boundedness of generalized first order riesz transforms obtained as conditional expectations of martingale transforms a la gundy varopoulos for quite general diffusions on manifolds and vector bundles several specific examples and applications are presented lie groups of compact type the heisenberg group mathbb su 2 and riesz transforms on forms and spinors

contractive inequalities for bergman spaces and multiplicative hankel forms we consider sharp inequalities for bergman spaces of the unit disc establishing analogues of the inequality in carleman s proof of the isoperimetric inequality and of weissler s inequality for dilations by contractivity and a standard tensorization procedure the unit disc inequalities yield corresponding inequalities for the bergman spaces of dirichlet series we use these results to study weighted multiplicative hankel forms associated with the bergman spaces of dirichlet series reproducing most of the known results on multiplicative hankel forms associated with the hardy spaces of dirichlet series in addition we find a direct relationship between the two type of forms which does not exist in lower dimensions finally we produce some counter examples concerning carleson measures on the infinite polydisc

erratum to a nonlinear elliptic system for maps from hermitian to riemannian manifolds and rigidity theorems in hermitian geometry it is the purpose of this paper to introduce and study a nonlinear elliptic system of equations imposed on a map from a hermitian into a riemannian manifold which seems to be more appropriate to hermitian geometry than the harmonic map system thus let x be a complex manifold with hermitian metric 7 in local coordinates n a riemannian manifold with metric gij and christoffel symbols f k a harmonic map f x n then has to satisfy

integrable g strands on semisimple lie groups the present paper derives systems of partial differential equations that admit a quadratic zero curvature representation for an arbitrary real semisimple lie algebra it also determines the general form of hamilton s principles and hamiltonians for these systems and analyzes the linear stability of their equilibrium solutions in the examples of so 3 and sl 2 r

hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n sphere a logarithmic sobolev inequality with respect to the probability measure a 1 x2 12 dx on 1 1 is proved for all 12 from this inequality sharp hypercontractive estimates are derived for the heat semigroups for ultraspherical polynomials and on the n sphere

reverse poincare inequalities isoperimetry and riesz transforms in carnot groups abstract we prove an optimal reverse poincare inequality for the heat semigroup generated by the sub laplacian on a carnot group of any step as an application we give new proofs of the isoperimetric inequality and of the boundedness of the riesz transform in carnot groups

a functional form of the isoperimetric inequality for the gaussian measure abstract letgbe a smooth function on r nwith values in 0 1 using the isoperimetric property of the gaussian measure itis proved that 1 eg e 1 g e g conversely this inequality implies the isoperimetric property of the gaussian measure

some integral identities and inequalities for entire functions and their application to the coherent state transform let be an entire function on cn and for any h 0 and r 0 define fr z r e 2 z 2h let d h denote h n times lebesgue measure on cn frs2 2 d h n sh frsd h from this and a logarithmic sobolev inequality we easily deduce qnq fr q pnp fr p for all 0 p q t8 where the lp norms are taken with respect to the measure d h above we apply these results to the study of the spaces ap consisting of all entire functions satisfying z pe 2 z 2h d h obtaining sharp bounds for some associated operators and proving denseness of analytic polynomials in ap for 1 p we then apply our results to the coherent state transform extending and simplifying some previously known results

degenerate sdes with singular drift and applications to heisenberg groups abstract by using the ultracontractivity of a reference diffusion semigroup krylov s estimate is established for a class of degenerate sdes with singular drifts which leads to existence and pathwise uniqueness by means of zvonkin s transformation the main result is applied to singular sdes on generalized heisenberg groups

from u bounds to isoperimetry with applications to h type groups abstract in this paper we study u bounds in relation to l 1 type coercive inequalities and isoperimetric problems for a class of probability measures on a general metric space r n d we prove the equivalence of an isoperimetric inequality with several other coercive inequalities in this general framework the usefulness of our approach is illustrated by an application to the setting of h type groups and an extension to infinite dimensions

a parametrix for step two hypoelliptic diffusion equations in this paper i construct a parametrix for the hypoelliptic diffusion equations 0 ot l u 0 where l ea 1 9a2 and where the ga are vector fields which satisfy the property that they together with all of the commutators ga gbi for a b are at each point linearly independent and span the tangent space

symplectic structure lagrangian and involutiveness of first integrals of the principal chiral field equation we deal with a form of the chiral equation for which first integrals can be written explicitly for these equations we find a symplectic structure the lagrangian and first integrals in involution

small time gaussian estimates of heat diffusion kernels ii the theory of large deviations la deuxieme partie d un article etudiant le noyau de chaleur d un operateur differentiel d ordre 2 sur une variete c

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

duality on gradient estimates and wasserstein controls abstract we establish a duality between l p wasserstein control and l q gradient estimate in a general framework our result extends a known result for a heat flow on a riemannian manifold especially we can derive a wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound by applying our result to a subelliptic heat flow on a lie group we obtain a coupling of heat distributions which carries a good control of their relative distance

stochastic analysis on sub riemannian manifolds with transverse symmetries we prove a geometrically meaningful stochastic representation of the derivative of the heat semigroup on sub riemannian manifolds with tranverse symmetries this representation is obtained from the study of bochner weitzenbock type formulas for sub laplacians on 1 forms as a consequence we prove new hypoelliptic heat semigroup gradient bounds under natural global geometric conditions the results are new even in the case of the heisenberg group which is the simplest example of a sub riemannian manifold with transverse symmetries

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

an isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in gauss space we prove an isoperimetric inequality on the discrete cube which is the precise analog of a logarithmic inequality due to talagrand as a consequence the gaussian isoperimetric inequality is derived

asymptotics of the spectral gap with applications to the theory of simulated annealing soit m une variete de riemann compacte connexe a n dimensions de produit interieur et soit u m r 1 une fonction lisse donnee pour 0 on definit 7b l c m c m par 7b l o exp u exp u o ou et sont le gradient et la divergence sur m soit la taille de l ecart entre 0 et le reste du spectre de 7b l on etudie le comportement asymptotique de

an inequality for fourier laplace transforms of entire functions and the existence of exponential frames in fock space abstract we study mapping properties of the fourier laplace transform between certain spaces of entire functions we introduce a variant of the classical fock space by integrating against the monge ampere measure of the weight function and show that the norm of the fourier laplace transform in a dual fock space dominates the norm of the function equality holds when the weight function is an hermitean form as an application we get a criterium for the existence of frames of exponential functions in fock space

gaussian kernels have only gaussian maximizers a gaussian integral kernel g x y on rn x rn is the exponential of a quadratic form in x and y the fourier transform kernel is an example the problem addressed here is to find the sharp bound of g as an operator from l p rp to l q rn and to prove that the l p rn functions that saturate the bound are necessarily gaussians this is accomplished generally for 1 q in some special cases besides greatly extending previous results in this area the proof technique is also essentially different from earlier ones a corollary of these results is a fully multidimensional multilinear generalization of young s inequality

singular operators on boson fields as forms on spaces of entire functions on hilbert space abstract invariant scales of entire analytic functions on hilbert space are introduced and applied singular operators represented by sesquilinear forms on spaces of regular vectors are given explicit integral representations via kernels that are entire functions on the direct sum of the hilbert space with its dual the weyl or exponentiated boson field operators act smoothly and irreducibly on corresponding spaces of entire functions arbitrary symplectic operators on a single particle hilbert space are shown to be implementable on the corresponding boson field by appropriate generalized operators

a one parameter family of third order methods to solve nonlinear equations in this paper we present a new one parameter family of iterative methods to solve nonlinear equations which includes some well known third order methods as particular ones the convergence analysis shows that the order of convergence of each method of the family is three numerical examples are given to illustrate the efficiency and performance of the presented methods

an optimal and efficient general eighth order derivative free scheme for simple roots abstract the main motivation of this study is to present an optimal scheme in a general way that can be applied to any existing optimal multipoint fourth order iterative scheme whose first substep employs steffensen s method or steffensen like method to further produce optimal eighth order iterative schemes a rational function approximation approach is used in the construction of proposed scheme in addition we also discussed the theoretical and computational properties of our scheme each member of the presented scheme satisfies the optimality conjecture for multipoint iterative methods without memory which was given by kung and traub in 1970 finally we also concluded on the basis of obtained numerical results that our methods have faster convergence in contrast to the existing methods of same order because they have minimum residual errors minimum error difference between two consecutive iterations and minimum asymptotic error constants corresponding to the considered test function

ornstein uhlenbeck semi groups on stratified groups we consider in the setting of stratified groups g two analogues of the ornstein uhlenbeck semi group namely markovian diffusion semi groups acting on lq pd whose invariant density p is a heat kernel at time 1 on g both generators have the same carre du champ the first semi group is symmetric on l2 pd with generator i 1nxi xi where xi i 1n is a basis of the first layer of the lie algebra of g the second one is compact on lq pd 1 q non symmetric on l2 pd and the formal real part of its generator n is i 1nxi xi the spectrum of n is the set of non negative integers if polynomials are dense in l2 pd i e if g has at most 4 layers we determine in this case the eigenspaces when g is step two we give another description of these eigenspaces similar to the classical definition of hermite polynomials by their generating function

pseudo differential operators and markov semigroups on compact lie groups abstract we extend the ruzhansky turunen theory of pseudo differential operators on compact lie groups into a tool that can be used to investigate group valued markov processes in the spirit of the work in euclidean spaces of n jacob and collaborators feller semigroups their generators and resolvents are exhibited as pseudo differential operators and the symbols of the operators forming the semigroup are expressed in terms of the fourier transform of the transition kernel the symbols are explicitly computed for some examples including the feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by levy processes we study a family of levy type linear operators on general lie groups that are pseudo differential operators when the group is compact and find conditions for them to give rise to symmetric dirichlet forms

g strands on symmetric spaces we study the g strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces these equations are simple field theory models whose configuration space is a lie group or in this case a symmetric space in this class of systems we derive several models that are completely integrable on finite dimensional lie group g and we treat in more detail examples with symmetric space su 2 s1 and so 4 so 3 the latter model simplifies to an apparently new integrable nine dimensional system we also study the g strands on the infinite dimensional group of diffeomorphisms which gives together with the sobolev norm systems of 1 2 camassa holm equations the solutions of these equations on the complementary space related to the witt algebra decomposition are the odd function solutions

the subelliptic heat kernel on su 2 representations asymptotics and gradient bounds the lie group su 2 endowed with its canonical subriemannian structure appears as a three dimensional model of a positively curved subelliptic space the goal of this work is to study the subelliptic heat kernel on it and some related functional inequalities

log sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality let m be a smooth connected manifold endowed with a smooth measure mu and a smooth locally subelliptic diffusion operator l which is symmetric with respect to mu we assume that l satisfies a generalized curvature dimension inequality as introduced by baudoin garofalo cite bg1 our goal is to discuss functional inequalities for mu like the poincare inequality the log sobolev inequality or the gaussian logarithmic isoperimetric inequality

a family of third order methods to solve nonlinear equations by quadratic curves approximation a one parameter family of iteration functions for finding the simple roots of nonlinear equations is presented the iteration process is based on one point approximation by the quadratic equation x 2 ay 2 bx cy d 0 where the unknowns b c and d are determined in terms of a different choices of a correspond to different approximating quadratic curves viz parabola circle ellipse and hyperbola euler chebyshev halley super halley methods and as an exceptional case newton method are seen as the special cases of the family all the methods of the family are cubically convergent except newton s which is quadratically convergent

hypercontractivity for log subharmonic functions abstract we prove strong hypercontractivity shc inequalities for logarithmically subharmonic functions on r n and different classes of measures gaussian measures on r n symmetric bernoulli and symmetric uniform probability measures on r as well as their convolutions surprisingly a slightly weaker strong hypercontractivity property holds for any symmetric measure on r a log sobolev inequality lsi is deduced from the shc for compactly supported measures on r n still for log subharmonic functions an analogous lsi is proved for gaussian measures on r n and for other measures for which we know the shc holds our log sobolev inequality holds in the log subharmonic category with a constant smaller than the one for gaussian measure in the classical context