a simple algorithm for generating discrete prolate spheroidal sequences the discrete prolate spheroidal sequences are optimum waveforms in many communication and signal processing applications because they comprise the most spectral efficient set of orthogonal sequences possible generation of the sequences has proven to be difficult in the past due to the absence of a closed form solution a new method of easily generating any single discrete prolate spheroidal sequence including sequences of very long length is presented also shown are some example sequences generated using the algorithm presented

on the index for array optimization and the discrete prolate spheroidal functions a class of array optimization problems is considered in which we seek to optimize the array response in a specified angular sector the optimization of array directivity is shown to be a special limiting case of these problems as the width of the specified angular sector approaches zero the optimum array patterns are also shown to be related to the well known prolate spheroidal functions

slepian spatial spectral concentration problem on the sphere analytical formulation for limited colatitude longitude spatial region in this paper we develop an analytical formulation for the slepian spatial spectral concentration problem on the sphere for a limited colatitude longitude spatial region on the sphere defined as the cartesian product of a range of positive colatitudes and longitudes the solution of the slepian problem is a set of functions that are optimally concentrated and orthogonal within a spatial or spectral region these properties make them useful for applications where measurements are taken within a spatially limited region of the sphere and or a signal is only to be analyzed within a region of the sphere to support localized spectral spatial analysis and estimation and sparse representation of localized data in these applications we exploit the expansion of spherical harmonics in the complex exponential basis to develop an analytical formulation for the slepian concentration problem for a limited colatitude longitude spatial region we also extend the analytical formulation for spatial regions that are comprised of a union of rotated limited colatitude longitude subregions by exploiting various symmetries of the proposed formulation we design a computationally efficient algorithm for the implementation of the proposed analytical formulation such a reduction in computation time is demonstrated through numerical experiments we present illustrations of our results with the help of numerical examples and show that the representation of a spatially concentrated signal is indeed sparse in the slepian basis

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

the topological trace formula the topological trace formula is a computation of the lefschetz number of a hecke correspondence c acting on the weighted cohomology groups defined in ghm of a locally symmetric space x it expresses this lefschetz number as a sum of contributions from fixed point components of c on the reductive borel serre compactifi cation of x the proof uses the lefschetz fixed point formula of gm2

hodge cohomology of gravitational instantons we study the space of l2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type these metrics generalize the geometric structures at infinity of several different well known classes of metrics including asymptotically locally euclidean manifolds the known types of gravitational instantons and also poincare metrics on q rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in kahler manifolds the answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold the l2 signature formula implied by our result is closely related to the one proved by dai 25 and more generally by vaillant 67 and identifies dai s invariant directly in terms of intersection cohomology of differing perversities this work is also closely related to a recent paper of carron 12 and the forthcoming paper of cheeger and dai 17 we apply our results to a number of examples gravitational instantons among them arising in predictions about l2 harmonic forms in duality theories in string theory

the boden hu conjecture holds precisely up to rank 8 we consider moduli schemes of vector bundles over a smooth projective curve endowed with parabolic structures over a marked point boden and hu observed that a slight variation of the weights leads to a desingularisation of the moduli scheme and they conjectured that one can always obtain a small resolution this way the present text proves this conjecture in some cases including all bundles of rank up to eight and gives counterexamples in all other cases in particular in every rank beyond eight the main tool is a generalisation of ext groups involving more than two quasiparabolic bundles

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

a new optimal window signal processing an optimal window that can provide the best tradeoff between the peak sidelobe level and the total sidelobe level is presented necessary and sufficient conditions are established for the optimal window a design algorithm is presented and examples are discussed the examples demonstrate that the window is capable of much better performance than previously available windows such as the hamming window relationships between the maximum error and the mean squared error are explored for the window optimization problem it is concluded that the minimax and least squares approximations are both fundamentally inefficient this conclusion is supported by experimental evidence and mathematical analyses

eigenfilter design of higher order digital differentiators an effective approach is proposed for designing higher order differentiators by the eigenfilter method by minimizing a quadratic measure of the error in the frequency band an eigenvector of an appropriate matrix is computed to get the filter coefficients this method is not only simple and fast but also optimal in the least squares sense comparison to the mcclellan parks algorithm for minimax equiripple filters shows that both are optimal in the sense of different minimum norms of the error function but much better performance is obtained with the proposed approach in most of the frequency band except in the narrowband region near the cutoff edge

optimal all pole filter design based on discrete prolate spheroidal sequences a new constrained optimization method for all pole digital filter design is presented the design philosophy employed minimizes a function of the area between the ideal low pass filter response in the passband and the actual filter response subject to a quadratic constraint which ensures filter realizability it is shown that unique solutions are obtained which are related to eigenvectors of a toeplitz matrix whose elements form a scaled discrete prolate spheroidal wave sequence dpss it is therefore possible to exploit the properties of dpss and discrete prolate spheroidal wave functions dpswf s to reduce the filter design effort the ratio of passband energy to total energy over the entire filter bandwidth is optimal for these designs

discrete exact solutions of modified volterra and volterra lattice equations via the new discrete sine gordon expansion algorithm abstract recently we have presented a sine gordon expansion method to construct new exact solutions of a wide of continuous nonlinear evolution equations in this paper we further develop the method to be the discrete sine gordon expansion method in nonlinear differential difference equations in particular discrete soliton equations we choose the modified volterra lattice and volterra lattice equation to illustrate the new method such as many types of new exact solutions are obtained moreover some figures display the profiles of the obtained solutions our method can be also applied to other discrete soliton equations

on the uniqueness of bivariant chern class and bivariant riemann roch transformations the existence of bivariant chern classes was conjectured by w fulton and r macpherson and proved by j p brasselet for cellular morphisms of analytic varieties however its uniqueness has been unsolved since then in this paper we show that restricted to morphisms whose target varieties are possibly singular but rational homology manifolds such as orbifolds the bivariant chern classes with rational coefficients are uniquely determined we also discuss some related results and problems

the decomposition theorem perverse sheaves and the topology of algebraic maps we give a motivated introduction to the theory of perverse sheaves culminating in the decomposition theorem of beilinson bernstein deligne and gabber a goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties while most proofs are omitted we discuss several approaches to the decomposition theorem indicate some important applications and examples

on the reductive borel serre compactification l p cohomology of arithmetic groups for large p the l 2 cohomology of an arithmetic quotient of an hermitian symmetric space i e of a locally symmetric variety is known to have the topological interpretation as the intersection homology of its baily borel satake compactification in this article we observe that even without the hermitian hypothesis the l p cohomology of an arithmetic quotient for p finite and sufficiently large is isomorphic to the ordinary cohomology of its reductive borel serre compactification we use this to generalize a theorem of mumford concerning homogeneous vector bundles their invariant chern forms and the canonical extensions of the bundles here though we are referring to canonical extensions to the reductive borel serre compactification of any arithmetic quotient to achieve that we give a systematic discussion of vector bundles and chern classes on stratified spaces

infinitely many conservation laws and integrable discretizations for some lattice soliton equations in this paper by means of the lax representations we demonstrate the existence of infinitely many conservation laws for the general toda type lattice equation the relativistic volterra lattice equation the suris lattice equation and some other lattice equations the conserved density and the associated flux are given formulaically we also give an integrable discretization for a lattice equation with n dependent coefficients

becker gottlieb transfer in etale homotopy introduction in this paper we establish a version of the beckergottlieb transfer in the context of etale homotopy the transfer map we obtain generalises the becker gottlieb transfer to smooth quasi projective varieties over any algebraically closed field of arbitrary characteristic we next recall the becker gottlieb transfer in topology for fiber bundles with compact fiber see b g 3 p 107 let

superperverse intersection cohomology stratification in dependence within its traditional range of perversity parameters intersection cohomology is a topological invariant of pseudomanifolds this is no longer true once one allows superperversities perversities with open image in new window in this case intersection cohomology may depend on the choice of the stratification by which it is defined topological invariance also does not hold if one allows stratifications with codimension one strata nonetheless both errant situations arise in important situations the former in the cappell shaneson superduality theorem and the latter in any discussion of pseudomanifold bordism we show that while full invariance of intersection cohomology under restratification does not hold in this generality it does hold up to restratifications that fix the the top stratum

integrable discretizations of the 2 1 dimensional sinh gordon equation in this paper we propose two semi discrete equations and one fully discrete equation and study them by hirota s bilinear method these equations have continuum limits into a system which admits the 2 1 dimensional generalization of the sinh gordon equation as a result two integrable semi discrete versions and one fully discrete version for the sinh gordon equation are found backlund transformations nonlinear superposition formulae determinant solution and lax pairs for these discrete versions are presented

intersection homology ii in 19 20 we introduced topological invariants ih x called intersection homology groups for the study of singular spaces x these groups depend on the choice of a perversity p a perversity is a function from 2 3 to the non negative integers such that both c and c 2 c are positive and increasing functions of c 2 1 the group ihr is defined for spaces x called pseudomanifolds a pseudomanifold of dimension n is a space that admits a stratification x x n x n z x n 3 x 1 x 0

the discrete prolate spheroidal filter as a digital signal processing tool the discrete prolate spheroidal dps filter is one of the class of nonrecursive finite impulse response fir filters the dps filter first introduced by tufts and francis 1 is superior to other filters in this class in that it has maximum energy concentration in the frequency passband and minimum ringing in the time domain slepian 2 gives a complete discussion of dps function properties we give a mathematical development of the dps filter properties provide information required to easily and accurately construct even many coefficient filters and compare the properties of this filter with those of the more commonly used filters of the same class we note that use of the dps filter allows for particularly useful statements of data time frequency resolution cell values and that overall it forms an especially useful though little known tool for digital signal processing

local cohomology of modules of covariants abstract let g be a connected reductive algebraic group over an algebraically closed field of characteristic zero and let w u be two finite dimensional representations of g in this paper we compute the local cohomology of u sw g provided a certain relatively weak technical condition is true

equivariant sheaves on flag varieties we show that the borel equivariant derived category of sheaves on the flag variety of a complex reductive group is equivalent to the perfect derived category of differential graded modules over the extension algebra of the direct sum of the simple equivariant perverse sheaves this proves a conjecture of soergel and lunts in the case of flag varieties

integrable discretizations of derivative nonlinear schrodinger equations we propose integrable discretizations of derivative nonlinear schrodinger dnls equations such as the kaup newell equation the chen lee liu equation and the gerdjikov ivanov equation by constructing lax pairs the discrete dnls systems admit the reduction of complex conjugation between two dependent variables and possess bi hamiltonian structure through transformations of variables and reductions we obtain novel integrable discretizations of the nonlinear schrodinger nls modified kdv mkdv mixed nls matrix nls matrix kdv matrix mkdv coupled nls coupled hirota coupled sasa satsuma and burgers equations we also discuss integrable discretizations of the sine gordon equation the massive thirring model and their generalizations

hodge genera of algebraic varieties i the aim of this paper is to study the behavior of hodge theoretic intersection homology genera and their associated characteristic classes under proper morphisms of complex algebraic varieties we obtain formulae that relate parametrized families of global invariants of a complex algebraic variety x to such invariants of singularities of proper algebraic maps defined on x such formulae severely constrain both topologically and analytically the singularities of complex maps even between smooth varieties similar results were announced by the first and third author in 13 32 2007 wiley periodicals inc

efficient computation of slepian functions for arbitrary regions on the sphere in this paper we develop a new method for the fast and memory efficient computation of slepian functions on the sphere slepian functions which arise as the solution of the slepian concentration problem on the sphere have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and or a function is required to be analyzed over the spatially limited region slepian functions are currently not easily computed for large band limits for an arbitrary spatial region due to high computational and large memory storage requirements for the special case of a polar cap the symmetry of the region enables the decomposition of the slepian concentration problem into smaller subproblems and consequently the efficient computation of slepian functions for large band limits by exploiting the efficient computation of slepian functions for the polar cap region on the sphere we develop a formulation supported by a fast algorithm for the approximate computation of slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band limits for the proposed algorithm we carry out accuracy analysis of the approximation computational complexity analysis and review of memory storage requirements we illustrate through numerical experiments that the proposed method enables faster computation and has smaller storage requirements while allowing for sufficiently accurate computation of the slepian functions

integrable discretizations and soliton solution of kdv and mkdv equations abstract a method of integrable discretizations for soliton equations is presented which is based on the bilinear formalism taking the kdv equation and mkdv equation for example we discretize the corresponding bilinear equations through properly replacing hirota operator by the hyperbolic operator we also get the soliton solutions of these discrete equations by hirota s bilinear method

an integrable coupled hybrid lattice and its exact solutions a coupled hybrid lattice system is proposed the model can be considered as one of the discrete forms of the coupled integrable korteweg de vries system which is a special case of some physically significant models the lax integrability of the coupled hybrid lattice system is proved by using a simple function expansion approach various novel types of exact solutions such as the kink soliton lattice the bell staggered soliton lattice the kink kink breather lattice and the soliton soliton breather lattice are found

whitney stratified chains and cochains this paper contains the technical constructions necessary for a geometric cycle definition of cohomology and homology in the context of whitney stratifications cup and cap products are interpreted as the transverse intersection of geometric cocycles and cycles