detection of signals by information theoretic criteria a new approach is presented to the problem of detecting the number of signals in a multichannel time series based on the application of the information theoretic criteria for model selection introduced by akaike aic and by schwartz and rissanen mdl unlike the conventional hypothesis testing based approach the new approach does not requite any subjective threshold settings the number of signals is obtained merely by minimizing the aic or the mdl criteria simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented

blind deconvolution using convex programming we consider the problem of recovering two unknown vectors w and x of length l from their circular convolution we make the structural assumption that the two vectors are members of known subspaces one with dimension n and the other with dimension k although the observed convolution is nonlinear in both w and x it is linear in the rank 1 matrix formed by their outer product wx this observation allows us to recast the deconvolution problem as low rank matrix recovery problem from linear measurements whose natural convex relaxation is a nuclear norm minimization program we prove the effectiveness of this relaxation by showing that for generic signals the program can deconvolve w and x exactly when the maximum of n and k is almost on the order of l that is we show that if x is drawn from a random subspace of dimension n and w is a vector in a subspace of dimension k whose basis vectors are spread out in the frequency domain then nuclear norm minimization recovers wx without error we discuss this result in the context of blind channel estimation in communications if we have a message of length n which we code using a random l x n coding matrix and the encoded message travels through an unknown linear time invariant channel of maximum length k then the receiver can recover both the channel response and the message when l n k to within constant and log factors

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

a new perturbation analysis for signal enumeration in rotational invariance techniques the esprit algorithm is a subspace based high resolution method used in source localization and spectral analysis which provides very accurate estimates of the signal parameters however the underlying theory assumes a known model order which is usually not the case in many applications in particular it is well known that underevaluating the model order biases the estimation in this paper we analyze the perturbation induced by an erroneous model order and we present an error bound for the estimated parameters based on this theoretical framework we propose a new method for selecting an appropriate modeling order which consists in minimizing the error bound this approach is applied to both synthetic and musical signals and its performance is compared to that of existing methods such as the information theoretic criteria

two parameter generalization of the logarithm and exponential functions and boltzmann gibbs shannon entropy the q sum x qy x y 1 q xy x 1y x y and the q product x qy x1 q y1 q 1 1 1 q x 1y xy emerge naturally within nonextensive statistical mechanics we show here how they lead to two parameter namely q and q generalizations of the logarithmic and exponential functions noted respectively lnq q x and eq q x as well as of the boltzmann gibbs shannon entropy sbgs k i 1wpilnpi noted sq q the remarkable properties of the q q generalized logarithmic function make the entropic form sq q k i 1wpilnq q 1 pi satisfy for large regions of q q important properties such as expansibility concavity and lesche stability but not necessarily composability

root numbers and ranks in positive characteristic abstract for a global field k and an elliptic curve e over k t silverman s specialization theorem implies rank e k t rank e t k for all but finitely many t p 1 k if this inequality is strict for all but finitely many t the elliptic curve e is said to have elevated rank all known examples of elevated rank for k q rest on the parity conjecture for elliptic curves over q and the examples are all isotrivial some additional standard conjectures over q imply that there does not exist a non isotrivial elliptic curve over q t with elevated rank in positive characteristic an analogue of one of these additional conjectures is false inspired by this for the rational function field k u over any finite field with characteristic 2 we construct an explicit 2 parameter family e c d of non isotrivial elliptic curves over k t depending on arbitrary c d such that under the parity conjecture each e c d has elevated rank

tsallis entropy maximization procedure revisited abstract the proper way of averaging is an important question with regards to tsallis thermostatistics three different procedures have been thus far employed in the pertinent literature the third one i e the tsallis mendes plastino tmp physica a 261 1998 534 normalization procedure exhibits clear advantages with respect to earlier ones in this work we advance a distinct from the tmp one way of handling the lagrange multipliers involved in the extremization process that leads to tsallis statistical operator it is seen that the new approach considerably simplifies the pertinent analysis without losing the beautiful properties of the tsallis mendes plastino formalism

the role of constraints in tsallis nonextensive treatment revisited abstract we revisit the nonextensive treatment advanced by tsallis et al physica a 261 1998 534 in order to perform a rather exhaustive study of its properties and compare them with the earlier formalism devised by curado and tsallis j phys a 24 1991 l69 some interesting features are in this way revealed

on the behavior of information theoretic criteria for model order selection the akaike 1974 information criterion aic and the minimum description length mdl are two well known criteria for model order selection in the additive white noise case our aim is to study the influence on their behavior of a large gap between the signal and the noise eigenvalues and of the noise eigenvalue dispersion our results are mostly qualitative and serve to explain the behavior of the aic and the mdl in some cases of great practical importance we show that when the noise eigenvalues are not clustered sufficiently closely then the aic and the mdl may lead to overmodeling by ignoring an arbitrarily large gap between the signal and the noise eigenvalues for fixed number of data samples overmodeling becomes more likely for increasing the dispersion of the noise eigenvalues for fixed dispersion overmodeling becomes more likely for increasing the number of data samples undermodeling may happen in the cases where the signal and the noise eigenvalues are not well separated and the noise eigenvalues are clustered sufficiently closely we illustrate our results by using simulations from the effective channel order determination area

robustness of least squares and subspace methods for blind channel identification equalization with respect to effective channel undermodeling overmodeling the least squares and the subspace methods are two well known approaches for blind channel identification equalization when the order of the channel is known the algorithms are able to identify the channel under the so called length and zero conditions furthermore in the noiseless case the channel can be perfectly equalized less is known about the performance of these algorithms in the practically inevitable cases in which the channel possesses long tails of small impulse response terms we study the performance of the mth order least squares and subspace methods using a perturbation analysis approach we partition the true impulse response into the mth order significant part and the tails we show that the mth order least squares or subspace methods estimate an impulse response that is close to the mth order significant part the closeness depends on the diversity of the mth order significant part and the size of the tails furthermore we show that if we try to model not only the large terms but also some small ones then the quality of our estimate may degrade dramatically thus we should avoid modeling small terms finally we present simulations using measured microwave radio channels highlighting potential advantages and shortcomings of the least squares and subspace methods

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

blind channel approximation effective channel order determination a common assumption of blind channel identification methods is that the order of the true channel is known this information is not available in practice and we are obliged to estimate the channel order by applying a rank detection procedure to an overmodeled data covariance matrix information theoretic criteria have been widely suggested approaches for this task we check the quality of their estimates in the context of order estimation of measured microwave radio channels and confirm that they are very sensitive to variations in the snr and the number of data samples this fact has prohibited their successful application for channel order estimation and hits created some confusion concerning the classification into under and over modeled cases recently it has been shown that blind channel approximation methods should attempt to model only the significant part of the channel composed of the large impulse response terms because efforts toward modeling small leading and or trailing terms lead to effective overmodeling which is generically ill conditioned and thus should be avoided this can be achieved by applying blind identification methods with model order equal to the order of the significant part of the true channel called the effective channel order toward developing an efficient approach for the detection of the effective channel order we use numerical analysis arguments the derived criterion provides a maximally stable decomposition of the range space of an overmodeled data covariance matrix into signal and noise subspaces it is shown to be robust to variations in the snr and the number of data samples furthermore it provides useful effective channel order estimates leading to sufficiently good blind approximation equalization of measured real world microwave radio channels

generalized ensemble theory with non extensive statistics abstract the non extensive canonical ensemble theory is reconsidered with the method of lagrange multipliers by maximizing tsallis entropy with the constraint that the normalized term of tsallis q average of physical quantities the sum p j q is independent of the probability p i for tsallis parameter q the self referential problem in the deduced probability and thermal quantities in non extensive statistics is thus avoided and thermodynamical relationships are obtained in a consistent and natural way we also extend the study to the non extensive grand canonical ensemble theory and obtain the q deformed bose einstein distribution as well as the q deformed fermi dirac distribution the theory is further applied to the generalized planck law to demonstrate the distinct behaviors of the various generalized q distribution functions discussed in literature

a least squares approach to blind channel identification conventional blind channel identification algorithms are based on channel outputs and knowledge of the probabilistic model of channel input in some practical applications however the input statistical model may not be known or there may not be sufficient data to obtain accurate enough estimates of certain statistics in this paper we consider the system input to be an unknown deterministic signal and study the problem of blind identification of multichannel fir systems without requiring the knowledge of the input statistical model a new blind identification algorithm based solely on the system outputs is proposed necessary and sufficient identifiability conditions in terms of the multichannel systems and the deterministic input signal are also presented

ideal gas provides q entropy abstract a mathematical procedure is suggested to obtain deformed entropy formulas of type k s k p i k ln p i by requiring zero mutual k s k information between a finite subsystem and a finite reservoir the use of this method is first demonstrated on the ideal gas equation of state with finite constant heat capacity c where it delivers the renyi and tsallis formulas a novel interpretation of the q 2 q duality arises from the comparison of canonical subsystem and total microcanonical partition approaches in the sequel a new generalized deformed entropy formula is constructed for the linear c s c 0 c 1 s relation

on the rank of an elliptic surface nagao has recently given a conjectural limit formula for the rank of an elliptic surface e in terms of a weighted average of fibral frobenius trace values we show that tate s conjecture on the order of vanishing of l 2 e s essentially implies nagao s formula in particular we prove nagao s formula for rational elliptic surfaces in the case that e is a twist we reduce nagao s and tate s conjectures to the case of products of curves and we verify the conjectures for many new classes of elliptic surfaces of kodaira dimension 0 and 1

fundamental limits of blind deconvolution part i ambiguity kernel blind deconvolution is an ubiquitous non linear inverse problem in applications like wireless communications and image processing this problem is generally ill posed and there have been efforts to use sparse models for regularizing blind deconvolution to promote signal identifiability part i of this two part paper characterizes the ambiguity space of blind deconvolution and shows unidentifiability of this inverse problem for almost every pair of unconstrained input signals the approach involves lifting the deconvolution problem to a rank one matrix recovery problem and analyzing the rank two null space of the resultant linear operator a measure theoretically tight parametric and recursive representation of the key rank two null space is stated and proved this representation is a novel foundational result for signal and code design strategies promoting identifiability under convolutive observation models part ii of this paper analyzes the identifiability of sparsity constrained blind deconvolution and establishes surprisingly strong negative results on scaling laws for the sparsity ambiguity trade off

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

generalized statistical mechanics connection with thermodynamics the authors show the manner through which the recent generalization by tsallis 1988 of the boltzmann gibbs statistics becomes consistent with a generalized thermodynamics preserving the legendre transformation framework of standard thermodynamics in addition to that they generalize the shannon additivity

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators