periodic potentials and pt symmetry we enlarge the limited class of analytically solvable one dimensional periodic potentials with a finite number of band gaps by considering the complex pt symmetric generalized associated lame gal potential with four parameters and applying supersymmetry we obtain three duality relations which relate the band edge eigenstates of the twenty four gal potentials obtained by permutation of the four parameters

graphical representations and cluster algorithms i discrete spin systems abstract graphical representations similar to the fk representation are developed for a variety of spin systems in several cases it is established that these representations have fkg monotonicity properties which enables characterization theorems for the uniqueness phase and the low temperature phase of the spin system certain systems with intermediate phases and or first order transitions are also described in terms of the percolation properties of the representations in all cases these representations lead in a natural fashion to swendsen wang type algorithms hence at least in the above mentioned instances these algorithms realize the program described by kandel and domany phys rev b 43 1991 8539 8548 all of the algorithms are shown to satisfy a li sokal bound which at least for systems with a divergent specific heat implies critical slowing down however the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points

the classical schrodinger equation prufer s spectral algorithm is applied to the classical analogue of schrodinger s equation the discrete spectrum is interpreted as a bifurcation phenomenon caused by two simultaneous classical motions rotation and squeezing the energy eigenvalues coincide with the bifurcation parameters for the classical orbits

a number of quasiexactly solvable n body problems we present several examples of quasiexactly solvable n body problems in one two and higher dimensions we study various aspects of these problems in some detail in particular we show that in some of these examples the corresponding polynomials form an orthogonal set and many of their properties are similar to those of the bender dunne polynomials we also discuss qes problems where the polynomials do not form an orthogonal set

a new effective mass hamiltonian and associated lame equation bound states a new quantum model with rational functions for the potential and effective mass is proposed in a stretchable region outside which both are constant starting from a generalized effective mass kinetic energy operator the matching and boundary conditions for the envelope wavefunctions are derived it is shown that in a mapping to an auxiliary constant mass schrodinger picture one obtains one period associated lame well bounded by two wells or barriers depending on the values of the ordering parameter the results for bound states of this new solvable model are provided for a wide variation of the parameters

discrete nonlinear schrodinger equations with arbitrarily high order nonlinearities a class of discrete nonlinear schrodinger equations with arbitrarily high order nonlinearities is introduced these equations are derived from the same hamiltonian using different poisson brackets and include as particular cases the saturable discrete nonlinear schrodinger equation and the ablowitz ladik equation as a common property these equations possess three kinds of exact analytical stationary solutions for which the peierls nabarro barrier is zero several properties of these solutions including stability discrete breathers and moving solutions are investigated

random cluster representation for the blume capel model we present in this paper a way to perform the mapping of the spin 1 blume capel model into a random cluster model and analyze thermodynamic properties of the former model in terms of geometric properties of clusters generated in the random cluster representation it is shown that there are two different relevant types of cluster and that one of them is the exact analogue of the type of cluster generated in the ising model we use this result to derive expressions for thermodynamical properties on the second order transition line which are equivalent to the ones found in the ising model the other type of cluster is responsible for the first order transitions and we may see the tricritical point as a point where both types of cluster compete on the same footing

analytical results of zero gap states in periodic potentials we develop a method to construct various classes of one dimensional periodic potentials with two intersecting energy bands analytical exact results for the zero gap states are presented in an explicit form under certain parameter conditions the position of the energies of these zero gap states in the energy bands is identified numerically

on the families of orthogonal polynomials associated to the razavy potential we show that there are two different families of weakly orthogonal polynomials associated to the quasi exactly solvable razavy potential v x cos 2x m 2 0 m one of these families encompasses the four sets of orthogonal polynomials recently found by khare and mandal while the other one is new these results are extended to the related periodic potential u x cos 2x m 2 for which we also construct two different families of weakly orthogonal polynomials we prove that either of these two families yields the ground state when m is odd and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the m 1 th gap and having the same parity as m 1 moreover we show that the algebraic eigenfunctions obtained in this way are the well known finite solutions of the whittaker hill or hill s three term periodic differential equation thus the foregoing results provide a lie algebraic justification of the fact that the whittaker hill equation unlike for instance mathieu s equation admits finite solutions

new families of isospectral hydrogen like potentials by applying algebraic techniques we construct a two parametric family of strictly isospectral hydrogen like potentials as well as some of its one parametric limits an additional one parametric almost isospectral family of hydrogen like potentials is also investigated it is argued that the construction of a susy partner hamiltonian using a factorization energy less than the ground state energy of the departure hamiltonian is unnecessarily restrictive

complex periodic potentials with a finite number of band gaps we obtain several new results for the complex generalized associated lame potential v x a a 1 msn2 y m b b 1 msn2 y k m m f f 1 msn2 y k m ik m m g g 1 msn2 y ik m m where y x k m 2 ik m 2 sn y m is the jacobi elliptic function with modulus parameter m and there are four real parameters a b f g first we derive two new duality relations which when coupled with a previously obtained duality relation permit us to relate the band edge eigenstates of the 24 potentials obtained by permutations of the parameters a b f g second we pose and answer the question how many independent potentials are there with a finite number a of band gaps when a b f g are integers and a b f g 0 for these potentials we clarify the nature of the band edge eigenfunctions we also obtain several analytic results when at least one of the four parameters is a half integer as a by product we also obtain new solutions of heun s differential equation

periodic solutions of nonlinear equations obtained by linear superposition we show that a type of linear superposition principle works for several nonlinear differential equations using this approach we find periodic solutions of the kadomtsev petviashvili equation the nonlinear schrodinger equation the 4 model the sine gordon equation and the boussinesq equation by making appropriate linear superpositions of known periodic solutions this unusual procedure for generating solutions of nonlinear differential equations is successful as a consequence of some powerful recently discovered cyclic identities satisfied by the jacobi elliptic functions

discontinuity of the magnetization in one dimensional 1 x y 2 ising and potts models results from percolation theory are used to study phase transitions in one dimensional ising and q state potts models with couplings of the asymptotic form j sub x y approx const absolute value x y sup 2 for translation invariant systems with well defined lim sub x infinity x sup 2 j sub x j sup possibly o or infinity we establish 1 there is no long range order at inverse temperatures beta with beta j sup less than or equal to1 2 if beta j sup q then by sufficiently increasing j sub 1 the spontaneous magnetization m is made positive 3 in models with 0 j sup infinity the magnetization is discontinuous at the transition point as originally predicted by thouless and obeys m beta sub c less than or equal to 1 beta sub c j sup sup 1 2 4 for ising q 2 models with j sup infinity it is noted that the correlation function decays as beta approx c beta absolute value x y sup 2 whenever beta beta sub c points 1 3 are deduced from previous percolation results by utilizing the fortuin kasteleyn representation which also yields other results of independent interest relating potts models with different values of q

quasi exactly solvable extensions of the lame equation quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors for a few of these equations such as the lame equation the number of algebraic solutions is larger than predicted by the general theory a class of quasi exactly solvable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed families of equations which by themselves are not quasi exactly solvable but can be transformed into systems of coupled quasi exactly solvable equations are discussed these results suggest a scheme for the classification of quasi exactly solvable systems in one variable

extremal laplacian energy of directed trees unicyclic digraphs and bicyclic digraphs abstract let a g and d g be the adjacency matrix of a digraph g with n vertices and the diagonal matrix of vertex outdegrees of g respectively then the laplacian matrix of the digraph g is l g d g a g the laplacian energy of a digraph g is defined as l e g i 1 n i 2 by using second spectral moment where 1 2 n are all the eigenvalues of l g of g in this paper by using arc shifting operation and out star shifting operation we determine the directed trees unicyclic digraphs and bicyclic digraphs which attain maximal and minimal laplacian energy among all digraphs with n vertices respectively

the skew energy of a digraph we are interested in the energy of the skew adjacency matrix of a directed graph d which is simply called the skew energy of d in this paper properties of the skew energy of d are studied in particular a sharp upper bound for the skew energy of d is derived in terms of the order of d and the maximum degree of its underlying undirected graph an infinite family of digraphs attaining the maximum skew energy is constructed moreover the skew energy of a directed tree is independent of its orientation and interestingly it is equal to the energy of the underlying undirected tree skew energies of directed cycles under different orientations are also computed some open problems are presented

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

anti isospectral transformations orthogonal polynomials and quasi exactly solvable problems we consider the double sinh gordon potential which is a quasi exactly solvable problem and show that in this case one has two sets of bender dunne orthogonal polynomials we study in some detail the various properties of these polynomials and the corresponding quotient polynomials in particular we show that the weight functions for these polynomials are not always positive we also study the orthogonal polynomials of the double sine gordon potential which is related to the double sinh gordon case by an anti isospectral transformation finally we discover a new quasi exactly solvable problem by making use of the anti isospectral transformation

linear superposition in nonlinear equations even though the kdv and modified kdv equations are nonlinear we show that suitable linear combinations of known periodic solutions involving jacobi elliptic functions yield a large class of additional solutions this procedure works by virtue of some remarkable new identities satisfied by the elliptic functions

a sharp transition for the two dimensional ising percolation we show that the percolation transition for the two dimensional ising model is sharp namely we show that for every reciprocal temperature 0 there exists a critical valueh c of external magnetic fieldh such that the following two statements hold we also shows that the percolation probability is continuous in h except on the half line 0 c

quasi exact solvability and the direct approach to invariant subspaces we propose a more direct approach to constructing differential operators that preserve polynomial subspaces than the one based on considering elements of the enveloping algebra of this approach is used here to construct new exactly solvable and quasi exactly solvable quantum hamiltonians on the line which are not lie algebraic it is also applied to generate potentials with multiple algebraic sectors we discuss two illustrative examples of these two applications we show that the generalized lame potential possesses four algebraic sectors and describe a quasi exactly solvable deformation of the morse potential which is not lie algebraic

on the uniqueness of the infinite occupied cluster in dependent two dimensional site percolation we consider dependent site percolation on the two dimensional square lattice the underlying probability measure being invariant and ergodic under each of the translations and invariant under axis reflections if this measure satisfies the fkg condition and if percolation occurs then we show that the infinite occupied cluster is unique with probability 1 and that all vacant star clusters are finite

5 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be the corresponding oriented graph of g with the orientation the skew energy of g denoted by e s g is defined as the sum of the singular values of the skew adjacency matrix s g in 2010 adiga et al certified that s g n where is the maximum degree of g of order n it has been shown that every 5 regular oriented graph with optimum skew energy has even neighborhood property that is each pair of neighborhoods of a graph have even number of common vertices in this paper we characterize all connected 5 regular graphs of order n with this property moreover we determine all connected 5 regular oriented graphs of order n with maximum skew energy

pt symmetry and its spontaneous breakdown in three dimensions the mechanism of introducing non hermiticity to non central symmetric potentials through both the azimuth and polar angles is discussed generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials such as the scarf i or rosen morse i potentials are used in the eigenvalue equation of the polar component it is shown that the spontaneous breakdown of symmetry can also occur in these non central potentials several potentials are proposed in the azimuthal eigenvalue equation too where the use of periodic boundary conditions is essential possible generalizations of the results are outlined

nonlocal supersymmetric deformations of periodic potentials irreducible second order darboux transformations are applied to the periodic schrodinger operators it is shown that for the pairs of factorization energies inside the same forbidden band they can create new nonsingular potentials with periodicity defects and bound states embedded in the spectral gaps the method is applied to the lame and periodic piece wise transparent potentials an interesting phenomenon of translational darboux invariance reveals nonlocal aspects of the supersymmetric deformations

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

exact static solutions to a translationally invariant discrete phi4 model for a discrete translationally invariant 4 model introduced by barashenkov et al 2005 phys rev e 72 35602r we provide the momentum conservation law and demonstrate how the first integral of the static version of the discrete model can be constructed from a jacobi elliptic function jef solution the first integral can be written in the form of a nonlinear map from which the static solution supported by the model can be constructed a set of jef solutions including the staggered ones is derived we also report on the stability analysis for the static bounded solutions and exemplify the dynamical behaviour of the unstable solutions this work provides a road map through this illustrative example on how to fully analyse translationally invariant models in terms of their static problem its first integral their full set of static solutions and associated conservation laws

dynamical blume capel model competing metastable states at infinite volume this paper concerns the microscopic dynamical description of competing metastable states we study at infinite volume and very low temperature metastability and nucleation for kinetic blume capel model a ferromagnetic lattice model with spins taking three possible values 1 0 1 in a previous paper mo we considered a simplified irreversible nucleation growth model in the present paper we analyze the full blume capel model we choose a region u of the thermodynamic parameters such that everywhere in u b 1 all minuses corresponds to the highest in energy metastable state b 0 all zeroes corresponds to an intermediate metastable state and b 1 all pluses corresponds to the stable state we start from b 1 and look at a local observable like in mo we find that when crossing a special line in u there is a change in the mechanism of transition towards the stable state b 1 we pass from a situation 1 where the intermediate phase b 0 is really observable before the final transition with a permanence in b 0 typically much longer than the first hitting time to b 0 to the situation 2 where b 0 is not observable since the typical permanence in b 0 is much shorter than the first hitting time to b 0 and moreover large growing 0 droplets are almost full of 1 in their interior so that there are only relatively thin layers of zeroes between 1 and 1

supersymmetry and potentials with bound states at arbitrary energies ii for pt i see ibid vol 19 p 2297 1986 it has been shown previously that a potential v0 chi in one dimension which supports no bound states may be used as a reference potential from which by successive applications of the concept of a supersymmetric partner to a given hamiltonian it is possible to find a potential vn chi which supports any specified number n of bound states at any chosen energies ej j 1 n the reflection coefficient of vn is related to the reflection coefficient of v0 various alternative representations of the potentials constructed by this procedure are presented an illustrative example in which vn is constructed by using a sech2 chi barrier as the reference potential is discussed

random cluster representation of the blume capel model the so called diluted random cluster model may be viewed as a random cluster representation of the blume capel model it has three parameters a vertex parameter a an edge parameter p and a cluster weighting factor q stochastic comparisons of measures are developed for the vertex marginal when q 1 2 and the edge marginal when q 1 taken in conjunction with arguments used earlier for the random cluster model these permit a rigorous study of part of the phase diagram of the blume capel model

exact moving and stationary solutions of a generalized discrete nonlinear schrodinger equation we obtain exact moving and stationary spatially periodic and localized solutions of a generalized discrete nonlinear schrodinger equation more specifically we find two different moving periodic wave solutions and a localized moving pulse solution we also address the problem of finding exact stationary solutions and for a particular case of the model when stationary solutions can be expressed through the jacobi elliptic functions we present a two point map from which all possible stationary solutions can be found numerically we demonstrate the generic stability of the stationary pulse solutions and also the robustness of moving pulses in long term dynamics

exactly solvable hydrogen like potentials and the factorization method a set of factorization energies is introduced giving rise to a generalization of the schrodinger or infeld and hull factorization for the radial hydrogen like hamiltonian an algebraic intertwining technique involving such factorization energies leads us to derive n parametric families of potentials which in general are almost isospectral to the hydrogen like radial hamiltonians the construction of susy partner hamiltonians with ground state energies greater than the corresponding ground state energy of the initial hamiltonian is also explicitly performed

mean field driven first order phase transitions in systems with long range interactions we consider a class of spin systems on d with vector valued spins sx that interact via the pair potentials jx ysx sy the interactions are generally spread out in the sense that the jx y s exhibit either exponential or power law fall off under the technical condition of reflection positivity and for sufficiently spread out interactions we prove that the model exhibits a first order phase transition whenever the associated mean field theory signals such a transition as a consequence e g in dimensions d 3 we can finally provide examples of the 3 state potts model with spread out exponentially decaying interactions which undergoes a first order phase transition as the temperature varies similar transitions are established in dimensions d 1 2 for power law decaying interactions and in high dimensions for next nearest neighbor couplings in addition we also investigate the limit of infinitely spread out interactions specifically we show that once the mean field theory is in a unique state then in any sequence of translation invariant gibbs states various observables converge to their mean field values and the states themselves converge to a product measure

hierarchy of qm susys on a bounded domain we systematically formulate a hierarchy of isospectral hamiltonians in one dimensional supersymmetric quantum mechanics on an interval and on a circle in which two successive hamiltonians form supersymmetry we find that boundary conditions compatible with supersymmetry are severely restricted in the case of an interval a hierarchy of at most three isospectral hamiltonians is possible with unique boundary conditions while in the case of a circle an infinite tower of isospectral hamiltonians can be constructed with a two parameter family of boundary conditions

new classes of quasi solvable potentials their exactly solvable limit and related orthogonal polynomials we have generated using an sl 2 r lie algebraic formalism several new classes of quasi solvable elliptic potentials which in the appropriate limit go over to the exactly solvable forms we have obtained exact solutions of the corresponding spectral problem for some real values of the potential parameters we have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable

3 regular digraphs with optimum skew energy abstract the skew energy of a digraph d is defined as the sum of the singular values of its skew adjacency matrix s d in this paper we first interpret the entries of the power of the skew adjacency matrix of a digraph in terms of the number of its walks and then focus on the question posed by adiga et al c adiga r balakrishnan wasin so the skew energy of a graph linear algebra appl 432 2010 1825 1835 of determining all 3 regular connected digraphs with optimum skew energy

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

some exact results for mid band and zero band gap states of associated lame potentials applying certain known theorems about one dimensional periodic potentials we show that the energy spectrum of the associated lame potentials a a 1 msn2 x m b b 1 m cn2 x m dn2 x m consists of a finite number of bound bands followed by a continuum band when both a and b take integer values further if a and b are unequal integers we show that there must exist some zero band gap states i e doubly degenerate states with the same number of nodes more generally in case a and b are not integers but either a b or a b is an integer a b we again show that several of the band gaps vanish due to degeneracy of states with the same number of nodes finally when either a or b is an integer and the other takes a half integral value we obtain exact analytic solutions for several mid band states

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations for a number of nonlocal nonlinear equations such as nonlocal nonlinear schrodinger equation nlse nonlocal ablowitz ladik al nonlocal saturable discrete nlse dnlse coupled nonlocal nlse coupled nonlocal al and coupled nonlocal saturable dnlse we obtain periodic solutions in terms of jacobi elliptic functions as well as the corresponding hyperbolic soliton solutions remarkably in all the six cases we find that unlike the corresponding local cases all the nonlocal models simultaneously admit both the bright and the dark soliton solutions further in all the six cases not only the elliptic functions dn x m and cn x m with modulus m but also their linear superposition is shown to be an exact solution finally we show that the coupled nonlocal nlse not only admits solutions in terms of lame polynomials of order 1 but also admits solutions in terms of lame polynomials of order 2 even though they are not the solution of the uncoupled nonlocal problem we also remark on the possible int

cyclic identities for jacobi elliptic and related functions identities involving cyclic sums of terms composed from jacobi elliptic functions evaluated at p equally shifted points on the real axis were recently found these identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations we derive four master identities from which the identities discussed earlier are derivable as special cases master identities are also obtained which lead to cyclic identities with alternating signs we discuss an extension of our results to pure imaginary and complex shifts as well as to the ratio of jacobi theta functions

uniform spanning forests we study uniform spanning forest measures on infinite graphs which are weak limits of uniform spanning tree measures from finite subgraphs these limits can be taken with free fsf or wired wsf boundary conditions pemantle proved that the free and wired spanning forests coincide in z d and that they give a single tree iff d 4 in the present work we extend pemantle s alternative to general graphs and exhibit further connections of uniform spanning forests to random walks potential theory invariant percolation and amenability the uniform spanning forest model is related to random cluster models in statistical physics but because of the preceding connections its analysis can be carried further among our results are the following the fsf and wsf in a graph g coincide iff all harmonic dirichlet functions on g are constant the tail fields of the wsf and the fsf are trivial on any graph on any cayley graph that is not a finite extension of z all component trees of the wsf have one end this is new in z d for d 5 on any tree as well as on any graph with spectral radius less than 1 a s all components of the wsf are recurrent the basic topology of the free and the wired uniform spanning forest measures on lattices in hyperbolic space h d is analyzed a cayley graph is amenable iff for all e 0 the union of the wsf and bernoulli percolation with parameter e is connected harmonic measure from infinity is shown to exist on any recurrent proper planar graph with finite codegrees we also present numerous open problems and conjectures

new quasi exactly solvable periodic potentials we present two new classes of quasi exactly solvable periodic potentials it is shown that their solutions are given in terms of a known special function when the parameters in the two periodic potentials satisfy a certain condition some of the bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first brillouin zone two fundamental sets of solutions have been constructed to calculate all the bloch waves and energy bands

periodic potentials and supersymmetry we review the current status of one dimensional periodic potentials and also present several new results it is shown that using the formalism of supersymmetric quantum mechanics one can considerably enlarge the limited class of analytically solvable one dimensional periodic potentials further using the landen transformations as well as cyclic identities for jacobi elliptic functions discovered by us recently it is shown that a linear superposition of lame as well as associated lame potentials is also analytically solvable finally using anti isospectral transformations we also obtain a class of analytically solvable complex pt invariant periodic potentials having real band spectra

second order darboux displacements the potentials for a one dimensional schrodinger equation that are displaced along the x axis under second order darboux transformations called 2 susy invariant are characterized in terms of a differential difference equation the solutions of the schrodinger equation with such potentials are given analytically for any value of the energy the method is illustrated by a two soliton potential it is proved that a particular case of the periodic lame ince potential is 2 susy invariant both bloch solutions of the corresponding schrodinger equation are found for any value of the energy a simple analytic expression for a family of two gap potentials is derived

decay of correlations in random cluster models we prove exponential decay for the tail of the radius r of the cluster at the origin for subcritical random cluster models under an assumption slightly weaker than that here d is the number of dimensions specifically if throughout the subcritical phase then for some 0 this implies the exponential decay of the two point correlation function of subcritical potts models subject to a hypothesis of at least polynomial decay of this function similar results are known already for percolation and ising models and for potts models when the number q of available states is sufficiently large indeed the hypothesis of polynomial decay has been proved rigorously for these cases in two dimensions the hypothesis that is weaker than requiring that the susceptibility be finite i e that the two point function be summable the principal new technique is a form of russo s formula for random cluster models reported by bezuidenhout grimmett and kesten for the current application this leads to an analysis of a first passage problem for random cluster models and a proof that the associated time constant is strictly positive if and only if the tail of r decays exponentially

coexistence of infinite clusters ii ising percolation in two dimensions we show a strong type of conditionally mixing property for the gibbs states ofd dimensional ising model when the temperature is above the critical one by using this property we show that there is always coexistence of infinite and clusters when is smaller than c andh 0 in two dimensions it is also possible to show that this coexistence region extends to some non zero external field case i e for every c there exists someh c 0 such that h h c implies the coexistence of infinite clusters with respect to the gibbs state for h

the asymmetric random cluster model and comparison of ising and potts models we introduce the asymmetric random cluster or arc model which is a graphical representation of the potts lattice gas and establish its basic properties the arc model allows a rich variety of comparisons in the fkg sense between models with different parameter values we give for example values h for which the 0 s configuration in the potts lattice gas is dominated by the configuration of the h ising model the potts model with possibly an external field applied to one of the spins is a special case of the potts lattice gas which allows our comparisons to yield rigorous bounds on the critical temperatures of potts models for example we obtain 0 571 1 exp c 0 600 for the 9 state potts model on the hexagonal lattice another comparison bounds the movement of the critical line when a small potts interaction is added to a lattice gas which otherwise has only interparticle attraction arc models can also be compared to related models such as the partial fk model obtained by deleting a fraction of the nonsingleton clusters from a realization of the fortuin kasteleyn random cluster model this comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the potts model

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

4 regular oriented graphs with optimum skew energy abstract let g be a simple undirected graph and g be an oriented graph of g with the orientation and skew adjacency matrix s g the skew energy of the oriented graph g denoted by e s g is defined as the sum of the absolute values of all the eigenvalues of s g in this paper we characterize the underlying graphs of all 4 regular oriented graphs with optimum skew energy and give orientations of these underlying graphs such that the skew energy of the resultant oriented graphs indeed attain optimum it should be pointed out that there are infinitely many 4 regular connected optimum skew energy oriented graphs while the 3 regular case only has two graphs k 4 the complete graph on 4 vertices and q 3 the hypercube

reply to the comment on generalization of the darboux transformation and generalized harmonic oscillators in the comment on our paper schulze halberg introduces a gauge transformation to gauge away the linear term in momentum he explicitly shows that the gauge transformation works for n fold darboux transformed system the comment is however entirely on the formalism of the darboux transformation while most pages of our paper are devoted to the applications of the formalism in this reply we add some considerations so that his gauge transformation method can be used to reproduce the solvable hermitian hamiltonian systems given in our paper

superposition of elliptic functions as solutions for a large number of nonlinear equations for a large number of nonlinear equations both discrete and continuum we demonstrate a kind of linear superposition we show that whenever a nonlinear equation admits solutions in terms of both jacobi elliptic functions cn x m and dn x m with modulus m then it also admits solutions in terms of their sum as well as difference we have checked this in the case of several nonlinear equations such as the nonlinear schrodinger equation mkdv a mixed kdv mkdv system a mixed quadratic cubic nonlinear schrodinger equation the ablowitz ladik equation the saturable nonlinear schrodinger equation 4 the discrete mkdv as well as for several coupled field equations further for a large number of nonlinear equations we show that whenever a nonlinear equation admits a periodic solution in terms of dn2 x m it also admits solutions in terms of dn 2 x m m cn x m dn x m even though cn x m dn x m is not a solution of these nonlinear equations finally we also obtain superposed solutions of variou

partition function zeros at first order phase transitions pirogov sinai theory this paper is a continuation of our previous analysis 2 of partition functions zeros in models with first order phase transitions and periodic boundary conditions here it is shown that the assumptions under which the results of ref 2 were established are satisfied by a large class of lattice models these models are characterized by two basic properties the existence of only a finite number of ground states and the availability of an appropriate contour representation this setting includes for instance the ising potts and blume capel models at low temperatures the combined results of ref 2 and the present paper provide complete control of the zeros of the partition function with periodic boundary conditions for all models in the above class

cyclic identities involving jacobi elliptic functions we derive a number of local identities involving jacobi elliptic functions and use them to obtain several new results first we present an alternative simpler derivation of the cyclic identities discovered by us recently along with an extension to several new cyclic identities second we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp 2i s wheres is any integer third we systematize the local identities by deriving four local master identities analogous to the master identities for the cyclic sums discussed by us previously fourth we point out that many of the local identities can be thought of as exact discretizations of standard non linear differential equations satisfied by the jacobi elliptic functions finally we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving jacobi elliptic functions

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

quasi exactly solvable potentials on the line and orthogonal polynomials in this paper we show that a quasi exactly solvable normalizable or periodic one dimensional hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three term recursion relation in particular we prove that normalizable exactly solvable one dimensional systems are characterized by the fact that their associated polynomials satisfy a two term recursion relation we study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one dimensional quasi exactly solvable hamiltonian showing in particular that its associated stieltjes measure is supported on a finite set from this we deduce that the corresponding moment problem is determined and that the kth moment grows like the kth power of a constant as k tends to infinity we also show that the moments satisfy a constant coefficient linear difference equation and that this property actually characterizes weakly orthogonal polynomial systems

new exactly solvable isospectral partners for symmetric potentials we examine in detail the possibility of applying the darboux transformation to non hermitian hamiltonians in particular we propose a simple method of constructing exactly solvable symmetric potentials by applying darboux transformation to higher states of an exactly solvable symmetric potential it is shown that the resulting hamiltonian and the original one are pseudo supersymmetric partners we also discuss application of the darboux transformation to hamiltonians with spontaneously broken symmetry

on the critical behavior of the magnetization in high dimensional ising models we derive rigorously general results on the critical behavior of the magnetization in ising models as a function of the temperature and the external field for the nearest neighbor models it is shown that ind 4 dimensions the magnetization is continuous att c and its critical exponents take the classical values 3 and 1 2 with possible logarithmic corrections atd 4 the continuity and other explicit bounds formally extend tod 3 1 2 other systems to which the results apply include long range models ind 1 dimension with 1 x y couplings for which 2 1 replacesd in the above summary the results are obtained by means of differential inequalities derived here using the random current representation which is discussed in detail for the case of a nonvanishing magnetic field

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

on the skew energy of orientations of hypercubes abstract let d be a digraph of order n and 1 2 n denote all the eigenvalues of the skew adjacency matrix of d the skew energy e s d of d is defined as e s d i 1 n i in this paper it is proved that for any positive integer k 3 there exists a k regular graph of order n having an orientation d with e s d n k this work positively answers a problem proposed by adiga et al c adiga r balakrishnan wasin so the skew energy of a digraph linear algebra appl 432 2010 1825 1835 in addition a digraph is also constructed such that its skew energy is the same as the energy of its underlying graph

on the singularity of the free energy at a first order phase transition at first order phase transition the free energy does not have an analytic continuation in the thermodynamical variable which is conjugate to an order parameter for the transition this result is proved at low temperature for lattice models with finite range interaction and two periodic ground states under the only condition that they satisfy the peierls condition

the stochastic random cluster process and the uniqueness of random cluster measures the random cluster model is a generalization of percolation and ferromagnetic potts models due to fortuin and kasteleyn not only is the random cluster model a worthwhile topic for study in its own right but also it provides much information about phase transitions in the associated physical models this paper serves two functions first we introduce and survey random cluster measures from the probabilist s point of view giving clear statements of some of the many open problems second we present new results for such measures as follows we discuss the relationship between weak limits of random cluster measures and measures satisfying a suitable dlr condition using an argument based on the convexity of pressure we prove the uniqueness of random cluster measures for all but at most countably many values of the parameter p related results concerning phase transition in two or more dimensions are included together with various stimulating conjectures the uniqueness of the infinite cluster is employed in an intrinsic way in part of these arguments in the second part of this paper is constructed a markov process whose level sets are reversible markov processes with random cluster measures as unique equilibrium measures this construction enables a coupling of random cluster measures for all values of p furthermore it leads to a proof of the semicontinuity of the percolation probability and provides a heuristic probabilistic justification for the widely held belief that there is a first order phase transition if and only if the cluster weighting factor q is sufficiently large

exact solutions of the saturable discrete nonlinear schrodinger equation exact solutions to a nonlinear schrodinger lattice with a saturable nonlinearity are reported for finite lattices we find two different standing wave like solutions and for an infinite lattice we find a localized soliton like solution the existence requirements and stability of these solutions are discussed and we find that our solutions are linearly stable in most cases we also show that the effective peierls nabarro barrier potential is nonzero thereby indicating that this discrete model is quite likely nonintegrable