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

a modification of the blowup technique for variational integrals with subquadratic growth we present a version of the blowup technique which applies to local minimizersu rn open in rn of strictly convex variational integralsj u f u dx wherefis of p growth for some 1 p 2 this provides an alternative approach towards the partial regularity theorem of anzellotti and giaquinta for two dimensional problems we obtain everywherec1 regularity

poisson law for the number of lattice points in a random strip with finite area let a smooth curve be given by a functionr f in polar coordinate system in the plane and letr be a uniformly distributed random variable on the interval a 1 l a 2 l with somea 2 a 1 0 and a largel 0 ya g sinai has conjectured that given some real numbersc 2 c 1 the number of lattice points in the domain between the curves left r frac c 1 r right and left r frac c 2 r right is asymptotically poisson distributed for good functionsf we cannot prove this conjecture but we show that if a probability measure with some nice properties is given on the space of smooth functions then almost all functions with respect to this measure satisfy sinai s conjecture this is an improvement of an earlier result of sinai 9 and actually the proof also contains many ideas of that paper

multifractal analysis of chinese stock volatilities based on the partition function approach abstract we have performed a detailed multifractal analysis on the 1 min volatility of two indexes and 1139 stocks in the chinese stock markets based on the partition function approach the partition function q s scales as a power law with respect to the box size s the scaling exponents q form a nonlinear function of q statistical tests based on bootstrapping show that the extracted multifractal nature is significant at the 1 significance level the individual securities can be well modeled by the p model in turbulence with p 0 40 0 02 based on the idea of ensemble averaging including quenched and annealed average we treat each stock exchange as a whole and confirm the existence of multifractal nature in the chinese stock markets

structural changes and out of sample prediction of realized range based variance in the stock market abstract this paper aims to examine the effects of structural changes on forecasting the realized range based variance in the stock market considering structural changes in variance in the stock market we develop the har rrv sc model on the basis of the har rrv model subsequently the har rrv and har rrv sc models are used to forecast the realized range based variance of s p 500 index we find that there are many structural changes in variance in the u s stock market and the period after the financial crisis contains more structural change points than the period before the financial crisis the out of sample results show that the har rrv sc model significantly outperforms the har bv model when they are employed to forecast the 1 day 1 week and 1 month realized range based variances which means that structural changes can improve out of sample prediction of realized range based variance the out of sample results remain robust across the alternative rolling fixed window the alternative threshold value in icss algorithm and the alternative benchmark models more importantly we believe that considering structural changes can help improve the out of sample performances of most of other existing har rrv type models in addition to the models used in this paper

spacing between phase shifts in a simple scattering problem we prove a scattering theoretical version of the berry tabor conjecture for an almost every surface in a class of cylindrical surfaces of revolution the large energy limit of the pair correlation measure of the quantum phase shifts is poisson that is it is given by the uniform measure

the pair correlation function of fractional parts of polynomials we investigate the pair correlation function of the sequence of fractional parts of n d n 1 2 n where d 2 is an integer and an irrational we conjecture that for badly approximable the normalized spacings between elements of this sequence have poisson statistics as n

distribution of energy levels of quantum free particle on the liouville surface and trace formulae we consider the weyl asymptotic formula left e n leqq r 2 right frac area q 4 pi r 2 n r for eigenvalues of the laplace beltrami operator on a two dimensional torusq with a liouville metric which is in a sense the most general case of an integrable metric we prove that if the surfaceq is non degenerate then the remainder termn r has the formn r r1 2 r where r is an almost periodic function of the besicovitch classb1 and the fourier amplitudes and the fourier frequencies of r can be expressed via lengths of closed geodesics onq and other simple geometric characteristics of these geodesics we prove then that if the surfaceq is generic then the limit distribution of r has a densityp t which is an entire function oft possessing an asymptotics on a real line logp t c t4 ast an explicit expression for the fourier transform ofp t via fourier amplitudes of r is also given we obtain the analogue of the guillemin duistermaat trace formula for the liouville surfaces and discuss its accuracy

multifractal detrended fluctuation analysis of nonstationary time series abstract we develop a method for the multifractal characterization of nonstationary time series which is based on a generalization of the detrended fluctuation analysis dfa we relate our multifractal dfa method to the standard partition function based multifractal formalism and prove that both approaches are equivalent for stationary signals with compact support by analyzing several examples we show that the new method can reliably determine the multifractal scaling behavior of time series by comparing the multifractal dfa results for original series with those for shuffled series we can distinguish multifractality due to long range correlations from multifractality due to a broad probability density function we also compare our results with the wavelet transform modulus maxima method and show that the results are equivalent

on the c1 γ ω w2 2 ω regularity for a class of electro rheological fluids abstract we are concerned with a system of nonlinear partial differential equations with p x structure 1 p p x p 0 and no slip boundary conditions we prove the existence and uniqueness of a c 1 w 2 2 solution corresponding to small data without further restrictions on the bounds p p 0 in particular this result is applicable to the steady motion of shear dependent electro rheological fluids

the model confidence set this paper introduces the model confidence set mcs and applies it to the selection of models a mcs is a set of models that is constructed such that it will contain the best model with a given level of confidence the mcs is in this sense analogous to a confidence interval for a parameter the mcs acknowledges the limitations of the data such that uninformative data yield a mcs with many models whereas informative data yield a mcs with only a few models the mcs procedure does not assume that a particular model is the true model in fact the mcs procedure can be used to compare more general objects beyond the comparison of models we apply the mcs procedure to two empirical problems first we revisit the inflation forecasting problem posed by stock and watson 1999 and compute the mcs for their set of inflation forecasts second we compare a number of taylor rule regressions and determine the mcs of the best regression in terms of in sample likelihood criteria

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

analysis of the efficiency and multifractality of gold markets based on multifractal detrended fluctuation analysis abstract in this paper we investigate the efficiency and multifractality of a gold market based on multifractal detrended fluctuation analysis our evidence shows that the gold return series are multifractal both for time scales smaller than a month and for time scales larger than a month for time scales smaller than a month the main contribution of multifractality is fat tail distribution for time scales larger than a month both long range correlations and fat tail distribution play important roles in the contribution of multifractality using the method of rolling windows we find that the gold market became more and more efficient over time especially after 2001 the abnormal points of scaling exponents can also be related to some occasional events by defining a new inefficiency measure related to the multifractality we find that the gold market is more efficient during the upward periods than during the downward periods

modeling returns volatility realized garch incorporating realized risk measure abstract this study applies realized garch models by introducing several risk measures of intraday returns into the measurement equation to model the daily volatility of e mini s p 500 index futures returns besides using the conventional realized measures realized volatility and realized kernel as our benchmarks we also use generalized realized risk measures realized absolute deviation and two realized tail risk measures realized value at risk and realized expected shortfall the empirical results show that realized garch models using the generalized realized risk measures provide better volatility estimation for the in sample and substantial improvement in volatility forecasting for the out of sample in particular the realized expected shortfall performs best for all of the alternative realized measures our empirical results reveal that future volatility may be more attributable to present losses risk measures the results are robust to different sample estimation windows

predictability of multifractal analysis of hang seng stock index in hong kong in this paper the daily hang seng index in hong kong stock market is studied by multifractal analysis the main parameter of multifractal spectra used is f which can be used to characterize the ratio of number of highest index moments to that of lowest ones the dependence of today s gain probability g and the day s index increase probability n with f of the previous 3 days are studied it is found that g and n can reach 70 80 at the large positive f region and can be very close to 20 at the big negative f region the predictability decreases with the increasing number of the previous days

the uniform central limit theorem for theta sums sommes gaussiennes sommes de kloosterman theoreme de limite centrale uniforme pour des series theta theoreme de limite centrale uniformes pour des sommes theta moments sur des intervalles arbitraires

on non newtonian p fluids the pseudo plastic case abstract in the following we study a class of stationary navier stokes equations with shear dependent viscosity under the non slip dirichlet boundary condition we consider pseudo plastic fluids a fluid is said pseudo plastic or shear thinning if in eq 1 1 below one has p 2 we are interested in global i e up to the boundary regularity results in dimension n 3 for the second order derivatives of the velocity and the first order derivatives of the pressure we consider a cubic domain and impose the non slip boundary condition only on two opposite faces on the other faces we assume periodicity as a device to avoid effective boundary conditions this choice is made so that we work in a bounded domain and simultaneously with a flat boundary

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

spectral statistics on zoll surfaces the subject of this article is a family of partly conjectural spectral invariants of a laplace operator with a discrete spectrum j known as its k level correlation functions k and level spacings distribution the 2 level or pair correlation function 2 for instance measures the asymptotic number of spacings i j between normalized eigenvalues i as the eigenvalues tend to infinity the normalization is so that the mean level spacing i 1 i is one these invariants have been studied by many physicists including wigner dyson and berry and are conjectured to have certain universal forms depending on the qualitative behaviour of the corresponding classical geodesic flow

multifractal analysis of hang seng index in hong kong stock market abstract in this paper the daily hang seng index in hong kong stock market is analyzed by multifractal the correlation of the parameters of the multifractal spectra with the variation of close return z is studied statistically it is found that the amount of the variation of return is correlated with the amount of of that day the increase or decrease of the return is related to the positive or negative value of f the gain probability and the increasing day s probability can be higher than 70 at the larger f region and be lower than 20 at the more negative region

measuring daily value at risk of ssec index a new approach based on multifractal analysis and extreme value theory abstract recent studies in the econophysics literature reveal that price variability has fractal and multifractal characteristics not only in developed financial markets but also in emerging markets taking high frequency intraday quotes of the shanghai stock exchange component ssec index as example this paper proposes a new method to measure daily value at risk var by combining the newly introduced multifractal volatility mfv model and the extreme value theory evt method two var backtesting techniques are then employed to compare the performance of the model with that of a group of linear and nonlinear generalized autoregressive conditional heteroskedasticity garch models the empirical results show the multifractal nature of price volatility in chinese stock market var measures based on the multifractal volatility model and evt method outperform many garch type models at high risk levels

forecasting volatility in shanghai and shenzhen markets based on multifractal analysis abstract this paper analyzes the multifractality in shanghai and shenzhen stock markets using multifractal spectrum analysis and multifractal detrended fluctuation analysis we find that the main source of multifractality is long range correlations of large and small fluctuations then we introduce a multifractal volatility measure mv and find that by taking mv as daily conditional volatility the simulated series displayed similar stylized facts to the original daily return series by capturing the dynamics of mv using the arfima model we find that the out of sample forecasting performance of the arfima mv model is better than some garch class models and the arfima rv model under some criteria of loss function

forecasting volatility of fuel oil futures in china garch type sv or realized volatility models abstract in most previous works on forecasting oil market volatility squared daily returns were taken as the proxy of unobserved actual volatility however as demonstrated by andersen and bollerslev 1998 22 this proxy with too high measurement noise could be perfectly outperformed by a so called realized volatility rv measure calculated by the cumulative sum of squared intraday returns with this motivation we further extend earlier works by employing intraday high frequency data to compare the performance of three typical volatility models in the daily out of sample volatility forecasting of fuel oil futures on the shanghai futures exchange shfe the garch type stochastic volatility sv and realized volatility models by taking rv as the proxy of actual daily volatility and then computing forecasting errors we find that the realized volatility model based on intraday high frequency data produces significantly more accurate volatility forecasts than the garch type and sv models based on daily returns furthermore the sv model outperforms many linear and nonlinear garch type models that capture long memory volatility and or the asymmetric leverage effect in volatility these results also prove that abundant volatility information is available in intraday high frequency data and can be used to construct more accurate oil volatility forecasting models

distribution of the error term for the number of lattice points inside a shifted circle define begin array 20 c n x r left n in z d x n leqslant r right x in 0 1 d end array 1 1 and f x r r frac d 1 2 n x r r d text vol d b 1 1 2 where b 1 is the unit ball in r d thus n x r is the number of lattice points in z d in the r ball centered at x and f x r is the normalized difference with the volume of the ball we are interested in the problem of the distribution of f x r as a function of r for given x

multifractality in stock indexes fact or fiction abstract multifractal analysis and extensive statistical tests are performed upon intraday minutely data within individual trading days for four stock market indexes including hsi szsc s p 500 and nasdaq to check whether the indexes instead of the returns possess multifractality we find that the mass exponent q is linear and the singularity q is close to 1 for all trading days and all indexes furthermore we find strong evidence showing that the scaling behaviors of the original data sets cannot be distinguished from those of shuffled time series hence the so called multifractality in the intraday stock market indexes is merely an illusion

boundary regularity for the steady stokes type flow with shear thickening viscosity abstract we work on the boundary regularity for weak solutions to the stationary stokes type equations with shear dependent viscosity using a weighted estimate near the boundary we obtain the holder continuity of the solution for the shear thickening fluid without the convection term

the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous dirichlet boundary condition abstract in this paper we show the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous dirichlet boundary condition under the condition p x 2 n n 2 n 2 3 in particular we improve a priori estimate for weak solutions to the problem with min p x 2 which is an improvement even for constant p

values of indefinite quadratic forms at integral points and flows on spaces of lattices this mostly expository paper centers on recently proved conjectures in two areas a a conjecture of a oppenheim on the values of real indefinite quadratic forms at integral points b conjectures of dani raghunathan and margulis on closures of orbits in spaces of lattices such as sl r sl z at first sight a belongs to analytic number theory and b belongs to ergodic and lie theory and they seem to be quite unrelated they are discussed together here because of a very interesting connection between the two pointed out by m s raghunathan namely a special case of b yields a proof of a the first main goal of this talk is to describe the oppenheim conjecture and various refinements and to derive them from one statement about closures of orbits in the space of unimodular lattices in r3 see proposition 2 in 2 3 in 3 and 4 we put this statement in context and describe more general conjectures and results on orbit closures and invariant probability measures on quotients of lie groups by discrete subgroups 5 gives some brief comments on the proofs and further developments 6 7 and 8 are devoted to the so called s arithmetic setting where we consider products of real and p adic groups 6 is concerned with a generalized oppenheim conjecture 7 with a generalization of the orbit closure theorem proved by m ratner r8 and 8 with applications to quadratic forms since the subject matter of that last section has not been so far discussed elsewhere we take this opportunity to present proofs obtained jointly with g prasad finally 9 gives the proof of a lemma on symmetric simple lie algebras a special case of which is used in 8 i am glad to thank m ratner and g prasad for a number of remarks on and corrections to a preliminary version of this paper thanks to which many typos and inaccuracies have been eliminated

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

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

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

the generic spacing distribution of the two dimensional harmonic oscillator the level spacing distribution of the two dimensional harmonic oscillator is studied by choosing a suitable average a stable distribution is obtained this is then shown to have a fixed form for the generic frequency ratio a delta function is also found for a dense class of ratio with hausdorff dimension thus the distribution is unstable under perturbation of the ratio all of this differs from the universal poissonian statistics associated with integrable systems 1

forecasting volatility of ssec in chinese stock market using multifractal analysis abstract in this paper taking about 7 years high frequency data of the shanghai stock exchange composite index ssec as an example we propose a daily volatility measure based on the multifractal spectrum of the high frequency price variability within a trading day an arfima model is used to depict the dynamics of this multifractal volatility mfv measures the one day ahead volatility forecasting performances of the mfv model and some other existing volatility models such as the realized volatility model stochastic volatility model and garch are evaluated by the superior prediction ability spa test the empirical results show that under several loss functions the mfv model obtains the best forecasting accuracy

forecasting the realized volatility of csi 300 abstract according to the characteristics of realized volatility existing in shanghai and shenzhen 300 index china securities index 300 csi 300 the garch family model is introduced to describe the arch effect of error sequences in har arfima and arfimax models then the har garch family the arfima garch family and the arfimax garch family models set is proposed which contains 33 kinds of models using the quasi maximum likelihood method the parameters of the all models are estimated with normal n and skewed student t skst distributions by rolling window technology one step ahead rolling prediction of realized volatility for csi 300 is conducted the results from prediction accuracy by model confidence set mcs test show that the realized volatility prediction models with skewed student t distribution possess higher precision than those of normal distribution in general the arfimax and other six models in arfimax garch family form the optimal prediction models set which can forecast the realized volatility of csi 300 with better accuracy

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

energy level statistics of model quantum systems universality and scaling in a lattice point problem we investigate the statistics of the numbern r s of lattice pointsn z 2 in an annular domain r w r w a ra wherer w 0 herea is a fixed convex set with smooth boundary andw is chosen so that the area of r w iss the statistics comes fromr being taken as random with a smooth density in some interval c 1 t c 2 t c 2 c 1 0 we find that in the limitt the variance and distribution of n n r s s depend strongly on hows grows witht there is a saturation regimes t ast in which the fluctuations in n coming from the two boundaries of are independent then there is a scaling regime s t z 0 z in which the distribution depends onz in an almost periodic way going to a gaussian asz 0 the variance in this limit approachesz for generic a but can be larger for degenerate cases the former behavior is what one would expect from the poisson limit of a distribution for annuli of finite area

which is the better forecasting model a comparison between har rv and multifractality volatility abstract in this paper by taking the 5 min high frequency data of the shanghai composite index as example we compare the forecasting performance of har rv and multifractal volatility realized volatility realized bipower variation and their corresponding short memory model with rolling windows forecasting method and the model confidence set which is proved superior to spa test the empirical results show that for six loss functions har rv outperforms other models moreover to make the conclusions more precise and robust we use the mcs test to compare the performance of their logarithms form models and find that the har log rv has a better performance in predicting future volatility furthermore by comparing the two models of har rv and har log rv we conclude that in terms of performance forecasting the har log rv model is the best model among models we have discussed in this paper

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

scaling behaviors in differently developed markets scaling properties of four di erent stock market indices are studied in terms of a generalized hurst exponent approach we nd that the deviations from pure brownian motion behavior are associated with the degrees of development of the markets and we observe strong di erentiations in the scaling properties of markets at di erent development stage c

forecasting stock market volatility do realized skewness and kurtosis help abstract in this study we investigate the predictability of the realized skewness rsk and realized kurtosis rku to stock market volatility that has not been addressed in the existing studies out of sample results show that rsk which can significantly improve forecast accuracy in mid and long term is more powerful than rku in forecasting volatility whereas these variables are useless in short term forecasting furthermore we employ the realized kernel rk for the robustness analysis and the conclusions are consistent with the rv measures our results are of great importance for portfolio allocation and financial risk management

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

a multiple regime smooth transition heterogeneous autoregressive model for long memory and asymmetries in this paper we propose a flexible model to describe nonlinearities and long range dependence in time series dynamics the new model is a multiple regime smooth transition extension of the heterogeneous autoregressive har model which is specifically designed to model the behavior of the volatility inherent in financial time series the model is able to simultaneously approximate long memory behavior as well as describe sign and size asymmetries a sequence of tests is developed to determine the number of regimes and an estimation and testing procedure is presented monte carlo simulations evaluate the finite sample properties of the proposed tests and estimation procedures we apply the model to several dow jones industrial average index stocks using transaction level data from the trades and quotes database that covers ten years of data we find strong support for long memory and both sign and size asymmetries furthermore the new model when combined with the linear har model is viable and flexible for purposes of forecasting volatility

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

econometric analysis of realized volatility and its use in estimating stochastic volatility models summary the availability of intraday data on the prices of speculative assets means that we can use quadratic variation like measures of activity in financial markets called realized volatility to study the stochastic properties of returns here under the assumption of a rather general stochastic volatility model we derive the moments and the asymptotic distribution of the realized volatility error the difference between realized volatility and the discretized integrated volatility which we call actual volatility these properties can be used to allow us to estimate the parameters of stochastic volatility models without recourse to the use of simulation intensive methods

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 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

correlations of eigenvalues on multi dimensional flat tori we show that for almost all k dimensional flat tori the eigenvalues of the laplacian follow a uniform distribution with respect to their 2 3 and k 2 level correlations

chasing volatility a persistent multiplicative error model with jumps abstract persistence and unpredictable large increments characterize the volatility of financial returns we propose the multiplicative error model with volatility jumps mem j to describe and predict the probability and the size of these extreme events under the mem j the conditional density of the realized measure is a countably infinite mixture of gamma and kappa distributions with closed form conditional moments we derive stationarity conditions and the asymptotic theory for the maximum likelihood estimation estimates of the volatility jump component confirm that the probability of jumps dramatically increases during the financial crises the mem j improves over other models with fat tails

global regularity of weak solutions for steady motions of electrorheological fluids in 3d smooth domain abstract in this paper we first show higher integrability of the gradient of weak solutions for steady motions of electrorheological fluids under a further assumption on regularity of external force f in 3d smooth domain based on the higher integrability result we then prove that any weak solution to the problem becomes global strong one under such an assumption on f moreover we improve the lower bound on p x for global regularity of solutions to the problem and their stokes setting even if p x is a constant