a generalized semiclassical expression for the eigenvalues of multiple well potentials from the poles of a generalized semiclassical green s function we derive expressions for the eigenvalues of 1d multiple well potentials in the case of asymmetric and symmetric double wells we also obtain analytical formulae for respectively the shift and splitting of energies our results are better than some approximations in the literature because they take more properly into account the tunnelling through the barriers forming the multiple well and depend on energy dependent maslov indices we illustrate the good numerical precision of the method by discussing some case tests on double wells

a polynomial time approximation scheme for the multiple knapsack problem the multiple knapsack problem mkp is a natural and well known generalization of the single knapsack problem and is defined as follows we are given a set of n items and m bins knapsacks such that each item i has a profit p i and a size s i and each bin j has a capacity c j the goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins mkp is a special case of the generalized assignment problem gap where the profit and the size of an item can vary based on the specific bin that it is assigned to gap is apx hard and a 2 approximation for it is implicit in the work of shmoys and tardos math program a 62 1993 pp 461 474 and thus far this was also the best known approximation for mkp the main result of this paper is a polynomial time approximation scheme ptas for mkp apart from its inherent theoretical interest as a common generalization of the well studied knapsack and bin packing problems it appears to be the strongest special case of gap that is not apx hard we substantiate this by showing that slight generalizations of mkp are apx hard thus our results help demarcate the boundary at which instances of gap become apx hard an interesting aspect of our approach is a ptas preserving reduction from an arbitrary instance of mkp to an instance with o log n distinct sizes and profits

new exact solutions for polynomial oscillators in large dimensions a new type of exact solvability is reported the schrodinger equation is considered in a very large spatial dimension d 1 and its central polynomial potential is allowed to depend on many 2q coupling constants in a search for its bound states possessing an exact and elementary wavefunction proportional to a harmonic oscillator like polynomial of a freely varying i e not just small degree n the solvability conditions are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of d q and n assuming that d is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all n and up to q 5 at least

equivalence of unstable anharmonic oscillators and double wells we give a direct and general proof of the equality of the energy levels of a pair of different quantum problems the double well and the unstable anharmonic oscillator defined by complex translation the resonances of the two problems are also equal and are the analytic continuations of the anharmonic oscillator energy levels

non borel summable φ4 theory in zero dimension a toy model for testing numerical and analytical methods the exact analytical solution for the generating functional of the zero dimensional 4 theory with degenerate minima is obtained in the whole complex coupling parameter plane for testing purposes the efficiency and precision of different computing tools proposed in non borel summable field theories to obtain approximate solutions in both perturbative and nonperturbative regimes are analyzed furthermore a new resummation approach is proposed in order to successfully deal with factorially divergent series it provides a representation of the generating function in terms of an unambiguously defined laplace borel integral on the other hand a recent approach called the generalized borel transform is shown to be an accurate and robust technique to capture non perturbative contributions in the coupling parameter an extension of this approach to path integrals is proposed

exponentially small corrections in the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator the asymptotic expansion of the eigenvalues of the cubic anharmonic oscillator is studied in a region of the coupling constant plane in which there is a sequence of exponentially small subseries beyond the standard rayleigh schrodinger perturbation theory rspt power series we give a simple algorithm for the calculation of these subseries to any desired order in terms of the rspt coefficients expressed as polynomials in the quantum number and illustrate our results with numerical borel pade summations of the expansion up to third exponentially small order

anharmonic oscillator discontinuity formulae up to second exponentially small order the eigenvalues of the quartic anharmonic oscillator as functions of the anharmonicity constant satisfy a once subtracted dispersion relation in turn this dispersion relation is driven by the purely imaginary discontinuity of the eigenvalues across the negative real axis in this paper we calculate explicitly the asymptotic expansion of this discontinuity up to second exponentially small order

fragile pt symmetry in a solvable model one of the simplest pseudo hermitian models with real spectrum viz square well on a real interval i of coordinates is re examined a pt symmetric complex deformation c of i is introduced and shown tractable via an innovated approach to matching conditions the result is surprising an arbitrarily small deformation i c implies a sudden collapse i e the spontaneous pt symmetry breaking of virtually all the spectrum i e up to its low energy part

pt symmetric pseudo perturbation recipe an imaginary cubic oscillator with spikes the pseudo perturbative shifted l expansion technique pslet is shown to be applicable in the non hermitian symmetric context the construction of bound states for several symmetric potentials is presented with special attention paid to v r ir3 ir oscillators

double wells perturbation series summable to the eigenvalues and directly computable approximations we give a rigorous proof of the analyticity of the eigenvalues of the double well schrodinger operators and of the associated resonances we specialize the rayleigh schrodinger perturbation theory to such problems obtaining an expression for the complex perturbation series uniquely related to the eigenvalues through a summation method by an approximation we obtain new series expansions directly computable still summable which in the case of the herbst simon model can be given in an explicit form

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

matching method and exact solvability of discrete symmetric square wells discrete symmetric square wells are studied their wavefunctions are found to be proportional to classical tshebyshev polynomials of complex argument the compact secular equations for energies are derived giving the real spectra in certain intervals of non hermiticity strengths z it is amusing to note that although the known square well re emerges in the usual continuum limit a twice as rich upside down symmetric spectrum is exhibited by all its present discretized predecessors

analytical solutions of the fokker planck equation for generalized morse and hulthen potentials in the present contribution we analytically calculate solutions of the transition probability of the fokker planck equation fpe for both the generalized morse potential and the hulthen potential the method is based on the formal analogy of the fpe with the schrodinger equation using techniques from supersymmetric quantum mechanics

regularization of the wkb integrals a simple general method of regularization of the higher order wentzel kramers brillouin wkb integrals for the bound state problem of the one dimensional schrodinger equation with a real analytic potential is proposed the method is based upon the explicit separation of the integrand singularities as an example a contribution of the higher order wkb corrections to the energy eigenvalues for the quasi exactly solvable potential v x 10x2 10x4 x6 is considered

some optimal inapproximability results we prove optimal up to an arbitrary e 0 inapproximability results for max e k sat for k 3 maximizing the number of satisfied linear equations in an over determined system of linear equations modulo a prime p and set splitting as a consequence of these results we get improved lower bounds for the efficient approximability of many optimization problems studied previously in particular for max e2 sat max cut max di cut and vertex cover

gadgets approximation and linear programming we present a linear programming based method for finding gadgets i e combinatorial structures reducing constraints of one optimization problem to constraints of another a key step in this method is a simple observation which limits the search space to a finite one using this new method we present a number of new computer constructed gadgets for several different reductions this method also answers a question posed by bellare goldreich and sudan siam j comput 27 1998 pp 804 915 of how to prove the optimality of gadgets linear programming duality gives such proofs the new gadgets when combined with recent results of ha stad proceedings of the 29th acm symposium on theory of computing 1997 pp 1 10 improve the known inapproximability results for max cut and max dicut showing that approximating these problems to within factors of 16 17 epsilon and 12 13 epsilon respectively is np hard for every epsilon 0 prior to this work the best known inapproximability thresholds for both problems were 71 72 m bellare o goldreich and m sudan siam j comput 27 1998 pp 804 915 without using the gadgets from this paper the best possible hardness that would follow from bellare goldreich and sudan and ha s tad is 18 19 we also use the gadgets to obtain an improved approximation algorithm for max3 sat which guarantees an approximation ratio of 801 this improves upon the previous best bound implicit from m x goemans and d p williamson j acm 42 1995 pp 1115 1145 u feige and m x goemans proceedings of the third israel symposium on theory of computing and systems 1995 pp 182 189 of 7704

expansion around instantons in quantum mechanics we calculate numerically a few terms of the corrections to the large order behavior of the ground state energy of the o n anharmonic oscillator by analyzing the perturbation series we then generate 94 terms of the perturbative expansion of the difference between the energies of the two low lying states of the double well potential and analyze their large order behavior

two parametric pt symmetric quartic family we describe a parametrization of the real spectral locus of the two parametric family of pt symmetric quartic oscillators for this family we find a parameter region where all eigenvalues are real extending the results of dorey et al 2007 j phys a math theor 40 r205 83 and shin 2005 j phys a math gen 38 6147 66 2002 commun math phys 229 543 64

symmetry and approximability of submodular maximization problems a number of recent results on optimization problems involving submodular functions have made use of the multilinear relaxation of the problem these results hold typically in the value oracle model where the objective function is accessible via a black box returning f s for a given s we present a general approach to deriving inapproximability results in the value oracle model based on the notion of symmetry gap our main result is that for any fixed instance that exhibits a certain symmetry gap in its multilinear relaxation there is a naturally related class of instances for which a better approximation factor than the symmetry gap would require exponentially many oracle queries this unifies several known hardness results for submodular maximization e g the optimality of 1 1 e approximation for monotone submodular maximization under a cardinality constraint and the impossibility of frac12 epsilon approximation for unconstrained nonmonotone submodular maximization as a new applica

discrete cal pt symmetric models of scattering one dimensional scattering mediated by non hermitian hamiltonians is studied a schematic set of models is used which simulates two point interactions at a variable strength and distance the feasibility of the exact construction of the amplitudes is achieved via the discretization of the coordinate by direct construction it is shown that in all our models the probability is conserved this feature is tentatively attributed to the space and time reflection symmetry also known as symmetry of our specific hamiltonians

dirac equation with position dependent effective mass and solvable potentials in the schrodinger equation we study the one dimensional non hermitian imaginary potential with a real energy spectrum in the framework of the position dependent effective mass dirac equation the dirac equation is mapped into the exactly solvable schrodinger like equation endowed with position dependent effective mass that we present a new procedure to solve it the point canonical transformation in non relativistic quantum mechanics is applied as an algebraic method to obtain the mass function and then by using the obtained mass function the imaginary potential can be obtained the spinor wavefunctions for some of the obtained electrostatic potentials are given in terms of orthogonal polynomials we also obtain the relativistic bound state spectrum for each case in terms of the bound state spectrum of the solvable potentials

improving the variational path integral approach to the quantum double well potential an improved variational path integral approach is developed and applied to the quantum double well potential in which part of the quartic term of the potential is included in the trial action the expression of the effective classical potential ecp under a non gaussian expectation is obtained here the frequency and fourth order derivative of the potential are treated as two variational parameters determined by the minimization of the ecp at each point we calculate the ecp the free energy and the level splitting of a symmetrical double well potential it is shown that the present results are better than those of the feynman kleinert gaussian variational method

pade summability of the cubic oscillator we prove the pade stieltjes summability of the perturbation series of any energy level en 1 of the cubic anharmonic oscillator as suggested by the numerical studies of bender and weniger at the same time we give a simple proof of the positivity of every level of the symmetric hamiltonian h1 for positive bessis zinn justin conjecture the n zeros of a state n 1 stable at 0 are confined for on the cut complex plane and are related to the level en 1 by the bohr sommerfeld quantization rule semiclassical phase integral condition we also prove the absence of non perturbative eigenvalues and the simplicity of the spectrum of our hamiltonians

factorization of quasiclassical integrable systems abstract using asymptotic methods one calculates the first temperature correction to the free energy of non linear kink bearing scalar fields the method allows to make a distinct separation between the contribution of the anharmonic optical phonon field and that of the dilute gas of kinks the same technique can be used to calculate the multi kink contributions to the free energy

complexified von roos hamiltonian s η weak pseudo hermiticity isospectrality and exact solvability a complexified von roos hamiltonian is considered and a hermitian first order intertwining differential operator is used to obtain the related position dependent mass weak pseudo hermitian hamiltonians using a liouvillean type change of variables the weak pseudo hermitian von roos hamiltonians hx are mapped into the traditional schr dinger hamiltonian form hq where exact isospectral correspondence between hx and hq is obtained under a user friendly position dependent mass setting it is observed that for each exactly solvable weak pseudo hermitian reference hamiltonian hq there is a set of exactly solvable weak pseudo hermitian isospectral target hamiltonians hx a non hermitian symmetric scarf ii and a non hermitian periodic type symmetric samsonov roy potentials are used as reference models and the corresponding weak pseudo hermitian isospectral target hamiltonians are obtained

langer cherry derivation of the multi instanton expansion for the symmetric double well the multi instanton expansion for the eigenvalues of the symmetric double well is derived using a langer cherry uniform asymptotic expansion of the solution of the corresponding schrodinger equation the langer cherry expansion is anchored to either one of the minima of the potential and by construction has the correct asymptotic behavior at large distance while the quantization condition amounts to imposing the even or odd parity of the wave function this method leads to an efficient algorithm for the calculation to virtually any desired order of all the exponentially small series of the multi instanton expansion and with trivial modifications can also be used for nonsymmetric double wells

complex calogero model with real energies recently the symmetry of many single particle non hermitian hamiltonians has been conjectured to be sufficient to keep their spectrum real we demonstrate this and how the similar concept of a weakened hermiticity can be extended to some exactly solvable two and three particle models

optimal inapproximability results for max cut and other 2 variable csps in this paper we show a reduction from the unique games problem to the problem of approximating max cut to within a factor of alpha text tiny gw epsilon for all epsilon 0 here alpha text tiny gw approx 878567 denotes the approximation ratio achieved by the algorithm of goemans and williamson in j assoc comput mach 42 1995 pp 1115 1145 this implies that if the unique games conjecture of khot in proceedings of the 34th annual acm symposium on theory of computing 2002 pp 767 775 holds then the goemans williamson approximation algorithm is optimal our result indicates that the geometric nature of the goemans williamson algorithm might be intrinsic to the max cut problem our reduction relies on a theorem we call majority is stablest this was introduced as a conjecture in the original version of this paper and was subsequently confirmed in e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 a stronger version of this conjecture called plurality is stablest is still open although e mossel r o donnell and k oleszkiewicz proceedings of the 46th annual ieee symposium on foundations of computer science 2005 pp 21 30 contains a proof of an asymptotic version of it our techniques extend to several other two variable constraint satisfaction problems in particular subject to the unique games conjecture we show tight or nearly tight hardness results for max 2sat max q cut and max 2lin q for max 2sat we show approximation hardness up to a factor of roughly 943 this nearly matches the 940 approximation algorithm of lewin livnat and zwick in proceedings of the 9th annual conference on integer programming and combinatorial optimization springer verlag berlin 2002 pp 67 82 furthermore we show that our 943 factor is actually tight for a slightly restricted version of max 2sat for max q cut we show a hardness factor which asymptotically for large q matches the approximation factor achieved by frieze and jerrum improved approximation algorithms for max k cut and max bisection in integer programming and combinatorial optimization springer verlag berlin pp 1 13 namely 1 1 q 2 rm ln q q 2 for max 2lin q we show hardness of distinguishing between instances which are 1 epsilon satisfiable and those which are not even roughly q epsilon 2 satisfiable these parameters almost match those achieved by the recent algorithm of charikar makarychev and makarychev proceedings of the 38th annual acm symposium on theory of computing 2006 pp 205 214 the hardness result holds even for instances in which all equations are of the form x i x j c at a more qualitative level this result also implies that 1 epsilon vs e hardness for max 2lin q is equivalent to the unique games conjecture

maximal couplings in pt symmetric chain models with the real spectrum of energies the domain of all the coupling strengths compatible with the reality of the energies is studied for a family of non hermitian n n matrix hamiltonians h n with tridiagonal and symmetric structure at all dimensions n the coordinates are found of the extremal points at which the boundary hypersurface touches the circumscribed sphere for odd n 2m 1 or ellipsoid for even n 2k

spiked potentials and quantum toboggans even if the motion of a quantum quasi particle proceeds along a left right symmetric symmetric curved path in complex plane the spectrum of bound states may remain physical i e real and bounded below a generalization is outlined first we show how the topologically less trivial tobogganic contours may be allowed to live on several sheets of a riemann surface second the specification of a scattering regime is formulated for such a class of models

instantons in quantum mechanics numerical evidence for a conjecture in previous articles we have given a proper definition of multi instanton contributions in quantum mechanics and calculated these contributions to leading order we have also presented a conjecture about the form of the expansion of these multi instanton contributions to all orders in powers of the coupling constant we give here some numerical results which support this conjecture

higher order corrections to instantons the energy levels of the double well potential receive beyond perturbation theory contributions which are non analytic in the coupling strength these are related to instanton effects for example the separation between the energies of odd and even parity states is given at leading order by the one instanton contribution however to determine the energies more accurately multi instanton configurations have also to be taken into account we investigate here the two instanton contributions first we calculate analytically higher order corrections to multi instanton effects we then verify that the difference betweeen numerically determined energy eigenvalues and the generalized borel sum of the perturbation series can be described to very high accuracy by two instanton contributions we also calculate higher order corrections to the leading factorial growth of the perturbative coefficients and show that these are consistent with analytic results for the two instanton effect and with exact data for the first 200 perturbative coefficients

exact wkb and abelianization for the t_3 equation we describe the exact wkb method from the point of view of abelianization both for schrodinger operators and for their higher order analogues opers the main new example which we consider is the t 3 equation an order 3 equation on the thrice punctured sphere with regular singularities at the punctures in this case the exact wkb analysis leads to consideration of a new sort of darboux coordinate system on a moduli space of flat mathrm sl 3 connections we give the simplest example of such a coordinate system and verify numerically that in these coordinates the monodromy of the t 3 equation has the expected asymptotic properties we also briefly revisit the schrodinger equation with cubic potential and the mathieu equation from the point of view of abelianization

the spectrum of the cubic oscillator we prove the simplicity and analyticity of the eigenvalues of the cubic oscillator hamiltonian begin array ll h beta frac d 2 dx 2 x 2 i sqrt beta x 3 end array for in the cut plane mathcal c c mathcal c backslash mathcal r moreover we prove that the spectrum consists of the perturbative eigenvalues en n 0 labeled by the constant number n of nodes of the corresponding eigenfunctions in addition for all beta in mathcal c c e n beta can be computed as the stieltjes pade sum of its perturbation series at 0 this also gives an alternative proof of the fact that the spectrum of h is real when is a positive number this way the main results on the repulsive pt symmetric and on the attractive quartic oscillators are extended to the cubic case

asymptotic solvability of an imaginary cubic oscillator with spikes for complex potentials v x ix 3 2 ix 2 2 ix 1 2 which are symmetric we show that in 1 strong coupling regime the low lying bound states almost coincide with harmonic oscillators whenever the spectrum remains real this means at all critical 1

poschl teller paradoxes in the well known poschl teller trigonometric potential well a symmetric regularization x x ie of the impenetrable end point barriers is performed this leads to the four different solvable generalizations of the model as a byproduct the scheme clarifies certain apparent paradoxes encountered in the classically forbidden coupling regime

high precision numerical determination of eigenvalues for a double well potential related to the zinn justin conjecture a numerical method of high precision is used to calculate the energy eigenvalues and eigenfunctions for a symmetric double well potential the method is based on enclosing the system within two infinite walls with a large but finite separation and developing a power series solution for the schrodinger equation the obtained numerical results are compared with those obtained on the basis of the zinn justin conjecture and found to be in excellent agreement

properties of diffusive systems near a saddle point application to a quartic double well this paper aims at the analysis of diffusive properties of unidimensional mechanical systems in the environment of maxima and minima of the potential it begins with a study of the properties of the singular solutions of the hamilton jacobi yasue equation in the above mentioned environment in both strong or very small frictional forces for the quartic symmetrical double well potential approximate solutions are found for local validity and the diffusion operator is then calculated in the limits of deep wells and small temperature the regime being supposed to be aperiodic with high or moderate values of frictional coefficient this equation is proved to be nonunique this operator is then reduced to second order by imposing suitable boundary conditions thus an appropriate eigenvalue equation is obtained to describe stationary states in the environment of extremal points of the potential energy function the main interest of this work relies upon the fact that transition times between wells mainly depend upon fluctuations near the saddle point

relative projectivity and ideals in cohomology rings abstract in this paper we explore one aspect of the relationship between group cohomology and representation theory for a finite groupgand a fieldkin characteristicp 0 ideals in the cohomology ringh g k can sometimes be characterized by exact sequences ofkg modules in much the same way that elements of extkgare classically represented by exact sequences furthermore the maps on the stable category which represent the exact sequences have functorial properties which mimic the structure of the ideas in the ring the natural setting for this study is the relative homological algebra for the module category relative to the subcategory generated by a single module using the tensor product operation

solvable non hermitian discrete square well with closed form physical inner product a new hermitizable quantum model is proposed in which the bound state energies are real and given as roots of an elementary trigonometric expression while the wave function components are expressed as superpositions of two chebyshev polynomials as an site lattice version of square well with complex robin type two parametric boundary conditions the model is unitary with respect to the hilbert space metric which becomes equal to the most common dirac s metric in the conventional textbook hermitian hamiltonian limit this metric is constructed in closed form at all

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

regularized scarf potentials energy band structure and supersymmetry the singular one dimensional periodic scarf potential is regularized by means of one parameter square well counter terms it is shown that the regularized spectrum converges formally to the conventional scarf energy bands for specific values of the parameter the behaviour of the regularizations under supersymmetric transformations is also investigated this is a key point for the algebraic solvability of the regularized scarf potential

maximizing non monotone submodular functions submodular maximization generalizes many important problems including max cut in directed and undirected graphs and hypergraphs certain constraint satisfaction problems and maximum facility location problems unlike the problem of minimizing submodular functions the problem of maximizing submodular functions is np hard in this paper we design the first constant factor approximation algorithms for maximizing nonnegative non monotone submodular functions in particular we give a deterministic local search frac 1 3 approximation and a randomized frac 2 5 approximation algorithm for maximizing nonnegative submodular functions we also show that a uniformly random set gives a frac 1 4 approximation for symmetric submodular functions we show that a random set gives a frac 1 2 approximation which can also be achieved by deterministic local search these algorithms work in the value oracle model where the submodular function is accessible through a black box returning f s for a given set s we show that in this model a frac 1 2 epsilon approximation for symmetric submodular functions would require an exponential number of queries for any fixed epsilon 0 in the model where f is given explicitly as a sum of nonnegative submodular functions each depending only on a constant number of elements we prove np hardness of frac 5 6 epsilon approximation in the symmetric case and np hardness of frac 3 4 epsilon approximation in the general case

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

virtual projectivity and the stable module category abstract we introduce the concept of a nearly null map in the stable module category and relate it to the notion of virtual projectivity we show that the trivial module k is virtually m projective if and only if the map f n k in the triangle n f k m m is nearly null if these conditions hold then m generates the stable module category in fewer than l steps where l is the radical length of the group algebra we give examples to show that if m generates the stable module category then k is not necessarily virtually m projective we also give examples to show that for a given group the degree of virtual projectivity of k with respect to a module m is not bounded finally we develop a theory of virtual m variety of a module x this is a subset of the spectrum of the cohomology ring which controls virtual projectivity of x with respect to m

large order perturbation theory for the o 2 anharmonic oscillator with negative anharmonicity and for the double well potential for the o 2 anharmonic oscillator with negative anharmonicity and one dimensional double well potential u x x2 r2 2 8r2 r is a large parameter for both problems conventional rayleigh schrodinger expansions in power series of 1 r2 for the energy eigenvalues agree the authors studied asymptotic expansions for the eigenvalues which contain in addition to the perturbation series exponentially small terms in r2 due to the tunnelling through corresponding potential barriers numerical calculations are also performed and comparisons with the asymptotic formulae are given

non hermitian matrix schrodinger equation backlund darboux transformation weyl functions and script pscript t symmetry the non hermitian matrix schrodinger equation obtained by the backlund darboux transformation bdt is treated the potentials fundamental solutions and weyl functions are constructed explicitly a symmetric reduction of the bdt is introduced and this case is studied in greater detail including potentials fundamental solutions bound states the reality of the discrete spectrum and spontaneous break of the symmetry the sign indefinite scalar product and examples

virtual turning points and bifurcation of stokes curves for higher order ordinary differential equations for a higher order linear ordinary differential operator p its stokes curve bifurcates in general when it hits another turning point of p this phenomenon is most neatly understandable by taking into account stokes curves emanating from virtual turning points together with those from ordinary turning points this understanding of the bifurcation of a stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth painleve equation

some spectral equivalences between schrodinger operators spectral equivalences of the quasi exactly solvable sectors of two classes of schrodinger operators are established using gaudin type bethe ansatz equations in some instances the results can be extended leading to full isospectrality in this manner we obtain equivalences between pt symmetric problems and hermitian problems we also find equivalences between some classes of hermitian operators

complete set of inner products for a discrete pt symmetric square well hamiltonian a discrete n point runge kutta version h n of one of the simplest non hermitian square well hamiltonians with real spectrum is studied its possible hermitizations mediated by nontrivial often called non dirac metrics i are considered as a source of nonequivalent standard probabilistic interpretations of this quantum model a complete set of these alternative multiparametric metrics a b n defining all the eligible hamiltonian dependent representations of the physical hilbert space of states is constructed in closed form for any coupling 1 1 and for any matrix dimension n

kurt symanzik a stable fixed point beyond triviality in 1970 kurt symanzik proposed a precarious 4 theory with a negative quartic coupling constant as a valid candidate for an asymptotically free theory of strong interactions symanzik s deep insight into the non trivial properties of this theory has been overruled since then by the hermitian intuition of generations of scientists who considered or consider this actually non hermitian highly important theory to be unstable this short certainly controversial communication tries to shed some light on the historical and formalistic context of symanzik s theory in order to sharpen our quantum intuition about non perturbative theoretical physics between non triviality and asymptotic freedom

calculation of energy eigenvalues for the quantum anharmonic oscillator with a polynomial potential we choose the squeezed vacuum state as a one parameter trial wavefunction to minimize the energy of an anharmonic oscillator with a polynomial perturbation potential using the optimal vacuum state obtained we generate the optimal generalized number state basis and set up the hamiltonian in the normal ordered form then truncating the basis we produce finite dimensional matrices in a simple manner and find their eigenvalues by standard methods we will apply our method to the morse potential whose exact eigenvalues are known as a check on the accuracy of the results that we have derived two models of the double well and the quartic potential are considered as examples

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

exact quantization condition for anharmonic oscillators in one dimension an exact quantization condition is given for the one dimensional schrodinger operator with a homogeneous anharmonic potential q2m it has the form of an explicit mapping from level sequences to level sequences involving a bohr sommerfeld like quantization step and having the exact spectrum as fixed point numerical tests and an approximate linear theory both suggest at least for the few lowest m that the mapping has a contractive region when an initial level sequence is only asymptotically correct to lowest order its iterates are seen to converge term by term towards the exact eigenvalues this type of approach ought to extend to general polynomial potentials

comparing apples and oranges query trade off in submodular maximization fast algorithms for submodular maximization problems have a vast potential use in applicative settings such as machine learning social networks and economics though fast algorithms were known for some special cases only recently such algorithms were considered in the general case of maximizing a monotone submodular function subject to a matroid independence constraint the known fast algorithm matches the best possible approximation guarantee while trying to reduce the number of value oracle queries the algorithm performs our main result is a new algorithm for this general case that establishes a surprising trade off between two seemingly unrelated quantities the number of value oracle queries and the number of matroid independence queries performed by the algorithm specifically one can decrease the former by increasing the latter and vice versa while maintaining the best possible approximation guarantee such a trade off is very useful since various applications might incur significantly diff

exact semiclassical expansions for one dimensional quantum oscillators a set of rules is given for dealing with wkb expansions in the one dimensional analytic case whereby such expansions are not considered as approximations but as exact encodings of wave functions thus allowing for analytic continuation with respect to whichever parameters the potential function depends on with an exact control of small exponential effects these rules which include also the case when there are double turning points are illustrated on various examples and applied to the study of bound state or resonance spectra in the case of simple oscillators it is thus shown that the rayleigh schrodinger series is borel resummable yielding the exact energy levels in the case of the symmetrical anharmonic oscillator one gets a simple and rigorous justification of the zinn justin quantization condition and of its solution in terms of multi instanton expansions

non hermitian hamiltonian versus e 0 localized states we analyze the zero energy solutions of a two dimensional system which undergoes a non radial symmetric complex potential v r by virtue of the coherent states concept the localized states are constructed and the consequences of the imaginary part of the potential are found both analytically and schematically

exact resolution method for general 1d polynomial schrodinger equation the stationary 1d schrodinger equation with a polynomial potential v q of degree n is reduced to a system of complex exact quantization conditions of bohr sommerfeld form they arise from bilinear wronskian functional relations pairing spectral determinants of n 2 generically distinct operators all the transforms of one quantum hamiltonian under a cyclic group of complex scalings the determinants zeros define n 2 semi infinite chains of points in the complex spectral plane and these encode the original quantum problem now each chain can be assigned an exact quantization condition which constrains it relative to its neighbours resulting in closed equilibrium conditions for the global chain system these are supplemented by the standard bohr sommerfeld quantization conditions which bind the infinite tail of each chain asymptotically this reduced problem is then probed numerically for effective solvability upon test cases mostly symmetric quartic oscillators we find that the iterative enforcement of all the quantization conditions generates discrete chain dynamics which appear to converge geometrically towards the correct eigenvalues eigenfunctions we conjecture that the exact quantization then acts by specifying reduced chain dynamics which can be stable contractive and thus reconstruct the exact quantum data as their fixed point to date this statement is verified only empirically and in a vicinity of purely quartic or sextic potentials v q

transfer maps and virtual projectivity abstract when g is a finite group and h is a subgroup of g the concept of relative h projectivity is of fundamental importance in modular representation theory unfortunately in some ways the properties of relatively h projective modules are not as closely related to other ideas of representation theory as one might hope for example the variety of a module bears only a weak connection to relative projectivity the purpose of this paper is to introduce and study a related concept known as virtual h projectivity that seems to be much more closely related to the theory of varieties the study is actually carried out in the more general setting of virtual projectivity with respect to a finitely generated module w and the main tool is a transfer map tr w that generalizes the usual transfer map tr g h with respect to a subgroup h

maximizing a monotone submodular function subject to a matroid constraint let f 2 x rightarrow cal r be a monotone submodular set function and let x cal i be a matroid we consider the problem rm max s in cal i f s it is known that the greedy algorithm yields a 1 2 approximation m l fisher g l nemhauser and l a wolsey math programming stud no 8 1978 pp 73 87 for this problem for certain special cases e g rm max s leq k f s the greedy algorithm yields a 1 1 e approximation it is known that this is optimal both in the value oracle model where the only access to f is through a black box returning f s for a given set s g l nemhauser and l a wolsey math oper res 3 1978 pp 177 188 and for explicitly posed instances assuming p neq np u feige j acm 45 1998 pp 634 652 in this paper we provide a randomized 1 1 e approximation for any monotone submodular function and an arbitrary matroid the algorithm works in the value oracle model our main tools are a variant of the pipage rounding technique of ageev and sviridenko j combin optim 8 2004 pp 307 328 and a continuous greedy process that may be of independent interest as a special case our algorithm implies an optimal approximation for the submodular welfare problem in the value oracle model j vondrak proceedings of the 38 th acm symposium on theory of computing 2008 pp 67 74 as a second application we show that the generalized assignment problem gap is also a special case although the reduction requires x to be exponential in the original problem size we are able to achieve a 1 1 e o 1 approximation for gap simplifying previously known algorithms additionally the reduction enables us to obtain approximation algorithms for variants of gap with more general constraints