gaussian multiplicative chaos for symmetric isotropic matrices motivated by isotropic fully developed turbulence we define a theory of symmetric matrix valued isotropic gaussian multiplicative chaos our construction extends the scalar theory developed by j p kahane in 1985

asymptotics of unitary and orthogonal matrix integrals abstract in this paper we prove that in small parameter regions arbitrary unitary matrix integrals converge in the large n limit and match their formal expansion secondly we give a combinatorial model for our matrix integral asymptotics and investigate examples related to free probability and the hciz integral our convergence result also leads us to new results of smoothness of microstates we finally generalize our approach to integrals over the orthogonal group

construction of optimal order nonlinear solvers using inverse interpolation abstract there is a vast literature on finding simple roots of nonlinear equations by iterative methods these methods can be classified by order by the information used or by efficiency there are very few optimal methods that is methods of order 2 m requiring m 1 function evaluations per iteration here we give a general way to construct such methods by using inverse interpolation and any optimal two point method the presented optimal multipoint methods are tested on numerical examples and compared to existing methods of the same order of convergence

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

a family of three point methods of optimal order for solving nonlinear equations a family of three point iterative methods for solving nonlinear equations is constructed using a suitable parametric function and two arbitrary real parameters it is proved that these methods have the convergence order eight requiring only four function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 3 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class of root solvers possesses very high computational efficiency numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

universal tail profile of gaussian multiplicative chaos in this article we study the tail probability of the mass of gaussian multiplicative chaos with the novel use of a tauberian argument and goldie s implicit renewal theorem we provide a unified approach to general log correlated gaussian fields in arbitrary dimension and derive precise first order asymptotics of the tail probability resolving a conjecture of rhodes and vargas the leading order is described by a universal constant that captures the generic property of gaussian multiplicative chaos and may be seen as the analogue of the liouville unit volume reflection coefficients in higher dimensions

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

a new eighth order iterative method for solving nonlinear equations abstract in this paper we present an improvement of the fourth order newton type method for solving a nonlinear equation the new newton type method is shown to converge of the order eight per iteration the new method requires three evaluations of the function and one evaluation of its first derivative and therefore the new method has the efficiency index of 8 4 which is better than the well known newton type methods of lower order we shall examine the effectiveness of the new eighth order newton type method by approximating the simple root of a given nonlinear equation numerical comparisons are made with several other existing methods to show the performance of the presented method

a family of optimal sixteenth order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth step weighting function a new family of four step optimal multipoint iterative methods of order sixteen for solving nonlinear equations are developed along with their convergence properties numerical experiments with comparison to some existing methods are demonstrated to support the underlying theory

a semilocal convergence analysis for directional newton methods a semilocal convergence analysis for directional newton methods in n variables is provided in this study using weaker hypotheses than in the elegant related work by y levin and a ben israel and introducing the center lipschitz condition we provide under the same computational cost as in levin and ben israel a semilocal convergence analysis with the following advantages weaker convergence conditions larger convergence domain finer error estimates on the distances involved and an at least as precise information on the location of the zero of the function a numerical example where our results apply to solve an equation but not the ones in levin and ben israel is also provided in this study

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

an analysis of a new family of eighth order optimal methods a new family of eighth order optimal methods is developed and analyzed numerical experiments show that our family of methods perform well and in many cases some members are superior to other eighth order optimal methods it is shown how to choose the parameters to widen the basin of attraction

an extension of the harishchandra itzykson zuber integral the harishchandra itzykson zuber integral over the unitary group u k 2 is present in numerous problems involving hermitian random matrices it is well known that the result is semi classically exact this simple result does not extend to other symmetry groups such as the symplectic or orthogonal groups in this article the analysis of this integral is extended first to the symplectic group sp k 4 there the semi classical approximation has to be corrected by a wkb expansion it turns out that this expansion stops after a finite number of terms in other words the wkb approximation is corrected by a polynomial in the appropriate variables the analysis is based upon new solutions to the heat kernel differential equation we have also investigated arbitrary values of the parameter which characterizes the symmetry group closed formulae are derived for arbitrary and k 3 and also for large and arbitrary k

liouville metric of star scale invariant fields tails and weyl scaling we study the liouville metric associated to an approximation of a log correlated gaussian field with short range correlation we show that below a parameter gamma c 0 the left right length of rectangles for the riemannian metric e gamma phi 0 n ds 2 with various aspect ratio is concentrated with quasi lognormal tails that the renormalized metric is tight when gamma min gamma c 0 4 and that subsequential limits are consistent with the weyl scaling

on a general class of optimal order multipoint methods for solving nonlinear equations abstract we develop a class of n point iterative methods with optimal 2 n order of convergence for solving nonlinear equations newton s second order and ostrowski s fourth order methods are special cases corresponding to n 1 and n 2 eighth and sixteenth order methods that correspond to n 3 and n 4 of the class are special cases of the eighth and sixteenth order methods proposed by sharma et al 25 the methodology is based on employing the previously obtained n 1 step scheme and modifying the n th step by using rational hermite interpolation unlike that of existing higher order techniques the proposed technique is attractive since it leads to a simple implementation local convergence analysis is provided to show that the iterations are locally well defined and convergent theoretical results are verified through numerical experimentations the performance is also compared with already established methods in literature it is observed that new algorithms are more accurate than existing counterparts and very effective in high precision computations

on a general class of multipoint root finding methods of high computational efficiency a general class of n point iterative methods for solving nonlinear equations is constructed by combining methods of newton s type and an arbitrary two point method of the fourth order of convergence it is proved that these methods have the convergence order 2 n requiring only n 1 function evaluations per iteration in this way it is demonstrated that the proposed class of methods supports the kung traub hypothesis 1974 on the upper bound 2 n of the order of multipoint methods based on n 1 function evaluations consequently this class possess as high as possible computational efficiency in the sense of the kung traub hypothesis numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations

statistical properties of the volatility of price fluctuations we study the statistical properties of volatility measured by locally averaging over a time window t the absolute value of price changes over a short time interval ensuremath delta t we analyze the s 500 stock index for the 13 year period jan 1984 to dec 1996 we find that the cumulative distribution of the volatility is consistent with a power law asymptotic behavior characterized by an exponent ensuremath mu ensuremath approx 3 similar to what is found for the distribution of price changes the volatility distribution retains the same functional form for a range of values of t further we study the volatility correlations by using the power spectrum analysis both methods support a power law decay of the correlation function and give consistent estimates of the relevant scaling exponents also both methods show the presence of a crossover at approximately 1 5 days in addition we extend these results to the volatility of individual companies by analyzing a data base comprising all trades for the largest 500 u s companies over the two year period jan 1994 to dec 1995

the witten complex and the degenerate morse inequalities on utilise le complexe introduit par witten pour demontrer des inegalites de morse dans le cas ou les points critiques de la fonction de morse sont isoles et aussi les inegalites de morse degenerees de bott

scaling exponents and multifractal dimensions for independent random cascades this paper is concerned with mandelbrot s stochastic cascade measures the problems of i scaling exponents of structure functions of the measure q and ii multifractal dimensions are considered for cascades with a generator vector w1 wc of the general type these problems were previously studied for independent strongly bounded variableswi 0 a wi c consequently a broad class of models used in applications including kolmogorov s long normal model in turbulence long stable universal cascades in atmospheric dynamics has not been covered roughly speaking problems i ii are here solved under the condition that the scaling exists the function is calculated for all arguments previously this was done for positiveq and a new effect emerges the function can generally involve discontinuities in the first derivative as well as in the second

the fixed points of the multivariate smoothing transform let mathbf t 1 mathbf t 2 ldots be a sequence of random d times d matrices with nonnegative entries and let q be a random vector with nonnegative entries consider random vectors x with nonnegative entries satisfying begin aligned x mathop limits mathcal l sum i ge 1 mathbf t i x i q end aligned where mathop limits mathcal l denotes equality of the corresponding laws x i i ge 1 are i i d copies of x and independent of q mathbf t 1 mathbf t 2 ldots for d 1 this equation known as fixed point equation of the smoothing transform has been intensively studied under assumptions similar to the one dimensional case we obtain a complete characterization of all solutions x to in the non critical case and existence results in the critical case

a stochastic approach to the poincare hopf theorem in this paper the poincare hopf theorem hopf 8 is proved by a probabilistic method the proof of this theorem is reduced to an estimate of a supertrace of the fundamental solution we use the malliavian calculus to estimate the fundametal solution our proof can be applied to the case that the zero point set is the union of the submanifolds under some conditions

a family of optimal three point methods for solving nonlinear equations using two parametric functions abstract using an interactive approach which combines symbolic computation and taylor s series a wide family of three point iterative methods for solving nonlinear equations is constructed these methods use two suitable parametric functions at the second and third step and reach the eighth order of convergence consuming only four function evaluations per iteration this means that the proposed family supports the kung traub hypothesis 1974 on the upper bound 2 m of the order of multipoint methods based on m 1 function evaluations providing very high computational efficiency different methods are obtained by taking specific parametric functions the presented numerical examples demonstrate exceptional convergence speed with only few function evaluations

gaussian multiplicative chaos revisited in this article we extend the theory of multiplicative chaos for positive definite functions in rd of the form f x 2 ln t x g x where g is a continuous and bounded function the construction is simpler and more general than the one defined by kahane in 1985 as main application we give a rigorous mathematical meaning to the kolmogorov obukhov model of energy dissipation in a turbulent flow

a glimpse of the conformal structure of random planar maps we present a way to study the conformal structure of random planar maps the main idea is to explore the map along an sle schramm loewner evolution process of parameter kappa 6 and to combine the locality property of the sle6 together with the spatial markov property of the underlying lattice in order to get a non trivial geometric information we follow this path in the case of the conformal structure of random triangulations with a boundary

the atiyah singer theorems a probabilistic approach i the index theorem abstract the atiyah singer index theorem for classical elliptic complexes is proved by using probabilistic methods the general idea is to use a probabilistic construction of the heat equation kernel which permits the direct derivation of the index formulas without using the theory of invariants of gilkey stochastic calculus on the exterior algebra is then used to find the classical local formula for the index theorem the lefschetz fixed point formulas of atiyah singer will be proved in part ii

morse type inequalities for dynamical systems and the witten laplacian we provide an analytic proof of morse type inequalities for vector flelds determining a morse decomposition with normally hyperbolic dynamics in the demonstration we reduce the problem to the gradient case using a morse bott lyapunov function and

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

subcritical multiplicative chaos for regularized counting statistics from random matrix theory for an n times n haar distributed random unitary matrix u n we consider the random field defined by counting the number of eigenvalues of u n in a mesoscopic arc centered at the point u on the unit circle we prove that after regularizing at a small scale epsilon n 0 the renormalized exponential of this field converges as n to infty to a gaussian multiplicative chaos measure in the whole subcritical phase we discuss implications of this result for obtaining a lower bound on the maximum of the field we also show that the moments of the total mass converge to a selberg like integral and by taking a further limit as the size of the arc diverges we establish part of the conjectures in ostrovsky nonlinearity 29 2 426 464 2016 by an analogous construction we prove that the multiplicative chaos measure coming from the sine process has the same distribution which strongly suggests that this limiting object should be universal our approach to the l1 phase is based on a generalization of the construction in berestycki electron commun probab 22 27 12 2017 to random fields which are only asymptotically gaussian in particular our method could have applications to other random fields coming from either random matrix theory or a different context

continuity and estimates of the liouville heat kernel with applications to spectral dimensions the liouville brownian motion lbm recently introduced by garban rhodes and vargas and in a weaker form also by berestycki is a diffusion process evolving in a planar random geometry induced by the liouville measure m gamma formally written as m gamma dz e gamma x z gamma 2 mathbb e x z 2 2 dz gamma in 0 2 for a massive gaussian free field x it is an m gamma symmetric diffusion defined as the time change of the two dimensional brownian motion by the positive continuous additive functional with revuz measure m gamma in this paper we provide a detailed analysis of the heat kernel p t x y of the lbm specifically we prove its joint continuity a locally uniform sub gaussian upper bound of the form p t x y le c 1 t 1 log t 1 exp bigl c 2 x y beta wedge 1 t frac 1 beta 1 bigr for t in bigl 0 frac 1 2 bigr for each beta frac 1 2 gamma 2 2 and an on diagonal lower bound of the form p t x x ge c 3 t 1 bigl log t 1 bigr eta for t in 0 t eta x with t eta x in bigl 0 frac 1 2 bigr heavily dependent on x for each eta 18 for m gamma almost every x as applications we deduce that the pointwise spectral dimension equals 2 m gamma a e and that the global spectral dimension is also 2

some properties of angular integrals we find new representations for itzykson zuber like angular integrals for arbitrary in particular for the orthogonal group o n the unitary group u n and the symplectic group sp 2n we rewrite the haar measure integral as a flat lebesge measure integral and we deduce some recursion formulae on n the same methods give also shatashvili s type moments finally we prove that in agreement with brezin and hikami s observation the angular integrals are linear combinations of exponentials whose coefficients are polynomials in the reduced variables xi xj yi yj

intermittent process analysis with scattering moments scattering moments provide nonparametric models of random processes with stationary increments they are expected values of random variables computed with a nonexpansive operator obtained by iteratively applying wavelet transforms and modulus nonlinearities which preserves the variance first and second order scattering moments are shown to characterize intermittency and self similarity properties of multiscale processes scattering moments of poisson processes fractional brownian motions levy processes and multifractal random walks are shown to have characteristic decay the generalized method of simulated moments is applied to scattering moments to estimate data generating models numerical applications are shown on financial time series and on energy dissipation of turbulent flows

lognormal star scale invariant random measures in this article we consider the continuous analog of the celebrated mandelbrot star equation with lognormal weights mandelbrot introduced this equation to characterize the law of multiplicative cascades we show existence and uniqueness of measures satisfying the aforementioned continuous equation these measures fall under the scope of the gaussian multiplicative chaos theory developed by kahane in 1985 or possibly extensions of this theory as a by product we also obtain an explicit characterization of the covariance structure of these measures we also prove that qualitative properties such as long range independence or isotropy can be read off the equation

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

multifractal random walk we introduce a class of multifractal processes referred to as multifractal random walks mrws to our knowledge it is the first multifractal process with continuous dilation invariance properties and stationary increments mrws are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio the mrws are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments we briefly explain how in the same way one can build stationary multifractal processes or positive random measures

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

three step iterative methods with optimal eighth order convergence in this paper based on ostrowski s method a new family of eighth order methods for solving nonlinear equations is derived in terms of computational cost each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative so that their efficiency indices are 1 682 which is optimal according to kung and traub s conjecture numerical comparisons are made to show the performance of the new family