a linear code for the sawtooth and cat maps abstract the sawtooth maps are a one parameter set of piecewise linear area preserving maps on the torus for positive integer values of the parameter k they are automorphisms of the torus known as the cat maps we present a symbolic dynamics for these maps in which the symbols are integers this code is related to a practical problem of the stabilisation of a system which is perturbed by impulses the code is linear in the sense that an orbit and its code are linearly related so it is not difficult to obtain a good approximation to one from the other in practice a stationary stochastic process for generating the code is given explicitly the theory uses green function methods which are also used to study ordered periodic orbits and cantori the problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed

information theoretic characterization of quantum chaos hypersensitivity to perturbation is a criterion for chaos based on the question of how much information about a perturbing environment is needed to keep the entropy of a hamiltonian system from increasing we demonstrate numerically that hypersensitivity to perturbation is present in the following quantum maps the quantum kicked top the quantum baker s map the quantum lazy baker s map and the quantum sawtooth and cat maps

matrix coefficient realization theory of noncommutative rational functions abstract noncommutative rational functions i e elements of the universal skew field of fractions of a free algebra can be defined through evaluations of noncommutative rational expressions on tuples of matrices this interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas from free real algebraic geometry to systems and control theory if a noncommutative rational function is regular at the origin it can be described by a linear object called a realization in this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rational functions and is well adapted for studying matrix evaluations of special interest are the minimal realizations which compensate the absence of a canonical form for noncommutative rational functions the non minimality of a realization is assessed by obstruction modules associated with it they enable us to devise an efficient method for obtaining minimal realizations with them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity using these results we determine concrete size bounds for rational identity testing construct minimal symmetric realizations and prove an effective local global principle for linear dependence of noncommutative rational functions

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

structured noncommutative multidimensional linear systems we introduce a class of multidimensional linear systems with evolution along a free semigroup the transfer function for such a system is a formal power series in noncommuting indeterminates standard system theoretic properties the operations of cascade parallel connection and inversion controllability observability kalman decomposition state space similarity theorem minimal state space realizations hankel operators realization theory are developed for this class of systems we also draw out the connections with the much earlier studied theory of rational and recognizable formal power series applications include linear fractional models for classical discrete time systems with structured time varying uncertainty dimensionless formulas in robust control multiscale systems and automata theory and the theory of formal languages

likelihood inference for exchangeable binary data with varying cluster sizes ce papier etudie l estimation au maximum de vraisemblance dans des modeles satures et insatures pour des donnees binaires correlees echangeables lorsqu on dispose d un echantillon de clusters independants de tailles variables nous discutons plusieurs parametrisations de ces modeles et nous proposons l utilisation de l algorithme em pour obtenir des estimations du maximum de vraisemblance la methodologie est illustree par des applications a une etude d agregation familiale de maladie et a la construction d un essai randomise par groupe pour la prevention du cancer

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

free pluriharmonic majorants and commutant lifting in this paper we initiate the study of sub pluriharmonic curves and free pluriharmonic majorants on the noncommutative open ball b h n 1 x1 xn b h n x1x1 xnxn 1 2 1 where b h is the algebra of all bounded linear operators on a hilbert space h several classical results from complex analysis have analogues in this noncommutative multivariable setting we present basic properties for sub pluriharmonic curves characterize the class of sub pluriharmonic curves that admit free pluriharmonic majorants and find in this case the least free pluriharmonic majorants we show that for any free holomorphic function on b h n 1 the map 0 1 c r1 rn r rr1 rrn rr1 rrn is a sub pluriharmonic curve in the cuntz toeplitz algebra generated by the right creation operators r1 rn on the full fock space with n generators we prove that is in the noncommutative hardy space hball2 if and only if has a free pluriharmonic majorant in this case we find herglotz riesz and poisson type representations for the least pluriharmonic majorant of moreover we obtain a characterization of the unit ball of hball2 and provide a parametrization and concrete representations for all free pluriharmonic majorants of when is in the unit ball of hball2 in the second part of this paper we introduce a generalized noncommutative commutant lifting gncl problem which extends to our noncommutative multivariable setting several lifting problems including the classical sz nagy foias commutant lifting problem and the extensions obtained by treil volberg foias frazho kaashoek and biswas foias frazho as well as their multivariable noncommutative versions we solve the gncl problem and using the results regarding sub pluriharmonic curves and free pluriharmonic majorants on noncommutative balls we provide a complete description of all solutions in particular we obtain a concrete schur type description of all solutions in the noncommutative commutant lifting theorem

chaos for liouville probability densities using the method of symbolic dynamics we show that a large class of classical chaotic maps exhibits exponential hypersensitivity to perturbation i e a rapid increase with time of the information needed to describe the perturbed time evolution of the liouville density the information attaining values that are exponentially larger than the entropy increase that results from averaging over the perturbation the exponential rate of growth of the ratio of information to entropy is given by the kolmogorov sinai entropy of the map these findings generalize and extend results obtained for the baker s map r schack and c m caves phys rev lett 69 3413 1992

noncommutative plurisubharmonic polynomials part ii local assumptions abstract we say that a symmetric noncommutative nc polynomial is nc plurisubharmonic nc plush on an nc open set if it has an nc complex hessian that is positive semidefinite when evaluated on open sets of matrix tuples of sufficiently large size in this paper we show that if an nc polynomial is nc plurisubharmonic on an nc open set then the polynomial is actually nc plurisubharmonic everywhere and has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic greene et al 2011 1 has shown that if p is nc plurisubharmonic everywhere then p has the form in eq 0 1 in other words 1 makes a global assumption while the current paper makes a local assumption but both reach the same conclusion we show that if p is nc plurisubharmonic on an nc open set local then p is in fact nc plurisubharmonic everywhere global and has the form expressed in eq 0 1 this paper requires a technique that is not used in 1 we use a gram like vector and matrix representation called the border vector and middle matrix for homogeneous degree 2 nc polynomials we then analyze this representation for the nc complex hessian on an nc open set and positive semidefiniteness forces a very rigid structure on the border vector and middle matrix this rigid structure plus the theorems in 1 ultimately force the form in eq 0 1

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

tests of homogeneity for the relative risk in multiply matched case control studies summary we examine several tests of homogeneity of the odds ratio in the analysis of 2 x 2 tables arising from epidemiologic 1 r matched case control studies the t4 and t5 statistics proposed by liang and self 1985 biometrika 72 353 358 are unable to detect obvious inhomogeneity in two numerical examples and in simulation studies the null hypothesis is rejected by the chi square statistic of ejigou and mchugh 1984 bioinetrika 71 408 411 and by a new proposed method whose significance level must be simulated

plurisubharmonic noncommutative rational functions abstract a noncommutative nc function in x 1 x g x 1 x g is called plurisubharmonic plush if its nc complex hessian takes only positive semidefinite values on an nc neighborhood of 0 the main result of this paper shows that an nc rational function is plush if and only if it is a composite of a convex rational function with an analytic no x j rational function the proof is entirely constructive further a simple computable necessary and sufficient condition for an nc rational function to be plush is given in terms of its minimal realization

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

pick interpolation for free holomorphic functions we give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron as an application we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function

noncommutative convexity arises from linear matrix inequalities this paper concerns polynomials in g noncommutative variables x x1 xg inverses of such polynomials and more generally noncommutative rational expressions with real coefficients which are formally symmetric and analytic near 0 the focus is on rational expressions r r x which are matrix convex near 0 i e those rational expressions r for which there is an 0 such that if x x1 xg is a g tuple of n n symmetric matrices satisfying in x12 xg2 is positive definite and y is also then the symmetric matrix tr x 1 t r y r tx 1 t y is positive semidefinite for all numbers t 0 t 1 this article gives a complete classification of matrix convex rational expressions see theorem 3 3 by representing such r in terms of a symmetric linear pencil l x id jajxj 0d 100 1 r 0 in the noncommuting variables xj where aj are symmetric d d matrices namely for a real number r is a schur complement of the linear pencil l moreover given a matrix convex r the set consisting of g tuples x of n n symmetric matrices 0 1 x r x iis negative definite has component containing 0 which is the same as the negativity set 0 2 x l x is negative definite for l conditions like l x is negative definite are known as linear matrix inequalities lmis in the engineering literature and arguably the main advance in linear systems theory in the 1990s was the introduction of lmi techniques in this language what we have shown in 0 1 vs 0 2 is that the set of solutions to a convex matrix inequality with noncommutative unknowns is the same as the set of solutions to some lmi in many engineering systems problems convexity would have all of the advantages of lmis indeed convexity guarantees that solutions are global and convexity bodes well for reliability of the numerics since lmis have a structure which is seemingly much more rigid than convexity there is the continual hope that a convexity based theory will be more far reaching than lmis but will it there are two natural situations one where the unknowns are scalars and one where the unknowns are matrices appearing in formulas which respect matrix multiplication these latter problems mathematically yield expressions with noncommutative unknowns and they arise in engineering systems problems which are dimensionless in the sense that they scale automatically with dimension as do most of the classics of control theory that is the case we study here and the result stated above suggests the surprising conclusion that for dimensionless systems problems convexity offers no greater generality than lmis indeed the result proves this for a class of model problems furthermore we show that existing algorithms together with algorithms described here construct the lmis above which are equivalent to the matrix inequalities based on the given matrix convex rational function r in a very different direction we prove that a symmetric polynomial p in g noncommutative symmetric variables has a symmetric determinantal representation namely there are symmetric matrices a0 ag in srd d with a0 invertible such that 0 3 detp x det a0 la x for each x a g tuple of symmetric n n matrices of course taking n 1 implies immediately that a commuting variables polynomial p on rg has a symmetric determinantal representation for g 2 much stronger commutative results can be obtained using tools of algebraic geometry but these do not seem to generalize to the higher dimensional case on the other hand a nonsymmetric commutative determinantal representation for any g is due to valiant universality of determinant in algebraic complexity theory our determinantal representation theorem is a bi product of the theory of systems realizations of noncommutative rational functions and can be read independently of much of the rest of the paper while the notion of noncommutative rational functions is standard the equivalence relation we use on rational expressions in our construction based on evaluating rational expressions on matrices is new and gives a new approach to noncommutative rational functions

unknown quantum states the quantum de finetti representation we present an elementary proof of the quantum de finetti representation theorem a quantum analog of de finetti s classical theorem on exchangeable probability assignments this contrasts with the original proof of hudson and moody z wahrschein verw geb 33 343 1976 which relies on advanced mathematics and does not share the same potential for generalization the classical de finetti theorem provides an operational definition of the concept of an unknown probability in bayesian probability theory where probabilities are taken to be degrees of belief instead of objective states of nature the quantum de finetti theorem in a closely analogous fashion deals with exchangeable density operator assignments and provides an operational definition of the concept of an unknown quantum state in quantum state tomography this result is especially important for information based interpretations of quantum mechanics where quantum states like probabilities are taken to be states of knowledge rather than

neutrino billiards time reversal symmetry breaking without magnetic fields a dirac hamiltonian describing massless spin half particles neutrinos moving in the plane r x y under the action of a 4 scalar not electric potential v r is in position representation h i h c v r z where x y and z are the pauli matrices acts on two component column spinor wavefunctions r 1 2 and has eigen values ck n does not possess time reversal symmetry t if v r describes a hard wall bounding a finite domain d billiards this is equivalent to a novel boundary condition for 2 1 t breaking is interpreted semiclassically as a difference of between the phases accumulated by waves travelling in opposite senses round closed geo desics in d with odd numbers of reflections the semiclassical large k asymptotics of the eigenvalue counting function spectral staircase n k are shown to have the weyl leading term ak 2 4 where a is the area of d but zero perimeter correction the dirac equation is transformed to an integral equation round the boundary of d and forms the basis of a numerical method for computing the k n when d is the unit disc geodesics are integrable and the eigenvalues which satisfy j l k n j l 1 k n are locally poisson distributed when d is an africa shape cubic conformal map of the unit disc the eigenvalues are locally distributed according to the statistics of the gaussian unitary ensemble of random matrix theory as predicted on the basis of t breaking and lack of geometric symmetry

statistics of energy levels without time reversal symmetry aharonov bohm chaotic billiards a planar domain d contains a single line of magnetic flux phi switching on phi breaks time reversal symmetry t for quantal particles with charge q moving in d whilst preserving the geometry of classical billiard trajectories bouncing off the boundary delta d if delta d is such that these classical trajectories are chaotic the authors predict that t breaking will cause the local statistics of quantal energy levels to change their universality class from that of the gaussian orthogonal ensemble goe of random matrix theory to that of the gaussian unitary ensemble gue in the semiclassical limit this transition is abrupt for statistics involving the first n levels gue behaviour requires that the quantum flux alpha identical to q phi h 0 13n 14 the special flux alpha 1 2 corresponds to false t breaking and for this case goe statistics are predicted these predictions are confirmed by numerical computation of spectral statistics for a classically chaotic billiard without symmetry for which delta d is a cubic conformal image of the unit disc

singularities of rational functions and minimal factorizations the noncommutative and the commutative setting we show that the singularities of a matrix valued noncommutative rational function which is regular at zero coincide with the singularities of the resolvent in its minimal state space realization the proof uses a new notion of noncommutative backward shifts as an application we establish the commutative counterpart of the singularities theorem the singularities of a matrix valued commutative rational function which is regular at zero coincide with the singularities of the resolvent in any of its fornasini marchesini realizations with the minimal possible state space dimension the singularities results imply the absence of zero pole cancellations in a minimal factorization both in the noncommutative and in the commutative setting

decoherence and linear entropy increase in the quantum baker s map we show that the coarse grained quantum baker s map exhibits a linear entropy increase at an asymptotic rate given by the kolmogorov sinai entropy of the classical chaotic baker s map the starting point of our analysis is a symbolic representation of the map on a string of n qubits i e an n bit register of a quantum computer to coarse grain the quantum evolution we make use of the decoherent histories formalism as a by product we show that the condition of medium decoherence holds asymptotically for the coarse grained quantum baker s map

free pick functions representations asymptotic behavior and matrix monotonicity in several noncommuting variables r nevanlinna showed that elements of the pick class the set of complex analytic functions taking the upper half plane into itself have certain integral representations which reflect their asymptotic behavior at infinity later lowner connected the pick class to matrix monotone functions we generalize the nevanlinna representation theorems and lowner s theorem on matrix monotone functions to the free pick class the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus

noncommutative plurisubharmonic polynomials part i global assumptions abstract we consider symmetric polynomials p in the noncommutative nc free variables x 1 x 2 x g we define the nc complex hessian of p as the second directional derivative replacing x t by y q x x t h h t 2 p s t x t h y s k t s 0 y x t k h t we call an nc symmetric polynomial nc plurisubharmonic nc plush if it has an nc complex hessian that is positive semidefinite when evaluated on all tuples of n n matrices for every size n i e q x x t h h t 0 for all x h r n n g for every n 1 in this paper we classify all symmetric nc plush polynomials as convex polynomials with an nc analytic change of variables i e an nc symmetric polynomial p is nc plush if and only if it has the form 0 1 p f j t f j k j k j t f f t where the sums are finite and f j k j f are all nc analytic in this paper we also present a theory of noncommutative integration for nc polynomials and we prove a noncommutative version of the frobenius theorem a subsequent paper j m greene preprint 6 proves that if the nc complex hessian q of p takes positive semidefinite values on an nc open set then q takes positive semidefinite values on all tuples x h thus p has the form in eq 0 1 the proof in j m greene preprint 6 draws on most of the theorems in this paper together with a very different technique involving representations of noncommutative quadratic functions

tests for 2 x k contingency tables with clustered ordered categorical data ordered categorical data summarized in a 2 x k table usually consist of two sample multinomial or k sample binomial observations in analysing these data we usually assign scores to the k columns and perform a testing for the equality of two multinomial distributions in the former case and no trend among k binomial proportions in the latter case among the most popular score tests are the wilcoxon rank sum test and the armitage s linear trend test in this paper we extend the score tests to be used for clustered data under diverse study designs our methods do not require correct specification of the dependence structure within clusters the proposed tests are based on the asymptotic normality for large number of clusters and are a generalization of the standard tests used for independent data simulation studies are conducted to investigate the finite sample performance of the new methods the proposed methods are applied to real life data

hodge decomposition theorem for abelian two form gauge theory we show that the becchi rouet stora tyutin brst anti brst invariant 3 1 dimensional two form gauge theory has further nilpotent symmetries dual brst anti dual brst that leave the gauge fixing term invariant the generator for the dual brst symmetry is analogous to the co exterior derivative of differential geometry there exists a bosonic symmetry which keeps the ghost terms invariant and it turns out to be the analogue of the laplacian operator the hodge duality operation is shown to correspond to a discrete symmetry in the theory the generators of all these continuous symmetries are shown to obey the algebra of the de rham cohomology operators of differential geometry we derive the extended brst algebra constituted by six conserved charges and discuss the hodge decomposition theorem in the quantum hilbert space of states

semiclassical properties and chaos degree for the quantum baker s map we study the chaotic behavior and the quantum classical correspondence for the baker s map correspondence between quantum and classical expectation values is investigated and it is numerically shown that it is lost at the logarithmic timescale the quantum chaos degree is computed and it is demonstrated that it describes the chaotic features of the model the correspondence between classical and quantum chaos degrees is considered

spectral zeta functions for aharonov bohm quantum billiards this is a study of the sum zeta s alpha identical to ej 1infinity ej alpha s over the eigenvalues ej alpha of schrodinger s equation in a billiard domain d with reflecting walls threaded by a single line of magnetic flux alpha for integer s zeta s alpha is calculated by generalising a green function technique of itzykson et al ibid vol 19 l111 5 1986 based on a conformal transformation between d and the unit disc when the transformation is generated by a polynomial of finite degree an explicit formula enables zeta 2 alpha to be easily computed with high accuracy in conjunction with a semiclassical approximation the exact values of zeta 2 alpha can be used to calculate the ground state e1 alpha for a non integrable billiard with an error of about one per cent

interlacing families ii mixed characteristic polynomials and the kadison singer problem we use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the kadison singer problem the rst is weaver s conjecture ks 2 which is known to imply kadison singer via a projection paving conjecture of akemann and anderson the second is a formulation due to casazza et al of anderson s original paving conjecture s for which we are able to compute explicit paving bounds the proof involves an analysis of the largest roots of a family of polynomials that we call the mixed characteristic polynomials of a collection of matrices

universal twinkling exponents for spectral fluctuations associated with mixed chaology for systems that are neither fully integrable nor fully chaotic bifurcations of periodic orbits give rise to semiclassically emergent singularities in the fluctuating part n fl of the energy level counting function the bifurcations dominate the spectral moments m m n fl 2 m in the limit 0 we argue that m m const vm and calculate the twinkling exponents v m as the result of a competition between bifurcations with different codimensions and repetition numbers

matrix convexity operator analogues of the bipolar and hahn banach theorems abstract several basic results of convexity theory are generalized to the quantized matrix convex sets of wittstock these include the bipolar theorem a gauge version of the hahn banach theorem and the existence theorem for support functionals

testing for linear trends in proportions using correlated otolaryngology or ophthalmology data this paper is concerned with the issue of testing for trend with correlated binary data we consider the problem where one has either one or two ears or eyes available for analysis at baseline and one wishes to look at changes over time in a dichotomous outcome taking into account the correlation between responses from two ears a reparameterization of rosner s 1982 biometrics 38 105 114 correlated binary data model is presented and applied to a test for trend where the stratifying variable is age or any other subject specific variable observed and expected values are calculated for the trend statistic separately for both unilateral and bilateral cases and are then summed to obtain an overall summary statistic the proposed method is illustrated by a reanalysis of data presented in a published study of the efficacy of antibiotics for the treatment of otitis media

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

multifractal eigenstates of quantum chaos and the thue morse sequence we analyze certain eigenstates of the quantum baker s map and demonstrate using the walsh hadamard transform the emergence of the ubiquitous thue morse sequence a simple sequence that is at the border between quasiperiodicity and chaos and hence is a good paradigm for quantum chaotic states we show a family of states that are also simply related to the thue morse sequence and are strongly scarred by short periodic orbits and their homoclinic excursions we give approximate expressions for these states and provide evidence that these and other generic states are multifractal

minimal and maximal matrix convex sets abstract for every convex body k r d there is a minimal matrix convex set w min k and a maximal matrix convex set w max k which have k as their ground level we aim to find the optimal constant k such that w max k k w min k for example if b p d is the unit ball in r d with the l p norm then we find that b p d d 1 1 p 1 2 this constant is sharp and it is new for all p 2 moreover for some sets k we find a minimal set l for which w max k w min l in particular we obtain that a convex body k satisfies w max k w min k only if k is a simplex these problems relate to dilation theory convex geometry operator systems and completely positive maps for example our results show that every d tuple of self adjoint contractions can be dilated to a commuting family of self adjoints each of norm at most d we also introduce new explicit constructions of these and other dilations

a parametric model for cluster correlated categorical data summary a fully parametric copula model for symmetric dependent clustered categorical data is discussed the model accommodates any marginal regression models of interest and admits a broad range of within cluster association the form of the distribution is independent of cluster size and may be used to model data with varying cluster sizes the model contains an association parameter that is estimated from the data to give a measure of the strength of the within cluster association and also a test of independence two examples are given to illustrate the methods

statistical methods in ophthalmology an adjusted chi square approach ophthalmologic studies often compare several groups of subjects for the presence or absence of some ocular finding where each subject may contribute two eyes to the analysis the values from the two eyes being highly correlated rosner 1982 biometrics 38 105 114 and dallal 1988 biometrics 44 253 257 proposed procedures for testing whether the proportion of affected eyes is the same among the different groups while accounting for the intrasubject correlation in this paper we propose an alternative approach based on a simple adjustment of the standard pearson chi square test for the equality of proportions the suggested approach utilizes information on subjects who supply only one eye to the analysis and readily generalizes to studies in which more than two units of analysis are provided by each subject

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

analytic representations in quantum mechanics various euclidean hyperbolic and elliptic analytic representations are introduced and relations among them are discussed the bargmann analytic representation in the complex plane is considered and its relation to other phase space methods for the harmonic oscillator is reviewed the general theory that relates the growth of analytic functions with the density of their zeros is applied to bargmann functions and it leads to theorems on the completeness of sequences of glauber coherent states two hyperbolic analytic representations in the unit disc based on su 1 1 coherent states and also on phase states are introduced a third analytic representation in the complex plane based on barut girardello states is also considered and transformations which relate it to the other ones are studied in the case of systems with finite dimensional hilbert space an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced the berezin formalism in the euclidean hyperbolic and elliptic cases is discussed contour analytic representations in these three cases are also presented

cohomological aspects of abelian gauge theory we discuss some aspects of cohomological properties of a two dimensional free abelian gauge theory in the framework of the brst formalism we derive the conserved and nilpotent brst and co brst charges and express the hodge decomposition theorem in terms of these charges and a conserved bosonic charge corresponding to the laplacian operator it is because of the topological nature of free u 1 gauge theory that the laplacian operator goes to zero when equations of motion are exploited we derive two sets of topological invariants which are related to each other by a certain kind of duality transformation and express the lagrangian density of this theory as the sum of terms that are brst and co brst invariants mathematically this theory captures together some of the key features of witten and schwarz type topological field theories

new topological field theories in two dimensions it is shown that two 1 1 dimensional 2d free abelian and self interacting non abelian gauge theories without any interaction with matter fields belong to a new class of topological field theories tfts these new theories capture together some of the key features of witten and schwarz types of tft because they are endowed with symmetries that are reminiscent of the schwarz type theories but their lagrangian density has the appearance of the witten type theories the topological invariants for these theories are computed on a 2d compact manifold and their recursion relations are obtained these new theories are shown to provide a class of tractable field theoretical models for the hodge theory in two dimensions of flat minkowski spacetime where there are no propagating degrees of freedom associated with the 2d gauge boson

bianalytic maps between free spectrahedra linear matrix inequalities lmis i d sum j 1 g a jx j sum j 1 g a j x j succeq 0 play a role in many areas of applications the set of solutions of an lmi is a spectrahedron lmis in dimension free matrix variables model most problems in linear systems engineering and their solution sets are called free spectrahedra free spectrahedra are exactly the free semialgebraic convex sets this paper studies free analytic maps between free spectrahedra and under certain generically valid irreducibility assumptions classifies all those that are bianalytic the foundation of such maps turns out to be a very small class of birational maps we call convexotonic the convexotonic maps in g variables sit in correspondence with g dimensional algebras if two bounded free spectrahedra mathcal d a and mathcal d b meeting our irreducibility assumptions are free bianalytic with map denoted p then p must after possibly an affine linear transform extend to a convexotonic map corresponding to a g dimensional algebra spanned by u i a 1 ldots u i a g for some unitary u furthermore b and ua are unitarily equivalent the article also establishes a positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from mathcal d a to mathcal d b not necessarily bianalytic another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron these theorems are needed for classifying free bianalytic maps

bianalytic free maps between spectrahedra and spectraballs abstract linear matrix inequalities lmis are ubiquitous in real algebraic geometry semidefinite programming control theory and signal processing lmis with dimension free matrix unknowns are central to the theories of completely positive maps and operator algebras operator systems and spaces and serve as the paradigm for matrix convex sets the matricial feasibility set of an lmi is called a free spectrahedron in this article the bianalytic maps between a very general class of ball like free spectrahedra examples of which include row or column contractions and tuples of contractions and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form they are all highly structured rational maps in the case that both the domain and codomain are ball like these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point in particular this result leads to a classification of automorphism groups of ball like free spectrahedra the proofs depend on a novel free nullstellensatz established only after new tools in free analysis are developed and applied to obtain fine detail geometric in nature locally and algebraic in nature globally about the boundary of ball like free spectrahedra

periodic orbits and a correlation function for the semiclassical density of states through the rule that there is one quantum state per planck cell in phase space classical mechanics supplies an average semiclassical density of states for a system with bound motion can it supply more refined averages such as the correlation function for the density of states i shall argue that for the two extreme cases of integrable classical motion and ergodic classical motion information can indeed be obtained on not quite this but a closely related correlation function the results for the two types of motion are markedly different

every convex free basic semi algebraic set has an lmi representation the matricial solution set of a linear matrix inequality lmi is a convex free basic open semi algebraic set the main theorem of this paper is a converse each such set arises from some lmi the result has implications for semi definite programming and systems engineering as well as for free semi algebraic geometry