sharp generalizations of the multilinear bohnenblust hille inequality we prove that the multilinear bohnenblust hille inequality is a particular case of a quite general family of optimal inequalities

the inclusion problem for mixed norm spaces given two mixed norm lebesgue spaces on an n fold product of arbitrary finite measure spaces when is one contained in the other if so what is the norm of the inclusion map these questions are answered completely for a large range of lebesgue indices and all measure spaces when the measure spaces are atomless both questions are settled for all indices when the measure spaces are not purely atomic the first question is settled for all indices some complete and some partial results are given in the remaining cases but a wide variety of behaviour is observed in particular the norm problem for purely atomic measure spaces is seen to be intractable for certain ranges of the lebesgue indices it is equivalent to an optimization problem that includes a known np hard problem as a special case

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

lower bounds for the constants of the hardy littlewood inequalities abstract given an integer m 2 the hardy littlewood inequality for real scalars says that for all 2 m p there exists a constant c m p r 1 such that for all continuous m linear forms a l p n l p n r and all positive integers n j 1 j m 1 n a e j 1 e j m 2 m p m p p 2 m m p p 2 m 2 m p c m p r a the limiting case p is the well known bohnenblust hille inequality the behavior of the constants c m p r is an open problem in this note we provide nontrivial lower bounds for these constants

there exist multilinear bohnenblust hille constants cn n 1 with limn cn 1 cn 0 abstract the n linear bohnenblust hille inequality asserts that there is a constant c n 1 such that the l 2 n n 1 norm of u e i 1 e i n i 1 i n 1 n is bounded above by c n times the supremum norm of u for any n linear form u c n c n c and n n the same holds for real scalars we prove what we call fundamental lemma which brings new information on the optimal constants k n n 1 for both real and complex scalars for instance k n 1 k n 0 87 n 0 473 for infinitely many n s for complex scalars we give a formula of surprisingly low growth in which e and the famous euler mascheroni constant appear k n 1 4 1 e 2 1 2 j 1 n 1 j log 2 e 2 1 2 1 n 2 we study the interplay between the kahane salem zygmund and the bohnenblust hille polynomial and multilinear inequalities and provide estimates for bohnenblust hille type inequality constants for any exponent q 2 n n 1

multiple rademacher means and their applications abstract we evaluate multiple rademacher means for various classes of multilinear operators under the assumption that the range has cotype q these results show that the class of multiple q 2 summing operators plays an important role further we get new extensions of bu s theorem in the multilinear case

a subexponential vector valued bohnenblust hille type inequality abstract bayart pellegrino and seoane recently proved that the polynomial bohnenblust hille inequality for complex scalars is subexponential we show that a vector valued polynomial bohnenblust hille inequality on complex banach lattices is also subexponential for some special cases our main result recovers the best known constants of the classical polynomial inequality provided in 4

the bohr radius of the n dimensional polydisk is equivalent to sqrt frac log n n we show that the bohr radius of the polydisk mathbb d n behaves asymptotically as sqrt log n n our argument is based on a new interpolative approach to the bohnenblust hille inequalities which allows us to prove that the polynomial bohnenblust hille inequality is subexponential

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

mathematical construction of chiral anomaly abstract chiral anomaly is constructed with mathematical rigor by means of the lattice regularization to this end the quantized dirac fermion on a torus coupled with an external gauge field with an arbitrary topology is analyzed it is claimed that in order to complete the rigorous lattice approach to the chiral anomaly the heat kernel regularization for the integrand should be introduced independently of the lattice regularization for the path measure

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

multiple summing operators on banach spaces abstract in this paper we improve some previous results about multiple p summing multilinear operators by showing that every multilinear form from l 1 spaces is multiple p summing for 1 p 2 the proof is based on the existence of a predual for the banach space of multiple p summing multilinear forms we also show the failure of the inclusion theorem in this class of operators and improve some results of y melendez and a tonge about dominated multilinear operators

exact boundary conditions for the initial value problem of convex conservation laws the initial value problem of convex conservation laws which includes the famous burgers inviscid equation plays an important rule not only in theoretical analysis for conservation laws but also in numerical computations for various numerical methods for example the initial value problem of the burgers equation is one of the most popular benchmarks in testing various numerical methods but in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations in this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries which contain a finite computational domain by using the lax s exact formulas for the convex conservation laws the well posedness of the initial boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described numerical tests with the proposed artificial boundary conditions are carried out by using the lax friedrichs monotone difference schemes

coordinatewise multiple summing operators in banach spaces we invent the new notion of coordinatewise multiple summing operators in banach spaces and use it to study various vector valued extensions of the well know bohnenblust hille inequality which originally extended littlewood s 4 3 inequality our results have application on the summability of monomial coefficients of m homogeneous polynomials p l lp as well as for the convergence theory of products of vector valued dirichlet series

on the convergence of finite difference methods for pde under temporal refinement abstract in this note we present an issue in the verification of order of convergence for finite difference approximations to partial differential equations discretized in time and space when one refines both space and time together the convergence rate for a correct implementation will match that expected from an analysis of the asymptotic convergence rate or local truncation error of the method sometimes however only the time step size is reduced while the spatial mesh is held fixed the observed rate of convergence in this case may then differ from the formal order of accuracy of the method in space time in particular one class of methods second order lax wendroff time differencing produces only first order when the time step is refined at fixed spatial mesh resolution because the time and space differencing are intrinsically linked our method to arrive at this result is through analyzing the error at a defined point in time associated with many time steps we demonstrate our results computationally

factorization of absolutely continuous polynomials abstract in this paper we study the ideal of dominated p continuous polynomials that extend the nowadays well known ideal of p dominated polynomials to the more general setting of the interpolated ideals of polynomials we give the polynomial version of pietsch s factorization theorem for this new ideal although based on botelho et al in press 16 our factorization theorem requires new techniques inspired in the theory of banach lattices

improving the closed formula for subpolynomial constants in the multilinear bohnenblust hille inequalities abstract for k r or c the bohnenblust hille inequality asserts that there exists a sequence of scalars c k m m 1 such that i 1 i m 1 n u e i 1 e i m 2 m m 1 m 1 2 m c k m sup z 1 z m d n tu z 1 z m for all m linear forms u k n k n k and every positive integer n where e i i 1 n denotes the canonical basis of k n and d n represents the open unit polydisc in k n very recently 2012 it was shown that there exist constants c k m m 1 with subpolynomial growth satisfying this inequality however these constants were obtained via a complicated recursive formula we improve the best known closed non recursive approximation for these constants

error estimate for the finite volume scheme applied to the advection equation we study the convergence of a finite volume scheme for the linear advection equation with a lipschitz divergence free speed in r d we prove a h 1 2 error estimate in the l infty 0 t l 1 norm for bv data this result was expected from numerical experiments and is optimal

a convergent scheme for a non local hamilton jacobi equation modelling dislocation dynamics we study dislocation dynamics with a level set point of view the model we present here looks at the zero level set of the solution of a non local hamilton jacobi equation as a dislocation in a plane of a crystal the front has a normal speed depending on the solution itself we prove existence and uniqueness for short time in the set of continuous viscosity solutions we also present a first order finite difference scheme for the corresponding level set formulation of the model the scheme is based on monotone numerical hamiltonian proposed by osher and sethian the non local character of the problem makes it not monotone we obtain an explicit convergence rate of the approximate solution to the viscosity solution we finally provide numerical simulations

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

a new multilinear insight on littlewood s 4 3 inequality abstract we unify littlewood s classical 4 3 inequality a forerunner of grothendieck s inequality together with its m linear extension due to bohnenblust and hille which originally settled bohr s absolute convergence problem for dirichlet series with a scale of inequalties of bennett and carl in l p spaces which are of fundamental importance in the theory of eigenvalue distribution of power compact operators as an application we give estimates for the monomial coefficients of homogeneous l p valued polynomials on c 0

the sharpness of kuznetsov s o d x l 1 error estimate for monotone difference schemes we derive a lower error bound for monotone difference schemes to the solution of the linear advection equation with bv initial data a rigorous analysis shows that for any monotone difference scheme the lower l 1 error bound is o x where x is the spatial stepsize

the asymptotic growth of the constants in the bohnenblust hille inequality is optimal the search of sharp estimates for the constants in the bohnenblust hille inequality besides its challenging nature has quite important applications in different fields of mathematics and physics for homogeneous polynomials it was recently shown that the bohnenblust hille inequality for complex scalars is hypercontractive this result interesting by itself has found direct striking applications in the solution of several important problems for multilinear mappings precise information on the asymptotic behavior of the constants of the bohnenblust hille inequality is of particular importance for applications in quantum information theory and multipartite bell inequalities in this paper using elementary tools we prove a quite surprising result the asymptotic growth of the constants in the multilinear bohnenblust hille inequality is optimal besides its intrinsic mathematical interest and potential applications to different areas the mathematical importance of this result also lies in the fact that all previous estimates and related results for the last 80 years such as for instance the multilinear version of the famous grothendieck theorem for absolutely summing operators always present constants c m s growing at an exponential rate of certain power of m

on the geometry of projective tensor products abstract in this work we study the volume ratio of the projective tensor products l p n l q n l r n with 1 p q r we obtain asymptotic formulas that are sharp in almost all cases as a consequence of our estimates these spaces allow for a nearly euclidean decomposition of kasin type whenever 1 p q r 2 or 1 p 2 r and q 2 also from the bourgain milman bound on the volume ratio of banach spaces in terms of their cotype 2 constant we obtain information on the cotype of these 3 fold projective tensor products our results naturally generalize to k fold products l p 1 n l p k n with k n and 1 p 1 p k

absolutely summing multilinear operators via interpolation abstract we use an interpolative technique from 1 to introduce the notion of multiple n separately summing operators our approach extends and unifies some recent results for instance we recover the best known estimates of the multilinear bohnenblust hille constants due to f bayart d pellegrino and j seoane sepulveda more precisely as a consequence of our main result for 1 t 2 and m 1 we prove that i 1 i m 1 u e i 1 e i m 2 t m 2 m 1 t 2 m 1 t 2 t m j 2 m 2 2 t j t 2 t 2 t j 2 2 2 t 2 j t u for all complex m linear forms u c 0 c 0 c

one sided stability and convergence of the nessyahu tadmor scheme non oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws the nessyahu tadmor nt scheme is an example of a second order scheme that is both robust and simple in this paper we prove a new stability property of the nt scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law this property is similar to the one sided lipschitz condition for first order schemes using this new stability we derive the convergence of the nt scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method we also derive an error estimate for monotone initial data

index of a family of dirac operators on loop space we use methods of constructive field theory to generalize index theory to an infinite dimensional setting we study a family of dirac operatorsq on loop space these operators arise in the context of supersymmetric nonlinear quantum field models with hamiltoniansh q2 in these modelsq is self adjoint and fredholm a natural grading operator exists such that q q 0 we studyq p qp wherep 1 2 1 are the orthogonal projections onto the eigenspaces of we calculate the indexi q for wess zumino models defined by a superpotentialv herev is a polynomial of degreen 2 we establish thati q n 1 deg v in particular the field theory models have unbroken supersymmetry and forn 3 they have degenerate vacua we believe that this is the first index theorem for a dirac operator that couples infinitely many degrees of freedom

multiple summing maps coordinatewise summability inclusion theorems and p sidon sets we discuss the multiple summability of a multilinear map t x1 xm y when we have informations on the summability of the maps it induces on each coordinate our methods have applications to inclusion theorems for multiple summing multilinear mappings and to the product of p sidon sets

error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws we consider a general class of finite volume schemes on unstructured but quasi uniform meshes for first order systems of hyperbolic balance laws on unstructured meshes provided the system is equipped with at least one entropy entropy flux tuple and the associated cauchy problem allows for a classical solution u we give conditions such that the finite volume approximation u h converges to u if the mesh parameter h tends to zero in fact we prove an error estimate of the form u u h l 2 le c sqrt h where c is independent of h the proof relies on a stability result for classical solutions in the class of entropy solutions due to dafermos arch rational mech anal 94 1979 pp 373 389 and diperna indiana univ math j 28 1979 pp 137 188 finally we present examples such that the conditions to apply the general convergence estimate can be satisfied at least in part the examples cover general scalar equations weakly coupled systems and the system of elastodynamics in one dimension moreover we generalize the concept of entropy conservative methods due to tadmor math comp 49 1987 pp 91 103 and show how this can be used to establish the convergence of finite volume methods for the system s case

rankine hugoniot riemann solver considering source terms and multidimensional effects a new approach for a flux solver is introduced which takes into account source terms viscous terms and multidimensional effects the basic idea is to distribute the source terms which also contain the viscous terms and multidimensional effects from the cells to the cell interfaces then the fluxes on both sides of a cell interface are determined by the rankine hugoniot conditions and a linearized riemann solver the resulting rankine hugoniot riemann rhr solver yields much more accurate results than conventional riemann solvers for steady premixed laminar flames in 1d and 2d and a steady 2d inviscid channel flow with injection unsteady flow simulations of two colliding flames producing sound and of acoustic oscillations flattening a 2d bunsen flame demonstrate that the new flux solver is able to compute acoustic effects in flames accurately this approach for a flux solver is more general and can also be applied to solve other partial differential equations which can be expressed as hyperbolic systems with source terms ex or including higher spatial derivatives e g for the shallow water equations and for the magnetohydrodynamical equations

on the upper bounds for the constants of the hardy littlewood inequality abstract the best known upper estimates for the constants of the hardy littlewood inequality for m linear forms on l p spaces are of the form 2 m 1 we present better estimates which depend on p and m an interesting consequence is that if p m 2 then the constants have a subpolynomial growth as m tends to infinity

on summability of nonlinear mappings a new approach the main goal of this paper is to characterize arbitrary nonlinear non multilinear mappings f x 1 times cdots times x n rightarrow y between banach spaces that satisfy a quite natural pietsch domination type theorem around a given point a 1 ldots a n in x 1 times cdots times x n as a consequence of our approach a notion of weighted summability arises naturally which may be an interesting topic for further investigation

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

error estimate for finite volume scheme we study the convergence of a finite volume scheme for the linear advection equation with a lipschitz divergence free speed in r d we prove a h1 2 error estimate in the l 0 t l1 norm for bv data this result was expected from numerical experiments and is optimal

on convergence of minmod type schemes a class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered this class of methods contains the classical lax friedrichs and the second order nessyahu tadmor schemes in the case of linear flux new l2 stability results and error estimates for the methods are proved numerical experiments confirm that these methods are one sided l2 stable for convex flux instead of the usual lip stability

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

bohr s strip for vector valued dirichlet series bohr showed that the width of the strip in the complex plane on which a given dirichlet series sum a n n s s in mathbb c converges uniformly but not absolutely is at most 1 2 and bohnenblust hille that this bound in general is optimal we prove that for a given infinite dimensional banach space y the width of bohr s strip for a dirichlet series with coefficients an in y is bounded by 1 1 cot y where cot y denotes the optimal cotype of y this estimate even turns out to be optimal and hence leads to a new characterization of cotype in terms of vector valued dirichlet series

a priori error estimates for approximate solutions to convex conservation laws summary we introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite difference approximation calculated with the scheme of engquistosher lax friedrichs or godunov this technique is a discrete counterpart of the duality technique introduced by tadmor siam j numer anal 1991 the error is related to the consistency error of cell averages of the entropy weak solution this consistency error can be estimated by exploiting a regularity structure of the entropy weak solution one ends up with optimal error estimates

on the bohnenblust hille inequality and a variant of littlewoodʼs 4 3 inequality the search for sharp constants for inequalities of the type littlewood s 4 3 and bohnenblust hille has lately shown unexpected applications in many fields such as analytic number theory quantum information theory or in results on n dimensional bohr radii recent estimates obtained for the multilinear bohnenblust hille inequality for real scalars have been used as a crucial tool by a montanaro in order to solve problems in quantum xor games here among other results we obtain new upper bounds for the bohnenblust hille constants for complex scalars for bilinear forms we provide optimal constants of variants of littlewood s 4 3 inequality for real scalars when the exponent 4 3 is replaced by any r 4 3 we also prove that the optimal constants in real case are always strictly greater than those from the complex case

verification of fluid dynamic codes in the presence of shocks and other discontinuities the verification that computer codes correctly solve their model equations is critical to the continued success of numerical simulation the method of manufactured solutions mms is the best method currently available for this kind of verification for differential equations however it cannot be used directly with discontinuous solutions as is required for the verification of high speed aerodynamic codes with shocks an integrative approach can extend the applicability of mms to both discontinuous solutions such as shocks or material interfaces as well as integral equations we present an implementation of integrative mms based on intelligent subdivision of integration domains that is both highly accurate and fast and results in a rigorous one step verification procedure for shock capturing codes numerical integration is found to be accurate to machine precision when tested on exact solutions of the linear heat equation and the euler equations even in the presence of discontinuous flow features intelligent subdivision of integration domains also improves computational performance by approximately 60i compared to the same algorithm without intelligent subdivisions we demonstrate the use of mms in the verification of the bacl streamer inviscid gas dynamics code integral mms is found to compute convergence rates that are equivalent to those computed using differential mms and comparable to those computed using discontinuous exact solutions suggesting integral mms is a valid method for verification of both integral and shock capturing codes

summing multilinear operators by blocks the isotropic and anisotropic cases abstract unifying several directions of the development of the study of summing multilinear operators between banach spaces we construct a general framework that studies under one single definition multilinear operators that are summing with respect to sums taken over any number of indices iterated and non iterated sums the isotropic and the anisotropic cases sums over arbitrary blocks and over several different sequence norms a large number of special classes of multilinear operators and of methods of generating classes of multilinear operators are recovered as particular instances ideal properties and coincidence theorems for the general classes are proved

volume estimates in spaces of homogeneous polynomials given m we derive upper and lower estimates for the volume of the unit ball of the banach space mathcal p m e n all m homogeneous polynomials defined on the nth section e n of a symmetric banach sequence space e with constants depending on m only we apply this result to obtain asymptotically correct orders of various important invariants of mathcal p m e n from local banach space theory as e g volume ratios cotype and type 2 constants projection constants or banach mazur distances

optimal exponents for hardy littlewood inequalities for m linear operators the hardy littlewood inequalities on lplp spaces provide optimal exponents for some classes of inequalities for bilinear forms on lplp spaces in this paper we investigate in detail the exponents involved in hardy littlewood type inequalities and provide several optimal results that were not achieved by the previous approaches our first main result asserts that for q1 qm 0q1 qm 0 and an infinite dimensional banach space y attaining its cotype cot ycot y if 1p1 1pm 1cot y then the following assertions are equivalent a there is a constant cp1 pmy 1 such that j1 1 j2 1 jm 1 a ej1 ejm qm qm 1qm q1q2 1q1 cp1 pmy a for all continuous m linear operators a lp1 lpm y a lp1 lpm y b the exponents q1 qmq1 qm satisfy q1 m cot yp1 pm q2 m 1 cot yp2 pm qm 1 cot ypm where for k 1 m k 1 m m k 1 cot ypk pm cot y1 1pk 1pm cot y as an application of the above result we generalize one of the classical hardy littlewood inequalities for bilinear forms to the m linear setting our result is sharp in a very strong sense the constants and exponents are optimal even if we consider mixed sums

some applications of hypercontractive inequalities in quantum information theory hypercontractive inequalities have become important tools in theoretical computer science and have recently found applications in quantum computation in this note we discuss how hypercontractive inequalities in various settings can be used to obtain fairly concise proofs of several results in quantum information theory a recent lower bound of lancien and winter on the bias achievable by local measurements which are 4 designs spectral concentration bounds for k local hamiltonians and a recent result of pellegrino and seoane sepulveda giving general lower bounds on the classical bias obtainable in multiplayer xor games

numerical integration techniques for discontinuous manufactured solutions when applying the method of manufactured solutions mms on computational fluid dynamic software determining the exact solutions and source terms for finite volume codes where the stored value is an integrated average over the control volume is non trivial and not frequently discussed mms with discontinuities further complicates the problem of determining these values in an effort to adapt the standard mms procedure to solutions that contain discontinuities we show that newton cotes and gauss quadrature numerical integration methods exhibit high error first order limitations we propose a new method for determining the exact solutions and source terms on a uniform structured grid containing shock discontinuities by performing linearly and quadratically exact transformations on split cells transformations are performed on triangular and quadrilateral elements of a systematically divided discontinuous cell using a quadratic transformation in conjunction with a nine point gauss quadrature method a minimum of 4th order accuracy is achieved for fully general solutions and shock shapes a linear approximation of curved shocks is also experimentally shown to be 2nd order accurate the numerical integration method is then applied to a cfd code using simple discontinuous manufactured solutions which return consistent 1st order convergence values the result is an important step towards being able to use mms to verify solutions with discontinuities this work also highlights the use of higher order numerical integration techniques for continuous and discontinuous solutions that are required for mms on higher order finite volume codes

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

approximation of mono dimensional hyperbolic systems a lattice boltzmann scheme as a relaxation method we focus on mono dimensional hyperbolic systems approximated by a particular lattice boltzmann scheme the scheme is described in the framework of the multiple relaxation times method and stability conditions are given an analysis is done to link the scheme with an explicit finite differences approximation of the relaxation method proposed by jin and xin several numerical illustrations are given for the transport equation burger s equation the p system and full compressible euler s system

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

lower bounds for the complex polynomial hardy littlewood inequality abstract the hardy littlewood inequality for complex homogeneous polynomials asserts that given positive integers m 2 and n 1 if p is a complex homogeneous polynomial of degree m on l p n with m p given by p x 1 x n m a x then there exists a constant c c m p pol 1 which does not depend on n such that m a 1 c c m p pol sup z b l p n p z with p p m if m p 2 m and 2 m p m p p 2 m if 2 m p in this short note we provide nontrivial lower bounds for the constants c c m p pol for instance we prove that for m 2 and m p c c m p pol 2 m p for m even and c c m p pol 2 m 1 p for m odd estimates for the case p this is the particular case of the complex polynomial bohnenblust hille inequality were recently obtained by d nunez alarcon in 2013

convergence of dirichlet polynomials in banach spaces both authors were supported by the mec project mtm2008 03211 the second author was partially supported by grants pr2007 0384 mec and upv paid 00 07

bohr radii of vector valued holomorphic functions abstract motivated by the scalar case we study bohr radii of the n dimensional polydisc d n for holomorphic functions defined on d n with values in banach spaces

rarefactions and large time behavior for parabolic equations and monotone schemes we consider the large time behavior of monotone semigroups associated with degenerate parabolic equations and monotone difference schemes for an appropriate class of initial data the solution is shown to converge to rarefaction waves at a determined asymptotic rate

the optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws we are interested in the rate of convergence in l1 of the approximate solution of a conservation law generated by a monotone finite difference scheme kuznetsov has proved that this rate is 1 2 ussr comput math math phys 16 1976 pp 105 119 and topics numer anal iii in proc roy irish acad conf dublin 1976 pp 183 197 and recently teng and zhang have proved this estimate to be sharp for a linear flux siam j numer anal 34 1997 pp 959 978 we prove by constructing appropriate initial data for the cauchy problem that kuznetsov s estimates are sharp for a nonlinear flux as well

hardy littlewood inequalities for norms of positive operators on sequence spaces abstract we consider estimates of hardy and littlewood for norms of operators on sequence spaces and we apply a factorization result of maurey to obtain improved estimates and simplified proofs for the special case of a positive operator

stochastic and variational approach to the lax friedrichs scheme we present a stochastic and variational aspect of the lax friedrichs scheme applied to hyperbolic scalar conservation laws this is a finite difference version of fleming s results 69 that the vanishing viscosity method is characterized by stochastic processes and calculus of variations we convert the difference equation into that of the hamilton jacobi type and introduce corresponding calculus of variations with random walks the stability of the scheme is obtained through the calculus of variations the convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks the main advantages due to our approach are the following our framework is basically pointwise convergence not l1 as usual which yields uniform convergence except small neighborhoods of shocks the convergence proof is verified for arbitrarily large time interval which is hard to obtain in the case of flux functions of general types depending on both space and time the approximation of characteristics curves is available as well as that of pde solutions which is particularly important for applications of the lax friedrichs scheme to the weak kam theory

reverse inclusions for multiple summing operators abstract multiple summing operators have been proven to be useful in several areas of analysis and mathematical physics in this paper we prove reverse inclusions for this class of operators completing the work already initiated in d perez garcia the inclusion theorem for multiple summing operators studia math 165 3 2004 275 290

on bounded multilinear forms on a class of lp spaces abstract in this paper we generalize an old result of littlewood and hardy about bilinear forms defined in a class of sequence spaces historically littlewood quart j math 1 1930 first proved a result on bilinear forms on bounded sequences and this result was then generalized by hardy and littlewood in a joint paper quart j math 5 1934 to bilinear forms on a class of lp spaces later davie and kaijser proved littlewood s results for multilinear forms in this paper theorems a and b generalize the results to multilinear forms on lp spaces all the results are stated at the end of section 1 theorems a and b are proved respectively in sections 2 and 3

on the piecewise smooth solutions to non homogeneous scalar conservation laws abstract we study the structure and smoothness of non homogeneous convex conservation laws the question regarding the number of smoothness pieces is addressed it is shown that under certain conditions on the initial data the entropy solution has only a finite number of discontinuous curves we also obtain some global estimates on derivatives of the piecewise smooth entropy solution along the generalized characteristics these estimates play important roles in obtaining the optimal rate of convergence for various approximation methods to conservation laws

a new estimate for the constants of an inequality due to hardy and littlewood abstract in this paper we provide a family of inequalities extending a recent result due to albuquerque et al

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche