coulomb plus power law potentials in quantum mechanics we study the discrete spectrum of the hamiltonian h v r for the coulomb plus power law potential v r 1 r sgn q rq where 0 q 2 and q 0 we show by envelope theory that the discrete eigenvalues enl of h may be approximated by the semiclassical expression enl q minr 0 1 r2 1 r sgn q r q values of and are prescribed which yield upper and lower bounds accurate upper bounds are also obtained by use of a trial function of the form r rl 1 e xr d we give detailed results for v r 1 r rq q 0 5 1 2 for n 1 l 0 1 2 along with comparison eigenvalues found by direct numerical methods

confluent hypergeometric equations and related solvable potentials in quantum mechanics the connection between the schrodinger and confluent hypergeometric equations is discussed it is shown that the factorization of the confluent hypergeometric equation gives a unifying powerful algebraic tool in order to study some quantum mechanical eigenvalue problems that description includes the linear and n dimensional harmonic oscillators as well as the coulomb and morse potentials

eigenvalue bounds for transformations of solvable potentials we study smooth transformations of potentials for which exact bound state solutions of schrodinger s equation are known eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave detailed results are presented for perturbed coulomb potentials of the form and

perfectly secure index coding in this paper we investigate the index coding problem in the presence of an eavesdropper messages are to be sent from one transmitter to a number of legitimate receivers who have side information about the messages and share a set of secret keys with the transmitter to do this the transmitter communicates to the legitimate receivers the public code c which is also heard by the eavesdropper we assume perfect secrecy meaning that the eavesdropper should not be able to retrieve any information about the message set from the public communication we study the minimum key lengths for zero error and perfectly secure index coding problem on one hand this problem is a generalization of the index coding problem and thus a difficult one on the other hand it is a generalization of the shannon s cipher system we show that a generalization of shannon s one time pad strategy is optimal up to a multiplicative constant meaning that it obtains the entire boundary of the cone formed by looking at the secure rate region from the origin this shows the optimality of the generalized one time pad for minimizing the consumption of shared secret keys per message bits when public communication is free the transmitter is not charged for the rate of the public communication finally we consider relaxation of the perfect secrecy and zero error constraints to weak secrecy and asymptotically vanishing probability of error and provide a secure version of the result obtained by langberg and effros on the equivalence of zero error and epsilon error regions in the conventional index coding problem

on block diagonal and schur complement preconditioning we study symmetric positive definite linear systems with a 2 by 2 block matrix preconditioned by inverting directly one of the diagonal blocks and suitably preconditioning the other using an approximate version of young s property a we show that the condition number of the schur complement is smaller than the condition number obtained by the block diagonal preconditioning we also get bounds on both condition numbers from a strengthened cauchy inequality for systems arising from the finite element method the bounds do not depend on the number of elements and can be obtained from element by element computations the results are applied to thep version finite element method where the first block of variables consists of degrees of freedom of a low order

approximating the independence number via the j function we describe an approximation algorithm for the independence number of a graph if a graph onn vertices has an independence numbern k m for some fixed integerk 3 and somem 0 the algorithm finds in random polynomial time an independent set of size tilde omega m 3 mathord left vphantom 3 k 1 right kern nulldelimiterspace k 1 improving the best known previous algorithm of boppana and halldorsson that finds an independent set of size m1 k 1 in such a graph the algorithm is based on semi definite programming some properties of the lovasz function of a graph and the recent algorithm of karger motwani and sudan for approximating the chromatic number of a graph if the function of ann vertex graph is at leastmn1 2 k for some absolute constantm we describe another related efficient algorithm that finds an independent set of sizek several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between the function and the independence number of a graph onn vertices 1998 the mathematical programming society inc published by elsevier science b v

envelope theory in spectral geometry it is shown that the discrete spectrum of schrodinger hamiltonians of the form h vf may be represented by the semiclassical expression enl minr 0 k f nl r vf r the k functions are found to be invariant with respect to coupling and shifts k af b k f for pure power laws f r sgn q rq and the log potential they are also invariant with respect to scale and have the simple forms pnl q r 2 and lnl r 2 respectively k functions are also derived for sech squared and hulthen potentials if f g h where g is a smooth transformation then the envelope approximation is expressed in terms of k by the relation k f k h when the transformation g has definite convexity then the approximation immediately yields eigenvalue bounds for all n and l the theory is used to prove the log power theorem lnl pnl 0 which in turn generates a simple eigenvalue formula for the log potential

necessary and sufficient conditions for existence of bound states in a central potential we obtain using the birman schwinger method a series of necessary conditions for the existence of at least one bound state applicable to arbitrary central potentials in the context of nonrelativistic quantum mechanics these conditions yield a monotonic series of lower limits on the critical value of the strength of the potential for which a first bound state appears which converges to the exact critical strength we also obtain a sufficient condition for the existence of bound states in a central monotonic potential which yield an upper limit on the critical strength of the potential

remarks on turschner s eigenvalue formula compares turschner s approximation to the eigenvalues of hamiltonians del 2 c mod x mod nu with previously published numerical results the authors examine evidence that turschner s approximation yields a lower bound to the ground state energy both for boson and fermion systems they discuss the convergence of turschner s approximation to the exact energies en nu as n to infinity and derive a simple expression for turschner s approximate eigenvalues in terms of the hypergeometric function

the contraction number of a multigrid method for solving the poisson equation the treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence for the linear equations which arise from the discretization of the poisson equation a convergence factor 0 5 is established independent of the shape of the domain and of the regularity of the solution

any l state solutions of the hulthen potential in arbitrary dimensions the energy spectra of the hulthen effective potential in d dimensions are obtained within the new quantization rule approach for any l states the interdimensional degeneracies among states are also presented in the case of d 3 the agreement between our results and those obtained by other methods is excellent

on two ways of stabilizing the hierarchical basis multilevel methods a survey of two approaches for stabilizing the hierarchical basis hb multilevel preconditioners both additive and multiplicative is presented the first approach is based on the algebraic extension of the two level methods exploiting recursive calls to coarser discretization levels these recursive calls can be viewed as inner iterations at a given discretization level exploiting the already defined preconditioner at coarser levels in a polynomially based inner iteration method the latter gives rise to hybrid type multilevel cycles this is the so called hybrid algebraic multilevel iteration amli method the second approach is based on a different direct multilevel splitting of the finite element discretization space this gives rise to the so called wavelet multilevel decomposition based on l 2 projections which in practice must be approximated both approaches the amli one and the one based on approximate wavelet decompositions lead to optimal relative condition numbers of the multilevel preconditioners

bounds on schrodinger eigenvalues for polynomial potentials in n dimensions if a single particle obeys nonrelativistic qm in rn and has the hamiltonian h q 0 a q rq a q 0 then the lowest eigenvalue e is given approximately by the semiclassical expression e minr 0 1 r2 q 0 a q p q n r q it is proved that this formula yields a lower bound when p q n ne 2 1 2 n qe 1 q 1 n 2 1 n q 1 n and an upper bound when p q n n 2 1 2 n q 2 n 2 1 q an extension is made to allow for a coulomb term when n 1 the general formula is applied to the examples v r r r2 r3 and v r r2 r4 r6 in dimensions 1 to 10 and the results are compared to accurate eigenvalues obtained numerically

lp bounds for rate distortion with variable side information we consider a rate distortion problem with side information at multiple decoders several upper and lower bounds have been proposed for this general problem or special cases of it we provide an upper bound for general instances of this problem which takes the form of a linear program by utilizing random binning and simultaneous decoding techniques 1 and compare it with the existing bounds we also provide a lower bound for the general problem which was inspired by a linear programming lower bound for index coding and show that it subsumes most of the lower bounds in literature using these upper and lower bounds we explicitly characterize the rate distortion function of a problem that can be seen as a gaussian analogue of the odd cycle index coding problem

variational analysis for a generalized spiked harmonic oscillator a variational analysis is presented for the generalized spiked harmonic oscillator hamiltonian operator d2 dx 2 bx 2 a x 2 x where is a real positive parameter the formalism makes use of a basis provided by exact solutions of schrodinger s equation for the gol dman and krivchenkov hamiltonian and the corresponding matrix elements that were previously found for all the discrete eigenvalues the method provides bounds which improve as the dimension d of the basis set is increased extension to the n dimensional case in arbitrary angular momentum subspaces is also presented by minimizing over the free parameter a we are able to reduce substantially the number of basis functions needed for a given accuracy

eigenvalues of schrodinger operators with definite and indefinite weights in this paper computational algorithms are presented to compute the eigenvalues of schrodinger operators with definite and indefinite weights the algorithms are based on titchmarsh weyl s theory and can be applied to solve a wide class of problems in quantum mechanics explicit examples are presented some of which recover the known exact solutions finally some analytical bounds have been reviewed for definite and indefinite singular problems weyl s spectral bisection algorithm has been proposed to solve definite problems as algebraic problems open problems have been presented based on the results of the presented algorithms

the perturbation of some exactly soluble problems in wave mechanics by the method of potential envelopes suppose the function epsilon k nu which represents the kth bound state eigenvalue of the hamiltonian h delta nu phi r u r is known exactly for all allowed values of nu or 0 the authors present the corresponding eigenvalue ek nu of the hamiltonian h delta nu f phi r u r where f phi is a smooth increasing and either convex or concave transformation of the potential phi r an application of the method of potential envelopes yields a simple formula for an upper or lower bound to ek nu according to whether the transformation f phi is concave or convex the example phi r r 1 omega r u r omega 2r2 and f phi lambda 1 elambda phi 1 for lambda 0 is discussed in detail

fourier analysis of a robust multigrid method for convection diffusion equations we consider a two grid method for solving 2d convection diffusion problems the coarse grid correction is based on approximation of the schur complement as a preconditioner of the schur complement we use the exact schur complement of modified fine grid equations we assume constant coefficients and periodic boundary conditions and apply fourier analysis we prove an upper bound for the spectral radius of the two grid iteration matrix that is smaller than one and independent of the mesh size the convection diffusion ratio and the flow direction i e we have a strong robustness result numerical results illustrating the robustness of the corresponding multigrid w cycle are given

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

networks matroids and non shannon information inequalities we define a class of networks called matroidal networks which includes as special cases all scalar linearly solvable networks and in particular solvable multicast networks we then present a method for constructing matroidal networks from known matroids we specifically construct networks that play an important role in proving results in the literature such as the insufficiency of linear network coding and the unachievability of network coding capacity we also construct a new network from the vamos matroid which we call the vamos network and use it to prove that shannon type information inequalities are in general not sufficient for computing network coding capacities to accomplish this we obtain a capacity upper bound for the vamos network using a non shannon type information inequality discovered in 1998 by zhang and yeung and then show that it is smaller than any such bound derived from shannon type information inequalities this is the first application of a non shannon type inequality to network coding we also compute the exact routing capacity and linear coding capacity of the vamos network finally using a variation of the vamos network we prove that shannon type information inequalities are insufficient even for computing network coding capacities of multiple unicast networks

a polynomial time algorithm for pliable index coding in pliable index coding we consider a server with m messages and n clients where each client has as side information a subset of the messages we seek to minimize the number of broadcast transmissions so that each client can recover any one unknown message she does not already have previous work has shown that the pliable index coding problem is np hard and requires at most mathcal o log 2 n broadcast transmissions which indicates exponential savings over the conventional index coding that requires in the worst case mathcal o n transmissions in this paper building on a decoding criterion that we propose we first design a deterministic polynomial time algorithm that can realize the exponential benefits by achieving in the worst case a performance upper bounded by mathcal o log 2 n broadcast transmissions we extend our algorithm to the t requests case where each client requires t unknown messages that she does not have and show that our algorithm requires at most mathcal o t log n log 2 n broadcast transmissions we construct lower bound instances that require at least omega log n transmissions for linear pliable index coding and at least omega t log n transmissions for the t requests case indicating that both our upper and lower bounds are polynomials of log n and differ within a factor of mathcal o log n we provide a probabilistic analysis over random instances and show that the required number of transmissions is almost surely theta log n as compared with the theta n log n for index coding in addition we show that these upper and lower bounds also hold for vector pliable index coding in the worst case instances and the random graph instances implying that vector coding does not provide benefits in terms of these bounds our numerical experiments show that our algorithm outperforms existing algorithms for pliable index coding by up to 50 less transmissions

constructive inversion of energy trajectories in quantum mechanics we suppose that the ground state eigenvalue e f v of the schrodinger hamiltonian h vf x in one dimension is known for all values of the coupling v 0 the potential shape f x is assumed to be symmetric bounded below and monotone increasing for x 0 a fast algorithm is devised which allows the potential shape f x to be reconstructed from the energy trajectory f v three examples are discussed in detail a shifted power potential the exponential potential and the sech squared potential are each reconstructed from their known exact energy trajectories

a rate distortion approach to index coding we approach index coding as a special case of rate distortion with multiple receivers each with some side information about the source specifically using techniques developed for the rate distortion problem we provide two upper bounds and one lower bound on the optimal index coding rate the upper bounds involve specific choices of the auxiliary random variables in the best existing scheme for the rate distortion problem the lower bound is based on a new lower bound for the general rate distortion problem the bounds are shown to coincide for a number of groupcast index coding instances including all instances for which the number of decoders does not exceed three

interlinked cycles for index coding generalizing cycles and cliques we consider a graphical approach to index coding as cycles have been shown to provide coding gain cycles and cliques a specific type of overlapping cycles have been exploited in an existing literature in this paper we define a more general form of overlapping cycles called the interlinked cycle mathsf ic structure that generalizes cycles and cliques we propose a scheme called the interlinked cycle cover mathsf icc scheme that leverages mathsf ic structures in digraphs to construct scalar linear index codes we characterize a class of infinitely many digraphs where our proposed scheme is optimal over all linear and non linear index codes consequently for this class of digraphs we indirectly prove that scalar linear index codes are optimal furthermore we show that the mathsf icc scheme can outperform all the existing graph based schemes including partial clique cover and fractional local chromatic number schemes and a random coding scheme namely composite coding for certain graphs

broadcasting with side information bounding and approximating the broadcast rate index coding has received considerable attention recently motivated in part by applications such as fast video on demand and efficient communication in wireless networks and in part by its connection to network coding optimal encoding schemes and efficient heuristics were studied in various settings while also leading to new results for network coding such as improved gaps between linear and non linear capacity as well as hardness of approximation the problem of broadcasting with side information a generalization of the index coding problem begins with a sender and sets of users and messages each user possesses a subset of the messages and desires an additional message from the set the sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message the fundamental parameter of interest is the broadcast rate the average communication cost for sufficiently long broadcasts though there have been many new nontrivial bounds on by bar yossef 2006 lubetzky and stav 2007 alon 2008 and blasiak 2011 there was no known polynomial time algorithm for approximating within a nontrivial factor and the exact value of remained unknown for all nontrivial instances using the information theoretic linear program introduced in blasiak 2011 we give a polynomial time algorithm for recognizing instances with 2 and pinpoint precisely for various classes of graphs e g various cayley graphs of cyclic groups further extending ideas from ramsey theory we give a polynomial time algorithm with a nontrivial approximation ratio for computing finally we provide insight into the quality of previous bounds by giving constructions showing separations between and the respective bounds in particular we construct graphs where is uniformly bounded while its upper bound derived from the naive encoding scheme is polynomially worse

is there a stable hydrogen atom in higher dimensions the schrodinger equation in higher dimensions is considered it consists of the kinetic energy part given by the corresponding laplace operator and a term describing the interaction with the electrostatic field of a point charge from rutherford type scattering experiments one can conclude that the potential of a point charge is 1 r irrespective of the dimension of the space where the experiment is carried through also the structure of the kinetic energy is unchanged in higher dimensions so that one is lead to the result that there exist stable atoms in higher spatial dimensions d 4 the solutions and energy eigenvalues to this schrodinger equation in higher dimensions are presented as a consequence the dimensionality of space can be read off from the spectral scheme of atoms the three dimensionality of space is a consequence of the existence of the lyman series another consequence is that the maxwell equations in higher dimensions must be modified in order to have the 1 r potential as solution for

a basic norm equivalence for the theory of multilevel methods subspace decompositions of finite element spaces based onl 2 like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods recently several authors have proved the equivalence of the associated discrete norms with theh 1 norm the present paper gives an elementary self contained derivation of this result which is based on the use ofk functionals known from the theory of interpolation spaces

optimal order nonnested multigrid methods for solving finite element equations iii on degenerate meshes in this paper we consider several model problems where finite element triangular meshes with arbitrarily small angles high aspect ratios are utilized to deal with anisotropy interfaces or singular perturbations the constant rate independent of the number of unknowns the smallest angle the interface discontinuity the singular perturbation parameter etc convergence of some special nonnested multigrid methods for solving the finite element systems on such degenerate meshes will be proved numerical data are provided to support the analysis in each case

nonsingular spiked harmonic oscillator a perturbative study of a class of nonsingular spiked harmonic oscillators defined by the hamiltonian h d2 dr 2 r 2 r in the domain 0 is carried out in the two extremes of a weak coupling and a strong coupling regimes a path has been found to connect both expansions for near 2

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

the contraction number of a multigrid method with mesh ratio 2 for solving poisson s equation abstract the convergence rate of a multigrid method for the solution of poisson s equation on a uniform grid is estimated in contrast to recent results of braess no intermediate grids are used refined estimates of gauss seidel relaxation by weak norms a strengthened cauchy inequality and a duality argument are central we obtain 0 273 as an upper bound for the contraction number of the two level procedure the results hold for arbitrary convex polygonal regions and are independent of the smoothness of the solution

the convergence rate of a multigrid method with gauss seidel relaxation for the poisson equation the numerical solution of the poisson equation is treated by a multigrid method for a uniform grid the convergence rate can be estimated even for the iteration with a v cycle independently of the shape of the domain as long as it is convex and polygonal the smoothing effect of the gaus seidel relaxation is described by a discrete seminorm which is weaker than the energy norm

generalized comparison theorems in quantum mechanics this paper is concerned with the discrete spectra of schrodinger operators h v where v r is an attractive potential in n spatial dimensions two principal results are reported for the bottom of the spectrum of h in each angular momentum subspace l i an optimized lower bound when the potential is a sum of terms v r v 1 r v 2 r and the bottoms of the spectra of v 1 r and v 2 r in l are known and ii a generalized comparison theorem which predicts spectral ordering when the graphs of the comparison potentials v 1 r and v 2 r intersect in a controlled way pure power law potentials are studied and an application of the results to the coulomb plus linear potential v r a r br is presented in detail for this problem an earlier formula for energy bounds is sharpened and generalized to n dimensions

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

convergence of multigrid methods for nonsymmetric indefinite problems a convergence theory is developed for multigrid methods for nonsymmetric indefinite problems in a variational setting we prove fast convergence with any bounded positive number of smoothing steps for v and w cycles under discrete analogues of the h 1 a regularity assumption the lower order nonsymmetric terms are treated as perturbations in addition we analyze a wide class of smoothers and get estimates of contraction numbers for multigrid methods with these smoothers

eigenvalue bounds for a class of singular potentials in n dimensions the eigenvalue bounds obtained earlier hall r l and saad n 1998 j phys a math gen 31 963 for smooth transformations of the form are extended to n dimensions in particular a simple formula is derived which bounds the eigenvalues for the spiked harmonic oscillator potential and is valid for all discrete eigenvalues arbitrary angular momentum l and spatial dimension n

a multigrid method for three dimensional elasticity and algebraic convergence estimates a two level iterative method for 3d elasticity problems discretized by quadratic elements is proposed the coarse level consists of linear elements we develop a sharpened algebraic convergence theory for two level methods with smoothing by gauss seidel which applies to more general situations as well the convergence estimates do not deteriorate for strongly discontinuous coefficients observed convergence rates are compared with theoretical estimates the method is competitive with a standard direct solver

ground state energy bounds for potentials x ν a theory is developed from which both upper and lower analytic bounds on schrodinger eigenvalues can be obtained we propose a recursion algorithm with which ground energies for certain potentials can be rigorously bounded to arbitrary precision these analytic and numerical methods together with existing techniques are applied to the ground state problem for power potentials x 0

spiked oscillators exact solution a procedure to obtain the eigenenergies and eigenfunctions of a quantum spiked oscillator is presented the originality of the method lies in an adequate use of asymptotic expansions of wronskians of algebraic solutions of the schrodinger equation the procedure is applied to three familiar examples of spiked oscillators

eigenvalue bounds for a class of singular potentials we study smooth transformations of the solvable potentials eigenvalue approximation formulae are obtained which provide lower or upper energy bounds for all the discrete energy eigenvalues accordingly as the transformation functions g and f are both convex or both concave detailed results are presented for the special case of two term singular potentials of the form and also for the potentials and for

an algebraic study of a local multigrid method for variational problems a two level iterative method for solving systems of equations arising from local refinement of finite element meshes is proposed the smoothing steps consists of gauss seidel relaxations applied only to the local mesh a direct solver is required on the global coarse mesh we analyze this scheme by modifying the standard algebraic convergence theory the estimates of the convergence factor do not require any regularity assumption the estimates can be evaluated element by element the evaluated estimates are compared with the actual convergence factor and with another bound in several numerical examples the theory also applies to a method with a direct solver on the local fine mesh in this case we have obtained very sharp and in fact optimal estimates of the convergence factors

the yukawa and hulthen potentials in quantum mechanics the concept of kinetic potentials is used to construct a global geometrical approximation theory for the spectra of schrodinger operators h delta vy in which the potential shape y is either i a transformation y r g h r of a soluble potential h r or ii a continuous mixture y r integral rho t h rt dt the case in which y is the yukawa potential and h er 1 1 is the hulthen potential is discussed in detail simple formulae are derived for eigenvalue bounds which are compared to accurate data obtained by the direct numerical integration of schrodinger s equation

convergence of algebraic multigrid based on smoothed aggregation we prove aconvergence estimate for the algebraic multigrid method with prolongations defined by aggregation using zero energy modes followed by a smoothing the method input is the problem matrix and a matrix of the zero energy modes the estimate depends only polylogarithmically on the mesh size and requires only a weak approximation property for the aggregates which can be a priori verified computationally construction of the prolongator in the case of a general second order system is described and the assumptions of the theorem are verified for a scalar problem discretized by linear conforming finite elements

a geometrical theory of energy trajectories in quantum mechanics suppose f r is an attractive central potential of the form f r ki 1 g i f i r where f i is a set of basis potentials powers log hulthen sech2 and g i is a set of smooth increasing transformations which for a given f are either all convex or all concave formulas are derived for bounds on the energy trajectories enl fnl v of the hamiltonian h vf r where v is a coupling constant the transform f f is carried out in two steps f f f where f s is called the kinetic potential of f and is defined by f s inf f subject to d l2 r3 where d is the domain of h 1 and s a table is presented of the basis kinetic potentials f i s the general trajectory bounds f v are then shown to be given by a legendre transformation of the form s f s v f v where f s ki 1g i f i s and f v mins 0 s v f s with the aid of this potential construction set a kind of schrodinger lego ground state trajectory bounds are derived for a vari

kinetic potentials in quantum mechanics suppose that the hamiltonian h vf r represents the energy of a particle which moves in an attractive central potential and obeys nonrelativistic quantum mechanics the discrete eigenvalues enl fnl v of h may be expressed as a legendre transformation fnl v mins 0 s vfnl s n 1 2 3 l 0 1 2 where the kinetic potentials fnl s associated with f r are defined by fnl s infdnl sup dnl 1 r f s 1 2r r d3r and dnl is an n dimensional subspace of l2 r3 labeled by ylm m 0 and contained in the domain d h of h if the potential has the form f r ni 1 g i f i r then in many interesting cases it turns out that the corresponding kinetic potentials can be closely approximated by ni 1 g i fnl i s this nice behavior of the kinetic potentials leads to a constructive global approximation theory for schrodinger eigenvalues as an illustration detailed recipes are provided for arbitrary linear combinations of power law potentials and the log potential fo

a note on mgr methods by mgr we denote a class of special highly efficient multigrid methods for solving h discrete elliptic differential equations unlike standard multigrid methods 4 9 18 mgr methods are characterized by intermediate grids between the given h grid and the 2h grid and by special fine to coarse and coarse to fine grid transfer techniques the mgr idea has been conceived 13 7 by the second author in trying to extend the range of applicability of the total reduction method 15 16 to more general problems described in a somewhat different way methods of mgr type have in the meantime also been considered by braess 1 3 and meis 11 the convergence properties of the simplest mgr method mgr 0 are essentially improved if it is combined with one step of checkered gauss seidel iteration mgr ch for the model problem of poisson s equation in the unit square for example the spectral radius decreases from 12 to 227 by such a modification whereas the computational effort is not essentially enlarged for poisson like equations mgr ch yields the fastest iterative solver known so far other mgr variants are particularly suitable for anisotropic operators

renormalization group solutions for yukawa potential the self similar renormalization group is used to obtain expressions for the spectrum of the hamiltonian with the yukawa potential the critical screening parameter above which there are no bound states is also obtained by this method the approach presented illustrates that one can achieve good accuracy without involving extensive numerical calculations but invoking instead the renormalization group techniques

asymptotic properties of eigenvalues in variational calculations for double well oscillators simple and accurate formulae for the eigenvalues of the hamiltonian with an anharmonic double well potential are presented special attention is paid to the following three regions of coupling parameters where difficulties arise i in the weak coupling limit the eigenvalues depend on the coupling parameter nonanalytically to obtain accurate explicit expressions in this limit a composite asymptotic expansion is applied ii in the strong coupling limit the derivation of asymptotic expansions is obstructed by the nonuniqueness of solutions to variational equations to overcome this obstacle a restricted variational procedure is suggested iii there exist also critical values of the coupling parameter at which eigenvalues cross zero in the regions around these critical values a dramatic increase of the percentage error occurs a remedy against this critical behaviour is the positive definite shift removing the critical regions the formulae obtained for the eigenvalues provide an accuracy within the errors not exceeding 3 for all values of the coupling parameter from zero to infinity

refining the comparison theorem of quantum mechanics if two potentials v1 x and v2 x are ordered v1 v2 then the comparison theorem tells the author that the corresponding schrodinger eigenvalues are ordered e1 e2 he presents some simple conditions which allow him to predict such spectral ordering for the ground state even when the graphs of the potentials cross over as illustrations the truncated quartic oscillator and the yukawa potential are studied by allowing coulomb tangents to cross over the yukawa in various ways he is able to improve earlier energy upper bounds which he had obtained by representing the yukawa potential as a smooth transformation of the coulomb potential and also to augment these results with lower bounds

an inversion inequality for potentials in quantum mechanics we suppose 1 that the ground state eigenvalue e f v of the schrodinger hamiltonian h vf x in one dimension is known for all values of the coupling v 0 and 2 that the potential shape can be expressed in the form f x g x2 where g is monotone increasing and convex the inversion inequality f x f 1 4x2 is established in which the kinetic potential f s is related to the energy function f v by the transformation f s f v s f v vf v as an example f is approximately reconstructed from the energy function f for the potential f x ax2 b c x2

index coding with side information motivated by a problem of transmitting supplemental data over broadcast channels birk and kol infocom 1998 we study the following coding problem a sender communicates with n receivers r1 rn he holds an input x 0 01l n and wishes to broadcast a single message so that each receiver ri can recover the bit xi each ri has prior side information about x induced by a directed graph grain nodes ri knows the bits of a in the positions j i j is an edge of g g is known to the sender and to the receivers we call encoding schemes that achieve this goal indexcodes for 0 1 n with side information graph g in this paper we identify a measure on graphs the minrank which exactly characterizes the minimum length of linear and certain types of nonlinear index codes we show that for natural classes of side information graphs including directed acyclic graphs perfect graphs odd holes and odd anti holes minrank is the optimal length of arbitrary index codes for arbitrary index codes and arbitrary graphs we obtain a lower bound in terms of the size of the maximum acyclic induced subgraph this bound holds even for randomized codes but has been shown not to be tight

improving the performance guarantee for approximate graph coloring the performance guarantee of a graph coloring algorithm is the worst case ratio between the number of colors it uses on the input graph and the chromauc number of this graph the previous best known polynomial time algorithm had a performance guarantee o n logn for graphs on n vertices this result stood unchallenged for eight years this paper presents an efficient algorithm with performance guarantee of o n loglog n 2 logn 2

an optimal order process for solving finite element equations a k level iterative procedure for solving the algebraic equations which arise from the finite element approximation of elliptic boundary value problems is presented and analyzed the work estimate for this procedure is proportional to the number of unknowns an optimal order result general geometry is permitted for the underlying domain but the shape plays a role in the rate of convergence through elliptic regularity finally a short discussion of the use of this method for parabolic problems is presented

mixtures of potentials in quantum mechanics a global geometrical approximation theory for the spectra of schrodinger operators h d2 vf x is discussed the potential f x is composed either of sums or of continuous mixtures of soluble potentials in both cases it is proved that the kinetic potential formalism j math phys 24 324 1983 25 2078 1984 automatically yields optimal energy lower bounds the examples f x x x2 and f x basech2 tx dt are treated in detail

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

nonlinear index coding outperforming the linear optimum the following source coding problem was introduced by birk and kol a sender holds a word x isin 0 1 n and wishes to broadcast a codeword to n receivers rn rn the receiver ri is interested in xi and has prior side information comprising some subset of the n bits this corresponds to a directed graph g on n vertices where i j is an edge ri ri knows the bit xj an index code for g is an encoding scheme which enables each ri to always reconstruct xi given his side information the minimal word length of an index code was studied by bar yossef birk jayram and kol focs 06 they introduced a graph parameter minrk2 g which completely characterizes the length of an optimal linear index code for g they showed that in various cases linear codes attain the optimal word length and conjectured that linear index coding is in fact always optimal in this work we disprove the main conjecture of bar yossef birk jayram and kol in the following strong sense for any epsiv 0 and sufficiently large n there is an n vertex graph g so that every linear index code for g requires codewords of length at least nepsiv and yet a nonlinear index code for g has a word length of ne this is achieved by an explicit construction which extends alon s variant of the celebrated ramsey construction of frankl and wilson in addition we study optimal index codes in various less restricted natural models and prove several related properties of the graph parameter minrk g

index coding capacity how far can one go with only shannon inequalities an interference alignment perspective is used to identify the simplest instances minimum possible number of edges in the alignment graph not more than 2 interfering messages at any destination of index coding problems where non shannon information inequalities are necessary for capacity characterization in particular this includes the first known example of a multiple unicast one destination per message index coding problem where non shannon information inequalities are shown to be necessary the simplest multiple unicast example has 7 edges in the alignment graph and 11 messages the simplest multiple groupcast multiple destinations per message example has 6 edges in the alignment graph 6 messages and 10 receivers for both the simplest multiple unicast and multiple groupcast instances the best outer bound based on only shannon inequalities is 2 5 which is tightened to 11 28 by the use of the zhang yeung non shannon type information inequality and the linear capacity is shown to be 5 13 using the ingleton inequality conversely identifying the minimal challenging aspects of the index coding problem allows an expansion of the class of solved index coding problems up to but not including these instances

optimal finite length and asymptotic index codes for five or fewer receivers index coding models broadcast networks in which a sender sends different messages to different receivers simultaneously where each receiver may know some of the messages a priori the aim is to find the minimum normalized index codelength that the sender sends this paper considers unicast index coding where each receiver requests exactly one message and each message is requested by exactly one receiver each unicast index coding instances can be fully described by a directed graph and vice versa where each vertex corresponds to one receiver for any directed graph representing a unicast index coding instance we show that if a maximum acyclic induced subgraph mais is obtained by removing two or fewer vertices from the graph then the minimum index codelength equals the number of vertices in the mais and linear codes are optimal for the corresponding index coding instance using this result we solved all unicast index coding instances with up to five receivers which correspond to all graphs with up to five vertices for 9818 non isomorphic graphs among all graphs up to five vertices we obtained the minimum index codelength for all message alphabet sizes for the remaining 28 graphs we obtained the minimum index codelength if the message alphabet size is k 2 for any positive integer k this work complements the result by arbabjolfaei et al isit 2013 who solved all unicast index coding instances with up to five receivers in the asymptotic regime where the message alphabet size tends to infinity

the contraction number of a multigrid method with mesh ratio two for solving model problems abstract the algorithm proposed in 6 is analyzed with the aid of model problem analysis the explicit expression for the convergence factor and the exact values of the contraction number are obtained

algebraic multilevel preconditioning methods ii a recursive way of constructing preconditioning matrices for the stiffness matrix in the discretization of selfadjoint second order elliptic boundary value problems is proposed it is based on a sequence of nested finite element spaces with the usual nodal basis functions using a nodeordering corresponding to the nested meshes the finite element stiffness matrix is recursively split up into two level block structures and is factored approximately in such a way that any successive schur complement is replaced approximated by a matrix defined recursively and thereform only implicitely given to solve a system with this matrix we need to perform a fixed number v of iterations on the preceding level using as an iteration matrix the preconditioning matrix already defined on that level it is shown that by a proper choice of iteration parameters it suffices to use v left 1 gamma 2 right tfrac 1 2 iterations for the so constructedv foldv cycle wherev 2 corresponds to aw cycle preconditioning matrices to be spectrally equivalent to the stiffness matrix the conditions involve only the constant in the strengthened c b s inequality for the corresponding two level hierarchical basis function spaces and are therefore independent of the regularity of the solution for instance if we use successive uniform refinements of the meshes the method is of optimal order of computational complexity if gamma 2 tfrac 8 9 under reasonable assumptions of the finite element mesh the condition numbers turn out to be so small that there are in practice few reasons to use an accelerated iterative method like the conjugate gradient method for instance