on the construction of equiangular frames from graphs abstract we give details of the 1 1 correspondence between equiangular frames of n vectors for r d and graphs with n vertices this has been studied recently for tight equiangular frames because of applications to signal processing and quantum information theory the nontight examples given here which correspond to graphs with more than 2 eigenvalues have the potential for similar applications e g the frame corresponding to the 5 cycle graph is the unique grassmannian frame of 5 vectors in openr 3 further the associated canonical tight frames have a small number of angles in many cases

analysis of d2d underlaid cellular networks sir meta distribution and mean local delay we study the performance of device to device d2d communication underlaying cellular wireless network in terms of the meta distribution of the signal to interference ratio sir which is the distribution of the conditional sir distribution given the locations of the wireless nodes modeling d2d transmitters and base stations as poisson point processes ppps moments of the conditional sir distribution are derived in order to calculate analytical expressions for the meta distribution and the mean local delay of the typical d2d receiver and cellular downlink user it turns out that for d2d users the total interference from the d2d interferers and base stations is equal in distribution to that of a single ppp while for downlink users the effect of the interference from the d2d network is more complicated we also derive the region of transmit probabilities for the d2d users and base stations that result in a finite mean local delay and give a simple inner bound on that region finally the impact of increasing the base station density on the mean local delay the meta distribution and the density of users reliably served is investigated with numerical results

the symmetry group of a finite frame abstract we define the symmetry group of a finite frame as a group of permutations on its index set this group is closely related to the symmetry group of vale and waldron 2005 12 for tight frames they are isomorphic when the frame is tight and has distinct vectors the symmetry group is the same for all similar frames in particular for a frame its dual and canonical tight frames it can easily be calculated from the gramian matrix of the canonical tight frame further a frame and its complementary frame have the same symmetry group we exploit this last property to construct and classify some classes of highly symmetric tight frames

hilbert space frames containing a riesz basis and banach spaces which have no subspace isomorphic toc0 abstract we prove that a hilbert space frame fi i icontains a riesz basis if every subfamily fi i j j i is a frame for its closed span secondly we give a new characterization of banach spaces which do not have any subspace isomorphic toc0 this result immediately leads to an improvement of a recent theorem of holub concerning frames consisting of a riesz basis plus finitely many elements

burst erasures and the mean square error for cyclic parseval frames this paper investigates the performance of frames for the linear redundant encoding of vectors when consecutive frame coefficients are lost due to the occurrence of random burst errors we assume that the distribution of bursts is invariant under cyclic shifts and that the burst length statistics are known in analogy with rate distortion theory we wish to find frames of a given size which minimize the mean square reconstruction error for the encoding of vectors in a complex finite dimensional hilbert space we obtain an upper bound for the mean square reconstruction error for a given parseval frame and in the case of cyclic parseval frames we find a family of frames which minimizes this upper bound under certain conditions these minimizers are identical to complex bose chaudhuri hocquenghem codes discussed in the literature the accuracy of our upper bounds for the mean square error is substantiated by complementary lower bounds all estimates are based on convexity arguments and a discrete rearrangement inequality

complex hadamard matrices and equiangular tight frames abstract in this paper we give a new construction of parametric families of complex hadamard matrices of square orders and connect them to equiangular tight frames the results presented here generalize some of the recent ideas of bodmann et al 3 and extend the list of known equiangular tight frames in particular a 36 21 frame coming from a nontrivial cube root signature matrix is obtained for the first time

maximum robustness and surgery of frames in finite dimensions abstract we consider frames in a finite dimensional hilbert space h n where frames are exactly the spanning sets of the vector space we present a method to determine the maximum robustness of a frame we present results on tight subframes and surgery of frames we also answer the question of when length surgery resulting in a tight frame set for h n is possible

optimal reconstruction systems for erasures and for the q potential abstract in this paper we introduce the q potential as an extension of the benedetto fickus frame potential defined on general reconstruction systems and show that protocols are the minimizers of this potential under certain restrictions we extend recent results of b g bodmann on the structure of optimal protocols with respect to 1 and 2 lost packets where the worst normalized reconstruction error is computed with respect to a compatible unitarily invariant norm we finally describe necessary and sufficient spectral conditions that we call q fundamental inequalities for the existence of protocols with prescribed properties by relating this problem to klyachko s and fulton s theory on sums of hermitian operators

some properties of g frames in hilbert c modules in this paper we give some new results for g frames in hilbert c modules and then we introduce a bounded a linear operator l by means of this operator l we character the properties of the g frames and g riesz basis in hilbert c modules finally we establish some important equalities and inequalities for frames and g frames in hilbert c modules

frame theoretic analysis of robust filter bank frames to quantization and erasures this paper presents the theoretic analysis for the robustness of filter bank fb frames in infinite dimensional hilbert space l2 bbz to quantization and erasures as well as studies the design of such robust frames from the perspective of both frame and fb theory first a characterization of the eigenstructure for the frame operator and the induced gram matrix of general fb frames is presented the robustness of fb frames to erasures is investigated in detail especially on the necessary and sufficient condition maximally robust frames are further analyzed by explicitly constructive methods moreover the stability and even the possible fir reconstruction of the subframe by the pseudoinverse are studied thoroughly next we examine the optimal quantized fb frames introducing a novel notion named as frame energy the universal optimality of tight fb frames to quantization is established in contrast to the optimality of only equal norm tight frames shown in previous works the added design freedom obtained by removing the equal norm constraint is explained and illustrated with examples finally the effect of erasures on the quantized fb frames is studied by incorporating the probability of erasures which shows the universal optimality of uniform tight fb frames for one erasure the optimal fb frame is usually not of equal norm and actually its norm distribution follows the reverse waterfilling principle an example of two state erasure channel model is shown to further explain our result followed by an analysis of the successive reconstruction which shows its potential application to frame based multiple description coding

on frames for krein spaces abstract a definition of frames for krein spaces is proposed which extends the notion of j orthonormal bases of krein spaces a j frame for a krein space h is in particular a frame for h in the hilbert space sense but it is also compatible with the indefinite inner product meaning that it determines a pair of maximal uniformly j definite subspaces an analogue to the maximal dual pair associated to a j orthonormal basis also each j frame induces an indefinite reconstruction formula for the vectors in h which resembles the one given by a j orthonormal basis

aliasing and oblique dual pair designs for consistent sampling abstract in this paper we study some aspects of oblique duality between finite sequences of vectors f and g lying in finite dimensional subspaces w and v respectively we compute the possible eigenvalue lists of the frame operators of oblique duals to f lying in v we compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for f with norm restrictions as an application we show that these optimal duals are the closest to being tight frames and therefore have their spectrum as concentrated as possible among oblique duals with norm restrictions we obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces v and w has in oblique duality we apply this analysis to compute those rigid rotations u for w such that the canonical oblique dual of u f minimize every convex potential we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations u for w such that the canonical oblique dual pair associated to u f minimize the aliasing we point out that these two last problems are intrinsic to oblique duality within the context of consistent sampling

properties of real and complex etfs and their application to the design of low coherence frames certain properties of an equiangular tight frame are contained in the sign or phase of its corresponding gram matrix in this paper we convey new properties for gram matrices of equiangular tight frames and use them to construct new frames with low average and worst case coherences based on these results we also provide properties of the synthesis matrix of equiangular tight frames

harmonic equiangular tight frames comprised of regular simplices abstract an equiangular tight frame etf is a sequence of unit norm vectors in a euclidean space whose coherence achieves equality in the welch bound and thus yields an optimal packing in a projective space a regular simplex is a simple type of etf in which the number of vectors is one more than the dimension of the underlying space more sophisticated examples include harmonic etfs which equate to difference sets in finite abelian groups recently it was shown that some harmonic etfs are comprised of regular simplices in this paper we continue the investigation into these special harmonic etfs we begin by characterizing when the subspaces that are spanned by the etf s regular simplices form an equi isoclinic tight fusion frame eitff which is a type of optimal packing in a grassmannian space we shall see that every difference set that produces an eitff in this way also yields a complex circulant conference matrix next we consider a subclass of these difference sets that can be factored in terms of a smaller difference set and a relative difference set it turns out that these relative difference sets lend themselves to a second related and yet distinct construction of complex circulant conference matrices finally we provide explicit infinite families of etfs to which this theory applies

equiangular tight frames and unistochastic matrices we demonstrate that a complex equiangular tight frame composed of n vectors in dimension d denoted etf d n exists if and only if a certain bistochastic matrix univocally determined by n and d belongs to a special class of unistochastic matrices this connection allows us to find new complex etfs in infinitely many dimensions and to derive a method to introduce non trivial free parameters in etfs we present an explicit six parametric family of complex etf 6 16 which defines a family of symmetric povms minimal and maximal possible average entanglement of the vectors within this qubit qutrit family are described furthermore we propose an efficient numerical procedure to compute the unitary matrix underlying a unistochastic matrix which we apply to find all existing classes of complex etfs containing up to 20 vectors

kirkman equiangular tight frames and codes an equiangular tight frame etf is a set of unit vectors in a euclidean space whose coherence is as small as possible equaling the welch bound also known as welch bound equality sequences such frames arise in various applications such as waveform design and compressed sensing at the moment there are only two known flexible methods for constructing etfs harmonic etfs are formed by carefully extracting rows from a discrete fourier transform steiner etfs arise from a tensor like combination of a combinatorial design and a regular simplex these two classes seem very different the vectors in harmonic etfs have constant amplitude whereas steiner etfs are extremely sparse we show that they are actually intimately connected a large class of steiner etfs can be unitarily transformed into constant amplitude frames dubbed kirkman etfs moreover we show that an important class of harmonic etfs is a subset of an important class of kirkman etfs this connection informs the discussion of both types of frames some steiner etfs can be transformed into constant amplitude waveforms making them more useful in waveform design some harmonic etfs have low spark making them less desirable for compressed sensing we conclude by showing that real valued constant amplitude etfs are equivalent to binary codes that achieve the grey rankin bound and then construct such codes using kirkman etfs

the thermodynamics of fractals revisited with wavelets abstract the multifractal formalism originally introduced to describe statistically the scaling properties of singular measures is revisited using the wavelet transform this new approach is based on the definition of partition functions from the wavelet transform modulus maxima we demonstrate that very much like thermodynamic functions the generalized fractal dimensions d q and the f singularity spectrum can be readily determined from the scaling behavior of these partition functions we show that this method provides a natural generalization of the classical box counting techniques to fractal signals the wavelets playing the role of generalized boxes we illustrate our theoretical considerations on pedagogical examples e g devil s staircases and fractional brownian motions we also report the results of some recent application of the wavelet transform modulus maxima method to fully developed turbulence data that we emphasize the wavelet transform as a mathematical microscope that can be further used to extract microscopic informations about the scaling properties of fractal objects in particular we show that a dynamical system which leaves invariant such an object can be uncovered form the space scale arrangement of its wavelet transform modulus maxima we elaborate on a wavelet based tree matching algorithm that provides a very promising tool for solving the inverse fractal problem this step towards a statistical mechanics of fractals is illustrated on discrete period doubling dynamical systems where the wavelet transform is shown to reveal the renormalization operation which is essential to the understanding of the universal properties of this transition to chaos finally we apply our technique to analyze the fractal hierarchy of dla azimuthal cantor sets defined by intersecting the inner frozen region of large mass off lattice diffusion limited aggregates dla wit a circle this study clearly lets out the existence of an underlying multiplicative process that is likely to account for the fibonacci structural ordering recently discovered in the apparently disordered arborescent dla morphology

optimal frames for erasures we study frames from the viewpoint of coding theory we introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting

revisiting multifractality of high resolution temporal rainfall using a wavelet based formalism 1 we reexamine the scaling structure of temporal rainfall using wavelet based methodologies which as we demonstrate offer important advantages compared to the more traditional multifractal approaches such as box counting and structure function techniques in particular we explore two methods based on the continuous wavelet transform cwt and the wavelet transform modulus maxima wtmm the partition function method and the newer and more efficient magnitude cumulant analysis method we also report the results of a two point magnitude correlation analysis which is able to infer the presence or absence of multiplicativity as the underlying mechanism of scaling the diagnostic power of these methodologies for small samples signals with short ranges of scaling and signals for which high frequency fluctuations are superimposed on a low frequency component all common attributes of geophysical signals is carefully documented application of these methodologies to several midwestern convective storms sampled every 5 s over several hours provides new insights they reveal the presence of a very intermittent multifractal structure a wide spectrum of singularities in rainfall fluctuations between the scales of 5 min and the storm pulse duration of the order of 1 2 hours for the analyzed storms the two point magnitude statistical analysis suggests that this structure is consistent with a multiplicative cascading mechanism which however is local in nature that is it applies only within each storm pulse but not over the whole storm duration

duality in reconstruction systems we consider the notion of finite dimensional reconstructions systems rs s which includes the fusion frames as projective rs s we study erasures some geometrical properties of these spaces the spectral picture of the set of all dual systems of a fixed rs the spectral picture of the set of rs operators for the projective systems with fixed weights and the structure of the minimizers of the joint potential in this setting we give several examples

equiangular tight frames that contain regular simplices abstract an equiangular tight frame etf is a type of optimal packing of lines in euclidean space a regular simplex is a special type of etf in which the number of vectors is one more than the dimension of the space they span in this paper we consider etfs that contain a regular simplex that is have the property that a subset of its vectors forms a regular simplex as we explain such etfs are characterized as those that achieve equality in a certain well known bound from the theory of compressed sensing we then consider the so called binder of such an etf namely the set of all regular simplices that it contains we provide a new algorithm for computing this binder in terms of products of entries of the etf s gram matrix in certain circumstances we show this binder can be used to produce a particularly elegant naimark complement of the corresponding etf other times an etf is a disjoint union of regular simplices and we show this leads to a certain type of optimal packing of subspaces known as an equichordal tight fusion frame we conclude by considering the extent to which these ideas can be applied to numerous known constructions of etfs including harmonic etfs

binary parseval frames from group orbits abstract binary parseval frames share many structural properties with real and complex ones on the other hand there are subtle differences for example that the gramian of a binary parseval frame is characterized as a symmetric idempotent whose range contains at least one odd vector here we study binary parseval frames obtained from the orbit of a vector under a group representation in short binary parseval group frames in this case the gramian of the frame is in the algebra generated by the right regular representation we identify equivalence classes of such parseval frames with binary functions on the group that satisfy a convolution identity this allows us to find structural constraints for such frames we use these constraints to catalogue equivalence classes of binary parseval frames obtained from group representations as an application we study the performance of binary parseval frames generated with abelian groups for purposes of error correction we show that if p is an odd prime then the group z p q is always preferable to z p q when searching for best performing codes associated with binary parseval group frames

simple approximations of the sir meta distribution in general cellular networks compared to the standard success coverage probability the meta distribution of the signal to interference ratio sir provides much more fine grained information about the network performance we consider general heterogeneous cellular networks hcns with base station tiers modeled by arbitrary stationary and ergodic non poisson point processes the exact analysis of non poisson network models is notoriously difficult even in terms of the standard success probability let alone the meta distribution hence we propose a simple approach to approximate the sir meta distribution for non poisson networks based on the asappp approximate sir analysis based on the poisson point process method we prove that the asymptotic horizontal gap g 0 between its standard success probability and that for the poisson point process exactly characterizes the gap between the b th moment of the conditional success probability as the sir threshold goes to 0 the gap g 0 allows two simple approximations of the meta distribution for general hcns 1 the per tier approximation by applying the shift g 0 to each tier and 2 the effective gain approximation by directly shifting the meta distribution for the homogeneous independent poisson network given the generality of the model considered and the fine grained nature of the meta distribution these approximations work surprisingly well

causal compensation for erasures in frame representations in a variety of signal processing and communications contexts erasures occur inadvertently or can be intentionally introduced as part of a data reduction strategy this paper discusses causal compensation for erasures in frame representations of signals the approach described assumes linear synthesis of the signal using a prespecified frame but no specific generation mechanism for the coefficients under this assumption it is demonstrated that erasures can be compensated for using low complexity causal systems if the transmitter is aware of the occurrence of the erasure an optimal compensation is to project the erasure error to the remaining coefficients it is demonstrated that the same compensation can be executed using a transmitter receiver combination in which the transmitter is not aware of the erasure occurrence the transmitter precompensates using projections as if assuming erasures will occur the receiver undoes the compensation for the coefficients that have not been erased thus maintaining the compensation only of the erased coefficients the stability of the resulting systems is explored and stability conditions are derived it is shown that stability for any erasure pattern can be enforced by optimizing a constrained quadratic program at the system design stage the paper concludes with examples and simulations that verify the theoretical results and illustrate key issues in the algorithms

directed recursion models for fractal growth fractals constructed by recursive processes are introduced to model growth phenomena these fractals are simultaneously directed and self similar in analogy with patterns growing under diffusion limited conditions the multifractal nature of the harmonic measure associated with laplacian interfaces is qualitatively interpreted using the models calculation of the largest singularity exponent allows one to make conclusions about the behaviour of diffusion limited aggregates

signal reconstruction from the magnitude of subspace components we consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems in the deterministic setting a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions that require at least a quadratic number of subspaces moreover we address reconstruction under the erasure of a subset of the norms using the concepts of p fusion frames and list decoding we propose an algorithm that outputs a finite list of candidate signals one of which is the correct one in the random setting we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program

linearly connected sequences and spectrally optimal dual frames for erasures abstract in the case that a frame is prescribed for applications and erasures occur in the process of data transmissions we examine optimal dual frames for the recovery from single erasures in contrast to earlier papers we consider the spectral radius of the error operator instead of its operator norm as a measure of optimality this notion of optimality is natural when the neumann series is used to recover the original data in an iterative manner we obtain a complete characterization of spectrally one erasure optimal dual frames in terms of the redundancy distribution of the prescribed frame our characterization relies on the connection between erasure optimal frames and the linear connectivity property of the frame we prove that the linear connectivity property is equivalent to the intersection dependent property and is also closely related to the well known concept of a k independent set additionally we also establish several necessary and sufficient conditions for the existence of an alternate dual frame to make the iterative reconstruction work

p woven frames abstract the known family of woven collections of frames have interesting properties and applications yet they cannot be easily constructed the present paper introduces new collections of frames sharing many properties with the original ones and having simpler construction the frames i i i and i i i for a hilbert space h are called partition woven or briefly p woven if there exists a nontrivial subset of i such that the family g i i i represented as the triple i i and called a p weaven frame is a frame defined by g i i if i and g i i otherwise the paper also studies the class of cp woven collections of frames which is a subset of the p woven collections of frames and satisfy a stronger condition the set of all woven frames is a strict subset of the set of all cp woven frames which is further a strict subset of the set of all p woven frames several examples and counterexamples are studied to show whether or not a pair of frames is p woven or cp woven

asymptotic deployment gain a simple approach to characterize the sinr distribution in general cellular networks in cellular network models the base stations are usually assumed to form a lattice or a poisson point process ppp in reality however they are deployed neither fully regularly nor completely randomly accordingly in this paper we consider the very general class of motion invariant models and analyze the behavior of the outage probability the probability that the signal to interference plus noise ratio sinr is smaller than a threshold as the threshold goes to zero we show that remarkably the slope of the outage probability in db as a function of the threshold also in db is the same for essentially all motion invariant point processes the slope merely depends on the fading statistics using this result we introduce the notion of the asymptotic deployment gain adg which characterizes the horizontal gap between the success probabilities of the ppp and another point process in the high reliability regime where the success probability is near 1 to demonstrate the usefulness of the adg for the characterization of the sinr distribution we investigate the outage probabilities and the adgs for different point processes and fading statistics by simulations

equiangular tight frames from paley tournaments abstract we prove the existence of equiangular tight frames having n 2 d 1 elements drawn from either c d or c d 1 whenever n is either 2 k 1 for k n or a power of a prime such that n 3 mod 4 we also find a simple explicit expression for the prime power case by establishing a connection to a 2 d element equiangular tight frame based on quadratic residues

what controls the width function shape and can it be used for channel network comparison and regionalization 1 the width function captures the essential features of the catchment s giuh s response this paper aims to identify the morphometric properties which control the shape of the width function and assess whether these properties can be used as similarity indices for catchment comparison a new deterministic iterative model of the width function is proposed on the basis of a conceptualization of the topology of the channel network and exploiting the morphometric characteristics of internal and external nodes tests are carried out on eleven french catchments and compared to the reference peano catchment results show that the morphometric properties of three main internal and external nodes such as the drained area the distance to the outlet and the position on the channel network are useful descriptors for modeling the width function and for representing the scaling properties of a channel network while the giuhs based on horton strahler ratios are strongly related to the method used to extract the channel network from the dem the new indices are independent of the method used they are sufficient descriptors to reproduce the main shape of the width function the peak the time to peak and the main properties such as non negativity non stationarity and power law decay of the spectrum they may be used to establish catchment typology to compare catchments and to classify the width function peaks for catchment regionalization

the pythagorean theorem i the finite case the pythagorean theorem and variants of it are studied the variations evolve to a formulation in terms of noncommutative conditional expectations on von neumann algebras that displays the theorem as the basic result of noncommutative metric euclidean geometry the emphasis in the present article is finite dimensionality both discrete and continuous

upper bounds for s distance sets and equiangular lines abstract the set of points in a metric space is called an s distance set if pairwise distances between these points admit only s distinct values two distance spherical sets with the set of scalar products 0 1 are called equiangular the problem of determining the maximum size of s distance sets in various spaces has a long history in mathematics we suggest a new method of bounding the size of an s distance set in compact two point homogeneous spaces via zonal spherical functions this method allows us to prove that the maximum size of a spherical two distance set in r n n 7 is n n 1 2 with possible exceptions for some n 2 k 1 2 3 k n we also prove the universal upper bound 2 3 n a 2 for equiangular sets with 1 a and employing this bound prove a new upper bound on the size of equiangular sets in all dimensions finally we classify all equiangular sets reaching this new bound

new results for self similar trees with applications to river networks in a little known series of papers beginning in 1966 tokunaga introduced an infinite class of tree graphs based on the strahler ordering scheme as recognized by tokunaga 1984 these trees are characterized by a self similarity property so we will refer to them as self similar trees or ssts ssts are defined in terms of a generator matrix which acts as a blueprint for constructing different trees many familiar tree constructions are absorbed as special cases however in tokunaga s work an additional assumption is imposed which restricts from ssts to a much smaller class we will refer to this subclass as tokunaga s trees this paper presents several new and unifying results for ssts in particular the conditions under which ssts have well defined horton strahler stream ratios are given as well as a general method for computing these ratios it is also shown that the diameters of ssts grow like m where m is the number of leaves in contrast to many other tree constructions here need not equal 1 2 thus ssts offer an explanation for hack s law finally it is demonstrated that large discrepancies exist between the predictions of shreve s well known model and detailed measurements for large river networks while other ssts fit the data quite well other potential applications of the sst framework include diffusion limited aggregation dla lightning bronchial passages neural networks and botanical trees

on some identities and inequalities for frames in hilbert spaces we give new parseval type identities and inequalities for frames in hilbert spaces our results generalize the remarkable results obtained recently by r balan p g casazza d edidin and g kutyniok

a note on equiangular tight frames we settle a conjecture of joseph renes about the existence and construction of certain equiangular tight frames

unit hydrograph approximations assuming linear flow through topologically random channel networks the instantaneous unit hydrograph iuh of a drainage basin is derived in terms of fundamental basin characteristics z where parameterizes the link channel segment length distribution and is a vector of hydraulic parameters z is one of three basin topological properties n n d or n m where n is magnitude number of first order streams d is diameter mainstream length and m is order the iuh is derived based on assumptions that the links are independent and identically distributed random variables and that the network is a member of a topologically random population linear routing schemes including translation diffusion and general linear routing are used and constant drainage density is assumed by using n as the fundamental basin characteristics asymptotic for large n considerations lead to a weibull probability density function for the iuh with time to peak given by tp 2n where is mean link length and is a scalar hydraulic parameter usually average celerity this asymptotic iuh is identical for all linear routing schemes

filter bank fusion frames in this paper we characterize and construct novel oversampled filter banks implementing fusion frames a fusion frame is a sequence of orthogonal projection operators whose sum can be inverted in a numerically stable way when properly designed fusion frames can provide redundant encodings of signals which are optimally robust against certain types of noise and erasures however up to this point few implementable constructions of such frames were known we show how to construct them using oversampled filter banks in this work we first provide polyphase domain characterizations of filter bank fusion frames we then use these characterizations to construct filter bank fusion frame versions of discrete wavelet and gabor transforms emphasizing those specific finite impulse response filters whose frequency responses are well behaved

frames for vector spaces and affine spaces a finite frame for a finite dimensional hilbert space is simply a spanning sequence we show that the linear functionals given by the dual frame vectors do not depend on the inner product and thus it is possible to extend the frame expansion and other elements of frame theory to any finite spanning sequence for a vector space the corresponding coordinate functionals generalise the dual basis the case when the vectors are linearly independent and are characterised by the fact that the associated gramian matrix is an orthogonal projection existing generalisations of the frame expansion to banach spaces involve an analogue of the frame bounds and frame operator the potential applications of our results are considerable whenever there is a natural spanning set for a vector space computations can be done directly with it in an efficient and stable way we illustrate this with a diverse range of examples including multivariate spline spaces generalised barycentric coordinates and vector spaces over the rationals such as the cyclotomic fields

on the existence of equiangular tight frames abstract an equiangular tight frame etf is a d n matrix that has unit norm columns and orthogonal rows of norm n d its key property is that the absolute inner products between pairs of columns are i identical and ii as small as possible etfs have applications in communications coding theory and sparse approximation numerical experiments indicate that etfs arise for very few pairs d n and it is an important challenge to develop restrictions on the pairs for which they can exist this article uses field theory to provide detailed conditions on real and complex etfs in particular it describes restrictions on harmonic etfs a specific type of complex etf that appears in applications finally the article offers empirical evidence that these conditions are sharp or nearly sharp especially in the real case

a stochastic tokunaga model for stream networks the tokunaga cyclic model describes average network topology a stochastic generalization is proposed the stochastic model assumes that actual tributary numbers are random realizations from a negative binomial distribution whose mean is defined by the tokunaga parameters 1 and k these parameters can be interpreted as representing the effects of regional controls upon these regional controls is superimposed an inherent spatial variability in network topology a third parameter characterizes this spatial variability when becomes large the negative binomial model approaches a poisson model a goodness of fit test based on a 2 test statistic is developed and an inference framework for estimation of parameters and stream related statistics is described this methodology is illustrated on tributary data from three catchments one of the order of 5 and two of the order of 8 it is shown that the stochastic tokunaga model using the negative binomial distribution is not inconsistent with the tributary data whereas the poisson model is unambiguously rejected by the data monte carlo bayesian methods are used to evaluate the uncertainty in the tokunaga parameters and in stream number related statistics such as the bifurcation ratio it is shown that tributary data from the order 5 network provide little power for discriminating between model hypotheses the tributary data for the two order 8 basins are significantly different from the asymptotic stream number statistics predicted by shreve s random network model finally the problem of space filling or preservation of nontopological properties is considered in the context of the stochastic tokunaga model

multifractal formalism for fractal signals the structure function approach versus the wavelet transform modulus maxima method several attempts have been made recently to generalize the multifractal formalism originally introduced for singular measures to fractal signals we report on a systematic comparison between the structure function approach pioneered by parisi and frisch in 2 proceedings of the international school on turbulence and predictability in geophysical fluid dynamics and climate dynamics edited by m ghil r benzi and g parisi north holland amsterdam 1985 p 84 to account for the multifractal nature of fully developed turbulent signals and an alternative method we have developed within the framework of the wavelet transform analysis we comment on the intrinsic limitations of the structure function approach this technique has fundamental drawbacks and does not provide a full characterization of the singularities of a signal in many cases we demonstrate that our method based on the wavelet transform modulus maxima representation works in most situations and is likely to be the ground of a unified multifractal description of self affine distributions our theoretical considerations are both illustrated on pedagogical examples and supported by numerical simulations

nilpotent bridging for unions of two bases in a recent article the authors discovered a new efficient method for perfect reconstruction from frame erasures called nilpotent bridging in order to perform the reconstruction it is necessary to invert a matrix called the bridge matrix which consists of inner products of the frame vectors indexed by the erasure set with vectors from the corresponding dual frame indexed by a set known as the bridge set which is disjoint from the erasure set if no search procedure is required to find a bridge set of the same size as the erasure set for which the bridge matrix is invertible the dual frame pair is said to satisfy the full skew spark property equivalently a frame satisfies the full skew spark property if every subset of the complement of the erasure set with the same cardinality as the erasure set yields an invertible bridge matrix in this paper we consider dual frame pairs which consist of a union of two bases and the union of their corresponding dual bases in finite dimensions it is shown that if we restrict the class of bridge sets to block bridge sets then a bridge set search is also unnecessary for most unions of two bases and most unions of two orthonormal bases by most we mean the set of unions of two bases resp orthonormal bases for which no bridge set search is necessary contains an open dense set in the set of all unions of two bases resp orthonormal bases so while not every bridge set of the same cardinality as the erasure set yields an invertible bridge matrix the bridge sets which have the special structure of block bridge sets are very likely to yield an invertible bridge matrix these unions of two bases are said to satisfy the block skew spark property in infinite dimensions we also show that the block skew spark property holds for a dense but not necessarily open set of unions of two riesz resp orthonormal bases lastly applications to shannon whittaker sampling theory are discussed and several open questions are posed pertaining to sampling theory

scale invariance properties of the peak of the width function in topologically random networks some scaling properties of the topological width function for an infinite population of networks obeying the random model are analyzed a monte carlo procedure is applied to generate width functions according to the hypothesis of topological randomness the probability distributions of both peak and distance to peak of the topological width functions conditioned 1 on the network diameter and 2 on and parameter log 2 l log are studied the parameter can be considered a shape factor of the network indeed low values describe elongated networks while high values refer to fan like networks scale invariance for both random variables is established in the first case by using as a scale parameter also in the second case scale invariance is observed for both the peak and the distance to peak of the topological width function in particular the invariance property for the peak involves a scaling function which is directly related to the shape factor allowing determination of the statistical similarity between random networks indexed by the same then a coarse graining procedure is applied to a set of 15 000 width functions with 512 a scaling behavior of peaks of the original width function and aggregated ones is observed over a wide range of aggregation scales consequently a statistical self similarity for the peaks is also observed which involved the same related scaling function finally possible implication of the present results on the hydrologic response at the basin scale is discussed

steiner equiangular tight frames abstract we provide a new method for constructing equiangular tight frames etfs the construction is valid in both the real and complex settings and shows that many of the few previously known examples of etfs are but the first representatives of infinite families of such frames it provides great freedom in terms of the frame s size and redundancy this method also explicitly constructs the frame vectors in their native domain as opposed to implicitly defining them via their gram matrix moreover in this domain the frame vectors are very sparse the construction is extremely simple a tensor like combination of a steiner system and a regular simplex this simplicity permits us to resolve an open question regarding etfs and the restricted isometry property rip we show that the rip behavior of some etfs is unfortunately no better than their coherence indicates

geometry of the welch bounds abstract a geometric perspective involving grammian and frame operators is used to derive the entire family of welch bounds this perspective unifies a number of observations that have been made regarding tightness of the bounds and their connections to symmetric k tensors tight frames homogeneous polynomials and t designs in particular a connection has been drawn between sampling of homogeneous polynomials and frames of symmetric k tensors it is also shown that tightness of the bounds requires tight frames the lack of tight frames of symmetric k tensors in many cases however leads to consideration of sets that come as close as possible to attaining the bounds the geometric derivation is then extended in the setting of generalized or continuous frames the welch bounds for finite sets and countably infinite sets become special cases of this general setting

nullspaces and frames in this paper we give new characterizations of riesz and conditional riesz frames in terms of the properties of the nullspace of their synthesis operators on the other hand we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms

equiangular tight frames and fourth root seidel matrices in this paper we construct complex equiangular tight frames etfs in particular we study the grammian associated with an etf whose off diagonal entries consist entirely of fourth roots of unity these etfs are classified and we also provide some computational techniques which give rise to previously undiscovered etfs

a note on fractal channel networks this paper studies the relation between the structure of river networks and the features of their geomorphologic hydrologic response a mathematical formulation of connectivity of a drainage network is proposed to relate contributing areas and the network geometry in view of the connectivity conjecture horton s bifurcation ratio r b tends for high values of strahler s order omega of the basin to the area ratio r a and horton s length ratio r l equals in the limit the single order contributing area ratio r a the relevance of these arguments is examined by reference to data from real basins well known empirical results from the geomorphological literature melton s law hack s relation moon s conjecture are viewed as a consequence of connectivity it is found that in hortonian networks the time evolution of contributing areas exhibits a multifractal behavior generated by a multiplicative process of parameter 1 r b the application of the method of the most probable distribution in view of connectivity contributes new inroads toward a general formulation of the geomorphologic unit hydrograph in particular generalizing its width function formulation a quantitative example of multifractal hydrologic response of idealized networks based on peano s construct for which r b r a 4 r l 2 closes the paper

lattices from equiangular tight frames abstract we consider the set of all linear combinations with integer coefficients of the vectors of a unit equiangular tight k n frame and are interested in the question whether this set is a lattice that is a discrete additive subgroup of the k dimensional euclidean space we show that this is not the case if the cosine of the angle of the frame is irrational we also prove that the set is a lattice for n k 1 and that there are infinitely many k such that a lattice emerges for n 2 k we dispose of all cases in dimensions k at most 9 in particular we show that a 7 28 frame generates a strongly eutactic lattice and give an alternative proof of roland bacher s recent observation that this lattice is perfect

on quasi orthogonal signatures for cdma systems sum capacity optimal signatures in synchronous code division multiple access cdma systems are functions of the codebook length as well as the number of active users a new signature set must be assigned every time the number of active users changes this correspondence considers signature sets that are less sensitive to changes in the number of active users equiangular signature sequences are proven to solve a certain max min signal to interference plus noise problem which results from their interference invariance unions of orthonormal bases have subsets that come close to satisfying the welch bound bounds on the maximum number of bases with minimum maximum correlation are derived and a new construction algorithm is provided connections are made between these signature design problems grassmannian line packing frame theory and algebraic geometry

optimal dual frames for erasures for any given frame for the purpose of encoding in a finite dimensional hilbert space we investigate its dual frames that are optimal for erasures for the purpose of decoding we show that in general the canonical dual is not necessarily optimal moreover optimal dual frames are not necessarily unique we present some sufficient conditions under which the canonical dual is the unique optimal dual frame for the erasure problem as an application we get that the canonical dual is the only optimal dual when either the frame is induced by a group representation or the frame is uniform tight

sir meta distribution of k tier downlink heterogeneous cellular networks with cell range expansion heterogeneous cellular networks hcns constitute a necessary step in the evolution of cellular networks in this paper we apply the signal to interference ratio sir meta distribution framework for a refined sir performance analysis of hcns focusing on k tier heterogeneous cellular networks based on the homogeneous independent poisson point process ppp model with range expansion bias offloading bias in each tier expressions for the b th moment of the conditional success probability for both the entire network and each tier are derived based on which the exact meta distributions and the beta approximations are evaluated and compared key performance metrics including the mean success probability the variance of the conditional success probability the mean local delay and the asymptotic sir gains of each tier are obtained the results show that the biases are detrimental to the overall mean success probability of the whole network and that the b th moment curve of the conditional success probability of each tier can be tightly approximated by the horizontal shifted versions of the first moment curve of the single tier ppp network we also provide lower bounds for the region of the active probabilities of the base stations to keep the mean local delay of each tier finite

area and width functions of river networks new results on multifractal properties 1 this paper investigates the multiscale statistical structure of the area and width functions of simulated and real river networks via state of the art wavelet based multifractal mf formalisms first several intricacies in performing mf analysis of these signals are discussed and a robust framework for accurate estimation of the mf spectrum is presented second it considers the following three questions 1 does the topology of river networks leave a unique signature on the mf spectrum of area and width functions 2 how different are the mf properties of commonly used simulated trees and those of real river networks and 3 are there differences between the mf properties of width and area functions and what can these tell us about the topology of hillslope versus channelized drainage patterns in a river basin the results indicate discrepancies between the statistical scaling of the area functions of real networks found to be multifractal with a considerable spread of local singularities and the most prevailing singularity ranging from 0 4 to 0 8 and that of several commonly used stochastic self similar networks found to be monofractal with a single singularity exponent h in the range of 0 5 0 65 moreover differences are found between the mf properties of width and area functions of the same basin these differences may be the result of distinctly different branching topologies in the hillslope versus channelized drainage paths and need to be further investigated

frames of uniform subframe bounds with applications to erasures abstract we introduce a class of frames called usb frames frames of uniform subframe bounds which mean that any subsequence of the frame is a frame for the subspace it spans and the frame bounds depend only on the cardinality of the subsequence we give several necessary and sufficient conditions for a frame to be a usb frame as an application we show that a tight usb frame is optimal for erasures in the sense that the reconstruction error is independent of the position of erasures we give several explicit construction of usb frames

optimal dual frames for communication coding with probabilistic erasures assume that a frame is preselected for encoding in a communication system we investigate the optimal dual frames for signal reconstruction decoding which minimize the maximal error when the probabilistic erasures occur in the transmission process from the perspective of mathematical theory of frames we set up a probability model under which we define the probability optimal po dual frames for a given frame when the frame based coding involves probabilistic erasures we obtain a sufficient and necessary condition under which the canonical dual frame is the unique po dual frame additionally we derive some general sufficient conditions for which the canonical dual frame is either not optimal or it is optimal but not the unique optimal one we present two simulation examples to compare the reconstruction effects when both the po dual frames and the general optimal go dual frames are used for reconstruction