equilibrium in continuous time financial markets endogenously dynamically complete markets we prove existence of equilibrium in a continuous time securities market in which the securities are potentially dynamically complete the number of securities is at least one more than the number of independent sources of uncertainty we prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model our result is universal rather than generic dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a brownian filtration and from a mild exogenous nondegeneracy condition on the terminal security dividends the nondegeneracy condition which requires that finding one point at which a determinant of a jacobian matrix of dividends is nonzero is very easy to check we find that the equilibrium prices consumptions and trading strategies are well behaved functions of the stochastic process describing the evolution of information we prove that equilibria of discrete approximations converge to equilibria of the continuous time economy

positive definite solutions of the matrix equations x a x qa q abstract in this paper we investigate nonlinear matrix equations x a x q a q and x a x q a q where q 0 1 we derive necessary conditions and sufficient conditions for the existence of positive definite solutions for these equations we provide a sufficient condition for the equation x a x q a q to have two different positive definite solutions and a sufficient condition for the equation x a x q a q to have a unique positive definite solution we also propose iterative methods for obtaining positive definite solutions for these equations

the iterative method for solving nonlinear matrix equation xs a x ta q abstract nonlinear matrix equation x s a x t a q where q is an hermitian positive definite matrix a is the conjugate transpose of the matrix a and both of s and t are positive integers has widely applied background in this paper necessary conditions and sufficient conditions for the existence of hermitian positive definite solutions of this equation are derived an iterative method for obtaining hermitian positive definite solutions is proposed moreover a perturbation bound for the hermitian positive definite solutions is given

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

regularity and decay of solutions of nonlinear harmonic oscillators abstract we prove sharp analytic regularity and decay at infinity of solutions of variable coefficients nonlinear harmonic oscillators namely we show holomorphic extension to a sector in the complex domain with a corresponding gaussian decay according to the basic properties of the hermite functions in r d our results apply in particular to nonlinear eigenvalue problems for the harmonic oscillator associated to a real analytic scattering or asymptotically conic metric in r d as well as to certain perturbations of the classical harmonic oscillator

on the spectral side of arthur s trace formula combinatorial setup in arthur s trace formula a ubiquitous role is played by certain limiting expressions arising from piecewise smooth functions with respect to projections of the coxeter fan g m families these include terms resulting from intertwining operators on the spectral side and volumes of polytopes on the geometric side we introduce the combinatorial concept of a compatible family with respect to an arbitrary polyhedral fan and obtain new formulas for the corresponding limiting expressions in this general framework our formulas can be regarded as algebraic generalizations of certain volume formulas for convex polytopes in a companion paper the results are used to study the spectral side of the trace formula

on the inner product of truncated eisenstein series introduction 1 the problem 2 residues of cuspidal eisenstein series 83 exponents 4 a comparison between two groups 5 a property of the truncation operator 6 the constant terms of eisenstein series 7 the negative dual chamber 8 coefficients of the zero exponents 9 conclusion bibliography

on matrix equations x a x 2a i the two matrix equations x a x 2 a i and x a x 2 a i are studied we construct iterative methods for obtaining positive definite solutions of these equations sufficient conditions for the existence of two different solutions of the equation x a x 2 a i are derived sufficient conditions for the existence of positive definite solutions of the equation x a x 2 a i are given numerical experiments are discussed 2001 elsevier science

an accelerated cyclic reduction based solvent method for solving quadratic eigenvalue problem of gyroscopic systems abstract the quadratic eigenvalue problem qep 2 m g k x 0 with m t m being positive definite k t k being negative definite and g t g is associated with gyroscopic systems in guo 2004 a cyclic reduction based solvent crs method was proposed to compute all eigenvalues of the above mentioned qep firstly the problem is converted to find a suitable solvent of the quadratic matrix equation qme m x 2 g x k 0 then using a cayley transformation and a proper substitution the qme is transformed into the nonlinear matrix equation nme z a t z 1 a q with a m k g and q 2 m k the problem finally can be solved by applying the cr method to obtain the maximal symmetric positive definite solution of the nme as long as the qep has no eigenvalues on the imaginary axis or for some cases where the qep has eigenvalues on the imaginary axis however when all eigenvalues of the qep are far away from or near the origin the cayley transformation seems not to be the best one and the convergence rate of the crs method proposed in guo 2004 might be further improved in this paper inspired by using a doubling algorithm to solve the qme we use a mobius transformation instead of the cayley transformation to present an accelerated crs acrs method for solving the qep of gyroscopic systems in addition we discuss the selection strategies of optimal parameter for the acrs method numerical results demonstrate the efficiency of our method

on the existence of a positive definite solution of the matrix equation x atx 1a i the question is raised under which conditions on the real square matrix a the matrix equation x atx 1a i has a real symmetric positive definite solution x both necessary and sufficient solvability conditions on a are derived moreover we give an algorithm to calculate the solution for a number of special cases we also present an analytic solution

on the positive definite solutions of nonlinear matrix equation x a x δa q abstract in this paper we consider the positive definite solutions of nonlinear matrix equation x a x a q where 0 1 which appears for the first time in s m el sayed a c m ran on an iteration methods for solving a class of nonlinear matrix equations siam j matrix anal appl 23 2001 632 645 the necessary and sufficient conditions for the existence of a solution are derived an iterative algorithm for obtaining the positive definite solutions of the equation is discussed the error estimations are found

spectral asymptotics for arithmetic quotients of sl n r so n in this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space sl n r so n in particular we obtain weyl s law with an estimation on the remainder term this extends some of the main results of duistermaat kolk and varadarajan dkv1 to this setting

the spectral side of arthur s trace formula the trace formula is one of the most important tools in the theory of automorphic forms it was invented in the 1950 s by selberg who mostly studied the case of hyperbolic surfaces and was later on developed extensively by arthur in the generality of an adelic quotient of a reductive group over a number field here we provide an explicit expression for the spectral side improving arthur s fine spectral expansion as a result we obtain its absolute convergence for a wide class of test functions

solutions and perturbation estimates for the matrix equations x a x n a q in this paper we investigate the equations x a x na q for the existence of positive definite solutions and perturbation bounds for these solutions are derived a sufficient condition for uniqueness of a unique positive definite solution of the equation x a x na q is given the results are illustrated by using numerical examples

regularity in time of solutions to nonlinear schrodinger equations abstract in this paper we consider the regularity of solutions to nonlinear schrodinger equations nls i t u 1 2 u f u u t x r r n u 0 x r n where f is a polynomial of degree p with complex coefficients we prove that if the initial function is in some gevrey class then there exists a positive constant t such that the solution u of nls is in the gevrey class of the same order as in the initial data in time variable t t t 0 in particilar we show that if the initial function has an analytic continuation on the complex domain a 1 a 2 z c n z j x j iy j x j a 2 tan x j y j a 2 tan x j j 1 2 n a 2 0 see fig 1 where 0 1 a 1 a 1 t and such that the solution u of nls is analytic in time variable t t t 0 and has an analytic continuation on z 0 t i arg z 0 t t where sin a 1 1 2 a 1 2 a 2 3 a 2 formula 1 r when x r

on the hermitian positive definite solutions of nonlinear matrix equation x s σ m i 1 a i x t i a i q in this paper the hermitian positive definite solutions of nonlinear matrix equation x email protected i 1 ma i x t ia i q are considered where q is a hermitian positive definite matrix a i are nonsingular complex matrices s m are positive numbers and 0

global existence of small analytic solutions to nonlinear schrodinger equations in this paper we consider the following nonlinear schrodinger equation in r n n 2 i u 1 2 u f u u u t x er r n u 0 x x xer n

space time analyticity of weak solutions to linear parabolic systems with variable coefficients analytic smoothing properties of a general strongly parabolic linear cauchy problem of second order in rn 0 t with analytic coefficients in space and time variables are investigated they are expressed in terms of holomorphic continuation of global weak l2 type solutions to the system given 0 0 are constants depending upon t the proof uses the extension of a solution to an l2 type solution in a domain in cn c such that this extension satisfies the cauchy riemann equations the holomorphic extension is thus obtained in a hardy space h2 applications include market completion by european options in finance

computing the extremal positive definite solutions of a matrix equation an efficient and numerically stable implementation of a known algorithm is suggested for finding the extremal positive definite solutions of the matrix equation x a x 1 a i if such solutions exist the convergence rate is analyzed a new algorithm that avoids matrix inversion is presented numerical examples are given to illustrate the effectiveness of the algorithms

space analyticity for the navier stokes and related equations with initial data inlp abstract we introduce a method of estimating the space analyticity radius of solutions for the navier stokes and related equations in terms of l p and l norms of the initial data the method enables us to express the space analyticity radius for 3d navier stokes equations in terms of the reynolds number of the flow also for the kuramoto sivashinsky equation we give a partial answer to a conjecture that the radius of space analyticity on the attractor is independent of the spatial period

an iterative method for obtaining the least squares solutions of quadratic inverse eigenvalue problems over generalized hamiltonian matrix with submatrix constraints abstract in this paper we consider a class of constrained matrix quadratic inverse eigenvalue problem and its optimal approximation problem it is proved that the proposed algorithm always converge to the generalized hamiltonian solutions with a submatrix constraint of problem 1 1 within finite iterative steps in the absence of roundoff error in addition by choosing a special kind of initial matrices it is shown that the minimum norm solution of problem 1 1 can be obtained consequently at last for a given matrix group in the solution set of problem 1 1 it is proved that the unique optimal approximation solution of problem 1 2 can be also obtained some numerical results are reported to demonstrate the efficiency of our algorithm

intertwining operators and residues i weighted characters operateurs d entrelacement normalisation groupes reels groupes p adiques representations standard les distributions j m la distribution j m x residus changements de contour les espaces 7b h ac g f s et 7b f ac g f s l application m

on twisted lifting if a is a generator of the galois group of a finite cyclic extension e f of local or global fields and e is a character of ce ex or ex ax whose restriction to cf has order n then the irreducible admissible or automorphic representations 1 cf denotes the multiplicative group fx if f is local and the idele class group fx a x if f is global put g gl and fix a character e of ce whose restriction to cf is of order n it determines a character e of g e or g e g ae by e g e det g the restriction of e to cf determines by class field theory a cyclic extension fe of f of degree n fix a generator a of the galois group gal e f denote by r g the set of equivalence classes of irreducible admissible or automorphic representations of g e or g ae then a acts on r g by v g vr a g g in g e or g a and we denote by re g the set of vr in r g with v n e theorem fftere ex s i a natural bijection between are g and the set of gal f f orbits of characters on nl fcl where l fee in the case e f this is due to kazhdan k in the local case and if it is cuspidal with two supercuspidal components kazhdan s proof is remarkably elegant com plete local and global results are derived from the study of the unramified case only our proof follows kazhdan s and uses kottwitz s base change lemma ko and arthur s explicit trace formula a we also show lemma 15 1 that the above 77 are induced ip r from a representation t t t e t e v e where e e 1 and t is a super cuspidal representation of gl w e n mm f is a parabolic subgroup with levi component gl m m then if l e n fe is an extension of f of degree we must have that divides m corollary 16 2 and m l is prime to e proposition 17 1 for example in the special case where e fe each it of the above type is induced from a character of a borel subgroup hence it is not in the discrete series we proved this result in f when e fe is a quadratic extension of f using the trace formula twisted as here in fact this motivated us to look for a twisted analogue of kazhdan s lifting further description of our vr is given in 16 and 17

iterative solution of two matrix equations we study iterative methods for finding the maximal hermitian positive definite solutions of the matrix equations x a x 1 a q and x a x 1 a q where q is hermitian positive definite general convergence results are given for the basic fixed point iteration for both equations newton s method and inversion free variants of the basic fixed point iteration are discussed in some detail for the first equation numerical results are reported to illustrate the convergence behaviour of various algorithms

improved methods and starting values to solve the matrix equations x a x 1a i iteratively the two matrix iterations x k 1 i a x 1 k a are known to converge linearly to a positive definite solution of the matrix equations x a x 1 a i respectively for known choices of x 0 and under certain restrictions on a the convergence for previously suggested starting matrices x 0 is generally very slow this paper explores different initial choices of x 0 in both iterations that depend on the extreme singular values of a and lead to much more rapid convergence further the paper offers a new algorithm for solving the minus sign equation and explores mixed algorithms that use newton s method in part

on the spectral side of arthur s trace formula absolute convergence we derive a renement of the spectral expansion of arthur s trace formula the expression is absolutely convergent with respect to the trace norm

properties of positive definite solutions of the equation x a x 2a i abstract in this paper we discuss some properties of a positive definite solution of the matrix equation x a x 2 a i two effective iterative methods for computing a positive definite solution of this equation are proposed necessary and sufficient conditions for existence of a positive definite solution are derived numerical experiments are executed with these methods

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

some investigation on hermitian positive definite solutions of the matrix equation xs a x ta q abstract in this paper the hermitian positive definite solutions of the matrix equation x s a x t a q are considered where q is an hermitian positive definite matrix s and t are positive integers necessary and sufficient conditions for the existence of an hermitian positive definite solution are derived a sufficient condition for the equation to have only two different hermitian positive definite solutions and the formulas for these solutions are obtained in particular the equation with the case aq 1 2 q 1 2 a is discussed a necessary condition for the existence of an hermitian positive definite solution and some new properties of the hermitian positive definite solutions are given which generalize the existing related results

on the matrix equation x atx 1a i simple necessary and sufficient conditions for the matrix equation x atx 1a i to have a positive definite solution are given these conditions are used to explain some properties of the equation and to obtain new information finally we prove a conjecture on the extremal solutions

gevrey regularity of the periodic gkdv equation abstract using the multilinear estimates which were derived for proving well posedness of the generalized korteweg de vries gkdv equation it is shown that if the initial data belongs to gevrey space g 1 in the space variable then the solution to the corresponding cauchy problem for gkdv belongs also to g in the space variable moreover the solution is not necessarily g in the time variable however it belongs to g 3 near 0 when 1 these are analytic regularity results for gkdv

quasi gradient based inversion free iterative algorithm for solving a class of the nonlinear matrix equations abstract inspired by the gradient based and inversion free iterations a new quasi gradient based inversion free iterative algorithm is proposed for solving the nonlinear matrix equation x a t x n a i the convergence proof of the suggested algorithm is given several matrix norm inequalities are established to depict the convergence properties of this algorithm three numerical examples are given to illustrate the effectiveness of the suggested algorithms

the selberg trace formula viii contribution from the continuous spectrum the purpose of this paper is to isolate the contribution from the continuous spectrum to the selberg trace formula

solutions to matrix equations x axb cy r and x axb cy r abstract the present work proposed an alternative approach to find the closed form solutions of the nonhomogeneous yakubovich matrix equation x a x b c y r based on the derived closed form solution to the nonhomogeneous yakubovich matrix equation the solutions to the nonhomogeneous yakubovich quaternion j conjugate matrix equation x a x b c y r are obtained by the use of the real representation of a quaternion matrix the existing complex representation method requires the coefficient matrix a to be a block diagonal matrix over complex field in contrast in this publication we allow a quaternion matrix of any dimension as an application eigenstructure assignment problem for descriptor linear systems is considered

on the hermitian positive definite solutions of nonlinear matrix equation xs a x ta q abstract nonlinear matrix equation xs a x ta q where a q are n n complex matrices with q hermitian positive definite has widely applied background in this paper we consider the hermitian positive definite solutions of this matrix equation with two cases s 1 0

on the existence of hermitian positive definite solutions of the matrix equation xs a x ta q abstract in this paper the existence of hermitian positive definite solutions of the general nonlinear matrix equation x s a x t a q is studied systematically and deeply a new estimate of hermitian positive definite solutions is derived based on a fixed point theorem some new sufficient conditions and new necessary conditions for the existence of hermitian positive definite solutions are obtained in the end a necessary and sufficient condition for the existence of a hermitian positive definite solution is proved

on a family of distributions obtained from eisenstein series i application of the paley wiener theorem introduction this is the first of two papers aimed at finding explicit formulas for certain distributions the distributions are obtained from eisenstein series and are important ingredients in the trace formula in this paper we shall resolve some analytic difficulties that center around an interchange of two limits the next paper will be devoted to the calculations which will culminate in the formulas let g be a reductive algebraic group defined over q as usual we will write g a for the intersection of the kernels of the maps

terms in the selberg trace formula for 3 ℒ 3 ℛ 3 ℛ associated to eisenstein series coming from a minimal parabolic subgroup in this paper we compute the contribution to the trace formula for sl 3 z of the integrals associated to inner products of eisenstein series we show these reduce to corresponding integrals for a lower rank trace formula plus a few residual terms

new lower bounds on the radius of spatial analyticity for the kdv equation abstract the radius of spatial analyticity for solutions of the kdv equation is studied it is shown that the analyticity radius does not decay faster than t 1 4 as time t goes to infinity this improves the works of selberg and da silva 2017 30 and tesfahun 2017 34 our strategy mainly relies on a higher order almost conservation law in gevrey spaces which is inspired by the i method

global solutions of the derivative schrodinger equation in a class of functions analytic in a strip lower bounds on the rate of decrease in time of a uniform radius of spatial analyticity for solutions of the derivative schrodinger equation are derived the bounds depend algebraically on time they are valid as long as the initial datum is of order one in l2 norm and satisfies suitable spatial decay requirements on the real axis

on the hermitian positive definite solution of the nonlinear matrix equation x i 1mai x 1ai i abstract in this paper we study the matrix equation x i 1 m a i x 1 a i i where ai i 1 2 m is square matrices and obtain some conditions for the existence of positive definite solution of this equation two iterative algorithms to find the maximal positive definite solution are given some numerical results are reported to illustrate the effectiveness of the algorithms

holomorphic extension of solutions of semilinear elliptic equations abstract we prove sharp analytic estimates and holomorphic extensions in sectors of c d for the solutions of a wide class of semilinear elliptic differential and pseudodifferential equations in r d improving earlier results where the extension was shown for a strip moreover we derive exponential decay estimates for such extended solutions the results presented apply in particular to solitary wave solutions of many classical nonlinear evolution equations as kdv type long wave type and schrodinger equations

iterative methods for the extremal positive definite solution of the matrix equation x a x αa q in this paper the inversion free variant of the basic fixed point iteration methods for obtaining the maximal positive definite solution of the nonlinear matrix equation x a x aa q with the case 0 1 are proposed some necessary conditions and sufficient conditions for the existence of positive definite solutions for the matrix equation are derived numerical examples to illustrate the behavior of the considered algorithms are also given

gevrey regularity for a class of water wave models abstract local well posedness for a class of higher order nonlinear dispersive partial differential equations is obtained in spaces of functions analytic on a strip around the real axis the proof relies on estimates in space time norms adapted to the linear part of the equation the class of equations in view contains a number of equations arising in the modeling of waves in fluids and also in other applications