wavevector dependent susceptibility in z invariant pentagrid ising model we study the q dependent susceptibility q of a z invariant ferromagnetic ising model on a penrose tiling as first introduced by korepin using de bruijn s pentagrid for the rapidity lines the pair correlation function for this model can be calculated exactly using the quadratic difference equations from our previous papers its fourier transform q is studied using a novel way to calculate the joint probability for the pentagrid neighborhoods of the two spins reducing this calculation to linear programming since the lattice is quasiperiodic we find that q is aperiodic and has everywhere dense peaks which are not all visible at very low or high temperatures more and more peaks become visible as the correlation length increases that is as the temperature approaches the critical temperature

partition function zeros of the one dimensional potts model the recursive method the yang lee fisher and potts zeros of the one dimensional q state potts model are studied using the theory of dynamical systems an exact recurrence relation for the partition function is derived it is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation further a general equation for zeros of the partition function is found and a classification of the yang lee fisher and potts zeros is given it is shown that the fisher zeros in a nonzero magnetic field are located on several lines in the complex temperature plane and that the number of these lines depends on the value of the magnetic field analytical expressions for the densities of the yang lee fisher and potts zeros are derived it is shown that densities of all types of zeros of the partition function are singular at the edge singularity points with the same critical exponent

orthogonal polynomials and measures on the unit circle the geronimus transformations in this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle this perturbation is the inverse of the christoffel transformation and is called the geronimus transformation we study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated hessenberg matrices finally we show an example of this kind of transformation

on orthogonal polynomials transformed by the qr algorithm abstract in this paper we study two equivalent ways of transforming a system of orthogonal polynomials one is the so called lifting of the recurrence relation associated with an orthogonal polynomial system ops and the other consists of applying the qr algorithm to the jacobi matrix of the ops these transformations are useful for among other things identifying the sobolev orthogonal polynomials with respect to certain measures which were previously investigated by iserles et al 1991 as a by product we obtain a new class of modified lommel polynomials

lee yang zeros in the scaling region of a two dimensional quasiperiodic ising model quasiperiodic planar ising models with ferromagnetic nearest neighbour interactions should show the same universal critical behaviour as the classical ising model on the square lattice we use the eightfold symmetric ammann beenker tiling to investigate this and employ the distribution of the lee yang and the temperature zeros of the partition function in the complex plane our results support as expected the existence of an onsager type phase transition i e a second order transition with critical exponents and

calculation of gauss quadratures with multiple free and fixed knots algorithms are derived for the evaluation of gauss knots in the presence of fixed knots by modification of the jacobi matrix for the weight function of the integral simple gauss knots are obtained as eigenvalues of symmetric tridiagonal matrices and a rapidly converging simple iterative process based on the merging of free and fixed knots of quadratic convergence is presented for multiple gauss knots the procedures also allow for the evaluation of the weights of the quadrature corresponding to the simple gauss knots a new characterization of simple gauss knots as a solution of a partial inverse eigenvalue problem is derived

optimality conditions in nonconvex optimization via weak subdifferentials abstract in this paper we study optimality conditions for optimization problems described by a special class of directionally differentiable functions the well known necessary and sufficient optimality condition of nonsmooth convex optimization given in the form of variational inequality is generalized to the nonconvex case by using the notion of weak subdifferentials the equivalent formulation of this condition in terms of weak subdifferentials and augmented normal cones is also presented

optimality conditions for lower semi continuous functions abstract let f be a lower semi continuous and bounded below function from a banach space x into where x is assumed to admit a lipschitz smooth bump function generalizing results of chaney we study optimality conditions for formula to be a local minimum point of f these conditions are described in terms of generalized chaney s subdifferentials and second order derivatives

multiple geronimus transformations abstract we consider multiple geronimus transformations and show that they lead to discrete non diagonal sobolev type inner products moreover it is shown that every discrete sobolev inner product can be obtained as a multiple geronimus transformation a connection with geronimus spectral transformations for matrix orthogonal polynomials is also considered

wavevector dependent susceptibility in quasiperiodic ising models using the various functional relations for correlation functions in planar ising models new results are obtained for the correlation functions and the q dependent susceptibility for ising models on a quadratic lattice with quasiperiodic coupling constants the effects are clearest if the interactions are both attractive and repulsive according to a quasiperiodic pattern in particular an exact scaling limit result for the two point correlation function of the z invariant inhomogeneous ising model is presented and the q dependent susceptibility is calculated for some cases where the coupling constants vary according to fibonacci rules it is found that the ferromagnetic case differs drastically from the case with both ferro and antiferromagnetic bonds in the mixed case the peaks of the q dependent susceptibility are everywhere dense for temperature t both above or below the critical temperature tc but due to overlap only a finite number of peaks is visible this number of visible peaks decreases as t moves away from tc in the ferromagnetic case there is typically only one single peak at q 0 in spite of the aperiodicity present in the lattice these results provide evidence that in real systems even if the atoms arrange themselves aperiodically there will be no dramatic difference in the diffraction pattern unless the pair correlation function has clear aperiodic oscillations the number of oscillations per correlation length determines the number of visible peaks

the directed and rubinov subdifferentials of quasidifferentiable functions part ii calculus abstract we continue the study of the directed subdifferential for quasidifferentiable functions started in r baier e farkhi v roshchina the directed and rubinov subdifferentials of quasidifferentiable functions part i definition and examples this journal calculus rules for the directed subdifferentials of sum product quotient maximum and minimum of quasidifferentiable functions are derived the relation between the rubinov subdifferential and the subdifferentials of clarke dini michel penot and mordukhovich is discussed important properties implying the claims of ioffe s axioms as well as necessary and sufficient optimality conditions for the directed subdifferential are obtained

on the existence of darboux transformations for banded matrices the darboux transformations provide joint with other applications a method for obtaining solutions of some integrable systems in this work the concepts of darboux factorization and darboux transformations for arbitrary hessenberg banded matrices are analyzed specifically the existence of this kind of factorization is studied and some sufficient conditions for the uniqueness are determined

partition function zeros for aperiodic systems the study of zeros of partition functions initiated by yang and lee provides an important qualitative and quantitative tool in the study of critical phenomena this has frequently been used for periodic as well as hierarchical lattices here we consider magnetic field and temperature zeros of ising model partition functions on several aperiodic structures in 1d we analyze aperiodic chains obtained from substitution rules the most prominent example being the fibonacci chain in 2d we focus on the tenfold symmetric triangular tiling which allows efficient numerical treatment by means of corner transfer matrices

spectral transformations for hermitian toeplitz matrices in this paper we deal with some perturbations of probability measures supported on the unit circle as well as in a more general framework with hermitian linear functionals we focus our attention in the hessenberg matrix associated with the multiplication operator in terms of an orthogonal basis in the linear space of polynomials with complex coefficients the lu and qr factorizations of such a matrix are introduced then the connection between the above mentioned perturbations and such factorizations is presented

on the functions with pseudoconvex sublevel sets and optimality conditions a new class of generalized convex functions called the functions with pseudoconvex sublevel sets is defined they include quasiconvex ones a complete characterization of these functions is derived further it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality equality and a set constraints are obtained in terms of the lower hadamard directional derivative in particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets

geronimus transformations of bivariate linear functionals abstract given a linear functional u in the linear space of polynomials in two variables with real coefficients and a polynomial x y in this contribution we deal with geronimus transformations of u i e those linear functionals v such that u x y v the connection formulas between the sequences of bivariate orthogonal polynomials with respect to u and v are given a matrix interpretation of such transformations by using lu and ul factorizations of the block jacobi matrices associated with such polynomials is given finally some illustrative examples of geronimus transformations of weight function supported in domains of r 2 are discussed

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

the partition function zeros of the anisotropic ising model with multisite interactions on a zigzag ladder abstract it is shown that the spin 1 2 anisotropic ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin 1 2 axial next nearest neighbor ising annni model with multisite interactions the partition function zeros of the annni model with multisite interactions are investigated a comprehensive analysis of the partition function zeros of the annni model with and without three site interactions on a zigzag ladder is done using the transfer matrix method analytical equations for the distribution of the partition function zeros in the complex magnetic field yang lee zeros and temperature fisher zeros planes are derived the yang lee and fisher zeros distributions are studied numerically for a variety of values of the model parameters the densities of the yang lee and fisher zeros are studied and the corresponding edge singularity exponents are calculated it is shown that the introduction of three site interaction terms in the annni model leads to a simpler distribution of the partition function zeros for example the yang lee zeros tend to a circular distribution when increasing by modulus the three site interactions term coefficient it is found that the yang lee and fisher edge singularity exponents are universal and equal to each other 1 2

high temperature expansion for ising models on quasiperiodic tilings we consider high temperature expansions for the free energy of zero field ising models on planar quasiperiodic graphs for the penrose and the octagonal ammann beenker tiling we compute the expansion coefficients up to 18th order as a by product we obtain exact vertex averaged numbers of self avoiding polygons on these quasiperiodic graphs in addition we analyse periodic approximants by computing the partition function via the kac ward determinant it turns out that the series expansions alone do not yield reliable estimates of the critical exponents this is due to the limitation on the order of the series caused by the number of graphs that have to be taken into account and more seriously to rather strong fluctuations in the behaviour of the coefficients nevertheless our results are compatible with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two dimensional lattices

topological properties of the approximate subdifferential abstract the approximate subdifferential introduced by mordukhovich has attracted much attention in recent works on nonsmooth optimization potential advantages over other concepts of subdifferentiability might be related to its nonconvexity this is motivation to study some topological properties more in detail as the main result it is shown that any weakly compact subset of any hilbert space may be obtained as the kuratowski painleve limit of approximate subdifferentials from a one parametric family of lipschitzian functions sharper characterizations are possible for strongly compact subsets as a consequence in any hilbert space the approximate subdifferential of a suitable lipschitzian function may be homeomorphic both in the strong and weak topology to the cantor set further results relate the approximate subdifferential to specific topological types to the one dimensional case which is extraordinary in some sense and to the value function of a c 1 optimization problem

darboux transformations for cmv matrices abstract we develop a theory of darboux transformations for cmv matrices canonical representations of the unitary operators in perfect analogy with their self adjoint version the darboux transformations of jacobi matrices they are equivalent to laurent polynomial modifications of the underlying measures we address other questions which emphasize the similarities between darboux transformations for jacobi and cmv matrices like their almost isospectrality or the relation that they establish between the corresponding orthogonal polynomials showing also that both transformations are connected by the szeg mapping nevertheless we uncover some features of the darboux transformations for cmv matrices which are in striking contrast with those of the jacobi case in particular when applied to cmv matrices the matrix realization of the inverse darboux transformations what we call darboux transformations with parameters leads to spurious solutions whose interpretation deserves future research such spurious solutions are neither unitary nor band matrices so darboux transformations for cmv matrices are much more subject to the subtleties of the algebra of infinite matrices than their jacobi counterparts a key role in our theory is played by the cholesky factorizations of infinite matrices actually the darboux transformations introduced in this paper are based on the cholesky factorizations of degree one hermitian laurent polynomials evaluated on cmv matrices these transformations are also generalized to higher degree laurent polynomials as well as to the extension of cmv matrices to quasi definite functionals what we call quasi cmv matrices furthermore we show that this cmv version of darboux transformations plays a role in integrable systems like the schur flows or the ablowitz ladik model which parallels that of darboux for jacobi matrices in the toda lattice

comparison principles for viscosity solutions of elliptic equations via fuzzy sum rule abstract a comparison principle for viscosity sub and super solutions of second order elliptic partial differential equations is derived using the fuzzy sum rule of non smooth calculus this method allows us to weaken the assumptions made on the function f when the equation f x u d u d 2 u 0 is under consideration

second order necessary and sufficient conditions in nonsmooth optimization in this paper we generalize and sharpen r w chaney s results on unconstrained and constrained second order necessary and sufficient optimality conditions 5 7 for general lipschitz functions without the semismoothness assumption

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

weak directional closedness and generalized subdifferentials abstract spingarn introduced the notion of a submonotone operator and showed that the clarke subdifferential is submonotone if and only if it is semismooth in the sense of mifflin and regular in the sense of clarke in this article a property of operators referred to as weak directional closedness wdc is introduced the wdc property is used to extend spingarn s result to a broad class of generalized subdifferentials for locally lipschitz functions two members of this class of subdifferentials are the clarke subdifferential which is always wdc and the michel penot subdifferential which may or may not be wdc a subdifferential that is wdc and is contained in the clarke subdifferential is constructed it is shown that this subdifferential coincides with the michel penot subdifferential whenever the michel penot subdifferential is wdc and submonotone

partition function zeros of the antiferromagnetic spin 1 2 ising heisenberg model on a diamond chain abstract the partition function zeros of the antiferromagnetic spin 1 2 ising heisenberg model on a diamond chain are studied using the transfer matrix method analytical equations for the distributions of yang lee and fisher zeros are derived the yang lee zeros are located on an arc of the unit circle and on the negative real axis in the complex magnetic fugacity plane in the limit t 0 the distribution pinches the positive real axis precipitating a phase transition fisher zeros manifest more complicated distributions depending on the values of the exchange parameters and external field densities of both categories of zeros are also studied the distributions of fisher zeros are investigated for different values of model parameters the yang lee and fisher edge singularity exponents are shown to be identical they are universal in nature and are calculated to be 1 2

darboux transformation and perturbation of linear functionals abstract let l be a quasi definite linear functional defined on the linear space of polynomials with real coefficients in the literature three canonical transformations of this functional are studied xl l c x and 1 x l c x where x denotes the linear functional x xk k 0 and k 0 is the kronecker symbol let us consider the sequence of monic polynomials orthogonal with respect to l this sequence satisfies a three term recurrence relation whose coefficients are the entries of the so called monic jacobi matrix in this paper we show how to find the monic jacobi matrix associated with the three canonical perturbations in terms of the monic jacobi matrix associated with l the main tools are darboux transformations in the case that the lu factorization of the monic jacobi matrix associated with xl does not exist and darboux transformation does not work we show how to obtain the monic jacobi matrix associated with x 2 l as a limit case we also study perturbations of the functional l that are obtained by combining the canonical cases finally we present explicit algebraic relations between the polynomials orthogonal with respect to l and orthogonal with respect to the perturbed functionals

epidifferentiability of the map of infima in hilbert spaces this article deals with derivatives for set valued maps that take values in ordered vector spaces in particular it concerns about the relationship between the epiderivatives of a set valued map and its associated map of infima when the image space is a real separable hilbert space ordered by an orthonormal basis by using a variational technique based on a decoupling of the ordering cone into half spaces we show that both epiderivatives coincide under certain hypothesis of compactness and stability furthermore we obtain some computation formulas for these derivatives in terms of associated scalar set valued maps

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

continuous symmetrized sobolev inner products of order n i abstract given a symmetrized sobolev inner product of order n the corresponding sequence of monic orthogonal polynomials q n satisfies that q 2 n x p n x 2 q 2 n 1 x x r n x 2 for certain sequences of monic polynomials p n and r n in this paper we deduce the integral representation of the inner products such that p n and r n are the corresponding sequences of orthogonal polynomials moreover we state a relation between both inner products which extends the classical result for symmetric linear functionals

noncoincidence of approximate and limiting subdifferentials of integral functionals for a locally lipschitz integral functional i f on l 1 t mathbf r n associated with a measurable integrand f the limiting subdifferential and the approximate subdifferential never coincide at a point x 0 where f t cdot is not subdifferentially regular at x 0 t for a e t in t the coincidence of both subdifferentials occurs on a dense set of l 1 t mathbf r n if and only if f t cdot is convex for a e t in t our results allow us to characterize aubin s lipschitz like property as well as the convexity of multivalued mappings between l 1 spaces new necessary optimality conditions for some bolza problems are also obtained

on the equivalence of some basic principles in variational analysis the primary goal of this paper is to study relationships between certain basic principles of variational analysis and its applications to nonsmooth calculus and optimization considering a broad class of banach spaces admitting smooth renorms with respect to some bornology we establish an equivalence between useful versions of a smooth variational principle for lower semicontinuous functions an extremal principle for nonconvex sets and an enhanced fuzzy sum rule formulated in terms of viscosity normals and subgradients with controlled ranks further refinements of the equivalence result are obtained in the case of a frechet differentiable norm based on the new enhanced sum rule we provide a simplified proof for the refined sequential description of approximate normals and subgradients in smooth spaces

spectral transformations self similar reductions and orthogonal polynomials we study spectral transformations in the theory of orthogonal polynomials which are similar to darboux transformations for the schr dinger equation linear transformations of the stieltjes function with coefficients that are rational in the argument are constructed as iterations of the christoffel and geronimus transformations we describe a characteristic property of semi classical orthogonal polynomials scop on the uniform and the exponential lattice namely that all these polynomials can be obtained through simple quasi periodic and q periodic self similar closures of the chain of linear spectral transformations in the self similar setting a characterization of the laguerre hahn polynomials on linear and q linear lattices is obtained by considering rational transformations of the stieltjes function generated by transitions to the associated polynomials

subdifferentiation of integral functionals we examine how the subdifferentials of nonconvex integral functionals can be deduced from the subdifferentials of the corresponding integrand or at least be estimated with the help of them in fact assuming some regularity properties of the integrands we obtain exact expressions for the subdifferentials of the integral functionals we draw some consequences in terms of duality for such integral functionals extending in this way the early work of rockafellar to the nonconvex case

geronimus spectral transforms and measures on the complex plane we analyze a special spectral transform of a measure m supported on a compact subset c of the complex plane a perturbation m 1 of m is said to be a geronimus spectral transform if d m 1 d m z a 2 where a c we focus our attention in the analysis of the hessenberg matrix associated with the multiplication operator in terms of the orthogonal polynomial basis defined by the measure m 1

polynomial perturbations of bilinear functionals and hessenberg matrices this paper deals with symmetric and non symmetric polynomial perturbations of symmetric quasi definite bilinear functionals we establish a relation between the hessenberg matrices associated with the initial and the perturbed functionals using lu and qr factorizations moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals

an extended extremal principle with applications to multiobjective optimization we develop an extended version of the extremal principle in variational analysis that can be treated as a variational counterpart to the classical separation results in the case of nonconvex sets and which plays an important role in the generalized differentiation theory and its applications to optimization related problems the main difference between the conventional extremal principle and the extended version developed below is that instead of the translation of sets involved in the extremal systems we allow deformations the new version seems to be more flexible in various applications and covers in particular multiobjective optimization problems with general preference relations in this way we obtain new necessary optimality conditions for constrained problems of multiobjective optimization with nonsmooth data and also for multiplayer multiobjective games

on minimizing and critical sequences in nonsmooth optimization let f be a bounded below lower semicontinuous function from a banach space into r cup infty we study the relationships between minimizing and critical sequences of f where the criticality condition is given in terms of some subdifferential partial here the objective function f is not supposed to be convex or smooth our work extends that of auslender and crouzeix and that of chou ng and pang