dwork s conjecture on unit root zeta functions annals of mathematics 150 1999 867 927 arxiv math 9911270v1 math nt 1 nov 1999 dwork s conjecture on unit root zeta functions by daqing wan 1 introduction in this article we introduce a systematic new method to investigate the conjectural p adic meromorphic continuation of professor bernard dwork s unit root zeta function attached to an ordinary family of algebraic varieties defined over a finite field of characteristic p after his pioneer p adic investigation of the weil conjectures on the zeta function of an algebraic variety over a finite field dwork went on to study the p adic analytic variation of a family of such zeta functions when the variety moves through an algebraic family in the course of doing so he was led to a new zeta function called the unit root zeta function which goes beyond the reach of the existing theory he conjectured 8 that such a unit root zeta function is p adic meromorphic everywhere these unit root zeta functions contain important arithmetic information about a family of algebraic varieties they are truly p adic in nature and are transcendental functions sometimes seeming quite mysterious in fact no single nontrivial example has been proved to be true about this conjecture other than the trivial overconvergent or log convergent case for which dwork s classical p adic theory already applies see 6 11 26 for various attempts in this article we introduce a systematic new method to study such unit root zeta functions our method can be used to prove the conjecture in the case when the involved unit root f crystal has rank one in particular this settles the first nontrivial case the rank one unit root f crystal coming from the family of higher dimensional kloosterman sums our method further allows us to understand reasonably well about analytic variation of an arithmetic family of such rank one unit root zeta functions motivated by the gouv ea mazur conjecture about dimension variation of classical and p adic modular forms we shall introduce another this work was partially supported by nsf the author wishes to thank p deligne and n katz for valuable discussions at one stage of this work during 1993 and 1994 the author would also like to thank s sperber for his careful reading of the manuscript and for his many detailed comments which led to a significant improvement of the exposition

improvements of the weil bound for artin schreier curves for the artin schreier curve y q y f x defined over a finite field mathbb f q of q elements the celebrated weil bound for the number of mathbb f q r rational points can be sharp especially in super singular cases and when r is divisible in this paper we show how the weil bound can be significantly improved using ideas from moment l functions and katz s work on l adic monodromy calculations roughly speaking we show that in favorable cases which happens quite often one can remove an extra sqrt q factor in the error term

signaling over arbitrarily permuted parallel channels the problem of transmission of information over arbitrarily permuted parallel channels is studied here the transmitter does not know over which channel a certain code sequence will actually be transmitted however the receiver knows how the sequences are permuted the permutation is arbitrary but constant during the simultaneous transmission of the code sequences via the parallel channels it is shown first that the sum of the capacities of each channel is achievable for such a communication system in the special case where the capacity achieving input distributions of all channels are identical more important is that this sum capacity can also be achieved using a single channel code for all channels combined with a sequential decoding method the construction of a rate matching code based on maximum distance separable mds codes turns out to be crucial finally the case where the parallel channels have different capacity achieving input distributions is investigated also for this case the capacity is determined again this capacity is achievable with a sequential decoding procedure

zeta functions of an infinite family of k3 surfaces we identify a parameterized family of k3 surfaces with generic picard number 19 and we employ elementary methods to determine their local zeta functions in addition we explicitly determine those surfaces which are modular 1 introduction given a projective variety defined over a number field a central problem in number theory is to write down all of its local zeta functions the case of elliptic curves over q has recently been settled and here we study the problem for k3 surfaces a natural two dimensional analog of elliptic curves this is in general a difficult problem since at any prime the nontrivial factor of the local zeta function of a k3 surface is a polynomial of degree as high as 22 in theory one can compute the zeta function of varieties whose cohomology splits into 1 dimensional pieces for instance using p adic hodge theory livne li computed the zeta function of a certain k3 surface which arose in the work of peters top and van der vlugt p t vdv another important work in this area is due to shioda and inose if a k3 surface has picard number 20 the largest it can be they show in i s that its l function is essentially the symmetric square of the l function of an elliptic curve with complex multiplication generally we call a surface defined over q modular if its l function is up to simple factors the l function of a weight 3 hecke eigenform although the shioda and inose result is general the problem of computing explicit examples is nontrivial in fact only a few examples exist in the literature for other examples see b s k t l and they were obtained by a wide variety of methods both geometric and transcendental there are several natural difficulties for instance there is the problem of identifying suitable candidates for such surfaces then there is the issue of demonstrating that the l function of an alleged modular surface agrees with the l function of an explicit weight 3 modular form work of morrison m and van geemen and top vg t suggests that one can obtain similar arithmetic results for k3 surfaces with picard number 19 in this vein we show for a certain family of k3 surfaces that the general problem of computing zeta functions can be quite simple and involve little more than

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

polar codes for broadcast channels polar codes are introduced for discrete memoryless broadcast channels for m user deterministic broadcast channels polarization is applied to map uniformly random message bits from m independent messages to one codeword while satisfying broadcast constraints the polarization based codes achieve rates on the boundary of the private message capacity region for two user noisy broadcast channels polar implementations are presented for two information theoretic schemes 1 cover s superposition codes and 2 marton s codes due to the structure of polarization constraints on the auxiliary and channel input distributions are identified to ensure proper alignment of polarization indices in the multiuser setting the codes achieve rates on the capacity boundary of a few classes of broadcast channels e g binary input stochastically degraded the complexity of encoding and decoding is o n log n where n is the block length in addition polar code sequences obtain a stretched exponential decay of o 2 n beta of the average block error probability where 0 reproducible experiments for finite block lengths n 512 1024 2048 corroborate the theory

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

higher rank case of dwork s conjecture journal of the american mathematical society volume 00 number 0 xxxx xxxx pages 000 000 s 0894 0347 xx 0000 0 arxiv math 0005309v1 math nt 9 may 2000 higher rank case of dwork s conjecture daqing wan dedicated to the memory of bernard dwork 1 introduction in this series of two papers we prove the p adic meromorphic continuation of the pure slope l functions arising from the slope decomposition of an overconvergent f crystal as conjectured by dwork 6 more precisely we prove a suitable extension of dwork s conjecture in our more general setting of modules see section 2 for precise definitions of the various notions used in this introduction our main result is the following theorem theorem 1 1 let x be a smooth affine variety defined over a finite field f q of characteristic p 0 let m be a finite rank overconvergent module over x f q then for each rational number s the pure slope s l function l s t attached to is p adic meromorphic everywhere the proof of this theorem will be completed in two papers in the present higher rank paper we introduce a reduction approach which reduces theorem 1 1 to the special case when the slope s s 0 part of has rank one and the base space x is the simplest affine space a n this part is essentially algebraic it depends on monsky s trace formula grothendieck s specialization theorem the hodge newton decomposition and katz s isogeny theorem in our next paper 23 we will handle the rank one case over the affine space a n the rank one case is very much analytic in nature and forces us to work in a more difficult infinite rank setting generalizing and improving the limiting approach introduced in 19 dwork s conjecture grew out of his attempt to understand the p adic analytic variation of the pure pieces of the zeta function of a variety when the variety moves through an algebraic family to give an important geometric example let us con sider the case that f y x is a smooth and proper morphism over f q with a smooth and proper lifting to characteristic zero berthelot s result 1 says that the relative crystalline cohomology r i f crys z p modulo torsion is an overconvergent f crystal m i over x applying theorem 1 1 we conclude that the pure l functions arising from these geometric overconvergent f crystals m i are p adic meromorphic in particular this implies the existence of an exact p adic formula for geometric p adic character sums and a suitable p adic equi distribution theorem for the roots of zeta functions for more detailed arithmetic motivations and further open prob lems see the expository papers 20 21 1991 mathematics subject classification primary 11g40 11s40 secondary 11m41 14g15 key words and phrases l functions p adic meromorphic continuation modules this work was partially supported by nsf c american mathematical society

the generalized triangular decomposition given a complex matrix h we consider the decomposition h qrp where r is upper triangular and q and p have orthonormal columns special instances of this decomposition include the singular value decomposition svd and the schur decomposition where r is an upper triangular matrix with the eigenvalues of h on the diagonal we show that any diagonal for r can be achieved that satisfies weyl s multiplicative majorization conditions k k k k i 1 r i i 1 1 k k k i 1 r i i 1 i where k is the rank of h i is the i th largest singular value of h and r i is the i th largest in magnitude diagonal element of r given a vector r which satisfies weyl s conditions we call the decomposition h qrp where r is upper triangular with prescribed diagonal r the generalized triangular decomposition gtd a direct nonrecursive algorithm is developed for computing the gtd this algorithm starts with the svd and applies a series of permutations and givens rotations to obtain the gtd the numerical stability of the gtd update step is established the gtd can be used to optimize the power utilization of a communication channel while taking into account quality of service requirements for subchannels another application of the gtd is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values

l functions for symmetric products of kloosterman sums the classical kloosterman sums give rise to a galois representation of the function field unramified outside 0 and we study the local monodromy of this representation at using l adic method based on the work of deligne and katz as an application we determine the degrees and the bad factors of the l functions of the symmetric products of the above representation our results generalize some results of robba obtained through p adic method

mmse decision feedback equalizers and coding ii coding results for pt i see ibid vol 43 no 10 p 2582 1995 the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure in systems that combine equalization with coded modulation with a given symbol rate 1 t and transmit spectrum the output signal to noise ratio snr sub mmse dfe u of a mmse dfe with an unbiased decision rule is a single parameter that characterizes the channel for coding purposes indeed the transmit spectrum that maximizes snr sub mmse dfe u is the capacity achieving water pouring spectrum and the capacity c t in bits per two dimensions is given by c t log sub 2 1 snr sub mmse dfe u regardless of the channel characteristics the performance of a coded system with a mmse dfe equalization structure may be accurately estimated using the gain of the coding scheme at a given pr e this performance is shown to be approximately the same as that of a multicarrier system using the same transmit spectrum and similar coding such systems are known to be able to approach capacity arbitrarily closely the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels simulation results indicate that performance of the mmse dfe is surprisingly insensitive to transmit spectral shaping as long as the transmit spectrum exceeds the capacity achieving band but that there is an optimal symbol rate that should approximately be used

functions of sheaves and drinfeld modules in this paper we apply dwork s p adic methods to study the meromorphic continuation and rationality of various l functions arising from 7r adic galois representations drinfeld modules and p sheaves as a consequence we prove some conjectures of goss about the rationality of the local l function and the meromorphic continuation of the global l function attached to a drinfeld module let fq be a finite field of q elements with characteristic p let 7r be a prime of the polynomial ring a fq t let a be the completion of the ring a at 7r this is an analogue of the classical ring 7p of p adic integers let x be an irreducible algebraic variety defined over fq and let 7r x be the arithmetic fundamental group of x ifq with respect to some base point the group 7r x may be regarded as the galois groiip of a separable closure of the function field of x fq modulo the inertia groups at the closed points of x lq suppose now that we are given a continuous 7r adic representation

families of modular eigenforms this article is an expansion of the notes to a one hour lecture for an msri workshop on computational number theory the editors of mathematics of computation kindly asked us to submit these notes for publication and we are enormously pleased to do so our original audience did not consist of experts in the field of modular forms and we have tried to keep this article accessible to nonexperts we have made an experimental investigation of certain arithmetic conjec tures using macsyma this investigation requires a search for certain mod ular eigenforms of high weight these computations pose problems which we feel may be interesting on their own we are novices here and we seek advice from people more experienced in making computations of an analogous sort we are in fact deeply grateful to j f mestre who came to our aid and who vastly extended our computations using pari 1 mestre has graciously allowed us to present his computations in this article the kind of families we have in mind as in the title of this lecture has as its prototypical example the standard family of classical eisenstein series of level 1 and weight k for k 4 6 8 whose fourier expansions are given by oo

the geometric mean decomposition given a complex matrix h we consider the decomposition h qrp where q and p have orthonormal columns and r is a real upper triangular matrix with diagonal elements equal to the geometric mean of the positive singular values of h this decomposition which we call the geometric mean decomposition has application to signal processing and to the design of telecommunication networks the unitary matrices correspond to information lossless filters applied to transmitted and received signals that minimize the maximum error rate of the network another application is to the construction of test matrices with specified singular values

the moments of a kloosterman sum and the weight distribution of a zetterberg type binary cyclic code in this correspondence we give the moments of a kloosterman sum over fq in terms of the frequencies of weights in the binary zetterberg code of length q 1 which are known by the work of schoof and van der vlugt the method is illustrated by giving explicit formulae for the moments up to the tenth moment as a corollary the weight distribution of a zetterberg type binary cyclic code is obtained

joint unitary triangularization for gaussian multi user mimo networks the problem of transmitting a common message to multiple users over the gaussian multiple input multiple output broadcast channel is considered where each user is equipped with an arbitrary number of antennas a closed loop scenario is assumed for which a practical capacity approaching scheme is developed by applying judiciously chosen unitary operations at the transmit and receive nodes the channel matrices are triangularized so that the resulting matrices have equal diagonals up to a possible multiplicative scalar factor this along with the utilization of successive interference cancellation reduces the coding and decoding tasks to those of coding and decoding over the single antenna additive white gaussian noise channel over the resulting effective channel any off the shelf code may be used for the two user case it was recently shown that such joint unitary triangularization is always possible in this paper it is shown that for more than two users it is necessary to carry out the unitary linear processing jointly over multiple channel uses i e space time processing is employed it is further shown that exact triangularization where all resulting diagonals are equal is still not always possible and appropriate conditions for the existence of such are established for certain cases when exact triangularization is not possible an asymptotic construction is proposed that achieves the desired property of equal diagonals up to edge effects that can be made arbitrarily small at the price of processing a sufficiently large number of channel uses together

orthogonal designs with maximal rates orthogonal designs have been used as space time block codes for wireless communications with multiple transmit antennas which can achieve full transmit diversity and have a very simple decoupled maximum likelihood decoding algorithm the rate of an orthogonal design is defined as the ratio of the number of transmitted information symbols in a block of channel uses to the length of the given block which reflects the bandwidth efficiency of the employed space time block code constructed from the orthogonal design this paper focuses on the analysis and synthesis of orthogonal designs with the maximum possible rates which may be real or complex and square or rectangular matrices we first provide several representations of orthogonal designs and their characterizations in terms of hurwitz radon matrix equations next we observe that the real orthogonal designs square or rectangular and the complex square orthogonal designs with maximal rates have been well understood from the existing results in the mathematics literature which can be dated back to 1890s however unfortunately it is not the case for the complex rectangular orthogonal designs with rates as high as possible we then construct a class of complex orthogonal designs for any number of transmit antennas the proposed complex orthogonal designs for the number of transmit antennas n 2m 1 and 2m have the same rate m 1 and 2m where m is any natural number finally we demonstrate that for n 2m 1 and 2m with any given natural number m the value m 1 and 2m is the maximum possible rate that the complex orthogonal designs square or rectangular with n transmit antennas can achieve

exponential sums and newton polyhedra cohomology and estimates the basic objects of this study are exponential sums on a variety defined over a finite field fq q pa p char fq as we have remarked in some earlier articles 1 2 we find it more natural to begin with exponential sums on the torus gm n extend via the usual toric decomposition of an to exponential sums on affine n space and finally proceed via a standard character argument 4 to exponential sums on an affine variety defined over fq while this is the natural order of the work what we do in fact in the first part of this article is to combine the first two steps and deal with exponential sums on varieties v of the form v gm r x as r s n let f e fq xl xn xl xr be an arbitrary regular function on v then f is a sum of monomials and as such has a well defined newton polyhedron a ff at infinity this is the convex closure in rn of the lattice points which occur as exponents of the terms of f together with the origin we have indicated in our previous work 1 2 the description of some of the invariants of the associated l function in terms of properties of this polyhedron for example in 1 we showed how bounds for the degree and total degree of the l function associated with a general exponential sum on v can be expressed in terms of the volumes of a f and the intersections of a f and the various coordinate spaces in the present article assuming f is nondegenerate and commode with respect to af f we show these estimates are sharp our methods are p adic and are based on the work of dwork 11 12 our main accomplishment from which our other results follow is the extension of dwork s cohomology theory from smooth projective hypersurfaces in characteristic p to a general class of exponential sums given f regular on v we construct a complex of p adic banach spaces on which frobenius acts the alternating product of characteristic polynomials of frobenius describes the associated l function in fact when f is nondegenerate and commode with respect to a f the complex is acyclic in dimensions other than 0 and the characteristic

radii of convergence and index for p adic differential operators we study the radii of p adic convergence of solutions at a generic point of homogeneous linear differential operators whose coefficients are analytic elements as an application we prove a conjecture of p robba for a certain class of operators concerning the relation between radii of convergence and index on analytic elements we also give an explicit factorization theorem for p adic differential operators based on the radii of generic convergence and the slopes of the associated newton polygon

meromorphic continuation of l functions of p adic representations x det i td x b xqd x l b xq b x where x is the teichmiiller lifting of zt and d x denotes the degree of x over fq this infinite product is a well defined power series with coefficients in rp because there are only finitely many closed points of a given degree if v fq is an affine variety contained in an then one defines l b v t to be the product in equation 1 1 except that we now restrict x to run only over the closed points of v fq using a standard reduction argument and dwork s

equal diagonal qr decomposition and its application to precoder design for successive cancellation detection in multiple input multiple output mimo multiuser detection theory the qr decomposition of the channel matrix h can be used to form the back cancellation detector in this paper we propose an optimal qr decomposition which we call the equal diagonal qr decomposition or briefly the qrs decomposition we apply the decomposition to precoded successive cancellation detection where we assume that both the transmitter and the receiver have perfect channel knowledge we show that for any channel matrix h there exists a unitary precoder matrix s such that hs qr where the nonzero diagonal entries of the upper triangular matrix r in the qr decomposition of hs are all equal to each other the precoder and the resulting successive cancellation detector have the following properties a the minimum euclidean distance between two signal points at the channel output is equal to the minimum euclidean distance between two constellation points at the precoder input up to a multiplicative factor that equals the diagonal entry in the r factor b the superchannel hs naturally exhibits an optimally ordered column permutation i e the optimal detection order for the vertical bell labs layered space time v blast detector is the natural order c the precoder s minimizes the block error probability of the qr successive cancellation detector d a lower and an upper bound for the free distance at the channel output is expressible in terms of the diagonal entries of the r factor in the qr decomposition of a channel matrix e the precoder s maximizes the lower bound of the channel s free distance subject to a power constraint f for the optimal precoder s the performance of the qr detector is asymptotically at large signal to noise ratios snrs equivalent to that of the maximum likelihood detector mld that uses the same precoder further we consider two multiplexing schemes time division multiple access tdma and orthogonal frequency division multiplexing ofdm we d

rank one case of dwork s conjecture journal of the american mathematical society volume 00 number 0 xxxx xxxx pages 000 000 s 0894 0347 xx 0000 0 arxiv math 0005308v1 math nt 9 may 2000 rank one case of dwork s conjecture daqing wan 1 introduction in the higher rank paper 17 we reduced dwork s conjecture from higher rank case over any smooth affine variety x to the rank one case over the simplest affine space a n in the present paper we finish our proof by proving the rank one case of dwork s conjecture over the affine space a n which is called the key lemma in 17 the key lemma had already been proved in 16 in the special case when the frobenius lifting is the simplest q th power map x x q thus the aim of the present paper is to treat the general frobenius lifting case our method here is an improvement of the limiting method in 16 it allows us to move one step further and obtain some explicit information about the zeros and poles of the unit root l function as in 16 to handle the rank one case we are forced to work in the more difficult infinite rank setting see section 2 for precise definitions of the various basic infinite rank notions let f q denote the finite field of characteristic p 0 our main result of this paper is the following theorem theorem 1 1 let be a nuclear overconvergent module over the affine n space a n f q ordinary at the slope zero side let unit be the unit root slope zero part of assume that unit has rank one let be another nuclear overconvergent module over a n f q then for each integer k the l function l k unit t is p adic meromorphic furthermore the family l k t of l functions unit parametrized by integers k in each residue class modulo q 1 is a strong family of meromorphic functions with respect to the p adic topology of k a finite rank module is automatically nuclear thus theorem 1 1 includes the key lemma of 17 over a n as a special case the basic ideas of the present paper are the same as the limiting approach in 16 the details are however quite different in the simplest q th power frobenius lifting case one has the fundamental dwork trace formula available which is completely explicit for uniform estimates this makes it easy to extend the dwork trace formula to infinite rank setting it also makes it possible to see the various analytic subtleties involved in a concrete case as a result we were able to prove analytically optimal results in 16 for a general frobenius lifting even over the simplest affine n space a n as we shall work in this paper one has to use the much more difficult monsky trace formula which is a generalization of dwork s trace formula thus the first task of this paper is to extend the monsky trace formula to infinite rank setting and to make it sufficiently 1991 mathematics subject classification primary 11g40 11s40 secondary 11m41 14g15 key words and phrases l functions fredholm determinants p adic meromorphic continua tion nuclear modules and banach modules this work was partially supported by nsf c american mathematical society

perfect space time codes for any number of antennas in a recent paper perfect n times n space time codes were introduced as the class of linear dispersion space time st codes having full rate nonvanishing determinant a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted energy per antenna consequence of these conditions include optimality of perfect codes with respect to the zheng tse diversity multiplexing gain tradeoff dmt as well as excellent low signal to noise ratio snr performance yet perfect space time codes have been constructed only for two three four and six transmit antennas in this paper we construct perfect codes for all channel dimensions present some additional attributes of this class of st codes and extend the notion of a perfect code to the rectangular case

an efficient chebyshev lanczos method for obtaining eigensolutions of the schrodinger equation on a grid a grid method for obtaining eigensolutions of bound systems is presented in this the block lanczos method is applied to a cheby shev approximation of exp h where is the range of eigenvalues we are interested in with this choice a preferential convergence of the eigenvectors corresponding to low lying eigenvalues ofhis achieved the method is used to solve a variety of one two and three dimensional problems to apply the kinetic energy operator we use the fast sine transform instead of the fast fourier transform thus fullfilling a priori the box boundary conditions we further extend the chebyshev approximation to treat general functions of matrices thus allowing its application to cases for which no analytical expressions of the expansion coefficients are available

polar codes are optimal for lossy source coding we consider lossy source compression of a binary symmetric source using polar codes and a low complexity successive encoding algorithm it was recently shown by arikan that polar codes achieve the capacity of arbitrary symmetric binary input discrete memoryless channels under a successive decoding strategy we show the equivalent result for lossy source compression i e we show that this combination achieves the rate distortion bound for a binary symmetric source we further show the optimality of polar codes for various multiterminal problems including the binary wyner ziv and the binary gelfand pinsker problems our results extend to general versions of these problems

some first excited energy levels for the generalized killingbeck potential with the differential quadratic method the ground energy and some low lying excited energies and the corresponding wavefunctions for the generalized killingbeck potential are investigated within the context of the recently introduced differential quadratic method highly accurate energy eigenvalues and an excellent agreement were found compared with the power series expansion and the supersymmetric perturbation theory

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

joint unitary triangularization for mimo networks this paper considers communication networks where individual links can be described as those based on multiple input multiple output channels unlike orthogonal modulation methods such as the singular value decomposition we allow interference between subchannels which can be removed by the receivers via successive cancellation the degrees of freedom earned by this relaxation are used for obtaining a basis and corresponding decomposition which are simultaneously good for more than one link specifically we derive necessary and sufficient conditions for shaping the ratio vector of subchannel gains of two broadcast channel receivers we then apply this decomposition to two scenarios first in digital multicasting we present a practical capacity achieving scheme which uses only scalar codes and linear processing then we consider the joint source channel problem of transmitting a gaussian source over a two user multiple input multiple output channel where we show the existence of nontrivial cases where the optimal distortion pair which for high signal to noise ratios snrs equals the optimal point to point distortions of the individual users may be achieved by employing a hybrid digital analog scheme over the induced equivalent channel these scenarios demonstrate the advantage of choosing a modulation basis based upon multiple links in the network thus we coin the approach network modulation

numerically complemented analytic method for solving the time independent one dimensional schrodinger equation a general method of solving the one dimensional schrodinger equation is developed the first step is to construct an exactly solvable reference potential of several smoothly joined morse type components which should be a good approximation to a given potential the exact solutions for that reference hamiltonian are then combined with a nonperturbative approach r g gordon j chem phys 51 14 1969 which enables us to numerically solve the energy eigenvalue problem for the original potential to any desired accuracy a full description of the analytical procedures is given and examples of both exact and numerical solutions are presented

l functions of symmetric powers of cubic exponential sums for each positive integer k we investigate the l function attached to the k th symmetric power of the f crystal associated to the family of cubic exponential sums of x 3 x where runs over fp we explore its rationality fleld of deflnition degree trivial factors functional equation and newton polygon the paper is essentially self contained due to the remarkable and attractive nature of dwork s p adic theory a novel feature of this paper is an extension of dwork s efiective decomposition theory when k p this allows for explicit computations in the associated p adic cohomology in particular the action of frobenius on the primitive cohomology spaces may be explicitly studied

achieving marton s region for broadcast channels using polar codes this paper presents polar coding schemes for the two user discrete memoryless broadcast channel dm bc which achieve marton s region with both common and private messages this is the best achievable rate region known to date and it is tight for all classes of two user dm bcs whose capacity regions are known to accomplish this task we first construct polar codes for both the superposition as well as binning strategy by combining these two schemes we obtain marton s region with private messages only finally we show how to handle the case of common information the proposed coding schemes possess the usual advantages of polar codes i e they have low encoding and decoding complexity and a superpolynomial decay rate of the error probability we follow the lead of goela abbe and gastpar who recently introduced polar codes emulating the superposition and binning schemes to align the polar indices for both schemes their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and channel outputs to remove these constraints we consider the transmission of k blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices the techniques described in this paper are quite general and they can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned

p adic banach spaces and families of modular forms let p be a prime cp the completion of an algebraic closure of the p adic numbers qp and k a nite extension of qp contained in cp let v be the valuation on cp such that v p 1 and let be the absolute value on cp such that x p v x for x cp suppose n is a positive integer prime to p let x1 np denote the modular curve over k which represents elliptic curves with 1 np structure and let up be the hecke operator on modular forms on x1 np which takes a form with q expansion n anq n to the modular form with q expansion n anpq n a modular form f is said to have slope q if there is a polynomial r t over cp such that r up f 0 and such that the newton polygon of r t has only one side and its slope is

newton slopes for artin schreier witt towers we fix a monic polynomial f x in mathbb f q x over a finite field and consider the artin schreier witt tower defined by f x this is a tower of curves cdots rightarrow c m rightarrow c m 1 rightarrow cdots rightarrow c 0 mathbb a 1 with total galois group mathbb z p we study the newton slopes of zeta functions of this tower of curves this reduces to the study of the newton slopes of l functions associated to characters of the galois group of this tower we prove that when the conductor of the character is large enough the newton slopes of the l function form arithmetic progressions which are independent of the conductor of the character as a corollary we obtain a result on the behavior of the slopes of the eigencurve associated to the artin schreier witt tower analogous to the result of buzzard and kilford

equidistribution of gauss sums and kloosterman sums let m be a positive integer fix a nontrivial additive character for each finite field f q to state the first result of this paper we also fix r distinct multiplicative characters 1 r for each finite field f q with more than r elements we shall prove that when varies over multiplicative characters of f q other than the m th roots of open image in new window the r tuples open image in new window of angles of gauss sums are asymptotically equidistributed on the r dimensional torus s1 r as q goes to infinity

asymmetric lee distance codes for dna based storage we introduce a new family of codes termed asymmetric lee distance ald codes designed to correct errors arising in dna based storage systems and systems with parallel string transmission protocols ald codes are defined over a quaternary alphabet and analyzed in this particular setting but the derived results hold for other alphabet sizes as well our technical contributions are twofold first we derive upper bounds on the size of the codes under the ald metric based on linear programming techniques second we propose a number of code constructions which imply lower bounds

dimension variation of classical and p adic modular forms a quadratic bound is obtained for a conjecture of gouvea mazur on arithmetic variation of dimensions of classical and p adic modular forms

the eigenenergies of the wave function through the non variational galerkin b spline approach the non variational galerkin b spline method has been applied to the solution of the schrodinger equation for the vibration and rotation bound states of diatomic molecules this method allows the construction of the full hamiltonian matrix of relatively modest size the eigenspectra obtained are compared with the analytic and modified shifted 1 n expansion cases

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

l functions of symmetric products of the kloosterman sheaf over z the classical n variable kloosterman sums over the finite field f p give rise to a lisse overline bf q l sheaf kl n 1 on bf g m bf f p bf p 1 bf f p 0 infty which we call the kloosterman sheaf let l p g m f p sym k kl n 1 s be the l function of the k fold symmetric product of kl n 1 we construct an explicit virtual scheme x of finite type over spec z such that the p euler factor of the zeta function of x coincides with l p g m f p sym k kl n 1 s we also prove similar results for otimes k rm kl n 1 and bigwedge k rm kl n 1

basis spline collocation method for the lattice solution of boundary value problems abstract we study a particular utilization of the basis spline collocation method bscm for the lattice solution of boundary value problems we demonstrate the implementation of a general set of boundary conditions among the selected problems are the schrodinger equation in radial coordinates the poisson and the generalized helmholtz equations in radial and three dimensional cartesian coordinates

l functions of symmetric powers of kloosterman sums unit root l functions and p adic estimates the l function of symmetric powers of classical kloosterman sums is a polynomial whose degree is now known as well as the complex absolute values of the roots in this paper we provide estimates for the p adic absolute values of these roots our method is indirect we first develop a dwork type p adic cohomology theory for the two variable infinite symmetric power l function associated to the kloosterman family and then study p adic estimates of the eigenvalues of frobenius a continuity argument then provides the desired p adic estimates

newton polygons of zeta functions and l functions in this article we give a systematic treatment of newton polygons of exponential sums the newton polygon is a nice way to describe p adic values of the zeroes or poles of zeta functions and l functions our main objective is to show that the adolphson sperber conjecture 12 which asserts that under a simple condition the generic newton polygon of l functions coincides with its lower bound is false in its full form but true in a slightly weaker form we also show that the full form is true in various important special cases for example we show that for a generic projective hypersurface of degree d the newton polygon of the interesting part of the zeta function coincides with its lower bound the hodge polygon this gives a p adic proof of a recent theorem of illusie conjectured by dwork and mazur for more examples let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f xi x7n of degree di with respect to xi 1 i n where the di are fixed positive integers then for all large prime numbers p the generic newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound we obtain our main results namely several decomposition theorems using certain maximizing functions from linear programming our work suggests a possible connection between newton polygons and the resolution of singularities of toric varieties let p be a prime q pa and let fq be the finite field of q elements and fqm its extension of degree m fix a nontrivial additive character qp of fp for any laurent polynomial f xi xn e fq xl xj1 x xi1 we form

functional equations of l functions for symmetric products of the kloosterman sheaf we determine the arithmetic local monodromy at 0 and at of the kloosterman sheaf using local fourier transformations and laumon s stationary phase principle we then calculate e factors for symmetric products of the kloosterman sheaf using laumon s product formula we get functional equations of l functions for these symmetric products and prove a conjecture of evans on signs of constants of functional equations 2010 american mathematical society

capacity achieving polar codes for arbitrarily permuted parallel channels channel coding over arbitrarily permuted parallel channels was first studied by willems and coworkers this paper introduces capacity achieving polar coding schemes for arbitrarily permuted parallel channels where the component channels are memoryless binary input and output symmetric

approximately universal codes over slow fading channels performance of reliable communication over a coherent slow fading multiple input multiple output mimo channel at high signal to noise ratio snr is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains this paper studies the problem of designing codes that optimally tradeoff the diversity and multiplexing gains the main contribution is a precise characterization of codes that are universally tradeoff optimal i e they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel this characterization is referred to as approximate universality the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains respectively the characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space time coding literature

rateless coding for gaussian channels a rateless code i e a rate compatible family of codes has the property that codewords of the higher rate codes are prefixes of those of the lower rate ones a perfect family of such codes is one in which each of the codes in the family is capacity achieving we show by construction that perfect rateless codes with low complexity decoding algorithms exist for additive white gaussian noise channels our construction involves the use of layered encoding and successive decoding together with repetition using time varying layer weights as an illustration of our framework we design a practical three rate code family we further construct rich sets of near perfect rateless codes within our architecture that require either significantly fewer layers or lower complexity than their perfect counterparts variations of the basic construction are also developed including one for time varying channels in which there is no a priori stochastic model

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

improved bounds on the finite length scaling of polar codes improved upper bounds on the blocklength required to communicate over binary input channels using polar codes below some given error probability are derived for that purpose an improved bound on the number of non polarizing channels is obtained the main result is that the blocklength required to communicate reliably scales at most as o i w r 5 702 where r is the code rate and i w is the symmetric capacity of the channel w the results are then extended to polar lossy source coding at rate r of a source with symmetric distortion rate function d the blocklength required scales at most as o d 0 5 702 where d 0 is the maximal allowed gap between the actual average or typical distortion and d r