on devising boussinesq type models with bounded eigenspectra the propagation of water waves in the nearshore region can be described by depth integrated boussinesq type equations the dispersive and nonlinear characteristics of the equations are governed by tuneable parameters we examine the associated linear eigenproblem both analytically and numerically using a spectral element method of arbitrary spatial order p it is shown that existing sets of parameters found by optimising the linear dispersion relation give rise to unbounded eigenspectra which govern stability for explicit time stepping schemes the global cfl time step restriction typically requires t p 2 we derive and present conditions on the parameters under which implicitly implicit boussinesq type equations will exhibit bounded eigenspectra two new bounded versions having comparable nonlinear and dispersive properties as the equations of nwogu 1993 and schaffer and madsen 1995 are introduced using spectral element simulations of stream function waves it is illustrated that i the bounded equations capture the physics of the wave motion as well as the standard unbounded equations and ii the bounded equations are computationally more efficient when explicit time stepping schemes are used thus the bounded equations were found to lead to more robust and efficient numerical schemes without compromising the accuracy

atoms and regularity for measures in a partially defined free convolution semigroup consider a borel probability measure on the real line and denote by t t 1 the free additive convolution semigroup defined by nica and speicher we show that the singular part of t is purely atomic and the density of t is locally analytic provided that t 1 the main ingredient is a global inversion theorem for analytic functions on a half plane

the measurable kesten theorem we give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d regular graphs it follows that the a finite d regular ramanujan graph g contains a negligible number of cycles of size less than c log log g we prove that infinite d regular ramanujan unimodular random graphs are trees through benjamini schramm convergence this leads to the following rigidity result if most eigenvalues of a d regular finite graph g fall in the alon boppana region then the eigenvalue distribution of g is close to the spectral measure of the d regular tree in particular g contains few short cycles in contrast we show that d regular unimodular random graphs with maximal growth are not necessarily trees institute of mathematical statistics 2016

explicit combinatorial interpretation of kerov character polynomials as numbers of permutation factorizations abstract we find an explicit combinatorial interpretation of the coefficients of kerov character polynomials which express the value of normalized irreducible characters of the symmetric groups s n in terms of free cumulants r 2 r 3 of the corresponding young diagram our interpretation is based on counting certain factorizations of a given permutation

jack polynomials and free cumulants abstract we study the coefficients in the expansion of jack polynomials in terms of power sums we express them as polynomials in the free cumulants of the transition measure of an anisotropic young diagram we conjecture that such polynomials have nonnegative integer coefficients this extends recent results about normalized characters of the symmetric group

asymptotic expansions in the clt in free probability we prove edgeworth type expansions for distribution functions of sums of free random variables under minimal moment conditions the proofs are based on the analytic definition of free convolution

fast exact multiplication by the hessian just storing the hessian h the matrix of second derivatives 2e wi wj of the error e with respect to each pair of weights of a large neural network is difficult since a common use of a large matrix like h is to compute its product with various vectors we derive a technique that directly calculates hv where v is an arbitrary vector to calculate hv we first define a differential operator rv f w r f w rv r 0 note that rv w hv and rv w v and then apply rv to the equations used to compute w the result is an exact and numerically stable procedure for computing hv which takes about as much computation and is about as local as a gradient evaluation we then apply the technique to a one pass gradient calculation algorithm backpropagation a relaxation gradient calculation algorithm recurrent backpropagation and two stochastic gradient calculation algorithms boltzmann machines and weight perturbation finally we show that this technique can be used at the heart of many iterative techniques for computing various properties of h obviating any need to calculate the full hessian

free probability for pairs of faces i we consider a notion of bi freeness for systems of non commutative random variables with two faces one of left variables and another of right variables this includes bi free convolution operations bi free cumulants the bi free central limit and bi freeness with amalgamation over an algebra b

multiplicative functions on the lattice of non crossing partitions and free convolution the lattice of non crossing partitions was introduced by kreweras kre in 1972 and examined further by poupard pou and edelman edel ede2 in the last time there has been an increased interest in this lattice on one hand from the purely combinatorial point of view siu eds sire bss and on the other hand from a quantum probabilistic point of view gss spel spe2 spe3 we became interested in this lattice when we noticed that it is connected with some new form of convolution for probability measures the free convolution introduced by voiculescu voil voi2 voi3 this connection between the free convolution and the lattice of non crossing partitions is exactly analogous to the connection between the usual convolution and the lattice of all partitions we shall work out this connection in the last part of this work the intention of this article is to show that both the purely combinatorial and the probabilistic point of view can benefit from each other motivated by voiculescu s formula for the r series of free convolution we examine multiplicative functions on the lattice of non crossing partitions and prove our main theorem which is much in the spirit of voiculescu s formula we shall then use this main theorem for deriving some known results on non crossing partitions in a unified way this part of our work is purely combinatorial the free convolution serves only as a motivation for our theorem in the last section we shall switch to the probabilistie side and show how the lattice of non crossing partitions determines the structure of the free convolution voiculescu s formula will then follow as an easy corollary of our main theorem in this respect we shall get a purely combinatorial proof of this formula without any operator algebraic or functional analytic tools

relations between cumulants in noncommutative probability abstract we express classical free boolean and monotone cumulants in terms of each other using combinatorics of heaps pyramids tutte polynomials and permutations we completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed

eigenvalue distributions of sums and products of large random matrices via incremental matrix expansions this paper uses an incremental matrix expansion approach to derive asymptotic eigenvalue distributions a e d s of sums and products of large random matrices we show that the result can be derived directly as a consequence of two common assumptions and matches the results obtained from using and transforms in free probability theory we also give a direct derivation of the a e d of the sum of certain random matrices which are not free this is used to determine the asymptotic signal to interference ratio of a multiuser code division multiple access cdma system with a minimum mean square error linear receiver

an analogue of hincin s characterization of infinite divisibility for operator valued free probability abstract let b be a finite separable von neumann algebra we prove that a b valued distribution that is the weak limit of an infinitesimal array is infinitely divisible the proof of this theorem utilizes the steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories we also develop weak topologies for this theory and prove the corresponding compactness and convergence results

on the limit distributions of random matrices with independent or free entries we state limit distribution results for random matrices with independent or free entries also addressing when we get freeness in the limit and semicircular and circular limits the results generalize some already known results about asymptotic freeness of large random matrices but our goal is to get a more optimal flavour on these results when having matrices with identically distributed entries we show that freeness in the limit is natural when we have free entries but unnatural when the entries are independent and restricted to the case of circular limits

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

asymptotic mean density of sub unitary ensembles the asymptotic behaviour of the mean spectral density for the product of n n random unitary matrices distributed according to the haar measure and a fixed diagonal matrix is investigated the large n limit of the exact expression derived by wei and fyodorov 2008 j phys a math theor 41 502001 is calculated by a modification of the saddle point method it is shown that in the bulk the result coincides with the one obtained within the free probability theory by haagerup and larsen 2000 j funct anal 176 331 close to the edge points of the bulk asymptotics the mean density for large n is described by a universal function depending on a certain scaling variable the large deviation formulae valid outside the bulk are also derived obtained formulae agree well with the results of direct numerical calculations

a one parameter family of transforms linearizing convolution laws for probability distributions we study a family of transforms depending on a parameterq 0 1 which interpolate in an algebraic framework between a relative namely iz log iz of the logarithm of the fourier transform for probability distributions and its free analogue constructed by d voiculescu 16 17 the classical case corresponds toq 1 and the free one toq 0

free point processes and free extreme values we continue here the study of free extreme values begun in ben arous and voiculescu ann probab 34 2037 2059 2006 we study the convergence of the free point processes associated with free extreme values to a free poisson random measure voiculescu in lecture notes in mathematics springer heidelberg pp 279 349 1998 barndorff nielsen and thorbjornsen in probab theory relat fields 131 197 228 2005 we relate this convergence to the free extremal laws introduced in ben arous and voiculescu ann probab 34 2037 2059 2006 and give the limit laws for free order statistics

cogrowth and amenability of discrete groups abstract let g be a group and g1 gt a set of generators there are approximately 2t 1 n reduced words in g1 gt of length n let gg n be the number of those which represent 1g we show that lim n gg n 1 n exists clearly 1 2t 1 log log 2t 1 is the cogrowth 0 1 in fact 0 1 2 1 the entropic dimension of g is shown to be 1 it is then proved that d g 1 if and only if g is free on g1 gt and d g 0 if and only if g is amenable

reinforcement learning in continuous time and space this article presents a reinforcement learning framework for continuous time dynamical systems without a priori discretization of time state and action based on the hamilton jacobi bellman hjb equation for infinite horizon discounted reward problems we derive algorithms for estimating value functions and improving policies with the use of function approximators the processof value function estimation is formulated asthe minimization of a continuous time form of the temporal difference td error update methods based on backward euler approximation and exponential eligibility traces are derived and their correspondences with the conventional residual gradient td 0 and td lambda algorithms are shown for policy improvement two methods a continuous actor critic method and a value gradient based greedy policy are formulated as a special case of the latter a nonlinear feedback control law using the value gradient and the model of the input gain is derived the advantage updating a model free algorithm derived previously is also formulated in the hjbbased framework the performance of the proposed algorithms is first tested in a nonlinear control task of swinging a pendulum up with limited torque it is shown in the simulations that 1 the task is accomplished by the continuous actor critic method in a number of trials several times fewer than by the conventional discrete actor critic method 2 among the continuous policy update methods the value gradient based policy with a known or learned dynamic model performs several times better than the actor critic method and 3 a value function update using exponential eligibility traces is more efficient and stable than that based on euler approximation the algorithms are then tested in a higher dimensional task cartpole swing up this task is accomplished in several hundred trials using the value gradient based policy with a learned dynamic model

superconvergence to the central limit and failure of the cramer theorem for free random variables we show that convergence of the semicircle law in the free central limit theorem for bounded random variables is much better than expected thus the distributions which tend to the semicircle become absolutely continuous in finite time and the densities converge in a very strong sense we also show that the semicircle law is the free convolution of laws which are not semicircular thus proving that cramer s classical result for the normal distribution does not have a free counterpart

the levy ito decomposition in free probability in this paper we prove the free analog of the levy ito decomposition for levy processes a significant part of the proof consists of introducing free poisson random measures proving their existence and developing a theory of integration with respect to such measures the existence of free poisson random measures also yields via the free levy ito decomposition an alternative proof of the general existence of free levy processes in law

spectral hp discontinuous galerkin methods for modelling 2d boussinesq equations we present spectral hp discontinuous galerkin methods for modelling weakly nonlinear and dispersive water waves described by a set of depth integrated boussinesq equations on unstructured triangular meshes when solving the equations two different formulations are considered directly solving the coupled momentum equations and the scalar method in which a wave continuity equation is solved as an intermediate step we demonstrate that the approaches are fully equivalent and give identical results in terms of accuracy convergence and restriction on the time step however the scalar method is shown to be more cpu efficient for high order expansions in addition to requiring less storage

a double layer boussinesq type model for highly nonlinear and dispersive waves we derive and analyse in the framework of the mild slope approximation a new double layer boussinesq type model that is linearly and nonlinearly accurate up to deep water assuming the flow to be irrotational we formulate the problem in terms of the velocity potential thereby lowering the number of unknowns the model derivation combines two approaches namely the method proposed by agnon et al agnon et al 1999 j fluid mech 399 319 333 and enhanced by madsen et al madsen et al 2003 proc r soc lond a 459 1075 1104 which consists of constructing infinite series taylor solutions to the laplace equation to truncate them at a finite order and to use pade approximants and the double layer approach of lynett liu lynett liu 2004 a proc r soc lond a 460 2637 2669 which allows lowering the order of derivatives we formulate the model in terms of a static dirichlet neumann operator translated from the free surface to the still water level and we derive an approximate inverse of this operator that can be built once and for all the final model consists of only four equations both in one and two horizontal dimensions and includes only second order derivatives which is a major improvement in comparison with so called high order boussinesq models a linear analysis of the model is performed and its properties are optimized using a free parameter determining the position of the interface between the two layers excellent dispersion and shoaling properties are obtained allowing the model to be applied up to the deep water value k h 10 finally numerical simulations are performed to quantify the nonlinear behaviour of the model and the results exhibit a nonlinear range of validity reaching at least k h 3

gaussian fluctuations of characters of symmetric groups and of young diagrams we study asymptotics of reducible representations of the symmetric groups s q for large q we decompose such a representation as a sum of irreducible components or alternatively young diagrams and we ask what is the character of a randomly chosen component or what is the shape of a randomly chosen young diagram our main result is that for a large class of representations the fluctuations of characters and fluctuations of the shape of the young diagrams are asymptotically gaussian in this way we generalize kerov s central limit theorem the considered class consists of representations for which the characters almost factorize and this class includes for example the left regular representation plancherel measure irreducible representations and tensor representations this class is also closed under induction restriction outer product and tensor product of representations our main tool in the proof is the method of genus expansion well known from the random matrix theory

two parameter families of quantum symmetry groups we introduce and study natural two parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups specifically for each pair p q of non negative integers we define and investigate quantum groups o p q b p q s p q and h p q corresponding to respectively orthogonal groups bistochastic groups symmetric groups and hyperoctahedral groups in the first three cases the new quantum groups turn out to be related to the dual free products of free quantum groups studied earlier for h p q the situation is different and we show that h p 0 qiso fp where the latter can be viewed as a liberation of the classical isometry group of the p dimensional torus

eigenvectors and eigenvalues in a random subspace of a tensor product given two positive integers n and k and a parameter t 0 1 we choose at random a vector subspace v n k n of dimension n tnk we show that the set of k tuples of singular values of all unit vectors in v n fills asymptotically as n tends to infinity a deterministic convex set k k t that we describe using a new norm in k

classical and quantum probability we follow the development of probability theory from the beginning of the last century emphasizing that quantum theory is really a generalization of this theory the great achievements of probability theory such as the theory of processes generalized random fields estimation theory and information geometry are reviewed their quantum versions are then described

levy type characterization of stable laws for free random variables we give a description of stable probability measures relative to free additive convolution the definition of domain of attraction is given and a proof is provided of the noncommutative analogue of levy theorem

commutators of free random variables let a be a unital c algebra given together with a specified state a c consider two selfadjoint elements a b of a which are free with respect to in the sense of the free probability theory of voiculescu let us denote c i ab ba where the i in front of the commutator is introduced to make c selfadjoint in this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b some properties of the corresponding operation on probability measures are also discussed the methods we use are combinatorial based on the description of freeness in terms of non crossing partitions an important ingredient is the notion of r diagonal pair introduced and studied in our previous paper 12

solitary wave transformation on the underwater step asymptotic theory and numerical experiments abstract the transformation of a solitary wave on an underwater step is studied analytically and numerically the theoretical model includes the linear potential description of the wave transformation on a step and the weakly nonlinear theory of long waves based on the korteweg de vries equation for reflected and transmitted waves far from a step numerical simulation of solitary wave transformation on an underwater step is performed in the framework of an extended 1d boussinesq like system and fully nonlinear fully dispersive 2d navier stokes equations the results of numerical simulations for the incident solitary wave of weak amplitude are in agreement with the theoretical predictions for the wave shapes of the secondary solitons but not with the predictions for travel times

chordal loewner families and univalent cauchy transforms we study chordal loewner families in the upper half plane and show that they have a parametric representation we show one that to every chordal loewner family there corresponds a unique measurable family of probability measures on the real line and two that to every measurable family of probability measures on the real line there corresponds a unique chordal loewner family in both cases the correspondence is being given by solving the chordal loewner equation we use this to show that any probability measure on the real line with finite variance and mean zero has univalent cauchy transform if and only if it belongs to some chordal loewner family if the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the cauchy transform the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal cauchy transform and the second in terms of moment inequalities corresponding to the grunsky inequalities

an explicit hybridized discontinuous galerkin method for serre green naghdi wave model abstract simulation of coastal water waves requires a relevant mathematical model and a proper numerical technique which can be used in irregular geometries in this paper we study the numerical solution to a class of irrotational green naghdi gn equation using the hybridized discontinuous galerkin method hdg we first use strang splitting to decompose this problem into a hyperbolic and a dispersive sub problem for the hyperbolic sub problem we use an explicit hdg with lax friedrichs numerical fluxes we also include different types of boundary conditions in our formulation similar to other hdg implementations our global unknowns are the numerical traces i e the unknowns on the mesh skeleton this choice can significantly reduce the number of global degrees of freedom solving this part of the problem involves newton iterations at each time step however unlike the implicit versions of hdg the element equations are decoupled and we do not need to solve a large global system of equations on the other hand for the dispersive sub problem we need to solve a global system of equations implicitly since hdg only involves unknowns on the mesh skeleton it can be a very efficient alternative for this implicit solve step by coupling these two solvers we obtain the solution to the green naghdi equation next we use this technique to solve a set of verification and validation examples in the first example we show the convergence properties of the numerical method next we compare our results with a set of experimental data for the nonlinear water waves in different situations finally we present a two dimensional simulation result for a solitary wave passing over a cosine bump we observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data

a two layer approach to wave modelling a set of model equations for water wave propagation is derived by piecewise integration of the primitive equations of motion through two arbitrary layers within each layer an independent velocity profile is derived with two separate velocity profiles matched at the interface of the two layers the resulting set of equations has three free parameters allowing for an optimization with known analytical properties of water waves the optimized model equations show good linear wave characteristics up to kh 6 while the second order nonlinear behaviour is captured to kh 6 as well a numerical algorithm for solving the model equations is developed and tested against one and two horizontal dimension cases agreement with laboratory data is excellent

limit distributions of random matrices abstract we study limit distributions of independent random matrices as well as limit joint distributions of their blocks under normalized partial traces composed with classical expectation in particular we are concerned with the ensemble of symmetric blocks of independent hermitian random matrices which are asymptotically free asymptotically free from diagonal deterministic matrices and whose norms are uniformly bounded almost surely this class contains symmetric blocks of unitarily invariant hermitian random matrices whose asymptotic distributions are compactly supported probability measures on the real line our approach is based on the concept of matricial freeness which is a generalization of freeness in free probability we show that the associated matricially free gaussian operators provide a unified framework for studying the limit distributions of sums and products of independent rectangular random matrices including non hermitian gaussian matrices and matrices of wishart type

convergence rates for weighted sums in noncommutative probability space abstract we study convergence rates for weighted sums of pairwise independent random variables in a noncommutative probability space of which the weights are in a von neumann algebra as applications we first study convergence rates for weighted sums of random variables in the noncommutative lorentz space and second we study convergence rates for weighted sums of probability measures with respect to the free additive convolution

ito formula for free stochastic integrals abstract the objects under investigation are the stochastic integrals with respect to free levy processes we define such integrals for square integrable integrands as well as for a certain general class of bounded integrands using the product form of the ito formula we prove the full functional ito formula in this context

low complexity polynomial expansion detector with deterministic equivalents of the moments of channel gram matrix for massive mimo uplink we consider a low complexity polynomial expansion pe detector in a massive multiple input multiple output mimo uplink channel in contrast to most massive mimo systems in the literature where single antenna user equipments ues are assumed multiple antenna ues are employed in this paper moreover the channel between a base station bs and a ue is a jointly correlated rician fading channel the pe detector reduces the computational complexity of the minimum mean square error mmse detector by replacing the matrix inversion with an approximate matrix polynomial the coefficients of the approximate matrix polynomial are computed from the deterministic equivalents of the moments of the channel gram matrix we use operator valued free probability which is a more general version of free probability to derive the deterministic equivalents in particular we use the operator valued moment cumulant formula the proposed low complexity pe detector is easy to compute simulation results show that the proposed detector can achieve performance close to the mmse detector

cumulants for random matrices as convolutions on the symmetric group in this paper we show that free cumulants can be naturally seen as the limiting value of cumulants of matrices we define these objects as functions on the symmetric group by some convolution relations involving the generalized moments we state that some characteristic properties of the free cumulants already hold for these cumulants

r transforms of free joint distributions and non crossing partitions abstract we introduce an extension of the notion of r transform defined by d voiculescu to free joint distributions i e normalized linear functionals on algebras of non commutative polynomials in several indeterminates we point out that the r transform has good behavior with respect to the operations of free product and free convolution of joint distributions we prove that the explicit computation of the inverse r transform of a joint distribution is done via a formula of summation over the lattice of non crossing partitions which shows that the r transform is the operator theoretic counterpart of the free cumulants considered in the combinatorial approach of r speicher moreover in connection to the equivariance of the multidimensional r transform under rotations we show that a natural free analogue of a classical result about rotations of independent random variables is holding

practical use of variational principles for modeling water waves this paper describes a method for deriving approximate equations for irrotational water waves the method is based on a relaxed variational principle i e on a lagrangian involving as many variables as possible this formulation is particularly suitable for the construction of approximate water wave models since it allows more freedom while preserving the variational structure the advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters as well as arbitrary depths using subordinate constraints e g irrotationality or free surface impermeability in various combinations several model equations are derived some being well known other being new the models obtained are studied analytically and exact travelling wave solutions are constructed when possible

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

free stein kernels and an improvement of the free logarithmic sobolev inequality we introduce a free version of the stein kernel relative to a semicircular law we use it to obtain a free counterpart of the hsi inequality of ledoux peccatti and nourdin which is an improvement of the free logarithmic sobolev inequality of biane and speicher as well as a rate of convergence in the multivariate entropic free central limit theorem we also compute the free stein kernels for several relevant families of self adjoint operators

quantum bernstein s theorem and the hyperoctahedral quantum group we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature we study an extension of bernstein theorem to the setting of quantum groups for a d tuple of free identically distributed random variables we consider a problem of preservation of freeness under the transformations from the free orthogonal quantum group for a subset not contained in the hyperoctahedral quantum group we prove that preservation of freeness characterizes wigner s semicircle law we show that freeness is always preserved if the quantum subset is contained in the hyperoctahedral quantum group we provide examples of quantum subsets which show that our result is an extension of results known in the literature

a splitting approach for the fully nonlinear and weakly dispersive green naghdi model the fully nonlinear and weakly dispersive green naghdi model for shallow water waves of large amplitude is studied the original model is first recast under a new formulation more suitable for numerical resolution an hybrid finite volume and finite difference splitting approach is then proposed which could be adapted to many physical models that are dispersive corrections of hyperbolic systems the hyperbolic part of the equations is handled with a high order finite volume scheme allowing for breaking waves and dry areas the dispersive part is treated with a classical finite difference approach extensive numerical validations are then performed in one horizontal dimension relying both on analytical solutions and experimental data the results show that our approach gives a good account of all the processes of wave transformation in coastal areas shoaling wave breaking and run up

a unified fluctuation formula for one cut β ensembles of random matrices using a coulomb gas approach we compute the generating function of the covariances of power traces for one cut ensembles of random matrices in the limit of large matrix size this formula depends only on the support of the spectral density and is therefore universal for a large class of models this allows us to derive a closed form expression for the limiting covariances of an arbitrary one cut ensemble as particular cases of the main result we consider the classical gaussian wishart and jacobi ensembles for which we derive previously available results as well as new ones within a unified simple framework we also discuss the connections between the problem of trace fluctuations for the gaussian unitary ensemble and the enumeration of planar maps

a fourier view on the r transform and related asymptotics of spherical integrals abstract we estimate the asymptotics of spherical integrals of real symmetric or hermitian matrices when the rank of one matrix is much smaller than its dimension we show that it is given in terms of the r transform of the spectral measure of the full rank matrix and give a new proof of the fact that the r transform is additive under free convolution these asymptotics also extend to the case where one matrix has rank one but complex eigenvalue a result related with the analyticity of the corresponding spherical integrals

successive galerkin approximation algorithms for nonlinear optimal and robust control nonlinear optimal control and nonlinear h infinity control are two of the most significant paradigms in nonlinear systems theory unfortunately these problems require the solution of hamilton jacobi equations which are extremely difficult to solve in practice to make matters worse approximation techniques for these equations are inherently prone to the so called curse of dimensionality while there have been many attempts to approximate these equations solutions resulting in closed loop control with well defined stability and robustness have remained elusive this paper describes a recent breakthrough in approximating the hamilton jacobi bellman and hamilton jacobi isaacs equations successive approximation and galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing closed loop control laws with well defined stability regions in addition we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po

an efficient hyperbolic relaxation system for dispersive non hydrostatic water waves and its solution with high order discontinuous galerkin schemes abstract in this paper we propose a novel set of first order hyperbolic equations that can model dispersive non hydrostatic free surface flows the governing pde system is obtained via a hyperbolic approximation of the family of non hydrostatic free surface flow models recently derived by sainte marie et al in 1 our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth averaged non hydrostatic pressure similar to the hyperbolic divergence cleaning applied in generalized lagrangian multiplier methods glm for magnetohydrodynamics mhd we suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing pde we then use arbitrary high order accurate ader discontinuous galerkin dg finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed pde system numerically the final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet dry fronts it is well balanced making use of a path conservative formulation of hll type riemann solvers based on the straight line segment path furthermore the proposed ader dg scheme with a posteriori subcell finite volume limiter adapts very well to modern gpu architectures resulting in a very accurate robust and computationally efficient computational method for non hydrostatic free surface flows the new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions in all cases the achieved agreement with analytical solutions or experimental data is very good thus showing the validity of both the proposed mathematical model and the numerical solution algorithm

on a neural approximator to odes the object of this brief is to present and analyze the training of a single layer neural network in order to solve ordinary differential equations odes properties of the approximator are derived and some examples of its application are shown

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

the q t gaussian process the q t fock space fq t h introduced in this paper is a deformation of the q fock space of bozejko and speicher the corresponding creation and annihilation operators now satisfy the commutation relation aq t f aq t g qaq t g aq t f f g htn turn mathjax off a defining relation of the chakrabarti jagannathan deformed quantum oscillator algebra the moments of the deformed gaussian element sq t h aq t h aq t h are encoded by the joint statistics of crossings and nestings in pair partitions the q 0 t specialization yields a natural single parameter deformation of the full boltzmann fock space of free probability with the corresponding semicircular measure variously encoded via the rogers ramanujan continued fraction the t airy function the t catalan numbers of carlitz riordan and the first order statistics of the reduced wigner process

weighted interior penalty discretization of fully nonlinear and weakly dispersive free surface shallow water flows in this paper we further investigate the use of a fully discontinuous finite element discrete formulation for the study of shallow water free surface flows in the fully nonlinear and weakly dis persive flow regime we consider a decoupling strategy in which we approximate the solutions of the classical shallow water equations supplemented with a source term globally accounting for the non hydrostatic effects this source term can be computed through the resolution of elliptic second order linear sub problems which only involve second order partial derivatives in space we then introduce an associated symmetric weighted internal penalty discrete bilinear form allowing to deal with the discontinuous nature of the elliptic problem s coefficients in a stable and consistant way similar discrete formulations are also introduced for several recent optimized fully nonlinear and weakly dis persive models these formulations are validated again several benchmarks involving h convergence p convergence and comparisons with experimental data showing optimal convergence properties

online optimal control of affine nonlinear discrete time systems with unknown internal dynamics by using time based policy update in this paper the hamilton jacobi bellman equation is solved forward in time for the optimal control of a class of general affine nonlinear discrete time systems without using value and policy iterations the proposed approach referred to as adaptive dynamic programming uses two neural networks nns to solve the infinite horizon optimal regulation control of affine nonlinear discrete time systems in the presence of unknown internal dynamics and a known control coefficient matrix one nn approximates the cost function and is referred to as the critic nn while the second nn generates the control input and is referred to as the action nn the cost function and policy are updated once at the sampling instant and thus the proposed approach can be referred to as time based adp novel update laws for tuning the unknown weights of the nns online are derived lyapunov techniques are used to show that all signals are uniformly ultimately bounded and that the approximated control signal approaches the optimal control input with small bounded error over time in the absence of disturbances an optimal control is demonstrated simulation results are included to show the effectiveness of the approach the end result is the systematic design of an optimal controller with guaranteed convergence that is suitable for hardware implementation

least squares solutions of the hjb equation with neural network value function approximators in this paper we present an empirical study of iterative least squares minimization of the hamilton jacobi bellman hjb residual with a neural network nn approximation of the value function although the nonlinearities in the optimal control problem and nn approximator preclude theoretical guarantees and raise concerns of numerical instabilities we present two simple methods for promoting convergence the effectiveness of which is presented in a series of experiments the first method involves the gradual increase of the horizon time scale with a corresponding gradual increase in value function complexity the second method involves the assumption of stochastic dynamics which introduces a regularizing second derivative term to the hjb equation a gradual reduction of this term provides further stabilization of the convergence we demonstrate the solution of several problems including the 4d inverted pendulum system with bounded control our approach requires no initial stabilizing policy or any restrictive assumptions on the plant or cost function only knowledge of the plant dynamics in the appendix we provide the equations for first and second order differential backpropagation

simple maps hurwitz numbers and topological recursion we introduce the notion of fully simple maps which are maps with non self intersecting disjoint boundaries in contrast maps where such a restriction is not imposed are called ordinary we study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder we show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks we also obtain an elegant formula for cylinders these relations reproduce the relation between moments and higher order free cumulants established by collins et al 22 and implement the symplectic transformation x leftrightarrow y on the spectral curve in the context of topological recursion we conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y we propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the 1 hermitian matrix model which is believed to be true but has not been proved yet our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion our argument relies on an unconditional matrix model interpretation of fully simple maps via the formal hermitian matrix model with external field we also deduce a universal relation between generating series of fully simple maps and of ordinary maps which involves double monotone hurwitz numbers in particular ordinary maps without internal faces which are generated by the gaussian unitary ensemble and with boundary perimeters lambda 1 ldots lambda n are strictly monotone double hurwitz numbers with ramifications lambda above infty and 2 ldots 2 above 0 combining with a recent result of dubrovin et al 24 this implies an elsv like formula for these hurwitz numbers

the linearization of the central limit operator in free probability theory we interpret the central limit theorem as a fixed point theorem for a certain operator and consider the problem of linearizing this operator in classical as well as in free probability theory vdn92 we consider two methods giving such a linearization and interpret the result as a weak form of the clt in the classical case the analysis involves dilation operators in the free case more general composition operators appear

zhang neural network for online solution of time varying linear matrix inequality aided with an equality conversion in this paper for online solution of time varying linear matrix inequality lmi such an lmi is first converted to a time varying matrix equation by introducing a time varying matrix of which each element is greater than or equal to zero then by employing zhang s neural dynamic method a special recurrent neural network termed zhang neural network znn is proposed and investigated for solving online the converted time varying matrix equation as well as the time varying lmi such a znn model showed in an explicit dynamics exploits the time derivative information of time varying coefficients in addition theoretical analysis and results of the proposed znn model are discussed and presented to show its excellent performance on solving the time varying lmi computer simulation results further demonstrate the efficacy of the proposed znn model for online solution of the time varying lmi and the converted time varying matrix equation

boussinesq type formulations for fully nonlinear and extremely dispersive water waves derivation and analysis boussinesq formulations valid for highly dispersive and highly nonlinear water waves are derived with the objective of improving the accuracy of the vertical variation of the velocity field as well

critical ratio between the amplitudes of two overtaking solitary water waves in this paper we study the critical ratio between the amplitudes of two overtaking solitary waves on a layer of water with uniform depth at the center of encounter the wave profile is fore and aft symmetric but it could have a single peak or double peaks the critical ratio separates these two regimes at the critical point the wave peak is flat with zero slope and curvature we solve the full water wave problem numerically by using a fully nonlinear and highly dispersive boussinesq model the model is numerically justified to be a good approximation of the euler equations for solitary waves with very large amplitude for small amplitude water waves our calculated critical ratio reduces to the well known result of 3 predicted by the korteweg de vries equation a weakly nonlinear and weakly dispersive model for large amplitude water waves the nonlinear effect is significant we find that the critical ratio deviates significantly from 3 for water waves with very high amplitude e g 0 6 relative wave height the critical ratio could be as large as 4 our result suggests that higher order nonlinear and dispersive effects are important when modeling the strong interaction between large amplitude water waves

analytic subordination for bi free convolution in this paper we study some analytic properties of bi free additive convolution both scalar and operator valued we show that using properties of voiculescu s subordination functions associated to free additive convolution of operator valued distributions simpler formulas for bi free convolutions can be derived we use these formulas in order to prove a result about atoms of bi free additive convolutions

dual lukacs regressions for non commutative variables abstract dual lukacs type characterizations of random variables in free probability are studied here first we develop a freeness property satisfied by lukacs type transformations of free poisson and free binomial non commutative variables which are free second we give a characterization of non commutative free poisson and free binomial variables by properties of first two conditional moments which mimic lukacs type assumptions known from classical probability more precisely our result is a non commutative version of the following result known in classical probability if u v are independent real random variables such that e v 1 u u v and e v 2 1 u 2 u v are non random then v has a gamma distribution and u has a beta distribution

second order cumulants of products we derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors this extends to the second order case the formula of krawczyk and speicher we apply our result to the problem of calculating the second order cumulants of a semi circular and haar unitary operator

sample variance in free probability abstract let x 1 x 2 x n denote i i d centered standard normal random variables then the law of the sample variance q n i 1 n x i x 2 is the 2 distribution with n 1 degrees of freedom it is an open problem in classical probability to characterize all distributions with this property and in particular whether it characterizes the normal law in this paper we present a solution of the free analogue of this question and show that the only distributions whose free sample variance is distributed according to a free 2 distribution are the semicircle law and more generally so called odd laws by which we mean laws with vanishing higher order even cumulants in the way of proof we derive an explicit formula for the free cumulants of q n which shows that indeed the odd cumulants do not contribute and which exhibits an interesting connection to the concept of r cyclicity

kesten s theorem for invariant random subgroups an invariant random subgroup of the countable group is a ran dom subgroup of whose distribution is invariant under conjuga tion by all elements of we prove that for a nonamenable invariant random subgroup h the spectral radius of every finitely supported random walk on i s strictly less than the spectral radius of the corresponding random walk on h this generalizes a result of kesten who proved this for normal subgroups as a byproduct we show that for a cayley graph g of a linear group with no amenable normal subgroups any sequence of finite quotients of g that spectrally approximates g converges to g in benjamini schramm convergence in particular this implies that infinite sequences of finite d regular ramanujan schreier graphs have essentially large girth

a noncommutative extended de finetti theorem abstract the extended de finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i i d and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability our main result is a noncommutative version of this theorem in contrast to the classical result of ryll nardzewski exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables out of our investigations emerges noncommutative conditional independence in terms of a von neumann algebraic structure closely related to popa s notion of commuting squares and kummerer s generalized bernoulli shifts our main result is applicable to classical probability quantum probability in particular free probability braid group representations and jones subfactors

on the multiplication of free n tuples of noncommutative random variables let a1 a n b 1 b n be random variables in a noncommutative probability space such that a1 a n is free from b1 b n we show how the joint distribution of the n tuple a1b1 a n b n can be described in terms of the joint distributions of a1 a n a nd b 1 b n by using the combinatorics of the n dimensional r transform we point out a few applications that can be easily derived from our result concerning the left and right translation with a semicircular element see sections 1 6 1 10 and the compression with a projection see sections 1 11 1 14 of an n tuple of noncommutative random variables a different approach to two of these applications is presented by dan voiculescu in an appendix to the paper introduction the theory of free random variables was developed in a se quence of papers of d voiculescu see 17 or the recent survey in 16 as an instrument for approaching free products of operator algebras its particular aspect addressed in the present paper is the one concerning the addition and multiplica tion of free random variables as shown by voiculescu in 13 14 a powerful method in the study of these operations is the use of transforms that convert them respectively into addition and multiplication of complex analytic func tions or in an algebraic framework of formal power series in an indeterminate z the precise definitions of these transforms called r transform for the addition problem and s transform for the multiplication problem will be reviewed in the sections 1 2 1 3 below in the present paper we are pursuing a combination of two ideas that have appeared recently in the study of the r and s transforms the first idea is that the connection between the r and the s transform is closer than one might suspect at first glance a way of making this precise was pointed out in our paper 6 in the form of the equation s r for a distribution with nonvan ishing mean and where is a combinatorial object with a precise significance the fourier transform for multiplicative functions on noncrossing partitions a byproduct of our result in 6 is the remark that the multiplication of free

convolution operations arising from vandermonde matrices different types of convolution operations involving large vandermonde matrices are considered the convolutions parallel those of large gaussian matrices and additive and multiplicative free convolution and include additive and multiplicative convolution of vandermonde matrices and deterministic diagonal matrices and cases where two independent vandermonde matrices are involved it is also shown that the convergence of any combination of vandermonde matrices is almost sure the convolutions are divided into two types those which depend on the phase distribution of the vandermonde matrices and those which depend only on the spectra of the matrices a general criterion is presented to find which type applies for any given convolution a simulation is presented verifying the results implementations of the presented convolutions are provided and discussed the implementation is based on the technique of fourier motzkin elimination and is quite general as it can be applied to virtually any combination of vandermonde matrices connections with related matrices such as toeplitz and hankel matrices are also discussed

free probability for pairs of faces iii 2 variables bi free partial s and t transforms abstract we introduce two 2 variables transforms the partial bi free s transform and the partial bi free t transform these transforms are the analogues for the bi multiplicative and respectively for the additive multiplicative bi free convolution of the 2 variables partial bi free r transform in our previous paper in this series

combined hybridizable discontinuous galerkin hdg and runge kutta discontinuous galerkin rk dg formulations for green naghdi equations on unstructured meshes abstract in this paper we introduce some new high order discrete formulations on general unstructured meshes especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations working with a recent family of optimized asymptotically equivalent equations we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms which globally account for the non hydrostatic effects through the introduction of auxiliary coupling variables high order discrete approximations of the main flow variables are obtained with a rk dg method while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second order linear elliptic sub problems with high order hdg formulations the combined use of hybrid unknowns and local post processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches the proposed formulation is then extended to a more complex family of three parameters enhanced green naghdi equations the resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies showing good convergence properties and very good agreements with several sets of experimental data

n player statistical nash game control m th cost cumulant optimization in this note we study a statistical nash game control problem where multiple players optimize system performance by shaping the distribution of their cost functions through cost cumulants our focus is on the nonzero sum dynamic statistical nash game which generalizes the risk sensitive nash game the necessary and sufficient conditions for the nash equilibrium solution are given for the n player m th cost cumulant optimization using the hamilton jacobi bellman hjb equations in addition we derive the optimal control strategies and provide simulation results to demonstrate the performance of our method