remarks on the multiplier operators associated with a cylindrical distance function abstract in this note we consider l p and maximal l p estimates for the generalized riesz means which are associated with the cylindrical distance function max d 1 d 1 r d r we prove these estimates up to the currently known range of the spherical bochner riesz and its maximal operators this is done by establishing implications between the corresponding estimates for the spherical bochner riesz and the cylindrical multiplier operators

an algorithm to compute averages on matrix lie groups averaging is a common way to alleviate errors and random fluctuations in measurements and to smooth out data averaging also provides a way to merge structured data in a smooth manner the present paper describes an algorithm to compute averages on matrix lie groups in particular we discuss the case of averaging over the special orthogonal group of matrices the unitary group of matrices and the group of symmetric positive definite matrices

irreducible representations of nongraded witt type lie algebras abstract from his classification of quadratic conformal algebras corresponding to certain hamiltonian pairs in integrable systems xu gave an explicit construction of seven types of nongraded infinite dimensional simple lie algebra related to a finite set of locally finite derivations on certain commutative associative algebras in this paper we construct a family of irreducible generalized weight modules for the nongraded witt type lie algebras based on finite dimensional irreducible modules of general linear lie algebras our results are nongraded generalizations of rao s work on irreducible representations of the derivation lie algebra of the algebra of laurent polynomials in several variables

new generalized simple lie algebras of cartan type over a field with characteristic 0 abstract we construct four new series of generalized simple lie algebras of cartan type using the mixtures of grading operators and down grading operators our results in this paper are further generalizations of those in osborn s work j algebra 185 1996 820 835

tensor operators and projection techniques in infinite dimensional representations of semi simple lie algebras polynomial identities satisfied by the infinitesimal generators of a semi simple lie group are employed to construct projection operators which project a tensor product representation v lambda x v v lambda finite dimensional and irreducible v an infinite dimensional representation admitting an infinitesimal character onto a primary subrepresentation such projection operators project out the generalised shift components of a tensor operator acting on an infinite dimensional representation v in direct analogy with the finite dimensional analysis of bracken and green 1971 applications of these methods to evaluating the matrix elements of the generators of the locally compact groups u p q and o p q in unitary discrete irreducible representations are discussed

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

stability of an elastic rod buckling into a soft wall the conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field we show that the operator approach presented by manning mannin

the static deformation of a twisted elastic rod constrained to lie on a cylinder the cosserat director theory is used to formulate the problem of a long thin weightless rod constrained by suitable distributed forces to lie on a cylinder while being held by end tension and twisting moment applications of this problem are found for instance in the buckling of drill strings inside a cylindrical hole in the case of a rod of isotropic cross section the equilibrium equations can be reduced to those of a one degree of freedom oscillator in terms of the angle that the local tangent to the rod makes with the axis of the cylinder depending on the radius of the cylinder and the applied load the oscillator has several fixed points each of which corresponds to a different helical solution of the rod more complicated shapes are also possible and special attention is given to localized configurations described by homoclinic orbits of the oscillator heteroclinic saddle connections are found to play an important role in the post buckling behaviour by defining critical loads at which a straight rod may coil up into a helix

on the cone multiplier in r3 abstract we prove the sharp l 3 bounds for the cone multiplier in r 3 and the associated square function which is known as mockenhaupt s square function

extensions of the conformal representations for orthogonal lie algebras abstract the conformal transformations with respect to the metric defining o n c give rise to an inhomogeneous polynomial representation of o n 2 c using shen s technique of mixed product we generalize the above representation to an inhomogeneous representation of o n 2 c on the tensor space of any finite dimensional irreducible o n c module with the polynomial space where a hidden central transformation is involved moreover we find a condition on the constant value taken by the central transformation such that the generalized representation is irreducible in our approach pieri s formulas invariant operators and the idea of kostant s characteristic identities play key roles the result could be useful in understanding higher dimensional conformal field theory with the constant value taken by the central transformation as the central charge our representations virtually provide natural extensions of the conformal transformations on a riemannian manifold to its vector bundles

radial fourier multipliers in high dimensions given a fixed p 2 we prove a simple and effective characterization of all radial multipliers of mathcal f l p left mathbb r d right provided that the dimension d is sufficiently large the method also yields new l q space time regularity results for solutions of the wave equation in high dimensions

weak type estimates for maximal operators with a cylindric distance function we establish sharp weak type estimates for the maximal operators t associated with cylindric riesz means for functions on h p 3 when 4 5 3 p 5 2

on the maximal bochner riesz conjecture in the plane for p 2 we give a new estimate on the maximal bochner riesz operator in the plane for p 2 as a corollary we obtain an almost everywhere convergence result for certain bochner riesz means this work was inspired by discussions with michael christ and chris sogge

application of optimal control method in buckling analysis of constrained elastica problems abstract main concern of the paper is to illustrate the generic application of the optimal control method in the analysis of a constrained buckling problem and its simple and elegant formulation when compared to other techniques thus an optimal control methodology is adopted to investigate the buckling behavior of thin elastic structures under the presence of unilateral constraints we particularly explore the post buckling response of a constant or variable length elastica constrained by rigid walls the equivalence between the calculus of variations and the optimal control is first demonstrated with a main focus on the post buckling behavior of a constant length elastica which is constrained by horizontal walls the gradual construction of the constrained buckling problem as an optimal control problem from its equivalent lagrangian form clearly shows its advantage when compared to the calculus of variations the necessary optimality conditions which constitute the pontryagin s minimum or maximum principle are also derived by considering a direct adjoining approach the solution of the optimal control problem is then performed by applying a direct method validation of the methodology is first achieved by reproducing examples available in the literature then the effects of factors such as the geometry of the walls and the variability in the bending stiffness of the elastica on its buckling response are analyzed these constrained buckling problems are investigated for the first time while through them it is also readily shown that the presence of geometric or material nonlinearity does not introduce any essential complexity in the buckling analysis when the optimal control methodology is adopted

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

generalized projective representations for sl n 1 it is well known that n dimensional projective group gives rise to a non homogeneous representation of the lie algebra sl n 1 on the polynomial functions of the projective space using shen s mixed product for witt algebras also known as larsson functor we generalize the above representation of sl n 1 to a non homogeneous representation on the tensor space of any finite dimensional irreducible gl n module with the polynomial space moreover the structure of such a representation is completely determined by employing projection operator techniques and well known kostant s characteristic identities for certain matrices with entries in the universal enveloping algebra in particular we obtain a new one parameter family of infinite dimensional irreducible sl n 1 modules which are in general not highest weight type for any given finite dimensional irreducible sl n module the results could also be used to study the quantum field theory with the projective group as the symmetry

linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces abstract we obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces as applications we improve the known estimates for oscillatory integrals

bochner riesz means with respect to a rough distance function the generalized bochner riesz operator s r may be defined as s r f x f 1 1 r f x where p is an appropriate distance function and f 1 is the inverse fourier transform the behavior of s r on l p r d x r is described for p d 1 max d 1 a rough distance function we conjecture that this operator is bounded on r d x r when max d 1 2 1 p 1 2 0 and p 2 6 d 3 and unbounded when p 2 6 d 3 this conjecture is verified for large ranges of p

isoperimetric conjugate points with application to the stability of dna minicircles a conjugate point test determining an index of the constrained second variation in one dimensional isoperimetric calculus of variations problems is described the test is then implemented numerically to determine stability properties of equilibria within a continuum mechanics model of dna minicircles

extended hamiltonian learning on riemannian manifolds theoretical aspects this paper introduces a general theory of extended hamiltonian second order learning on riemannian manifolds as an instance of learning by constrained criterion optimization the dynamical learning equations are derived within the general framework of extended hamiltonian stationary action principle and are expressed in a coordinate free fashion a theoretical analysis is carried out in order to compare the features of the dynamical learning theory with the features exhibited by the gradient based ones in particular gradient based learning is shown to be an instance of dynamical learning and the classical gradient based learning modified by a momentum term is shown to resemble discrete time dynamical learning moreover the convergence features of gradient based and dynamical learning are compared on a theoretical basis this paper discusses cases of learning by dynamical systems on manifolds of interest in the scientific literature namely the stiefel manifold the special orthogonal group the grassmann manifold the group of symmetric positive definite matrices the generalized flag manifold and the real symplectic group of matrices

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

representations of the lie algebra of derivations for quantum torus abstract let der c q be the derivation lie algebra of the quantum torus c q we construct a functor from gld modules to der c q modules which generalizes the constructions obtained by shen scientia sinica a 29 1986 570 larsson internat j modern phys a 7 1992 6493 and rao j algebra 182 1996 401 we also give a complete description of the structure of the der c q modules when all the entries of the quantum torus matrix q qij d d are roots of unity

conjugate points revisited and neumann neumann problems the theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite dimensional optimization the object of central importance is the spectrum of the second variation operator analogous to the eigenvalues of the hessian matrix in finite dimensions with a few basic properties of this spectrum one can gain a new perspective on the classic result that stability requires the lack of conjugate points furthermore we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature such as problems with neumann neumann boundary conditions

extensions of inhomogeneous polynomial representations for sl m 1 n we extend an inhomogeneous polynomial representation of special linear lie superalgebras sl m 1 n to an inhomogeneous representation on the tensor space of any irreducible tensor representation of the general linear lie superalgebras gl m n with this polynomial space via shen s mixed product we find a sufficient and necessary condition for irreducibility of these inhomogeneous representations and get the jordan holder series of those which are not irreducible furthermore we point out that all these irreducible modules are lowest weight modules their lowest weights and character formulas are obtained

irreducible representations of the lie algebra of the diffeomorphisms of ad dimensional torus in this paper we investigate the irreducibility of certain modules for the lie algebra of diffeomorphisms of torus t the lie algebra of diffeomorw 1 1 x phisms can be described as the derivations of a s c t t see 1 d w x w x rss we denote this lie algebra by der a larsson l1 constructed a functor from gl modules to der a modules in this paper we prove that d the image of a finite dimensional irreducible gl module is most often d irreducible the only exceptions are fundamental modules and the onedimensional modules in these cases we describe the sub modules and the quotients surprisingly the same class of modules image of finite dimen w x sional gl module are also constructed in e without any reference to d gl modules they are motivated by the vertex operator constructions in d w x em more details near the end of this introduction we will now describe our results in more detail let gl be the d lie algebras of d d matrices over complex numbers c let gl s sl d d c id where sl is the finite dimensional simple lie algebra of trace zero d matrices and id the identity matrix which is central it is well known that finite dimensional irreducible modules and dominant integral weights are in one one correspondence for a finite dimensional simple lie algebra so let v c be finite dimensional irreducible sl module corresponding to a d dominant integral weight c let id the central element act by a complex number b and denote the corresponding gl module by v c b d a d d larsson defined see 1 6 a functor f for a belonging to c rz from w x gl modules to der a modules in l1 these modules are weight modules d

constrained euler buckling an interplay of computation and analysis we consider elastic buckling of an inextensible beam with hinged ends and fixed end displacements confined to the plane and in the presence of rigid frictionless sidewalls which constrain overall lateral displacements we formulate the geometrically nonlinear euler problem and develop global search and path following algorithms to find equilibria in various classes satisfying different contact patterns and hence boundary conditions we derive complete analytical results for the case of line contacts with the sidewalls and partial results for point contact and mixed cases the analysis is essential to understanding the numerical results for in contrast to the unconstrained problem we find a very rich bifurcation structure with the cardinality of branches growing exponentially with mode number