solution to a conjecture on the maximal energy of bipartite bicyclic graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let c n denote the cycle of order n and p n 6 6 the graph obtained from joining two cycles c 6 by a path p n 12 with its two leaves let b n denote the class of all bipartite bicyclic graphs but not the graph r a b which is obtained from joining two cycles c a and c b a b 10 and a b 2 mod 4 by an edge in i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005 gutman and vidovic conjectured that the bicyclic graph with maximal energy is p n 6 6 for n 14 and n 16 in x li j zhang on bicyclic graphs with maximal energy linear algebra appl 427 2007 87 98 li and zhang showed that the conjecture is true for graphs in the class b n however they could not determine which of the two graphs r a b and p n 6 6 has the maximal value of energy in b furtula s radenkovic i gutman bicyclic molecular graphs with the greatest energy j serb chem soc 73 4 2008 431 433 numerical computations up to a b 50 were reported supporting the conjecture so it is still necessary to have a mathematical proof to this conjecture this paper is to show that the energy of p n 6 6 is larger than that of r a b which proves the conjecture for bipartite bicyclic graphs for non bipartite bicyclic graphs the conjecture is still open

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

monte carlo studies of three dimensional two step restricted self avoiding walks two step restricted walk tsrw models are a class of restricted self avoiding walk saw where in addition to the self avoidance constraint certain restrictions are placed upon each pair of successive steps in this paper we explore the relationship between the restrictions and the scaling of the average size of walks in three dimensional models we use the pruned enriched rosenbluth method algorithm to perform monte carlo studies in five representative tsrw models in three dimensions the results present strong numerical evidence that all non trivial tsrw models in three dimensions have the same size scaling behaviour as unrestricted saws this is in contrast to two dimensions where several universality classes are accepted to exist in particular we find no rule analogous to the spiral walk of two dimensions

trees with minimal laplacian coefficients let g be a simple undirected graph with the characteristic polynomial of its laplacian matrix l g p g m email protected k 0 n 1 kc email protected n k it is well known that for trees the laplacian coefficient c n 2 is equal to the wiener index of g while c n 3 is equal to the modified hyper wiener index of the graph in this paper we characterize n vertex trees with given matching number m which simultaneously minimize all laplacian coefficients the extremal tree a n m is a spur obtained from the star graph s n m 1 with n m 1 vertices by attaching a pendant edge to each of certain m 1 non central vertices of s n m 1 in particular a n m minimizes the wiener index the modified hyper wiener index and the recently introduced incidence energy of trees defined as ie g email protected k 0 email protected k where m k are the eigenvalues of signless laplacian matrix q g d g a g we introduced a general r transformation which decreases all laplacian coefficients simultaneously in conclusion we illustrate on examples of wiener index and incidence energy that the opposite problem of simultaneously maximizing all laplacian coefficients has no solution

analyzing lattice networks through substructures abstract analyzing the topology of network structures is an important topic studied from many different aspects of science and mathematics the wiener polarity index number of unordered pairs of vertices at distance 3 from each other is one of the representative descriptors of graph structures it was computed for several lattice networks by chen et al 11 in an effort to understand the properties of these networks the wiener polarity index is a variation of the classic distance based graph invariant the wiener index sum of distances between all pairs of vertices which is known to be closely related to the number of substructures in this paper we examine the numbers of various subgraphs of order 4 for these lattice networks in addition to confirming their symmetric nature comparing the numbers of various substructures leads to insights on other less trivial characteristics of these network structures of common interest

frequency operator for anharmonic oscillators by perturbation theory we obtain approximate analytical expressions for the frequency operator for anharmonic oscillators by means of perturbation theory and a unitary transformation

partition functions and the continuum limit in penner matrix models we present an implementation of the method of orthogonal polynomials which is particularly suitable to study the partition functions of penner random matrix models to obtain their explicit forms in the exactly solvable cases and to determine the coefficients of their perturbative expansions in the continuum limit the method relies on identities satisfied by the resolvent of the jacobi matrix in the three term recursion relation of the associated families of orthogonal polynomials these identities lead to a convenient formulation of the string equations as an application we show that in the continuum limit the free energy of certain exactly solvable models like the linear and double penner models can be written as a sum of gaussian contributions plus linear terms to illustrate the one cut case we discuss the linear double and cubic penner models and for the two cut case we discuss theoretically and numerically the existence of a double branch structure of the free energy for the gaussian penner model

fractal weyl laws in discrete models of chaotic scattering we analyse simple models of quantum chaotic scattering namely quantized open baker s maps we numerically compute the density of quantum resonances in the semiclassical regime this density satisfies a fractal weyl law where the exponent is governed by the fractal dimension of the set of trapped trajectories this type of behaviour is also expected in the physically more relevant case of hamiltonian chaotic scattering within a simplified model we are able to rigorously prove this weyl law and compute quantities related to the coherent transport through the system namely the conductance and shot noise the latter is close to the prediction of random matrix theory

lie symmetries of multidimensional difference equations a method is presented for calculating the lie point symmetries of a scalar difference equation on a two dimensional lattice the symmetry transformations act on the equations and on the lattice they take solutions into solutions and can be used to perform symmetry reduction the method generalizes the one presented in a recent publication for the case of ordinary difference equations in turn it can easily be generalized to difference systems involving an arbitrary number of dependent and independent variables

universal microscopic correlation functions for products of truncated unitary matrices we investigate the spectral properties of the product of m complex non hermitian random matrices that are obtained by removing l rows and columns of larger unitary random matrices uniformly distributed on the group u n l such matrices are called truncated unitary matrices or random contractions we first derive the joint probability distribution for the complex eigenvalues of the product matrix for fixed n l and m given by a standard determinantal point process in the complex plane the weight however is non standard and can be expressed in terms of the meijer g function the explicit knowledge of all eigenvalue correlation functions and the corresponding kernel allows us to take various large n and l limits at fixed m at strong non unitarity with l n finite the eigenvalues condense on a domain inside the unit circle at the edge and in the bulk we find the same universal microscopic kernel as for a single complex non hermitian matrix from the ginibre ensemble at the origin we find the same new universality classes labeled by m as for the product of m matrices from the ginibre ensemble keeping a fixed size of truncation l when n goes to infinity leads to weak non unitarity with most eigenvalues on the unit circle as for unitary matrices here we find a new microscopic edge kernel that generalizes the known results for m 1 we briefly comment on the case when each product matrix results from a truncation of different size lj

on the laplacian coefficients of tricyclic graphs abstract let g d e t i n l g k 0 n 1 k c k n k be the characteristic polynomial of the laplacian matrix of a graph g of order n in this paper we show that among all connected tricyclic graphs of order n the k th coefficient c k is smallest when the graph is b n 7 1 3 3 3 obtained from the complete graph k 4 by adding n 4 pendent vertices attached to the vertex of degree 3 and for some lemmas in c x he h y shan on the laplacian coefficients of bicyclic graphs discrete math 310 2010 3404 3412 we present a new method to prove them

solutions to unsolved problems on the minimal energies of two classes of trees the energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph let t n p t n d be the set of all trees of order n with p pendent vertices diameter d respectively in this paper we completely characterize the trees with second minimal and third minimal energy in t n p t n d respectively for 4 p n 9 10 d n 3 respectively which solves the problems left in ma 2014

correlation property of length sequences based on global structure of the complete genome this paper considers three kinds of length sequences of the complete genome detrended fluctuation analysis spectral analysis and the mean distance spanned within time l are used to discuss the correlation property of these sequences the values of the exponents from these methods of these three kinds of length sequences of bacteria indicate that the long range correlations exist in most of these sequences the correlations have a rich variety of behaviors including the presence of anti correlations furthermore using the exponent gamma it is found that these correlations are all linear gamma 1 0 0 03 it is also found that these sequences exhibit l f noise in some interval of frequency f l the length of this interval of frequency depends on the length of the sequence the shape of the periodogram in f l exhibits some periodicity the period seems to depend on the length and the complexity of the length sequence

joint probability distributions for projection probabilities of random orthonormal states the quantum chaos conjecture applied to a finite dimensional quantum system implies that such a system has eigenstates that show similar statistical properties as the column vectors of random orthogonal or unitary matrices here we consider the different probabilities for obtaining a specific outcome in a projective measurement provided the system is in one of its eigenstates we then give analytic expressions for the joint probability density for these probabilities with respect to the ensemble of random matrices in the case of the unitary group our results can be applied also to the phenomenon of universal conductance fluctuations where the same mathematical quantities describe partial conductances in a two terminal mesoscopic scattering problem with a finite number of modes in each terminal

long range correlations in the electric signals that precede rupture the smoluchowski chapman kolmogorov functional equation is applied to the electric signals that precede rupture the results suggest a non markovian character of the analyzed data the rescaled range hurst and detrended fluctuation analyses as well as that related with the mean distance a walker spanned lead to power law exponents which are consistent with the existence of long range correlations a universality in the power spectrum characteristics of these signals emerges if an analysis is made not in the conventional time frame but in the natural time domain within this frame it seems that certain power spectrum characteristics of ion current fluctuations in membrane channels distinguish them from the electric signals preceding rupture the latter exhibit a behavior compatible with that expected from a model based on the random field ising hamiltonian at the critical point

the transfer matrices of the self similar fractal potentials on the cantor set on the basis of an exact formalism a system of functional equations for the tunnelling parameters of self similar fractal potentials ssfps is obtained three different families of solutions are found for these equations two of them having one parameter and one being free of parameters both one parameter solutions are shown to be described in the long wave limit by a fractal dimension at the same time the third solution yields transfer matrices which are analytical in this region similar to the case of structures with the euclidean geometry we have revealed some manifestations of scale invariance in the physical properties of ssfps nevertheless in the common case these potentials do not possess strictly speaking this symmetry the point is that ssfps in the common case are specified in contrast to the cantor set by two length scales but not one a particular case when ssfps are exactly scale invariant to an electron with well defined energy is found

real space renormalization group approach to driven diffusive systems we introduce a real space renormalization group procedure for driven diffusive systems which predicts both steady state and dynamic properties we apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram as well as the crossovers in the relaxation dynamics for each phase

evaluation of the coulomb force via the fredholm integral equation the electrostatic force between two conducting spheres is solved exactly from coulomb s law the force is evaluated in two steps i the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the fredholm integral equation ii the effect of the known charge distribution is integrated to obtain the electrostatic force evaluation of the electrostatic force is fast because the series expression for the charge distribution is super converging a characteristic trait for some fredholm type equations

random matrix ensembles with column row constraints ii we analyze statistical properties of a complex system subjected to conditions which manifests through specific constraints on the column row sum of the matrix elements of its hermitian operators the presence of additional constraints besides real symmetric nature leads to new correlations among their eigenfunctions hinders a complete delocalization of dynamics and affects the eigenvalues too the statistical analysis of the latter indicates the presence of a new universality class analogous to that of a special type of brownian ensemble appearing between poisson and gaussian orthogonal ensemble

on the maximal energy tree with two maximum degree vertices abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix for 3 and t 3 denote by t a t or simply t a the tree formed from a path p t on t vertices by attaching 1 p 2 s on each end of the path p t and t b t or simply t b the tree formed from p t 2 by attaching 1 p 2 s on an end of the p t 2 and 2 p 2 s on the vertex next to the end in li et al 2009 16 proved that among trees of order n with two vertices of maximum degree the maximal energy tree is either the graph t a or the graph t b where t n 4 4 3 however they could not determine which one of t a and t b is the maximal energy tree this is because the quasi order method is invalid for comparing their energies in this paper we use a new method to determine the maximal energy tree it turns out that things are more complicated we prove that the maximal energy tree is t b for 7 and any t 3 while the maximal energy tree is t a for 3 and any t 3 moreover for 4 the maximal energy tree is t a for all t 3 but one exception that t 4 for which t b is the maximal energy tree for 5 the maximal energy tree is t b for all t 3 but 44 exceptions that t is both odd and 3 t 89 for which t a is the maximal energy tree for 6 the maximal energy tree is t b for all t 3 but three exceptions that t 3 5 7 for which t a is the maximal energy tree one can see that for most cases of t b is the maximal energy tree 5 is a turning point and 3 and 4 are exceptional cases which means that for all chemical trees whose maximum degrees are at most 4 with two vertices of maximum degree at least 3 t a has maximal energy with only one exception t a 4 4

distributions for the eigenvalues of large random matrices generated from four manifolds in this paper we will first study the spectrum of certain large euclidean random matrices the entries of these matrices are functions of n random points lp norm uniformly sampled in n dimensional lp ball or lp sphere under the setting n n 0 as n and n tend to it is shown that the empirical distributions of eigenvalues of normalized euclidean random matrices generated from the above two manifolds both converge weakly to semicircle law we also investigate certain sample covariance type matrices whose entries are determined by n random positions from an n dimensional lp ellipsoid or its surface when n and n with n n 0 we prove that their empirical spectral distribution converge to the same fixed distribution

motions in a bose condensate x new results on the stability of axisymmetric solitary waves of the gross pitaevskii equation the stability of the axisymmetric solitary waves of the gross pitaevskii gp equation is investigated the implicitly restarted arnoldi method for banded matrices with shift invert is used to solve the linearized spectral stability problem the rarefaction solitary waves on the upper branch of the jones roberts dispersion curve are shown to be unstable to axisymmetric infinitesimal perturbations whereas the solitary waves on the lower branch and all two dimensional solitary waves are linearly stable the growth rates of the instabilities on the upper branch are so small that an arbitrarily specified initial perturbation of a rarefaction wave at first usually evolves towards the upper branch as it acoustically radiates away its excess energy this is demonstrated through numerical integrations of the gp equation starting from an initial state consisting of an unstable rarefaction wave and random non axisymmetric noise the resulting solution evolves towards and remains for a significant time in the vicinity of an unperturbed unstable rarefaction wave it is shown however that ultimately or for an initial state extremely close to the upper branch the solution evolves onto the lower branch or is completely dissipated as sound it is shown how density depletions in uniform and trapped condensates can generate rarefaction waves and a simple method is suggested by which such waves can be created in the laboratory

on the tree with diameter 4 and maximal energy the energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t this paper generalizes the result we determine the structure of the general tree of diameter 4 and maximal energy

gauge transformation of the third kind for u 1 invariant coupled schrodinger equations we consider a large class of canonical coupled nonlinear schrodinger equations invariant over the action of the u 1 group the most general nonlinearity is taken into account through a matrix that without loss of generality can be separated into the sum of a hermitian matrix and an anti hermitian matrix the u 1 symmetry implies the existence of a set of continuity equations for the conserved densities where the corresponding currents have in general a nonlinear structure for this class of coupled schrodinger equations we introduce a nonlinear gauge transformation which changes the nonlinear matrix into another one purely hermitian consequently the currents are transformed in the standard bilinear form generalization to noncanonical systems is also discussed some examples are presented to illustrate the applicability of the method

a generic intermittency model and its 1 d meta map power laws invariants and the succession of laminar sequences abstract intermittent behaviour has been found in many systems able to switch between two different dynamic states e g between long laminar phases and short chaotic bursts despite the apparently high dimensional complexity certain one dimensional 1 d maps are known to mimic properties of such dynamics to these belongs the iterative map x n 1 i x n i x n i z mod 1 giving rise to long laminar lengths the statistics of the laminar lengths are of special interest starting from this map we are interested in the values of x 0 i which arise after passing through the modulo operation these determine the laminar lengths uniquely a 1 d meta map x 0 i f x 0 i 1 is derived heuristically it is used to calculate statistical properties of the laminar phases our results show an improvement in the behaviour of short and very long laminar phases as compared to earlier analytical results introducing the concept of the generic starting value we find laminar phases not to be strictly independent of their predecessors

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

extremal laplacian energy of threshold graphs let g be a connected threshold graph of order n with m edges and trace t in this paper we give a lower bound on laplacian energy in terms of n m and t of g from this we determine the threshold graphs with the first four minimal laplacian energies moreover we obtain the threshold graphs with the largest and the second largest laplacian energies

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

semiclassical expansions in the toda hierarchy and the hermitian matrix model an iterative algorithm for determining a class of solutions of the dispersionful 2 toda hierarchy characterized by string equations is developed this class includes the solution which underlies the large n limit of the hermitian matrix model in the one cut case it is also shown how the double scaling limit can be naturally formulated in this scheme

statistics of biophysical signal characteristics and state specificity of the human eeg abstract the analysis of records of biophysical signals like eeg have become increasingly interesting to physicists since the development of new algorithms for time series analysis this analysis however requires both an adequate methodology assessing the time series characteristics and an appropriate statistical assessment of the measures resulting from these algorithms the latter should separate inherent effects from those appearing only by chance e g in terms of an appropriate test statistics as a step towards such a statistical assessment we present here a data resampling approach data resampling techniques allow for the estimation of the variance of an f value as obtained through variance analysis a test statistics based on the so called f ratio 2 f f 2 is derived and the simulation of outcomes allows for the determination of its quantiles a multivariate formulation allows for the simultaneous inclusion of linear and nonlinear time series measures into the analysis we applied these concepts to the problem of distinguishing eeg of healthy probands from the eeg of patients diagnosed as schizophrenic together with an approach involving bayes formula to weight the variables it reveals a reliable discriminance performance of 81 when the nonlinear measure and the linear measures and power are included

fluctuation analysis of the time dynamics of laser distance data measured in the medieval jeroným mine czech republic abstract fractal tools are used to investigate the hourly time fluctuations of the ceiling height of a chamber located in the medieval jeron m mine which is one of the cultural national heritage sites of the czech republic the data were measured from 2008 to 2010 by a laser distance meter leica disto t m a 4 the scaling behavior is revealed by means of different statistics the power spectrum method the detrended fluctuation analysis the higuchi analysis and the mean distance spanned within time l the values of the scaling exponents estimated by means of these methods indicate that the temporal fluctuations of the ceiling height are not typical of a purely random stochastic process i e white noise but evidence the presence of long range correlations furthermore it is found that these correlations are linear

on the energy of some circulant graphs we give an explicit construction of circulant graphs of very high energy this construction is based on gauss sums we also show the littlewood conjecture can be used to establish new result for a certain class of circulant graphs

the lattice schwarzian kdv equation and its symmetries in this paper we present a set of results on the symmetries of the lattice schwarzian korteweg de vries 1skdv equation we construct the lie point symmetries and using its associated spectral problem an infinite sequence of generalized symmetries and master symmetries we finally show that we can use master symmetries of the 1skdv equation to construct non autonomous non integrable generalized symmetries

two matrix model with semiclassical potentials and extended whitham hierarchy we consider the two matrix model with potentials whose derivatives are arbitrary rational functions of fixed pole structure and the support of the spectra of the matrices are union of intervals hard edges we derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows us to compute derivatives of arbitrarily high order by extending classical rauch s variational formulae the four point correlation functions are explicitly worked out the formalism extends naturally to the computation of residue formulae for the tau function of the so called universal whitham hierarchy studied mainly by i krichever our setting extends the moduli space in that there are certain extra data

the rpa equation embedded into infinite dimensional fock space f to clear up both algebraic and geometric structures for integrable systems derived from self consistent field theory in particular a geometrical aspect of the random phase approximation rpa equation is presented from the viewpoint of symmetry of the evolution equation the rpa equation for an infinite dimensional grassmannian is constructed it gives us a simple geometrical interpretation that the collective submanifold is a rotator on a curved surface

the quantum probability equation iii perturbing in the energy time independent perturbation theory in the scattering domain developed on the basis of the quantum probability equation milward g c and wilkin c 2001 j phys a math gen 34 5101 is extended to study the energy dependence of the s wave scattering phase shifts and probability density to second order in the energy change at low energies the bethe formula for the effective range is reproduced and an analogous formula for the shape dependent term is also derived the energy dependence of the probability density at short distances is important in final state interaction theory and the method allows the investigation of some of the prescriptions used there results from the present approach are illustrated with soluble models

partition function for multi cut matrix models we consider the partition function z n of a random matrix model with polynomial potential v t 1 t 2 2 t 2d 2d it is known that the second logarithmic derivative of z n with respect to the times t k can be expressed in terms of the recurrence coefficients of the related orthogonal polynomials an explicit formula for the recurrence coefficients of the orthogonal polynomials in the limit n oo for multi cut regular v has been derived in 10 through the riemann hilbert approach the expression for z n in the limit n has been derived in 7 through a mean field approach we show that the above asymptotic formulae satisfy the same relations that hold for finite n

finite correlation dimension for stochastic systems with power law spectra abstract we discuss a counter example to the traditional view that stochastic processes lead to a non convergence of the correlation dimension in computed or measured time series specifically we show that a simple class of colored random noises characterized by a power law power spectrum have a finite and predictable value for the correlation dimension these results have implications on the experimental study of deterministic chaos as they indicate that the soie observation of a finite fractal dimension from the analysis of a time series is not sufficient to infer the presence of a strange attractor in the system dynamics we demonstrate that the types of random noises considered herein may be given an interpretation in terms of their fractal properties the consequent exploitation of the non gaussian behavior of these random noises leads us to the introduction of a new time series analysis method which we call multivariate scaling analysis we apply this approach to characterize several global properties of random noise

elastic scattering by deterministic and random fractals self affinity of the diffraction spectrum the diffraction spectrum of coherent waves scattered from fractal supports is calculated exactly the fractals considered are of the class generated iteratively by successive dilations and translations and include generalizations of the cantor set and sierpinski carpet as special cases also randomized versions of these fractals are treated the general result is that the diffraction intensities obey a strict recursion relation and become self affine in the limit of large iteration number with a self affinity exponent related directly to the fractal dimension of the scattering object applications include neutron scattering x rays optical diffraction magnetic resonance imaging electron diffraction and he scattering which all display the same universal scaling textcopyright 1996 the american physical society

on tricyclic graphs of a given diameter with minimal energy abstract the energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph let g n d be the class of tricyclic graphs g on n vertices with diameter d and containing no vertex disjoint odd cycles c p c q of lengths p and q with p q 2 mod 4 in this paper we characterize the graphs with minimal energy in g n d

the riemann zeta function and the one dimensional weyl berry conjecture for fractal drums based on his earlier work on the vibrations of drums with fractal boundary the first author has refined m v berry s conjecture that extended from the smooth to the fractal case h weyl s conjecture for the asymptotics of the eigenvalues of the laplacian on a bounded open subset of w see 16 we solve here in the one dimensional case that is when n 1 this modified weyl berry conjecture we discover in the process some unexpected and intriguing connections between spectral geometry fractal geometry and the riemann zeta function we therefore show that one can hear that is recover from the spectrum not only the minkowski fractal dimension of the boundary as was established previously by the first author but also under the stronger assumptions of the conjecture its minkowski content a fractal analogue of its length we also prove still in dimension one a related conjecture of the first author as well as its converse which characterizes the situation when the error estimates of the aforementioned paper are sharp

double scaling limit in random matrix models and a nonlinear hierarchy of differential equations we study the critical behaviour of a random hermitian one matrix model with nonsymmetric interaction at a critical point in which the eigenvalue density function has a zero of degree 2m m 1 inside a cut we prove that in the generic case m 1 the model exhibits a third order phase transition in temperature we formulate an ansatz for the double scaling limit of recurrence coefficients which is consistent with the quasiperiodic asymptotics of recurrence coefficients in the low temperature region and from this ansatz we derive the painleve ii hierarchy of ordinary differential equations for the recurrence coefficients in addition we derive an integral kernel which governs the double scaling limit of correlation functions

a new type of solution of the schrodinger equation on a self similar fractal potential scattering of a quantum particle by a self similar fractal potential on a cantor set is investigated we present a new type of solution of the functional equation for the transfer matrix of this potential which was derived earlier from the schrodinger equation

random matrices close to hermitian or unitary overview of methods and results the paper discusses recent progress in understanding statistical properties of eigenvalues of weakly non hermitian and non unitary random matrices the first type of ensembles is of the form i with being a large random n n hermitian matrix with independent entries deformed by a certain anti hermitian n n matrix i satisfying in the limit of large dimension n the condition tr 2 n tr 2 here can be either a random or just a fixed given hermitian matrix ensembles of such a type with 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics related models are used to mimic complex spectra of the dirac operator with chemical potential in the context of quantum chromodynamics ensembles of the second type arising naturally in scattering theory of discrete time systems are formed by n n matrices with complex entries such that i for 0 this coincides with the circular unitary ensemble and 0 i describes deviation from unitarity our result amounts to answering statistically the following old question given the singular values of a matrix describe the locus of its eigenvalues we systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non trivial crossover from wigner dyson statistics of real unimodular eigenvalues typical of hermitian unitary matrices to ginibre statistics in the complex plane typical of ensembles with strong non hermiticity tr 2 tr 2 when n finally we discuss scarce results available on eigenvector statistics for weakly non hermitian random matrices

temporal patterns of human behaviour are there signs of deterministic 1 f scaling abstract temporal patterns apparently exhibiting scaling properties may originate either from fractal stochastic processes or from causal i e deterministic dynamics in general the distinction between the possible two origins remains a non trivial task this holds especially for the interpretation of properties derived from temporal patterns of various types of human behaviour which were reported repeatedly we propose here a computational scheme based on a generic intermittency model to test predictability thus determinism of a part of a time series with knowledge gathered from another part the method is applied onto psychodynamic time series related to turns from non psychosis to psychosis a nonrandom correlation 0 76 between prediction and real outcome is found our scheme thus provides a particular kind of fractal risk assessment for this possibly deterministic process we briefly discuss possible implications of these findings to evaluate the risk to undergo a state transition in our case a patients risk to enter a next psychotic state we further point to some problems concerning data sample pecularities and equivalence between data and model setup

uncertainty relations in deformation quantization robertson and hadamard robertson theorems on non negative definite hermitian forms are generalized to an arbitrary ordered field these results are then applied to the case of formal power series fields and the heisenberg robertson robertson schrodinger and trace uncertainty relations in deformation quantization are found some conditions under which the uncertainty relations are minimized are also given

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

a unified fluctuation formula for one cut β ensembles of random matrices using a coulomb gas approach we compute the generating function of the covariances of power traces for one cut ensembles of random matrices in the limit of large matrix size this formula depends only on the support of the spectral density and is therefore universal for a large class of models this allows us to derive a closed form expression for the limiting covariances of an arbitrary one cut ensemble as particular cases of the main result we consider the classical gaussian wishart and jacobi ensembles for which we derive previously available results as well as new ones within a unified simple framework we also discuss the connections between the problem of trace fluctuations for the gaussian unitary ensemble and the enumeration of planar maps

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

graphs and matrices with maximal energy abstract given a complex m n matrix a we index its singular values as 1 a 2 a and call the value e a 1 a 2 a the energy of a thereby extending the concept of graph energy introduced by gutman koolen and moulton proved that e g n 2 1 n for any graph g of order n and exhibited an infinite family of graphs with e g v g 2 1 v g we prove that for all sufficiently large n there exists a graph g g n with e g n 3 2 2 n 11 10 this implies a conjecture of koolen and moulton we also characterize all square nonnegative matrices and all graphs with energy close to the maximal one in particular such graphs are quasi random

on the laplacian coefficients of unicyclic graphs abstract let g be a graph of order n and let p g k 0 n 1 k c k n k be the characteristic polynomial of its laplacian matrix generalizing an approach of mohar on graph transformations we show that among all connected unicyclic graphs of order n the k th coefficient c k is largest when the graph is a cycle c n and smallest when the graph is the a s n with an additional edge between two of its pendent vertices a relation to the recently established laplacian like energy of a graph is discussed

reply to comment on integrable kondo impurity in one dimensional q deformed t j models this is a reply to the comment by p schlottmann and a a zvyagin pacs numbers 71 10 fd 71 27 75 10 jm in their comment 1 schlottmann and zvyagin raise several issues regarding the nature of integrable impurities in one dimensional quantum lattice models and claim to expose false statements in our recent paper 2 in order to address these issues in a pedagogical manner we feel that it is appropriate to discuss these questions in terms of models based on the gl 2 1 invariant solution of the yang baxter equation however it is important from the outset to make it clear that the arguments we present below are general and apply to other classes of models the first point that we would like to make is that it is claimed in 1 that there are two approaches to the algebraic bethe ansatz however it appears to us that approach i described in point 2 of 1 is the coordinate bethe ansatz and we do not understand why this should be referred to as an algebraic bethe ansatz in the coordinate bethe ansatz approach one starts with a prescribed hamiltonian and then solves the schrodinger equation to obtain the two particle scattering matrices and the particle impurity scattering matrix together these scattering matrices form the monodromy matrix it is impossible to infer the hamiltonian from such a monodromy matrix in this context we do not feel that point 1 of 1 answers our query about the existence of the impurity monodromy matrix in the algebraic approach ii of point 2 in 1 hereafter we focus our attention on this case the solution of the yang baxter equation r12 u v r13 u r23 v r23 v r13 u r12 u v 1 associated with the lie superalgebra gl 2 1 is an operator r u end v v where v is a three dimensional z2 graded space with one bosonic and two fermionic degrees of freedom explicitly this operator takes the form r u u i i p 2 where p is the z2 graded permutation operator for the purposes of constructing integrable one dimensional quantum systems on a closed lattice it is usual to introduce the yang baxter 0305 4470 02 296197 05 30 00 2002 iop publishing ltd printed in the uk 6197

an integrating factor matrix method to find first integrals in this paper we develop an integrating factor matrix method to derive conditions for the existence of first integrals we use this novel method to obtain first integrals along with the conditions for their existence for two and three dimensional lotka volterra systems with constant terms the results are compared to previous results obtained by other methods

resurgence in quasi classical scattering in quasi classical studies of closed systems e g a billiard resurgence means that the contribution of long periodic orbits to the spectral determinant can be expressed in terms of composites of short orbits and the resulting expression for the determinant is manifestly real the question has thus long been posed whether something like resurgence applies to a scattering system with its resonances we find here a resurgent expression for wigner s r matrix which gives the s matrix by a cayley transform in which long scattering pseudo orbits are expressed in terms of composites of short pseudo orbits both scattering and periodic and the result is manifestly hermitian giving a unitary expression for the s matrix this is particularly useful in the case that the resonance width is comparable with the resonance spacing the pseudo orbits are defined in terms of a fictitious and to some extent arbitrary closed reference system we relate the results to other formulations we give a simple but non trivial approximation for a particular example which illustrates the phenomenon of resonance trapping

scattering on fractal measures we study the one dimensional potential scattering problem when the potential is a radon measure with compact support we show that the usual reflection and transmission amplitude r p and t p of an incoming wave are well defined we also show that the scattering problem on fractal potentials can be obtained as a limit case of scattering on smooth potentials we then explain how to retrieve the fractal 2 wavelet dimension and or the correlation dimension of the potential by means of the reflexion amplitude r p we study the particular case of self similar measures and show that under some general conditions r p has a large scale renormalization a numerical application is presented

lower bounds of nikiforov s energy over digraphs abstract the energy of a graph g is defined as e g i 1 n i where 1 2 n are the eigenvalues of the adjacency matrix of g this concept was extended by nikiforov 8 to digraphs as n d i 1 n i where d is a digraph with n vertices and singular values 1 n upper bounds of n were found by kharaghani and tayfeh rezaie 4 in this work we find lower bounds of n over the set of digraphs

replica equivalence in the edwards anderson model after introducing and discussing the link overlap between spin configurations we show that the edwards anderson model has a replica equivalent quenched equilibrium state a property introduced by parisi in the description of the mean field spin glass phase which generalizes ultrametricity our method is based on the control of fluctuations through the property of stochastic stability and works for all the finite dimensional spin glass models

eigenvalues distribution for products of independent spherical ensembles we consider the product of independent spherical ensembles by the special structure of eigenvalues as a rotation invariant determinant point process we show that the empirical spectral distribution of the product converges with probability one to a non random distribution and the limiting eigenvalue distribution is a power of spherical law we also present an interesting correspondence between the eigenvalues of three classes of random matrix ensembles and zeros of random polynomials

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

the diamagnetic coulomb problem an eigenvalue problem with two singularities by using the angular oblate spheroidal functions as basis functions a bounded wavefunction is constructed in the singularities of the schrodinger equation of the diamagnetic coulomb problem with infinite nuclear mass the expansion in terms of these functions is a model to resolve singularities in an eigenvalue problem of non separable partial differential equations of non relativistic quantum mechanics a comprehensive asymptotic analysis reveals the complete set of asymptotic solutions makes possible a uniform numerical treatment of the bound autoionizing continuum and continuum levels and indicates how to find hitherto unknown low lying stationary levels an example the splitting of the ground level has been found numerically by an iterative shooting method

lie discrete symmetries of lattice equations we extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential difference equations as an example of the application of the methods we construct the discrete symmetries of the discrete painleve i equation and of the toda lattice equation

on the integrability of a new discrete nonlinear schrodinger equation we consider the nonlinear schrodinger equation on the lattice introduced by leon and manna two years ago to describe the slowly varying envelope approximation of some nonlinear differential difference equations onlinear klein gordon chain etc we show that this equation does not admit local generalized symmetries of order greater than three in such a way we prove that the leon and manna discrete nonlinear schrodinger equation does not have the same integrability properties as the toda lattice equation from which it has been derived at the end we provide some reasoning to justify the result obtained

continuous symmetries of the lattice potential kdv equation in this paper we present a set of results on the integration and on the symmetries of the lattice potential korteweg de vries lpkdv equation using its associated spectral problem we construct the soliton solutions and the lax technique enables us to provide infinite sequences of generalized symmetries finally using a discrete symmetry of the lpkdv equation we construct a large class of non autonomous generalized symmetries

asymptotics of finite system lyapunov exponents for some random matrix ensembles for products pn of n random matrices of size there is a natural notion of finite n lyapunov exponents in the case of standard gaussian random matrices with real complex or real quaternion elements and extended to the general variance case for methods known for the computation of are used to compute the large n form of the variances of the exponents analogous calculations are performed in the case that the matrices making up pn are products of sub blocks of random unitary matrices with haar measure furthermore we make some remarks relating to the coincidence of the lyapunov exponents and the stability exponents relating to the eigenvalues of pn

note on unicyclic graphs with given number of pendent vertices and minimal energy abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let g n p denote the set of unicyclic graphs with n vertices and p pendent vertices in h hua m wang unicyclic graphs with given number of pendent vertices and minimal energy linear algebra appl 426 2007 478 489 hua and wang discussed the graphs that have minimal energy in g n p and determined the minimal energy graphs among almost all different cases of n and p in their work certain case of the values was left as an open problem in which the minimal energy species have to be chosen in two candidate graphs but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials this paper aims at solving the problem completely by using the well known coulson integral formula

remarks on the energy of regular graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix this note is about the energy of regular graphs it is shown that graphs that are close to regular can be made regular with a negligible change of the energy also a k regular graph can be extended to a k regular graph of a slightly larger order with almost the same energy as an application it is shown that for every sufficiently large n there exists a regular graph g of order n whose energy g satisfies g 1 2 n 3 2 n 13 10 several infinite families of graphs with maximal or submaximal energy are given and the energy of almost all regular graphs is determined

on ordering of complements of graphs with respect to matching numbers we consider an earlier much studied quasi order defined in terms of matching numbers of graphs and apply it to graph complements we establish four transformations on the complements of graphs that increase or decrease the matching numbers accordingly several applications of these results are put forward

propagation of dark solitons in a system of coupled higher order nonlinear schrodinger equations in this paper we consider the higher order linear and nonlinear effects in optics and analyse a coupled system of higher order nonlinear schrodinger equations to identify the conditions for dark soliton propagation through painleve analysis to supplement the known bright soliton conditions we also construct the explicit lax pair and the hirota bilinear form is used to generate one and two dark solitons

scaling properties in temporal patterns of schizophrenia investigations into the patterns of schizophrenia reveal evidence of scaling properties in temporal behaviour this is shown in the spectral properties of mid range and long range up to two years daily recordings from a sample of patients drawn at the therapeutic dwelling soteria ambuhl et al in springer series in synergetics vol 58 eds tschacher et al springer berlin 1992 pp 195 203 and references therein of the psychiatric university hospital in bern the therapeutic setting is unique in that it tries to avoid treatment by medication power law behaviour has been found within fractal walk analysis and fourier spectra for the daily fluctuations a simple dynamic principle based on a generic intermittency model is put in relation to these time series thus predicting an additional scaling law for the distribution p t of time spans t between successive hospitalizations testing this hypothesis with our data shows only insignificant deviations a possible role of this dynamic principle in the risk assignment of psychotic phases is explored with the help of an example

complex curve of the two matrix model and its tau function we study the hermitian and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports its planar graph interpretation is given the study reveals a general structure of the underlying analytic complex curve different from the hyperelliptic curve of the one matrix model the matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution as a function of filling numbers and coefficients of the matrix potential is shown to be a quasiclassical tau function the relation to 1 supersymmetric yang mills theories is discussed a general class of solvable multi matrix models with tree like interactions is considered

matrix models with hard walls geometry and solutions we discuss various aspects of most general multisupport solutions to matrix models in the presence of hard walls i e in the case where the eigenvalue support is confined to subdomains of the real axis the structure of the solution at the leading order is described by semiclassical or generalized whitham krichever hierarchies as in the unrestricted case derivatives of tau functions for these solutions are associated with families of riemann surfaces with possible double points and satisfy the witten dijkgraaf verlinde verlinde equations we then develop the diagrammatic technique for finding free energy of this model in all orders of the t hooft expansion in the reciprocal matrix size generalizing the feynman diagrammatic technique for the hermitian one matrix model due to eynard