image reconstruction from zero crossings this study is concerned with the information in zero crossings zc of images logan s conditions specifying when a one dimensional signal may be recovered within a multiplicative constant from its zc are extended for various cases of two dimensional signals an algorithm is implemented in reconstruction of context free band pass images it is also successfully applied in reconstruction of contextual images by first dissecting the image into appropriate bandpass band limited two dimensional signals in almost all cases of spectrum confined to less than an octave in one dimension the reconstruction algorithm converges whereas it does not converge for any signal exceeding one octave in bandwidth this paper substantiates the proposition that images are well represented by the partial information confined to zc

estimates for the extremal sections of lpn balls abstract the problem of finding the maximal hyperplane section of b p n where p 2 has been open for a long time it is known that the answer depends on both p and n in this paper using the well known equivalence between hyperplane sections and the isotropic constant of a body we give an upper bound estimate for the volume of hyperplane sections of normalized l p n balls that does not depend on n and p in addition on the basis of results of meyer pajor and schmuckenschlager we show further the corresponding extremal body and hyperplane section when this volume attains its minimum

carl s inequality for quasi banach spaces we prove that for any two quasi banach spaces x and y and any 0 there exists a constant 0 such that sup1 k n k ek t sup1 k n k ck t holds for all linear and bounded operators t x y here ek t is the k th entropy number of t and ck t is the k th gelfand number of t for banach spaces x and y this inequality is widely used and well known as carl s inequality for general quasi banach spaces it is a new result

windowed fourier phase completeness and signal reconstruction it is established that a signal either continuous or discrete is determined up to a multiplicative constant by its windowed fourier phase wfp at any frequency this result indicates that the wfp contains richer information than zero crossings and peaks it also provides insight into why wfp based image matching may achieve highly accurate and stable results to experimentally demonstrate the completeness of the wfp an algorithm is developed to reconstruct signals from the wfp

eigenvalue distribution of some fractal semi elliptic differential operators we consider differential operators of type au x u x 1 t 1 frac partial 2t 1 u x partial x 1 2t 1 1 t 2 frac partial 2t 2 u x partial x 2 2t 2 x x 1 x 2 in r 2 and sierpinski carpets g subset r 2 the aim of the paper is to investigate spectral properties of the fractal differential operator a 1 circ tr gamma acting in the anisotropic sobolev space w t 1 t 2 2 bbb r 2 where tr gamma is closely related to the trace operator tr gamma

geometry of sets of quantum maps a generic positive map acting on a high dimensional system is not completely positive we investigate the set a of positive trace preserving maps acting on density matrices of size n and a sequence of its nested subsets the sets of maps which are b decomposable c completely positive and d extended by identity impose positive partial transpose and e are superpositive working with the hilbert schmidt euclidean measure we derive tight explicit two sided bounds for the volumes of all five sets a sample consequence is the fact that as n increases a generic positive map becomes not decomposable and a fortiori not completely positive due to the jamiolkowski isomorphism the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets such as states with positive partial transpose or separable states or supersets our approach depends on the systematic use of duality to derive quantitative estimates and on various tools of classical convexity high dimensional probability and geometry of banach spaces some of which are not standard

image representation by one bit fourier phase theory sampling and coherent image model the problem of recovering an image from its fourier transform phase quantized to one bit or equivalently finding the locations of the zero crossings of the real part of the fourier transform is addressed theoretical results are presented that set an algebraic condition under which real zero crossings uniquely specify a band limited image it is then shown that sampling in the frequency domain presents a major obstacle to obtaining good reconstruction results the one bit fourier phase reconstruction problem is then considered when the original image is coherent i e the image phase is random and highly uncorrelated examples are given which demonstrate that the information loss produced by frequency sampling is not as severe as that in the nonclassical problem motivated by digital phase only holograms a known random diffuser is used as the image phase and a well known iterative reconstruction procedure is extended to incorporate the knowledge of the image phase at each stage of the iteration this reconstruction method produces good image quality by using a few iterations unlike its noncoherent counterpart

geometry of the l_q centroid bodies of an isotropic log concave measure we study some geometric properties of the lq centroid bodies zq of an isotropic log concave measure on r for any 2 6 q 6 n and for e e0 q n 1 we determine the inradius of a random 1 e n dimensional projection of zq up to a constant depending polynomially on e using this fact we obtain estimates for the covering numbers n qb 2 tzq t 1 thus showing that zq is a regular convex body as a consequence we also get an upper bound for m zq

small ball probability estimates ψ2 behavior and the hyperplane conjecture we introduce a method which leads to upper bounds for the isotropic constant we prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the euclidean norm on isotropic convex bodies as a consequence of our method we obtain an alternative proof of the result of j bourgain that every 2 body has bounded isotropic constant with a slightly better estimate if k is a symmetric convex body in rn such that q 2 for every sn 1 and every q 2 then lk c log where c 0 is an absolute constant

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

entropy numbers s numbers and eigenvalue problems abstract we establish inequalities between entropy numbers and approximation numbers for operators acting between banach spaces furthermore we derive inequalities between eigenvalues and entropy numbers for operators acting on a banach space the results are compared with the classical inequalities of bernstein and jackson

image matching using the windowed fourier phase a theoretical framework is presented in which windowed fourier phase wfp is introduced as the primary matching primitive zero crossings and peaks correspond to special values of the phase the wfp is quasi linear and dense and its spatial period and slope are controlled by the scale this framework has the following important characteristics 1 matching primitives are available almost everywhere to convey dense disparity information in every channel either coarse or fine 2 the false target problem is significantly mitigated 3 the matching is easier uniform and can be performed by a network suitable for parallel computer architecture 4 the matching is fast since very few iterations are needed in fact the wfp is so informative that the original signal can be uniquely determined up to a multiplicative constant by the wfp in any channel the use of phase as matching primitive is also supported by some existing psychophysical and neurophysiological studies an implementation of the proposed theory has shown good results from synthesized and natural images

entropy and asymptotic geometry of non symmetric convex bodies abstract we extend to the general not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very essentially the symmetry in their proofs some of these extensions look surprising the main additional tool is a study of volume behavior around the centroid of the body

structure theorem and isomorphism test for graphs with excluded topological subgraphs we generalize the structure theorem of robertson and seymour for graphs excluding a fixed graph h as a minor to graphs excluding h as a topological subgraph we prove that for a fixed h every graph excluding h as a topological subgraph has a tree decomposition where each part is either almost embeddable to a fixed surface or has bounded degree with the exception of a bounded number of vertices furthermore we prove that such a decomposition is computable by an algorithm that is fixed parameter tractable with parameter h we present two algorithmic applications of our structure theorem to illustrate the mechanics of a typical application of the structure theorem we show that on graphs excluding h as a topological subgraph partial dominating set find k vertices whose closed neighborhood has maximum size can be solved in time f h k cdot n o 1 more significantly we show that on graphs excluding h as a topological subgraph graph isomorphism can be solved in time n f h this result unifies and generalizes two previously known important polynomial time solvable cases of graph isomorphism bounded degree graphs e m luks j comput system sci 25 1982 pp 42 65 and h minor free graphs i n ponomarenko zap nauchn sem leningrad otdel mat inst steklov lomi 174 1988 pp 147 177 182 the proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition

signal reconstruction from signed fourier transform magnitude in this paper we show that a one dimensional or multidimensional sequence is uniquely specified under mild restrictions by its signed fourier transform magnitude magnitude and 1 bit of phase information in addition we develop a numerical algorithm to reconstruct a one dimensional or multidimensional sequence from its fourier transform magnitude reconstruction examples obtained using this algorithm are also provided

gelfand numbers of operators with values in a hilbert space in the paper we prove two inequalities involving gelfand numbers of operators with values in a hilbert space the first inequality is a rademacher version of the main result in pa to 1 which relates the gelfand numbers of an operator from a banach spacex intol 2 n with a certain rademacher average for the dual operator the second inequality states that the gelfand numbers of an operatoru froml 1 n into a hilbert space satisfy the inequality k 1 2 c k u leqq c parallel u parallel log 1 n k 1 2 wherec is a universal constant several applications of these inequalities in the geometry of banach spaces are given

intrinsic entropies of log concave distributions the entropy of a random variable is well known to equal the exponential growth rate of the volumes of its typical sets in this paper we show that for any log concave random variable x the sequence of the lfloor n theta rfloor text th intrinsic volumes of the typical sets of x in dimensions n geq 1 grows exponentially with a well defined rate we denote this rate by h x theta and call it the theta text th intrinsic entropy of x we show that h x theta is a continuous function of theta over the range 0 1 thereby providing a smooth interpolation between the values 0 and h x at the endpoints 0 and 1 respectively

scaling theorems for zero crossings we characterize some properties of the zero crossings of the laplacian of signals in particular images filtered with linear filters as a function of the scale of the filter extending recent work by witkin 16 we prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the gaussian this result can be generalized to apply to level crossings of any linear differential operator it applies in particular to ridges and ravines in the image intensity in the case of the second derivative along the gradient there is no filter that avoids creation of zero crossings unless the filtering is performed after the derivative is applied

rogers shephard inequality for log concave functions abstract in this paper we prove different functional inequalities extending the classical rogers shephard inequalities for convex bodies the original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body such as its difference body we characterize the equality cases in all these inequalities our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log concave functions and the study of some geometrical properties of these new sets

number of nodal domains of eigenfunctions on non positively curved surfaces with concave boundary it is an open problem in general to prove that there exists a sequence of delta g eigenfunctions varphi j k on a riemannian manifold m g for which the number n varphi j k of nodal domains tends to infinity with the eigenvalue our main result is that n varphi j k rightarrow infty along a subsequence of eigenvalues of density 1 if m g is a non positively curved surface with concave boundary i e a generalized sinai or lorentz billiard unlike the recent closely related work of ghosh reznikov sarnak and of the authors on the nodal domain counting problem the surfaces need not have any symmetries

isoperimetry of waists and local versus global asymptotic convex geometries existence of nicely bounded sections of two symmetric convex bodies k and l implies that the intersection of random rotations of k and l is nicely bounded for l subspace this main result immediately yields the unexpected phenomenon if k has one nicely bounded section then most sections of k are nicely bounded this existence implies randomness consequence was proved independently in giannopoulos milman and tsolomitis the main result represents a new connection between the local asymptotic convex geometry study of sections of convex bodies and the global asymptotic convex geometry study of convex bodies as a whole the method relies on the new isoperimetry of waists on the sphere due to gromov

reverse brunn minkowski and reverse entropy power inequalities for convex measures we develop a reverse entropy power inequality for convex measures which may be seen as an anegeometric inverse of the entropy power inequality of shannon and stam the specialization of this inequality to log concave measures may be seen as a version of milman s reverse brunn minkowski inequality the proof relies on a demonstration of new relationships between the entropy of high dimensional random vectors and the volume of convex bodies and on a study of eective supports of convex measures both of which are of independent interest as well as on milman s deep technology of m ellipsoids and on certain information theoretic inequalities as a by product we also give a

when random proportional subspaces are also random quotients we discuss when a generic subspace of some fixed proportional dimension of a finite dimensional normed space can be isomorphic to a generic quotient of some proportional dimension of another space we show in theorem 4 1 that if this happens for some natural random structures then for any proportion arbitrarily close to 1 the first space has a lot of euclidean subspaces and the second space has a lot of euclidean quotients

on a characterization of operators from lq into a banach space of type p with some applications to eigenvalue problems abstract let s be an operator admitting a factorization where d is a diagonal operator and t an arbitrary operator with the image in a banach space of type p we shall characterize these operators by entropy numbers using a well known inequality between eigenvalues and entropy numbers we establish some results about distributions of eigenvalues

entanglement thresholds for random induced states for a random quantum state on h c d otimes c d obtained by partial tracing a random pure state on h otimes c s we consider the whether it is typically separable or typically entangled for this problem we show the existence of a sharp threshold s 0 s 0 d of order roughly d 3 more precisely for any a 0 and for d large enough such a random state is entangled with very large probability when s 1 a s 0 one consequence of this result is as follows for a system of n identical particles in a random pure state there is a threshold k 0 k 0 n sim n 5 such that two subsystems of k particles each typically share entanglement if k k 0 and typically do not share entanglement if k k 0 our methods work also for multipartite systems and for unbalanced systems such as c d otimes c d d neq d the arguments rely on random matrices classical convexity high dimensional probability and geometry of banach spaces some of the auxiliary results may be of reference value a high level non technical overview of the results of this paper and of a related article arxiv 1011 0275 can be found in arxiv 1112 4582

high dimensional random sections of isotropic convex bodies we study two properties of random high dimensional sections of convex bodies in the first part of the paper we estimate the central section function k f n k1 k for random f gn k and k rn a centrally symmetric isotropic convex body this partially answers a question raised by v d milman and a pajor see v d milman a pajor isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n dimensional space in lecture notes in math vol 1376 springer 1989 p 88 in the second part we show that every symmetric convex body has random high dimensional sections f gn k with outer volume ratio bounded by ovr k f cnn klog 1 nn k

effects of additive noise on signal reconstruction from fourier transform phase the effects of additive noise in the given phase on signal reconstruction from the fourier transform phase are experimentally studied specifically the effects on the sequence reconstruction of different methods of sampling the degraded phase of the number of nonzero points in the sequence and of the noise level are examined a sampling method that significantly reduces the error in the reconstructed sequence is obtained and the error is found to increase as the number of nonzero points in the sequence increases and as the noise level increases in addition an averaging technique is developed which reduces the effects of noise when the continuous phase function is known finally as an illustration of how the results in this paper may be applied in practice fourier transform signal coding is considered coding only the fourier transform phase and reconstructing the signal from the coded phase is found to be considerably less efficient i e a higher bit rate is required for the same mean square error than reconstructing from both the coded phase and magnitude

euclidean embeddings in spaces of finite volume ratio via random matrices let rn be the space rn equipped with a norm whose unit ball has a bounded volume ratio with respect to the euclidean unit ball let be any random n n matrix with n n whose entries are independent random variables satisfying some moment assumptions we show that with high probability is a good isomorphism from the n dimensional euclidean space rn onto its image in rn i e there exist 0 such that for all x rn n x x n x this solves a conjecture of schechtman on random embeddings of 2 into n 1

the weisfeiler leman dimension of planar graphs is at most 3 we prove that the weisfeiler leman wl dimension of the class of all finite planar graphs is at most 3 in particular every finite planar graph is definable in first order logic with counting using at most 4 variables the previously best known upper bounds for the dimension and number of variables were 14 and 15 respectively first we show that for dimension 3 and higher the wl algorithm correctly tests isomorphism of graphs in a minor closed class whenever it determines the orbits of the automorphism group of every arc colored 3 connected graph belonging to this class then we prove that apart from several exceptional graphs which have wl dimension at most 2 the individualization of two appropriately chosen vertices of a colored 3 connected planar graph followed by the one dimensional wl algorithm produces the discrete vertex partition this implies that the three dimensional wl algorithm determines the orbits of arc colored 3 connected planar graphs as a byproduct of the proof we get a classification of the 3 connected planar graphs with fixing number 3

positions of convex bodies associated to extremal problems and isotropic measures abstract we show that there are close relations between extremal problems in dual brunn minkowski theory and isotropic type properties for some borel measures on the sphere the methods we use allow us to obtain similar results in the context of firey brunn minkowski theory we also study reverse inequalities for dual mixed volumes which are related with classical positions such as l position or isotropic position

isomorphism of graphs which are pairwise k separable a polynomial time algorithm for testing isomorphism of graphs which are pairwise k separable for fixed k is given the pairwise k separable graphs are those graphs where each pair of distinct vertices are k separable this is a natural generalization of the bounded valence test of luks the subgroup of automorphisms of a hypergraph whose restriction to the vertices is in a given k group for fixed k is constructed in polynomial time

on stability of reconstruction from fourier transform modulus the lower bound on the condition number of reconstruction of two classes of sequences from their fourier transform magnitude ftm is derived the lower bound for one class is shown to be 1 and for the other class to be 1 2 n sup q 2 where q is the dimensionality of the sequence and n is the number of nonzero elements in each dimension stability of reconstruction from the ftm and the space domain phase is discussed it is found that randomizing the space domain phase improves reconstruction robustness experimental results are presented to verify theoretical predictions

thomas bayes walk on manifolds convergence of the bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold or more generally on a compact metric space verifying some conditions a natural geometric prior based on randomly rescaled solutions of the heat equation is considered upper and lower bound posterior contraction rates are derived

computational complexity of quantum satisfiability we connect both discrete and algebraic complexity theory with the satisfiability problem in certain non boolean lattices specifically quantum logic was introduced in 1936 by garrett birkhoff and john von neumann as a framework for capturing the logical peculiarities of quantum observables in the 1d case it coincides with boolean propositional logic but starting with dimension two violates the distributive law we introduce the weak and strong satisfiability problem for quantum logic propositional formulae it turns out that in dimension two both are also np complete for higher dimensional spaces rd and cd with d 3 fixed on the other hand we show both problems to be complete for the nondeterministic blum shub smale bss model of real computation this provides a unified view on both turing and real bss complexity theory and extends the still relatively scarce list of problems established npr complete with one perhaps closest in spirit to the classical cook levin theorem more precisely strong satisfiability of terms is complete while that of terms i e those in conjunctive form can be decided in polynomial time in dimensions d 2 the decidability of the infinite dimensional case being still open we proceed to investigate the case of indefinite finite dimensions here weak satisfiability still belongs to npr and strong satisfiability is still hard the latter in fact turns out as polynomial time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings

s numbers of integral operators with holder continuous kernels over metric compacta abstract we determine the asymptotic behaviour of kolmogorov and approximation numbers of integral operators which act in c x where x is a compact metric space and which are defined by means of holder continuous kernels the results show the connection of the decay of these s numbers with the entropy and in this way with the dimension of x

mean width and diameter of proportional sections of a symmetric convex body let k be a symmetric convex body in r the purpose of this paper is to provide upper and lower bounds for the diameter of a random n dimensional central section of k where the proportion 0 1 is arbitrary but fixed there are several aspects of our approach to this question that should be clarified right away 1 we are interested in bounds expressed in terms of average parameters of the body k which can be efficiently computed in a simple way therefore being useful from the computational geometry point of view 2 the dimension n is understood to tend to infinity then we say that our bounds hold for a random n dimensional section of k if they are satisfied by all k where is in a subset of the appropriate grassmannian with haar probability measure greater than 1 h n and h n 0 as n tends to infinity 3 we say that our estimates are tight for a class of bodies and a fixed proportion if the ratio of our upper and lower bounds depends only on it is clear that one cannot obtain tight bounds for the class of all symmetric convex bodies it is not hard to describe almost degenerated bodies in r for example an ellipsoid with highly incomparable semiaxes for which the diameter of n sections does not concentrate around some value so it is an important question to see under what conditions on k the estimates obtained by a method are tight

centroid bodies and the logarithmic laplace transform a unified approach abstract we unify and slightly improve several bounds on the isotropic constant of high dimensional convex bodies in particular a linear dependence on the body s 2 constant is obtained along the way we present some new bounds on the volume of l p centroid bodies and yet another equivalent formulation of bourgain s hyperplane conjecture our method is a combination of the l p centroid body technique of paouris and the logarithmic laplace transform technique of the first named author

duality of metric entropy abstractfor two convex bodies kand t in r n the covering number of kbyt denoted n k t is de ned as the minimal number of translates oft needed to cover k let us denote by k the polar body of k andby d the euclidean unit ball in r n we prove that the two functionsof t n k td and n d tk are equivalent in the appropriate sense uniformly over symmetric convex bodies k r n and over n n inparticular this veri es the duality conjecture for entropy numbers of linearoperators posed by pietsch in 1972 in the central case when either thedomain or the range of the operator is a hilbert space 1 introduction for two convex bodies kand t in r n the covering number of kby t denoted n k t is de ned as the minimal number of translates of tneeded to cover kn k t min n x 1 x n r n k i n x i t we denote by dthe euclidean unit ball in r n in this paper we prove thefollowing duality result for covering numbers theorem 1 main theorem there exist two universal constants

image restoration by the method of convex projections part 1ߞtheory a projection operator onto a closed convex set in hilbert space is one of the few examples of a nonlinear map that can be defined in simple abstract terms moreover it minimizes distance and is nonexpansive and therefore shares two of the more important properties of ordinary linear orthogonal projections onto closed linear manifolds in this paper we exploit the properties of these operators to develop several iterative algorithms for image restoration from partial data which permit any number of nonlinear constraints of a certain type to be subsumed automatically their common conceptual basis is as follows every known property of an original image f is envisaged as restricting it to lie in a well defined closed convex set thus m such properties place f in the intersection e0 ei of the corresponding closed convex sets e1 e2 em given only the projection operators pi onto the individual ei s i 1 m we restore f by recursive means clearly in this approach the realization of the pi s in a hilbert space setting is one of the major synthesis problems section i describes the geometrical significance of the three main theorems in considerable detail and most of the underlying ideas are illustrated with the aid of simple diagrams section ii presents rules for the numerical implementation of 11 specific projection operators which are found to occur frequently in many signal processing applications and the appendix contains proofs of all the major results

polynomial system of equations and its applications to the study of the effect of noise on multidimensional fourier transform phase retrieval from magnitude in this paper we deal with the problem of retrieving a finite extent signal from the magnitude of its fourier transform we will present a brief review of the algebraic problem of the uniqueness of the solution for both discrete and continuous phase retrieval models several important issues which are yet unresolved will be pointed out and discussed we will then consider the discrete phase retrieval problem as a special case of a more general problem which consists of recovering a real valued signal x from the magnitude of the output of a linear distortion hx j j 1 n an important result concerning the conditioning of this problem will be obtained for this general setting by means of algebraic geometric techniques in particular the problems of the existence of a solution for phase retrieval conditioning of the problem and stability of the essentially unique solution will be addressed

spectral theory of some degenerate elliptic operators with local singularities abstract this paper is based on our previous results haroske and skrzypczak 2008 23 haroske and skrzypczak in press 25 on compact embeddings of muckenhoupt weighted function spaces of besov and triebel lizorkin type with example weights of polynomial growth near infinity and near some local singularity our approach also extends haroske and triebel 1994 21 in various ways we obtain eigenvalue estimates of degenerate pseudodifferential operators of type b 2 p x d b 1 where b i l r i r n w i w i a i 1 2 and p x d 1 0 0 finally we deal with the negative spectrum of some operator h a v for where the potential v may have singularities in terms of muckenhoupt weights and a is a positive elliptic pseudodifferential operator of order 0 self adjoint in l 2 r n this part essentially relies on the birman schwinger principle we conclude this paper with a number of examples also comparing our results with preceding ones

an isomorphic version of the slicing problem abstract here we show that any centrally symmetric convex body k r n has a perturbation t r n which is convex and centrally symmetric such that the isotropic constant of t is universally bounded t is close to k in the sense that the banach mazur distance between t and k is o log n if k is a body of a non trivial type then the distance is universally bounded the distance is also universally bounded if the perturbation t is allowed to be non convex our technique involves the use of mixed volumes and alexandrov fenchel inequalities some additional applications of this technique are presented here

convergence of iterative nonexpansive signal reconstruction algorithms iterative algorithms for signal reconstruction from partial time and frequency domain knowledge have proven useful in a number of application areas in this paper a general convergence proof applicable to a general class of such iterative reconstruction algorithms is presented the proof relies on the concept of a nonexpansive mapping in both the time and frequency domains two examples studied in detail are time limited extrapolation equivalently band limited extrapolation and phase only signal reconstruction the proof of convergence for the phase only iteration is a new result obtained by this method of proof the generality of the approach allows the incorporation of nonlinear constraints such as time or space domain positivity or minimum and maximum value constraints finally the underrelaxed form of these iterations is also shown to converge even when the solution is not guaranteed to be unique

signal reconstruction from fourier transform sign information in this paper we present new results on the reconstruction of signals from only the sign of the real part of the fourier transform specifically we develop new theoretical results which state conditions under which two dimensional signals are uniquely specified to within a scale factor with this information and show that these conditions include a broad class of signals furthermore we apply this result to the problem of reconstructing two dimensional signals from their zero crossings we also present two algorithms for reconstructing a signal from sign information in either the time or frequency domain

diameters of sections and coverings of convex bodies abstract we study the diameters of sections of convex bodies in r n determined by a random n n matrix either as kernels of or as images of entries of are independent random variables satisfying some boundedness conditions and typical examples are matrices with gaussian or bernoulli random variables we show that if a symmetric convex body k in r n has one well bounded k codimensional section then for any m ck random sections of k of codimension m are also well bounded where c 1 is an absolute constant it is noteworthy that in the gaussian case when determines randomness in sense of the haar measure on the grassmann manifold we can take c 1

tropical varieties for exponential sums we study the complexity of approximating complex zero sets of certain n variate exponential sums we show that the real part r of such a zero set can be approximated by the n 1 dimensional skeleton t of a polyhedral subdivision of mathbb r n in particular we give an explicit upper bound on the hausdorff distance delta r t o left t 3 5 delta right where t and delta are respectively the number of terms and the minimal spacing of the frequencies of g on the side of computational complexity we show that even the n 2 case of the membership problem for r is undecidable in the blum shub smale model over mathbb r whereas membership and distance queries for our polyhedral approximation t can be decided in polynomial time for any fixed n

relational queries computable in polynomial time we characterize the polynomial time computable queries as those expressible in relational calculus plus a least fixed point operator and a total ordering on the universe we also show that even without the ordering one application of fixed point suffices to express any query expressible with several alternations of fixed point and negation this proves that the fixed point query hierarchy suggested by chandra and harel collapses at the first fixed point level it is also a general result showing that in finite model theory one application of fixed point suffices

on the volume of non central sections of a cube abstract let q n be the cube of side length one centered at the origin in r n and let f be an affine n d dimensional subspace of r n having distance to the origin less than or equal to 1 2 where 0 d n we show that the n d dimensional volume of the section q n f is bounded below by a value c d depending only on the codimension d but not on the ambient dimension n or a particular subspace f in the case of hyperplanes d 1 we show that c 1 1 17 is a possible choice we also consider a complex analogue of this problem for a hyperplane section of the polydisc

volume ratio and other s numbers of operators related to local properties of banach spaces abstract volume ratio numbers are studied and inequalities relating them to other s numbers of operators are established characterizations of spaces of weak cotype 2 in terms of s numbers and of operator ideal norms are obtained

asymptotic formulas for the diameter of sections of symmetric convex bodies sharpening work of the first two authors for every proportion 0 1 we provide exact quantitative relations between global parameters of n dimensional symmetric convex bodies and the diameter of their random n dimensional sections using recent results of gromov and vershynin we obtain an asymptotic formula for the diameter of random proportional sections

fixed point definability and polynomial time on graphs with excluded minors we give a logical characterization of the polynomial time properties of graphs embeddable in some surface for every surface s a property p of graphs embeddable in s is decidable in polynomial time if and only if it is definable in fixed point logic with counting it is a consequence of this result that for every surface s there is a k such that a simple combinatorial algorithm namely the k dimensional weisfeiler lehman algorithm decides isomorphism of graphs embeddable in s in polynomial time we also present without proof generalizations of these results to arbitrary classes of graphs with excluded minors

isomorphic steiner symmetrization this paper proves that there exist 3n steiner symmetrizations that transform any convex set k n into an isomorphic euclidean ball i e if vol k vol dn where dn is the standard euclidean unit ball then k can be transformed into a body k such that c1dn k c2dn where c1 c2 are numerical constants moreover for any c 2 cn symmetrizations are also enough