scattering by stripe grating when bismuth is substituted at certain lattice points of rare earth garnet crystals a material is formed which exhibits an almost ideal magnetic stripe domain structure the structure consists of a one dimensional periodic array of regions of nearly constants magnetization so that the magnitudes of the magnetization vectors in two adjacent regions are the same but their directions are opposite this condition produces the alternating sign magnetic permeability tensors shown in section 1 7 3 when a beam of light is passed through such a film it splits into many diffracted beams all sharing a plane defined by the periodicity direction i e the y component of all the wave vectors is the same see figure 1 1 the angles of the diffracted beam directions relative to the incident beam direction are determined by the periodicity length

generalized aor method for solving a class of generalized saddle point problems abstract in this paper by constructing appropriate splitting a generalized aor gaor method for solving a class of generalized saddle point problems has been presented and the convergence of the method is studied the numerical experiments results show the efficiency and robustness of the new method

cohomology of toric line bundles via simplicial alexander duality we give a rigorous mathematical proof for the validity of the toric sheaf cohomology algorithm conjectured in the recent paper by blumenhagen jurke rahn and roschy arxiv 1003 5217 we actually prove not only the original algorithm but also a speed up version of it our proof is independent from in fact appeared earlier on the arxiv than the proof by roschy and rahn arxiv 1006 2392 and has several advantages such as being shorter and cleaner and can also settle the additional conjecture on serre duality for betti numbers which was raised but unresolved in arxiv 1006 2392

rings modules and algebras in infinite loop space theory we give a new construction of the algebraic k theory of small permutative categories that preserves multiplicative structure and therefore allows us to give a unified treatment of rings modules and algebras in both the input and output this requires us to define multiplicative structure on the category of small permutative categories the framework we use is the concept of multicategory elsewhere also called colored operad a generalization of symmetric monoidal category that precisely captures the multiplicative structure we have present at all stages of the construction our method ends up in the hovey shipley smith category of symmetric spectra with an intermediate stop at a category of functors out of a particular wreath product

a note on spectrum analysis of augmentation block preconditioned generalized saddle point matrices in this note we consider a set of augmentation block preconditioners for solving generalized saddle point systems whose coefficient matrices have singular 1 1 blocks results concerning the eigenvalue distribution and forms of the eigenvectors of the augmentation block preconditioned generalized saddle point matrix and its minimal polynomial are given and an optimal augmentation block preconditioner of the set is derived these results extend previous ones in the literature numerical experiments that show the very effective performance of the optimal augmentation block preconditioner are reported

a generalized shift splitting preconditioner for singular saddle point problems recently some authors cao et al 2014 chen and ma 2015 salkuyeh et al 2014 discussed the generalized shift splitting preconditioner for nonsingular saddle point problems in this paper we further study the generalized shift splitting preconditioner for solving singular saddle point problems with symmetric positive definite 1 1 block theoretical analysis shows that the generalized shift splitting iteration method is unconditionally semi convergent numerical experiments are given to illustrate the efficiency of the proposed preconditioner with appropriate parameters

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

cech cohomology of semiring schemes abstract a semiring scheme generalizes a scheme in such a way that the underlying algebra is that of semirings we generalize cech cohomology theory and invertible sheaves to semiring schemes in particular when x p m n a projective space over a totally ordered idempotent semifield m we show that h m x o x is in agreement with the classical computation for all m finally we classify all invertible sheaves on x p m n by computing pic x explicitly

eigenvalue estimates of an indefinite block triangular preconditioner for saddle point problems in this paper an indefinite block triangular preconditioner for symmetric saddle point problems is discussed the bounds for all the real eigenvalues of the preconditioned matrix are provided and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided the corresponding theoretical results of paper by bergamaschi 2012 are improved numerical experiments are reported to confirm our theoretical results

symmetric sor method for augmented systems in this paper we develop symmetric successive overrelaxation ssor method to find solution of augmented systems these systems have appeared in many different applications of scientific computing e g the finite element approximation to solve the navier stokes equation and the constrained optimization the convergence of ssor method are studied finally we apply the new method to solve augmented systems the iteration number of this method to receive the solution is very better than sor like method

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

a relaxed dimensional factorization preconditioner for the incompressible navier stokes equations in this paper we introduce a relaxed dimensional factorization rdf preconditioner for saddle point problems properties of the preconditioned matrix are analyzed and compared with those of the closely related dimensional splitting ds preconditioner recently introduced by benzi and guo 7 numerical results for a variety of finite element discretizations of both steady and unsteady incompressible flow problems indicate very good behavior of the rdf preconditioner with respect to both mesh size and viscosity

preconditioned hermitian and skew hermitian splitting methods for non hermitian positive semidefinite linear systems for the positive semidefinite system of linear equations of a block two by two structure by making use of the hermitian skew hermitian splitting iteration technique we establish a class of preconditioned hermitian skew hermitian splitting iteration methods theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system moreover the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly numerical examples further confirm the correctness of the theory and the effectiveness of the method

the limiting absorption and amplitude principles for the diffraction problem with two unbounded media we consider the classical diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant and the surface separating two media is unbounded the validity of the limiting absorption and amplitude principles is proved

on the spectra and eigenfunctions of the schrodinger and maxwell operators the new sufficient conditions of the exponential decay of eigenfunctions and the absence of eigenvalues are obtained for the second order elliptic operator in l2 rn generalizing the schrodinger operator and for the maxwell operator with the matrix medium characteristics the crystal optics operator in l2 rn

on generalized symmetric sor method for augmented systems in this paper we establish the generalized symmetric sor method gssor for solving the large sparse augmented systems of linear equations which is the extension of the ssor iteration method the convergence of the gssor method for augmented systems is studied numerical resume shows that this method is effective

sheaves and k theory for f1 schemes this paper is devoted to the open problem in f1 geometry of developing k theory for f1 schemes we provide all necessary facts from the theory of monoid actions on pointed sets and we introduce sheaves for m0 schemes and f1 schemes in the sense of connes and consani a wide range of results hopefully lies the background for further developments of the algebraic geometry over f1 special attention is paid to two aspects particular to f1 geometry namely normal morphisms and locally projective sheaves which occur when we adopt quillen s q construction to a definition of g theory and k theory for f1 schemes a comparison with waldhausen s s construction yields the ring structure of k theory in particular we generalize deitmar s k theory of monoids and show that k specf1 realizes the stable homotopy of the spheres as a ring spectrum

theorie spectrale de la propagation des ondes acoustiques dans un milieu stratifie perturbe abstract in this article the spectral analysis of the self adjoint operator governing the propagation acoustic waves in a perturbed stratified medium is given both a medium whose sound speed is a short range perturbation of that of a stratified medium and a stratified medium exterior to a compact set are considered the basic tool is a division theorem for the self adjoint operator governing the propagation of acoustic waves in a pure stratified medium

remark on essential selfadjointness of dirac operators with coulomb potentials on considere des operateurs de dirac a j 1 3 j d j p x p x j 1 n p j x a j d j i j on demontre que toutes les informations sur le caractere autoadjoint et l invariance du spectre essentiel sont contenues dans l inegalite r 1 u c j d j p u

algebraic groups over the field with one element in this paper we provide a first realization of an idea of jacques tits from a 1956 paper which first mentioned that there should be a field of characteristique une which is now called mathbb f 1 the field with one element this idea was that every split reductive group scheme over mathbb z should descend to mathbb f 1 and its group of mathbb f 1 rational points should be its weyl group we connect the notion of a torified scheme to the notion of mathbb f 1 schemes as introduced by connes and consani this yields models of toric varieties schubert varieties and split reductive group schemes as mathbb f 1 schemes we endow the class of mathbb f 1 schemes with two classes of morphisms one leading to a satisfying notion of mathbb f 1 rational points the other leading to the notion of an algebraic group over mathbb f 1 such that every split reductive group is defined as an algebraic group over mathbb f 1 furthermore we show that certain combinatorics that are expected from parabolic subgroups of gl n and grassmann varieties are realized in this theory

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

the spectral properties of the hermitian and skew hermitian splitting preconditioner for generalized saddle point problems in this paper we consider the hermitian and skew hermitian splitting hss preconditioner for generalized saddle point problems with nonzero 2 2 blocks the spectral property of the preconditioned matrix is studied in detail under certain conditions all eigenvalues of the preconditioned matrix with the original system being non hermitian will form two tight clusters one is near 0 0 and the other is near 2 0 as the iteration parameter approaches to zero from above so do all eigenvalues of the preconditioned matrix with the original system being hermitian numerical experiments are given to demonstrate the results

cohomology of line bundles proof of the algorithm we present a proof of the algorithm for computing line bundle valued cohomology classes over toric varieties conjectured by blumenhagen et al e print arxiv 1003 5217 and suggest a kind of serre duality for combinatorial betti numbers that we observed when computing examples

energy concentration and explicit sommerfeld radiation condition for the electromagnetic helmholtz equation we study the electromagnetic helmholtz equation ib x 2u x n x u x f x x rd with the magnetic vector potential b x and n x a variable index of refraction that does not necessarily converge to a constant at infinity but can have an angular dependence like n x n x x as x we prove an explicit sommerfeld radiation condition rd bu in 1 2x x u 2dx1 x for solutions obtained from the limiting absorption principle and we also give a new energy estimate rd n x x 2 u 21 x dx which explains the main physical effect of the angular dependence of n at infinity and deduces that the energy concentrates in the directions given by the critical points of the potential

on the strong unique continuation principle for inequalities of maxwell type our goal of this paper is to prove carleman type inequalities for solutions a of maxwell type inequalities curl a div a cre 1 a assuming ij ec 0 1 in a neighborhood of 0 such inequalities also arise by nonlinear maxwell equations with lipschitz nonlinearity f a provided that a is locally bounded our investigation was inspired by the study of the corresponding problem for the elliptic dirac system

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

unitary equivalence and scattering theory for stark like hamiltonians let h0 v0 x1 h h0 v x be self adjoint in l2 rn v0 depends only on one coordinate and tends monotonically to as x1 v is a real h0 compact potential short range with respect to v0 in particular the cases v0 x1 sgn x1 x1 0 2 and v x c x 1 are included 1 being the stark effect it is shown that a h is spectrally absolutely continuous over the entire real axis apart from a possible discrete sequence of eigenvalues of finite multiplicity and rapidly decaying eigenfunctions h0 has no eigenvalues and b the wave operators w h h0 exist and are complete

cohomology of line bundles a computational algorithm we present an algorithm for computing line bundle valued cohomology classes over toric varieties this is the basic starting point for computing massless modes in both heterotic and type iib f theory compactifications where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles

a limiting absorption principle for schrodinger operators with generalized von neumann wigner potentials i construction of approximate phase abstract in this series of papers we prove the limiting absorption principle over a given interval for a class of hamiltonians which contains the original one of von neumann and wigner more specifically the hamiltonians are of the form c sin b x x v x where 2 3 v x is a short range potential and i is a given compact subinterval of the open positive axis which does not contain the point b 2 4

scattering theory for the dirac operator with a long range electromagnetic potential abstract we consider the dirac operator with a long range potential v x scalar pseudo scalar and vector components of v x may have arbitrary power like decay at infinity we introduce wave operators with time independent modifiers these modifiers are pseudo differential operators whose symbols are roughly speaking constructed in terms of approximate eigenfunctions of the stationary problem we derive and solve eikonal and transport equations for the corresponding phase and amplitude functions from an analytical point of view our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper this allows us to fit the long range scattering theory for the dirac operator into the framework of smooth perturbations finally we find the asymptotics for large times t of solutions u x t of the time dependent dirac equation

torified varieties and their geometries over mathbb f _1 this paper invents the notion of torified varieties a torification of a scheme is a decomposition of the scheme into split tori a torified variety is a reduced scheme of finite type over mathbb z that admits a torification toric varieties split chevalley schemes and flag varieties are examples of this type of scheme given a torified variety whose torification is compatible with an affine open covering we construct a gadget in the sense of connes consani and an object in the sense of soule and show that both are varieties over mathbb f 1 in the corresponding notion since toric varieties and split chevalley schemes satisfy the compatibility condition we shed new light on all examples of varieties over mathbb f 1 in the literature so far furthermore we compare connes consani s geometry soule s geometry and deitmar s geometry and we discuss to what extent chevalley groups can be realized as group objects over mathbb f 1 in the given categories

deitmar s versus toen vaquie s schemes over mathbb f _ 1 deitmar introduced schemes over mathbb f 1 the so called field with one element as certain spaces with an attached sheaf of monoids generalizing the definition of schemes as ringed spaces on the other hand toen and vaquie defined them as particular zariski sheaves over the opposite category of monoids generalizing the definition of schemes as functors of points we show the equivalence between deitmar s and toen vaquies notions and establish an analog of the classical case of schemes over mathbb z this result has been assumed by the leading experts on mathbb f 1 but no proof was given during the proof we also conclude some new basic results on commutative algebra of monoids such as a characterization of local flat epimorphisms and of flat epimorphisms of finite presentation we also inspect the base change functors from the category of schemes over mathbb f 1 to the category of schemes over mathbb z

resolvent estimates for a certain class of schrodinger operators with exploding potentials let h ve x v x be a schrodinger operator in rn here ve x is an exploding radially symmetric potential which is at least c2 monotone nonincreasing and o r2 as r v is a general potential which is short range with respect to ve in particular ve 0 leads to the classical short range case v being an agmon potential let limr ve r and r z h z 1 0 im z re z it is shown that r z can be extended continuously to im z 0 except possibly for a discrete subset n in a suitable operator topology b l l and l l2 rn is a weighted l2 space h is then absolutely continuous over except possibly for a discrete set of eigenvalues the corresponding eigenfunctions are shown to be rapidly decreasing

an application of asymptotic techniques to certain problems of spectral and scattering theory of stark like hamiltonians let lo a v x1 l lo vp x be selfadjoint in l2 r here v vp are real functions v x1 depends only on the first coordinate existence of the wave operators w l lo s 1imt ur exp itl exp itl0 is proved using the stationary phase method for this an asymptotic technique is applied to the study of d2 dt2 v t in l2 r its absolute continuity is proved as well as a suitable eigenfunction expansion v is a stark like potential in particular the cases

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

on hermitian and skew hermitian splitting iteration methods for the linear matrix equation axb c in this paper a hermitian and skew hermitian splitting hss iteration method for solving the linear matrix equation axb c is proposed this method is formed by extending the corresponding hss iteration method for solving ax b the analysis shows that the hss iteration method will converge under certain assumptions moreover the optimal parameter of this iteration method is presented numerical results show that this new method is more efficient and robust than existing methods

structured preconditioners for nonsingular matrices of block two by two structures for the large sparse block two by two real nonsingular matrices we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations for the specific versions such as modified block jacobi type modified block gauss seidel type and modified block unsymmetric symmetric gauss seidel type preconditioners we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices also we show that when these structured preconditioners are employed to precondition the krylov subspace methods such as gmres and restarted gmres fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two by two coefficient matrices in particular these structured preconditioners can lead to efficient and high quality preconditioning matrices for some typical matrices from the real world applications

the limiting absorption principle for the discrete wigner von neumann operator abstract we apply weighted mourre commutator theory to prove the limiting absorption principle for the discrete schrodinger operator perturbed by the sum of a wigner von neumann and long range type potential in particular this implies a new result concerning the absolutely continuous spectrum for these operators even for the one dimensional operator we show that methods of classical mourre theory based on differential inequalities and on the generator of dilation cannot apply to the aforementioned schrodinger operators

on positive definite and skew hermitian splitting iteration methods for continuous sylvester equation ax xb c in this paper we present a positive definite and skew hermitian splitting pss iteration method for continuous sylvester equations ax xb c with positive definite semi definite matrices the theoretical analysis shows that the pss iteration method will converge unconditionally and the optimal parameter of the new method is presented moreover to reduce the computing cost an inexact variant of the pss iteration method ipss and the analysis of its convergence property in detail have been established numerical results show that this new method and its inexact invariant are efficient and robust solvers for this class of continuous sylvester equations

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

new preconditioners for saddle point problems we present new preconditioners based on matrix splittings for the saddle point problems the spectral property of one of the preconditioned matrix is studied in detail numerical examples are used to illustrate the efficiency of the new preconditioners

regularity and decay of solutions to the stark evolution equation long time behavior and regularity are studied for solutions of the stark equationut i x1 v x u u 0 x l2 rn it is shown that for a class of short range potentialsv x the gain of local smoothness and the decay as t are close to those of the corresponding schrodinger equationut i v x u

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

radiation condition and limiting amplitude principle for acoustic propagators with two unbounded media consider the diffraction problem for perturbed acoustic propagators with perturbations decreasing slowly at infinity the propagation speed is discontinuous at the interface of two unbounded media and the interface may be an arbitrary and smooth surface locally a sommerfeld radiation condition is introduced for the acoustic propagator and is then used to establish the limiting absorption principle and the resolvent estimate at low frequencies for such an operator furthermore we prove the existence of a unique solution to the diffraction problem and the validity of the limiting amplitude principles for the acoustic propagator

morrey campanato estimates for helmholtz equations with two unbounded media we prove uniform morrey campanato estimates for helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface they imply weighted l 2 estimates for the solution we prove also a uniform l 2 estimate em without weight for the trace of the solution on the interface

a limiting absorption principle for schrodinger operators with oscillating potentials part ii δ c sin b x x v x for short range v and small coupling constant c

preconditioned conjugate gradient method for generalized least squares problems abstract a variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented if the problem is min ax b t w 1 ax b with a r m n and w r m m symmetric and positive definite the method needs only a preconditioner a 1 r n n but not the inverse of matrix w or of any of its submatrices freund s comparison result for regular least squares problems is extended to generalized least squares problems an error bound is also given

cech cohomology over f12 abstract in this text we generalize cech cohomology to sheaves f with values in blue b modules where b is a blueprint with 1 if x is an object of the underlying site then the cohomology sets h l x f turn out to be blue b modules for a locally free o x module f on a monoidal scheme x we prove that h l x f h l x f where x is the scheme associated with x and f is the locally free o x module associated with f in an appendix we show that the naive generalization of cohomology as a right derived functor is infinite dimensional for the projective line over f 1

uzawa low method and preconditioned uzawa low method for three order block saddle point problem in this paper uzawa low method for three order block saddle point problem is presented and the corresponding convergence conditions are established furthermore by introducing a preconditioner we propose centered preconditioned uzawa low method cpu low method and obtain its convergence conditions as well experimental results show that the two proposed methods are more efficient than uzawa low method for the original saddle point problem and the cpu low method is superior to the uzawa low method for three order block saddle point problem

absolutely continuous spectrum of dirac operators for long range potentials abstract using the resolvent method and the technique of weighted l2 estimates we deduce the non existence of the singular continuous spectrum of the dirac operator p x on the interval 1 we assume that the hermitian matrix potential p x p1 x p2 x is divided into a long range part p 1 x o x e r r p 1 x o x e and a short range part p 2 x o x 1 e x with local singularities p x o x a j 1 of coulomb type for n nuclei a1 an r 3

solutions of the time harmonic wave equation in periodic waveguides asymptotic behaviour and radiation condition in this paper we give the expression and the asymptotic behaviour of the physical solution of a time harmonic wave equation set in a periodic waveguide this enables us to define a radiation condition and show well posedness of the helmholtz equation set in a periodic waveguide

limiting amplitude principle for acoustic propagators in perturbed stratified fluids abstract by use of the commutator method we study the resolvent estimate at low frequencies of acoustic operators c x 2 in perturbed stratified fluids and prove the principle of limiting amplitude for such operators

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

blue schemes semiring schemes and relative schemes after toen and vaquie abstract it is a classical insight that the yoneda embedding defines an equivalence of schemes as locally ringed spaces with schemes as sheaves on the big zariski site similarly the yoneda embedding identifies monoid schemes or f 1 schemes in the sense of deitmar with schemes relative to sets in the sense of toen and vaquie in this paper we investigate the generalization to blue schemes and to semiring schemes we establish yoneda functors for both scheme theories these functors fail however to be equivalences in both situations the reason for this failure is a divergence in the grothendieck pretopologies coming from schemes as topological spaces and schemes as sheaves restricted to blue schemes that are locally of finite type over a blue field we construct an inverse to the yoneda functor which establishes an equivalence for this subclass of blue schemes moreover we verify the compatibility of the yoneda functors with the base extension from blue schemes to semiring schemes and with the base extension from semiring schemes to usual schemes

on the hall algebra of semigroup representations over mathbb f _1 let mathrm a be a finitely generated semigroup with 0 an mathrm a module over mathbb f 1 also called an mathrm a set is a pointed set m together with an action of mathrm a we define and study the hall algebra mathbb h mathrm a of the category mathcal c mathrm a of finite mathrm a modules mathbb h mathrm a is shown to be the universal enveloping algebra of a lie algebra mathfrak n mathrm a called the hall lie algebra of mathcal c mathrm a in the case of langle t rangle the free monoid on one generator langle t rangle the hall algebra or more precisely the hall algebra of the subcategory of nilpotent langle t rangle modules is isomorphic to kreimer s hopf algebra of rooted forests this perspective allows us to define two new commutative operations on rooted forests we also consider the examples when mathrm a is a quotient of langle t rangle by a congruence and the monoid g cup 0 for a finite group g