on the cartan matrix of mackey algebras let k be a field of characteristic p 0 and let g be a finite group the first result of this paper is an explicit formula for the determinant of the cartan matrix of the mackey algebra k g of g over k the second one is a formula for the rank of the cartan matrix of the cohomological mackey algebra co k g of g over k and a characterization of the groups g for which this matrix is nonsingular the third result is a generalization of this rank formula and characterization to blocks of co k g in particular if b is a block of kg the cartan matrix of the corresponding block co k b of co k g is nonsingular if and only if b is nilpotent with cyclic defect groups

pair correlation densities of inhomogeneous quadratic forms under explicit diophantine conditions on 2 w eprove that the local two point correlations of the sequence given by the values m 2 n 2 with m n 2 are those of a poisson process this partly confirms a conjecture of berry and tabor 2 on spectral statistics of quantized integrable systems and also establishes a particular case of the quantitative version of the oppenheim conjecture for inhomogeneous quadratic forms of signature 2 2 the proof uses theta sums and ratner s classification of measures invariant under unipotent flows

quantum properties of irrational triangular billiards triangles with sides given by consecutive integers n n 1 n 2 are fully irrational all angles irrational with pi if 3 120 and eventually the occurrence of gaps in the distribution for n 180 define the onset of a long crossover towards the sequence observed in the integrable limit of the equilateral triangle n infinity

boundary controllability of parabolic coupled equations abstract this paper is concerned with the boundary controllability of non scalar linear parabolic systems more precisely two coupled one dimensional parabolic equations are considered we show that in this framework boundary controllability is not equivalent and is more complex than distributed controllability in our main result we provide necessary and sufficient conditions for the null controllability

boundary controllability of a cascade system coupling fourth and second order parabolic equations abstract a control system coupling fourth and second order parabolic equations is considered in this paper the main topic is the study of the control properties of this system when we only control the second order partial differential equation through a boundary condition depending on the choice of the diffusion coefficients we obtain positive and negative results for approximate and null controllability in particular we prove that for any given positive time t 0 we can find some diffusion coefficients such that the system is null controllable in time t if t t 0 and is not null controllable if t t 0

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

minimal time for the null controllability of parabolic systems the effect of the condensation index of complex sequences let a d a be a diagonalizable generator of a c0 semigroup of contractions on a complex hilbert space x b l c y y being some suitable extrapolation space of x and u l2 0 t c under some assumptions on the sequence of eigenvalues k k 1 c of a d a we prove the existence of a minimal time t0 0 depending on bernstein s condensation index of and on b such that y ay bu is null controllable at any time t t0 and not null controllable for t t0 as a consequence we solve controllability problems of various systems of coupled parabolic equations in particular new results on the boundary controllability of one dimensional parabolic systems are derived these seem to be difficult to achieve using other classical tools

the energy level spacing for two harmonic oscillators with golden mean ratio of frequencies the limit distribution of energy level spacing is studied for the system of two harmonic oscillators with generic ratio of frequencies it is proved that for any fixed generic ratio no limit distribution exists but for random with any absolutely continuous distributionp d on 0 1 a universal random limit distribution of the energy level spacing exists some properties of the random limit distribution are discussed

boundary controllability of a one dimensional phase field system with one control force abstract in this paper we present some controllability results for linear and nonlinear phase field systems of caginalp type considered in a bounded interval of r when the scalar control force acts on the temperature equation of the system by means of the dirichlet condition on one of the endpoints of the interval in order to prove the linear result we use the moment method providing an estimate of the cost of fast controls using this estimate and following the methodology developed in 19 we prove a local exact boundary controllability result to constant trajectories of the nonlinear phase field system to the authors knowledge this is the first nonlinear boundary controllability result in the framework of non scalar parabolic systems framework in which some hyperbolic behaviors could arise

a detailed investigation of the spectral rigidity for integrable hamiltonian systems the authors carry out a detailed analysis of the spectral rigidity for integrable systems with a view towards understanding the deviations from the universal behaviour l 15 in systems with degeneracies in orbit actions in the process they also investigate the saturation region their studies show that the energy dependence of delta 3 infinity is a better indicator of the underlying universality in integrable systems

the local structure of the spectrum of the one dimensional schrodinger operator let h v frac d text 2 dt text 2 q t omega be an one dimensional random schrodinger operator in 2 v v with the classical boundary conditions the random potentialq t has a formq t f x t wherex t is a brownian motion on the compact riemannian manifoldk andf k r 1 is a smooth morse function mathop min limits k f 0 let n v delta sum limits ei v in delta 1 where 0 e i v are the eigenvalues ofh v the main result theorem 1 of this paper is the following ifv e 0 0 k z anda 0 a is a fixed constant then p left n v left e 0 frac a 2v e 0 frac a 2v right k right xrightarrow v to infty e an e 0 an e 0 k k wheren e 0 is a limit state density ofh v v this theorem mean that there is no repulsion between energy levels of the operatorh v v the second result theorem 2 describes the phenomen of the repulsion of the corresponding wave functions

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces

diagonal p permutation functors abstract let k be an algebraically closed field of positive characteristic p and let f be an algebraically closed field of characteristic 0 we consider the f linear category f p p k of finite groups in which the set of morphisms from g to h is the f linear extension f t h g of the grothendieck group t h g of p permutation k h k g bimodules with twisted diagonal vertices the f linear functors from f p p k to f mod are called diagonal p permutation functors they form an abelian category f p p k we study in particular the functor f t sending a finite group g to the grothendieck group f t g of p permutation kg modules and show that f t is a semisimple object of f p p k equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs p s of a finite p group p and a generator s of a p subgroup acting faithfully on p this leads to a precise description of the evaluations of these simple functors in particular we show that the simple functor indexed by the trivial pair 1 1 is isomorphic to the functor sending a finite group g to f k 0 k g where k 0 k g is the grothendieck group of projective kg modules

the primitive idempotents of the p permutation ring let g be a finite group let p be a prime number and let k be a field of characteristic 0 and k be a field of characteristic p both large enough in this note we state explicit formulae for the primitive idempotents of the ring of p permutation kg modules tensored with k

the control transmutation method and the cost of fast controls in this paper the null controllability in any positive time t of the first order equation 1 dot x t e i theta ax t bu t theta pi 2 fixed is deduced from the null controllability in some positive time l of the second order equation 2 ddot z t az t bv t the differential equations 1 and 2 are set in a banach space b is an admissible unbounded control operator and a is a generator of cosine operator function the control transmutation method makes explicit the input function u of 1 in terms of the input function v of 2 u t int mathbb r k t s v s ds where the compactly supported kernel k depends only on t and l this method proves roughly that the norm of a u steering the system 1 from an initial state x 0 x 0 to the final state x t 0 grows at most like x 0 exp alpha l 2 t as the control time t tends to zero the rate alpha is characterized independently by a one dimensional controllability problem in applications to the cost of fast controls for the heat equation l is roughly the length of the longest ray of geometric optics which does not intersect the control region

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

insensitizing controls for a heat equation with a nonlinear term involving the state and the gradient in this paper we present two results on the existence of insensitizing controls for a heat equation in a bounded domain of ir n we rst consider a semilinear heat equation involving gradient terms with homogeneous dirichlet boundary conditions then a heat equation with a nonlinear term f y and linear boundary conditions of fourier type is considered the nonlinearities are assumed to be globally lipschitz continuous in both cases we prove the existence of controls insensitizing the l 2 norm of the observation of the solution in an open subset o of the domain under suitable assumptions on the data each problem boils down to a special type of null controllability problem general observability inequalities are proved for linear systems similar to the linearized problem the proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate xed point arguments

level repulsion in the spectrum of two dimensional harmonic oscillators the asymptotic energy limit of the density p s of spacings between adjacent levels of the two dimensional harmonic oscillator tdho spectrum is studied it is shown that in any integer segment m m 1 containing approximately m levels of the tdho spectrum m n p s has the form wi s si where i takes on at most three values for large m p s displays strong level repulsion for irrational but it does not settle on a stationary form nor does its average over m this is in marked contrast with the behaviour of generic integrable systems for which the poisson statistics p s exp s is known to apply

a small scale density of states formula let m g be a c compact riemann manifold with classical hamiltonian h c t m assume that the corresponding quantization p1 op h is quantum completely integrable we establish an microlocal weyl law on short spectral intervals of size 2 e e 0 for various families of operators p1u u i containing p1 both in the mean and pointwise a e for u i the microlocalization refers to a small tubular neighbourhood of a non degenerate stable periodic bicharacteristic t m 0

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

two point spectral correlations for the square billiard we investigate the two point correlations in the quantum spectrum of the square billiard this system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states hence for example the standard level spacings distribution does not tend to the poissonian limit expected for integrable systems in this paper we calculate the leading order asymptotic form of a degeneracy weighted two point correlation function using a combination of probabilistic techniques and classical number theory the result exhibits number theoretical fluctuations about a mean which is a sum of two terms one having the usual constant poissonian form and the second representing a small correction which decays as the inverse of the correlation distance this is confirmed by numerical computations

new phenomena for the null controllability of parabolic systems minimal time and geometrical dependence abstract we consider the null controllability problem for two coupled parabolic equations with a space depending coupling term we analyze both boundary and distributed null controllability in each case we exhibit a minimal time of control that is to say a time t 0 0 such that the corresponding system is null controllable at any time t t 0 and is not if t t 0 in the distributed case this minimal time depends on the relative position of the control interval and the support of the coupling term we also prove that for a fixed control interval and a time 0 0 there exist coupling terms such that the associated minimal time is 0

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

sharp estimates of the one dimensional boundary control cost for parabolic systems and application to the n dimensional boundary null controllability in cylindrical domains in this paper we consider the boundary null controllability of a system of n parabolic equations on domains of the form omega 0 pi times omega 2 with omega 2 a smooth domain of mathbb r n 1 n 1 when the control is exerted on 0 times omega 2 with omega 2 subset omega 2 we obtain a necessary and sufficient condition that completely characterizes the null controllability this result is obtained through the lebeau robbiano strategy and requires an upper bound of the cost of the one dimensional boundary null control on 0 pi the latter is obtained using the moment method and it is shown to be bounded by ce c t when t goes to 0

quadratic forms of signature 2 2 and eigenvalue spacings on rectangular 2 tori the oppenheim conjecture proved by margulis mar1 see also mar2 asserts that for a nondegenerate indefinite irrational quadratic form q in n 3 variables the set q zn is dense in emm where we used the lower bounds established in dm a quantitative version of the conjecture was established namely let be a continuous positive function on the sphere v rn v 1 and let v rn v v v we denote by t the dilate of by t for an indefinite quadratic form q in n variables let nq a b t denote the cardinality of the set

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

spectral analysis of discrete elliptic operators and applications in control theory in this paper we propose an analysis of discrete spectral properties for a finite difference discretization of quite general 1d second order self adjoint elliptic operators we particularly investigate some uniform in the discretization parameter qualitative behavior of eigenfunctions and eigenvalues with those estimates we manage to obtain new results for the construction of bounded families of controls for semi discrete parabolic pdes in particular for boundary controls of a coupled parabolic system with fewer controls than equations

on spectral statistics of classically integrable systems this work is an extensive study of the spectral statistics of three representative classically integrable systems namely rectangle torus and circle billiards we analyse the statistics and focus on the related level spacing distribution p s and the delta statistics the agreement with the poissonian model is typically found to be perfect up to the outer unfolded energy scale beyond which the saturation is observed in agreement with berry s dynamical theory of the spectral rigidity where as the untypical systems are those where for example p s is not a smooth distribution but a sum of the delta functions due to the granularity of the energy scale for example in the rectangle with the rational squared sides ratio however even there we find reasonable trend towards poissonian statistics for large ranges l but we describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement also in irrational rectangle billiards we show and explain the existence of large fluctuations by one order of magnitude bigger than the statistical ones whose origin is in the closeness to some rational billiard shape these fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy finally we tested the mode distribution i e distribution of the reduced mode fluctuation number w and found that it was not poissonian in agreement with steiner s conjecture and in fact follows the prediction by bleher et al that its tail behaves as the general reason for non universal behaviour of p w is that at largest energy scales we are always in the saturation regime

pair correlation of sums of rationals with bounded height for each positive integer q let f q denote the farey sequence of order q we prove the existence of the pair correlation measure associated to the sum f q f q modulo 1 as q tends to infinity and compute the corresponding limiting pair correlation function

statistical properties of spectral fluctuations for a quantum system with infinitely many components extending the idea formulated in makino phys rev e 67 066205 2003 that is based on the berry robnik approach m v berry and m robnik j phys a 17 2413 1984 we investigate the statistical properties of a two point spectral correlation for a classically integrable quantum system the eigenenergy sequence of this system is regarded as a superposition of infinitely many independent components in the semiclassical limit we derive the level number variance lnv in the limit of infinitely many components and discuss its deviations from poisson statistics the slope of the limiting lnv is found to be larger than that of poisson statistics when the individual components have a certain accumulation this property agrees with the result from the semiclassical periodic orbit theory that is applied to a system with degenerate torus actions d biswas m azam and s v lawande phys rev a 43 5694 1991

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

resonant harmonic oscillators and eigenvalue multiplicity explicit formulae are worked out for the eigenvalue multiplicity of a system of n independent quantum harmonic oscillators in the general case of 1 or s or n 1 resonance relations among the frequencies omega 1 omega n as a particular case we prove that even though the quantum numbers are always less than the degrees of freedom the eigenvalues are in general intrinsically degenerate only in the completely resonant case s n 1

level spacing statistics of classically integrable systems investigation along the lines of the berry robnik approach by extending the approach of berry and robnik the limiting level spacing distribution of a system consisting of infinitely many independent components is investigated the limiting level spacing distribution is characterized by a single monotonically increasing function mu s of the level spacing s three cases are distinguished 1 poissonian if mu infinity 0 2 poissonian for large s but possibly not for small s if 0 mu infinity 1 and 3 sub poissonian if mu infinity 1 this implies that even when energy level distributions of individual components are statistically independent non poissonian level spacing distributions are possible

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

unfolding of the spectrum for chaotic and mixed systems abstract random matrix theory rmt is capable of making predictions for the spectral fluctuations of a physical system only after removing the influence of the level density by unfolding the spectra when the level density is known unfolding is done by using the integrated level density to transform the eigenvalues into dimensionless variables with unit mean spacing when it is not known as in most practical cases one usually approximates the level staircase function by a polynomial we here study the effect of unfolding procedure on the spectral fluctuation of two systems for which the level density is known asymptotically the first is a time reversal invariant chaotic system which is modeled in rmt by a gaussian orthogonal ensemble goe the second is the case of chaotic systems in which m quantum numbers remain almost undistorted in the early stage of the stochastic transition the hamiltonian of a system may be represented by a block diagonal matrix with m blocks of the same size in which each block is a goe unfolding is done once by using the asymptotic level densities for the eigenvalues of the m blocks and once by representing the integrated level density in terms of polynomials of different orders we find that the spacing distribution of the eigenvalues shows a little sensitivity to the unfolding method on the other hand the variance of level number 2 l is sensitive to the choice of the unfolding function unfolding that utilizes low order polynomials enhances 2 l relative to the theoretical value while the use of high order polynomial reduces it the optimal value of the order of the unfolding polynomial depends on the dimension of the corresponding ensemble

controllability of two coupled wave equations on a compact manifold we consider the exact controllability problem on a compact manifold for two coupled wave equations with a control function acting on one of them only action on the second wave equation is obtained through a coupling term first when the two waves propagate with the same speed we introduce the time t omega rightarrow mathcal o rightarrow omega for which all geodesics traveling in go through the control region then through the coupling region mathcal o and finally come back in we prove that the system is controllable if and only if both and mathcal o satisfy the geometric control condition and the control time is larger than t omega rightarrow mathcal o rightarrow omega second we prove that the associated hum control operator is a pseudodifferential operator and we exhibit its principal symbol finally if the two waves propagate with different speeds we give sharp sufficient controllability conditions on the functional spaces the geometry of the sets and mathcal o and the minimal time

null controllability of some reaction diffusion systems with one control force this work is concerned with the null controllability of semilinear parabolic systems by a single control force acting on a subdomain

comment on energy level statistics in the mixed regime we comment on the recent paper by abul magd 1996 29 1 concerning the energy level statistics in the mixed regime i e such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space we point out that his basic assumption on the additive property of the level repulsion function r s conditional probability density in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes and is not justified since among other things it relies on the type of berry s ergodic assumption which however is only correct in a homogeneous ensemble of ergodic systems not in the neighbourhood of an integrable system thus his resulting distribution cannot be regarded as a theoretically well founded object we point out that the semiclassical limiting energy level spacing distribution must be of berry robnik 1984 type and explain what transitional behaviour of the brody type with fractional power law energy level repulsion we observe in the near semiclassical regime where effective is not yet small enough thus we refer to the derivation arguments and conclusions in our previous paper prosen and robnik 1994 26 8059 and explain again the behaviour in this double transition region

local null controllability of the three dimensional navier stokes system with a distributed control having two vanishing components in this paper we prove a local null controllability result for the three dimensional navier stokes equations on a smooth bounded domain of mathbb r 3 with null dirichlet boundary conditions the control is distributed in an arbitrarily small nonempty open subset and has two vanishing components lions and zuazua proved that the linearized system is not necessarily null controllable even if the control is distributed on the entire domain hence the standard linearization method fails we use the return method together with a new algebraic method inspired by the works of gromov and previous results by gueye