noise sensitivity of portfolio selection in constant conditional correlation garch models abstract this paper investigates the efficiency of minimum variance portfolio optimization for stock price movements following the constant conditional correlation garch process proposed by bollerslev simulations show that the quality of portfolio selection can be improved substantially by computing optimal portfolio weights from conditional covariances instead of unconditional ones measurement noise can be further reduced by applying some filtering method on the conditional correlation matrix such as random matrix theory based filtering as an empirical support for the simulation results the analysis is also carried out for a time series of s p500 stock prices

estimated correlation matrices and portfolio optimization abstract correlations of returns on various assets play a central role in financial theory and also in many practical applications from a theoretical point of view the main interest lies in the proper description of the structure and dynamics of correlations whereas for the practitioner the emphasis is on the ability of the models to provide adequate inputs for the numerous portfolio and risk management procedures used in the financial industry the theory of portfolios initiated by markowitz has suffered from the curse of dimensions from the very outset over the past decades a large number of different techniques have been developed to tackle this problem and reduce the effective dimension of large bank portfolios but the efficiency and reliability of these procedures are extremely hard to assess or compare in this paper we propose a model simulation based approach which can be used for the systematical testing of all these dimensional reduction techniques to illustrate the usefulness of our framework we develop several toy models that display some of the main characteristic features of empirical correlations and generate artificial time series from them then we regard these time series as empirical data and reconstruct the corresponding correlation matrices which will inevitably contain a certain amount of noise due to the finiteness of the time series next we apply several correlation matrix estimators and dimension reduction techniques introduced in the literature and or applied in practice as in our artificial world the only source of error is the finite length of the time series and in addition the true model hence also the true correlation matrix are precisely known therefore in sharp contrast with empirical studies we can precisely compare the performance of the various noise reduction techniques one of our recurrent observations is that the recently introduced filtering technique based on random matrix theory performs consistently well in all the investigated cases based on this experience we believe that our simulation based approach can also be useful for the systematic investigation of several related problems of current interest in finance

lp zd estimates for discrete operators of radon type maximal functions and vector valued estimates abstract we prove l p z d bounds for p 1 of discrete maximal functions corresponding to averaging operators and truncated singular integrals of radon type and their applications to pointwise ergodic theory our new approach is based on a unified analysis of both types of operators and also yields an extension to the vector valued form of these results

on sums of powers of almost equal primes we investigate the waring goldbach problem of representing a positive integer n as the sum of s kth powers of almost equal prime numbers define sk 2k k 1 when k 3 and put s2 6 suppose that n satisfies the necessary congruence conditions and put x n s 1 k we show that whenever s sk and 0 and n is sufficiently large thenn is represented as the sum of s kth powers of prime numbers p with p x 6 x 19 24 this conclusion is based on a new estimate of weyl type specific to exponential sums having variables constrained to short intervals

noise dressing of financial correlation matrices we show that results from the theory of random matrices are potentially of great interest to understand the statistical structure of the empirical correlation matrices appearing in the study of multivariate time series the central result of the present study which focuses on the case of financial price fluctuations is the remarkable agreement between the theoretical prediction based on the assumption that the correlation matrix is random and empirical data concerning the density of eigenvalues associated to the time series of the different stocks of the s 500 or other major markets in particular the present study raises serious doubts on the blind use of empirical correlation matrices for risk management

the bulk of the stock market correlation matrix is not pure noise abstract we analyse the structure of the distribution of eigenvalues of the stock market correlation matrix with increasing length of the time series representing the price changes we use 100 highly capitalized stocks from the american market and relate the result to the corresponding ensemble of wishart random matrices it turns out that systematically more eigenvalues remain beyond the borders prescribed by this variant of the random matrix theory rmt this may indicate that even the bulk of the spectrum of the stock market correlation matrix carries some sort of correlations that are masked by a measurement noise when the time series used to construct the matrix are short we also study some other characteristics of the noisy eigensignals like their return distributions temporal correlations or their multifractal spectra and the results support the above conclusions

a new iterative method in waring s problem on introduit une nouvelle methode iterative dans le probleme de waring on ameliore toutes les bornes superieures anterieures pour g k lorsque k est superieur ou egal a 5

spectral methods and cluster structure in correlation based networks abstract we investigate how in complex systems the eigenpairs of the matrices derived from the correlations of multichannel observations reflect the cluster structure of the underlying networks for this we use daily return data from the nyse and focus specifically on the spectral properties of weight w i j c i j i j and diffusion matrices d i j w i j s j i j where c i j is the correlation matrix and s i j w i j the strength of node j the eigenvalues and corresponding eigenvectors of the weight matrix are ranked in descending order as in the earlier observations the first eigenvector stands for a measure of the market correlations its components are to first approximation equal to the strengths of the nodes and there is a second order roughly linear correction the high ranking eigenvectors excluding the highest ranking one are usually assigned to market sectors and industrial branches our study shows that both for weight and diffusion matrices the eigenpair analysis is not capable of easily deducing the cluster structure of the network without a priori knowledge in addition we have studied the clustering of stocks using the asset graph approach with and without spectrum based noise filtering it turns out that asset graphs are quite insensitive to noise and there is no sharp percolation transition as a function of the ratio of bonds included thus no natural threshold value for that ratio seems to exist we suggest that these observations can be of use for other correlation based networks as well

noisy covariance matrices and portfolio optimization ii abstract recent studies inspired by results from random matrix theory galluccio et al physica a 259 1998 449 laloux et al phys rev lett 83 1999 1467 risk 12 3 1999 69 plerou et al phys rev lett 83 1999 1471 found that covariance matrices determined from empirical financial time series appear to contain such a high amount of noise that their structure can essentially be regarded as random this seems however to be in contradiction with the fundamental role played by covariance matrices in finance which constitute the pillars of modern investment theory and have also gained industry wide applications in risk management our paper is an attempt to resolve this embarrassing paradox the key observation is that the effect of noise strongly depends on the ratio r n t where n is the size of the portfolio and t the length of the available time series on the basis of numerical experiments and analytic results for some toy portfolio models we show that for relatively large values of r e g 0 6 noise does indeed have the pronounced effect suggested by galluccio et al 1998 laloux et al 1999 and plerou et al 1999 and illustrated later by laloux et al int j theor appl finance 3 2000 391 plerou et al phys rev e e print cond mat 0108023 and rosenow et al europhys lett e print cond mat 0111537 in a portfolio optimization context while for smaller r around 0 2 or below the error due to noise drops to acceptable levels since the length of available time series is for obvious reasons limited in any practical application any bound imposed on the noise induced error translates into a bound on the size of the portfolio in a related set of experiments we find that the effect of noise depends also on whether the problem arises in asset allocation or in a risk measurement context if covariance matrices are used simply for measuring the risk of portfolios with a fixed composition rather than as inputs to optimization the effect of noise on the measured risk may become very small

variational estimates for discrete operators modeled on multi dimensional polynomial subsets of primes we prove the extensions of birkhoff s and cotlar s ergodic theorems to multi dimensional polynomial subsets of prime numbers mathbb p k we deduce them from ell p big mathbb z d big boundedness of r variational seminorms for the corresponding discrete operators of radon type where p 1 and r 2

random matrix theory and portfolio optimization in moroccan stock exchange abstract in this work we use random matrix theory to analyze eigenvalues and see if there is a presence of pertinent information by using marcenko pastur distribution thus we study cross correlation among stocks of casablanca stock exchange moreover we clean correlation matrix from noisy elements to see if the gap between predicted risk and realized risk would be reduced we also analyze eigenvectors components distributions and their degree of deviations by computing the inverse participation ratio this analysis is a way to understand the correlation structure among stocks of casablanca stock exchange portfolio

sharp bounds for the cubic parsell vinogradov system in two dimensions abstract we prove a sharp decoupling for a certain two dimensional surface in r 9 as an application we obtain the full range of expected estimates for the cubic parsell vinogradov system in two dimensions

not all that glitters is rmt in the forecasting of risk of portfolios in the brazilian stock market abstract using stocks of the brazilian stock exchange bm f bovespa we build portfolios of stocks based on markowitz s theory and test the predicted and realized risks this is done using the correlation matrices between stocks and also using random matrix theory in order to clean such correlation matrices from noise we also calculate correlation matrices using a regression model in order to remove the effect of common market movements and their cleaned versions using random matrix theory this is done for years of both low and high volatility of the brazilian stock market from 2004 to 2012 the results show that the use of regression to subtract the market effect on returns greatly increases the accuracy of the prediction of risk and that although the cleaning of the correlation matrix often leads to portfolios that better predict risks in periods of high volatility of the market this procedure may fail to do so the results may be used in the assessment of the true risks when one builds a portfolio of stocks during periods of crisis

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

on the universality of the probability distribution of the product b 1x of random matrices consider random matrices a of dimension m x m n drawn from an ensemble with probability density f tr aa with f x a given appropriate function break a b x into an m x m block b and the complementary m x n block x and define the random matrix z b 1 x we calculate the probability density function p z of the random matrix z and find that it is a universal function independent of f x the universal probability distribution p z is a spherically symmetric matrix variate t distribution universality of p z is essentially a consequence of rotational invariance of the probability ensembles we study as an application we study the distribution of solutions of systems of linear equations with random coefficients and extend a classic result due to girko

on large values of weyl sums abstract a special case of the menshov rademacher theorem implies for almost all polynomials x 1 z x d z d r z of degree d for the weyl sums satisfy the upper bound n 1 n exp 2 i x 1 n x d n d n 1 2 o 1 n here we investigate the exceptional sets of coefficients x 1 x d with large values of weyl sums for infinitely many n and show that in terms of the baire categories and hausdorff dimension they are quite massive in particular of positive hausdorff dimension in any fixed cube inside of 0 1 d we also use a different technique to give similar results for sums with just one monomial x n d we apply these results to show that the set of poorly distributed modulo one polynomials is rather massive as well

the cubic case of the main conjecture in vinogradov s mean value theorem abstract we apply a variant of the multigrade efficient congruencing method to estimate vinogradov s integral of degree 3 for moments of order 2 s establishing strongly diagonal behaviour for 1 s 6 consequently the main conjecture is now known to hold for the first time in a case of degree exceeding 2

decouplings for surfaces in r4 abstract we prove a sharp decoupling for nondegenerate surfaces in r 4 this puts the progress in 1 on the lindelof hypothesis into a more general perspective

randomly generating portfolio selection covariance matrices with specified distributional characteristics abstract in portfolio selection there is often the need for procedures to generate realistic covariance matrices for security returns for example to test and benchmark optimization algorithms for application in portfolio optimization such a procedure should allow the entries in the matrices to have distributional characteristics which we would consider realistic for security returns deriving motivation from the fact that a covariance matrix can be viewed as stemming from a matrix of factor loadings a procedure is developed for the random generation of covariance matrices a whose off diagonal covariance entries possess a pre specified expected value and standard deviation and b whose main diagonal variance entries possess a likely different pre specified expected value and standard deviation the paper concludes with a discussion about the futility one would likely encounter if one simply tried to invent a valid covariance matrix in the absence of a procedure such as in this paper

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

vinogradov s mean value theorem via efficient congruencing ii we apply the efficient congruencing method to estimate vino gradov s integral for moments of order 2s with 1 6 s 6 k 2 1 thereby we show that quasi diagonal behaviour holds when s o k 2 we obtain near optimal estimates for 1 6 s 6 1 k 2 k and optimal estimates for s k 2 1 in this way we come half way to proving the main conjecture in two different directions there are consequences for estimates of weyl type and in several allied applications thus for example the anticipated asymptotic formula in waring s problem is established for sums of s kth powers of natural numbers whenever s 2k 2 2k 8 k 6 estimates stemming from vinogradov s mean value theorem deliver bounds for exponential sums of large degree both in mean and pointwise beyond the competence of alternate approaches the ubiquity of such exponential sums in analytic number theory in the analysis for example of the riemann zeta function in waring s problem and beyond accounts for the high pro file of vinogradov s methods in the associated literature in recent work we established a version of vinogradov s mean value theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible see 19 for systems of degree k previous estimates missed such a bound by a factor of order logk our earlier approach provides no upper bounds when the number of variables is smaller precluding the possibility of applications involving the finer features of these mean values our goal in this paper is to remedy this deficiency at the same time strengthening our previous conclusions it transpires that we are able to come within a hair s breadth of proving the main conjecture concerning vino gradov s mean value theorem in half of the basic interval of relevant moments such developments illustrate the flexibility of the new efficient congruencing method introduced in 19 we now introduce some notation when k n and r k define

universal and nonuniversal properties of cross correlations in financial time series we use methods of random matrix theory to analyze the cross correlation matrix c of stock price changes of the largest 1000 u s companies for the 2 year period 1994 1995 we find that the statistics of most of the eigenvalues in the spectrum of c agree with the predictions of random matrix theory but there are deviations for a few of the largest eigenvalues we find that c has the universal properties of the gaussian orthogonal ensemble of random matrices furthermore we analyze the eigenvectors of c through their inverse participation ratio and find eigenvectors with large ratios at both edges of the eigenvalue spectrum a situation reminiscent of localization theory results

emerging spectra of singular correlation matrices under small power map deformations correlation matrices are a standard tool in the analysis of the time evolution of complex systems in general and financial markets in particular yet most analysis assume stationarity of the underlying time series this tends to be an assumption of varying and often dubious validity the validity of the assumption improves as shorter time series are used if many time series are used this implies an analysis of highly singular correlation matrices we attack this problem by using the so called power map which was introduced to reduce noise its nonlinearity breaks the degeneracy of the zero eigenvalues and we analyze the sensitivity of the so emerging spectra to correlations this sensitivity will be demonstrated for uncorrelated and correlated wishart ensembles

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

on integer solutions of parsell vinogradov systems we prove a sharp upper bound on the number of integer solutions of the parsell vinogradov system in every dimension d ge 2

non hermitian random matrix theory summation of planar diagrams the single ring theorem and the disc annulus phase transition i review aspects of work done in collaboration with a zee and r scalettar 1997 nucl phys b 504 579 1997 nucl phys b 501 643 2001 j math phys 42 5718 on complex non hermitian random matrices i open by explaining why the bag of tools used regularly in analysing hermitian random matrices cannot be applied directly to analyse non hermitian matrices and then introduce the method of hermitization which solves this problem then for rotationally invariant ensembles i derive a master equation for the average density of eigenvalues in the complex plane in the limit of infinitely large matrices this is achieved by resumming all the planar diagrams which appear in the perturbative expansion of the hermitized green function remarkably this resummation can be carried out explicitly for any rotationally invariant ensemble i prove that in the limit of infinitely large matrices the shape of the eigenvalue distribution is either a disc or an annulus this is the celebrated single ring theorem which of these shapes is realized is determined by the parameters coupling constants which determine the ensemble by varying these parameters a phase transition may occur between the two possible shapes i briefly discuss the universal features of this transition as the analysis of this problem relies heavily on summation of planar feynman diagrams i make a special effort at presenting a pedagogical exposition of the diagrammatic method which some readers may find useful

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

a generalization of vinogradov s mean value theorem we obtain new upper bounds for the number of integral solutions of a complete system of symmetric equations which may be viewed as a multi dimensional version of the system considered in vinogradov s mean value theorem we then use these bounds to obtain weyl type estimates for an associated exponential sum in several variables finally we apply the hardy littlewood method to obtain asymptotic formulas for the number of solutions of the vinogradov type system and also of a related system connected to the problem of finding linear spaces on hypersurfaces

restriction estimates of e removal type for k th powers and paraboloids we obtain restriction estimates of varepsilon removal type for the set of k th powers of integers and for discrete d dimensional surfaces of the form begin aligned n 1 dots n d n 1 k cdots n d k n 1 dots n d leqslant n end aligned which we term k paraboloids for these surfaces we obtain a satisfying range of exponents for large values of d k we also obtain estimates of varepsilon removal type in the full supercritical range for k th powers and for k paraboloids of dimension d k k 2 we rely on a variety of techniques in discrete harmonic analysis originating in bourgain s works on the restriction theory of the squares and the discrete parabola

large rectangular random matrices models with a multiplet of field variables arranged into rectangular matrices in the limit of infinite dimensions of the matrices are studied in zero dimensional space where the problem is a combinatorial one a closed solution is given that improves the one previously known in arbitrary space dimension a symmetry is described that connects rectangular models with vector models

subconvexity for additive equations pairs of undenary cubic forms we investigate pairs of diagonal cubic equations with integral coefficients for a class of such diophantine systems with 11 or more variables we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude

on vinogradov s mean value theorem strongly diagonal behaviour via efficient congruencing we enhance the efficient congruencing method for estimating vinogradov s integral for moments of order 2s with 1 leqslant s leqslant k 2 1 in this way we prove the main conjecture for such even moments when 1 leqslant s leqslant tfrac 1 4 k 1 2 showing that the moments exhibit strongly diagonal behaviour in this range there are improvements also for larger values of s these finding application to the asymptotic formula in waring s problem

on the density of integer points on generalised markoff hurwitz and dwork hypersurfaces we use bounds of mixed character sums modulo a square free integer q of a special structure to estimate the density of integer points on the hypersurface begin aligned f 1 x 1 cdots f n x n a x 1 k 1 ldots x n k n end aligned for some polynomials f i in mathbb z x and nonzero integers a and k i i 1 ldots n in the case of begin aligned f 1 x cdots f n x x 2 quad text and quad k 1 cdots k n 1 end aligned the above hypersurface is known as the markoff hurwitz hypersurface while for begin aligned f 1 x cdots f n x x n quad text and quad k 1 cdots k n 1 end aligned it is known as the dwork hypersurface our results are substantially stronger than those known for general hypersurfaces

decouplings for three dimensional surfaces in mathbb r 6 r 6 we obtain the sharp l p decoupling for three dimensional nondegenerate surfaces in mathbb r 6 this can be thought of as a generalization of bourgain and demeter s result which is the sharp l p decoupling for two dimensional nondegenerate surfaces in mathbb r 4

random diophantine equations i abstract we consider additive diophantine equations of degree k in s variables and establish that whenever s 3 k 2 then almost all such equations satisfy the hasse principle the equations that are soluble form a set of positive density and among the soluble ones almost all equations admit a small solution our bound for the smallest solution is nearly best possible

decoupling for moment manifolds associated to arkhipov chubarikov karatsuba systems abstract we prove l p l p decoupling inequalities for a class of moment manifolds these inequalities imply optimal mean value estimates for multidimensional weyl sums of the kind considered by arkhipov chubarikov and karatsuba and by parsell in our proofs we take a new point of view on the bourgain demeter guth induction on scales argument this point of view substantially simplifies even the proof of l 2 l p decoupling for the moment curve

restricted weak type endpoint estimates for k spherical maximal functions in this paper we use the approximation formula for the fourier transform of the solution set of lattice points on k spheres and methods of bourgain and ionescu to refine the ell p mathbb z d boundedness results for discrete k spherical maximal functions to a restricted weak type result at the endpoint we introduce a density parameter which may be viewed as a discrete version of minkowski dimension used in related works on the continuous analgoue in order to exploit recent progress of wooley and bourgain demeter guth on the vinogradov mean value conjectures via a novel approximation formula for a single average and obtain improved bounds for lacunary discrete k spherical maximal functions when k ge 3

local normalization uncovering correlations in non stationary financial time series abstract the measurement of correlations between financial time series is of vital importance for risk management in this paper we address an estimation error that stems from the non stationarity of the time series we put forward a method to rid the time series of local trends and variable volatility while preserving cross correlations we test this method in a monte carlo simulation and apply it to empirical data for the s p 500 stocks

the asymptotic estimates and hasse principle for multidimensional waring s problem abstract motivated by the asymptotic estimates and hasse principle for multidimensional waring s problem via the circle method we prove for the first time that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption the number of variables we need is near optimal by proving a more general uniform density result over certain complete discrete valuation rings with finite residue fields we also establish uniform lower bounds for both singular series and singular integral in f q t we thus obtain asymptotic formulas and the hasse principle for multidimensional waring s problem in f q t via a variant of the circle method

further improvements in waring s problem 1 i n t r o d u c t i o n 147 2 i t e ra t ive schemes 149 3 e s t i m a t e s for the n u m b e r of so lu t ions of auxi l ia ry equa t ions 155 4 m a j o r and mino r arc e s t ima tes 168 5 t he i t e ra t ive scheme for fifth powers i 182 6 t h e i t e ra t ive scheme for fifth powers ii 185 7 t he proof of t h e o r e m 1 1 for fifth powers 191 8 t he i te ra t ive scheme for k exceeding 5 second differences 197 9 t he i t e ra t ive scheme for k exceeding 5 t h i r d differences 202 10 t he i t e ra t ive scheme for k exceeding 6 four th differences 206 11 t h e i t e ra t ive scheme for k exceeding 7 fifth and s ix th differences 212 12 t h e proof of t h e o r e m 1 1 for s ix th powers 216 13 t h e h a r d y l i t t l e w o o d dissec t ion for larger k 230 14 t h e p roof of t h e o r e m 1 1 for s even th powers 232 15 t he proof of t h e o r e m 1 1 for e igh th powers 234 16 t he proof of t h e o r e m 1 1 for n i n t h powers 234 append ix numer ica l values of p a r a m e t e r s 236 references 240

probability density of determinants of random matrices in this brief paper the probability density of a random real complex and quaternion determinant is rederived using the singular values the behaviour of suitably rescaled random determinants is studied in the limit of infinite order of the matrices

l p regularity of averages over curves and bounds for associated maximal operators we prove that for a finite type curve in r 3 the maximal operator generated by dilations is bounded on if for sufficiently large p we also show the endpoint lp l p i p regularity result for the averaging operators for large p the proofs make use of a deep result of thomas wolff about decompositions of cone multipliers

multigrade efficient congruencing and vinogradov s mean value theorem we develop a multigrade enhancement of the efficient congru encing method to estimate vinogradov s integral of degree k for moments of order 2s thereby obtaining near optimal estimates for 5 k 2 1 543k 2 the asymptotic formula is also established for sums of 28 fifth powers

quasi diagonal behaviour in certain mean value theorems of additive number theory of fundamental significance in many problems of additive number theory are estimates for mean values of exponential sums over polynomial functions in this paper we shall show that the exponential sums of greatest interest in additive number theory demonstrate quasi diagonal behaviour which is to say that by taking the degree of the polynomial argument sufficiently large we can obtain upper bounds for low moments of the exponential sums arbitrarily close in a suitable sense to the diagonal estimate let us illustrate these notions by considering the situation in vinogradov s mean value theorem let s and k be positive integers and write

additive representation in short intervals ii sums of two like powers we establish that for almost all natural numbers n there is a sum of two positive integral cubes lying in the interval n n 7 18 varepsilon n here the exponent 7 18 lies half way between the trivial exponent 4 9 stemming from the greedy algorithm and the exponent 1 3 constrained by the number of integers not exceeding x that can be represented as the sum of two positive integral cubes we also provide analogous conclusions for sums of two positive integral k th powers when k ge 4

rational decisions random matrices and spin glasses we consider the problem of rational decision making in the presence of nonlinear constraints by using tools borrowed from spin glass and random matrix theory we focus on the portfolio optimisation problem we show that the number of optimal solutions is generally exponentially large and each of them is fragile rationality is in this case of limited use in addition this problem is related to spin glasses with levy like long ranged couplings for which we show that the ground state is not exponentially degenerate

an analysis of cross correlations in an emerging market abstract we apply random matrix theory to compare correlation matrix estimators c obtained from emerging market data the correlation matrices are constructed from 10 years of daily data for stocks listed on the johannesburg stock exchange jse from january 1993 to december 2002 we test the spectral properties of c against random matrix predictions and find some agreement between the distributions of eigenvalues nearest neighbour spacings distributions of eigenvector components and the inverse participation ratios for eigenvectors we show that interpolating both missing data and illiquid trading days with a zero order hold increases agreement with rmt predictions for the more realistic estimation of correlations in an emerging market we suggest a pairwise measured data correlation matrix for the data set used this approach suggests greater temporal stability for the leading eigenvectors an interpretation of eigenvectors in terms of trading strategies is given as opposed to classification by economic sectors

vinogradov s integral and bounds for the riemann zeta function the main result is an upper bound for the riemann zeta function in the critical strip zeta sigma it le a t b 1 sigma 3 2 log 2 3 t with a 76 2 and b 4 45 valid for frac12 le sigma le 1 and t ge 3 the previous best constant b was 18 5 tools include a variant of the korobov vinogradov method of bounding exponential sums an explicit version of t d wooley s bounds for vinogradov s integral and explicit bounds for mean values of exponential sums over numbers without small prime factors also using methods of wooley an auxiliary result is the exponential sum bound s n t le 9 463 n 1 1 133 66 lambda 2 where n is a positive integer t is a real number lambda log t log n and s n t max 0 2000 mathematical subject classification primary 11m06 11n05 11l15 secondary 11d72 11m35