symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

lie symmetry analysis and some new exact solutions of the wu zhang equation the lie symmetry analysis and the basic similarity reductions are performed for the wu zhang equation a 2 1 dimensional nonlinear dispersive wave equation some new exact solutions generated from the similarity transformation are provided they demonstrate some new three dimensional features of a single solitary wave and two interacting solitary waves

new results on hermitian matrix rank one decomposition in this paper we present several new rank one decomposition theorems for hermitian positive semidefinite matrices which generalize our previous results in huang and zhang math oper res 32 3 758 768 2007 ai and zhang siam j optim 19 4 1735 1756 2009 the new matrix rank one decomposition theorems appear to have wide applications in theory as well as in practice on the theoretical side for example we show how to further extend some of the classical results including a lemma due to yuan math program 47 53 63 1990 the classical results on the convexity of the joint numerical ranges pang and zhang in unpublished manuscript 2004 au yeung and poon in southeast asian bull math 3 85 92 1979 and the so called finsler s lemma bohnenblust in unpublished manuscript au yeung and poon in southeast asian bull math 3 85 92 1979 on the practical side we show that the new results can be applied to solve two typical problems in signal processing and communication one for radar code optimization and the other for robust beamforming the new matrix decomposition theorems are proven by construction in this paper and we demonstrate that the constructive procedures can be implemented efficiently stably and accurately the url of our matlab programs is given in this paper we strongly believe that the new decomposition procedures as a means to solve non convex quadratic optimization with a few quadratic constraints are useful for many other potential engineering applications

extension of ambarzumyan s theorem to general boundary conditions abstract we extend the classical ambarzumyan s theorem for the sturm liouville equation which is concerned only with neumann boundary conditions to the general boundary conditions by imposing an additional condition on the potential function our result supplements the poschel trubowitz inverse spectral theory we also have parallel results for vectorial sturm liouville systems

pfaffianization of the two dimensional toda lattice pfaffianization procedure due to hirota and ohta is applied to the two dimensional toda lattice as a result a pfaffianized version of the two dimensional toda lattice is found

robust beamforming for cognitive multi antenna relay networks with bounded channel uncertainties in cognitive relay networks the interferences from secondary users sus and relays to primary users are constrained to be lower than a threshold the interference constraints are difficult to satisfy when the channel state information csi is imperfect in this paper we propose a robust beamforming scheme for the multi antenna non regenerative cognitive relay network where the multi antenna relay with imperfect csis helps the communication of single antenna su our objective is to design a robust beamforming scheme which maximizes the system capacity subject to transmit power constraint and interference constraints the bounded channel uncertainties are modeled using the worst case model the robust beamforming problem neglecting the correlation of channel uncertainties is reformulated as a convex semidefinite programming sdp by rank one relaxation this convex sdp is related with the worst case relay transmit power minimization problem which is further reformulated as a convex sdp whose rank one solution is proved to exist thus we propose the suboptimal solution to the robust beamforming problem which is found effectively by solving two convex sdps simulation results are provided to demonstrate the effectiveness of the proposed scheme

complex matrix decomposition and quadratic programming this paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex hermitian quadratic functions over specific domains in the complex space in its real case analog such studies were conducted in sturm and zhang sturm j f s zhang 2003 on cones of nonnegative quadratic functions math oper res 28 246 267 in this paper it is shown that stronger results can be obtained for the complex hermitian case in particular we show that the matrix rank one decomposition result of sturm and zhang sturm j f s zhang 2003 on cones of nonnegative quadratic functions math oper res 28 246 267 can be strengthened for the complex hermitian matrices as a consequence it is possible to characterize several new matrix co positive cones over specific domains by means of linear matrix inequality as examples of the potential application of the new rank one decomposition result we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming sdp problem and offer alternative proofs for a result of hausdorff hausdorff f 1919 der wertvorrat einer bilinearform mathematische zeitschrift3 314 316 and a result of brickman brickman l 1961 on the field of values of a matrix proc amer math soc 12 61 66 on the joint numerical range

an efficient global algorithm for single group multicast beamforming consider the single group multicast beamforming problem where multiple users receive the same data stream simultaneously from a single transmitter the problem is np hard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions in this paper we propose an efficient branch and bound algorithm for the problem that is guaranteed to find its global solution to the best of our knowledge our proposed algorithm is the first tailored global algorithm for the single group multicast beamforming problem simulation results show that our proposed algorithm is computationally efficient albeit its theoretical worst case iteration complexity is exponential with respect to the number of receivers and it significantly outperforms a state of the art general purpose global optimization solver called baron our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem by using it as the benchmark we show that two state of the art algorithms semidefinite relaxation algorithm and successive linear approximation algorithm work well when the problem dimension i e the number of antennas at the transmitter and the number of receivers is small but their performance deteriorates quickly as the problem dimension increases

some constraints and solutions of the kadomtsev petviashvili equation we illustrate the relations among the symmetry invariant group and constraint for a differential equation applied to the kadomtsev petviashvili equation some constraints and solutions are given in particular the equation associated with symmetry sigma of the kp equation is introduced and discussed

lie symmetries of a generalised non linear schrodinger equation ii exact solutions for pt i see ibid vol 21 p 1493 1988 the authors obtain group invariant solutions of the non linear equation i psi t delta psi a0 psi a1 psi mod psi mod 2 a2 psi mod psi mod 4 for which the symmetry group was previously shown to be the extended galilei group for a1a2 not 0 and the extended galilei similitude group for a1 0 or a2 0 they use the symmetry subgroups to reduce the equation to ordinary differential equations which are solved whenever possible with the help of a singularity analysis solutions are obtained in terms of elementary functions jacobi elliptic functions and painleve transcendents

robust optimization for amplify and forward mimo relaying from a worst case perspective in this paper we consider robust optimization of amplify and forward af multiple input multiple output mimo relay precoders in presence of deterministic imperfect channel state information csi when the csi uncertainty lies in a norm bounded region two widely used performance metrics mutual information mi and mean square error mse are adopted as design objectives according to the philosophy of worst case robustness the robust optimization problems with respect to maximizing the worst case mi and minimizing the worst case mse are formulated as maximin and minimax problems respectively due to the fact that these two problems do not have a concave convex or convex concave structure we cannot rely on the conventional saddle point theory to find the robust solutions nevertheless by exploiting majorization theory we show that the formulated maximin and minimax problems both admit saddle points we further analytically characterize the saddle points and provide closed form solutions to robust relay precoder designs interestingly we find that under both mi and mse metrics the robust relay optimization leads to a channel diagonalizing structure meaning that eigenmode transmission is optimal from the worst case robustness perspective the proposed robust designs can improve the spectral efficiency and reliability of af mimo relaying against csi uncertainties at the similar cost of computational complexity as the existing non robust schemes

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

negative flows of the potential kp hierarchy we construct a grassmannian like formulation for the potential kp hierarchy including additional negative flows our approach will generalize the notion of a r function to include negative flows we compare the resulting hierarchy with results by hirota satsuma and bogoyavlenskii

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

robust multicast beamforming for spectrum sharing based cognitive radios we consider a robust downlink beamforming optimization problem for secondary multicast transmission in a multiple input multiple output mimo spectrum sharing cognitive radio cr network the minimization of transmit power is formulated subject to both quality of service qos constraints on the secondary receivers and interference temperature constraints on the primary users under the assumption of imperfect channel state information csi the problem is a nonconvex quadratically constrained quadratic program qcqp and in general it is hard to achieve the global optimality as a compromise we present two randomized approximation algorithms for the problem via convex optimization techniques apart from the general setting of the robust beamforming problem we identify one interesting special case the robust problem of which can be solved efficiently simulation results are presented to demonstrate the performance gains of the proposed algorithms over an existing robust design

randomized algorithms for optimal solutions of double sided qcqp with applications in signal processing quadratically constrained quadratic programming qcqp with double sided constraints has plenty of applications in signal processing as have been addressed in recent years qcqp problems are hard to solve in general and they are typically approached by solving a semidefinite programming sdp relaxation followed by a postprocessing procedure existing postprocessing schemes include gaussian randomization to generate an approximate solution rank reduction procedure the so called purification and some specific rank one matrix decomposition techniques to yield a globally optimal solution in this paper we propose several randomized postprocessing methods to output not an approximate solution but a globally optimal solution for some solvable instances of the double sided qcqp i e instances with a small number of constraints we illustrate their applicability in robust receive beamforming radar optimal code design and broadcast beamforming for multiuser communications as a byproduct we derive an alternative shorter proof for the sturm zhang rank one matrix decomposition theorem

superinvariant chiral and vector fields this work is concerned with the characterization of field supermultiplets on four dimensional minkowski space that are stable under the action of subgroups of the superconformal group su 2 2 1 the most general scalar and vector superfields whose lie derivative with respect to a fermionic tangent vector vanishes are determined invariance under subgroups of su 2 2 1 with more than one odd generator is also discussed

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

symmetries of einstein yang mills fields and dimensional reduction let be a manifold on which a compact lie group s acts simply all orbits of the same type e can be written locally as mxs i m being the manifold of orbits space time and a typical isotropy group for the s action we study the geometrical structure given by an s invariant metric and an s invariant yang mills field on e with gauge group r we show that there is a one to one correspondence between such structures and quadruplets w aa a ha of fields defined solely on m y v is a metric on m ha are scalar fields characterizing the geometry of the orbits internal spaces are other scalar fields higgs fields characterizing the s invariance of the lie r valued yang mills field and is a yang mills field for the gauge group n i i x z l 7 n i being the normalizer of in s is a homomorphism of into r associated to the s action and z i is the centralizer o i in r we express the einstein yang mills lagrangian of e in terms of the component fields on m examples and model building recipes are given

robust minimum variance beamforming this paper introduces an extension of minimum variance beamforming that explicitly takes into account variation or uncertainty in the array response sources of this uncertainty include imprecise knowledge of the angle of arrival and uncertainty in the array manifold in our method uncertainty in the array manifold is explicitly modeled via an ellipsoid that gives the possible values of the array for a particular look direction we choose weights that minimize the total weighted power output of the array subject to the constraint that the gain should exceed unity for all array responses in this ellipsoid the robust weight selection process can be cast as a second order cone program that can be solved efficiently using lagrange multiplier techniques if the ellipsoid reduces to a single point the method coincides with capon s method we describe in detail several methods that can be used to derive an appropriate uncertainty ellipsoid for the array response we form separate uncertainty ellipsoids for each component in the signal path e g antenna electronics and then determine an aggregate uncertainty ellipsoid from these we give new results for modeling the element wise products of ellipsoids we demonstrate the robust beamforming and the ellipsoidal modeling methods with several numerical examples

on the infinite dimensional symmetry group of the davey stewartson equations the lie algebra of the group of point transformations leaving the davey stewartson equations dse s invariant is obtained the general element of this algebra depends on four arbitrary functions of time the algebra is shown to have a loop structure a property shared by the symmetry algebras of all known 2 1 dimensional integrable nonlinear equations subalgebras of the symmetry algebra are classified and used to reduce the dse s to various equations involving only two independent variables

lorentz positive maps and quadratic matrix inequalities with applications to robust miso transmit beamforming consider a unicast downlink beamforming optimization problem with robust signal to interference plus noise ratio constraints to account for imperfect channel state information at the base station in a multiple input single output miso communication system the convexity of this robust beamforming problem remains unknown a slightly conservative version of the robust beamforming problem is thus studied herein as a compromise it is in the form of a semi infinite second order cone program socp and more importantly it possesses an equivalent and explicit convex reformulation due to a linear matrix inequality lmi description of the cone of lorentz positive maps hence the conservative robust beamforming problem can be efficiently solved by an optimization solver additional robust shaping constraints can also be easily handled to control the amount of interference generated on other co existing users such as in cognitive radio systems

ambarzumyan type theorems on graphs with loops and double edges abstract in this work we consider inverse spectral problems for two types of graphs with loops and or double edges it is shown that analogs of ambarzumyan s theorem are true for sturm liouville problems on graphs with neumann boundary conditions at the pendant vertices and kirchhoff s conditions at the interior vertices

symmetry reduction of the wess zumino model and supersymmetric reduced systems an investigation of the supersymmetric reduction of the wess zumino model is carried out for the subgroups of the poincare group the necessary conditions for residual supersymmetry derived from super poincare transformations and a classification of the subalgebras of the poincare algebra which might give rise to supersymmetric residual systems are obtained two supersymmetric reductions of the off shell model with interactions by bidimensional subgroups are presented

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

high performance adaptive algorithms for single group multicast beamforming the single group multicast beamforming problem is np hard and the available approximations do not always achieve favorable performance complexity tradeoffs this paper introduces a new class of adaptive multicast beamforming algorithms that features guaranteed convergence and state of the art performance at low complexity each update takes a step in the direction of an inverse signal to noise ratio snr weighted linear combination of the snr gradient vectors of all the users convergence of this update to a karush kuhn tucker kkt point of proportionally fair beamforming is established simulations show that the proposed approach can enable better performance than the prior state of art in terms of multicast rate at considerably lower complexity this reveals an interesting link between max min fair and proportionally fair multicast beamforming formulations for cases where there is no initial channel state information at the transmitter an online algorithm is developed that simultaneously learns the user channel correlation matrices and adapts the beamforming vector to maximize the minimum long term average snr among the users using only periodic binary snr feedback from each receiver the online algorithm uses the analytic center cutting plane method to quickly learn the user correlation matrices with limited signaling overhead

precoding for mimo channels in cognitive radio networks with csi uncertainties and for mimo compound capacity inaccurate knowledge of channel state information csi limits the performance offered by multiple input multiple output mimo communications the design of a mimo precoder needs to take the csi uncertainty into consideration to mitigate its effects this paper proposes a method to design the precoder for a secondary user in an underlay cognitive radio framework that maximizes its transmission performance where the direct link to its receiver and the interference link to a primary user have csi uncertainties we model the csi uncertainties through the schatten norm that encompasses most commonly used norm measures such as the spectral and frobenious norms the proposed method solves iteratively a minimax problem by deriving the optimal solution for the worst case interference csi uncertainty applying the alternate iterate technique for the worst case direct link csi deviation and developing a feasible direction projected subgradient technique for the precoder simpler solutions for the precoder are also derived under some specific norm measure of csi uncertainty and certain requirement of transmission power simulations corroborate the expected performance of the proposed design

analysis of the symmetry group and exact solutions of the dispersionless kp equation in n 1 dimensions the lie algebra of the symmetry group of the n 1 dimensional generalization of the dispersionless kadomtsev petviashvili equation is obtained and identified as a semi direct sum of a finite dimensional simple lie algebra and an infinite dimensional nilpotent subalgebra group transformation properties of solutions under the subalgebra sl 2 r are presented known explicit analytic solutions in the literature are shown to be actually group invariant solutions corresponding to certain specific infinitesimal generators of the symmetry group

multi cell beamforming with decentralized coordination in cognitive and cellular networks this paper considers a downlink beamforming problem in a cognitive radio network where multiple primary and secondary cells coexist each multiantenna primary and secondary transmitter serves its own set of single antenna users the optimization objective is to minimize the sum transmission power over secondary transmitters while guaranteeing the minimum sinr for each secondary user and satisfying the maximum aggregate interference power constraint for each primary user we propose a decentralized beamforming algorithm where the original centralized problem is decomposed via primal decomposition method into two levels i e transmitter level subproblems managed by a network level master problem the master problem is solved independently at each secondary transmitter using a projected subgradient method requiring limited backhaul signaling among secondary transmitters to solve the independent transmitter level subproblems we propose three alternative approaches which are based on second order cone programming semidefinite programming and uplink downlink duality special emphasis is put on the last approach which is also considered in a multi cell miso cellular network numerical results show that the proposed algorithm achieves close to optimal solution even after a few iterations in quasi static channel conditions moreover near centralized performance is demonstrated in time correlated channels

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

equations of arbitrary order invariant under the kadomtsev petviashvili symmetry group by means of a simple new approach a general kadomtsev petviashvili kp family with an arbitrary function of group invariants of arbitrary order is proposed it is proved that the general kp family possesses a common infinite dimensional kac moody virasoro lie point symmetry algebra the known fourth order one can be re obtained as a special example the finite transformation group is presented in a clearer form the kac moody virasoro group invariant solutions and the kac moody group invariant solutions of the kp family are determined by the boussinesq and kdv families respectively

painleve analysis lie symmetries and invariant solutions of potential kadomstev petviashvili equation with time dependent coefficients in this paper the painleve analysis of 2 1 dimensional variable coefficients potential kadomstev petviashvili vcpkp equation s x email protected t s email protected x email protected t s x x x email protected t s y y 0 has been performed to study the painleve properties the vcpkp equation is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and reduced equations are further studied to obtain certain general solutions

kdv extensions with painleve property by means of the conformal invariance mobious transformation invariance the well known kdv equation is extended to i a 1 1 dimensional space time symmetric form ii two 2 1 dimensional space isotropic forms and iii general 3 1 dimensional and n dimensional forms the extensions are proven to be integrable under the meaning that they possess the painleve property