correction induced by irrelevant operators in the correlators of the two dimensional ising model in a magnetic field we investigate the presence of irrelevant operators in the two dimensional ising model perturbed by a magnetic field by studying the corrections induced by these operators in the spin spin correlator of the model to this end we perform a set of high precision simulations for the correlator both along the axes and along the diagonal of the lattice by comparing the numerical results with the predictions of a perturbative expansion around the critical point we find unambiguous evidence of the presence of such irrelevant operators it turns out that among the irrelevant operators the one which gives the largest correction is the spin 4 operator t 2 2 which accounts for the breaking of the rotational invariance due to the lattice this result agrees with what was already known for the correlator evaluated exactly at the critical point and also with recent results obtained in the case of the thermal perturbation of the model

high precision estimate of g4 in the 2d ising model we compute the renormalized four point coupling in the 2d ising model using transfer matrix techniques we greatly reduce the systematic uncertainties which usually affect this type of calculation by using the exact knowledge of several terms in the scaling function of the free energy our final result is g4 14 697 35 3

minimization of a closed loop response to a fixed input for siso systems it is shown that the problem of minimizing a regulated response of a single input single output system due to a fixed bounded input can be converted via polynomial techniques to a linear infinite dimensional chebyshev data fitting problem approximating feasible solutions within any specified degree of accuracy can be obtained by converting the original problem into a sequence of increasingly large finite dimensional chebyshev approximation problems for which solution stable and efficient numerical methods exist a direct formula for calculating tight upper bounds to the approximation error is provided the link between the present algebraic approach and the dahleh and pearson functional analytic one 1988 is also discussed

minimization of a regulated response to a fixed input the problem of minimizing a regulated output due to a specific bounded input is solved in the siso single input single output discrete time case it is shown that in most situations the optimal solution will not exist but given any epsilon 0 it is possible to find a suboptimal solution with performance within epsilon of the optimal performance the proposed approach is extended to the problem of shaping the error transient response it is shown how time domain templates can be used in this problem in the same way that frequency weighting functions are used in the h sup infinity and h sup 2 problems

integrable field theory and critical phenomena the ising model in a magnetic field the two dimensional ising model is the simplest model of statistical mechanics exhibiting a second order phase transition while in absence of magnetic field it is known to be solvable on the lattice since onsager s work of the 1940s exact results for the magnetic case have been missing until the late 1980s when a zamolodchikov solved the model in a field at the critical temperature directly in the scaling limit within the framework of integrable quantum field theory in this paper we review this field theoretical approach to the ising universality class with particular attention to the results obtained starting from zamolodchikov s scattering solution and to their comparison with the numerical estimates on the lattice the topics discussed include scattering theory form factors correlation functions universal amplitude ratios and perturbations around integrable directions although we restrict our discussion to the ising model the emphasis is on the general methods of integrable quantum field theory which can be used in the study of all universality classes of critical behaviour in two dimensions

on the singularity structure of the 2d ising model susceptibility some simplifications of the integrals 2n 1 derived by wu et al 1976 13 316 that contribute to the zero field susceptibility of the 2d square lattice ising model are reported in particular several alternate expressions for the integrands in 2n 1 are determined which greatly facilitate both the generation of high temperature series and analytical analysis one can show that as series 2n 1 22n s 2 4n n 1 1 o s where s is the high temperature variable sin 2k with k the conventional normalized inverse temperature analysis of the integrals near symmetry points of the integrands shows that 2n 1 s is singular on the unit circle at sk exp ik where 2cos k cos 2 k 2n 1 cos 2 2n 1 n k n the singularities k 0 excepted are logarithmic branch points of order 2n n 1 1ln with 1 s sk there is numerical evidence from series that these van hove points in addition to the known points at s 1 and i exhaust the singularities on the unit circle barring cancellation from extra unobserved singularities one can conclude that s 1 is a natural boundary for the susceptibility

a heuristic approach to solving a class of bilinear matrix inequality problems abstract many canonical and modern control problems can be recast into the problem of a group of matrix inequalities some of them are in the form of linear matrix inequalities lmis which can be solved very efficiently by the powerful lmi toolbox in matlab but some others are in the form of bilinear matrix inequalities the characteristic of this latter class of problems is that when the so called communicating variables are fixed the overall problem will be reduced to the problem in lmis thus how to find the communicating variables is the key to solve the whole problem in this paper an optimal estimate for the communicating variables is presented we will illustrate our method by completely solving the problems of overshoot bound control and reachable set analysis for uncertain systems numerical examples are provided to show the effectiveness of the proposed method

l sub 1 optimality of feedback control systems the siso discrete time case controllers that optimally reject a class of disturbances are considered the problem of determining when a stabilizing control is l sub 1 optimal for a given plant is studied for some stable weighting function this problem belongs to the class of inverse problems in optimal control introduced by kalman it is shown that for a given plant the set of all the h sub infinity optimal controllers obtained by considering all stable weighting functions with no zeros on the unit circle is actually contained in the corresponding set of l sub 1 optimal controllers it is also demonstrated that an l sub 1 optimal controller unlike an h sub infinity optimal controller can remain l sub 1 optimal for the same plant for a wide range of nontrivially different weighting functions

ising field theory in a magnetic field analytic properties of the free energy we study the analytic properties of the scaling function associated with the 2d ising model free energy in the critical domain t tc h 0 the analysis is based on numerical data obtained through the truncated free fermion space approach we determine the discontinuities across the yang lee and langer branch cuts we confirm the standard analyticity assumptions and propose extended analyticity roughly speaking the latter states that the yang lee branching point is the nearest singularity under langer s branch cut we support the extended analyticity by evaluating numerically the associated extended dispersion relation

optimal pole placement design for siso discrete time systems in this paper we incorporate the placement of one real closed loop pole into a compensator design framework based upon the youla parameterization duality theory and linear programming this framework has been used to design discrete time compensators to solve the l sub 1 controller design problem as well as other related time domain optimization problems previous work on these problems has focused on deadbeat systems it is known that these can require high order controllers part of the motivation for this work is to improve the tradeoff between controller order and performance over that using deadbeat control

the susceptibility of the square lattice ising model new developments we have made substantial advances in elucidating the properties of the susceptibility of the square lattice ising model we discuss its analyticity properties certain closed form expressions for subsets of the coefficients and give an algorithm of complexity o n6 to determine its first n coefficients as a result we have generated and analyzed series with more than 300 terms in both the high and low temperature regime we quantify the effect of irrelevant variables to the scaling amplitude functions in particular we find and quantify the breakdown of simple scaling in the absence of irrelevant scaling fields arising first at order t tc 9 4 though high low temperature symmetry is still preserved at terms of order t tc 17 4 and beyond this symmetry is no longer present the short distance terms are shown to have the form t tc p log t tc q with p q2 conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments

analytic continuation at first order phase transitions we study the analytic structure of thermodynamic functions at first order phase transitions in systems with short range interactions and in particular in the two dimensional ising model we analyze the nature of the approximation of the d 2 system by ann strip investigation of the structure of the eigenvalues of the transfer matrix in the vicinity of h 0 in the complexh plane allows us to define a new function which provides rapidly convergent approximations to the stable free energyf and its derivatives for allh 0 this new function is used for numerical calculation of the coefficients cn in the power series expansions of the magnetizationm in the form m h 1 cn h h 0 n for various h0 0 the resulting series are studied by conventional methods we confirm recent series analysis results on the existence of the droplet model type essential singularity at h 0 evidence is found for a spinodal at h hsp ti 0

tau functions for the dirac operator on the cylinder the goal of the present paper is to calculate the determinant of the dirac operator with a mass in the cylindrical geometry the domain of this operator consists of functions that realize a unitary one dimensional representation of the fundamental group of the cylinder with n marked points the determinant represents a version of the isomonodromic function introduced by m sato t miwa and m jimbo it is calculated by comparison of two sections of the det bundle over an infinite dimensional grassmannian the latter is composed of the spaces of boundary values of some local solutions to the dirac equation the principal ingredients of the computation are the formulae for the green function of the singular dirac operator and for the so called canonical basis of global solutions on the 1 punctured cylinder we also derive a set of deformation equations satisfied by the expansion coefficients of the canonical basis in the general case and find a more explicit expression for the function in the simplest case n 2

minimum overshoot design for siso discrete time systems the problem of designing a controller to minimize the maximum negative error due to a specific bounded input is studied it is found that the solution can be obtained by solving an infinite linear programming problem in the special case of a step input a simple analytical formula for minimum overshoot is derived for a plant with one unstable pole and one nonminimum phase zero the solution technique has potential application to other control problems which seek to minimize quantities not directly representable as a norm

scaling and universality in the two dimensional ising model with a magnetic field the scaling function of the two dimensional ising model on the square and triangular lattices is obtained numerically via baxter s variational corner transfer matrix approach the use of aharony fisher nonlinear scaling variables allowed us to perform calculations sufficiently away from the critical point and to confirm all predictions of the scaling and universality hypotheses our results are in excellent agreement with quantum field theory calculations of fonseca and zamolodchikov as well as with many previously known exact and numerical calculations including susceptibility results by barouch mccoy tracy and wu

rational suboptimal continuous time controller design we obtain rational suboptimal continuous time solutions to some optimal control problems specified via time domain performance criteria these include spl lscr sub 1 and spl lscr sub spl infin norms and peak overshoot the approach uses linear semi infinite programming to compute weights for a given finite set of rational basis functions and makes the best possible use of the basis the closed loop transfer functions obtained satisfy appropriate interpolation constraints for internal stability and the formulation allows additional linear constraints to be incorporated

an application of lattice basis reduction to polynomial identities for algebraic structures abstract the authors recent classification of trilinear operations includes among other cases a fourth family of operations with parameter q q and weakly commutative and weakly anticommutative operations these operations satisfy polynomial identities in degree 3 and further identities in degree 5 for each operation using the row canonical form of the expansion matrix e to find the identities in degree 5 gives extremely complicated results we use lattice basis reduction to simplify these identities we compute the hermite normal form h of et obtain a basis of the nullspace lattice from the last rows of a matrix u for which uet h and then use the lll algorithm to reduce the basis

jordan quadruple systems abstract we define jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the jordan tetrad a b c d a b c d d c b a as a quadrilinear operation on associative algebras we find further identities in degree 10 which are not consequences of the defining identities we introduce four infinite families of finite dimensional jordan quadruple systems and construct the universal associative envelope for a small system in each family we obtain analogous results for the anti tetrad a b c d a b c d d c b a our methods rely on computer algebra especially linear algebra on large matrices the lll algorithm for lattice basis reduction representation theory of the symmetric group noncommutative grobner bases and wedderburn decompositions of associative algebras

fundamental limitations on the time domain shaping of response to a fixed input in this paper we examine time domain limitation of performance problems in feedback control system design the main result is a theorem which gives a dual formulation to the problem of determining absolute limits on the time domain shaping of the response to a fixed input the duals that arise are particularly illuminating they have a simple interpretation as optimizing the impulse response of a transfer function whose poles are readily constructed from the data much insight is obtainable from the dual for example in helping to classify classes of plants with the same quantitative behavior new results on classical performance measures such as overshoot and undershoot are presented we also examine minimization of the difference between the maximum and minimum values of the error response a quantity we term fluctuation

the universal associative envelope of the anti jordan triple system of n n matrices abstract we show that the universal associative enveloping algebra of the simple anti jordan triple system of all n n matrices n 2 over an algebraically closed field of characteristic 0 is finite dimensional we investigate the structure of the universal envelope and focus on the monomial basis the structure constants and the center we explicitly determine the decomposition of the universal envelope into matrix algebras we classify all finite dimensional irreducible representations of the simple anti jordan triple system and show that the universal envelope is semisimple we also provide an example to show that the universal enveloping algebras of anti jordan triple systems are not necessary to be finite dimensional

the structure of strongly prime quadratic jordan algebras abstract we extend the structure theory for strongly prime jordan algebras of arbitrary dimension to quadratic algebras of characteristic 2 we show that such j are either 1 hermitian forms h 0 a j h q a lying between an ample subspace of hermitian elements in a prime associative algebra a and those in its martindale ring of quotients q a 2 clifford forms with a scalar extension j j which is an algebra j j q 1 of a nondegenerate quadratic from q with basepoint 1 or 3 albert forms with a scalar extension j j which is a 27 dimensional split albert algebra j h 3 k of 3 3 hermitian matrices over an 8 dimensional split octonion algebra k as a consequence the simple jordan algebras of arbitrary dimension are either hermitian h 0 a for simple a or clifford forms in j q 1 or are albert algebras in particular our methods give an idempotent free classification of all simple finite dimensional algebras which are i special the key to this characterization is the existence of zel manov polynomials in the free special jordan algebra which are both hermitian polynomials whose values always look like hermitian elements and at the same time clifford polynomials nontrivial on 3 3 hermitian matrices hence on special algebras with 3 interconnected idempotents if a zel manov polynomial does not vanish on j then j is of hermitian type and if it does vanish on j then j is of clifford type one example of such a polynomial is q48 p16 x1 y1 z1 w1 p16 x2 y2 z2 w2 p16 x3 y3 z3 w3 for p16 d2x y z 2 dx y w dx y w dx y z x y z thus the structure of jordan algebras depends very directly on polynomial identities and commutators

minimizing a convex combination of the overshoot and undershoot abstract we study the problem of minimizing a parameterized convex combination of the overshoot and undershoot of siso continuous time system in response to a known input from a dual formulation we develop a condition for solution existence and specify the structure of optimal solution in addition an interrelation between the overshoot and undershoot in controller synthesis is analytically explained in our framework

critical and off critical conformal analysis of the ising quantum chain in an imaginary field the critical curve with central charge c 22 5 of the ising quantum chain in an imaginary longitudinal field is calculated by finite size extrapolation of the yang lee square root branch point position various power laws are found the field dependence of the off criticality mass scale and the four lowest off criticality scaling functions are calculated the latter turn out to be universal along the critical line and agree very well with the results of the truncated hilbert space method for the c 22 5 continuum theory the author obtains the universal bulk scaling coefficient in qualitative agreement with the thermodynamic bethe ansatz result the scattering phaseshift calculated by luscher s method 1988 reproduces the cardy mussardo minimal solution 1989 for the s matrix if a boundary defect is introduced the only non trivial primary field decouples so that the spectrum consists only of the tower of the unit operator in the antiferromagnetic regime with a non staggered imaginary field the author finds boundary dependent lines where the gap vanishes

nonhomogeneous subalgebras of lie and special jordan superalgebras abstract we consider polynomial identities satisfied by nonhomogeneous subalgebras of lie and special jordan superalgebras we ignore the grading and regard the superalgebra as an ordinary algebra the lie case has been studied by volichenko and baranov they found identities in degrees 3 4 and 5 which imply all the identities in degrees 6 we simplify their identities in degree 5 and show that there are no new identities in degree 7 the jordan case has not previously been studied we find identities in degrees 3 4 5 and 6 which imply all the identities in degrees 6 and demonstrate the existence of further new identities in degree 7 our proofs depend on computer algebra we use the representation theory of the symmetric group the hermite normal form of an integer matrix the lll algorithm for lattice basis reduction and the chinese remainder theorem

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

yang lee zeros of the yang lee model to understand the distribution of the yang lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the 2d yang lee model the grand canonical partition function of this quantum field theory as a function of the fugacity z and the inverse temperature can be computed in terms of the thermodynamics bethe ansatz based on its exact s matrix we extract the yang lee zeros in the complex plane by using a sequence of polynomials of increasing order n in z which converges to the grand canonical partition function we show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories there is though an important difference between the interactive theory and the free theories for the radius of the zeros in the interactive theory goes continuously to zero in the high temperature limit while in the free theories it remains close to 1 even for small values of jumping to 0 only at

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

sign weighted peak minimization problem for feedback systems this note studies the problem of minimizing a parameterized convex combination of the overshoot and undershoot of a single input single output siso discrete time system to a fixed input we provide a simple condition for the solution existence and characterize the structure of optimal solutions it is shown that a known l sub spl infin suboptimal scheme also works in our problem in addition we make a comparative analysis with the l sub spl infin problem

exact curie temperature for the ising model on archimedean and laves lattices using the feynman vdovichenko combinatorial approach to the two dimensional ising model we determine the exact curie temperature for all two dimensional archimedean lattices by means of duality we extend our results to cover all two dimensional laves lattices for those lattices where the exact critical temperatures are not yet exactly known we compare them with monte carlo simulations