staggered lattice models some staggered models are studied along three different lines exact solvability yang baxter equations the existence of other exact solutions called disorder solutions their group of symmetries the analysis based on the commutation of row to row transfer matrices of arbitrary size n suggests that the exactly solvable models are either of non staggered type or of free fermion staggered type in contrast disorder solutions can easily be exhibited for staggered models the example of the ashkin teller model is analysed finally the symmetry group of these models is seen to be a straightforward generalisation of the one of homogeneous models

weighted shift matrices unitary equivalence reducibility and numerical ranges abstract an n by n n 3 weighted shift matrix a is one of the form 0 a 1 0 a n 1 a n 0 where the a j s called the weights of a are complex numbers assume that all a j s are nonzero and b is an n by n weighted shift matrix with weights b 1 b n we show that b is unitarily equivalent to a if and only if b 1 b n a 1 a n and for some fixed k 1 k n b j a k j a n j a j for all j next we show that a is reducible if and only if a j j 1 n is periodic that is for some fixed k 1 k n 2 n is divisible by k and a j a k j for all j 1 j n k finally we prove that a and b have the same numerical range if and only if a 1 a n b 1 b n and s r a 1 2 a n 2 s r b 1 2 b n 2 for all 1 r n 2 where s r s are the circularly symmetric functions

disorder solutions for ising and potts models with a field the exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard ising and potts models with a magnetic field are obtained different specialisations and extensions of these exact results are examined

explicit solutions to poncelet s porism abstract solution to the following problem is considered for given conics c and k and an integer n 3 determine whether there exists a closed n sided polygon inscribed in c and circumscribed about k the case of c and k being circles is considered in detail equations are proposed with a relatively small number of arithmetic operations near log 2 n along the way the following result is obtained for circles with rational coefficients the polygons can only have the following number of sides n 3 4 5 6 8 10 and 12 a subset of the mazur s set of integers for rational elliptic curves the proposed solution may also be applied to determine whether a hankel determinant of order n 2 having special form used in the classical cayley criterion is equal to zero and for related problems possible generalizations for ellipses and hyperbolas are also presented particularly equations are proposed for parameters of concentric poncelet s ellipses billiard case

dimensional reduction and correlation functions on 3d lattice models the exact calculation of an infinite number of correlation functions for general cubic and hcp anisotropic ising models on their disorder variety is described the intra plane correlation functions are found to be exactly the same as for a two dimensional free fermion ising model this allows the authors to characterise the intersection between the disorder and critical variety which displays in general anisotropic scaling with non trivial exponents

spectrum of monodromy operator for a time delay system with application to stability analysis this note studies the spectral properties of monodromy operators which play an important role in stability analysis of linear time invariant time delay feedback systems the note is motivated by the fact that this operator can actually be defined naturally on four spaces where the difference stems from different choices for the function space on which the infinite dimensional state of such a time delay system is assumed to take its value it is first shown that the spectrum of the monodromy operator is independent of the spaces on which it is defined this implies that stability of time delay systems is independent of the underlying function spaces it is further shown that the operator spectrum is continuous at monodromy operators which justifies the spectrum computation of the monodromy operator through its approximation by any sort of tractable operators a numerical study relevant to the theoretical development is provided and a practical implication of our theoretical study is suggested

computing the l2 induced norm of a compression operator abstract a new computationally viable approach is derived for computing the induced norm of a state space compression operator that is an integral operator defined on the finite length space l2 0 h determining this norm is a crucial component in the control analysis of both delay systems and sampled data systems the technique developed may have application to a wider variety of integral operators

numerical ranges of companion matrices we show that an n by n companion matrix a can have at most n line segments on the boundary w a of its numerical range w a and it has exactly n line segments on w a if and only if for n odd a is unitary and for n even a is unitarily equivalent to the direct sum a1 a2 of two n 2 by n 2 companion matrices a1

susceptibility of the checkerboard ising model the authors show that at the disorder variety the n point correlation functions of the checkerboard potts model have a simple causal structure and decay as simple exponentials for space like separations an exact expression for the susceptibility in the ising case is obtained the result is additional evidence in favour of the conjectured s4 symmetry of the checkerboard potts model in the presence of a finite field

on geometric properties of the numerical range abstract we investigate the shape of the numerical range a criterion for the numerical range of a matrix to be an elliptical disk is given the result is applied to show that there exist neither 3 by 3 nor 4 by 4 nilpotent matrices whose numerical range is an elliptical noncircular disk sufficient conditions for n by n tridiagonal matrices to have elliptical numerical range are obtained the boundary of the numerical range near a sharp point is examined finally the numerical range of a reducible matrix is compressed and its geometric properties are discussed

weighted sensitivity minimization for delay systems in this note we discuss the h infty sensitivity minimization problem for linear time invariant delay systems while the unweighted case reduces to simple nevanlinna pick interpolation the weighted case turns out to be much more complicated and demands certain functional analytic techniques for its solution

a function space approach to sampled data control systems and tracking problems this paper presents a new framework for hybrid sampled data control systems instead of considering the state only at sampling instants this paper introduces a function piece during the sampling period as the state and gives an infinite dimensional model with such a state space this gives the advantage that sampled data systems with built in intersample behavior can be regarded as linear time invariant discrete time systems as a result the approach makes it possible to introduce such time invariant concepts as transfer functions poles and zeros to the sampled data systems even with the presence of the intersample behavior in particular tracking problems can be studied in this setting in a simple and unified way and ripples are completely characterized as a mismatch between the intersample reference signal and transmission zero directions this leads to the internal model principle for sampled data systems

sampled data control of continuous time systems with an h optimality criterion abstract the synthesis of sampled data controllers for continuous time systems is studied in the context of h control theory the sampled data h optimal control problem consists of finding a sampled data controller which minimizes the l2 induced norm from the continuous time disturbance to the continuous time output the problem is here solved via a discrete system representation this is achieved by describing the closed loop system as a discrete system with finite state dimension and with a representation of the disturbance and the output signals of the discrete system as sequences of elements of certain infinite dimensional spaces via finite rank approximations of a compact integral operator associated with the system the sampled data h control problem can be solved to any degree of accuracy from the structure of the problem it follows that irrespective of the order of the approximation a feasible sampled data controller can be found which has the same order as the system

l_ 1 discretization for sampled data controller synthesis via piecewise linear approximation this paper develops a new discretization method with piecewise linear approximation for the l 1 optimal controller synthesis problem of sampled data systems which is the problem of minimizing the l infty induced norm of sampled data systems we apply fast lifting on the top of the lifting technique by which the sampling interval 0 h is divided into m subintervals with an equal width the signals on each subinterval are then approximated by linear functions by introducing two types of linearizing operators for input and output which leads to piecewise linear approximation of sampled data systems by using the arguments of preadjoint operators we provide an important inequality that forms a theoretical basis for tackling the l 1 optimal controller synthesis problem of sampled data systems more efficiently than the conventional method more precisely a mathematical basis for the piecewise linear approximation method associated with the convergence rate is shown through this inequality and this suggests that the piecewise linear approximation method may drastically outperform the conventional method in the l 1 optimal controller synthesis problem of sampled data systems we then provide a discretization procedure of sampled data systems by which the l 1 optimal controller synthesis problem is converted to the discrete time l 1 optimal controller synthesis problem finally effectiveness of the proposed method is demonstrated through a numerical example

frequency response of sampled data systems this paper introduces the concept of frequency response for sampled data systems and explores some basic properties as well as its computational procedures it is shown that 1 by making use of the lifting technique the notion of frequency response can be naturally introduced to sampled data systems in spite of their time varying characteristics 2 it represents a frequency domain steady state behaviour and 3 it is also closely related to the original transfer function representation via an integral formula it is shown that the computation of the frequency response can be reduced to a finite dimensional eigenvalue problem and some examples are presented to illustrate the results

boundary generating curves of the c numerical range abstract let a be an n n matrix and c c 1 c 2 c n be a real n tuple the c numerical range of a is the set w c a j 1 n c j x j ax j x 1 x 2 x n is an orthonormal basis of c n we obtain parametric representations of the boundary generating curve of the c numerical range of a matrix applying this result we generalize the result of anderson to the c numerical range furthermore we give a description of the boundary generating curve of the c numerical range of certain types of nilpotent toeplitz matrices a sufficient condition for the boundary generating curve to be rational is obtained finally we explicitly compute the boundary generating curves of the numerical ranges for several concrete matrices and classify the rationality of the curves

disorder points of the irf and checkerboard potts models conditions are given for a general interactions round a face irf model in statistical mechanics to have a disorder point and expressions obtained for the free energy and intra low correlations these are applied to the checkerboard potts model thereby verifying a recent conjecture by jaeckel and maillard 1984

a disorder solution for a generalised mixed spin model on anisotropic kagome lattices a disorder solution is given for a generalised mixed spin model on an anisotropic kagome lattice because of its lower rotational symmetry the solution is obtained by decimation along four different directions in the case of a spin 1 2 and spin 1 mixed model the intra row correlations in both horizontal and vertical directions are zero when a relation imposed on the interactions is satisfied

on the h optimal sensitivity problem for systems with delays in this paper we extend some of the results of ieee trans automat control ac 31 1986 pp 763 766 to more general delay systems in particular we analyze the effect of the interaction of delays and nonminimum phase zeros on the h infty optimal weighted sensitivity

computing quasi lti robustness margins in sampled data systems the robustness test for sampled data systems with slowly time varying perturbations is known to be infinite dimensional in nature the note develops computationally explicit upper and lower bounds for the corresponding stability radius presenting them in terms of linear matrix inequalities lmis given by state space formulas derived the upper bound is shown to converge monotonically to the stability radius and so can be systematically tightened at the cost of increased computational effort the lower bound is monotonically increasing and is conjectured to also converge to the stability radius

the diameter and width of numerical ranges abstract we characterize the diameter and width of the numerical range present an algorithm for computing the diameter and width of the numerical range and formulate the diameters of the numerical ranges of unitary bordering matrices for lower dimensions we also determine the boundaries of the numerical ranges of certain nilpotent toeplitz matrices to be curves of constant width

uniform robust performance against quasi lti uncertainty in sampled data systems uniform robust performance of sampled data systems is studied in the context of arbitrarily slowly time varying structured perturbations exact conditions for robustness are obtained these conditions are convex but inherently infinite dimensional in nature

disorder solutions and the star triangle relation it is shown that the disorder solutions exhibited by two dimensional statistical lattice systems verify the star triangle relation these solutions of the star triangle relation are of a different type than those corresponding to completely integrable models

series analysis of the q state checkerboard potts model the low temperature expansion of the checkerboard potts model in a magnetic field is obtained up to order twelve from the so called disorder solutions various expansions and results known in the literature it is shown that this expansion drastically simplifies on the dual of the disorder variety the low temperature expansion of the magnetisation is seen to become equal to one up to order twelve when it is restricted to the dual of the disorder variety these results have to be seen as exact formal constraints on the analytic continuation of the low temperature expansion of the partition function per site similar simplifications occur for the susceptibility and higher derivatives with respect to the magnetic field these expansions are also analysed in the vicinity of these particular varieties

numerical ranges and poncelet curves abstract convex circuits which have the property of circles of the great poncelet theorem are introduced the circuits are generated as boundaries of numerical ranges of specially constructed matrices algebraic and geometric properties of such matrices and their numerical ranges are obtained these properties lead to a new proof of the poncelet theorem the proposed approach illustrates the fact that complex relations between geometrical figures in a plane may have simple interpretation in a space of higher dimension

s4 symmetry on the checkerboard potts model the large q expansion of the partition function z of the checkerboard q state potts model with a magnetic field is given up to sixth order in q 12 up to this order z is shown to exhibit an unexpected s4 symmetry using various arguments exact results expansion etc the authors suggest the general validity of this new symmetry property of the considered model an exact expression for the magnetisation jump is also proposed and checked on the large q expansion

the complex structured singular value a tutorial introduction to the complex structured singular value is presented with an emphasis on the mathematical aspects of the based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared and a simple synthesis problem is studied uncertain systems are represented using linear fractional transformations lfts which naturally unify the frequency domain and state space methods

lower bounds for the spread of a matrix a characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread comparisons are then made with several known bounds

order and disorder lines in systems with competing interactions ii the irf model systems with competing interactions can be often exactly solved on a restricted subspace of the parameter space called an order or disorder trajectory a simple method introduced within the transfer matrix formalism allows for the calculation of the free energy and spin spin correlation functions along the order and disorder lines of the ising model with all possible interactions around a face of the square lattice irf model the general eight vertex model is thoroughly examined and shows full analogy with the quantum spin chain results of the previous paper

on the weighted sensitivity minimization problem for delay systems we consider the h optimal sensitivity problem for delay systems in particular we consider computation of inf w q q h jr where w s is any function in rh jr and in h jr is any inner function we derive a new explicit solution in the pure delay case where e sh h 0

h 0e sensitivity minimization for delay systems this paper announces results on the problem of feedback compensator design for x norm weighted sensitivity minimization when the plant contains a delay in the input a complete solution is presented for the case of one pole zero weighting function and a single input single output plant for stable minimum phase rational part generalizations and proofs will be published elsewhere

a general framework for linear periodic systems with applications to h sup infinity sampled data control the authors present a framework for dealing with continuous time periodic systems the main tool is a lifting technique which provides a strong correspondence between continuous time periodic systems and certain types of discrete time time invariant systems with infinite dimensional input and output spaces despite the infinite dimensionality of the input and output spaces a lifted system has a finite dimensional state space if the original system does this fact permits rather constructive methods for analyzing these systems as a demonstration of the utility of this framework the authors use it to describe the continuous time i e intersample behavior of sampled data systems and to obtain a complete solution to the problem of parameterizing all controllers that constrain the l sup 2 induced norm of a sampled data system to within a certain bound

the vicinity of the disorder varieties a systematic expansion the authors show that it is possible to obtain information in the vicinity of disorder varieties this can be achieved through a new systematic type of diagrammatic expansion illustrated here on the anisotropic triangular ising model non perturbative methods can also be used to get an exact expression for some quantities such as finite field correlation functions magnetisation and susceptibility restricted to the disorder varieties this last result confirms an exact result of dhar and maillard ibid vol 18 p 1383 1985

determinantal representations of invariant hyperbolic plane curves abstract we study hyperbolic polynomials with nice symmetry and express them as the determinant of a hermitian matrix with special structure the goal of this paper is to answer a question posed by chien and nakazato in 2015 3 by properly modifying a determinantal representation construction of dixon 1902 5 we show for every hyperbolic polynomial of degree n invariant under the cyclic group of order n there exists a determinantal representation admitted via some cyclic weighted shift matrix moreover if the polynomial is invariant under the action of the dihedral group of order n the associated cyclic weighted shift matrix is unitarily equivalent to one with real entries

corrections to bisection algorithm for computing the frequency response gain of sampled data systems infinite dimensional congruent transformation approach this paper derives a bisection algorithm for computing the frequency response gain of sampled data systems with their intersample behavior taken into account the properties of the infinite dimensional congruent transformation i e the schur complement arguments and the sylvester law of inertia play a key role in the derivation specifically it is highlighted that counting up the numbers of the negative eigenvalues of self adjoint operators is quite important for the computation of the frequency response gain this contrasts with the well known arguments on the related issue of the sampled data h sub spl infin problem where the key role is played by the positivity of operators and the loop shifting technique the effectiveness of the derived algorithm is demonstrated through a numerical example

finite rank criteria for h infty control of infinite dimensional systems this paper studies the hinfin control of infinite dimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar possibly irrational inner function this class of systems is adequate for describing many control problems in practice when weighting functions are rational and plants have at most finitely many unstable modes we show that this problem can be reduced to solving two matrix valued riccati equations and an additional rank condition furthermore the obtained controller structure is characterized by the inner function and controllers for the finite dimensional part this result provides us with a solution that is easily implementable without much prior knowledge on infinite dimensional control theory a numerical example is given to illustrate the result

a disorder solution of the general 8 vertex model a disorder solution for the general 8 vertex model is found by applying a procedure recently developed by baxter 1984 the solution is valid when an adequate relation between the vertex weights of the model is satisfied