on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

random matrix theory and the zeros of ζ s we study the density of the roots of the derivative of the characteristic polynomial z u z of an n n random unitary matrix with distribution given by haar measure on the unitary group based on previous random matrix theory models of the riemann zeta function s this is expected to be an accurate description for the horizontal distribution of the zeros of s to the right of the critical line we show that as n the fraction of the roots of z u z that lie in the region 1 x n 1 z 1 tends to a limit function we derive asymptotic expressions for this function in the limits x and x 0 and compare them with numerical experiments

identifiability of uncontrolled nonlinear rational systems in this paper an approach for the identifiability analysis of uncontrolled rational systems is provided the method is based on the use of a local smooth state space transformation in particular it is shown that provided the model satisfies an observability rank condition the state trajectories of an uncontrolled system corresponding to parameter vectors with outputs that are identical locally in time are connected via a smooth transformation

vibration of circular plates on a free fluid surface effect of surface waves the effect of free surface waves on free vibrations of circular plates resting on a free fluid surface is studied the solution is achieved by using a perturbation technique and the hankel transformation method which give a couple of dual integral equations of titchmarsh type the fluid is considered inviscid and incompressible and the velocity potential describes its motion the kirchhoff theory of plates is used to model the elastic thin plate the theory is suitable for all axisymmetric plate boundary conditions numerical results are given in non dimensional form for modes up to five nodal circles and diameters for clamped simply supported and free edge plates to be ready to use in applications the effect of free surface waves on the plate s natural frequencies is significant only when bluging and sloshing modes of the system have close natural frequencies

hydroelastic vibration of two identical rectangular plates a theoretical study is presented on the hydroelastic vibration of two identical rectangular plates coupled with a bounded fluid it is assumed that the plates are clamped along the plate edges and an ideal fluid is surrounded by the two rectangular plates and a rigid rectangular container the velocity potential satisfying the fluid boundary conditions is expanded in terms of a finite fourier series and the modal displacements of the plates are also expanded by the finite fourier series for the compatibility requirement along the contacting surface between the plates and the fluid two transverse vibration modes in phase and out of phase are observed in the symmetric fluid coupled structure each in phase mode is assumed as a combination of the beam modes in air but every out of phase mode is assumed as a combination of polynomials satisfying the plate boundary condition and fluid volume conservation the coupled natural frequencies are obtained from the relationship between the reference kinetic energy of the structure including the fluid and the maximum strain energy of the two plates the proposed analytical method was found to be in good agreement with the results of a three dimensional finite element analysis

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

finite n fluctuation formulas for random matrices for the gaussian and laguerre random matrix ensembles the probability density function p d f for the linear statistic jn 1 xj x is computed exactly and shown to satisfy a central limit theorem asn for the circular random matrix ensemble the p d f s for the statistics jn 1 j and jn 1 log 2 sin j 2 are calculated exactly by using a constant term identity from the theory of the selberg integral and are also shown to satisfy a central limit theorem asn

the circular unitary ensemble and the riemann zeta function the microscopic landscape and a new approach to ratios we show in this paper that after proper scalings the characteristic polynomial of a random unitary matrix converges to a random analytic function whose zeros which are on the real line form a determinantal point process with sine kernel our scaling is performed at the so called microscopic level that is we consider the characteristic polynomial at points whose distance to 1 has order 1 n we prove that the rescaled characteristic polynomial does not even have a moment of order one hence making the classical techniques of random matrix theory difficult to apply in order to deal with this issue we couple all the dimensions n on a single probability space in such a way that almost sure convergence occurs when n goes to infinity the strong convergence results in this setup provide us with a new approach to ratios we are able to solve open problems about the limiting distribution of ratios of characteristic polynomials evaluated at points of the form exp 2 i pi alpha n and related objects such as the logarithmic derivative we also explicitly describe the dependence relation for the logarithm of the characteristic polynomial evaluated at several points on the microscopic scale on the number theory side inspired by the work by keating and snaith we conjecture some new limit theorems for the value distribution of the riemann zeta function on the critical line at the level of stochastic processes

structural indistinguishability between uncontrolled autonomous nonlinear analytic systems in this paper an approach for analysing the structural indistinguishability between two uncontrolled or autonomous analytic systems is presented the approach involves constructing if possible a smooth mapping between the trajectories of two candidate models if either of the models satisfies an observability criterion then such a transformation always exists when the models are indistinguishable from their outputs the approach is illustrated by examples from epidemiology and chemical reaction kinetics one important outcome is that the susceptible infectious recovered sir and sir with temporary immunity sirs models are shown to be indistinguishable when a proportion of the number of infectives is measured

axisymmetric vibration of circular plates in contact with fluid abstract this paper is concerned with the effect of fluid on the natural frequencies of circular plates vibrating axisymmetrically in contact with fluid the ratio between the natural frequencies in fluid and the natural frequencies in air is a function of so called added virtual mass incremental avmi factor which reflects the increase of inertia due to the presence of fluid in this paper the non dimensionalized added virtual mass incremental navmi factor for circular plates having simply supported clamped and free edges is obtained by employing the integral transformation technique in conjunction with the dual integral equation method it is shown that the effect of fluid on the natural frequencies decreases with order

hydroelastic vibration of a circular plate submerged in a bounded compressible fluid abstract an analytical method for the linear free vibration of a circular plate submerged in a fluid was developed by the rayleigh ritz method based on the fourier bessel series expansion it is assumed that the plate is clamped at an offcenter location of a rigid cylindrical container and the fluid bisected by the plate in the container is non viscous and compressible since a combination of the dry modal functions of the circular plate can approximate the wet vibration modes the functions were used to form a set of linearly independent functions of the rayleigh ritz method it was found that the theoretical results could predict well the fluid coupled natural frequencies with excellent accuracy compared with the finite element analysis results the effects of the fluid compressibility and the offcenter distance on the natural frequencies were also observed

free vibration of two identical circular plates coupled with bounded fluid abstract this paper examines the hydroelastic vibration of two identical circular plates coupled with a bounded fluid an analytical method based on the finite fourier bessel series expansion and the rayleigh ritz method is suggested in the theory it is assumed that a rigid cylindrical container is filled with the ideal fluid and the two plates are clamped along the container edges the proposed method is verified by finite element analysis using commercial software with excellent accuracy two transverse vibration modes in phase and out of phase are observed alternately in the fluid coupled system when the number of nodal circles increases for the fixed nodal diameter it is found that the normalized natural frequency of the system monotonically increases with an increase in the number of nodal diameters and circles by virtue of a decrease in relative hydrodynamic mass

dynamics of a rectangular plate resting on an elastic foundation and partially in contact with a quiescent fluid abstract in this study a method of analysis is presented for investigating the effects of elastic foundation and fluid on the dynamic response characteristics natural frequencies and associated mode shapes of rectangular kirchhoff plates for the interaction of the kirchhoff plate pasternak foundation a mixed type finite element formulation is employed by using the g teaux differential the plate finite element adopted in this study is quadrilateral and isoparametric having four corner nodes and at each node four degrees of freedom are present one transverse displacement two bending moments and one torsional moment therefore a total number of 16 degrees of freedom are assigned to each element a consistent mass formulation is used for the eigenvalue solution in the mixed finite element analysis the plate structure considered is assumed clamped or simply supported along its edges and resting on a pasternak foundation furthermore the plate is fully or partially in contact with fresh water on its one side for the calculation of the fluid structure interaction effects generalized fluid structure interaction forces a boundary element method is adopted together with the method of images in order to impose an appropriate boundary condition on the fluid s free surface it is assumed that the fluid is ideal i e inviscid incompressible and its motion is irrotational it is also assumed that the plate elastic foundation system vibrates in its in vacuo eigenmodes when it is in contact with fluid and that each mode gives rise to a corresponding surface pressure distribution on the wetted surface of the structure at the fluid structure interface continuity considerations require that the normal velocity of the fluid is equal to that of the structure the normal velocities on the wetted surface of the structure are expressed in terms of the modal structural displacements obtained from the finite element analysis by using the boundary integral equation method the fluid pressure is eliminated from the problem and the fluid structure interaction forces are calculated in terms of the generalized hydrodynamic added mass coefficients due to the inertial effect of fluid to asses the influences of the elastic foundation and fluid on the dynamic behavior of the plate structure the natural frequencies and associated mode shapes are presented furthermore the influence of the submerging depth on the dynamic behavior is also investigated

gaussian fluctuation for the number of particles in airy bessel sine and other determinantal random point fields we prove the central limit theorem clt for the number of eigenvalues near the spectrum edge for certain hermitian ensembles of random matrices to derive our results we use a general theorem essentially due to costin and lebowitz concerning the gaussian fluctuation of the number of particles in random point fields with determinantal correlation functions as another corollary of the costin lebowitz theorem we prove the clt for the empirical distribution function of the eigenvalues of random matrices from classical compact groups

the forced vibration of a thin plate floating on an infinite liquid abstract in this paper a solution is presented for the harmonically forced vibration of an arbitrary thin plate floating on the surface of an infinite liquid the full linear potential problem for the liquid is solved by the use of the appropriate green s function a variational equation which the plate liquid system must satisfy is derived and a solution by the rayleigh ritz method is presented examples of possible calculations are given for a square and a rectangular plate

large deviations for wigner s law and voiculescu s non commutative entropy we study the spectral measure of gaussian wigner s matrices and prove that it satisfies a large deviation principle we show that the good rate function which governs this principle achieves its minimum value at wigner s semicircular law which entails the convergence of the spectral measure to the semicircular law as a conclusion we give some further examples of random matrices with spectral measure satisfying a large deviation principle and argue about voiculescu s non commutative entropy

eigenvalue distributions of random unitary matrices let u be an n n random matrix chosen from haar measure on the unitary group for a fixed arc of the unit circle let x be the number of eigenvalues of m which lie in the specified arc we study this random variable as the dimension n grows using the connection between toeplitz matrices and random unitary matrices and show that x e x var x 1 2 is asymptotically normally distributed in addition we show that for several fixed arcs i1 im the corresponding random variables are jointly normal in the large n limit

high powers of random elements of compact lie groups if a random unitary matrix u is raised to a sufficiently high power its eigenvalues are exactly distributed as independent uniform phases we prove this result and apply it to give exact asymptotics of the variance of the number of eigenvalues of u falling in a given arc as the dimension of u tends to infinity the independence result it turns out extends to arbitrary representations of arbitrary compact lie groups we state and prove this more general theorem paying special attention to the compact classical groups and to wreath products this paper is excerpted from the author s doctoral thesis 9

central limit theorems for sequences of multiple stochastic integrals we characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1 some applications are given in particular to study the limiting behavior of quadratic functionals of gaussian processes

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

effect of fluid depth on the hydroelastic vibration of free edge circular plate abstract this paper is concerned with the effect of water depth on the free vibration of free edge circular plates resting on a free fluid surface the problem addressed is formulated by using the hankel transformation method which leads to dual integral equations the solution of dual integral equations is solved numerically by using fourier bessel series the fluid is assumed to be inviscid and incompressible the kirchhoff theory of plates is used to model the elastic thin plate numerical results are given in non dimensional form for free edge circular plates in order to be ready to use in applications to validate the theoretical results experiments were carried out experimental results are in good agreement with theoretical results it is found that the effect of the fluid depth can be neglected when the fluid depth is greater than the diameter of the circular plates but becomes significant as the water depth decreases it is also observed in the experimental results that the fluid damping increases as the water depth decreases

the winding of stationary gaussian processes this paper studies the winding of a continuously differentiable gaussian stationary process f mathbb r rightarrow mathbb c in the interval 0 t we give formulae for the mean and the variance of this random variable the variance is shown to always grow at least linearly with t and conditions for it to be asymptotically linear or quadratic are given moreover we show that if the covariance function together with its second derivative are in l 2 mathbb r then the winding obeys a central limit theorem these results correspond to similar results for zeroes of real valued stationary gaussian functions by malevich cuzick slud and others

asymptotic formulas for the acoustic radiation impedance of an elastically supported annular plate this paper deals with the normalized acoustic radiation impedance of an elastically supported annular plate in axisymmetric vibrations cauchy s theorem about residues and the stationary phase method have been used to approximate the corresponding integrals as a result elementary asymptotic formulas valid for axisymmetric boundary configurations of clamped guided simply supported and free annular plates as well as for all the intermediate boundary configurations have been obtained

on barnes beta distributions and applications to the maximum distribution of the 2d gaussian free field a new family of barnes beta distributions on 0 infty is introduced and its infinite divisibility moment determinacy scaling and factorization properties are established the morris integral probability distribution is constructed from barnes beta distributions of types 1 0 and 2 2 and its moment determinacy and involution invariance properties are established for application the maximum distributions of the 2d gaussian free field on the unit interval and circle with a non random logarithmic potential are conjecturally related to the critical selberg and morris integral probability distributions respectively and expressed in terms of sums of barnes beta distributions of types 1 0 and 2 2

spectral functions special functions and the selberg zeta function the functional determinant of an eigenvalue sequence as defined by zeta regularization can be simply evaluated by quadratures we apply this procedure to the selberg trace formula for a compact riemann surface to find a factorization of the selberg zeta function into two functional determinants respectively related to the laplacian on the compact surface itself and on the sphere we also apply our formalism to various explicit eigenvalue sequences reproducing in a simpler way classical results about the gamma function and the barnesg function concerning the latter our method explains its connection to the selberg zeta function and evaluates the related glaisher kinkelin constanta

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

free vibration analysis of a circular plate partially in contact with a liquid abstract this paper provides a theoretical method for a hydroelastic vibration of a clamped circular plate partially in contact with a liquid it is assumed that the liquid is inviscid and incompressible while the thickness of the plate is small compared to the plate radius and the gravity effect on the system is neglected the wet dynamic displacement of the plate is approximated by combining the orthogonal modal functions of a dry circular plate with a clamped boundary condition the liquid displacement potential satisfying the boundary conditions is derived and the wet dynamic modal functions of the plate are derived by using a compatibility requirement along the contacting surface between the plate and the liquid the eigenvalue equation for the system is obtained by using the rayleigh ritz method in order to calculate the wet natural frequencies and the mode shapes for the symmetric and the asymmetric modes the proposed method is verified by observing a good agreement with three dimensional finite element analysis results the effects of the liquid thickness and the liquid depth on the natural frequencies are also discussed

fluctuations of the increment of the argument for the gaussian entire function the gaussian entire function is a random entire function characterised by a certain invariance with respect to isometries of the plane we study the fluctuations of the increment of the argument of the gaussian entire function along planar curves we introduce an inner product on finite formal linear combinations of curves with real coefficients that we call the signed length which describes the limiting covariance of the increment we also establish asymptotic normality of fluctuations

nonlinear compartmental model indistinguishability a version of the local state isomorphism theorem is used to show how nonlinear model structures within some specified class can be tested for indistinguishability from some given structure via a suitable choice of their respective parameters this theory is applied to a two compartmental structure with an assumed michaelis menten elimination from compartment 1 models tested for indistinguishability include structures with michaelis menten elimination from compartment 2 in addition to or instead of that from compartment 1

an arithmetic model for the total disorder process we prove a multidimensional extension of selberg s central limit theorem for the logarithm of the riemann zeta function on the critical line the limit is a totally disordered process whose coordinates are all independent and gaussian

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

correlation functions for random complex zeroes strong clustering and local universality we prove strong clustering of k point correlation functions of zeroes of gaussian entire functions in the course of the proof we also obtain universal local bounds for k point functions of zeroes of arbitrary nondegenerate gaussian analytic functions in the second part of the paper we show that strong clustering yields the asymptotic normality of fluctuations of some linear statistics of zeroes of gaussian entire functions in particular of the number of zeroes in measurable domains of large area this complements our recent results from the paper fluctuations in random complex zeroes