nonlinear aspects of the renormalization group flows of dyson s hierarchical model we review recent results concerning the renormalization group rg transformation of dyson s hierarchical model hm this model can be seen as an approximation of a scalar field theory on a lattice we introduce the hm and show that its large group of symmetry simplifies drastically the blockspinning procedure several equivalent forms of the recursion formula are presented with unified notations rigourous and numerical results concerning the recursion formula are summarized it is pointed out that the recursion formula of the hm is inequivalent to both wilson s approximate recursion formula and polchinski s equation in the local potential approximation despite the very small difference with the exponents of the latter we draw a comparison between the rg of the hm and functional rg equations in the local potential approximation the construction of the linear and nonlinear scaling variables is discussed in an operational way we describe the calculation of non universal critical amplitudes in terms of the scaling variables of two fixed points this question appears as a problem of interpolation between these fixed points universal amplitude ratios are calculated we discuss the large n limit and the complex singularities of the critical potential calculable in this limit the interpolation between the hm and more conventional lattice models is presented as a symmetry breaking problem we briefly introduce models with an approximate supersymmetry one important goal of this review is to present a configuration space counterpart suitable for lattice formulations of functional rg equations formulated in momentum space often called exact rg equations and abbreviated erge

precise determination of the energy levels of the anharmonic oscillator from the quantization of the angle variable using an ansatz motivated by the classical form of el phi where phi is the angle variable we construct operators which satisfy the commutation relations of the creation annihilation operators for the anharmonic oscillator the matrix elements of these operators can be expressed in terms of entire functions in the position complex plane these functions provide solutions of the ricatti equation associated with the time independent schrodinger equation we relate the normalizability of the eigenstates to the global properties of the flows of this equation these exact results yield approximations which complement the wkb approximation and allow an arbitrarily precise determination of the energy levels we give numerical results for the first 10 levels with 30 digits we address the question of the quantum integrability of the system

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

averaging for fundamental solutions of parabolic equations abstract herein an averaging theory for the solutions to cauchy initial value problems of arbitrary order e dependent parabolic partial differential equations is developed indeed by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the fundamental solution of an averaged parabolic equation we bring forth a novel approach to comparing x derivatives of t u e x t k 2p a k x t e k x u e x t f e x t u e x 0 e x on r d 0 t to like derivatives of t u x t k 2p a 0 k x k x u e x t f x t u x x as e 0 under general regularity conditions and our basic hypothesis that t 0 a k x s e a 0 k x ds e 0 0 for each x t i e pointwise the flexibility afforded by studing fundamental vis a vis specific solutions of these equations not only permits e dependent cauchy data and provides a unified method of treating all x derivatives of u e up to order 2 p 1 but also proves an invaluable tool when considering related problems of stochastic averaging our development was motivated by and retains a strong resemblance to the classical theory of parabolic partial differential equations however it will turn out that the classical conditions under which fundamental solutions are known to exist are somewhat unsuitable for our purposes and a modified set of conditions must be used

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

simple method to make asymptotic series of feynman diagrams converge we show that for two nontrivial lambda phi 4 problems the anharmonic oscillator and the landau ginzburg hierarchical model improved perturbative series can be obtained by cutting off the large field contributions the modified series converge to values exponentially close to the exact ones for lambda larger than some critical value the method outperforms pade s approximants and borel summations the method can also be used for series which are not borel summable such as the double well potential series we show that semiclassical methods can be used to calculate the modified feynman rules estimate the error and optimize the field cutoff

quantum anharmonic oscillators a new approach the determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy namely the wronskian of two solutions of the schr dinger equation which are regular respectively at the origin and at infinity we show in this paper how to evaluate that wronskian exactly except for numerical rounding errors the procedure is illustrated by application to the gx2 x2n n a positive integer oscillator

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

storage capacity and optimal learning of potts model perceptrons by a cavity method by means of a general formulation for the optimal learning capacity of perceptrons with multi state neurons and real valued couplings with spherical constraints which we derive by a cavity method we calculate the optimal learning capacity alpha c q k pmax n q 1 for perceptrons with a q resp q state potts model input resp output neurons as a function of q and the stability parameter k among other results the asymptote for q to infinity is found and it is shown that for k 0 the information gain per coupling delta i alpha c in q q 1 converges slowly to 1 2 in this limit moreover for q to infinity the same asymptotics also apply for the simple case of hebbian learning

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

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

phase transitions in the generalization behaviour of multilayer neural networks we study the generalization ability of multilayer neural networks with tree architecture for the case of random training sets the first layer consists of k spherical perceptrons with binary output a boolean function b computes the final output from the k values produced by the first layer we first calculate the learning behaviour of gibbs learning in the case of learnable rules where teacher and student have the same architecture and the boolean function b is permutation symmetric with respect to the hidden units in the asymptotic case of high loading alpha to infinity as usual alpha is the loading parameter we find that the generalization error vanishes in the same way for all b the reason being is that there are two effects cancelling each other in the opposite limit for small alpha we find qualitatively different behaviour i e some networks undergo a phase transition we show how these differences in the behaviour depend on certain characteristics of the boolean function b we then study the bayes algorithm which generalizes according to the majority decision of a certain ensemble of machines and find that the parity function plays a special role if the teacher is a parity machine there always exists a single student parity machine that generalizes as well as the bayes algorithm

a kuhn tucker cavity method for generalization with applications to perceptrons with ising and potts neurons within the framework of statistical physics we derive a cavity method for generalization by perceptrons where the kuhn tucker conditions for optimal stability are built into the cavity fields in this way the calculation of the generalization ability for learning processes leading to optimal stability is simplified within our approach the degrees of freedom of the neurons can be rather arbitrary for perceptrons with ising neurons we relate our method to the traditional replica approach new results are obtained for e state ports model perceptrons including the asymptotic behaviour for alpha to infinity and general q

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

conformal mappings versus other power series methods for solving ordinary differential equations illustration on anharmonic oscillators the simplicity and the efficiency of a quasi analytical method for solving nonlinear ordinary differential equations ode is illustrated on the study of anharmonic oscillators ao with a potential v x x2 x2m m 0 the method bervillier 2008 nucl phys b801 296 applies a priori to any ode with two point boundaries one being located at infinity the solution of which has fixed singularities in the complex plane of the independent variable x a conformal mapping of a suitably chosen angular sector of the complex plane of x upon the unit disc centered at the origin makes convergent the transformed taylor series of the generic solution so that the boundary condition at infinity can be easily imposed in principle this constraint when applied on the logarithmic derivative of the wavefunction determines the eigenvalues to an arbitrary level of accuracy in practice for 0 or slightly negative the accuracy of the results obtained is astonishingly large with regard to the modest computing power used various aspects of the method and comparisons with some seemingly similar methods based also on expressing the solution as a taylor series are shortly reviewed presented and discussed

arbitrarily accurate eigenvalues for one dimensional polynomial potentials we show that the riccati form of the one dimensional schrodinger equation can be reformulated in terms of two linear equations depending on an arbitrary function g when g and the potential as for anharmonic oscillators are polynomials the solutions of these two equations are entire functions l and k and the zeros of k are identical to those of the wavefunction requiring such a zero at a large but finite value of the argument yields low energy eigenstates with exponentially small errors approximate formulae for these errors are provided we explain how to choose g in order to dramatically improve the numerical treatment the method yields many significant digits with modest computer means we discuss the extension of this method in the case of several variables

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

a tractable example of perturbation theory with a field cutoff the anharmonic oscillator for 4 models the introduction of a large field cutoff improves significantly the accuracy that can be reached with perturbative series but the calculation of the modified coefficients remains a challenging problem we show that this problem can be solved numerically and analytically in the limits of large and small field cutoffs for the ground state energy of the anharmonic oscillator for the two lowest orders in the approximate formulae obtained in the large field cutoff limit extend unexpectedly far in the low field cutoff region and there is a significant overlap with the small field cutoff approximation for the higher orders this is not the case however the shape of the transition between the small field cutoff regime and the large field cutoff regime is approximately order independent

invariance principles for parabolic equations with random coefficients a general hilbert space based stochastic averaging theory is brought forth herein for arbitrary order parabolic equations with possibly long range dependent random coefficients we use regularity conditions on tue t x 0 k 2pak t e x kxue t x ue 0 x x 1 which are slightly stronger than those required to prove pathwise existence and uniqueness for 1 equation 1 can be obtained from the singularly perturbed system ve x 0 k 2peak x kxve x ve 0 x x 2 through time change next we impose on the coefficients of 1 a pointwise inxandt weak law of large numbers and a weak invariance principleeh te 10ak x s a0k x ds k 2p k k 2p 3 inc 0 t h1 h1being a separable hilbert space of functions andh 0 1 denoting a constant h 1 2 allows for long range time dependence then under these extraordinarily general conditions we infer the weak invariance principleeh 1 ue u uis the non random e homogeneous solution of tu t x 0 k 2pa0k x kxu t x u 0 x x 4 and mildly satisfies the linear stochastic partial differential equation ty t x k 2pa0k x kxy t x dt k 2p k dt x kxu t x 5