discrete time wavelet extrema representation design and consistent reconstruction the paper studies wavelet transform extrema and zero crossings representations within the framework of convex representations in spl lscr z wavelet zero crossings representation of two dimensional signals is introduced as a convex multiscale edge representation as well one appealing property of convex representations is that the reconstruction problem can be solved at least theoretically using the method of alternating projections onto convex sets it turns out that in the case of the wavelet extrema and wavelet zero crossings representations this method yields simple and practical reconstruction algorithms nonsubsampled filter banks that implement the wavelet transform for the two representations are also studied in the paper relevant classes of nonsubsampled perfect reconstruction fir filter banks are characterized this characterization gives a broad class of wavelets for the representations which are derived from those of the filter banks which satisfy a regularity condition

on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

congruences between modular forms raising the level and dropping euler factors abstract we discuss the relationship among certain generalizations of results of hida ribet and wiles on congruences between modular forms hida s result accounts for congruences in terms of the value of an l function and ribet s result is related to the behavior of the period that appears there wiles theory leads to a class number formula relating the value of the l function to the size of a galois cohomology group the behavior of the period is used to deduce that a formula at nonminimal level is obtained from one at minimal level by dropping euler factors from the l function

simultaneous pseudo timestepping for aerodynamic shape optimization problems with state constraints this paper presents a numerical method for constrained aerodynamic shape optimization problems it is based on simultaneous pseudo timestepping in which stationary states are obtained by solving the pseudo stationary system of equations representing the state costate and design equations the main advantages of this method are that it blends in nicely with a previously existing pseudo timestepping method for the states only and that a preconditioner can be used for convergence acceleration which stems from the reduced sqp methods design examples for drag reduction with constant lift and drag reduction with constant pitching moment for an rae2822 airfoil are included the overall cost of computation is 4 8 times that of the forward simulation runs depending on the grid size and the number of design parameters

refined error analysis in second order spl sigma spl delta modulation with constant inputs although the technique of sigma delta spl sigma spl delta modulation is well established in practice for performing high resolution analog to digital a d conversion theoretical analysis of the error between the input signal and the reconstructed signal has remained partial for modulators of order higher than 1 the only rigorous error analysis currently available that matches practical and numerical simulation results is only applicable to a very special configuration namely the standard and ideal k bit k loop spl sigma spl delta modulator moreover the error measure involves averaging over time as well as possibly over the input value at the second order it is known in practice that the mean squared error decays with the oversampling ratio spl lambda at the rate o spl lambda sup 5 in this paper we introduce two new fundamental results in the case of constant input signals we first establish a framework of analysis that is applicable to all second order modulators provided that the built in quantizer has uniformly spaced output levels and that the noise transfer function has its two zeros at the zero frequency in particular this includes the one bit case a rigorous and deterministic analysis of which is still not available this generalization has been possible thanks to the discovery of the mathematical tiling property of the state variables of such modulators the second aspect of our contribution is to perform an instantaneous error analysis that avoids infinite time averaging until now only an o spl lambda sup 4 type error bound was known to hold in this setting under our generalized framework we provide two types of squared error estimates one that is statistically averaged over the input and another that is valid for almost every input in the sense of lebesgue measure in both cases we improve the error bound to o spl lambda sup 4 5 up to a logarithmic factor for a general class of modulators including some specific ones that are covered in this paper in detail in the particular case of the standard and ideal two bit double loop configuration our methods provide a previously unavailable instantaneous error bound of o spl lambda sup 5 again up to a logarithmic factor

direct projection decoding algorithm for sigma delta modulated signals usually sigma delta modulators spl sigma spl delta m are modeled by replacing the quantizer with an additive white noise source based on this linear model linear decoder structures can be designed however the spl sigma spl delta m is a nonlinear system and corresponding nonlinear decoder structures may be able to achieve a better signal to quantization noise ratio sqnr performance in this paper a new nonlinear decoding algorithm is presented it is based on the projection onto complex sets pocs algorithm developed by hein and zakhor our algorithm reduces the decoding problem to the solution of a single quadratic programming problem based on the state equations of the modulator and the condition that the modulator input signal is bandlimited the bandlimitation constraint is directly applied to the state equations thus the resulting quadratic programming problem needs to be solved only once

triple product p adic l functions for balanced weights we construct p adic triple product l functions that interpolate square roots of central critical l values in the balanced region thus our construction complements that of harris and tilouine there are four central critical regions for the triple product l functions and two opposite settings according to the sign of the functional equation in the first case three of these regions are of interpolation having positive sign they are called the unbalanced regions and one gets three p adic l functions one for each region of interpolation this is the harris tilouine setting in the other setting there is only one region of interpolation called the balanced region we produce the corresponding p adic l function our triple product p adic l function arises as p adic period integrals interpolating normalizations of the local archimedean period integrals the latter encode information about classical representation theoretic branching laws the main step in our construction of p adic period integrals is showing that these branching laws vary in a p adic analytic fashion this relies crucially on the ash stevens theory of highest weight representations over affinoid algebras

weighted median filters with sigma delta modulation encoding digital decimation filters play a fundamental role in oversampled sigma delta a d decoders in this paper we first show that weighted median wm filtering of a demodulated sequence at the nyquist rate can be implemented concurrently in the a d decoder through a simple modification of the binary time series outputted by the a d modulator the sequence obtained after the sigma delta modulation sdm decoder is shown to be equivalent to wm filtering the multilevel sequence at the nyquist rate second we show that wm filters can be used for sdm decimation filters and that these filters are readily implemented in the sdm binary domain a very promising characteristic of sdm converters equipped with wm decimating filters is that sharp discontinuities edges can be preserved and acquired thus the bandlimited constraint imposed on the input signals can be relaxed making sdm more attractive to a d conversion of signals containing sharp transitions the proposed signal processing algorithms in essence combine a d sigma delta converters and wm filters into a single programmable system

computing the degree of the modular parametrization of a modular elliptic curve the weil taniyama conjecture states that every elliptic curve e q of conductor n can be parametrized by modular functions for the congruence subgroup t0 n of the modular group r f5l 2 z equivalently there is a nonconstant map ip from the modular curve xq n to e we present here a method of computing the degree of such a map for arbitrary n our method which works for all subgroups of finite index in i and not just r0 ar is derived from a method of zagier published in 1985 by using those ideas together with techniques which have recently been used by the author to compute large tables of modular elliptic curves we are able to derive an explicit and general formula which is simpler to implement than zagier s we discuss the results obtained including a table of degrees for all the modular elliptic curves of conductors up to 200

aerodynamic shape optimization using simultaneous pseudo timestepping the paper deals with a numerical method for aerodynamic shape optimization it is based on simultaneous pseudo timestepping in which stationary states are obtained by solving the non stationary system of equations representing the state costate and design equations the main advantages of this method are that it requires no additional globalization techniques and that a preconditioner can be used for convergence acceleration which stems from the reduced sqp method a design example for drag reduction for an rae2822 airfoil keeping its thickness fixed is included the overall cost of computation is less than four times that of the forward simulation run

multigrid one shot method for aerodynamic shape optimization the paper deals with a numerical method for aerodynamic shape optimization using simultaneous pseudo timestepping we have recently developed a method for the optimization problem in which stationary states are obtained by solving the pseudo stationary system of equations representing state costate and design equations the method requires no additional globalization techniques in the design space and a reduced sequential quadratic programming rsqp methods based preconditioner can be used for convergence acceleration the overall cost of computation is reduced up to 70the number of optimization iterations is comparatively large since we update the design parameters in each time step of state and costate runs in this paper we use a multigrid strategy to reduce the total number of optimization iterations keeping the idea of the one shot method design examples of drag reduction together with geometrical constraints for an rae2822 airfoil and an sct wing are included the total effort required for the convergence of the optimization problem is less than 2 times 4 times that of the forward simulation runs in two dimensions in three dimensions

on the robustness of single loop sigma delta modulation sigma delta modulation a widely used method of analog to digital a d signal conversion is known to be robust to hardware imperfections i e bit streams generated by slightly imprecise hardware components can be decoded comparably well we formulate a model for robustness and give a rigorous analysis for single loop sigma delta modulation applied to constant signals dc inputs for n time cycles with an arbitrary small enough initial condition u sub o and a quantizer that may contain an offset error the mean square error mse of any decoding scheme for this quantizer with u sub o and the offset error known is bounded below by 1 96n sup 3 we also determine the asymptotically best possible mse as n spl rarr spl infin for perfect decoding when u sub o 0 and u sub o 1 2 the robustness result is the upper bound that a triangular linear filter decoder with both u sub o and the offset error unknown achieves an mse of 40 3n sup 3 these results establish the known result that the o 1 n sup 3 decay of the mse with n is optimal in the single loop case under weaker assumptions than previous analyses and show that a suitable linear decoder is robust against offset error these results are obtained using methods from number theory and fourier analysis

model problems for the multigrid optimization of systems governed by differential equations we discuss a multigrid approach to the optimization of systems governed by differential equations such optimization problems appear in many applications and are of a different nature than systems of equations our approach uses an optimization based multigrid algorithm in which the multigrid algorithm relies explicitly on nonlinear optimization models as subproblems on coarser grids our goal is not to argue for a particular optimization based multigrid algorithm but instead to demonstrate how multigrid can be used to accelerate nonlinear programming algorithms furthermore using several model problems we give evidence both theoretical and numerical that the optimization setting is well suited to multigrid algorithms some of the model problems show that the optimization problem may be more amenable to multigrid than the governing differential equation in addition we relate the multigrid approach to more traditional optimization methods as further justification for the use of an optimization based multigrid algorithm

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

hauteur de faltings de quotients de j 0 n discriminants d algebres de hecke et congruences entre formes modulaires in this paper we compute the faltings height of the jacobian of the modular curve j 0 n we explain how it is related to the discriminant of the hecke algebra this discriminant measures congruences between modular forms as a corollary we obtain a lower bound for this discriminant the faltings height of a quotient a of j 0 n is also computed

non linear reconstruction of delta sigma modulated signals randomized surrogate constraint decoding algorithm delta sigma a d conversion is a popular technique used to achieve high resolution data conversion for low to moderate bandwidth applications it is commonly accepted that the digital circuitry of the a d converters should perform a linear time invariant lti filtering on the output of the modulator in order to reconstruct the input signal however it has been shown that when higher order a d converters are operated at higher oversampling ratios osrs non linear reconstruction algorithms extract more information about the input signal than lti methods still use of non linear algorithms has been limited due to their complexity and stability issues two practical non linear methods presented in the literature are fast projection onto convex sets pocs and direct projection dp in this work we show that the non linear reconstruction process for modulated sequences can be treated as a linear feasibility problem lfp we then show that if the reconstruction process is defined as an lfp its implementation is less complex than in the case of the dp method we further describe how the lfp can be well conditioned and demonstrate the reconstruction process by using the surrogate constraint algorithm with random constraint selection the described algorithm considers a small number of constraints at a time which makes it suitable for serial implementation

a generalization of greenberg s l invariant using the theory of phi gamma modules we generalize greenberg s construction of the cal l invariant to p adic representations which are semistable at p this allows us to formulate a quite general conjecture about the behavior of p adic l functions at trivial zeros

the iwasawa main conjectures for gl2 we prove the one two and three variable iwasawa greenberg main conjectures for a large class of modular forms that are ordinary with respect to an odd prime p the method of proof involves an analysis of an eisenstein ideal for ordinary hida families for gu 2 2

congruence of siegel modular forms and special values of their standard zeta functions in this paper we consider the relationship between the congruence of cuspidal hecke eigenforms with respect to sp n z and the special values of their standard zeta functions in particular we propose a conjecture concerning the congruence between saito kurokawa lifts and non saito kurokawa lifts and prove it under certain condition

kummer s criterion for the special values of hecke l functions of imaginary quadratic fields and congruences among cusp forms here arrow i refers to our previous papers 9 12 and 13 in which we have shown that a fixed primitive cusp form f of sk f i n has congruences with other cusp forms of sk f n modulo the special value at s k of a certain zeta function off in this paper we will construct the second arrow for the cusp form f associated with a hecke character of an imaginary quadratic field k and consequently obtain the third namely in w167 6 we will show as in the works of coates wiles 3 and robert 23 that the split primes of k which divide this special value are irregular in an appropriate sense in this case the zeta function o f f mentioned above is a hecke l function of k such a criterion of irregularity for hurwitz numbers has already been obtained in 3 and 233 however our result together with those of 3 and 23 does not cover all the l values of hecke l functions of k whose algebraicity was given by damerell 5 for details see below our method is analogous to that of ribet 21 see also wiles 35 who used congruences between cusp forms and eisenstein series to prove the non vanishing of certain eigenspaces relative to the action of gal q q of the p primary part of the class group of the field of p th roots of unity by adopting this approach we will obtain such an information of eigenspaces again in our case see theorem 0 1 below to be more specific let 2 be a hecke character of k satisfying

a fast block based nonlinear decoding algorithm for spl sigma spl delta modulators previous work has established that the digital output of a spl sigma spl delta modulator as an a d converter contains more information about the analog input than is extracted with conventional linear filtering under reasonable mathematical assumptions optimal nonlinear decoding of the digital output can achieve significantly larger signal to noise ratios than linear filtering however the hitherto proposed decoding algorithms only demonstrate conceptual feasibility and are impractical from a computational point of view we present a new block based decoding algorithm that like previous work employs projections onto convex sets the algorithm owes its speed to a change of projection norm an accelerated convergence scheme and a decimation like subsampling it is on the order of 10 sup 4 10 sup 5 times faster than one previously published algorithm for typical parameter values and about 2 10 times slower than linear decoding the new algorithm is applicable to all currently popular spl sigma spl delta architectures

oversampled a d conversion and error rate dependence of nonbandlimited signals with finite rate of innovation we study the problem of a d conversion and error rate dependence of a class of nonbandlimited signals with finite rate of innovation in particular we analyze a continuous periodic stream of diracs characterized by a finite set of time positions and weights previous research has only considered sampling of this type of signals ignoring the presence of quantization necessary for any practical implementation to this end we first define the concept of consistent reconstruction and introduce corresponding oversampling in both time and frequency high accuracy in a consistent reconstruction is achieved by enforcing the reconstructed signal to satisfy three sets of constraints related to low pass filtering quantization and the space of continuous periodic streams of diracs we provide two schemes to reconstruct the signal for the first one we prove that the estimation mean squared error of the time positions is o 1 r sub t sup 2 r sub f sup 3 where r sub t and r sub f are the oversampling ratios in time and frequency respectively for the second scheme it is experimentally observed that at the cost of higher complexity the estimation accuracy lowers to o 1 r sub t sup 2 r sub f sup 5 our experimental results show a clear advantage of consistent over nonconsistent reconstruction regarding the rate we consider a threshold crossing based scheme where as opposed to previous research both oversampling in time and in frequency influence the coding rate we compare the error rate behavior resulting on the one hand from increasing the oversampling in time and or frequency and on the other hand from decreasing the quantization stepsize

deterministic analysis of oversampled a d conversion and decoding improvement based on consistent estimates discusses the deterministic analysis of oversampled a d conversion adc the properties derivable from such an analysis and the consequences on reconstruction using nonlinear decoding given a band limited input x producing a quantized version c the authors consider the set of all input signals that are band limited and produce c they call any element of this set a consistent estimate of x regardless of the type of encoder simple predictive or noise shaping they show that this set is convex and as a consequence any nonconsistent estimate can be improved they also show that the classical linear decoding estimates are not necessarily consistent numerical tests performed on simple adc single loop and multiloop spl sigma spl delta modulation show that consistent estimates yield a mean square error mse that decreases asymptotically with the oversampling ratio faster than the linear decoding mse by approximately 3 db octave this implies an asymptotic mse of the order of spl oscr r sup spl minus 2n 2 instead of spl oscr r sup spl minus 2n 1 in linear decoding where r is the oversampling ratio and n the order of the modulator methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple adc predictive adc single loop and multiloop spl sigma spl delta modulation

single bit oversampled a d conversion with exponential accuracy in the bit rate a scheme for simple oversampled analog to digital a d conversion using single bit quantization is presented the scheme is based on recording positions of zero crossings of the input signal added to a deterministic dither function this information can be represented in a manner such that the bit rate increases only logarithmically with the oversampling factor r the input band limited signal can be reconstructed from this information locally with o 1 r pointwise error resulting in an exponentially decaying distortion rate characteristic in the course of studying the accuracy of the proposed a d conversion scheme some new results are established about reconstruction of band limited signals from irregular samples using linear combination of functions with fast decay schemes for local interpolation of band limited signals from quantized irregular samples are also proposed

entropy of highly correlated quantized data this paper considers the entropy of highly correlated quantized samples two results are shown the first concerns sampling and identically scalar quantizing a stationary continuous time random process over a finite interval it is shown that if the process crosses a quantization threshold with positive probability then the joint entropy of the quantized samples tends to infinity as the sampling rate goes to infinity the second result provides an upper bound to the rate at which the joint entropy tends to infinity in the case of an infinite level uniform threshold scalar quantizer and a stationary gaussian random process specifically an asymptotic formula for the conditional entropy of one quantized sample conditioned on the previous quantized sample is derived at high sampling rates these results indicate a sharp contrast between the large encoding rate in bits sec required by a lossy source code consisting of a fixed scalar quantizer and an ideal sampling rate adapted lossless code and the bounded encoding rate required by an ideal lossy source code operating at the same distortion

recursive consistent estimation with bounded noise estimation problems with bounded uniformly distributed noise arise naturally in reconstruction problems from over complete linear expansions with subtractive dithered quantization we present a simple recursive algorithm for such bounded noise estimation problems the mean square error mse of the algorithm is almost o 1 n sup 2 where n is the number of samples this rate is faster than the o 1 n mse obtained by standard recursive least squares estimation and is optimal to within a constant factor

sampling and reconstruction of shapes with algebraic boundaries we present a sampling theory for a class of binary images with finite rate of innovation fri every image in our model is the restriction of mathbb 1 p leq 0 to the image plane where mathbb 1 denotes the indicator function and p is some real bivariate polynomial this particularly means that the boundaries in the image form a subset of an algebraic curve with the implicit polynomial p we show that the image parameters i e the polynomial coefficients satisfy a set of linear annihilation equations with the coefficients being the image moments the inherent sensitivity of the moments to noise makes the reconstruction process numerically unstable and narrows the choice of the sampling kernels to polynomial reproducing kernels as a remedy to these problems we replace conventional moments with more stable generalized moments that are adjusted to the given sampling kernel the benefits are threefold 1 it relaxes the requirements on the sampling kernels 2 produces annihilation equations that are robust at numerical precision and 3 extends the results to images with unbounded boundaries we further reduce the sensitivity of the reconstruction process to noise by taking into account the sign of the polynomial at certain points and sequentially enforcing measurement consistency we consider various numerical experiments to demonstrate the performance of our algorithm in reconstructing binary images including low to moderate noise levels and a range of realistic sampling kernels

theoretical and numerical aspects of an svd based method for band limiting finite extent sequences the authors present an svd based method for band limiting over sampled discrete time finite extent sequences for this purpose they show that finite extent band limitation is best defined in terms of the discrete prolate spheroidal sequences rather than complex exponentials their method has maximum energy concentration as defined in the paper its dimension agrees asymptotically with slepian s 1978 dimension result and the method specializes correctly to the discrete time fourier transform as the sample size tends to infinity they propose an efficient computational method based on the lanczos algorithm for computing only the necessary singular vectors the svd is signal independent only needs to be done once and can be precomputed the svd based band limitation itself is not necessarily much slower than the fast fourier transform for sample sizes on the order of 4096

systematic dft frames principle eigenvalues structure and applications motivated by a host of recent applications requiring some amount of redundancy frames are becoming a standard tool in the signal processing toolbox in this paper we study a specific class of frames known as discrete fourier transform dft codes and introduce the notion of systematic frames for this class this is encouraged by a new application of frames namely distributed source coding that uses dft codes for compression studying their extreme eigenvalues we show that unlike dft frames systematic dft frames are not necessarily tight then we come up with conditions for which these frames can be tight in either case the best and worst systematic frames are established in the minimum mean squared reconstruction error sense eigenvalues of dft frames and their subframes play a pivotal role in this work particularly we derive some bounds on the extreme eigenvalues dft subframes which are used to prove most of the results these bounds are valuable independently

vector quantization analysis of spl sigma spl delta modulation when considering a class of finite dimensional input signals such as bandlimited signals in the discrete fourier sense within a finite time window o t of observation we show that a single loop spl sigma spl delta modulator behaves like the encoder of a vector quantizer the intrinsic behavior of the modulator can be studied by analyzing the partition it generates in the input space we show that this partition yields a particular structure that we call the hyperplane wave structure as one consequence to this analysis it can be proved for the considered class of bandlimited signals that the mean squared error mse of reconstruction cannot asymptotically decrease with the oversampling ratio r faster than o r sup 4 regardless of the type of reconstruction used we then generalize this analysis to n loop spl sigma spl delta modulators and show that the mse cannot decrease faster than o r sup 2n 2

unobstructed modular deformation problems let be a newform of weight k 3 with fourier coefficients in a number field k we show that the universal deformation ring of the mod a galois representation associated to is unobstructed and thus isomorphic to a power series ring in three variables over the witt vectors for all but finitely many primes a of k we give an explicit bound on such a for the 6 known cusp forms of level 1 trivial character and rational fourier coefficients we also prove a somewhat weaker result for weight 2

canonical periods and congruence formulae the purpose of this article is to show how congruences between the fourier coefficients of hecke eigenforms give rise to corresponding congruences between the algebraic parts of the critical values of the associated l functions this study was initiated by b mazur in his fundamental work on the eisenstein ideal see maz77 and maz79 where it was made clear that congruences for analytic l values were closely related to the integral structure of certain hecke rings and cohomology groups the results of maz79 also showed that congruences were useful in the study of nonvanishing of l functions this idea was then further developed by stevens ste82 and rubin wiles rw82 the work of rubin and wiles in particular used congruences to study the behavior of elliptic curves in towers of cyclotomic fields a key ingredient here was a theorem of washington which states roughly that almost l values in certain families are nonzero modulo p this theme has recently been taken up again in the work of ono skinner osa osb james jam and kohnen koh97 while the earlier history was primarily concerned with cyclotomic twists the current emphasis is on families of twists by quadratic characters here one wants quantitative estimates for the number of quadratic twists of a given modular form which have nonvanishing l function at s 1 we continue this trend in the present work by using our general results to obtain a strong nonvanishing theorem for the quadratic twists of modular elliptic curves with rational points of order three this generalizes a beautiful example due to kevin james and provides new evidence for a conjecture of goldfeld gol79 it should however be pointed out that even the study of quadratic twists may be traced back to mazur the reader is urged to look at pages 212 213 of maz79 and especially at the footnote at the bottom of page 213 the theorems of davenport heilbronn dh71 and washington was78 which are crucial in this paper are both mentioned in mazur s article we want to begin by discussing the congruences that lie at the heart of this article thus let f anq n be an elliptic modular cuspform of level m and weight k 2 assume that f is a simultaneous eigenform for all the hecke operators and that a1 f 1 the l function associated to f is defined by the dirichlet series l s f ann s which converges for the real part of s sufficiently large and has analytic continuation to s c a fundamental theorem of shimura shi76 states that l s f enjoys the following algebraicity property

resilience properties of redundant expansions under additive noise and quantization representing signals using coarsely quantized coefficients of redundant expansions is an interesting source coding paradigm the most important practical case of which is oversampled analog to digital a d conversion signal reconstruction from quantized redundant expansions and the accuracy of such representations are problems which are not well understood and we study them in this paper for uniform scalar quantization in finite dimensional spaces to give a more global perspective we first present an analysis of the resilience of redundant expansions to degradation by additive noise in general and then focus on the effects of uniform scalar quantization the accuracy of signal representations obtained by applying uniform scalar quantization to coefficients of redundant expansions measured as the mean squared euclidean norm of the reconstruction error has been previously shown to be lower bounded by an 1 r sup 2 expression we establish some general conditions under which the 1 r sup 2 accuracy can actually be attained and under those conditions prove a 1 r sup 2 upper error bound for a particular kind of structured expansions which includes many popular frame classes we propose reconstruction algorithms which attain the 1 r sup 2 accuracy at low numerical complexity these structured expansions moreover facilitate efficient encoding of quantized coefficients in a manner which requires only a logarithmic bit rate increase in redundancy resulting in an exponential error decay in the bit rate results presented in this paper are immediately applicable to oversampled a d conversion of periodic bandlimited signals

sigma delta spl sigma spl delta quantization and finite frames the k level sigma delta spl sigma spl delta scheme with step size spl delta is introduced as a technique for quantizing finite frame expansions for spl ropf sup d error estimates for various quantized frame expansions are derived and in particular it is shown that spl sigma spl delta quantization of a unit norm finite frame expansion in spl ropf sup d achieves approximation error where n is the frame size and the frame variation spl sigma f p is a quantity which reflects the dependence of the spl sigma spl delta scheme on the frame here spl par spl middot spl par is the d dimensional euclidean 2 norm lower bounds and refined upper bounds are derived for certain specific cases as a direct consequence of these error bounds one is able to bound the mean squared error mse by an order of 1 n sup 2 when dealing with sufficiently redundant frame expansions this represents a significant improvement over classical pulse code modulation pcm quantization which only has mse of order 1 n under certain nonrigorous statistical assumptions spl sigma spl delta also achieves the optimal mse order for pcm with consistent reconstruction

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

high resolution distributed sampling of bandlimited fields with low precision sensors the problem of sampling a discrete time sequence of spatially bandlimited fields with a bounded dynamic range in a distributed communication constrained processing environment is studied a central unit having access to the data gathered by a dense network of low precision sensors is required to reconstruct the field snapshots to maximum accuracy both deterministic and stochastic field models are considered for stochastic fields results are established in the almost sure sense the feasibility of having a flexible tradeoff between the oversampling rate sensor density and the analog to digital converter adc precision while achieving an exponential accuracy in the number of bits per nyquist interval per snapshot is demonstrated this exposes an underlying conservation of bits principle the bit budget per nyquist interval per snapshot the rate can be distributed along the amplitude axis sensor precision and space sensor density in an almost arbitrary discrete valued manner while retaining the same exponential distortion rate characteristics

on modular representations of bar q q arising from modular forms where g is the galois group gai i q and f is a finite field of characteristic i 3 suppose that p is modular of level n i e that it arises from a weight 2 newform of level dividing n and trivial nebentypus character then p is an odd representation the matrix p c where c is a complex conjugation in g has eigenvalues 1 1 since 1 and 1 are distinct in f p is absolutely irreducible and has a model over every subfield of f containing the set trace p we assume that f has been chosen so that it is generated by this set assume that p is a prime which exactly divides n we write p i n and restrict p to a decomposition group of g for the prime p view the restriction as a representation p p of gal t p qp we say that p is finite at p 34 or 35 w if there is a finite flat f vector space scheme h over zp for which the action of gal t p qp on the f vector space h 1 p gives pp i f p 9 is finite at p if and only if pp is unramified it is clear that p is finite at p whenever p is already modular of level n p in w of 34 cf 35 serre conjectured the converse i e that if p is finite then p is modular of level n p soon thereafter mazur 1 20 proved this conjecture in the case

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

shapes from pixels continuous domain visual signals are usually captured as discrete digital images this operation is not invertible in general in the sense that the continuous domain signal cannot be exactly reconstructed based on the discrete image unless it satisfies certain constraints e g bandlimitedness in this paper we study the problem of recovering shape images with smooth boundaries from a set of samples thus the reconstructed image is constrained to regenerate the same samples consistency as well as forming a shape bilevel image we initially formulate the reconstruction technique by minimizing the shape perimeter over the set of consistent binary shapes next we relax the non convex shape constraint to transform the problem into minimizing the total variation over consistent non negative valued images we also introduce a requirement called reducibility that guarantees equivalence between the two problems we illustrate that the reducibility property effectively sets a requirement on the minimum sampling density we also evaluate the performance of the relaxed alternative in various numerical experiments

congruence formula for certain dihedral twists in this article we prove a congruence formula for the special values of certain dihedral twists of two primitive modular forms of weight two with isomorphic residual galois representation at a prime p

sampling and reconstruction of spatial fields using mobile sensors spatial sampling is traditionally studied in a static setting where static sensors scattered around space take measurements of the spatial field at their locations in this paper we study the emerging paradigm of sampling and reconstructing spatial fields using sensors that move through space we show that mobile sensing offers some unique advantages over static sensing in sensing bandlimited spatial fields since a moving sensor encounters such a spatial field along its path as a time domain signal a time domain anti aliasing filter can be employed prior to sampling the signal received at the sensor such a filtering procedure when used by a configuration of sensors moving at constant speeds along equispaced parallel lines leads to a complete suppression of spatial aliasing in the direction of motion of the sensors we analytically quantify the advantage of using such a sampling scheme over a static sampling scheme by computing the reduction in sampling noise due to the filter we also analyze the effects of nonuniform sensor speeds on the reconstruction accuracy using simulation examples we demonstrate the advantages of mobile sampling over static sampling in practical problems we extend our analysis to sampling and reconstruction schemes for monitoring time varying bandlimited fields using mobile sensors we demonstrate that in some situations we require a lower density of sensors when using a mobile sensing scheme instead of the conventional static sensing scheme the exact advantage is quantified for a problem of sampling and reconstructing an audio field

complete characterization of stable bandlimited systems under quantization and thresholding in this paper we analyze the approximation behavior of sampling series where the sample values taken equidistantly at nyquist rate are disturbed either by the nonlinear threshold operator or the nonlinear quantization operator we perform the analysis for several spaces of bandlimited signals and completely characterize the spaces for which an approximation is possible additionally we study the approximation of outputs of stable linear time invariant systems by sampling series with disturbed samples for signals in pw pi 1 we show that there exist stable systems that become unstable under thresholding and quantization and that the approximation error is unbounded irrespective of how small the quantization step size is chosen further we give a necessary and sufficient condition for the pointwise and the uniform convergence of the series surprisingly this condition is the well known condition for bounded input bounded output bibo stability finally we discuss the special case of finite impulse response fir filters and give an upper bound for the approximation error

special values of adjoint l functions and congruences for automorphic forms on gl n over a number field we prove an integrality result for the value at s 1 of the adjoint l function associated to a cohomological cuspidal automorphic representation on rm gl n over any number field we then show that primes outside an exceptional set dividing this special value give rise to congruences between automorphic forms we also prove a non vanishing property at infinity for the relevant rankin selberg l functions on rm gl n times rm gl n

multigrid one shot method for state constrained aerodynamic shape optimization this paper presents applications of an optimization based multigrid method to state constrained aerodynamic shape optimization problems the one shot simultaneous pseudo time stepping method has been applied successfully to such problems reducing the cost of computation up to 70 from that of a traditional gradient method however the number of optimization iterations is comparatively large in this method since we update the design parameters in each time step of state and costate runs in this paper we use an optimization based multigrid strategy to reduce the total number of optimization iterations while keeping the idea of the one shot pseudo time stepping method design examples of drag reduction with constant lift together with geometrical constraints for an rae2822 airfoil are included the total effort required for convergence of the optimization problem is between four and six times that of forward simulation runs depending on computational grid size

adjoint modular galois representations and their selmer groups in the last 15 years many class number formulas and main conjectures have been proven here we discuss such formulas on the selmer groups of the three dimensional adjoint representation ad phi of a two dimensional modular galois representation phi we start with the p adic galois representation phi0 of a modular elliptic curve e and present a formula expressing in terms of l 1 ad phi0 the intersection number of the elliptic curve e and the complementary abelian variety inside the jacobian of the modular curve then we explain how one can deduce a formula for the order of the selmer group sel ad phi0 from the proof of wiles of the shimura taniyama conjecture after that we generalize the formula in an iwasawa theoretic setting of one and two variables here the first variable t is the weight variable of the universal p ordinary hecke algebra and the second variable is the cyclotomic variable s in the one variable case we let phi denote the p ordinary galois representation with values in gl2 zp t lifting phi0 and the characteristic power series of the selmer group sel ad phi is given by a p adic l function interpolating l 1 ad phik for weight k 2 specialization phik of phi in the two variable case we state a main conjecture on the characteristic power series in zp t s of sel ad phi symbol see text nu 1 where nu is the universal cyclotomic character with values in zp s finally we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p adic siegel modular forms

maps between jacobians of shimura curves and congruence kernels let p 5 be a prime and a place ofq above it we denote bygq the absolute galois group gal q q for a finite field k of characteristic p an absolutely irreducible representation gq gl2 k is said to arise from a newform of weight k and level a positive integer m if is isomorphic to the reduction mod of the adic representation associated to a newform of weight k and level m note that then it is known that also arises from a newform of weight 2 and level dividingmp2 thus we assume without loss of generality that arises from a newform of weight 2 and level a positive integer n the question of raising the level of i e determining when a arising from a newform of weight 2 and level n as above also arises from a newform of weight 2 and leveln wheren dividesn has been widely studied by numerous mathematicians cf 10 8 5 2 in the first paper that considers this kind of question 10 as well as in most of the papers dealing with similar questions e g 2 the cases considered correspond to n n being coprime to p some discussion for the case where

a use of double integration in sigma delta modulation sigma delta modulation is viewed as a technique that employs integration and feedback to move quantization noise out of baseband this technique may be iterated by placing feedback loop around feedback loop but when three or more loops are used the circuit can latch into undesirable overloading modes in the desired mode a simple linear theory gives a good description of the modulation even when the quantization has only two levels a modulator that employs double integration and two level quantization is easy to implement and is tolerant of parameter variation at sampling rates of 1 mhz it provides resolution equivalent to 16 bit pcm for voiceband signals digital filters that are suitable for converting the modulation to pcm are also described

behavior of the quantization operator for bandlimited nonoversampled signals the process of quantization generates a loss of information and thus the original signal cannot be reconstructed exactly from the quantized samples in general however it is desirable to keep the error as small as possible in this paper the quantization error is quantified in terms of several distortion measures all these measures employ the difference between the original signal and the reconstructed signal which is obtained by bandlimited interpolation of the quantized samples we assume that the signals are bandlimited and that the samples are taken at nyquist rate it is shown that for signals in the paley wiener space pw 1 the supremum of the reconstructed signal and hence the quantization error cannot be bounded in the sense that there exists a bounded subset of pw 1 on which both quantities can increase unboundedly this unexpected behavior is due to the nonlinearity of the quantization operator and the slow decay of the sinc function the nonlinearity is essential for this behavior because every linear operator that fulfills a certain property of the quantization operator would otherwise have to be bounded furthermore it is proven that for a fixed signal the possible quantization error increases as the quantization step size tends to zero the treatment of the quantization error in this paper is completely deterministic

reconstruction of oversampled band limited signals from spl sigma spl delta encoded binary sequences considers the application of spl sigma spl delta modulators to analog to digital conversion the authors have previously shown that for constant input signals optimal nonlinear decoding can achieve large gains in signal to noise ratio snr over linear decoding the present paper shows a similar result for band limited input signals the new nonlinear decoding algorithm is based on projections onto convex sets pocs and alternates between a time domain operation and a band limitation to find a signal invariant under both the time domain operation results in a quadratic programming problem the band limitation can be based on singular value decomposition of a certain matrix the authors show simulation results for the snr performance of a pocs based decoder and a linear decoder for the single loop double loop and two stage spl sigma spl delta modulators and for a specific fourth order interpolative modulator depending on the modulator and the oversampling ratio improvements in snr of up to 10 20 db can be achieved

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

reduction of the mse in r times oversampled a d conversion o 1 r to o 1 r sup 2 in oversampled analog to digital conversion the usual reconstruction method using lowpass filtering leads to a mean squared error mse inversely proportional to the oversampling ratio r the authors prove under certain assumptions and with periodic analog input signals that optimal reconstruction achieves an mse with an oversampling ratio dependence order of at least o 1 r sup 2 that is an mse slope of spl minus 6 db per octave of oversampling is obtained rather than the conventional spl minus 3 db octave slope of classical schemes

noise reduction in oversampled filter banks using predictive quantization we introduce two methods for quantization noise reduction in oversampled filter banks these methods are based on predictive quantization noise shaping or linear prediction it is demonstrated that oversampled noise shaping or linear predictive subband coders are well suited for subband coding applications where for technological or other reasons low resolution quantizers have to be used in this case oversampling combined with noise shaping or linear prediction improves the effective resolution of the subband coder at the expense of increased rate simulation results are provided to assess the achievable quantization noise reduction and resolution enhancement and to investigate the rate distortion properties of the proposed methods

quantization noise spectra several results describing the behavior of quantization noise in a unified and simplified manner are discussed exact formulas for quantizer noise spectra are developed they are applied to a variety of systems and inputs including scalar quantization pcm dithered pcm sigma delta modulation dithered sigma delta modulation two stage sigma delta modulation and second order sigma delta modulation

direct treatment of state constraints in aerodynamic shape optimization using simultaneous pseudo time stepping practical shape design problems involve additional state constraints e g shape optimization of aircraft for drag reduction with constant lift effectiveness of the optimization methods for aerodynamic shape design depends crucially on proper choice of the cost function the constraints and their treatment during the optimization one can treat the constraints indirectly e g by making some kind of transformation so that the constraints are added to the objective function with some weighting and the constrained problem is reduced to an unconstrained one this kind of treatment is termed as soft constraints in 49 indirect treatment of constraints using traditional gradient methods can be found among others in 103 142 143