a convergence rate theorem for finite difference approximations to delta functions we prove a rate of convergence theorem for approximations to certain integrals over codimension one manifolds in r n the type of manifold involved here is defined by the zero level set of a smooth mapping u r email protected r our approximations are based on the two finite difference methods for discretizing delta functions presented in 16 we included a convergence proof in that paper but only proved rates of convergence in some greatly simplified situations numerical experiments indicated that our two methods were at least first and second order accurate respectively in this note we prove those empirical convergence rates for the two algorithms under fairly general hypotheses

lineability and algebrability of pathological phenomena in analysis we show that in analysis many pathological phenomena occur more often than one could expect that is in a linear or algebraic way we show this by means of the construction of large algebraic structures infinite dimensional vector spaces or infinitely generated algebras enjoying some special or pathological properties

two methods for discretizing a delta function supported on a level set this paper presents two new methods for discretizing a dirac delta function which is concentrated on the zero level set of a smooth function u r n r the function u is only known at the discrete set of points belonging to a regular mesh covering r n these two methods are used to approximate integrals over the manifold defined by the level set both methods are conceptually simple and easy to implement we present the results of numerical experiments indicating that as the mesh size h goes to zero the rate of convergence is at least o h for the first method and o h 2 for the second method we perform a limited analysis of the proposed algorithms including a proof of convergence for both methods

independent bernstein sets and algebraic constructions abstract we present the method of constructing algebras and linear spaces of 2 c generators using independent bernstein sets as an application we obtain large algebras of special functions i strongly everywhere surjective functions which are not perfectly everywhere surjective ii nowhere continuous darboux functions iii nowhere continuous compact to compact functions iv functions which are continuous precisely on a fixed closed proper subset of r most conclusions obtained in this paper are improvements of some already known results

subsets of some families of real functions and their algebrability abstract we consider the following families of real valued functions darboux d quasi continuous q and wi tkowski functions s the aim of this paper is to study the level of algebrability for various intersections of families d q and s and their complements

a hybrid phase flow method for hamiltonian systems with discontinuous hamiltonians in this paper we propose a new phase flow method for hamiltonian systems with discontinuous hamiltonians in the original phase flow method introduced by ying and candes j comput phys 220 2006 pp 184 215 the phase map should be smooth to ensure the accuracy of the interpolation such an interpolation is inaccurate if the phase map is nonsmooth for example when the hamiltonian is discontinuous we modify the phase flow method using a discontinuous hamiltonian solver and establish the stability for piecewise constant potentials of such a solver this extends the applicability of the highly efficient phase flow method to singular hamiltonian systems with a mild increase of algorithm complexity such a particle method can be useful for the computation of high frequency waves through interfaces

nonseparable spaceability and strong algebrability of sets of continuous singular functions abstract let cbv denote the banach algebra of all continuous real valued functions of bounded variation defined in 0 1 we show that the set of strongly singular functions in cbv is nonseparably spaceable we also prove that certain families of singular functions constitute strongly c algebrable sets the argument is based on a criterion of strong c algebrability using compositions with exponential like functions

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

lineability spaceability and additivity cardinals for darboux like functions we introduce the concept of maximal lineability cardinal number ml m of a subset m of a topological vector space and study its relation to the cardinal numbers known as additivity a m homogeneous lineability hl m and lineability l m of m in particular we will describe in terms of l the lineability and spaceability of the families of the following darboux like functions on r n n 1 extendable jones and almost continuous functions

high order numerical methods to a type of delta function integrals we study second to fourth order numerical methods to a type of delta function integrals in one to three dimensions these delta function integrals arise from recent efficient level set methods for computing the multivalued solutions of nonlinear pdes we show that the natural quadrature approach with usual discrete delta functions and support size formulas to the two dimensional delta function integrals suffer from nonconvergence we then design high order numerical methods to this type of delta function integrals based on interpolation approach numerical examples are presented to verify the efficiency and accuracy of our methods

the ring of evenly weighted points on the line let m w mathbb p 1 n hbox sl 2 denote the geometric invariant theory quotient of mathbb p 1 n by the diagonal action of hbox sl 2 using the line bundle mathcal o w 1 w 2 ldots w n on mathbb p 1 n let r w be the coordinate ring of m w we give a closed formula for the hilbert function of r w which allows us to compute the degree of m w the graded parts of r w are certain kostka numbers so this hilbert function computes stretched kostka numbers if all the weights w i are even we find a presentation of r w so that the ideal i w of this presentation has a quadratic grobner basis in particular r w is koszul we obtain this result by studying the homogeneous coordinate ring of a projective toric variety arising as a degeneration of m w

spaceability of the set of continuous injections from blp into lp with nowhere continuous inverses abstract let p 1 we show that in the banach space of all bounded continuous mappings from b l p the open unit ball in l p into l p the subset consisting of all injections with nowhere continuous inverses is isometrically l p spaceable i e it contains an isometric copy of l p the proof uses some ideas of creswell s example

lineability and integrability in the sense of riemann lebesgue denjoy and khintchine abstract in this paper we continue the ongoing research on lineability related questions on this occasion we shall consider among others the classes of integrable functions in the sense of riemann lebesgue denjoy and khintchine improving some already known results and expanding the study of lineability to other famous integrable classes never considered before

structural aspects of the non uniformly continuous functions and the unbounded functions within c x abstract we prove in this paper that if a metric space supports a real continuous function which is not uniformly continuous then under appropriate mild assumptions there exists in fact a plethora of such functions in both topological and algebraical senses corresponding results are also obtained concerning unbounded continuous functions on a non compact metrizable space

geometric optics in a phase space based level set and eulerian framework we introduce a level set approach for ray tracing and the construction of wavefronts in geometric optics this is important in a wide variety of applications in wave propagation our approach automatically handles the multivalued solutions that appear and automatically resolves the wavefronts this is achieved through solving for the bicharacteristic strips whose projection to spatial space gives the wavefronts in a reduced phase space under a eulerian and partial differential equation based framework the bicharacteristic strips are represented using a level set approach for handling higher codimensional objects and the partial differential equations responsible for the evolution are reduced forms of the liouville equations results for the two dimensional case for constant and variable indices of refraction are shown and compared to those of other current methods in the field results are also introduced to show the ability to handle reflection and to extend the method to the three dimensional case

toric fiber products abstract we introduce and study the toric fiber product of two ideals in polynomial rings that are homogeneous with respect to the same multigrading under the assumption that the set of degrees of the variables form a linearly independent set we can explicitly describe generating sets and grobner bases for these ideals this allows us to unify and generalize some results in algebraic statistics

highly accurate surface and volume integration on implicit domains by means of moment fitting summary we introduce a new method for the numerical integration over curved surfaces and volumes defined by a level set function the method is based on the solution of a small linear system based on a simplified variant of the moment fitting equations numerical experiments suggest that the accuracy of the resulting quadrature rules exceeds the accuracy of traditional methods by orders of magnitude using moments up to an order of p the measured experimental orders of convergence exceed hp consequently their construction is very efficient because only coarse computational grids are required the conceptual simplicity allows for the application on very general grid types which is demonstrated by numerical experiments on quadrilateral triangular and hexahedral grids copyright 2013 john wiley sons ltd

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

phylogenetic complexity of the kimura 3 parameter model abstract in algebraic statistics the kimura 3 parameter model is one of the most interesting and classical phylogenetic models we prove that the ideals associated to this model are generated in degree four confirming a conjecture by sturmfels and sullivant

computing multi valued physical observables for the high frequency limit of symmetric hyperbolic systems we develop a level set method for the computation of multi valued physical observables density velocity energy etc for the high frequency limit of symmetric hyperbolic systems in any number of space dimensions we take two approaches to derive the method the first one starts with a weakly coupled system of an eikonal equation for phase s and a transport equation for density r ts h x s 0 t x r xr n email protected email protected x email protected ph x xs 0 the main idea is to evolve the density near the n dimensional bi characteristic manifold of the eikonal hamiltonian jacobi equation which is identified as the common zeros of n level set functions in phase space x k r 2 n these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity email protected t x k det email protected by solving again the liouville equation near the obtained zero level set f 0 but with initial density as initial data the multi valued density and higher moments are thus resolved by integrating f along the bi characteristic manifold in the phase directions the second one uses the high frequency limit of symmetric hyperbolic systems derived by the wigner transform this gives rise to liouville equations in the phase space with measure valued solution in its initial data due to the linearity of the liouville equation we can decompose the density distribution into products of function each of which solves the liouville equation with l initial data on any bounded domain it yields higher order moments such as energy and energy flux the main advantages of these new approaches in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these methods can be used to compute all physical observables for multi dimensional problems our method applies to the wave fields corresponding to simple eigenvalues of the dispersion matrix one such example is the wave equation which will be studied numerically in this paper

discrete singular convolution and its application to the analysis of plates with internal supports part 1 theory and algorithm this paper presents a novel computational approach the discrete singular convolution dsc algorithm for analysing plate structures the basic philosophy behind the dsc algorithm for the approximation of functions and their derivatives is studied approximations to the delta distribution are constructed as either bandlimited reproducing kernels or approximate reproducing kernels unified features of the dsc algorithm for solving differential equations are explored it is demonstrated that different methods of implementation for the present algorithm such as global local galerkin collocation and finite difference can be deduced from a single starting point the use of the algorithm for the vibration analysis of plates with internal supports is discussed detailed formulation is given to the treatment of different plate boundary conditions including simply supported elastically supported and clamped edges this work paves the way for applying the dsc approach in the following paper to plates with complex support conditions which have not been fully addressed in the literature yet

feedback control of combustion instabilities from within limit cycle oscillations using h loop shaping and the ν gap metric combustion instabilities arise owing to a two way coupling between acoustic waves and unsteady heat release oscillation amplitudes successively grow until nonlinear effects cause saturation into limit cycle oscillations feedback control in which an actuator modifies some combustor input in response to a sensor measurement can suppress combustion instabilities linear feedback controllers are typically designed using linear combustor models however when activated from within limit cycle the linear model is invalid and such controllers are not guaranteed to stabilize this work develops a feedback control strategy guaranteed to stabilize from within limit cycle oscillations a low order model of a simple combustor exhibiting the essential features of more complex systems is presented linear plane acoustic wave modelling is combined with a weakly nonlinear describing function for the flame the latter is determined numerically using a level set approach its implication is that the open loop transfer function oltf needed for controller design varies with oscillation level the difference between the mean and the rest of the oltfs is characterized using the gap metric providing the minimum required robustness margin for an formula see text loop shaping controller such controllers are designed and achieve stability both for linear fluctuations and from within limit cycle oscillations

on algebrability of nonabsolutely convergent series we prove that the set of all complex series which are nonabsolutely convergent is c algebrable we establish a similar result for the set of all divergent complex series with bounded partial sums

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

algebraic structures in the sets of surjective functions abstract in the paper we construct several algebraic structures vector spaces algebras and free algebras inside sets of different types of surjective functions among many results we prove that the set of everywhere but not strongly everywhere surjective complex functions is strongly c algebrable and that its 2 c algebrability is consistent with zfc under ch the set of everywhere surjective complex functions which are sierpinski zygmund in the sense of continuous but not borel functions is strongly c algebrable the set of jones complex functions is strongly 2 c algebrable

short note a proof that a discrete delta function is second order accurate it is proved that a discrete delta function introduced by smereka p smereka the numerical approximation of a delta function with application to level set methods j comput phys 211 2006 77 90 gives a second order accurate quadrature rule for surface integrals using values on a regular background grid the delta function is found using a technique of mayo a mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j numer anal 21 1984 285 299 it can be expressed naturally using a level set function

bloch decomposition based gaussian beam method for the schrodinger equation with periodic potentials the linear schrodinger equation with periodic potentials is an important model in solid state physics the most efficient direct simulation using a bloch decomposition based time splitting spectral method 18 requires the mesh size to be o e where e is the scaled semiclassical parameter in this paper we generalize the gaussian beam method introduced in jin et al 23 to solve this problem asymptotically we combine the technique of bloch decomposition and the eulerian gaussian beam method to arrive at an eulerian computational method that requires mesh size of o e the accuracy of this method is demonstrated via several numerical examples

computing multivalued physical observables for the semiclassical limit of the schrodinger equation we develop a level set method for the computation of multivalued physical observables density velocity etc for the semiclassical limit of the schrodinger equation this method uses an eulerian formulation and applies directly to arbitrary number of space dimensions the main idea is to evolve the density near an n dimensional manifold that is identified as the common zeros of n level set functions in phase space these level set functions are generated from solving the liouville equation with initial data chosen to embed the phase gradient simultaneously we track a new quantity f by solving again the liouville equation near the obtained zero level set but with initial density as initial data the multivalued density and higher moments are thus resolved by integrating f along the n dimensional manifold in the phase directions we show that this is equivalent to using the wigner approach but decomposing the velocity from the density each of which evolves by the same liouville equation the main advantages of this approach in contrast to the standard kinetic equation approach using the liouville equation with a dirac measure initial data include 1 the liouville equations are solved with l initial data and a singular integral involving the email protected function is evaluated only in the post processing step thus avoiding oscillations and excessive numerical smearing 2 a local level set method can be utilized to significantly reduce the computation in the phase space these advantages allow us to compute for the first time all physical observables for multidimensional problems in an eulerian framework

gaussian beam methods for the schrodinger equation with discontinuous potentials we propose eulerian and lagrangian gaussian beam methods for the schrodinger equation with discontinuous potentials at the quantum barriers where the potential is discontinuous we derive suitable interface conditions to account for quantum scattering information these scattering interface conditions are then built into the numerical fluxes in the eulerian level set formulation of the gaussian beam methods and are also used in the lagrangian formulation including an interface condition for the hessian matrix we carry both 1d and 2d numerical examples to verify the accuracy of the method

discretization of dirac delta functions in level set methods discretization of singular functions is an important component in many problems to which level set methods have been applied we present two methods for constructing consistent approximations to dirac delta measures concentrated on piecewise smooth curves or surfaces both methods are designed to be convenient for level set simulations and are introduced to replace the commonly used but inconsistent regularization technique that is solely based on a regularization parameter proportional to the mesh size the first algorithm is based on a tensor product of regularized one dimensional delta functions it is independent of the irregularity relative to the grid in the second method the regularization is constructed from a one dimensional regularization that is extended to multi dimensions with a variable support depending on the orientation of the singularity relative to the computational grid convergence analysis and numerical results are given

robust second order accurate discretizations of the multi dimensional heaviside and dirac delta functions we present a robust second order accurate method for discretizing the multi dimensional heaviside and the dirac delta functions on irregular domains the method is robust in the following ways 1 integrations of source terms on a co dimension one surface are independent of the underlying grid and therefore stable under perturbations of the domain s boundary 2 the method depends only on the function value of a level function not on its derivatives we present the discretizations in tabulated form to make their implementations straightforward we present numerical results in two and three spatial dimensions to demonstrate the second order accuracy in the l 1 norm in the case of the solution of pdes with singular source terms in the case of evaluating the contribution of singular source terms on interfaces the method is also second order accurate in the l norm

regular article a pde based fast local level set method we develop a fast method to localize the level set method of osher and sethian 1988 j comput phys 79 12 and address two important issues that are intrinsic to the level set method a how to extend a quantity that is given only on the interface to a neighborhood of the interface b how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface this fast local level set method reduces the computational effort by one order of magnitude works in as much generality as the original one and is conceptually simple and easy to implement our approach differs from previous related works in that we extract all the information needed from the level set function or functions in multiphase flow and do not need to find explicitly the location of the interface in the space domain the complexity of our method to do tasks such as extension and distance reinitialization is o n where n is the number of points in space not o n log n as in works by sethian 1996 proc nat acad sci 93 1591 and helmsen and co workers 1996 spie microlithography ix p 253 this complexity estimation is also valid for quite general geometrically based front motion for our localized method

computational high frequency waves through curved interfaces via the liouville equation and geometric theory of diffraction we construct a class of numerical schemes for the liouville equation of geometric optics coupled with the geometric theory of diffractions to simulate the high frequency linear waves with a discontinuous index of refraction in this work s jin x wen a hamiltonian preserving scheme for the liouville equation of geometric optics with partial transmissions and reflections siam j numer anal 44 2006 1801 1828 a hamiltonian preserving scheme for the liouville equation was constructed to capture partial transmissions and reflections at the interfaces this scheme is extended by incorporating diffraction terms derived from geometric theory of diffraction into the numerical flux in order to capture diffraction at the interface we give such a scheme for curved interfaces this scheme is proved to be positive under a suitable time step constraint numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation

high order numerical methods to two dimensional delta function integrals in level set methods in this paper we design and analyze a class of high order numerical methods to delta function integrals appearing in level set methods in two dimensional case the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the two dimensional delta function integral can be rewritten as a one dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral and thus is approximated by applying standard one dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals proposed in x wen high order numerical methods to a type of delta function integrals j comput phys 226 2007 1952 1967 we establish error estimates for the method which show that the method can achieve any desired accuracy by assigning the corresponding accuracy to the sub algorithms and has better accuracy under an assumption on the zero level set of the level set function which holds generally numerical examples are presented showing that the second to fourth order methods implemented in this paper achieve or exceed the expected accuracy and demonstrating the advantage of using our high order numerical methods

non lipschitz functions with bounded gradient and related problems let e be a topological vector space and let us consider a property p we say that the subset m of e formed by the vectors in e which satisfy p is lineable for certain cardinal finite or infinite if m 0 contains an infinite dimensional linear space of dimension in this note we prove that there exist uncountably infinite dimensional linear spaces of functions enjoying the following properties 1 being continuous on 0 1 a e differentiable with a e bounded derivative and not lipschitz 2 differentiable in r2 r and not enjoying the mean value theorem 3 real valued differentiable on an open connected and non convex set having bounded gradient non lipschitz and therefore not verifying the mean value theorem

linear subsets of nonlinear sets in topological vector spaces for the last decade there has been a generalized trend in mathematics on the search for large algebraic structures linear spaces closed subspaces or infinitely generated algebras composed of mathematical objects enjoying certain special properties this trend has caught the eye of many researchers and has also had a remarkable influence in real and complex analysis operator theory summability theory polynomials in banach spaces hypercyclicity and chaos and general functional analysis this expository paper is devoted to providing an account on the advances and on the state of the art of this trend nowadays known as lineability and spaceability

high order numerical quadratures to one dimensional delta function integrals we study high order numerical quadratures to one dimensional delta function integrals in this paper this is motivated by the fact that traditional numerical quadratures give only first order accuracy in general we provide criteria on discrete delta functions and support size formulas which ensure any desired accuracy of the numerical quadratures discrete delta functions and support size formulas satisfying these criteria are designed numerical examples are presented to verify the performed analysis and the high order accuracy of the proposed numerical quadratures

projective normality of the moduli space of rank 2 vector bundles on a generic curve we prove that the embedding to the projective space by the generalized theta divisors of the moduli space of rank 2 vector bundles on a generic curve is projective normal

geometric integration over irregular domains with application to level set methods we present a geometric approach for calculating integrals over irregular domains described by a level set function this procedure can be used to evaluate integrals over a lower dimensional interface and may be used to evaluate the contribution of singular source terms this approach produces results that are second order accurate and robust to the perturbation of the interface location on the grid moreover since we use a cell wise approach this procedure can be easily extended to quadtree and octree grids we demonstrate the second order accuracy and the robustness of the method in two and three spatial dimensions

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

peano curves on topological vector spaces the starting point of this paper is the existence of peano curves that is continuous surjections mapping the unit interval onto the unit square from this fact one can easily construct of a continuous surjection from the real line r to any euclidean space rn the algebraic structure of the set of these functions as well as extensions to spaces with higher dimensions is analyzed from the modern point of view of lineability and large algebras are found within the families studied we also investigate topological vector spaces that are continuous image of the real line providing an optimal lineability result

coordinate rings for the moduli stack of sl2 c quasi parabolic principal bundles on a curve and toric fiber products abstract we continue the program started in manon 2010 m1 to understand the combinatorial commutative algebra of the projective coordinate rings of the moduli stack m c p sl 2 c of quasi parabolic sl 2 c principal bundles on a generic marked projective curve we find general bounds on the degrees of polynomials needed to present these algebras by studying their toric degenerations in particular we show that the square of any effective line bundle on this moduli stack yields a koszul projective coordinate ring this leads us to formalize the properties of the polytopes used in proving our results by constructing a category of polytopes with term orders we show that many of results on the projective coordinate rings of m c p sl 2 c follow from closure properties of this category with respect to fiber products

the numerical approximation of a delta function with application to level set methods it is shown that a discrete delta function can be constructed using a technique developed by anita mayo the fast solution of poisson s and the biharmonic equations on irregular regions siam j sci comput 21 1984 285 299 for the numerical solution of elliptic equations with discontinuous source terms this delta function is concentrated on the zero level set of a continuous function in two space dimensions this corresponds to a line and a surface in three space dimensions delta functions that are first and second order accurate are formulated in both two and three dimensions in terms of a level set function the numerical implementation of these delta functions achieves the expected order of accuracy

solving poisson type equations with robin boundary conditions on piecewise smooth interfaces abstract we present two finite volume schemes to solve a class of poisson type equations subject to robin boundary conditions in irregular domains with piecewise smooth boundaries the first scheme results in a symmetric linear system and produces second order accurate numerical solutions with first order accurate gradients in the l norm for solutions with two bounded derivatives the second scheme is nonsymmetric but produces second order accurate numerical solutions as well as second order accurate gradients in the l norm for solutions with three bounded derivatives numerical examples are given in two and three spatial dimensions

spaceability of strict order integrability abstract in this paper sharp conditions on a measure space are provided in order that the subset of functions in the corresponding lebesgue space l p which are in no other l q contains except for zero an infinite dimensional closed subspace this completes earlier work of several authors the main result is derived from a new general criterion for spaceability on function spaces a further application to spaceability of the class of non absolutely continuous functions is supplied

an undecidable case of lineability in rr abstract recently it has been proved that assuming that there is an almost disjoint family of cardinality 2 c in c which is assured for instance by either martin s axiom or the continuum hypothesis or even 2 c c one has that the set of sierpinski zygmund functions is 2 c strongly algebrable and thus 2 c lineable here we prove that these two statements are actually equivalent and moreover that they both are undecidable this would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability and spaceability

high order quadrature methods for implicitly defined surfaces and volumes in hyperrectangles a high order accurate numerical quadrature algorithm is presented for the evaluation of integrals over curved surfaces and volumes which are defined implicitly via a fixed isosurface of a given function restricted to a given hyperrectangle by converting the implicitly defined geometry into the graph of an implicitly defined height function the approach leads to a recursive algorithm on the number of spatial dimensions which requires only one dimensional root finding and one dimensional gaussian quadrature the computed quadrature scheme yields strictly positive quadrature weights and inherits the high order accuracy of gaussian quadrature a range of different convergence tests demonstrate orders of accuracy up to 20th order also presented is an application of the quadrature algorithm to a high order embedded boundary discontinuous galerkin method for solving partial differential equations on curved domains

highly accurate computation of volume fractions using differential geometry abstract this paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso contour of a sufficiently smooth level set function after spatial discretization local approximation of the hypersurface and application of the gaussian divergence theorem the volume integrals are transformed to surface integrals application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature we discuss the theoretical foundations and provide details of the numerical algorithm finally we present numerical results for convex and non convex hypersurfaces embedded in cuboidal domains showing both high accuracy and third to fourth order convergence in space

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

a variational level set approach to multiphase motion a coupled level set method for the motion of multiple junctions of e g solid liquid and grain boundaries which follows the gradient flow for an energy functional consisting of surface tension proportional to length and bulk energies proportional to area is developed the approach combines the level set method of s osher and j a sethian with a theoretical variational formulation of the motion by f reitich and h m soner the resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions the gradient projection method leads to a coupled system of perturbed by curvature terms hamilton jacobi equations the coupling is enforced using a single lagrange multiplier associated with a constraint which essentially prevents a regions from overlapping and b the development of a vacuum the numerical implementation is relatively simple and the results agree with and go beyond the theory as given in 12 other applications of this methodology including the decomposition of a domain into subregions with minimal interface length are discussed finally some new techniques and results in level set methodology are presented

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

a high order discontinuous galerkin method for compressible flows with immersed boundaries summary we present a higher order discretization scheme for the compressible euler and navier stokes equations with immersed boundaries our approach makes use of a discontinuous galerkin dg discretization in a domain that is implicitly defined by means of a level set function the zero iso contour of this level set function is considered as an additional domain boundary where we weakly enforce boundary conditions in the same manner as in boundary fitted cells in order to retain the full order of convergence of the scheme it is crucial to perform volume and surface integrals in boundary cells with high accuracy this is achieved using a linear moment fitting strategy moreover we apply a non intrusive cell agglomeration technique that averts problems with very small and ill shaped cuts the robustness accuracy and convergence properties of the scheme are assessed in several two dimensional test cases for the steady compressible euler and navier stokes equations approximation orders range from zero to four even though the approach directly generalizes to even higher orders in all test cases with a sufficiently smooth solution the experimental order of convergence matches the expected rate for dg schemes this article is protected by copyright all rights reserved

motion of curves in three spatial dimensions using a level set approach the level set method was originally designed for problems dealing with codimension one objects where it has been extremely succesful especially when topological changes in the interface i e merging and breaking occur attempts have been made to modify it to handle objects of higher codimension such as vortex filaments while preserving the merging and breaking property we present numerical simulations of a level set based method for moving curves in r3 the model problem for higher codimension that allows for topological changes a vector valued level set function is used with the zero level set representing the curve our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking

holomorphic functions on non runge domains and related problems let u be a domain in cn that is not a runge domain we study the topological and algebraic properties of the family of holomorphic functions on u which cannot be approximated by polynomials

infinite dimensional holomorphic non extendability and algebraic genericity abstract in this note the linear structure of the family h e g of holomorphic functions in a domain g of a complex banach space that are not holomorphically continuable beyond the boundary of g is analyzed more particularly we prove that h e g contains except for zero a closed and a dense vector space having maximal dimension as well as a maximally generated free algebra the results obtained complete a number of previous ones by several authors

maximal spaceability in sequence spaces abstract in this paper we introduce a new technique to prove the existence of closed subspaces of maximal dimension inside sets of topological vector sequence spaces the results we prove cover some sequence spaces not studied before in the context of spaceability and settle some questions on classical sequence spaces that remained open

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy

high order numerical methods to three dimensional delta function integrals in level set methods in this paper we propose a class of high order numerical methods to delta function integrals appearing in level set methods in the three dimensional case by extending the idea for designing high order methods to two dimensional delta function integrals in x wen j comput phys 228 2009 pp 4273 4290 the methods comprise approximating the mesh cell restrictions of the delta function integral in each mesh cell the three dimensional delta function integral can be rewritten as a two dimensional ordinary integral with the smooth integrand being a one dimensional delta function integral a key idea in designing high order methods in this paper is that the high order accuracy of the methods is not ensured in each mesh cell and the methods are designed in a consistent way to foster error cancelations in mesh cells where high order accuracy is not ensured in the individual cell this issue is essentially related to the construction of integral area for the two dimensional ordinary integral in each mesh cell which is achieved by considering the intersection points between the zero level set and the edges or sides of the mesh cell the mesh cell restrictions of the three dimensional delta function integral are then approximated by applying standard two dimensional high order numerical quadratures and high order numerical methods to one dimensional delta function integrals numerical examples are presented showing that our methods in this paper achieve or exceed the expected second to fourth order accuracy and demonstrating the stability of shifting mesh for our methods and the advantage of using our high order numerical methods