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

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

grassmann invariants almost zeros and the determinantal zero pole assignment problems of linear multivariable systems the strucxpects of the rationxr spaces x c x fwhere x c is the columxof the transfer function matrix g s and x f is the space associated with right matrix fracxcriptions of g s are investigated for x f gcanonixqb s grassmgesentax g x c and g x f arxd and are shown xmplete basis free invariants for x c and x f respectively the almost zeros az and almost decoupling zeros adz of g s gdefinex local minima of a normgn defix g x c and g x f respectively the computation and certain aspects of the distribution in the complex plane of azs and adzs are examined the role of azs and adzs in the determinantal zero and pole assignment problems respectively is examined next two important families of systems are defined the strongly zero non assignable szna and the strongly pole non assignable spna systems for szna and spna systems minimal radius discs dem e z r em e z and dem f zb centred at an az and adz respectively are defined it is shown that dem e r em e z contains at least one zer

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

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

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

solution of the determinantal assignment problem using the grassmann matrices abstractthe paper provides a direct solution to the determinantal assignment problem dap which unifies all frequency assignment problems of the linear control theory the current approach is based on the solvability of the exterior equation v 1 v 2 v m z where z mu v i u u is an n dimensional vector space over f which is an integral part of the solution of dap new criteria for existence of solution and their computation based on the properties of structured matrices are referred to as grassmann matrices the solvability of this exterior equation is referred to as decomposability of z and it is in turn characterised by the set of quadratic plucker relations qprs describing the grassmann variety of the corresponding projective space alternative new tests for decomposability of the multi vector z are given in terms of the rank properties of the grassmann matrix nm z of the vector z which is constructed by the coordinates of z mu it is shown that the exterior equation is solvable z

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

decentralized determinantal assignment problem fixed and almost fixed modes and zeros the decentralized determinantal assignment problem ddap is defined as the unifying description for the study of pole and zero assignment problems under decentralized output state feedback dof dsf and decentralized squaring down dsd respectively ddap is reduced to a linear problem of zero assignment of polynomial combinants and a multilinear problem of restricted decomposability of multivectors the decentralization characteristic dc and the decentralized polynomial grassmann representative d s gr of ddap are defined the fixed zero polynomial of ddap is then determined as the zero polynomial of d s gr the canonical d s gr cd r s gr and the decentralized plucker matrix dpm of ddap are introduced and necessary conditions for arbitrary assignment of the non fixed zeros are given in terms of the dpm the family of strongly zero non assignable sna systems is defined and for such systems the notion of the fixed zero is extended to that of the almost fixed zero afz an

global asymptotic linearisation of the pole placement map a closed form solution for the constant output feedback problem abstract the problem of pole assignment by constant output feedback controllers is studied for minimal systems described by a proper transfer function matrix g s r m p s with mcmillan degree n a new method is presented based on asymptotic linearisation around a degenerate point of the pole placement map related to the problem the essence of the present approach is to reduce the multilinear nature of the problem to one of solving a linear set of equations and this is achieved without losing any of the degrees of freedom in the controller the solution is given in closed form in terms of a one parameter family of static feedback compensators for which the closed loop poles approach the required ones as 0 conditions for the generic as well as exact solvability of the arbitrary pole placement problem are given in terms of the numbers m p n and the rank of a new system invariant the d restricted plucker matrix it is shown that the method works generically when mp n which along with the boundary case mp n is the best possible condition as far as the number of states of the open loop system is concerned for achieving arbitrary pole placement via constant output feedback

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

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

quasi newton methods on grassmannians and multilinear approximations of tensors in this paper we proposed quasi newton and limited memory quasi newton methods for objective functions defined on grassmannians or a product of grassmannians specifically we defined bfgs and limited memory bfgs updates in local and global coordinates on grassmannians or a product of these we proved that when local coordinates are used our bfgs updates on grassmannians share the same optimality property as the usual bfgs updates on euclidean spaces when applied to the best multilinear rank approximation problem for general and symmetric tensors our approach yields fast robust and accurate algorithms that exploit the special grassmannian structure of the respective problems and which work on tensors of large dimensions and arbitrarily high order extensive numerical experiments are included to substantiate our claims

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

the approximate determinantal assignment problem the determinantal assignment problem dap has been introduced as the unifying description of all frequency assignment problems in linear systems and it is studied in a projective space setting this is a multi linear nature problem and its solution is equivalent to finding real intersections between a linear space associated with the polynomials to be assigned and the grassmann variety of the projective space this paper introduces a new relaxed version of the problem where the computation of the approximate solution referred to as the approximate dap is reduced to a distance problem between a point in the projective space from the grassmann variety gm rn the cases g2 rn and its hodge dual gn 2 rn are examined and a closed form solution to the distance problem is given based on the skew symmetric matrix description of multivectors via the gap metric a new algorithm for the calculation of the approximate solution is given and stability radius results are used to investigate the acceptability of the resulting perturbed solutions

central configurations with a quasihomogeneous potential function for the classical n body problem if the masses are paced at rest in the configuration that is central then the masses will tend to the origin and result in a simultaneous collision a winter the analytical foundations of celestial mechanics 1st ed princeton university press princeton 1941 however if the potential is quasihomogeneous in nature that is the potential is of the form u v w where w i jamimjrij and v i jbmimjrij then we will dramatically get different results this paper determines the orbits that will result when the potential function is quasihomogeneous it also shows when there existences a relative equilibrium solution for the regular n gon configuration this is one half of perko s theorem for a homogeneous potential finally we show the existence of a central configuration which is not central for either w or v this configuration is the placement of two mutual equal pairs of masses placed at the vertices of a square this shows that the reverse implication of perk