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

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

moments of ratios of characteristic polynomials of a certain class of random matrices we derive a new method of calculating the mean negative moments and ratios of squared modulus of characteristic polynomials of a certain class of random matrices new results obtained with this method are presented

analytical computation of differential equations involved in dynamical nonlinear optimal problems abstract we present two programs written in reduce for non constrained free endpoint nonlinear dynamical optimal problems in fixed time in closed loop and in open loop which compute analytically the optimal feedback laws in terms of differential equations the open loop case leads to ordinary differential equations odes and the closed loop leads to partial differential equations pdes in the case of closed loop problems the program uses nonstandard reduce programming for the declarations of dependencies of u and its partial derivatives algorithms are presented for the open loop and closed loop cases and the same example is computed in both these cases

on minimum fuel control of affine nonlinear systems the minimum fuel control problem is investigated for a class of multiinput affine nonlinear systems whose associated lie algebra is nilpotent after preliminary notions and definitions are presented emphasis is on the solution to the optimal control problem of the lie algebra generated by certain system vector fields is nilpotent consequences of the maximum principle are deduced for such systems

a perturbation result for prescribing mean curvature in this article we consider a boundary value problem associated with the conformal deformation of metrics we are interested in the case in which a noncompact group of conformal transformation acts on the equation so that the kazdanwarner condition gives rise to obstruction as in the nirenberg problem the simplest situation is the following let bn 1 be the unit ball in rn 1 with euclidean metric g0 its boundary will be denoted by s n bn 1 let h be a smooth function on s n we study the problem of finding a conformal metric gu u4 n 1 g0 whose scalar curvature vanishes in bn 1 and on the boundary bn 1 the mean curvature is given by h this problem is equivalent to solving the following boundary value problem

applications of the l functions ratios conjectures in upcoming papers by conrey farmer and zirnbauer there appear conjectural formulas for averages over a family of ratios of products of shifted l functions in this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory such as lower order terms in the zero statistics of l functions mollified moments of l functions and discrete averages over zeros of the riemann zeta function in particular using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations

on conformal deformations of metrics onsn abstract on s n there is a naturally metric defined n th order conformal invariant operator p n associated with this operator is a so called q curvature quantity when two metrics are pointwise conformally related their associated operators together with their q curvatures satisfy the natural differential equations this paper is devoted to the question of which function can be a q curvature candidate this is the so called prescribing q curvature problem our main result is that if q is positive nondegenerate and the naturally defined mapping associated with q has nonzero degree then our problem has a solution this is the natural generalization of prescribing gaussian curvature on s 2 into s n

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

the prescribed boundary mean curvature problem on b4 abstract in this paper we perform a fine blow up analysis for a boundary value elliptic equation involving the critical trace sobolev exponent related to the conformal deformation of the metrics on the standard ball namely the problem of prescribing the boundary mean curvature from this analysis some a priori estimates in low dimension are obtained with these estimates we prove the existence of at least one solution when an index counting formula associated to the prescribed mean curvature is different from zero

the profile of bubbling solutions of a class of fourth order geometric equations on 4 manifolds abstract we study a class of fourth order geometric equations defined on a 4 dimensional compact riemannian manifold which includes the q curvature equation we obtain sharp estimates on the difference near the blow up points between a bubbling sequence of solutions and the standard bubble

pair correlation and twin primes revisited we establish a connection between the conjectural two over two ratios formula for the riemann zeta function and a conjecture concerning correlations of a certain arithmetic function specifically we prove that the ratios conjecture and the arithmetic correlations conjecture imply the same result this casts a new light on the underpinnings of the ratios conjecture which previously had been motivated by analogy with formulae in random matrix theory and by a heuristic recipe

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

on absolute moments of characteristic polynomials of a certain class of complex random matrices the integer moments of the spectral determinant det zi w 2 of complex random matrices w are obtained in terms of the characteristic polynomial of the positive semidefinite matrix ww for the class of matrices w au where a is a given matrix and u is random unitary this work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context

constant q curvature metrics in arbitrary dimension working in a given conformal class we prove existence of constant q curvature metrics on compact manifolds of arbitrary dimension under generic assumptions the problem is equivalent to solving a nth order non linear elliptic differential or integral equation with variational structure where n is the dimension of the manifold since the corresponding euler functional is in general unbounded from above and below we use critical point theory jointly with a compactness result for the above equation

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

concentration compactness phenomena in the higher order liouville s equation we investigate different concentration compactness and blow up phenomena related to the q curvature in arbitrary even dimension we first treat the case of an open domain in r 2 then that of a closed manifold and finally the particular case of the sphere s 2m in all cases we allow the sign of the q curvature to vary and show that in the case of a closed manifold contrary to the case of open domains in r 2m blow up phenomena can occur only at points of positive q curvature as a consequence on a locally conformally flat manifold of non positive euler characteristic we always have compactness

random matrices generalized zeta functions and self similarity of zero distributions there is growing evidence for a connection between random matrix theories used in physics and the theory of the riemann zeta function and l functions the theory underlying the location of the zeros of these generalized zeta functions is one of the key unsolved problems physicists are interested because of the hilbert polya conjecture that the non trivial zeros of the zeta function correspond to the eigenvalues of some positive operator to complement the continuing theoretical work it would be useful to study empirically the locations of the zeros by different methods in this paper we use the rescaled range analysis to study the spacings between successive zeros of these functions over large ranges of the zeros the spacings have a hurst exponent of about 0 095 using sample sizes of 10 000 zeros this implies that the distribution has a high fractal dimension 1 9 and shows a lot of detailed structure the distribution is of the anti persistent fractional brownian motion type with a significant degree of anti persistence thus the high order zeros of these functions show a remarkable self similarity in their distribution over fifteen orders of magnitude for the riemann zeta function we find that the hurst exponents for the random matrix theories show a different behaviour a heuristic study of the effect of low order primes seems to show that this effect is a promising candidate to explain the results that we observe in this study we study the distribution of zeros for l functions of conductors 3 and 4 and find that the distribution is similar to that of the riemann zeta functions

scalar curvatures on sn soit m n g une variete de riemann compacte de dimension n 3 et r x une fonction lisse sur m n on etudie si r x peut etre la courbure scalaire d une metrique g qui est conforme point par point a la metrique originale g

moments of the derivative of characteristic polynomials with an application to the riemann zeta function we investigate the moments of the derivative on the unit circle of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the riemann function on the critical line we do the same for the analogue of hardy s z function the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle our formulae are expressed in terms of a determinant of a matrix whose entries involve the i bessel function and alternately by a combinatorial sum

the scalar curvature flow on s n perturbation theorem revisited for the problem of finding a geometry on s n for n 3 with a prescribed scalar curvature there is a well known result which is called the perturbation theorem it is due to chang and yang duke math j 64 27 69 1991 their key assumption is that the candidate f for the prescribed scalar curvature is sufficiently near the scalar curvature of the standard metric in the sup norm it is important to know how large that difference in sup norm can possibly be here we consider prescribing scalar curvature problem using the scalar curvature flow

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

scalar curvature equation on sn part ii analytic characterizations this is the second part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension by studying topological degrees for certain abstract maps we will give explicit analytic conditions on the scalar curvature function which verify the topological degree conditions given in the first part of the series to ensure the solvability of the problem general existence results for the prescribing scalar curvature equation will be given on both h symmetric and sub h symmetric solutions corresponding to h symmetric scalar curvature functions as well as on non symmetric solutions corresponding to symmetric like scalar curvature functions special axisymmetric and axisymmetric like cases will be also considered our analysis will be based on a general approach of dimension reductions and degree calculations by taking advantage of symmetries and symmetric like properties

conformal deformation of metrics on s 2 soit d un domaine lisse de s 2 et k une fonction lisse definie sur d alors pour que le probleme aux limites de neumann u k exp 2u 1 sur d u n 0 sur d ait une solution u on a a quand aire d max x d k x 0

derivatives of random matrix characteristic polynomials with applications to elliptic curves the value distribution of derivatives of characteristic polynomials of matrices from so n is calculated at the point 1 the symmetry point on the unit circle of the eigenvalues of these matrices we consider subsets of matrices from so n that are constrained to have at least n eigenvalues equal to 1 and investigate the first non zero derivative of the characteristic polynomial at that point the connection between the values of random matrix characteristic polynomials and values of l functions in families has been well established the motivation for this work is the expectation that through this connection with l functions derived from families of elliptic curves and using the birch and swinnerton dyer conjecture to relate values of the l functions to the rank of elliptic curves random matrix theory will be useful in probing important questions concerning these ranks

on the symmetric scalar curvature problem on sn abstract we discuss some existence results dealing with the scalar curvature problem on s n in the presence of various symmetries

conformal metrics on the unit ball with prescribed mean curvature this paper focuses on the study of the prescribed mean curvature problem on the unit ball if the difference between the mean curvature candidate f and mean curvature of the standard metric in the supremum norm is sufficiently small then the existence of positive solutions of conformal mean curvature equation has been known the purpose of the paper is to investigate quantitatively how large that difference can be by using a flow method

prescribing q curvature problem on sn let pn be the n th order paneitz operator on sn n 3 we consider the following prescribing q curvature problem on sn pnu n 1 q x enuon sn where q is a smooth positive function on sn satisfying the following non degeneracy condition q 2 q 2 0 let g sn rn 1 be defined by g x q x q x we show that if q 0 is non degenerate and deg g g sn 0 then the above equation has a solution when n is even this has been established in our earlier work j wei x xu on conformal deformation of metrics on sn j funct anal 157 1998 292 325 when n is odd pn becomes a pseudo differential operator here we develop a unified approach to treat both even and odd cases the key idea is to write it as an integral equation and use liapunov schmidt reduction method

generalizations of the borsuk theorem for any subgroup h of g we will denote by v the fixed point subspace of h action i e i xeir t x x dh h for xrz wn let o x t x 1 g e g be the g orbit of 3 if g is finite we let l x denote the order of o x suppose that 2 is an open bounded g invariant set in iwn andf s wn is a continuous g equivariant map if 0 f x the brouwer degree deg f 52 o is well defined it is known that we can obtain important informations on deg f q 0 if the g action on abn is of some special types for example in the case that g z 0 1 t i and t z where i is the identity transformation on iwn the borsuk theorem asserts that deg f 52 0 is an odd number provided 0 e q recently efforts have been made to generalize the borsuk theorem for the study of differential equations with various symmetries under the assumption that vg 0 marzantowicz 4 obtains the following results in the sequel we always assume that c2 is an open bounded g invariant subset of rn

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

optimal feedback control of non linear systems abstract a method is proposed for computing optimal nonlinear feedback control laws it is shown that the feedback loop satisfies a system of quasi linear partial differential equations this system is of first order when the dimensions of the state and the control vector are the same hereby it degenerates into algebraic equations when the performance index does not depend on the control these important results offer new ways for determining optimal feedback laws connections with the volterra series and the hamilton jacobi bellman equation are treated some examples are given to illustrate the advantages of this method

derivative moments for characteristic polynomials from the cue we calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer the calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity the latter asymptotic calculation allows us to prove a long standing conjecture from random matrix theory

freezing transition characteristic polynomials of random matrices and the riemann zeta function we argue that the freezing transition scenario previously explored in the statistical mechanics of 1 f noise random energy models also determines the value distribution of the maximum of the modulus of the characteristic polynomials of large n ifmmode times else texttimes fi n random unitary matrices we postulate that our results extend to the extreme values taken by the riemann zeta function ensuremath zeta s over sections of the critical line s 1 2 it of constant length and present the results of numerical computations in support our main purpose is to draw attention to possible connections between the statistical mechanics of random energy landscapes random matrix theory and the theory of the riemann zeta function

the computation of ck spline functions abstract we present in this paper an algorithm for computing ck spline functions i e spline functions which generate k time continuous and derivable approximations these functions seem to be very efficient in several applied nonlinear differential problems like nonlinear control or nonlinear simulation

autocorrelation of random matrix polynomials we calculate the autocorrelation functions or shifted moments of the characteristic polynomials of matrices drawn uniformly with respect to haar measure from the groups u n o 2n and usp 2n in each case the result can be expressed in three equivalent forms as a determinant sum and hence in terms of symmetric polynomials as a combinatorial sum and as a multiple contour integral these formulae are analogous to those previously obtained for the gaussian ensembles of random matrix theory but in this case are identities for any size of matrix rather than large matrix asymptotic approximations they also mirror exactly the autocorrelation formulae conjectured to hold for l functions in a companion paper this then provides further evidence in support of the connection between random matrix theory and the theory of l functions

on the nonlinear elliptic equations with symmetry has a smooth solution with norm r in the appropriate function space this theorem is based on an infinite dimensional analogue of the following theorem of borsuk if d c r is the unit disc in the euclidean space then for every odd mapping j d rk k 1 then there is no g equivariant mapf s v w o in case g is infinite we prove that if the fixed point set of some torus tc g consists only of the point 0 vt 0 then there is no g equivariant map f s v w o the main tools in the proofs of the above results are the properties of lefschetz numbers of g equivariant maps these properties are most conveniently formulated in terms of the burnside ring of the group g we end this section by giving a method of computing the number p v from the character of v in section ii we extend the results of the previous section to the infinite156

a new approach to nonlinear optimal feedback law abstract for a fixed time free endpoint optimal control problem it is shown that the optimal feedback control satisfies a system of ordinary differential equations they are obtained using an elimination procedure of the adjoint vector which appears linearly in a set of differential equations these equations involving lie brackets of vector fields are derived from the maximum principle an application of this approach to robotics is given

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

on the moments of the moments of the characteristic polynomials of random unitary matrices denoting by p n a theta det i ae i theta the characteristic polynomial on the unit circle in the complex plane of an n times n random unitary matrix a we calculate the kth moment defined with respect to an average over a in u n of the random variable corresponding to the 2 beta th moment of p n a theta with respect to the uniform measure frac d theta 2 pi for all k beta in mathbb n these moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log correlated gaussian fields our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions and an analysis of a second representation in terms of multiple contour integrals our main result is that the moments of moments are polynomials in n of degree k 2 beta 2 k 1 this resolves a conjecture of fyodorov and keating philos trans r soc a 372 2007 20120503 2014 concerning the scaling of the moments with n as n rightarrow infty for k beta in mathbb n indeed it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

problemes de neumann non lineaires sur les varietes riemanniennes nonlinear neumann problems on riemannian manifolds let m g be a c compact riemannian manifold of dimension n 2 whose boundary b is an n 1 dimensional submanifold and let m m b be the interior of m study of neumann problems of the form x 0 in m d dn g y 0 on b where for every t x y r m b t x and g t y are bounded by c 1 t a or c exp t a application to the determination of a conformal metric for which the scalar curvature of m and the mean curvature of b take prescribed values

scalar curvature equation on sn part i topological conditions abstract this is the first part of a series devoting to the study of the prescribing scalar curvature problem on the standard sphere of any dimension in the first part we will adopt the degree theoretic approach to give a topological condition and some general explicit conditions on the scalar curvature functions to ensure the solvability of the problem our topological condition is imposed on some of simple maps explicitly defined by the scalar curvature function which is derived from the asymptotic expansion of the boundary map introduced in a chang p yang a perturbation result in prescribing scalar curvature on s n duke math j 64 1991 27 69 our conditions particularly allowing non isolation and non degeneracy of the critical points of the scalar curvature functions can be easily verified in many situations in the second part of series we will make a detailed study on the verification of the topological condition our results will generalize almost all existing ones in the same direction and meanwhile provide a unified treatment for both symmetric and non symmetric cases of the scalar curvature functions

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

conformal metrics on the ball with zero scalar curvature and prescribed mean curvature on the boundary abstract this paper considers the prescribed zero scalar curvature and mean curvature problem on the n dimensional euclidean ball for n 3 we consider the limits of solutions of the regularization obtained by decreasing the critical exponent we characterize those subcritical solutions which blow up at the least possible energy level determining the points at which they can concentrate and their morse indices we show that when n 3 this is the only blow up which can occur for solutions we use this in combination with the morse inequalities for the subcritical problem to obtain a general existence theorem for the prescribed zero scalar curvature and mean curvature on the three dimensional euclidean ball in the higher dimensional case n 4 we give conditions on the function h to guarantee there is only one simple blow up point

convergence of the yamabe flow for arbitrary initial energy we consider the yamabe flow frac partial g partial t r g r g g where g is a riemannian metric on a compact manifold m r g denotes its scalar curvature and r g denotes the mean value of the scalar curvature we prove convergence of the yamabe flow if the dimension n satisfies 3 leq n leq 5 or the initial metric is locally conformally flat

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

universal correlations of trapped one dimensional impenetrable bosons we calculate the asymptotic behaviour of the one body density matrix of one dimensional impenetrable bosons in finite size geometries our approach is based on a modification of the replica method from the theory of disordered systems we obtain explicit expressions for oscillating terms similar to fermionic friedel oscillations these terms are universal and originate from the strong short range correlations between bosons in one dimension

control of nonlinear systems in singular problems abstract the minimum singular functional control problem is analyzed for a class of multi input affine nonlinear systems we develop an algorithm for solving the singular problem that is applicable whether or not singular sub arcs exist in the optimal control the optimal control of bilinear systems to obtain by considering the lie algebra generated by the system matrices

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

sums of divisor functions in fq t and matrix integrals we study the mean square of sums of the kth divisor function d k n over short intervals and arithmetic progressions for the rational function field over a finite field of q elements in the limit as q rightarrow infty we establish a relationship with a matrix integral over the unitary group evaluating this integral enables us to compute the mean square of the sums of d k n in terms of a lattice point count this lattice point count can in turn be calculated in terms of a certain piecewise polynomial function which we analyse our results suggest general conjectures for the corresponding classical problems over the integers which agree with the few cases where the answer is known

random matrix theory and the derivative of the riemann zeta function random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the riemann zeta function s evaluated at the complex zeros i n we also discuss the probability distribution of ln 1 2 i n proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations

remarks on prescribing gauss curvature we study the nonlinear partial differential equation for the problem of prescribing gauss curvature k on s 2 we give an example of a rotationally symmetric k for which the kazdan warner obstruction is satisfied but the equation has no rotationally symmetric solution on the other hand we give a simple sufficient condition for solvability of the equation when k is rotationally symmetric finally we give a sufficient condition for solvability when k is not necessarily rotationally symmetric

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

mixed moments of characteristic polynomials of random unitary matrices following the work of conrey rubinstein and snaith commun math phys 267 611 2006 and forrester and witte j phys a math gen 39 8983 2006 we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group u n also known as the cue and relate the moment to the solution of a painleve differential equation we also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle following the work of conrey rubinstein and snaith commun math phys 267 611 2006 and forrester and witte j phys a math gen 39 8983 2006 we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group u n also known as the cue and relate the moment to the solution of a painleve differential equation we also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle

on minimum quadratic functional control of affine nonlinear systems abstract the minimization control problem of the quadratic functionals for the class of affine nonlinear systems in the hypothesis of nilpotent associated lie algebra is analyzed the optimal control correspondent to the first second and third order nilpotent operator is determined h bourdache siguerdinljane has applied the method of lie algebras to the study of optimal control regulation of satellites s banks and m yew have studied the optimal control of energy consumption minimization for a class of bilinear systems and j s liu et al have generalized this result for the class of affine nonlinear systems the optimal control determination of nonlinear system with a nilpotent structure minimizing the quadratic functionals generalizes the results of j s liu k yuan w s lin and s p banks and m k yew regarding the energy minimization of the affine nonlinear and bilinear systems