effective algorithms with circulant block matrices abstract effective numerical algorithms for circulant block matrices a whose blocks are circulant are obtained the eigenvalues of such matrices are determined in terms of the eigenvalues of matrices of reduced dimension and systems of linear equations involving these matrices are solved efficiently using fast fourier transforms

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

an approximate solution for a system with a symmetric matrix abstract in this paper a large system with a symmetric and essentially 2 2 band matrix is reformulated as the product of two banded matrices for the purpose of obtaining a solution of a linear system more efficiently an error term for the approximate solution is justified in the spirit of the work by yan and chung 1

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

a new modification of the rojo method for solving symmetric circulant five diagonal systems of linear equations abstract a new effective and stable modification of the rojo method 1 for solving of real symmetric circulant five diagonal systems of linear equations is proposed this special kind of system appears in many applications spline approximation difference solution of partial differential equations etc the method presented in this paper a very efficient and stable method

integral crystalline cohomology over very ramified valuation rings in this paper we intend to show how fontaine s comparison theory fol fo2 between crystalline and p adic etale cohomology can be extented to very ramified base rings so far it had been developed either for qp coefficients or zp coefficients if the base is unramified over the witt vectors see fa2 for example it states that for any p adic discrete valuation ring v there exists a ring b v with the following property if x denotes a smooth and proper v scheme the p adic etale cohomology of the generic fiber x xv k is related to the crystalline cohomology of x v0 vo defined below by an isomorphism

a realization of generalized deformed oscillators with cyclic symmetry the algebra of generalized deformed oscillators that possess cyclic symmetry is studied the general set of multilinear operator relations is presented which characterizes the multilinear part of the algebra of cyclic extended deformed oscillators the realization scheme of generalized deformed oscillators with cyclic symmetry in terms of those without cyclic symmetry is established it is shown that all generalized deformed oscillators with cyclic symmetry constructed by the scheme are actually equivalent in the viewpoint of mathematics though each generalized deformed oscillator admits its own particular cyclic extended counterpart examples are given to illustrate this peculiar realization scheme

a new algorithm for solving nearly penta diagonal toeplitz linear systems a new numerical algorithm for solving nearly penta diagonal toeplitz linear systems is presented the algorithm is suited for implementation using computer algebra systems cass such as matlab macsyma and maple numerical examples are given in order to illustrate the efficiency of our algorithm

the solution of linear systems equations with circulant like coefficient matrices a direct method is proposed to get the solution of linear systems of equations with circulant like coefficient matrices that find important application in engineering the elements of solutions are functions of zero points of the characteristic polynomial g z and g z of circulant matrix three examples to get the inverse matrix and the solutions of linear system are presented in the paper

a computational algorithm for solving periodic penta diagonal linear systems in this paper we present a new efficient computational algorithm for solving periodic penta digonal linear systems the implementation of the algorithm using computer algebra systems cas such as maple macsyma mathematica and matlab is straightforward an example is given in order to illustrate the algorithm

a direct method for solving circulant tridiagonal block systems of linear equations this paper presents a modification of rojo s algorithm comput math appl 20 1990 61 to solve block circulant tridiagonal systems of linear equations which are toeplitz and hermitian this new approach gives us a general direct algorithm for solving the problem we will show how to choose a block matrix as a parameter to describe the method we employ the factorization of block toeplitz tridiagonal matrices as the product of two block toeplitz subdiagonal and superdiagonal matrices the algorithm is based on obtaining the solution of the nonlinear matrix equation email protected b c 1b finally some numerical results will be given

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

deforming semistable galois representations abstract let v be a p adic representation of gal q q one of the ideas of wiles s proof of flt is that if v is the representation associated to a suitable autromorphic form a modular form in his case and if v is another p adic representation of gal q q closed enough to v then v is also associated to an automorphic form in this paper we discuss which kind of local condition at p one should require on v and v in order to be able to extend this part of wiles s methods

modularity lifting theorems for ordinary galois representations we generalize results of clozel harris and taylor by proving modularity lifting theorems for ordinary l adic galois representations of any dimension of an imaginary cm or totally real number field the main theorems are obtained by establishing an r mathrm red mathbb t theorem over a hida family a key part of the proof is to construct appropriate ordinary lifting rings at the primes dividing l and to determine their irreducible components

solving banded and near symmetric systems in an earlier work by the authors 8 finite difference methods applied to obtain numerical approximations for the solution of nonlinear volterra integro differential equations with two point boundary conditions were considered in 8 the auxiliary conditions required to complete the system were chosen to preserve the symmetry of the banded system in general applying natural auxiliary conditions nac would give rise to a nonsymmetric banded system in this article we consider the efficient solving of a linear banded system with a near symmetric coefficient matrix

a parallel numerical algorithm for near symmetric and banded systems a numerical method of solution which by its nature provides approximate solutions for tridiagonal systems with toeplitz coefficient matrices is extended in two ways first the coefficient matrix is perturbed and replaced by a banded near toeplitz matrix then the algorithm is modified to allow for parallel processing the perturbations are then taken into account in two separate correction processes the convergence of the modified algorithm is examined

potentially semi stable deformation rings let k qp be a finite extension and gk gal k k the galois group of an alge braic closure k let f be a finite field of characteristic p and vf a finite dimensional f vector space equipped with a continuous action of gkthe study of the defor mation theory of galois representations was initiated by mazur ma who showed that if vf has no non trivial endomorphisms then it admits a universal deformation ring ryw after the work of wiles wi and the conjectures of fontaine mazur fm it became clear that for arithmetic applications it was important to understand cer tain quotients of ry corresponding to deformations satisfying certain conditions for example wiles uses deformations which arise from finite flat group schemes and the corresponding quotient of ryw was constructed by ramakrishna ra suppose that l k is a finite extension and let a b be integers it seems to be a kind of folklore conjecture that there should be a quotient of ry whose points in finite extensions of w f l p correspond to deformations of vf which become semi stable over l and have hodge tate weights in the interval a b this is closely related to the final conjecture of fo 3 special cases of this when vf is 2 dimensional are conjectured in the papers of fontaine mazur fm p 191 breuil conrad diamond taylor bcdt conj 1 1 1 and breuil m zard bm 2 2 2 4 the purpose of this paper is to prove such a result

on cscs based iteration methods for toeplitz system of weakly nonlinear equations abstract for toeplitz system of weakly nonlinear equations by using the separability and strong dominance between the linear and the nonlinear terms and using the circulant and skew circulant splitting cscs iteration technique we establish two nonlinear composite iteration schemes called picard cscs and nonlinear cscs like iteration methods respectively the advantage of these methods is that they do not require accurate computation and storage of jacobian matrix and only need to solve linear sub systems of constant coefficient matrices therefore computational workloads and computer storage may be saved in actual implementations theoretical analysis shows that these new iteration methods are local convergent under suitable conditions numerical results show that both picard cscs and nonlinear cscs like iteration methods are feasible and effective for some cases

a new method for solving symmetric circulant tridiagonal systems of linear equations in this paper a new method for computing the solution of a linear system having a symmetric circulant tridiagonal matrix is presented this special kind of system appears in many applications after an appropriate partition of the system and the elimination of the last unknown we apply the woodbury formula to arrive at a very efficient and stable method which is quite competitive with gaussian elimination and with the modified double sweep method

the inverse of circulant matrix abstract a direct method is proposed to get the inverse matrix of circulant matrix that find important application in engineering the elements of the inverse matrix are functions of zero points of the characteristic polynomial g z and g z of circulant matrix four examples to get the inverse matrix are presented in the paper

on the solution of circulant linear systems the solution of linear systems having circulant coefficient matrices iss considered in this paper this kind of system occurs in many applications prediction time series analysis spline approxim

low complexity bayesian estimation of cluster sparse channels this paper addresses the problem of channel impulse response estimation for cluster sparse channels under the bayesian estimation framework we develop a novel low complexity minimum mean squared error mmse estimator by exploiting the sparsity of the received signal profile and the structure of the measurement matrix it is shown that due to the banded toeplitz circulant structure of the measurement matrix a channel impulse response such as underwater acoustic channel impulse responses can be partitioned into a number of orthogonal or approximately orthogonal clusters the orthogonal clusters the sparsity of the channel impulse response and the structure of the measurement matrix all combined result in a computationally superior realization of the mmse channel estimator the mmse estimator calculations boil down to simpler in cluster calculations that can be reused in different clusters the reduction in computational complexity allows for a more accurate implementation of the mmse estimator the proposed approach is tested using synthetic gaussian channels as well as simulated underwater acoustic channels symbol error rate performance and computation time confirm the superiority of the proposed method compared to selected benchmark methods in systems with preamble based training signals transmitted over cluster sparse channels

preconditioning for vector valued cahn hilliard equations the solution of vector valued cahn hilliard systems is of interest in many applications we discuss strategies for the handling of smooth and nonsmooth potentials as well as for different types of constant mobilities concerning the nonsmooth systems the necessary bound constraints are incorporated via the moreau yosida regularization technique we develop effective preconditioners for the efficient solution of the linear systems in saddle point form numerical results illustrate the efficiency of our approach in particular we numerically show mesh and phase independence of the developed preconditioner in the smooth case the results in the nonsmooth case are also satisfying and the preconditioned version always outperforms the unpreconditioned one

parallel algorithms for solving tridiagonal and near circulant systems many problems in mathematics and applied science lead to the solution of linear systems having circulant coefficient matrices this paper presents a new stable method for the exact solution of non symmetric tridiagonal circulant linear systems of equations the method presented in this paper is quite competitive with gaussian elimination both in terms of arithmetic operations and storage requirements it is also competitive with the modified double sweep method this method can be applied to solve the near circulant tridiagonal system in addition the method is modified to allow for parallel processing

an efficient algorithm for a model with a bidiagonal coefficient matrix bidiagonal matrices arise directly in the discretization of models using one step methods and indirectly through the factoring of tridiagonal systems with special characteristics we first look at the solution of a bidiagonal system and then relate it to an experimental method that has been used to determine the overall heat transfer coefficient in a concentric tube parallel flow heat exchanger the discretization of the governing partial differential equation with its boundary conditions gives rise to a near toeplitz bidiagonal system here we consider a method for the parallel processing of the resulting systems to provide measurements at fixed spatial and time increments

low sampling rate ultra wideband channel estimation using equivalent time sampling in this paper a low sampling rate scheme for ultra wideband channel estimation is proposed the scheme exploits multiple observations generated by transmitting multiple pulses in the proposed scheme p pulses are transmitted to produce channel impulse response estimates at a desired sampling rate while the adc samples at a rate that is p times slower to avoid loss of fidelity the number of sampling periods based on the desired rate in the inter pulse interval is restricted to be co prime with p this condition is affected when clock drift is present and the transmitted pulse locations change to handle this case and to achieve an overall good channel estimation performance without using prior information we derive an improved estimator based on the bounded data uncertainty bdu model it is shown that this estimator is related to the bayesian linear minimum mean squared error lmmse estimator channel estimation performance of the proposed sub sampling scheme combined with the new estimator is assessed in simulation the results show that high reduction in sampling rate can be achieved the proposed estimator outperforms the least squares estimator in almost all cases while in the high snr regime it also outperforms the lmmse estimator in addition to channel estimation a synchronization method is also proposed that utilizes the same pulse sequence used for channel estimation

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

a parallel method for linear equations with tridiagonal toeplitz coefficient matrices abstract there are many articles on symmetric tridiagonal toeplitz and circulant systems such systems in areas including numerical methods for solving boundary value differential equations and in graph theory these matrices can often be written as the product of bidiagonal matrices in this article nonsymmetric toepliz systems and nonsymmetric circulant systems are examined the coefficient matrix is split into two bidiagonal matrices and the efficient solution of the resulting systems is considered

a structure preserving matrix factorization for solving general periodic pentadiagonal toeplitz linear systems abstract in the current article a new algorithm for solving linear systems with the periodic pentadiagonal toeplitz ppt coefficient matrix is presented this algorithm is based on a structure preserving matrix factorization and sherman morrison woodbury formula some numerical examples are given in order to illustrate the effectiveness of the proposed algorithms all of the experiments are performed on a computer with the aid of programs written in matlab