the continuum limit of critical random graphs we consider the erd s renyi random graph g n p inside the critical window that is when p 1 n n 4 3 for some fixed lambda in mathbb r we prove that the sequence of connected components of g n p considered as metric spaces using the graph distance rescaled by n 1 3 converges towards a sequence of continuous compact metric spaces the result relies on a bijection between graphs and certain marked random walks and the theory of continuum random trees our result gives access to the answers to a great many questions about distances in critical random graphs in particular we deduce that the diameter of g n p rescaled by n 1 3 converges in distribution to an absolutely continuous random variable with finite mean

a partition of unity finite element method for three dimensional transient diffusion problems with sharp gradients abstract an efficient partition of unity finite element method for three dimensional transient diffusion problems is presented a class of multiple exponential functions independent of time variable is proposed to enrich the finite element approximations as a consequence of this procedure the associated matrix for the linear system is evaluated once at the first time step and the solution is obtained at subsequent time step by only updating the right hand side of the linear system this results in an efficient numerical solver for transient diffusion equations in three space dimensions compared to the conventional finite element methods with h refinement the proposed approach is simple more efficient and more accurate the performance of the proposed method is assessed using several test examples for transient diffusion in three space dimensions we present numerical results for a transient diffusion equation with known analytical solution to quantify errors for the new method we also solve time dependent diffusion problems in complex geometries we compare the results obtained using the partition of unity finite element method to those obtained using the standard finite element method it is shown that the proposed method strongly reduces the necessary number of degrees of freedom to achieve a prescribed accuracy

a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

weighted complex projective 2 designs from bases optimal state determination by orthogonal measurements we introduce the problem of constructing weighted complex projective 2 designs from the union of a family of orthonormal bases if the weight remains constant across elements of the same basis then such designs can be interpreted as generalizations of complete sets of mutually unbiased bases being equivalent whenever the design is composed of d 1 bases in dimension d we show that for the purpose of quantum state determination these designs specify an optimal collection of orthogonal measurements using highly nonlinear functions on abelian groups we construct explicit examples from d 2 orthonormal bases whenever d 1 is a prime power covering dimensions d 6 10 and 12 for example where no complete sets of mutually unbiased bases have thus far been found

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

symmetric informationally complete positive operator valued measures a new computer study we report on a new computer study of the existence of d2 equiangular lines in d complex dimensions such maximal complex projective codes are conjectured to exist in all finite dimensions and are the underlying mathematical objects defining symmetric informationally complete measurements in quantum theory we provide numerical solutions in all dimensions d 67 and moreover a putatively complete list of weyl heisenberg covariant solutions for d 50 a symmetry analysis of this list leads to new algebraic solutions in dimensions d 24 35 and 48 which are given together with algebraic solutions for d 4 15 and 19

galois quantum systems irreducible polynomials and riemann surfaces finite quantum systems in which the position and momentum take values in the galois field gf pl are studied ideas from the subject of field extension are transferred in the context of quantum mechanics the frobenius automorphisms in galois fields lead naturally to the frobenius formalism in a quantum context the hilbert space splits into frobenius subspaces which are labeled with the irreducible polynomials associated with the ypl y the frobenius maps transform unitarily the states of a galois quantum system and leave fixed all states in some of its galois subsystems where the position and momentum take values in subfields of gf pl an analytic representation of these systems in the l sheeted complex plane shows deeper links between galois theory and riemann surfaces

geometrical approach to mutually unbiased bases we propose a unifying phase space approach to the construction of mutually unbiased bases for a two qubit system it is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness these consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties we also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the bloch sphere principal axes we suggest how to generalize the method to systems in dimensions that are powers of a prime

mesh independent p orthotropic enrichment using the generalized finite element method this paper is aimed at presenting a simple yet effective procedure to implement a mesh independent p orthotropic enrichment in the generalized finite element method the procedure is based on the observation that shape functions used in the gfem can be constructed from polynomials defined in any co ordinate system regardless of the underlying mesh or type of element used numerical examples where the solution possesses boundary or internal layers are solved on coarse tetrahedral meshes with isotropic and the proposed p orthotropic enrichment copyright 2002 john wiley sons ltd

on sic povms in prime dimensions the generalized pauli group and its normalizer the clifford group have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete povms sic povms to date almost every known sic povm fiducial vector is an eigenstate of a canonical unitary in the clifford group i show that every canonical unitary in prime dimensions p 3 lies in the same conjugacy class of the clifford group and give a class representative for all such dimensions it follows that if even one such sic povm fiducial vector is an eigenvector of such a unitary then all of them are for a given such dimension i also conjecture that in all dimensions d the number of conjugacy classes is bounded above by 3 and depends only on dmod9 and i support this claim with computer computations in all dimensions 48

transient analysis of sharp thermal gradients using coarse finite element meshes abstract this paper investigates the application of the generalized finite element method with global local enrichments gfemgl to problems of transient heat transfer involving localized features the gfemgl is utilized in order to numerically construct general specially tailored shape functions yielding high levels of accuracy on coarse fem meshes the use of time dependent shape functions requires that the system of equations be discretized temporally first and then spatially in order to properly account for the time dependency the standard method is used for the time integration scheme the transient three dimensional gfemgl is then applied to a laser heating example in order to demonstrate its ability to resolve localized transient features on a fixed coarse mesh convergence analysis of the proposed method as well as applications to heterogeneous materials and moving heat sources are also provided

analysis and improvements of global local enrichments for the generalized finite element method abstract partition of unity methods like the gfem rely on closed form analytical enrichment functions however only limited a priori knowledge about the solution is available for many problems of engineering relevance control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods this has led to the development of the gfem with global local enrichments gfem gl which uses ideas of the classical global local finite element method to numerically build enrichment functions for the gfem it involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh these local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework this paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems two strategies to control the error of the gfem gl solution due to these boundary conditions are also presented the first one is based on the use of a buffer or over sampling zone in the local problems the second strategy investigated is based on multiple global local iterations several numerical experiments performed on three dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the gfem gl solution the problems are selected such that the boundary conditions on the local problems range from smooth to singular functions

locally adapted tetrahedral meshes using bisection we present an algorithm for the construction of locally adapted conformal tetrahedral meshes the algorithm is based on bisection of tetrahedra a new data structure is introduced which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected we prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity and we bound the amount of additional refinement that is needed to achieve conformity numerical examples of the effectiveness of the algorithm are presented

local adaptive q enrichments and generalized finite elements for transient heat diffusion problems abstract a residual a posteriori error estimate is used in conjunction with a q adaptive procedure for selecting enrichment functions in the modelling of transient heat diffusion problems with the generalized finite element method gfem the error estimate allows to assess local as well as global errors of the gfem solutions and provides a tool to adaptively enrich the approximation space with the proposed adaptive algorithm the total number of degrees of freedom dofs is significantly reduced in comparison to uniform enrichment for a comparable accuracy a reduction of up to 80 in dofs is achieved moreover the error estimate captures the deterioration of the conditioning of the system matrix typically encountered in enrichment based methods and enables the possibility of keeping the condition number within an acceptable limit

time independent hybrid enrichment for finite element solution of transient conduction radiation in diffusive grey media we investigate the effectiveness of the partition of unity finite element method for transient conduction radiation problems in diffusive grey media the governing equations consist of a semi linear transient heat equation for the temperature field and a stationary diffusion approximation to the radiation in grey media the coupled equations are integrated in time using a semi implicit method in the finite element framework we show that for the considered problems a combination of hyperbolic and exponential enrichment functions based on an approximation of the boundary layer leads to improved accuracy compared to the conventional finite element method it is illustrated that this approach can be more efficient than using h adaptivity to increase the accuracy of the finite element method near the boundary walls the performance of the proposed partition of unity method is analyzed on several test examples for transient conduction radiation problems in two space dimensions

a fragmentation process connected to brownian motion let b s s 0 be a standard brownian motion and t 1 its first passage time at level 1 for every t 0 we consider ladder time set t of the brownian motion with drift t b t s b s ts and the decreasing sequence f t f 1 t f 2 t of lengths of the intervals of the random partition of 0 t 1 induced by t the main result of this work is that f t t 0 is a fragmentation process in the sense that for 0 t t f t is obtained from f t by breaking randomly into pieces each component of f t according to a law that only depends on the length of this component and independently of the others we identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by aldous and pitman 3

mixed enrichment for the finite element method in heterogeneous media summary problems of multiple scales of interest or of locally nonsmooth solutions may often involve heterogeneous media these problems are usually very demanding in terms of computations with the conventional finite element method on the other hand different enriched finite element methods such as the partition of unity which proved to be very successful in treating similar problems are developed and studied for homogeneous media in this work we present a new idea to extend the partition of unity finite element method to treat heterogeneous materials the idea is studied in applications to wave scattering and heat transfer problems where significant advantages are noted over the standard finite element method although presented within the partition of unity context the same enrichment idea can also be extended to other enriched methods to deal with heterogeneous materials copyright 2014 john wiley sons ltd

a generalized finite element method for polycrystals with discontinuous grain boundaries summary we present a generalized finite element method for the analysis of polycrystals with explicit treatment of grain boundaries grain boundaries and junctions understood as loci of possible displacement discontinuity are inserted into finite elements by exploiting the partition of unity property of finite element shape functions consequently the finite element mesh does not need to conform to the polycrystal topology the formulation is outlined and a numerical example is presented to demonstrate the potential and accuracy of the approach the proposed methodology can also be used for branched and intersecting cohesive cracks and comparisons are made to a related approach int j numer meth engng 2000 48 1741 copyright 2006 john wiley sons ltd

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

symmetric informationally complete quantum measurements we consider the existence in arbitrary finite dimensions d of a positive operator valued measure povm comprised of d2 rank one operators all of whose operator inner products are equal such a set is called a symmetric informationally complete povm sic povm and is equivalent to a set of d2 equiangular lines in cd sic povms are relevant for quantum state tomography quantum cryptography and foundational issues in quantum mechanics we construct sic povms in dimensions two three and four we further conjecture that a particular kind of group covariant sic povm exists in arbitrary dimensions providing numerical results up to dimension 45 to bolster this claim

special finite element methods for a class of second order elliptic problems with rough coefficients in this paper the approximate solution of a class of second order elliptic equations with rough coefficients is considered problems of the type considered arise in the analysis of unidirectional composites where the coefficients represent the properties of the material several methods for this class of problems are presented and it is shown that they have the same accuracy as usual methods have for problems with smooth coefficients the methods are referred to as special finite element methods because they are of finite element type but employ special shape functions chosen to accurately model the unknown solution

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

an enriched finite element method and level sets for axisymmetric two phase flow with surface tension a finite element method for axisymmetric two phase flow problems is presented the method uses an enriched finite element formulation in which the interface can move arbitrarily through the mesh without remeshing the enrichment is implemented by the extended finite element method x fem which models the discontinuity in the velocity gradient at the interface by a local partition of unity it provides an accurate representation of the velocity field at interfaces on an eulerian grid that is not conformal to the weak discontinuity the interface is represented by a level set which is also used in the construction of the element enrichment surface tension effects are considered and the interface curvature is computed from the level set field the method is demonstrated by several examples copyright 2003 john wiley sons ltd

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

conjugate variables in finite phase plane we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane which form sets of conjugate variables in such a plane the coordinate x is quantized with a step c and the momentum p with a step where mc is the size of the phase plane in the x direction the construction depends crucially on the possibility of writing m m1m2 with m1 and m2 relatively prime the conjugate variables are applied to harper like hamiltonians it is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels say m1 each of them having a prescribed degeneracy m2

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

feynman s path integral and mutually unbiased bases our previous work on quantum mechanics in hilbert spaces of finite dimension n is applied to elucidate the deep meaning of feynman s path integral pointed out by g svetlichny he speculated that the secret of the feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases we confirm the corresponding property of the short time propagator by using a specially devised n n approximation of quantum mechanics in applied to our finite dimensional analogue of a free quantum particle

parametric enrichment adaptivity by the extended finite element method an adaptive method within the extended finite element method xfem framework which adapts the enrichment function locally to the physics of a problem as opposed to polynomial or mesh refinement is presented the method minimizes a local residual and determines the parameters of the enrichment function we consider an energy form and a strong form of the residual as error measures to drive the algorithm numerical examples for boundary layers and solid mechanics problems illustrate that the procedure converges moreover when only the character of the solution is known a good approximation is obtained in the area of interest it is also shown that the method can be used to determine the order of singularities in solutions copyright 2007 john wiley sons ltd

factorizations and physical representations a hilbert space in m dimensions is shown explicitly to accommodate representations that reflect the decomposition of m into prime numbers representations that exhibit the factorization of m into two relatively prime numbers the kq representation zak j 1970 phys today 23 51 and related representations termed q1q2 representations together with their conjugates are analysed as well as a representation that exhibits the complete factorization of m in this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up m

well posedness of smoluchowski s coagulation equation for a class of homogeneous kernels the uniqueness and existence of measure valued solutions to smoluchowski s coagulation equation are considered for a class of homogeneous kernels denoting by 2 0 the degree of homogeneity of the coagulation kernel a measure valued solutions are shown to be unique under the sole assumption that the moment of order of the initial datum is finite a similar result was already available for the kernels a x y 2 x y and xy and is extended here to a much wider class of kernels by a different approach the uniqueness result presented herein also seems to improve previous results for several explicit kernels furthermore a comparison principle and a contraction property are obtained for the constant kernel

a generalized finite element method for the simulation of three dimensional dynamic crack propagation this paper is aimed at presenting a partition of unity method for the simulation of three dimensional dynamic crack propagation the method is a variation of the partition of unity finite element method and hp cloud method in the context of crack simulation this method allows for modeling of arbitrary dynamic crack propagation without any remeshing of the domain in the proposed method the approximation spaces are constructed using a partition of unity pu and local enrichment functions the pu is provided by a combination of shepard and finite element partitions of unity this combination of pus allows the inclusion of arbitrary crack geometry in a model without any modification of the initial discretization it also avoids the problems associated with the integration of moving least squares or conventional shepard partitions of unity used in several meshless methods the local enrichment functions can be polynomials or customized functions these functions can efficiently approximate the singular fields around crack fronts the crack propagation is modeled by modifying the partition of unity along the crack surface and does not require continuous remeshings or mappings of solutions between consecutive meshes as the crack propagates in contrast with the boundary element method the proposed method can be applied to any class of problems solvable by the classical finite element method in addition the proposed method can be implemented into most finite element data bases several numerical examples demonstrating the main features and computational efficiency of the proposed method for dynamic crack propagation are presented

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

the augmented multiplicative coalescent bounded size rules and critical dynamics of random graphs random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component one of the standard models are the achlioptas random graph processes on a fixed set of n vertices here at each step one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion an important class of such rules are the bounded size rules where for a fixed k ge 1 all components of size greater than k are treated equally while a great deal of work has gone into analyzing the subcritical and supercritical regimes the nature of the critical scaling window the size and complexity deviation from trees of the components in the critical regime and nature of the merging dynamics has not been well understood in this work we study such questions for general bounded size rules our first main contribution is the construction of an extension of aldous s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components we show that this process referred to as the standard augmented multiplicative coalescent amc is nearly feller with a suitable topology on the state space our second main result proves the convergence of suitably scaled component size and surplus vector for any bounded size rule to the standard amc this result is new even for the classical erd s renyi setting the key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from bhamidi et al the barely subcritical regime arxiv preprint 2012 on the size of the largest component in the barely subcritical regime

a two scale generalized finite element approach for modeling localized thermoplasticity summary predicting localized nonlinear thermoplastic behavior and residual stresses and deformations in structures subjected to intense heating is a prevalent challenge in a range of modern engineering applications the authors present a generalized finite element method targeted at this class of problems involving the solution of intrinsically parallelizable local boundary value problems to capture localized time dependent thermo elasto plastic behavior which is embedded in the coarse structural scale approximation via enrichment functions the method accommodates approximation spaces that evolve in between time or load steps while maintaining a fixed global mesh which avoids the need to map solutions and state variables on changing meshes typical of traditional adaptive approaches representative three dimensional examples exhibiting localized transient nonlinear thermal and thermomechanical effects are presented to demonstrate the advantages of the method with respect to available approaches especially in terms of its flexibility and potential for realistic future applications in this area parallelism of the approach is also discussed copyright 2016 john wiley sons ltd

generalized finite element analysis of three dimensional heat transfer problems exhibiting sharp thermal gradients abstract in this paper heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods the effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated convergence studies show that low order linear and quadratic elements require strongly refined meshes for acceptable accuracy we propose a generalized fem with global local enrichments for the class of problems investigated in the paper in this procedure a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized fem with solutions of local boundary value problems the local problems are defined using the same procedure as in the global local fem where boundary conditions are provided by a coarse scale global solution coarse uniform global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence the use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem the two way information transfer provided by the proposed generalized fem is appealing to several classes of problems especially those involving multiple spatial scales the proposed methodology brings the benefits of generalized fem to problems where limited or no information about the solution is known a priori

solution of nearly incompressible field problems using a generalized finite element approach abstract it is well known that for problems approaching incompressible limits standard finite element approaches suffer from suboptimal convergence due to a phenomenon known as locking we analyze this phenomenon and present a robust generalized finite element formulation that enables optimal solution convergence this approach is explored for the two dimensional incompressible elasticity equations with a variable poisson s ratio as well as the two dimensional lid driven cavity stokes flow problem using a penalty pressure formulation

a partition of unity fem for time dependent diffusion problems using multiple enrichment functions an enriched partition of unity fem is developed to solve time dependent diffusion problems in the present formulation multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space the resulting enrichment is in the form of a local asymptotic expansion unlike previous works in this area where the enrichment must be updated at each time step here the temporal decay in the solution is embedded in the asymptotic expansion thus the system matrix that is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right hand side of the linear system of equations the advantage is a significant saving in the computational effort where previously the linear system must be reevaluated and resolved at every time step in comparison with the traditional finite element analysis with p version refinements the present approach is much simpler more efficient and yields more accurate solutions for a prescribed number of dofs numerical results are presented for a transient diffusion equation with known analytical solution the performance of the method is analyzed on two applications the transient heat equation with a single source and the transient heat equation with multiple sources the aim of such a method compared with the classical fem is to solve time dependent diffusion applications efficiently and with an appropriate level of accuracy

the standard additive coalescent regard an element of the set x 1 x 2 x 1 x 2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent of evans and pitman is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j they showed that a version x t t of this process arises as a n weak limit of the process started at time 1 2 log n with n clusters of mass 1 n we show this standard additive coalescent may be constructed from the continuum random tree of aldous by poisson splitting along the skeleton of the tree we describe the distribution of x t on at a fixed time t we show that the size of the cluster containing a given atom as a process in t has a simple representation in terms of the stable subordinator of index 1 2 as t we establish a gaussian limit for centered and normalized cluster sizes and study the size of the largest cluster

time dependent shape functions for modeling highly transient geothermal systems this paper presents new time dependent finite element shape functions suitable for modeling high gradient transient conductive heat flow in geothermal systems the shape functions are made adaptive by enhancing the approximation functions with time dependent variables which may vary according to the transient process without adding extra degrees of freedom or applying mesh adaptation two different approaches are presented first an iterative method is proposed in which an exponential approximation function which is optimized continually during the transient process is incorporated in the shape function second an analytical method is suggested in which an analytical solution of a simplified process is incorporated in the shape function enabling an explicit update of the shape functions in each time step a methodology for modeling the variation of temperature in one and two dimensions is introduced the ability of the method to capture high gradient temperature profiles using relatively large elements is illustrated with numerical examples of cases in which equally large standard finite elements fail copyright 2008 john wiley sons ltd

on time integration in the xfem the extended finite element method xfem is often used in applications that involve moving interfaces examples are the propagation of cracks or the movement of interfaces in two phase problems this work focuses on time integration in the xfem the performance of the discontinuous galerkin method in time space time finite elements fes and time stepping schemes are analyzed by convergence studies for different model problems it is shown that space time fe achieve optimal convergence rates special care is required for time stepping in the xfem due to the time dependence of the enrichment functions in each time step the enrichment functions have to be evaluated at different time levels this has important consequences in the quadrature used for the integration of the weak form a time stepping scheme that leads to optimal or only slightly sub optimal convergence rates is systematically constructed in this work copyright 2009 john wiley sons ltd

inhomogeneous continuum random trees and the entrance boundary of the additive coalescent regard an element of the set of ranked discrete distributions x1 x2 x1 x2 0 i x i 1 as a fragmentation of unit mass into clusters of masses x i the additive coalescent is the valued markov process in which pairs of clusters of masses x i x j merge into a cluster of mass x i x j at rate x i x j aldous and pitman 1998 showed that a version of this process starting from time with infinitesimally small clusters can be constructed from the brownian continuum random tree of aldous 1991 1993 by poisson splitting along the skeleton of the tree in this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees

global local rom for the solution of parabolic problems with highly concentrated moving sources abstract problems characterised by steep moving gradients are challenging for any numerical technique and even more for the successful formulation of reduced order models roms the aim of this work is to study the numerical solution of problems with steep moving gradients by placing the focus on parabolic problems with highly concentrated moving sources more specifically a global local scheme well suited for reduction methods is formulated with this global localscheme the local nature of the steep moving gradients is exploited by modelling the neighbourhood of the heat source with a moving local domain this domain is coupled to the global domain without requiring any re mesh and preserving the meshes of both domains during the whole simulation then a rom based on the proper orthogonal decomposition pod technique is developed for the moving local domain the proposed technique establishes a valid approach for tackling the non separability of the space and time dimensions of these problems in order to assess the numerical performance of the proposed numerical techniques several evaluation tests are performed

an enriched finite element model with q refinement for radiative boundary layers in glass cooling radiative cooling in glass manufacturing is simulated using the partition of unity finite element method the governing equations consist of a semi linear transient heat equation for the temperature field and a stationary simplified p 1 approximation for the radiation in non grey semitransparent media to integrate the coupled equations in time we consider a linearly implicit scheme in the finite element framework a class of hyperbolic enrichment functions is proposed to resolve boundary layers near the enclosure walls using an industrial electromagnetic spectrum the proposed method shows an immense reduction in the number of degrees of freedom required to achieve a certain accuracy compared to the conventional h version finite element method furthermore the method shows a stable behaviour in treating the boundary layers which is shown by studying the solution close to the domain boundaries the time integration choice is essential to implement a q refinement procedure introduced in the current study the enrichment is refined with respect to the steepness of the solution gradient near the domain boundary in the first few time steps and is shown to lead to a further significant reduction on top of what is already achieved with the enrichment the performance of the proposed method is analysed for glass annealing in two enclosures where the simplified p 1 approximation solution with the partition of unity method the conventional finite element method and the finite difference method are compared to each other and to the full radiative heat transfer as well as the canonical rosseland model

factorization in finite quantum systems unitary transformations in an angular momentum hilbert space h 2j 1 are considered they are expressed as a finite sum of the displacement operators which play the role of su 2j 1 generators with the weyl function as coefficients the chinese remainder theorem is used to factorize large qudits in the hilbert space h 2j 1 in terms of smaller qudits in hilbert spaces h 2ji 1 all unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

hjelmslev geometry of mutually unbiased bases the basic combinatorial properties of a complete set of mutually unbiased bases mubs of a q dimensional hilbert space with p being a prime and r a positive integer are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective hjelmslev plane defined over a galois ring of characteristic p2 and rank r the q vectors of a basis of correspond to the q points of a so called neighbour class and the q 1 mubs answer to the total number of pairwise disjoint neighbour classes on the conic

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

brownian excursions critical random graphs and the multiplicative coalescent let b t s 0 s be reflecting inhomogeneous brownian motion with drift t s at time s started with b t 0 0 consider the random graph script g sign n n 1 tn 4 3 whose largest components have size of order n 2 3 normalizing by n 2 3 the asymptotic joint distribution of component sizes is the same as the joint distribution of excursion lengths of b t corollary 2 the dynamics of merging of components as t increases are abstracted to define the multiplicative coalescent process the states of this process are vectors x of nonnegative real cluster sizes x i and clusters with sizes x i and x j merge at rate x i x j the multiplicative coalescent is shown to be a feller process on l 2 the random graph limit specifies the standard multiplicative coalescent which starts from infinitesimally small clusters at time the existence of such a process is not obvious

projective ring line of an arbitrary single qudit as a continuation of our previous work havlicek and saniga 2007 j phys a math gen 40 f943 52 preprint 0708 4333 an algebraic geometrical study of a single d dimensional qudit is made with d being any positive integer the study is based on an intricate relation between the symplectic module of the generalized pauli group of the qudit and the fine structure of the projective line over the modular ring explicit formulae are given for both the number of generalized pauli operators commuting with a given one and the number of points of the projective line containing the corresponding vector of we find remarkably that a perp set is not a set theoretic union of the corresponding points of the associated projective line unless d is a product of distinct primes the operators are also seen to be structured into disjoint layers according to the degree of their representing vectors a brief comparison with some multiple qudit cases is made

arbitrary discontinuities in space time finite elements by level sets and x fem an enriched finite element method with arbitrary discontinuities in space time is presented the discontinuities are treated by the extended finite element method x fem which uses a local partition of unity enrichment to introduce discontinuities along a moving hyper surface which is described by level sets a space time weak form for conservation laws is developed where the rankine hugoniot jump conditions are natural conditions of the weak form the method is illustrated in the solution of first order hyperbolic equations and applied to linear first order wave and non linear burgers equations by capturing the discontinuity in time as well as space results are improved over capturing the discontinuity in space alone and the method is remarkably accurate implications to standard semi discretization x fem formulations are also discussed copyright 2004 john wiley sons ltd

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

identifying the wavenumber for the inverse helmholtz problem using an enriched finite element formulation abstract we investigate the inverse problem of identifying the wavenumber for the helmholtz equation the problem solution is based on measurements taken at few points from inside the computational domain or on its boundary a novel iterative approach is proposed based on coupling the secant and the descent methods with the partition of unity method starting from an initial guess for the unknown wavenumber the forward problem is solved using the partition of unity method then the secant descent methods are used to improve the initial guess by minimizing a predefined objective function based on the difference between the solution and a set of data points in the next round of iterations the improved wavenumber estimate is used for the forward problem solution and the partition of unity approximation is improved by adding more enrichment functions the iterative process is terminated when the objective function has converged and a set of two predefined tolerances are met to evaluate the estimate accuracy we propose to utilize extra data points to validate the approach and test its efficiency two wave applications with known analytical solutions are studied the results show that the proposed approach can achieve high accuracy for the studied applications even when the considered data is contaminated with noise despite the clear advantages that were previously shown in the literature for solving the forward helmholtz problem this work presents a first attempt to solve the inverse helmholtz problem with an enriched finite element approach

a new encoding of coalescent processes applications to the additive and multiplicative cases we revisit the discrete additive and multiplicative coalescents starting with n particles with unit mass these cases are known to be related to some combinatorial coalescent processes a time reversal of a fragmentation of cayley trees or a parking scheme in the additive case and the random graph process g n p p in the multiplicative case time being fixed encoding these combinatorial objects in real valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as n to infty we propose to use the prim order on the vertices instead of the classical breadth first or depth first traversal to encode the combinatorial coalescent processes in the additive case this yields interesting connections between the different representations of the process in the multiplicative case it allows one to answer to a stronger version of an open question of aldous ann probab vol 25 pp 812 854 1997 we prove that not only the sequence of rescaled masses seen as a process indexed by the time lambda converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a brownian motion with parabolic drift b t lambda t t 2 2 t 0 but we also construct a version of the standard augmented multiplicative coalescent of bhamidi budhiraja and wang probab theory rel to appear using an additional poisson point process

the generalized finite element method an example of its implementation and illustration of its performance the generalized finite element method gfem was introduced in reference 1 as a combination of the standard fem and the partition of unity method the standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data the geometry of the domain the loads and the boundary conditions the special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation the essential boundary conditions can be imposed exactly as in the standard fem adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation this paper gives an example of how the gfem can be developed for the laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the gfem versus the standard fem copyright 2000 john wiley sons ltd

geometrical approach to the discrete wigner function in prime power dimensions we analyse the wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field we separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non uniqueness of the representation of the wigner function explicit expressions for the wigner kernel are given in both cases

the isotropic lines of bb z _ d 2 we show that the isotropic lines in the lattice are the lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice the set of isotropic lines decomposes into orbits under the action of we give an explicit description of those orbits as well as their number and respective cardinalities we also develop two group actions on the group related to the topic

hamiltonians of quantum systems with positions and momenta in gf pl a quantum system with positions and momenta in gf pl is considered such a system can be constructed from l smaller systems in which the positions and momenta take values in zp if the hamiltonian of this l partite system is compatible with gf pl the concept of compatibility of a hamiltonian with gf pl allows the quantum formalism in the l partite system to be expressed in terms of galois arithmetic transformations of the basis in gf pl produce unitary transformations of the quantum states which form a representation of gl l zp they are used to define which subset of the general set of hamiltonians in the l partite system is compatible with gf pl

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

symmetric informationally complete positive operator valued measures and the extended clifford group we describe the structure of the extended clifford group defined to be the group consisting of all operators unitary and antiunitary which normalize the generalized pauli group or weyl heisenberg group as it is often called we also obtain a number of results concerning the structure of the clifford group proper i e the group consisting just of the unitary operators which normalize the generalized pauli group we then investigate the action of the extended clifford group operators on symmetric informationally complete positive operator valued measures or sic povms covariant relative to the action of the generalized pauli group we show that each of the fiducial vectors which has been constructed so far including all the vectors constructed numerically by renes et al is an eigenvector of one of a special class of order 3 clifford unitaries this suggests a strengthening of a conjecture of zauner s we give a complete characterization of the orbits and stability groups in dimensions 2 7 fina

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

bridging multiple structural scales with a generalized finite element method summary this paper presents a generalized fem based on the solution of interdependent coarse scale global and fine scale local problems in order to resolve multiscale effects due to fine scale heterogeneities overall structural behavior is captured by the global problem while local problems focus on the resolution of fine scale solution features in regions where material heterogeneities may govern the structural response fine scale problems are accurately solved in parallel and to address the intrinsic coupling of scales these solutions are embedded into the global solution space using a partition of unity approach this method is demonstrated on representative heat transfer examples in order to examine its accuracy efficiency and flexibility copyright 2014 john wiley sons ltd