optimal mean square state and parameter estimation for stochastic linear systems with poisson noises this paper presents the mean square state and parameter estimation problem for stochastic linear systems with unknown multiplicative and additive parameters over linear observations where unknown parameters are considered poisson processes the original problem is reduced to the filtering problem for an extended state vector that incorporates parameters as additional states the obtained optimal filter for the extended state vector also serves as the optimal identifier for the unknown parameters performance of the designed optimal state filter and parameter identifier is verified for both stable and unstable stochastic linear systems and compared against the mean square estimator designed for polynomial systems with white gaussian noises

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

criteria for solvable radical membership via p elements abstract guralnick kunyavskii plotkin and shalev have shown that the solvable radical of a finite group g can be characterized as the set of all x g such that x y is solvable for all y g we prove two generalizations of this result firstly it is enough to check the solvability of x y for every p element y g for every odd prime p secondly if x has odd order then it is enough to check the solvability of x y for every 2 element y g

equidistribution in shrinking sets and l 4 norm bounds for automorphic forms we study two closely related problems stemming from the random wave conjecture for maas forms the first problem is bounding the l 4 norm of a maas form in the large eigenvalue limit we complete the work of spinu to show that the l 4 norm of an eisenstein series e z 1 2 it g restricted to compact sets is bounded by sqrt log t g the second problem is quantum unique ergodicity in shrinking sets we show that by averaging over the centre of hyperbolic balls in gamma backslash mathbb h quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the planck scale this result is conditional on the generalised lindelof hypothesis for hecke maas eigenforms but is unconditional for eisenstein series we also show that equidistribution for hecke maas eigenforms need not hold at or below the planck scale finally we prove similar equidistribution results in shrinking sets for heegner points and closed geodesics associated to ideal classes of quadratic fields

characterization of finite soluble groups by laws in a small number of variables for k 1 of course 6 g gck so that g is soluble of derived length at most k if and only if 6 g 1 for all n 2 k the sequence 6 involves infinitely many variables indeed if soluble groups could be characterized as above by a sequence in a finite number n of variables then a group would be soluble if and only if every n generator subgroup is soluble however this is not so for each integer n 1 there is an n 1 generator insoluble group all of whose n generator subgroups are finite and nilpotent see golod 31 however the situation becomes rather different if we restrict attention to finite groups because a finite group is soluble if and only if each twogenerator subgroup is soluble see thompson 7 corollary 21 it follows 334 0021 8693 88 3 00

quantum ergodicity of c dynamical systems we define a notion of quantum or non commutative ergodicity for a class ofc dynamical systems a g which we callquantized gns systems such a system possesses a natural classical limit state which induces a classical limit system by the gns construction the criterion for quantum ergodicity is that the time average a of an observable a a equals the space average a i plus an errork which is negligible in the classical limit we prove that ergodicity of is a sufficient condition for quantum ergodicity of a g if the classical limit system is abelian give a conditional converse and discuss a number of applications

recursive parameter estimation using incomplete data abstract stochastic approximation procedures are considered for the estimation of parameters using incomplete data one procedure is stated and illustrated which often leads to asymptotically efficient estimators others are developed which although possibly not optimal in the above sense will be very much easier to apply this will be particularly advantageous when quick recursive estimates are required examples are given and a link is made between one of the sub optimal methods and the em algorithm author

exact finite dimensional filters for maximum likelihood parameter estimation of continuous time linear gaussian systems in this paper we derive a new class of finite dimensional filters for integrals and stochastic integrals of moments of the state for continuous time linear gaussian systems apart from being of significant mathematical interest these new filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the model parameters

refined and microlocal kakeya nikodym bounds of eigenfunctions in higher dimensions we prove a kakeya nikodym bound on eigenfunctions and quasimodes which sharpens a result of the authors blair and sogge in anal pde 8 747 764 2015 and extends it to higher dimensions as in the prior work the key intermediate step is to prove a microlocal version of these estimates which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic geodesic flow in a companion paper blair and sogge in j differ geom 2015 it will be seen that these sharpened estimates yield improved l q m bounds on eigenfunctions in the presence of nonpositive curvature when 2 q frac 2 d 1 d 1

reproducing gaussian densities and linear gaussian detection abstract it is shown that the expected value of a gaussian density function of a gaussian random variable is another gaussian density function this is used to determine which of m partially observed gaussian linear systems is the most likely given the observations

doubling inequality and nodal sets for solutions of bi laplace equations we investigate the doubling inequality and nodal sets for the solutions of bi laplace equations a polynomial upper bound for the nodal sets of solutions and their gradient is obtained based on the recent development of nodal sets for laplace eigenfunctions by logunov in addition we derive an implicit upper bound for the nodal sets of solutions we show two types of doubling inequalities for the solutions of bi laplace equations as a consequence the rate of vanishing is given for the solutions

asymptotically optimal discrete time nonlinear filters from stochastically convergent state process approximations we consider the problem of approximating optimal in the mmse sense nonlinear filters in a discrete time setting exploiting properties of stochastically convergent state process approximations more specifically we consider a class of nonlinear partially observable stochastic systems comprised by a possibly nonstationary hidden stochastic process the state observed through another conditionally gaussian stochastic process the observations under general assumptions we show that given an approximating process which for each time step is stochastically convergent to the state process in some appropriate sense an approximate filtering operator can be defined which converges to the true optimal nonlinear filter of the state in a strong and well defined sense i e compactly in time and uniformly in a completely characterized measurable set of probability measure almost unity also providing a purely quantitative justification of egoro s theorem for the problem at hand the results presented in this paper can form a common basis for the analysis and characterization of a number of heuristic approaches for approximating a large class of optimal nonlinear filters such as approximate grid based techniques known to perform well in a variety of applications

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

simple lie algebras of small characteristic iv solvable and classical roots abstract let l be a finite dimensional simple lie algebra over an algebraically closed field f of characteristic p 3 and t a torus of maximal dimension in the p envelope of l in der l in this paper we describe the t semisimple quotients of the 2 sections of l relative to t and prove that if all 1 sections of l relative to t are compositionally classical or solvable then l is either classical or a block algebra or a filtered lie algebra of type s

planck scale mass equidistribution of toral laplace eigenfunctions we study the small scale distribution of the l 2 mass of eigenfunctions of the laplacian on the two dimensional flat torus given an orthonormal basis of eigenfunctions lester and rudnick commun math phys 350 1 279 300 2017 showed the existence of a density one subsequence whose l 2 mass equidistributes more or less down to the planck scale we give a more precise version of their result showing equidistribution holds down to a small power of log above planck scale and also showing that the l 2 mass fails to equidistribute at a slightly smaller power of log above the planck scale this article rests on a number of results about the proximity of lattice points on circles much of it based on foundational work of javier cilleruelo

the quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

decay of correlations for normally hyperbolic trapping we prove that for evolution problems with normally hyperbolic trapping in phase space correlations decay exponentially in time normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it flows with this structure include contact anosov flows classical flows in molecular dynamics and null geodesic flows for black holes metrics the decay of correlations is a consequence of the existence of resonance free strips for green s functions cut off resolvents and polynomial bounds on the growth of those functions in the semiclassical parameter

state and mode estimation for discrete time jump markov systems in this article we compute new state and mode estimation algorithms for discrete time gauss markov models whose parameter sets switch according to a known markov law an important feature of our algorithms is that they are based upon the exact filter dynamics computed in r j elliott f dufour and d sworder ieee trans automat control 41 1996 pp 1807 1810 the fundamental and well known obstacle in estimation of jump markov systems is managing the geometrically growing history of candidate hypotheses in our scheme we address this issue by proposing an extension of an idea due to viterbi our scheme maintains a fixed number of candidate paths in a history each identified by an optimal subset of estimated mode probabilities we compute finite dimensional suboptimal filters and smoothers which estimate the hidden state process and the mode probability our smoothers are based upon a duality between forward and backward dynamics further our smoothing algorithms are general and can be configured into the standard forms of fixed point fixed lag and fixed interval smoothers a computer simulation is included to demonstrate performance

from thompson to baer suzuki a sharp characterization of the solvable radical abstract we prove that an element g of prime order 3 belongs to the solvable radical r g of a finite or more generally a linear group if and only if for every x g the subgroup generated by g x g x 1 is solvable this theorem implies that a finite or a linear group g is solvable if and only if in each conjugacy class of g every two elements generate a solvable subgroup

maximum likelihood estimation of the dynamic shock error model abstract in this paper an efficient estimation technique in the time domain for the dynamic shock error model is developed this approach utilizes a state space representation of the model and the kalman filtering technique to obtain the maximum likelihood estimates the resulting estimates are shown to be consistent and asymptotically normal with the inverse of the information matrix as their precision measure a consistent estimate of the information matrix is suggested to obtain the standard errors of the estimates

l p norms nodal sets and quantum ergodicity abstract for small range of p 2 we improve the l p bounds of eigenfunctions of the laplacian on negatively curved manifolds our improvement is by a power of logarithm for a full density sequence of eigenfunctions we also derive improvements on the size of the nodal sets our proof is based on a quantum ergodicity property of independent interest which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate

localized lp estimates of eigenfunctions a note on an article of hezari and riviere abstract we use a straightforward variation on a recent argument of hezari and riviere 8 to obtain localized l p estimates for all exponents larger than or equal to the critical exponent p c 2 n 1 n 1 we are able to do this directly by just using the l p bounds for spectral projection operators from our much earlier work 12 the localized bounds we obtain here imply for instance that for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic all of the l p 2 p bounds of the corresponding eigenfunctions are relatively small compared to the general ones in 12 which are saturated on round spheres the connection with quantum ergodicity was established for exponents 2 p p c in the recent results of the author 13 and blair and the author 3 however the article of hezari and riviere 8 was the first one to make this connection in the case of negatively curved manifolds for the critical exponent p c as is well known and we indicate here bounds for the critical exponent p c imply ones for all the other exponents 2 p the localized estimates involve l 2 norms over small geodesic balls b r of radius r and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature we shall also state a problem of determining when one can improve on the trivial o r 1 2 estimates for these l 2 b r bounds if r 1 one can improve on the trivial estimates if one has improved l p c m bounds just by using holder s inequality however obtaining improved bounds for r 1 seems to be subtle

quantum ergodicity and l p norms of restrictions of eigenfunctions we prove an analogue of sogge s local l p estimates for l p norms of restrictions of eigenfunctions to submanifolds and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of burq gerard tzvetkov hu and chen sogge the improvements are logarithmic on negatively curved manifolds without boundary and by o 1 for manifolds with or without boundary with ergodic geodesic flows in the case of ergodic billiards with piecewise smooth boundary we get o 1 improvements on l infty estimates of cauchy data away from a shrinking neighborhood of the corners and as a result using the methods of ghosh et al jung and zelditch jung and zelditch we get that the number of nodal domains of 2 dimensional ergodic billiards tends to infinity as lambda to infty these results work only for a full density subsequence of any given orthonormal basis of eigenfunctions we also present an extension of the l p estimates of burq gerard tzvetkov hu chen sogge for the restrictions of dirichlet and neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary this part does not assume ergodicity on the manifolds

a solvable version of the baer suzuki theorem suppose that g is a finite group and x g has prime order p 5 then x is contained in the solvable radical of g o g if and only if x x g is solvable for all g g if g is an almost simple group and x g has prime order p 5 then this implies that there exists g g such that x x g is not solvable in fact this is also true when p 3 with very few exceptions which are described explicitly

concerning the lp norm of spectral clusters for second order elliptic operators on compact manifolds in this paper we study the lp l2 mapping properties of a spectral projection operator for riemannian manifolds this operator is a generalization of the harmonic projection operator for spherical harmonics on s see c d sogge duke math j 53 1986 43 65 among other things we generalize the l2 restriction theorems of c fefferman e m stein and p tomas bull amer math sot 81 1975 477478 for the fourier transform in r to the setting of riemannian manifolds we obtain these results for the spectral projection operator as a corollary of a certain sobolev inequality involving d r2 for large r this sobolev inequality generalizes certain results for r of c kenig a ruiz and the author duke math j 55 1987 329 347 the main tools in the proof of the sobolev inequalities for riemannian manifolds are the hadamard parametrix cf l hbrmander acta mad 88 1968 341 370 and the analysis of linear partial differential equations vol iii springer verlag new york 1985 and oscillatory integral theorems of l carleson and p sjolin studia math 44 19x 287 299 and stein arm math stud 112 1986 307 357 0 1988 academic pnss iw

anosov flows and dynamical zeta functions we study the ruelle and selberg zeta functions for c r anosov ows r 2 on a compact smooth manifold we prove several results the most remarkable being a for c 1 ows the zeta function is meromorphic on the entire complex plane b for contact ows satisfying a bunching condition e g geodesic ows on manifolds of negative curvature better than 1 9 pinched the zeta function has a pole at the topological entropy and is analytic in a strip to its left c under the same hypotheses as in b we obtain sharp results on the number of periodic orbits our arguments are based on the study of the spectral properties of a transfer operator acting on suitable banach spaces of anisotropic currents

characterization of the soluble radical by a sequence of words abstract it is shown that there is a sequence w n x y of elements of the free group on x y with the following property if g is a finite group or more generally a linear group then the largest soluble normal subgroup of g consists of all elements g such that w n g h 1 for all h g and all sufficiently large indices n

improved critical eigenfunction restriction estimates on riemannian manifolds with constant negative curvature abstract we show that one can obtain logarithmic improvements of l 2 geodesic restriction estimates for eigenfunctions on 3 dimensional compact riemannian manifolds with constant negative curvature we obtain a log 1 2 gain for the l 2 restriction bounds which improves the corresponding bounds of burq gerard and tzvetkov 4 hu 10 chen and sogge 6 we achieve this by adapting the approaches developed by chen and sogge 6 blair and sogge 3 xi and the author 19 we derive an explicit formula for the wave kernel on 3d hyperbolic space which improves the kernel estimates from the hadamard parametrix in chen and sogge 6 we prove detailed oscillatory integral estimates with fold singularities by phong and stein 12 and use the poincare half space model to establish bounds for various derivatives of the distance function restricted to geodesic segments on the universal cover h 3

maximum likelihood localization of narrow band autoregressive sources via the em algorithm the authors derive an expectation maximization algorithm for the maximum likelihood estimation of the directions of arrival of multiple narrowband autoregressive ar signals embedded in noise the proposed algorithm simultaneously estimates the location parameters the ar coefficients and the signals the additional structural information that the sources are of the ar type enables one to resolve the case in which the number of sources is equal to or possibly larger than the number of sensors the authors provide examples demonstrating that the proposed algorithm can resolve two spectrally and spatially closely spaced sources by using a two sensor array

on line estimation of dynamic shock error models based on the kullback leibler information measure develops two sequential or on line estimation schemes in the time domain for dynamic shock error models which are special cases of errors in variables models the author s approach utilizes a state space representation of the model kalman filtering techniques and on line algorithms the first on line algorithm is based on the expectation maximization algorithm and uses a recursive gauss newton scheme to maximize the kullback leibler information measure the second on line algorithm the author proposes is a gradient based scheme and uses stochastic approximations to maximize the log likelihood in comparison to the off line maximum likelihood estimation scheme used in ghosh 1989 the author s on line algorithms have significantly reduced computational costs and negligible memory requirements simulations illustrate the satisfactory performance of the algorithms in estimating errors in variables systems with parameters that vary slowly with time or undergo infrequent jump changes

hidden markov processes an overview of statistical and information theoretic aspects of hidden markov processes hmps is presented an hmp is a discrete time finite state homogeneous markov chain observed through a discrete time memoryless invariant channel in recent years the work of baum and petrie 1966 on finite state finite alphabet hmps was expanded to hmps with finite as well as continuous state spaces and a general alphabet in particular statistical properties and ergodic theorems for relative entropy densities of hmps were developed consistency and asymptotic normality of the maximum likelihood ml parameter estimator were proved under some mild conditions similar results were established for switching autoregressive processes these processes generalize hmps new algorithms were developed for estimating the state parameter and order of an hmp for universal coding and classification of hmps and for universal decoding of hidden markov channels these and other related topics are reviewed

lower bounds for nodal sets of eigenfunctions we prove lower bounds for the hausdorff measure of nodal sets of eigenfunctions 0 introduction let m be a smooth closed riemannian manifold and the laplace operator a function u is said to be an eigenfunction with eigenvalue if u u 0 1 with our convention on the sign of the eigenvalues are non negative and go to infinity one of the most fundamental questions about eigenfunctions is to understand the sets where they vanish these sets are called nodal sets nodal sets are n 1 dimensional manifolds away from n 2 dimensional singular sets where the gradient also vanishes so it is natural to estimate the n 1 dimensional hausdorff measure hn 1 the main result of this short note is the following lower bound theorem 1 given a closed n dimensional riemannian manifold m there exists c so that hn 1 u 0 c 3 n 4 0 2 in particular theorem 1 gives a uniform lower bound in dimension n 3 in y s t yau conjectured the lower bound c in all dimensions this was proven for surfaces by bruning b and yau independently and for real analytic metrics by donnelly and fefferman in df but remains open in the smooth case the authors were partially supported by nsf grants dms 0606629 dms 0906233 and nsf frg grants dms 0854774 and dms 0853501 778 t h colding w p minicozzi ii the donnelly fefferman argument leads to exponentially decaying lower bounds in the smooth case see also hl in m mangoubi considered eigenfunctions on a ball in a manifold and proved lower bounds for the volume of the subset of the ball where the function is positive and the subset where it is negative combined with the isoperimetric inequality one can get the lower bound c 3 n 2 1 2n for the measure of the nodal set on the entire manifold recently sogge and zelditch sz proved the lower bound c 7 3n 8 their argument is completely different and is based in part on a beautiful new integral formula relating the l1 norm of u on the nodal set and the l1 norm of u on m finally note that some papers use 2 for the eigenvalue i e u 2 u with that convention our bound is c 3 n 2 1 finding good balls throughout m will be a fixed closed manifold with laplace operator and u will be an eigenfunction with eigenvalue and m u 2 1 we will always assume that 1 since our interest is in what happens when goes to infinity the first step is to fix a scale r depending on lemma 1 there exists a 0 so that u has a zero in every ball of radius a 3 this lemma is standard but we will include a proof since it is so short proof if u does not vanish on b r 3 then barta s theorem implies that the lowest dirichlet eigenvalue on this ball is at least let be a function that is identically one on b r 6 and that cuts off linearly to zero on the annulus b r 3 b r 6 it follows that 2 2 36r 2 vol b r 3 vol b r 6 c r2 1 1 where c comes from the bishop gromov volume comparison see p 275 of g and depends only on n the ricci curvature of m and an upper bound for r since this is impossible for r2 c 1 the lemma follows from now on we set r a 1 2 with a given by lemma 1 next we use a standard covering argument to decompose the manifold m into small balls of radius r if b is a ball in m then we write 2b for the ball with the same center as b and twice the radius lemma 2 there exists a collection bi of balls of radius r in m so that m i bi and each point of m is contained in at most cm cm m of the double balls 2bi s proof choose a maximal disjoint collection of balls b r 2 pi it follows immediately from maximality that the double balls bi br pi cover m suppose that p m is contained in balls b2r p1 b2r pk in particular the disjoint balls b r 2 p1 b 2 pk are all contained in b3r p so that k i 1 vol b r 2 pi vol b3r p 1 2

planck scale number of nodal domains for toral eigenfunctions abstract we study the number of nodal domains in balls shrinking slightly above the planck scale for generic toral eigenfunctions we prove that up to the natural scaling the nodal domains count obeys the same asymptotic law as the global number of nodal domains the proof on one hand uses new arithmetic information to refine bourgain s de randomisation technique at planck scale and on the other hand it requires a planck scale version of yau s conjecture which we believe to be of independent interest

thompson like characterizations of the solvable radical abstract we prove that the solvable radical of a finite group g coincides with the set of elements y having the following property for any x g the subgroup of g generated by x and y is solvable this confirms a conjecture of flavell we present analogues of this result for finite dimensional lie algebras and some classes of infinite groups we also consider a similar problem for pairs of elements

on the rate of quantum ergodicity i upper bounds one problem in quantum ergodicity is to estimate the rate of decay of the sums s k lambda a frac 1 n lambda sum limits sqrt lambda j leqq lambda left a varphi j varphi j bar sigma a right k on a compact riemannian manifold m g with ergodic geodesic flow here j j are the spectral data of the of m g a is a 0 th order do bar sigma a is the liouville average of its principal symbol and n lambda j sqrt lambda j leqq lambda thats k a o 1 is proved in s z 1 cv 1 our purpose here is to show thats k a o log k 2 on a manifold of possibly variable negative curvature the main new ingredient is the central limit theorem for geodesic flows on such spaces r si

some finite varieties of lie algebras in the class whenever all n generated subalgebras of l are in the class thus the classes jv of nilpotent lie algebras and d of abelian lie algebras are 2 recognizable the problems are to find if a given class is n recognizable and to find the smallest n we consider the classes jv of lie algebras which have an abelian ideal with nilpotent quotient jv of lie algebras which have a nilpotent ideal with abelian quotient of supersolvable lie algebras and y of solvable lie algebras the same notation is used for the corresponding classes of finite groups most notation is standard except where indicated

quantitative quantum ergodicity and the nodal domains of hecke maass cusp forms we prove a quantitative statement of the quantum ergodicity for hecke maass cusp forms on the modular surface as an application of our result along a density 1 subsequence of even hecke maass cusp forms we obtain a sharp lower bound for the l 2 norm of the restriction to a fixed compact geodesic segment of eta iy y 0 subset mathbb h we also obtain an upper bound of o epsilon left t phi 3 8 epsilon right for the l infty norm along a density 1 subsequence of hecke maass cusp forms for such forms this is an improvement over the upper bound of o epsilon left t phi 5 12 epsilon right given by iwaniec and sarnak in a recent work of ghosh reznikov and sarnak the authors proved for all even hecke maass forms that the number of nodal domains which intersect a geodesic segment of eta grows faster than t phi 1 12 epsilon for any epsilon 0 under the assumption that the lindelof hypothesis is true and that the geodesic segment is long enough upon removing a density zero subset of even hecke maass forms we prove without making any assumptions that the number of nodal domains grows faster than t phi 1 8 epsilon for any epsilon 0

technical communique a zakai equation derivation of the extended kalman filter a discrete time filter is considered where both the observation and signal process have non linear dynamics with additive gaussian noise using the reference probability framework a convolution type zakai equation is obtained which updates the unnormalized conditional density using first order approximations this equation can be solved recursively and the extended kalman filter can be derived

lower bounds for nodal sets of dirichlet and neumann eigenfunctions let phi be a dirichlet or neumann eigenfunction of the laplace beltrami operator on a compact riemannian manifold with boundary we prove lower bounds for the size of the nodal set phi 0

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

exponential decay of correlations for piecewise cone hyperbolic contact flows we prove exponential decay of correlations for a realistic model of piecewise hyperbolic flows preserving a contact form in dimension three this is the first time exponential decay of correlations is proved for continuous time dynamics with singularities on a manifold our proof combines the second author s version liverani in ann math 159 1275 1312 2004 of dolgopyat s estimates for contact flows and the first author s work with gouezel j mod dyn 4 91 137 2010 on piecewise hyperbolic discrete time dynamics

on the convergence properties of the em algorithm two convergence aspects of the em algorithm are studied i does the em algorithm find a local maximum or a stationary value of the incomplete data likelihood function ii does the sequence of parameter estimates generated by em converge several convergence results are obtained under conditions that are applicable to many practical situations two useful special cases are a if the unobserved complete data specification can be described by a curved exponential family with compact parameter space all the limit points of any em sequence are stationary points of the likelihood function b if the likelihood function is unimodal and a certain differentiability condition is satisfied then any em sequence converges to the unique maximum likelihood estimate a list of key properties of the algorithm is included

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

on decay of correlations in anosov flows there is some disagreement about the meaning of the phrase chaotic flow however there is no doubt that mixing anosov flows provides an example of such systems anosov systems were introduced and extensively studied in his classical memoir a among other things he proved the following fact known now as anosov alternative for flows either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an anosov diffeomorphism by a constant roof function if the first alternative holds gt is mixing with respect to every gibbs measure see pp2 now the natural question is what is the estimated rate of mixing this is certainly one of the simplest questions concerning correlation decay in continuous time systems nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure the easier case of anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of sinai ruelle and bowen b2 namely one uses markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing this method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations however the simplest example suspensions with locally constant roof functions never have such a property ri one can use the above observation to produce examples of axiom a flows with arbitrary slow correlation decay it became clear therefore that some additional geometric properties should be taken into account in recent work chernov chl ch2 has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature his method relies on the technique of markov approximation developed in chl the aim of this paper is to combine geometric considerations of chernov with the thermodynamic formalism approach the later method seems to be more appropriate than markov approximations since it gives simpler

generators of simple lie algebras in arbitrary characteristics in this paper we study the minimal number of generators for simple lie algebras in characteristic 0 or p 3 we show that any such algebra can be generated by 2 elements we also examine the one and a half generation property i e when every non zero element can be completed to a generating pair we show that classical simple algebras have this property and that the only simple cartan type algebras of type w which have this property are the zassenhaus algebras

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

iterative and sequential algorithms for multisensor signal enhancement in problems of enhancing a desired signal in the presence of noise multiple sensor measurements will typically have components from both the signal and the noise sources when the systems that couple the signal and the noise to the sensors are unknown the problem becomes one of joint signal estimation and system identification the authors specifically consider the two sensor signal enhancement problem in which the desired signal is modeled as a gaussian autoregressive ar process the noise is modeled as a white gaussian process and the coupling systems are modeled as linear time invariant finite impulse response fir filters the main approach consists of modeling the observed signals as outputs of a stochastic dynamic linear system and the authors apply the estimate maximize em algorithm for jointly estimating the desired signal the coupling systems and the unknown signal and noise spectral parameters the resulting algorithm can be viewed as the time domain version of the frequency domain approach of feder et al 1989 where instead of the noncausal frequency domain wiener filter the kalman smoother is used this approach leads naturally to a sequential adaptive algorithm by replacing the kalman smoother with the kalman filter and in place of successive iterations on each data block the algorithm proceeds sequentially through the data with exponential weighting applied to allow adaption to nonstationary changes in the structure of the data a computationally efficient implementation of the algorithm is developed an expression for the log likelihood gradient based on the kalman smoother filter output is also developed and used to incorporate efficient gradient based algorithms in the estimation process

new finite dimensional filters for parameter estimation of discrete time linear gaussian models the authors derive a new class of finite dimensional recursive filters for linear dynamical systems the kalman filter is a special case of their general filter apart from being of mathematical interest these new finite dimensional filters can be used with the expectation maximization em algorithm to yield maximum likelihood estimates of the parameters of a linear dynamical system important advantages of their filter based em algorithm compared with the standard smoother based em algorithm include 1 substantially reduced memory requirements and 2 ease of parallel implementation on a multiprocessor system the algorithm has applications in multisensor signal enhancement of speech signals and also econometric modeling

hidden gauss markov models for signal classification continuous state hidden markov models cs hmms are developed as a tool for signal classification analogs of the baum 1972 viterbi 1962 and baum welch algorithms are formulated for this class of models the cs hmm algorithms are then specialized to hidden gauss markov models hgmms with linear gaussian state transition and output densities a new gaussian refactorization lemma is used to show that the baum and viterbi algorithms for hgmms are implemented by two different formulations of the fixed interval kalman smoother the measurement likelihoods obtained from the forward pass of the hgmm baum algorithm and from the kalman filter innovation sequence are shown to be equal a direct link between the baum welch training algorithm and an existing expectation maximization em algorithm for gaussian models is demonstrated a new expression for the cross covariance between time adjacent states in hgmms is derived from the off diagonal block of the conditional joint covariance matrix a parameter invariance structure is noted for the hgmm likelihood function cs hmms and hgmms are extended to incorporate mixture densities for the a priori density of the initial state application of hgmms to signal classification is demonstrated with a three class test simulation

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

small scale equidistribution for a point scatterer on the torus we study the small scale distribution of the eigenfunctions of a point scatterer the laplacian perturbed by a delta potential on two and three dimensional flat tori in two dimensions we establish small scale equidistribution for the new eigenfunctions holding all the way down to the planck scale in three dimensions small scale equidistribution is established for all of the new eigenfunctions at certain scales

a characterisation of f2 g in 1 we obtained a short proof of the theorem of thompson that a finite group is soluble if and only if every two of its elements generate a soluble subgroup a natural next question to ask is what happens if we keep one of the generators fixed for a finite group we define sol g the soluble radical of g to be the largest normal soluble subgroup of g and sol g x g x g is soluble for all g g

intersection bounds for nodal sets of planar neumann eigenfunctions with interior analytic curves let r be a bounded piecewise smooth domain and be a neumann or dirichlet eigenfunction with eigenvalue and nodal set n x x 0 let h be an interior c curve consider the intersection number n h h n we first prove that for general piecewise analytic domains and under an appropriate goodness condition on h see theorem 1 1 n h oh as then using theorem 1 1 we prove in theorem 1 2 that the bound in is satisfied in the case of quantum ergodic qe sequences of interior eigenfunctions provided is convex and h has strictly positive geodesic curvature

finite groups in which every two elements generate a soluble subgroup this result was first obtained by john thompson as a by product of his classification of n groups 7 the proof we present is short and direct it does not use any classification theorem the possibility of obtaining results of this type by direct means was first realized by martin powell who proved that a finite group in which every three elements generate a soluble subgroup is soluble an account of his work can be found in 2 pages 473 476 powell s argument uses the hall higman theorem b we use a different strategy that we shall now describe let g be a soluble group and p an odd prime divisor of g if g contains an abelian p subgroup that normalizes no nontrivial p subgroup of g then op g 1 so as g is soluble we must have op g 6 1 if every abelian p subgroup of g normalizes a nontrivial p subgroup then a result of thompson 3 theorem 1 12 page 19 implies that op g 6 1 the point of these

exact adaptive filters for markov chains observed in gaussian noise abstract a discrete time finite state markov chain is observed through a real or vector valued function whose values are corrupted by gaussian noise by introducing a new measure exact unnormalized recursive estimates and smoothers are obtained for the state of the markov chain for the number of jumps from one state to another for the occupation time in any state and for processes related to the observation parameters using the em algorithm these allow estimates of all the parameters of the model to be revised including the variance of the gaussian noise in the observations the filters are therefore adaptive or self tuning

further solvable analogues of the baer suzuki theorem and generation of nonsolvable groups abstract let g be an almost simple group we prove that if x g has prime order p 5 then there exists an involution y such that x y is not solvable also if x is an involution then there exist three conjugates of x that generate a nonsolvable group unless x belongs to a short list of exceptions which are described explicitly we also prove that if x has order 6 or 9 then there exist two conjugates that generate a nonsolvable group

nodal sets of eigenfunctions on riemannian manifolds soit m n une variete connexe compacte a metrique riemannienne c soit le laplacien de m soit f une fonction propre reelle de de valeur propre on demontre que la fonction propre f s annule au plus a l ordre c 1 2 pour tout point dans m