finite size scaling and universality in the spectrum of the quantum ising chain i periodic and antiperiodic boundary condition the spectrum of the quantum ising chain is studied in the finite size scaling limit for periodic and antiperiodic boundary conditions the finite size corrections are computed for all energy gaps the results of the model are used to test the privman fisher universality hypothesis and conformal theory at the critical point it is shown how to build the spectrum from the irreducible representations of the virasoro algebra in addition the author finds evidence for an unsuspected symmetry for systems both non critical and of finite size

the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

surface critical behaviour of the smoothly inhomogeneous planar ising model the pfaffian method a semi infinite nearest neighbour square ising lattice is investigated whose couplings j l at a distance l from the boundary differ from homogeneity by an amount j l j infinity approximately a l on the basis of the pfaffian method the critical behaviour at the surface of this system is investigated the exponents eta11 nu11 beta1 gamma11 and delta11 all display rich non universal behaviour as a function of the amplitude a for a below a critical value there is a spontaneous surface magnetisation when the bulk l infinity is critical and an asymmetry between the exponents on either side of the critical point

correlation length in ising strips with free and fixed boundary conditions the correlation length of boundary spins in the ising model defined on strips of triangular lattice with free boundary conditions is determined with an efficient numerical procedure based on the star triangle transformation in the case of isotropic critical interactions the extrapolated amplitude of the correlation length is in excellent agreement with the value 2 pi eta sub predicted by conformal invariance an analytical formula for the amplitude in strips with anisotropic interactions is proposed fixing the spins on one edge reduces the amplitude of the correlation length on the other edge by a factor 1 2 the convergence of phenomenological renormalisation with free boundary conditions is studied

low lying excitations in the square triangle random tiling model we consider the sub dominant eigenstates of the transfer matrix for the square triangle random tiling model on an infinite strip of width l a numerical algorithm for generation of the corresponding solution of the bethe ansatz is developed numerical finite size scaling analysis of the associated eigenvalues reveals the presence of both integer and non integer critical exponents the analytical value of one of the non integer exponents is found it is also shown numerically that along with the leading correction to the free energy density for some excitations there is a term proportional to

critical exponents near a random fractal boundary the critical behaviour of correlation functions near a boundary is modified from that in the bulk when the boundary is smooth this is known to be characterized by the surface scaling dimension we consider the case when the boundary is a random fractal specifically a self avoiding walk or the frontier of a brownian walk in two dimensions and show that the boundary scaling behaviour of the correlation function is characterized by a set of multifractal boundary exponents given exactly by conformal invariance arguments to be this result may be interpreted in terms of a scale dependent distribution of opening angles of the fractal boundary on short distance scales these are sharply peaked around similar arguments give the multifractal exponents for the case of coupling to a quenched random bulk geometry

efficient quadrature rules based on spline quasi interpolation for application to iga bems abstract we propose a new class of quadrature rules for the approximation of weakly and strongly singular integrals based on the spline quasi interpolation scheme introduced in mazzia and sestini 2009 these integrals in particular occur in the entries of the stiffness matrix coming from isogeometric boundary element methods iga bems the presented formulas are efficient since they combine the locality of any spline quasi interpolation scheme with the capability to compute the modified moments for b splines i e the weakly or strongly singular integrals of such functions no global linear system has to be solved to determine the quadrature weights but just local systems whose size linearly depends on the adopted spline degree the rules are preliminarily tested in their basic formulation i e when the integrand is defined as the product of a singular kernel and a continuous function g then such basic formulation is compared with a new one specific for the approximation of the singular integrals appearing in the iga bem context where a b spline factor is explicitly included in g such a variant requires the usage of the recursive spline product formula given in morken 1991 and it is useful when the ratio between g and its b spline factor is smooth enough

marginally inhomogeneous semiinfinite two dimensional ising models the scaling dimensions of the semiinfinite two dimensional ising model with smoothly inhomogeneous couplings are determined in general by the eigenvalues of a schrodinger type equation the author obtains exact results for a marginal two parameter inhomogeneity that corresponds to a supersymmetric quantum mechanical potential the results are compared with predictions of conformal invariance

conformal off diagonal boundary density profiles on a semi infinite strip the off diagonal profile odb v associated with a local operator v order parameter or energy density close to the boundary of a semi infinite strip with width l is obtained at criticality using conformal methods it involves the surface exponent xs and displays a simple universal behaviour which crosses over from surface finite size scaling when v l is held constant to corner finite size scaling when v l 0

approximating the singular integrals of cauchy type with weight function on the interval it is known that the solutions of characteristic singular integral equations sies are expressed in terms of singular integrals of cauchy type with weight functions w x 1 x n 1 x m where n 12 m 12 new quadrature formulas qfs are presented to approximate the singular integrals sis of cauchy type for all solutions of characteristic sie on the interval 1 1 linear spline interpolation modified discrete vortex method and product quadrature rule are utilized to construct the qfs estimation of errors are obtained in the classes of functions h a 1 1 a and c 1 1 1 it is found that the numerical results are very stable even for the cases of semi bounded and unbounded solutions of singular integral equation of the first kind

random trimer tilings we study tilings of the square lattice by linear trimers for a cylinder of circumference m we construct a conserved functional of the base of the tilings and use this to block diagonalize the transfer matrix the number of blocks increases exponentially with m the dimension of the block corresponding to the largest eigenvalue is shown to grow as 32 13 m we numerically diagonalize this block for m or 27 obtaining the estimate s infinity 0 158520 0 000015 for the entropy per site in the thermodynamic limit we present numerical evidence that the continuum limit of the model has conformal invariance we measure several scaling dimensions including those corresponding to defects of monomers and l shaped trimers the trimer tilings of a plane admits a two dimensional height representation monte carlo simulations of the height variables show that the height height correlations grows logarithmically at large separation and the orientation orientation correlations decay as a power law

numerical integration based on approximating splines abstract convergence results are proved for product integration rules based on approximating splines these results are both for bounded and unbounded integrands pointwise and uniform convergence results are proved for sequences of cauchy principal values of these approximating splines

universal order parameter profiles in confined critical systems with boundary fields the authors consider systems at the bulk critical temperature with an ordering field applied at the boundary for various boundary geometries universal expressions for the spatial dependence of the order parameter are obtained by conformal mapping of the known results for the half space analytical results for strip wedge and other two dimensional geometries and for systems with spherical boundaries in arbitrary dimension are given

critical properties of the spin 1 2 heisenberg chain with frustration and bond alternation we study the heisenberg chain with frustration and alternation which is expressed by both analytically and numerically we focus on the case where there is no marginal operator which brings about the logarithmic corrections in various quantities by using the bosonization method we calculate the energy gap the change in the ground state energy due to the alternation so called energy gain the spin correlation and the string correlation not only their exponents but also their amplitudes and compare them with the results of the numerical diagonalization for finite systems we point out the existence of the logarithmic correction in the energy gain despite the absence of the marginal operator taking into account this logarithmic correction we can obtain a reasonable hyperscaling relation between the critical exponents of the energy gap and an energy gain from the numerical data

gradient critical phenomena in the ising quantum chain we consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings in the direction of the gradient an interface region separates the ordered phase from the disordered one we develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region the scaling predictions are tested in the framework of the mean field ginzburg landau theory then we consider the ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two dimensional ising model the quantum hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator the energy density the magnetization profile and the two point correlation function are studied either analytically or by exact numerical calculations their scaling behaviour is in agreement with the predictions of the scaling theory

an efficient method for approximate solution of a singular integral equation with cauchy kernel abstract in this paper the least squares method and the reproducing kernel space are employed to obtain approximate solution of a general form of singular integral equations with cauchy kernel this approach is based on the construction of the subspace which is dense in the reproducing kernel space the numerical examples illustrate the high accuracy of the method

on the convergence of some cubic spline interpolation schemes cubic spline interpolation schemes which require no derivative information at the end points are of great practical importance and have been included in several general purpose software libraries in this paper optimal order error estimates are developed for three popular schemes of this derivative free type the approximation of c 1 a b backslash c 2 a b functions by any such derivative free method that reproduces cubics necessarily displays some dependence on the local mesh ratio however for the spline interpolants studied here this dependence is restricted to the first and last subintervals

uniform convergence of optimal order quadrature rules for cauchy principal value integrals abstract for the numerical evaluation of cauchy principal value integrals of the form e 1 1 f e c s 1 1 we consider a quadrature method based on spline interpolation of odd degree 2 k 1 k n 0 we show that these rules converge uniformly for 1 1 in particular we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded s th derivative if s e s 2 k 1 2 k 2 finally we compare the rule to the well known quadrature rule of elliott and paget which only converges pointwise

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

finite size scaling of corner transfer matrices for the two dimensional ising model the eigenvalue spectra of generalised corner transfer matrices are calculated for an ising model on a square lattice the predictions of conformal invariance at the critical point are thereby verified

corner critical exponents from the mean field renormalisation group the previously introduced method of mean field renormalisation has provided a unified approach to bulk and surface critical behaviour the method is used for the computation of the critical exponent associated with the order parameter at corners applications to the square and triangular ising model are presented the angle dependence of the corner critical exponent is qualitatively reproduced the results for the bulk critical exponents are more accurate than in previously employed mean field renormalisation schemes

polynomial size frege and resolution proofs of st connectivity and hex tautologies a grid graph has rectangularly arranged vertices with edges permitted only between orthogonally adjacent vertices the st connectivity principle states that it is not possible to have a red path of edges and a green path of edges which connect diagonally opposite corners of the grid graph unless the paths cross somewhere we prove that the propositional tautologies which encode the st connectivity principle have polynomial size frege proofs and polynomial size tc0 frege proofs for bounded width grid graphs the st connectivity tautologies have polynomial size resolution proofs a key part of the proof is to show that the group with two generators both of order two has word problem in alternating logtime alogtime and even in tc0 conversely we show that constant depth frege proofs of the st connectivity tautologies require near exponential size the proof uses a reduction from the pigeonhole principle via tautologies that express a directed single source principle sink which is related to papadimitriou s search classes ppad and ppads or psk the st connectivity principle is related to urquhart s propositional hex tautologies and we establish the same upper and lower bounds on proof complexity for the hex tautologies in addition the hex tautology is shown to be equivalent to the sink tautologies and to the one to one onto pigeonhole principle

conformal invariance and line defects in the two dimensional ising model the quantum ising chain with p equidistant defects is studied the exact form of the hamiltonian is found for an infinite number of sites using conformal invariance generalised corner exponents are computed and found to depend continuously on the defect strengths realisations of extended conformal algebras in the hamiltonian spectrum are obtained

conformal invariance and the critical behavior of the triplet xy quantum chain we introduce and study the critical properties of the tripletxy quantum chain this system is described in terms of three spin interactions and is the generalization of the standardxy quantum chain we show that this model with periodic boundaries has a local gauge invariance and can be described by the composition of two triplet ising models with general toroidal boundary conditions from this composition the phase diagram as well the conformai anomaly and critical exponents are determined by exploring their relations with the mass gap amplitudes predicted by conformai invariance

star triangle approach to boundary behavior in the two dimensional ising model abstract hilhorst and van leeuwen showed how to calculate boundary properties of the ising model on the triangular lattice by iterating a mapping based on the star triangle transformation we apply this approach to the ising model with homogeneous initial couplings in both the semi infinite and strip geometries several exact results for the boundary correlation length and the magnetization are reproduced the correlation dimensionality transition for enhanced edge couplings dual of abraham s interface unbinding transition is also considered

the ashkin teller quantum chain and conformal invariance the authors study the finite size effects for the fourth state one dimensional quantum chain introduced by kohmoto et al 1981 using different boundary conditions from the corrections to the ground state energy for periodic boundary conditions they verify that the central charge of the virasoro algebra is equal to one in the whole domain of criticality their numerical study suggests a new relation between the corrections to the ground state energy with free boundary conditions and with periodic boundary conditions various critical exponents including the surface ones and those corresponding to the para fermions are determined

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

phase diagram of a random tiling quasicrystal we study the phase diagram of a two dimensional random tiling model for quasicrystals at proper concentrations the model has 8 fold rotational symmetry landau theory correctly gives most of the qualitative features of the phase diagram which is in turn studied in detail numerically using a transfer matrix approach we find that the system can enter the quasicrystal phase from many other crystalline and incommensurate phases through first order or continuous transitions exact solutions are given in all phases except for the quasicrystal phase and for the phase boundaries between them we calculate numerically the phason elastic constants and entropy density and confirm that the entropy density reaches its maximum at the point where phason strains are zero and the system possesses 8 fold rotational symmetry in addition to the obvious application to quasicrystals this study generalizes certain surface roughening models to two dimensional surfaces in four dimensions

operator dimensions and surface exponents for the non linear schrodinger model at t 0 using the euler maclaurin formula the authors compute finite size corrections to the ground and excited state energies and momenta this enables them to obtain all possible operator scaling dimensions at the critical point t 0 and surface exponents for a variety of boundary conditions they extend the predictions of conformal invariance to include green functions with oscillating terms

marginal extended perturbations in two dimensions and gap exponent relations the most general form of a marginal extended perturbation in a two dimensional system is deduced from scaling considerations it includes as particular cases extended perturbations decaying either from a surface a line or a point for which exact results have been previously obtained the first order corrections to the local exponents which are functions of the amplitude of the defect are deduced from a perturbation expansion of the two point correlation functions assuming covariance under conformal transformation the perturbed system is mapped onto a cylinder working in the hamiltonian limit the first order corrections to the lowest gaps are calculated for the ising model the results confirm the validity of the gap exponent relations for the perturbed system

on the uniform convergence of gaussian quadrature rules for cauchy principal value integrals and their derivatives the convergence of the aforementioned quadrature rules for integrands possessing h6lder continuous derivatives of an appropriate order is proved to be uniform and not only pointwise the rate of convergence is also established and an application to the numerical solution of singular integral equations is made

semi infinite two dimensional ising model with marginally inhomogeneous couplings and conformal invariance the authors consider a semi infinite two dimensional ising model with nearest neighbour couplings that deviate from the bulk critical couplings by al 2 where l is the distance from the surface the surface critical exponents of this system are non universal under a conformal mapping onto a strip of width l the al 1 inhomogeneity transforms into a l pi sin pi l l 1 for the square lattice the spectrum of the transfer matrix in the strip geometry is calculated exactly in the extreme anisotropic limit the analytical results and numerical results for the triangular lattice are compared with predictions of conformal invariance

critical behaviour in parabolic geometries the authors study two dimensional systems with boundary curves described by power laws using conformal mappings they obtain the correlations at the bulk critical point three different classes of behaviour are found and explained by scaling arguments which also apply to higher dimensions for an ising system of parabolic shape the behaviour of the order at the tip is also found

product integration of piecewise continuous integrands based on cubic splineinterpolation at equally spaced nodes in this paper we consider the approximate evaluation of int limits a b k x f x dx wherek x is a fixed lebesgue integrable function by product formulas of the form sum limits i 0 n w i f x i based on cubic spline interpolation of the functionf generally whenever it is possible product quadratures incorporate the bad behaviour of the integrand in the kernelk here however we allowf to have a finite number of jump discontinuities in a b convergence results are established and some numerical applications are given for a logarithmic singularity structure in the kernel

universal amplitudes in finite size scaling generalisation to arbitrary dimensionality the relationship between the correlation length and critical exponents in finite width strips in two dimensions is generalised to cylindrical geometries of arbitrary dimensionality d for d not 2 these correspond however to curved spaces the result is verified for the spherical model

logarithmic singularity in the surface free energy near commensurate incommensurate transitions the authors study finite size scaling behaviour of a two dimensional random tiling model this model exhibits commensurate incommensurate phase transitions of the pokrovsky talapov type where anisotropic scaling appears a simple scaling argument near this transition suggests an integral surface critical exponent which in turn implies a logarithmic singularity with a universal coefficient in the surface free energy they confirm the existence of this singularity numerically employing the transfer matrix method in the case of free boundary conditions

piecewise polynomial quadratures for cauchy singular integrals in this paper we propose piecewise polynomial methods for the approximation of cauchy principal value integrals and develop a simple efficient and numerically stable algorithm for the evaluation of the weights of the resulting piecewise polynomial quadratures we present two examples to illustrate the advantages of these quadratures versus the gauss jacobi quadratures

critical behaviour near multiple junctions and dirty surfaces in the two dimensional ising model the authors consider m two dimensional semi infinite planes of ising spins joined together through surface spins and study the critical behaviour near to the junction the m 0 limit of the model according to the replica trick corresponds to the semi infinite ising model in the presence of a random surface field rsfi using conformal mapping second order perturbation expansion around the weak and strong coupled planes limits and a differential renormalization group they show that the surface critical behaviour of the rsfi model is described by ising critical exponents with logarithmic corrections to scaling while at multiple junctions m 2 the transition is of first order there is a spontaneous junction magnetization at the bulk critical point

corner spontaneous magnetization details are given of a new method allowing an exact calculation of the spontaneous magnetization in the corner as well as along the edge at an arbitrary distance of the corner for a rectangular planar ising ferromagnet

product integration of singular integrands based on cubic spline interpolation at equally spaced nodes in this paper the convergence of product integration rules based on cubic spline interpolation at equally spaced nodes with not a knot end condition is investigated for integrand functions with a interior or endpoint singularity in the integration interval

a new efficient method with error analysis for solving the second kind fredholm integral equation with cauchy kernel the main objective of this paper is to give an efficient numerical method for the solution of the second kind fredholm integral equation with cauchy type kernel although numerical treatment of singular integral equations sies has been considered by many researchers and many numerical methods have been proposed for the solution of this equation strong singularity of the cauchy singular integral operator still remains as a challenge to numerical methods most of the previous methods rely on specific quadrature rule or suitable base functions for capturing the singularity here we focus on operator transformation and graded mesh in addition we study error analysis in this regard efficiency of reproducing kernel hilbert space rkhs method using smooth transformation on the graded mesh is improved and compared with some other numerical methods

conformal invariance and critical spectrum of corner transfer matrices the finite size scaling spectra of ising model corner transfer matrices and their generators are studied at criticality the generators are diagonalised using fermion algebra the low lying eigenvalues given by the zeros of meixner polynomials are equally spaced and collapse like 1 log n for large n as predicted by conformal invariance the asymptotics are obtained using a generalised euler maclaurin summation formula the shift in the largest eigenvalue is given analytically as pi c 6 log n with central charge c 1 2 the spectrum generating functions for both fixed and free boundary conditions are expressed simply in terms of the c 1 2 virasoro characters chi delta q with modular parameter q exp pi log n and conformal dimensions delta 0 1 2 1 16

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

anisotropic critical phenomena in parabolic geometries the directed self avoiding walk the critical behaviour of directed self avoiding walks is studied on parabolic like systems with a free boundary at x or ctalpha using a scaling argument 1 c is shown to be a marginal variable when alpha vperpendicular to v 1 2 i e on a parabola as a consequence the directed walk may display varying local exponents such a behaviour is indeed observed for restricted walks this generalizes a result of cardy 1983 showing that non universal behaviour occurs at corners for isotropic systems

a family of methods for the solution of lattice models abstract we discuss a family of methods for evaluating the thermodynamic functions for a class of lattice models that includes the ising model it is shown that the usual transfer matrix method can be considered as a member of the family some typical members of this family are also introduced furthermore we are concerned with the acceleration of these methods for this reason an extra linear extrapolation algorithm which is based on the member of this family related to the transfer matrix method is presented by this algorithm not only can the bulk physical quantities be evaluated more precisely but also an approximation to the boundary thermodynamic quantities can be obtained a new expression for the boundary free energy of the ising model is derived in addition the convergence to the thermodynamic limit is proven for the scaling method presented by chorin some numerical results for the bulk and boundary thermodynamic functions of the ising model are included

inhomogeneous ising chain in a transverse field finite size scaling and asymptotic conformal spectrum the quantum version of the hilhorst van leeuwen model i e an inhomogeneous lsing model in a transverse field with a marginal perturbation of the bulk interaction varying like i with the distance i from the surface is studied numerically on a finite chain with symmetric interactions using fermion techniques and finite size scaling at the bulk critical point the surface exponents for the energy and the magnetisation are respectively x 2a 2 xm a 4when a a andx a f oandx a when as a where a is the critical value of the perturbation amplitude below which the surface becomes ordered at the bulk critical point although the magnetic exponents can be translated from the known classical values the surface thermal exponents seem to be ne the excitation spectrum obtained in the continuum limit exhibits an asymptotic conformal behaviour an analytic calculation of the leading finite size corrections when a ac and a test for the consistency of the numerical results when a c a are given in the appendix

pair correlations in the antiperiodic ising strip and conformal invariance an explicit formula for the spatial dependence of the pair correlation function in an ising strip with antiperiodic boundary conditions at the bulk critical temperature is given it follows from the conformal invariance approach of belevin et al 1986 and cardy 1986

on the convergence of spline product quadratures for cauchy principal value integrals abstract in a recent paper this journal 1990 the authors proposed product integration formulas for the numerical evaluation of the cauchy principal value integral f1 1u x f x x dx based on cubic spline interpolation of f and obtained convergence results for functions f ck 1 1 k 1 2 or 3 in this report the same rules are considered and their convergence is investigated for a larger class of functions f an error bound and some uniform convergence results are established in the case of equally spaced quadrature nodes for functions f satisfying a holder condition of order on 1 1 0

on the convergence of product formulas for the numerical evaluation of derivatives of cauchy principal value integrals product rules of interpolatory type on the zeros of generalized smooth jacobi polynomials for the numerical approximation of cauchy principal value integrals and their derivatives are considered convergence results for these rules are given and some estimates of the remainder are established in relation to the regularity of the density function

logarithmic corrections in the two dimensional ising model in a random surface field in the two dimensional ising model weak random surface field is predicted to be a marginally irrelevant perturbation at the critical point we study this question by extensive monte carlo simulations for various strengths of disorder the calculated effective temperature or size dependent critical exponents fit with the field theoretical results and can be interpreted in terms of the predicted logarithmic corrections to the pure system s critical behaviour

critical behaviour at an edge the critical behaviour of a magnetic system with o n spin symmetry bounded by two d 1 dimensional hyperplanes meeting at an angle alpha is studied within mean field theory and in d 4 epsilon dimensions new exponents emerge for correlation functions and magnetisation and susceptibilities for spins close to the edge they can be expressed in terms of known bulk and surface exponents together with a single new edge exponent which depends however on the angle alpha this exponent is computed to first order in epsilon

magnetization at corners in two dimensional ising models we investigate the corner spin magnetization of two dimensional ferromagnetic ising models in various wedge geometries results are obtained for triangular and square lattices by numerical studies on finite wedges using the star triangle transformation as well as analytic calculations using corner transfer matrices and a fermionic representation of the row to row transfer matrix the corner magnetizations vanish at the bulk critical temperature tc with an exponent c differing from the bulk exponent for isotropic systems with free edges we find that c is given simply by c 2 where is the angle at the corner for apex magnetizations of conical lattices we obtain the strikingly similar result a 4 these formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle eff determined by the relative strengths of the interactions

scaling dimensions and conformal anomaly in anisotropic lattice spin models the effect of anisotropic interactions on the eigenvalue spectrum of the row to row transfer matrix of critical lattice spin models is investigated it is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta ay ax sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy for exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices this is done for the eight vertex model hard hexagons and interacting hard squares and it is found that theta pi u lambda 10u 3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter for each of these models the amplitude of the finite size corrections to the free energy at criticality is found to be of the form pi zeta c 6n2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c 1 4 5 and 7 10 respectively this is an analytic result for the eight vertex model for the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to n 48 finally the authors argue that the anisotropy angle for magnetic hard squares and the q state potts models is also given by theta pi u lambda so this result is quite general

bulk and surface properties in the critical phase of the two dimensional xy model monte carlo simulations of the two dimensional xy model are performed in a square geometry with various boundary conditions bc using conformal mappings we deduce the exponent t of the order parameter correlation function and its surface analogue t as a function of the temperature in the critical low temperature phase of the model the temperature dependence of both exponents is obtained numerically with a good accuracy up to the kosterlitz thouless transition temperature the bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields at very low temperatures t is found in pretty good agreement with the linear temperature dependence of berezinskii s spin wave approximation we also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the kosterlitz thouless kt transition temperature while these corrections exist for the correlation function at the kt transition the value tkt 1 4 is accurately recovered the exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions the correlation function is computed with free bc and the profile with mixed fixed free bc it exhibits a monotonic behaviour with temperature starting linearly according to the spin wave approximation and increasing up to a value tkt 1 2 at the kosterlitz thouless transition temperature the thermal exponent e t is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the kt transition

the spin 1 heisenberg chain with uniaxial anisotropy and its classical counterpart abstract the spin 1 heisenberg chain with uniaxial anisotropy is related to a two dimensional roughening model with anisotropic next nearest neighbor interactions using this relation some peculiar features of the phase diagram of the quantum chain can be explained naturally and predictions for the critical exponents are also obtained which agree with previous finite size scaling studies of the quantum model a numerical transfer matrix calculation of the free energy of the roughening model on infinite strips of small width n 6 yields a critical behavior consistent with the one found for the t 0 properties of the quantum chain

finite size scaling amplitudes in a random tiling model we study finite size corrections to the free energy in a two dimensional random tiling model this model is equivalent to an asymmetric six vertex model and exhibits commensurate incommensurate phase transitions of the pokrovsky talapov type we calculate finite size scaling amplitudes of the free energy inside the incommensurate phase analytically and numerically by employing a transfer matrix method both periodic and free boundary conditions are considered these amplitudes are consistent with predictions of the theory of conformal invariance with the conformal charge c 1

on interpolation and automatization for frege systems the interpolation method has been one of the main tools for proving lower bounds for propositional proof systems loosely speaking if one can prove that a particular proof system has the feasible interpolation property then a generic reduction can usually be applied to prove lower bounds for the proof system sometimes assuming a usually modest complexity theoretic assumption in this paper we show that this method cannot be used to obtain lower bounds for frege systems or even for tc0 frege systems more specifically we show that unless factoring of blum integers is feasible neither frege nor tc0 frege has the feasible interpolation property in order to carry out our argument we show how to carry out proofs of many elementary axioms theorems of arithmetic in polynomial sized tc0 frege as a corollary we obtain that tc0 frege as well as any proof system that polynomially simulates it is not automatizable under the assumption that factoring of blum integers is hard we also show under the same hardness assumption that the k provability problem for frege systems is hard

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

potts and percolation models on bowtie lattices we give the exact critical frontier of the potts model on bowtie lattices for the case of q 1 the critical frontier yields the thresholds of bond percolation on these lattices which are exactly consistent with the results given by ziff et al j phys a 39 15083 2006 for the q 2 potts model on a bowtie a lattice the critical point is in agreement with that of the ising model on this lattice which has been exactly solved furthermore we do extensive monte carlo simulations of the potts model on a bowtie a lattice with noninteger q our numerical results which are accurate up to seven significant digits are consistent with the theoretical predictions we also simulate the site percolation on a bowtie a lattice and the threshold is s c 0 5479148 7 in the simulations of bond percolation and site percolation we find that the shape dependent properties of the percolation model on a bowtie a lattice are somewhat different from those of an isotropic lattice which may be caused by the anisotropy of the lattice

corner exponents in the two dimensional potts model the critical behavior at a corner in two dimensional ising and three state potts models is studied numerically on the square lattice using transfer operator techniques the local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite size scaling results at the critical point for isotropic or anisotropic couplings the scaling dimensions compare quite well with the values expected from conformal invariance provided the opening angle is replaced by an effective one in anisotropic systems

finite size scaling of the quantum ising chain with periodic free and antiperiodic boundary conditions the authors give exact results for the energy spectrum of a chain of n ising spins in a transverse field with periodic free and antiperiodic boundary conditions the dependence of the energy gaps on boundary conditions is compatible with predictions of conformal invariance for correlation lengths in two dimensional strips the feasibility of calculating surface critical indices using phenomenological renormalisation with free boundary conditions and the convergence for large n are discussed

solving cauchy integral equations of the first kind by the adomian decomposition method a new approach is presented to resolve cauchy integral equations of the first kind in the general case by first considering a regularized integral equation and then transforming it into a canonical form suitable for applying the adomian decomposition method adm we obtain a decomposition solution f e of the regularized integral equation and prove the convergence of our new combined method as the regularization parameter e 1 the obtained solution is shown to be a sufficiently good approximate solution for the particular cauchy integral equation the proposed method has been tested for a variety of cauchy integral equations which are particularly important in engineering applications e g airfoil design etc

finite size scaling for directed bond percolation with and without cycles on a triangular lattice the authors apply finite size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic directed cyclic and undirected bond percolation on a triangular lattice the results are in good agreement with known estimates and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases as well as the distinction between locally directed and fully isotropic problems in addition the authors draw tentative comments on the extension to directed problems of the relation a pi eta where a is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point known to hold for various isotropic systems

on the convergence of an interpolatory product rule for evaluating cauchy principal value integrals etablissement des theoremes de convergence des regles de produits interpolatoires pour l evaluation d integrales singulieres de cauchy avec estimation asymptotique du reste amelioration et generalisation de resultats anterieurs

percolation and conduction in restricted geometries the finite size scaling behaviour for percolation and conduction is studied in two dimensional triangular shaped random resistor networks at the percolation threshold the numerical simulations are performed using an efficient star triangle algorithm the percolation exponents linked to the critical behaviour at corners are in good agreement with the conformal results the conductivity exponent t is found to be independent of the shape of the system its value is very close to recent estimates for the surface and bulk conductivity exponents

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

time space trade offs in resolution superpolynomial lower bounds for superlinear space we give the first time space trade off lower bounds for resolution proofs that apply to superlinear space in particular we show that there are formulas of size n that have resolution refutations of size and space t n n theta log n and like all formulas have another resolution refutation of space n but for which no resolution refutation can simultaneously have space s n t n o 1 and size t n o 1 in other words any substantial reduction in space results in a super polynomial increase in total size we also show somewhat stronger time space trade off lower bounds for regular resolution which are also the first to apply to superlinear space for any function t that is at most weakly exponential t n 2 o n 1 4 we give a tautology that has regular resolution proofs of size and space t n but no such proofs with space s n t n 1 omega 1 and size t n o 1 thus any polynomial reduction in space has a superpolynomial cost in size these tautologies

corner transfer matrices abstract for two dimensional lattice models with interactions only between nearest and diagonally nearest neighbour spins a well known concept is the row to row transfer matrix less well known is the corner transfer matrix ctm this has some very useful properties if it is normalized so that its largest eigenvalue is unity and the eigenvalues are arranged in numerically decreasing order then each eigenvalue tends to a limit as the lattice becomes large for those models which have been solved exactly notably the ising eight vertex and hard hexagon models this limiting eigenvalue distribution is very simple being basically that of a direct product of two by two matrices from it the order parameter can easily be obtained for all models one can write down formally exact matrix relations for the ctm but the matrices are of infinite size if one uses a representation in which the ctm is diagonal and then truncates these relations to finite size then one obtains a quite accurate approximation the larger the size the greater the accuracy i g enting and i have thereby obtained comparatively long series expansions for the ising model in a field and for the hard squares model

on the evaluation of cauchy principal value integrals by rules based on quasi interpolating splines abstract in this paper we consider the numerical evaluation of one dimensional cauchy principal value integrals of the form 1 1 k x f x x d x 1 by rules obtained by subtracting out the singularity and then applying quadrature formulas based on quasi interpolating splines

the use of spline on spline for the approximation of cauchy principal value integrals in this paper the spline on spline interpolatory polynomials are used to approximate cauchy principal value integrals and their derivatives this technique yields better results than using the derivatives of the traditional single spline functions which has been considered by orsi improved order of convergence for these approximations are discussed

corner transfer matrices and corner magnetization for the ising model the authors consider the corner magnetization of an ising square lattice with free edges along a diagonal direction using some remarkable properties of the corner transfer matrix they give a unified treatment for various opening angles the calculation of the magnetization is reduced to the solution of a matrix equation in the thermodynamic limit they construct an explicit form of this equation and discuss its numerical and asymptotic solution and the results obtained thereby

surface free energy of the critical six vertex model with free boundaries the bethe ansatz equations are derived for the six vertex model with general boundary weights on a lattice in a diagonal orientation these are solved in the thermodynamic limit finite size corrections to the free energy for a restricted class of boundary weights including those corresponding to a potts model with free boundaries are calculated in the critical region the first order term gives the surface free energy of the model the second order term is found to be pi okt tan pi nu 2 mu c 48n 2 where c 1 6 mu 2 pi 2 pi mu is the conformal anomaly this can be compared to pi kt sin pi nu mu c 6n 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions this difference can be explained geometrically using conformal invariance

radial fredholm perturbation in the two dimensional ising model and gap exponent relation we consider concentric circular defects in the two dimensional ising model which are distributed according to a generalized fredholm sequence i e at exponentially increasing radii this type of aperiodicity does not change the bulk critical behaviour but introduces a marginal extended perturbation the critical exponent of the local magnetization is obtained through finite size scaling using a corner transfer matrix approach in the extreme anisotropic limit it varies continuously with the amplitude of the modulation and is closely related to the magnetic exponent of the radial hilhorst van leeuwen model through a conformal mapping of the system onto a strip the gap exponent relation is shown to remain valid for such an aperiodic defect

critical phenomena at perfect and non perfect surfaces the effect of imperfections on surface critical properties is studied for ising models with nearest neighbour ferromagnetic couplings on simple cubic lattices in particular results of monte carlo simulations for flat perfect surfaces are compared to those for flat surfaces with random weak or strong interactions between neighbouring spins in the surface layer and for surfaces with steps of monoatomic height surface critical exponents at the ordinary transition in particular are found to be robust against these perturbations

critical off critical interface in the ising quantum chain and conformal invariance the low energy excitation spectrum of an ising quantum chain in a transverse field which is critical h j 1 for 1 or n or l 2 and off critical h not j for l 2 n or l is calculated up to o l 1 interface critical exponents are determined numerically in the odd sector and exactly in the even sector using finite size scaling one gets an ordinary surface transition when h j and an extraordinary surface transition when h j the gap exponent relation is verified the mass gaps and the level degeneracy are in agreement with conformal invariance

quadrature formula for approximating the singular integral of cauchy type with unbounded weight function on the edges new quadrature formulas qfs for evaluating the singular integral si of cauchy type with unbounded weight function on the edges is constructed the construction of the qfs is based on the modification of discrete vortices method mmdv and linear spline interpolation over the finite interval 1 1 it is proved that the constructed qfs converge for any singular point x not coinciding with the end points of the interval 1 1 numerical results are given to validate the accuracy of the qfs the error bounds are found to be of order o h a lnh and o h lnh in the classes of functions h a 1 1 and c 1 1 1 respectively

the use of piecewise quadratic polynomials for the solution of singular integral equations of cauchy type abstract a method for the numerical solution of singular integral equations of cauchy type is developed the unknown function is expressed as a product of a weight function and a continuous function t the continuous function t is approximated by piecewise quadratic polynomials and the singular integral equation is reduced to a linear algebraic system numerical examples are given and comparisons are made with the widely used gauss type methods

product integration of logarithmic singular integrands based on cubic splines abstract this paper is concerned with the practical evaluation of the product integral 1 1 f x k x d x for the case when k x in x 1 1 and f is bounded in 1 1 the approximation is a quadrature rule where the weights w n n i are chosen to be exact when f is given by a linear combination of a chosen set of functions n j in this paper the functions n j are chosen to be cubic b splines an error bound for product quadrature rules based on cubic splines is provided examples that test the performance of the product quadrature rules for different choices of the function are given a comparison is made with product quadrature rules based on first kind chebyshev polynomials