relative equilibria of point vortices on the hyperbolic sphere relative equilibria of point vortices on a hyperbolic sphere with constant negative curvature is formulated and considered known symmetries for the vortex motion on the hyperbolic sphere are expounded by setting up a geometrical formulation of dynamics in particular the configuration of rings of vortices is investigated in detail

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

random positive operator valued measures we introduce several notions of random positive operator valued measures povms and we prove that some of them are equivalent we then study statistical properties of the effect operators for the canonical examples starting from the limiting eigenvalue distribution we derive the large system limit for several quantities of interest in quantum information theory such as the sharpness the noise content and the probability range finally we study different compatibility criteria and we compare them for generic povms

thomson s heptagon a case of bifurcation at infinity abstract vortex modeling has a long history descartes 1644 used it as a model for the solar systems j j thomsom 1883 used it as a model for the atom we consider point vortex systems which can be regarded as discrete solutions of the euler equation their dynamics is described by a hamiltonian system of equations we are interested in polygonal configurations and how their stability depends upon various dynamical variables in the plane a polygon with seven vortices has been shown to be a special boundary case polygons with n 7 vortices are linearly and nonlinearly stable while polygons with n 7 vortices are unstable why should n 7 be special celestial mechanics helped us to simplify a problem that has been studied for over a century and to show that the case of thomson s heptagon is actually a case of bifurcation at infinity this becomes particularly clear when considering the corresponding problem of a ring on a sphere with two polar vortices of variable intensities n and s at the north and south pole respectively

point vortices on the hyperbolic plane we investigate the dynamical system of point vortices on the hyperboloid this system has non compact symmetry sl 2 r and a coadjoint equivariant momentum map the relative equilibrium conditions are found and the trajectories of relative equilibria with non zero momentum value are described we also provide the classification of relative equilibria and the stability criteria for a number of cases focusing on 2 and 3 vortices unlike the system on the sphere this system has relative equilibria with non compact momentum isotropy subgroup and these are used to illustrate the different stability types of relative equilibria

time optimal simultaneous control of two level quantum systems in this paper we solve the problem of simultaneously driving in minimum time to arbitrary final conditions n two level quantum systems subject to independent controls the solution of this problem is obtained via an explicit description of the reachable set of the associated control system on s u 2 the treatment generalizes previous results on the time optimal control of two level quantum systems and suggests that similar techniques could be used to solve the minimum time control problem for a larger class of right invariant systems on lie groups

integrable and chaotic motions of four vortices i the case of identical vortices it is shown that the three vortex problem in two dimensional hydrodynamics is integrable whereas the motion of four identical vortices is not a sequence of canonical transformations is obtained that reduces the n degree of freedom hamiltonian which describes the interaction of n identical vortices to one with n 2 degrees of freedom for n 3 a reduction to a single degree of freedom is obtained and this problem can be solved in terms of elliptic functions for n 4 the reduction procedure leads to an effective hamiltonian with two degrees of freedom of the form found in problems with coupled nonlinear oscillators resonant interaction terms in this hamiltonian suggest non integrable behaviour and this is verified by numerical experiments explicit construction of a solution that corresponds to a heteroclinic orbit in phase space is possible the relevance of the results obtained to fundamental problems in hydrodynamics such as the question of integrability of euler s equation in two dimensions is discussed the paper also contains a general exposition of the hamiltonian and poisson bracket formalism for point vortices

complete diagrammatics of the single ring theorem using diagrammatic techniques we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices the formalism allows us to map two distinct areas of free random variables hermitian positive definite operators and non normal r diagonal operators we also rederive the haagerup larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach

resolvents of r diagonal operators we consider the resolvent a 1 of any diagonal operator a in a ii 1 factor our main theorem theorem 1 1 gives a universal asymptotic formula for the norm of such a resolvent en route to its proof we calculate the transform of the operator c 2 where c is voiculescu s circular operator and we give an asymptotic formula for the negative moments of a 2 for any diagonal a we use a mixture of complex analytic and combinatorial techniques each giving finer information where the other can give only coarse detail in particular we introduce partition structure diagrams in section 4 a new combinatorial structure arising in free probability

the n vortex problem on a rotating sphere i multi frequency configurations the problem of n point vortices moving on a rotating unit sphere is considered through a sequence of linear coordinate transformations which takes into account the orientation of the centre of vorticity vector with respect to the axis of rotation we show how to reduce the problem to that on a non rotating sphere where the centre of vorticity vector is aligned with the z axis as a consequence we prove that integrability on the rotating sphere is the same as on the non rotating sphere namely the three vortex problem on the rotating sphere is integrable for all vortex strengths while the four vortex problem is integrable in the special case where the centre of vorticity is zero rigid multi frequency configurations that retain their shape while rotating about two independent axes with two independent frequencies are obtained and necessary conditions for one and two frequency motions are derived examples including dipoles which exhibit global wobbling and tumbling dynamics rings and platonic solid configurations are shown to undergo either periodic or quasi periodic evolution on the rotating sphere depending on the ratio of the solid body rotational frequency to the rotational frequency associated with the rigid structure

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

n body dynamics on closed surfaces the axioms of mechanics a major challenge for our understanding of the mathematical basis of particle dynamics is the formulation of n body and n vortex dynamics on riemann surfaces in this paper we show how the two problems are in fact closely related when considering the role played by the intrinsic geometry of the surface this enables a straightforward deduction of the dynamics of point masses using recently derived results for point vortices on general closed differentiable surfaces m endowed with a metric g we find generally that kepler s laws do not hold what is more even newton s first law the law of inertia fails on closed surfaces with variable curvature e g the ellipsoid

minimum time control of a pair of two level quantum systems with opposite drifts in this paper we solve two equivalent time optimal control problems on one hand we design the control field to implement in minimum time the swap or equivalent operator on a two level system assuming that it interacts with an additional uncontrollable two level system on the other hand we synthesize the swap operator simultaneously in minimum time on a pair of two level systems subject to opposite drifts we assume that it is possible to perform three independent control actions and that the total control strength is bounded these controls either affect the dynamics of the target system under the first perspective or simultaneously the dynamics of both systems in the second view we obtain our results by using techniques of geometric control theory on lie groups in particular we apply the pontryagin maximum principle and provide a complete characterization of singular and nonsingular extremals our analysis shows that the problem can be formulated as the motion of a material point in a central force a well known system in classical mechanics although we focus on obtaining the swap operator many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator

almost all quantum channels are equidistant in this work we analyze properties of generic quantum channels in the case of large system size we use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger as a measure of the distance we use the diamond norm in the case of a flat hilbert schmidt distribution on quantum channels we obtain that the distance converges to 12 2 giving also an estimate for the maximum success probability for distinguishing the channels we also consider the problem of distinguishing two random unitary rotations in this work we analyze properties of generic quantum channels in the case of large system size we use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger as a measure of the distance we use the diamond norm in the case of a flat hilbert schmidt distribution on quantum channels we obtain that the distance converges to 12 2 giving also an estimate for the maximum success probability for distinguishing the channels we also consider the problem of distinguishing two random unitary rotations

numerical description of the motion of a point vortex pair on ovaloids this paper presents a numerical study on the dynamics of point vortex pairs moving on convex closed surfaces ovaloids approximated by triangular icosahedral meshes firstly both a discretized conformal mapping between the ovaloid and the unit sphere and a least squares fitting for this approximated conformal map in terms of a spherical harmonic expansion are obtained then an approximation of the conformal spherical metric of the ovaloid is derived by obtaining a spherical harmonic expansion of the corresponding conformal factor and its gradient on the equations of motion for a point vortex pair on with this quasi conformal metric are integrated by using the classical gaussian collocation method for some particular models

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

three ways to solve the poisson equation on a sphere with gaussian forcing motivated by the needs of vortex methods we describe three different exact or approximate solutions to the poisson equation on the surface of a sphere when the forcing is a gaussian of the three dimensional distance 2 j exp 2 e 2 1 cos q c g a u s s e more precisely the forcing is a gaussian minus the gauss constraint constant c g a u s s this subtraction is necessary because j is bounded for any type of forcing only if the integral of the forcing over the sphere is zero y kimura h okamoto vortex on a sphere j phys soc jpn 56 1987 4203 4206 d g dritschel contour dynamics surgery on the sphere j comput phys 79 1988 477 483 the legendre polynomial series is simple and yields the exact value of the gauss constraint constant but converges slowly for large e the analytic solution involves nothing more exotic than the exponential integral but all four terms are singular at one or the other pole cancelling in pairs so that j is everywhere nice the method of matched asymptotic expansions yields simpler uniformly valid approximations as series of inverse even powers of e that converge very rapidly for the large values of e e 40 appropriate for geophysical vortex computations the series converges to a nonzero o exp 4 e 2 error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order

on the asymptotic distribution of the commutator and anticommutator of random matrices we study in the limit of infinite matrix order the normalized eigenvalue counting measures of the commutator and anticommutator of two hermitian or real symmetric matrices rotated independently one respect to another by the random unitary or orthogonal haar distributed matrix we establish the convergence with probability 1 to a limiting nonrandom measure we obtain and analyze the functional equations for the stieltjes transforms of the limiting measures

strong haagerup inequalities for free r diagonal elements abstract in this paper we generalize haagerup s inequality u haagerup an example of a nonnuclear c algebra which has the metric approximation property invent math 50 1978 1979 279 293 on convolution norm in the free group to a very general context of r diagonal elements in a tracial von neumann algebra moreover we show that in this holomorphic setting the inequality is greatly improved from its original form we give combinatorial proofs of two important special cases of our main result and then generalize these techniques en route we prove a number of moment and cumulant estimates for r diagonal elements that are of independent interest finally we use our strong haagerup inequality to prove a strong ultracontractivity theorem generalizing and improving the one in p biane free hypercontractivity comm math phys 184 1997 457 474

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 method of images and green s function for spherical domains motivated by problems in electrostatics and vortex dynamics we develop two general methods for constructing green s function for simply connected domains on the surface of the unit sphere we prove a riemann mapping theorem showing that such domains can be conformally mapped to the upper hemisphere we then categorize all domains on the sphere for which green s function can be constructed by an extension of the classical method of images we illustrate our methods by several examples such as the upper hemisphere geodesic triangles and latitudinal rectangles we describe the point vortex motion in these domains which is governed by a hamiltonian determined by the dirichlet green s function

minimum time optimal synthesis for two level quantum systems for the time optimal problem of an invariant system on su 2 with two independent controls and a bound on the norm of the control the extremals of the pontryagin maximum principle are explicit functions of time we use this fact here to perform the optimal synthesis for these systems i e to find all time optimal trajectories although the lie group su 2 is three dimensional time optimal trajectories can be described in the unit disk of the complex plane we find that a circular trajectory separates optimal trajectories that reach the boundary of the unit disk from the others inside this separatrix circle another trajectory the critical trajectory plays an important role in that all optimal trajectories end at an intersection with this curve the results allow us to find the minimum time needed to achieve a given evolution of a two level quantum system

brown s spectral distribution measure for r diagonal elements in finite von neumann algebras abstract in 1983 l g brown introduced a spectral distribution measure for non normal elements in a finite von neumann algebra m with respect to a fixed normal faithful tracial state in this paper we compute brown s spectral distribution measure in case t has a polar decomposition t uh where u is a haar unitary and u and h are free when ker t 0 this is equivalent to that t t is an r diagonal pair in the sense of nica and speicher the measure t is expressed explicitly in terms of the s transform of the distribution t t of the positive operator t t in case t is a circular element i e t x1 ix2 2 where x1 x2 is a free semicircular system then sp t d the closed unit disk and t has constant density 1 on d

vortex motion on surfaces with constant curvature vortex motion on two dimensional riemannian surfaces with constant curvature is formulated by way of the stereographic projection the relation and difference between the vortex motion on a sphere s 2 and on a hyperbolic plane h 2 can be clearly analysed the hamiltonian formalism is presented for the motion of point vortices on s 2 and h 2 the set of first integrals for each hamiltonian shows a corresponding algebraic property in terms of the poisson bracket defined respectively for s 2 and h 2 as an example of analytic solutions the motion of a vortex pair dipole is considered it is shown that a dipole draws a geodesic curve as its trajectory on s 2 and h 2

minimal time trajectories for two level quantum systems with two bounded controls in this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude the controls are modeled as real functions and we do not use the rotating wave approximation after projection on the bloch sphere we treat the time optimal control problem with techniques of optimal synthesis on 2d manifolds based on the pontryagin maximum principle we characterize a restricted set of candidate optimal trajectories properties on this set crucial for complete optimal synthesis are illustrated by numerical simulations furthermore when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the initial condition

spectrum of the product of independent random gaussian matrices we show that the eigenvalue density of a product x x1x2 xm of m independent nxn gaussian random matrices in the limit n infinity is rotationally symmetric in the complex plane and is given by a simple expression rho z z 1 mpisigma 2 m z 2 2 m for z sigma the parameter sigma corresponds to the radius of the circular support and is related to the amplitude of the gaussian fluctuations this form of the eigenvalue density is highly universal it is identical for products of gaussian hermitian non hermitian and real or complex random matrices it does not change even if the matrices in the product are taken from different gaussian ensembles we present a self contained derivation of this result using a planar diagrammatic technique additionally we conjecture that this distribution also holds for any matrices whose elements are independent centered random variables with a finite variance or even more generally for matrices which fulfill pastur lindeberg s condition we provide a numerical evidence supporting this conjecture

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

the k hbox p problem on tensor products of lie groups and time optimal control of n quantum bits with a bounded field we use properties of sub riemannian k hbox p problems and the optimal synthesis on su 2 to solve the time optimal control of n noninteracting two level quantum systems qubits controlled in parallel with a control field bounded in l 2 norm every qubit is controlled independently but the total energy of the controls used is bounded for this system we describe the reachable set at every time moreover we compute the diameter that is the worst case minimum time the mathematical problem is motivated by the control of two level systems with different resonance frequencies where the control fields are obtained from a single signal having bounded energy

eigenvalues and singular values of products of rectangular gaussian random matrices we derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions we show that they both have power law behavior at zero and determine the corresponding powers we also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions

the motion of point vortices on closed surfaces we develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere when the sum of the vortex circulations is non ze

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

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

r diagonal dilation semigroups this paper addresses extensions of the complex ornstein uhlenbeck semigroup to operator algebras in free probability theory if a 1 a k are free fancyscript r diagonal operators in a ii1 factor then d t a i 1 cdots a i n e nt a i 1 cdots a i n defines a dilation semigroup on the non self adjoint operator algebra generated by a 1 a k we show that d t extends in two different ways to a semigroup of completely positive maps on the von neumann algebra generated by a 1 a k moreover we show that d t satisfies an optimal ultracontractive property d t colon l 2 to l infty sim t 1 for small t 0