theta dichotomy and doubling epsilon factors for spn f in this work we aim to prove the theta dichotomy conjecture for dual pairs of type sp w o v with dim v odd the crux of the proof relies on the analysis of the reducibility of a certain degenerate principal series representation of the group sp w having double the rank of sp w this analysis is carried out by way of a normalized standard intertwining operator which is also used to define an epsilon factor coming from the doubling integral construction of piatetski shapiro and rallis in this paper we also illustrate the connection between the doubling epsilon factor and the theta dichotomy conjecture

h 2 optimal filtering for continuous time periodic linear stochastic systems with state dependent noise abstract this note addresses the problem of optimal h 2 filtering for a class of continuous time time varying stochastic systems the time variation character is considered to be periodic the state space representation of the optimal filter is designed based on the unique periodic solution of a suitable lyapunov differential equation and the periodic and stabilizing solution of a suitable generalized periodic riccati differential equation some numerical experiments are also added to show the effectiveness of the proposed method

linear matrix inequalities riccati equations and indefinite stochastic linear quadratic controls this paper deals with an optimal stochastic linear quadratic lq control problem in infinite time horizon where the diffusion term in dynamics depends on both the state and the control variables in contrast to the deterministic case we allow the control and state weighting matrices in the cost functional to be indefinite this leads to an indefinite lq problem which may still be well posed due to the deep nature of uncertainty involved the problem gives rise to a stochastic algebraic riccati equation sare which is however fundamentally different from the classical algebraic riccati equation as a result of the indefinite nature of the lq problem to analyze the sare we introduce linear matrix inequalities lmis whose feasibility is shown to be equivalent to the solvability of the sare moreover we develop a computational approach to the sare via a semi definite programming associated with the lmis finally numerical experiments are reported to illustrate the proposed approach

phase space methods and path integration the analysis and computation of scalar wave equations abstract the scalar helmholtz equation plays a significant role in studies of electromagnetic seismic and acoustic direct wave propagation phase space or microscopic methods and path functional integral representations provide the appropriate framework to extend homogeneous fourier methods to inhomogeneous environments the two complementary approaches to this analysis and computation of the n dimensional helmholtz propagator are reviewed for the factorization one way path integration invariant imbedding approach the exact solution of the helmholtz composition equation for the weyl square root operator symbol is presented in the quadratic case the filtered one way phase space marching algorithm is examined in detail and compared numerically with wide angle one way partial differential wave equations formally derived from approximation theory for the second approach which directly constructs approximate two way path functionals the feasibility of a monte carlo statistical evaluation of the feynman garrod propagator is discussed

gradient estimates via non linear potentials we present pointwise gradient bounds for solutions to p laplacean type non homogeneous equations employing non linear wolff type potentials and then prove similar bounds via suitable caloric potentials for solutions to parabolic equations

factorization and path integration of the helmholtz equation numerical algorithms the propagator for the reduced scalar helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation path functional integrals are taken to provide the principal representation of the propagator and are computed directly the path integral is the primary tool in extending the classical fourier methods so appropriate for wave propagation in homogeneous media to inhomogeneous media for transversely inhomogeneous environments the n dimensional helmholtz equation can be exactly factored into separate forward and backward one way wave equations a parabolic based one way phase space path integral construction provides the generalization of the tappert hardin split step fft algorithm to the full one way factored helmholtz wave equation these extended marching algorithms can readily accommodate density profiles and range updating and further in conjunction with imbedding methods provide the basis for incorporating backscatter effects in a complementary manner for general range dependent environments elliptic based two way path integral constructions lead to an approximate representation of the propagator feynman garrod and a natural statistical monte carlo means of evaluation taken together the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process the one way marching algorithms are applied to ocean acoustic environments seismological environments and extreme model environments designed to establish their range of validity and manner of breakdown

distributed h state estimation with stochastic parameters and nonlinearities through sensor networks the finite horizon case this paper deals with the distributed h state estimation problem for a class of discrete time varying nonlinear systems with both stochastic parameters and stochastic nonlinearities the system measurements are collected through sensor networks with sensors distributed according to a given topology the purpose of the addressed problem is to design a set of time varying estimators such that the average estimation performance of the networked sensors is guaranteed over a given finite horizon through available output measurements from not only the individual sensor but also its neighboring sensors a necessary and sufficient condition is established to achieve the h performance constraint and then the estimator design scheme is proposed via a certain h 2 type criterion the desired estimator parameters can be obtained by solving coupled backward recursive riccati difference equations rdes a numerical simulation example is provided to demonstrate the effectiveness and applicability of the proposed estimator design approach

the class of a torus in the grothendieck ring of varieties we establish a formula for the classes of certain tori in the grothendieck ring of varieties expressing them in terms of the natural lambda structure on the grothendieck ring more explicitly we will see that if l is the torus of invertible elements in the n dimensional separable k algebra l then the class of l can be expressed as an alternating sum of the images of the spectrum of l under the lambda operations multiplied by powers of the lefschetz class this formula is suggested from the cohomology of the torus illustrating a heuristic method that can be used in other situations to prove the formula will require some rather explicit calculations in the grothendieck ring to be able to perform these we introduce a homomorphism from the burnside ring of the absolute galois group of k to the grothendieck ring of varieties over k in the process we obtain some information about the structure of the subring generated by zero dimensional varieties

on the isoperimetric deficit in gauss space we prove a sharp quantitative version of the isoperimetric inequality in the space bbb r n endowed with the gaussian measure

stability and l_2 synthesis of a class of periodic piecewise time varying systems in this paper the stability stabilization and l 2 gain problems are investigated for periodic piecewise systems with time varying subsystems continuous lyapunov function with time varying lyapunov matrix is adopted a condition guaranteeing the negative definiteness of a matrix polynomial deriving from the lyapunov derivative is first obtained based on such a condition an exponential stability condition is provided moreover a state feedback controller with time varying gain is developed to stabilize the unstable periodic piecewise time varying system the l 2 gain criterion for periodic piecewise time varying system is also studied numerical examples are given to show the validity of the proposed techniques

some thoughts on the jacobian conjecture part ii in abhyankar s purdue lectures of 1971 the bivariate jacobian conjecture was settled for the case of two plus epsilon characteristic pairs in the published version the epsilon part got left out now we take care of the omission by preparing for a sharper result with full proof in part iii the jacobian method is applied to giving a new simple proof of jung s automorphism theorem a detailed description of the degreewise newton polygon is given some thoughts on the multivariate jacobian conjecture are included

nondispersive radial solutions to energy supercritical non linear wave equations with applications in this paper we establish optimal pointwise decay estimates for non dispersive compact radial solutions to non linear wave equations in 3 dimensions in the energy supercritical range as an application we show for the full energy supercritical range in the defocusing case that if the scale invariant sobolev norm of a radial solution remains bounded in its maximal interval of existence then the solution must exist for all times and scatter

a construction of elements in the bernstein center for quasi split groups the bernstein center of a reductive p adic group is the algebra of conjugation invariant distributions on the group which are essentially compact i e invariant distributions whose convolution against a locally constant compactly supported function is again locally constant compactly supported the center acts naturally on any smooth representation and if the representation is irreducible each element of the center acts as a scalar for a connected reductive quasi split group we show certain linear combinations of orbital integrals belong to the bernstein center furthermore when these combinations are projected to a bernstein component they form an ideal in the bernstein center which can be explicitly described and is often a principal ideal the elements constructed here should have applications to various questions in harmonic analysis

h filtering with randomly occurring sensor saturations and missing measurements in chap 5 the h filtering problem is studied for a class of nonlinear systems with randomly occurring incomplete information where the considered incomplete information includes both the sensor saturations and the missing measurements a new phenomenon of sensor saturation namely randomly occurring sensor saturation ross is first put forward in order to better reflect the reality in a networked environment then a novel sensor model is established to account for both the ross and missing measurement in a unified representation based on this sensor model a regional h filter with a certain ellipsoid constraint is designed such that the filtering error dynamics is locally mean square asymptotically stable and the h norm requirement is satisfied

propagation modelling based on wave field factorization and invariant imbedding the proposed propagation models address the propagation of cw acoustic fields through a transition region separating two different range independent oceans wave field factorization refers to the conceptual decoupling of the radiation fields in the two range independent oceans into positive and negative propagating components invariant imbedding refers to the reformulation of the propagation problem from a two point boundary value problem to an initial value problem in the reformulation the extent of the transition region is an independent variable the formulation is expressed in terms of reflection and transmission operators which apply to the transition region the proposed models are generalizations of existent models in reflection seismology these latter models allow for no cross range variability of wave speed in either the two now homogeneous oceans nor in the transition region the seismology models do accommodate p s coupling which is neglected in the acoustic application the incorporat

algebraic theory of dicritical divisors the topological theory of dicritical divisors is algebracized which makes it valid for nonzero characteristic as well as mixed characteristic

stochastic h infty we consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations our objective is to develop an h infty type theory for such systems we prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations this enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to be controlled output below a given threshhold gamma 0 in the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead the connection between h infty theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases

optimal filtering for a class of linear ito stochastic systems the dichotomic case this paper focuses on the problem of optimal h2 filtering for a class of continuous time stochastic systems without assuming their exponential stability in the mean square sense indeed the stability assumption is relaxed and we assume instead that the lyapunov operator associated to the dynamical stochastic system is exponentially dichotomic the optimal solution of the considered optimization problem is expressed in terms of the unique solution of a suitable algebraic lyapunov equation and the stabilizing solution of a certain algebraic riccati equation

local to global extensions for gln in non zero characteristic a characterization of yf s p sym2 v and yf s p a2 v let f be a locally compact field of positive characteristic in his thesis the second author used the langlands shahidi method to define factors for the symmetric square and exterior square representations of the dual group of gl n f we prove here that such factors are characterized by local properties including a multiplicativity property with respect to parabolic induction and their role in global functional equations for l functions for this we prove that a given cuspidal representation of gl n f is the component at some place of a global automorphic representation the ramification of which is controlled at other places in fact we use an equivalent result for l adic galois representations due to o gabber and n katz and transport it via the global langlands correspondence proved by l lafforgue

on the shape of possible counterexamples to the jacobian conjecture abstract we improve the algebraic methods of abhyankar for the jacobian conjecture in dimension two and describe the shape of possible counterexamples we give an elementary proof of the result of heitmann in 5 which states that gcd deg p deg q 16 for any counterexample p q we also prove that gcd deg p deg q 2 p for any prime p