properties of the voice transform of the blaschke group and connections with atomic decomposition results in the weighted bergman spaces abstract feichtinger and grochenig described a unified approach to atomic decomposition through integrable group representations in banach spaces studying the properties of a special voice transform of the blascke group outlined by the general theory developed by feichtinger and grochenig we obtain that every function from the minimal mobius invariant space will generate an atomic decomposition in the weighted bergman spaces in the unified approach of the atomic decomposition a useful tool is the q density the v separated property and the bounded uniform partitions of the unity of the locally compact group using the hyperbolic metric we can describe the q density from right and the separation from right in the blaschke group using this we can give an example of bounded uniform partitions of the unity from right in the general theory of atomic decomposition it is used the q density from the left this is the reason why we will make a small modification in the discretizing operator which corresponds to the q density from the right in order to obtain atomic decomposition in the weighted bergman spaces

frames arising from irreducible solvable actions i abstract in this work we provide a unified method for the construction of reproducing systems arising from unitary irreducible representations of some solvable lie groups in contrast to other well known techniques such as the coorbit theory the generalized coorbit theory and other discretization schemes we make no assumption on the integrability or square integrability of the representations of interest moreover our scheme produces explicit constructions of frames with precise frame bounds as an illustration of the scope of our results we highlight that a large class of representations which naturally occur in wavelet theory and time frequency analysis is handled by our scheme for example the affine group the generalized heisenberg groups the shearlet groups solvable extensions of vector groups and various solvable extensions of non commutative nilpotent lie groups are a few examples of groups whose irreducible representations are handled by our method the class of representations studied in this work is described as follows let g be a simply connected connected completely solvable lie group with lie algebra g p m next let be an infinite dimensional unitary irreducible representation of g obtained by inducing a character from a closed normal subgroup p exp p of g additionally we assume that g p m m exp m is a closed subgroup of g d m is a fixed haar measure on the solvable lie group m and there exists a linear functional p such that the representation ind p g is realized as acting in l 2 m d m making no assumption on the integrability of we describe explicitly a discrete subset of g and a vector f l 2 m d m such that f is a tight frame for l 2 m d m we also construct compactly supported smooth functions s and discrete subsets g such that s is a frame for l 2 m d m

mesh adaptive direct search algorithms for constrained optimization this paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available we introduce the mesh adaptive direct search mads class of algorithms which extends the generalized pattern search gps class by allowing local exploration called polling in an asymptotically dense set of directions in the space of optimization variables this means that under certain hypotheses including a weak constraint qualification due to rockafellar mads can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained the main gps convergence result is to identify limit points hat x where the clarke generalized derivatives are nonnegative in a finite set of directions called refining directions although in the unconstrained case nonnegative combinations of these directions span the whole space the fact that there can only be finitely many gps refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for gps the main result of this paper is that the general mads framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the clarke derivatives are nonnegative we propose an instance of mads for which the refining directions are dense in the hypertangent cone at hat x with probability 1 whenever the iterates associated with the refining directions converge to a single hat x the instance of mads is compared to versions of gps on some test problems we also illustrate the limitation of our results with examples

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

weyl product algebras and modulation spaces we discuss algebraic properties of the weyl product acting on modulation spaces for a certain class of weight functions omega we prove that m omega p q is an algebra under the weyl product if p epsilon 1 infinity and 1 q min p p for the remaining cases p epsilon 1 infinity and min p p q infinity we show that the unweighted spaces m p m q are not algebras under the weyl product c 2007 elsevier inc all rights reserved

modeling and optimization of an ammonia reactor using a penalty like method abstract ammonia is one of the most important chemicals produced in the world due to its broad applicability modeling and simulation of process of ammonia has received considerable attention among the industrial processes ammonia is produced from the reaction of hydrogen and nitrogen at high temperature and high pressure along with a catalyst its production depends on temperature of feed gas at the top of the reactor the partial pressures of the reactants and the reactor length the optimal design problem requires obtaining the optimal reactor length with maximum economic return subject to a number of equality constraints involving solution of coupled nonlinear differential equations a more efficient algorithm for the solution of optimization constrained differential equations is proposed by coupling numerical techniques barrier function with a direct search method with runge kutta method the global convergence of the proposed algorithm is established under some usual conditions numerical results showed that the resulting algorithm based only on function values is highly competitive with other global optimization methods

the space complexity analysis in the general number field sieve integer factorization the general number sieve is the most efficient algorithm for integer factorization it consists of polynomial selection sieving solving equations and finding square roots root lifting of polynomial is discussed in this paper the p adic evaluation provided by each root and the expected p value are also given then we gain the space complexity of sieving and building equations over the ring z 2 z

generalized coorbit space theory and inhomogeneous function spaces of besov lizorkin triebel type coorbit space theory is an abstract approach to function spaces and their atomic decom positions the original theory developed by feichtinger and grochenig in the late 1980ies heavily uses integrable representations of locally compact groups their theory covers in particular homogeneous besov lizorkin triebel spaces modulation spaces bergman spaces and the recent shearlet spaces however inhomogeneous besov lizorkin triebel spaces cannot be covered by their group theoretical approach later it was recognized by fornasier and the first named author 24 that one may replace coherent states related to the group representation by more general abstract continuous frames in the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces a unified approach towards atomic decompositions and banach frames with new results for general coorbit spaces is presented in the second part we apply the abstract setting to a specific framework and study coorbits of what we call peetre spaces they allow to recover inhomogeneous besov lizorkin triebel spaces of var ious types of interest as coorbits we obtain several old and new wavelet characterizations based on explicit smoothness decay and vanishing moment assumptions of the respective wavelet as main examples we obtain results for weighted spaces muckenhoupt doubling general 2 microlocal spaces besov lizorkin triebel morrey spaces spaces of dominating mixed smoothness and even mixtures of the mentioned ones due to the generality of our approach there are many more examples of interest where the abstract coorbit space theory is applicable

time frequency analysis of localization operators abstract we study a class of pseudodifferential operators known as time frequency localization operators anti wick operators gabor toeplitz operators or wave packets given a symbol a and two windows 1 2 we investigate the multilinear mapping from a 1 2 s r 2d s r d s r d to the localization operator aa 1 2 and we give sufficient and necessary conditions for aa 1 2 to be bounded or to belong to a schatten class our results are formulated in terms of time frequency analysis in particular we use modulation spaces as appropriate classes for symbols and windows

toeplitz operators and localization operators we show that for any localization operator on the fock space with polynomial window there exists a constant coecient linear partial dierential operator d such that the localization operator with symbol f coincides with the toeplitz operator with symbol df an analogous result also holds in the context of bergman spaces on bounded symmetric domains this verifies a recent conjecture of coburn and simplifies and generalizes recent results of lo

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

banach spaces of distributions having two module structures abstract after a discussion of a space of test functions and the corresponding space of distributions a family of banach spaces b b in standard situation is described these are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution the main results concern relations between the different spaces associated to b established by means of well known methods from the theory of banach modules among them b0 and b the closure of the test functions in b and the weak relative completion of b respectively the latter is shown to be always a dual banach space the main diagram given in theorem 4 7 gives full information concerning inclusions between these spaces showing also a complete symmetry a great number of corresponding formulas is established how they can be applied is indicated by selected examples in particular by certain segal algebras and the ap algebras of herz various further applications are to be given elsewhere

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

a new active set algorithm for box constrained optimization an active set algorithm asa for box constrained optimization is developed the algorithm consists of a nonmonotone gradient projection step an unconstrained optimization step and a set of rules for branching between the two steps global convergence to a stationary point is established for a nondegenerate stationary point the algorithm eventually reduces to unconstrained optimization without restarts similarly for a degenerate stationary point where the strong second order sufficient optimality condition holds the algorithm eventually reduces to unconstrained optimization without restarts a specific implementation of the asa is given which exploits the recently developed cyclic barzilai borwein cbb algorithm for the gradient projection step and the recently developed conjugate gradient algorithm cg descent for unconstrained optimization numerical experiments are presented using box constrained problems in the cuter and minpack 2 test problem libraries

a progressive barrier for derivative free nonlinear programming we propose a new constraint handling approach for general constraints that is applicable to a widely used class of constrained derivative free optimization methods as in many methods that allow infeasible iterates constraint violations are aggregated into a single constraint violation function as in filter methods a threshold or barrier is imposed on the constraint violation function and any trial point whose constraint violation function value exceeds this threshold is discarded from consideration in the new algorithm unlike the filter method the amount of constraint violation subject to the barrier is progressively decreased adaptively as the iteration evolves we test this progressive barrier pb approach versus the extreme barrier eb with the generalized pattern search gps and the lower triangular mesh adaptive direct search ltmads methods for nonlinear derivative free optimization tests are also conducted using the gps filter which uses a version of the fletcher leyffer filter approach we know that gps cannot be shown to yield kkt points with this strategy or the filter but we use the clarke nonsmooth calculus to prove clarke stationarity of the sequences of feasible and infeasible trial points for ltmads pb numerical experiments are conducted on three academic test problems with up to 50 variables and on a chemical engineering problem the new ltmads pb method generally outperforms our ltmads eb in the case where no feasible initial points are known and it does as well when feasible points are known which leads us to recommend ltmads pb thus the ltmads pb is a useful practical extension of our earlier ltmads eb algorithm particularly in the common case for real problems where no feasible point is known the same conclusions hold for gps pb versus gps eb

decay and smoothness for eigenfunctions of localization operators abstract we study decay and smoothness properties for eigenfunctions of compact localization operators a a 1 2 operators a a 1 2 with symbols a in the wide modulation space m p containing the lebesgue space l p p and windows 1 2 in the schwartz class s are known to be compact we show that their l 2 eigenfuctions with non zero eigenvalues are indeed highly compressed onto a few gabor atoms similarly for symbols a in the weighted modulation spaces m v s 1 r 2 d s 0 subspaces of m p r 2 d p 2 d s the l 2 eigenfunctions of a a 1 2 are actually schwartz functions an important role is played by quasi banach wiener amalgam and modulation spaces as a tool new convolution relations for modulation spaces and multiplication relations for wiener amalgam spaces in the quasi banach setting are exhibited

vector valued nonstationary gabor frames abstract as an extension of gabor frames nonstationary gabor nsg frames were recently introduced in adaptive signal analysis they allow for efficient reconstruction with flexible sampling and varying window functions in this paper we generalize the notion of nsg frames from l 2 r to the vector valued hilbert space l 2 r c l and investigate the resulting vector valued nsg frames we derive a walnut s representation of the mixed frame operator and provide some necessary sufficient conditions for a vector valued nsg system to be a frame for l 2 r c l furthermore we show the existence of painless vector valued nsg frames and of vector valued nsg frames with fast decaying window functions

coherent state lattices and square integrability of representations in this paper we discretize the continuous theory of coherent states on a general semidirect product group g v x s where v is a vector space and s subset of gl v is a semisimple connected lie group we show that it is always possible to construct a discrete frame associated with a unitary irreducible representation of g which is square integrable modulo an affine admissible section we also prove the converse result that the existence of a discrete frame associated with a unitary representation u of g indexed by a homogeneous space gamma implies the square integrability of u on gamma

shear madness new orthonormal bases and frames using chirp functions the proportional bandwidth and constant bandwidth time frequency signal decompositions of the wavelet gabor and wilson orthonormal bases have attracted substantial interest for representing nonstationary signals however these representations are limited in that they are based on rectangular tessellations of the time frequency plane while much effort has gone into methods for designing nice wavelet and window functions for these frameworks little consideration has been given to methods for constructing orthonormal bases employing nonrectangular time frequency tilings the authors take a first step in this direction by deriving two new families of orthonormal bases and frames employing elements that shear or chirp in the time frequency plane in addition to translate and scale the new scale shear fan bases and shift shear chevron bases are obtained by operating on an existing wavelet gabor 1946 or wilson basis set with two special unitary warping transformations in addition to the theoretical benefit of broadening the class of valid time frequency plane tilings these new bases could possibly also be useful for representing certain types of signals such as chirping and dispersed signals

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

the range of localization operators and lifting theorems for modulation and bargmann fock spaces we study the range of time frequency localization operators acting on modulation spaces and prove a lifting theorem as an application we also characterize the range of gabor multipliers and in t

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

continuity properties for modulation spaces with applications to pseudo differential calculus i we discuss continuity for weighted modulation spaces andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces we also discussthe trace operator a a 0 acting on modulationspaces the results are used to get inclusions betweenmodulation spaces and besov spaces and proving continuityfor pseudo differential operators and toeplitz operators

an improved parallel block lanczos algorithm over gf 2 for integer factorization abstract rsa algorithm is one of the most popular and secure public key cryptographic algorithms and has been widely used in many real life applications the security of the rsa algorithm lies in the difficulty of factoring large integers efficiently and the general number field sieve gnfs algorithm is the most efficient algorithm for factoring integers greater than 110 digits at present in this paper targeted to speed up the factorization process of rsa we discuss the current research about solving large and sparse linear systems over gf 2 which is one of the most time consuming steps of the gnfs algorithm with that we propose an improved parallel block lanczos ibl algorithm to reduce the communication cost of solving large and sparse linear systems over gf 2 more specifically we firstly re implement the parallel block lanczos algorithm from the bsp paradigm to open mpi to further improve the performance we then reorganize and redesign the algorithm to reduce the synchronization and communication costs during the outer product step after this we integrate the improved parallel block lanczos algorithm with the gnfs algorithm finally theoretical and experimental results demonstrate that the ibl algorithm greatly enhances the performance of gnfs compared with previous parallel block lanczos pbl algorithm in terms of both execution time and speedup

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

an integrated parallel gnfs algorithm for integer factorization based on linbox montgomery block lanczos method over gf 2 integer factorization is known to be one of the most important and useful methods in number theory and arithmetic it also has a very close relationship to some algorithms in cryptography such as rsa algorithm the rsa cryptosystem is one of the most popular and attractive public key cryptosystems in the world today its security is based on the difficulty of integer factorization solving a large and sparse linear system over gf 2 is one of the most time consuming steps in most modern integer factorization algorithms including the fastest one gnfs algorithm the montgomery block lanczos method from linbox 13 is for solving large and sparse linear systems over finite fields and it can be integrated into the general number field sieve gnfs algorithm which is the best known algorithm for factoring large integers over 110 digits this paper will present an improved montgomery block lanczos method integrated with parallel gnfs algorithm the experimental results show that the improved montgomery block lanczos method has a better performance compared with the original method it can find more solutions or dependencies than the original method with less time complexities implementation details and experimental results are provided in this paper as well

generalized metaplectic operators and the schrodinger equation with a potential in the sjostrand class it is well known that the matrix of a metaplectic operator with respect to phase space shifts is concentrated along the graph of a linear symplectic map we show that the algebra generated by metaplectic operators and by pseudodifferential operators in a sjostrand class enjoys the same decay properties we study the behavior of these generalized metaplectic operators and represent them by fourier integral operators our main result shows that the one parameter group generated by a hamiltonian operator with a potential in the sjostrand class consists of generalized metaplectic operators as a consequence the schrodinger equation preserves the phase space concentration as measured by modulation space norms

a two dimensional data distribution method for parallel sparse matrix vector multiplication a new method is presented for distributing data in sparse matrix vector multiplication the method is two dimensional tries to minimize the true communication volume and also tries to spread the computation and communication work evenly over the processors the method starts with a recursive bipartitioning of the sparse matrix each time splitting a rectangular matrix into two parts with a nearly equal number of nonzeros the communication volume caused by the split is minimized after the matrix partitioning the input and output vectors are partitioned with the objective of minimizing the maximum communication volume per processor experimental results of our implementation mondriaan for a set of sparse test matrices show a reduction in communication volume compared to one dimensional methods and in general a good balance in the communication work experimental timings of an actual parallel sparse matrix vector multiplication on an sgi origin 3800 computer show that a sufficiently large reduction in communication volume leads to savings in execution time

global well posedness of the one dimensional cubic nonlinear schrodinger equation in almost critical spaces abstract in this paper we first introduce a new function space m h p whose norm is given by the l p sum of modulated h norms of a given function in particular when 1 2 we show that the space m h p agrees with the modulation space m 2 p r on the real line and the fourier lebesgue space f l p t on the circle we use this equivalence of the norms and the galilean symmetry to adapt the conserved quantities constructed by killip visan zhang to the modulation space and fourier lebesgue space setting by applying the scaling symmetry we then prove global well posedness of the one dimensional cubic nonlinear schrodinger equation nls in almost critical spaces more precisely we show that the cubic nls on r is globally well posed in m 2 p r for any p while the renormalized cubic nls on t is globally well posed in f l p t for any p in appendix we also establish analogous global in time bounds for the modified kdv equation mkdv in the modulation spaces on the real line and in the fourier lebesgue spaces on the circle an additional key ingredient of the proof in this case is a galilean transform which converts the mkdv to the mkdv nls equation

continuity properties of the gabor frame operator abstract for a gabor frame with small lattice it is shown that the frame operator is continuous and invertible on many banach spaces defined by smoothness and decay properties as a consequence it is shown that there exist gabor type frame dual frame pairs and tight frames with good decay and smoothness properties also it is shown that the iteration scheme by which a function can be recovered from its frame coefficients converges robustly e g in lp and sobolev norms also explicit estimates on lattice sizes for which such results hold can be obtained

toeplitz operators on the segal bargmann space in this paper we give a complete characterization of those functions on 2n dimensional euclidean space for which the berezin toeplitz quantizations admit a symbol calculus modulo the compact operators the functions in question are characterized by a condition of small oscillation at infinity

geometric dynamics of a harmonic oscillator arbitrary minimal uncertainty states and the smallest step 3 nilpotent lie group the paper presents a new method of geometric solution of a schrodinger equation by constructing an equivalent first order partial differential equation with a bigger number of variables the equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform the method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three lie group the group also known under many names e g quartic group we obtain a geometric solution for an arbitrary minimal uncertainty state used as a fiducial vector in contrast it is shown that the well known fock segal bargmann transform and the heisenberg group require a specific fiducial vector to produce a geometric solution a technical aspect considered in this paper is that a certain modification of a coherent state transform is required although the irreducible representation of the group is square integrable modulo a subgroup the obtained dynamic is transverse to the homogeneous space