on the maslov index in this paper we give four definitions of maslov index and show that they all satisfy the same system of axioms and hence are equivalent to each other moreover relationships of several symplectic and differential geometric analytic and topological invariants including triple maslov indices eta invariants spectral flow and signatures of quadratic forms to the maslov index are developed and formulae relating them are given the broad presentation is designed with a view to applications both in geometry and in analysis 1994 john wiley sons inc

index and stability of symmetric periodic orbits in hamiltonian systems with application to figure eight orbit in this paper using the maslov index theory in symplectic geometry we build up some stability criteria for symmetric periodic orbits in a hamiltonian system which is motivated by the recent discoveries in the n body problem the key ingredient is a generalized bott type iteration formula for periodic solution in the presence of finite group action on the orbit for second order system we prove under general boundary conditions the close formula for the relationship between the morse index of an orbit in a lagrangian system and the maslov index of the fundamental solution for the corresponding orbit in its hamiltonian system counterpart and the boundary conditions cover the cases which appeared in the n body problem as an application we consider the stability problem of the celebrated figure eight orbit due to chenciner and montgomery in the planar three body problem with equal masses and we clarify the relationship between linear stability and its variational nature on various loop spaces the basic idea is as follows the variational characterization of the figure eight orbit provides information about its morse index based on its relation to the maslov index our stability criteria come into play

on the shannon cipher system with a capacity limited key distribution channel we consider the shannon cipher system in a setting where the secret key is delivered to the legitimate receiver via a channel with limited capacity for this setting we characterize the achievable region in the space of three figures of merit the security measured in terms of the equivocation the compressibility of the cryptogram and the distortion associated with the reconstruction of the plaintext source although lossy reconstruction of the plaintext does not rule out the option that the noisy decryption key would differ to a certain extent from the encryption key we show nevertheless that the best strategy is to strive for perfect match between the two keys by applying reliable channel coding to the key bits and to control the distortion solely via rate distortion coding of the plaintext source before the encryption in this sense our result has a flavor similar to that of the classical source channel separation theorem some variations and extensions of this model are discussed as well

high action orbits for tonelli lagrangians and superlinear hamiltonians on compact configuration spaces abstract multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds given by lagrangians or hamiltonians which have quadratic growth in the velocities or in the momenta such results are based on the richness of the topology of the space of curves satisfying the given boundary conditions in this note we show how these results can be extended to the classical setting of tonelli lagrangians lagrangians which are c 2 convex and superlinear in the velocities or to hamiltonians which are superlinear in the momenta and have a coercive action integrand

an index theory for asymptotic motions under singular potentials abstract we develop an index theory for parabolic and collision solutions to the classical n body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of trajectories ending with a total collapse or expanding with vanishing limiting velocities both problems suffer from a lack of compactness and can be brought in a similar form of a lagrangian system on the half time line by a regularising change of coordinates which preserve the lagrangian structure we then introduce a maslov type index which is suitable to capture the asymptotic nature of these trajectories as half clinic orbits by taking into account the underlying hamiltonian structure we define the appropriate notion of geometric index for this class of solutions and we develop the relative index theory

linear instability of relative equilibria for n body problems in the plane abstract following smale we study simple symmetric mechanical systems of n point particles in the plane in particular we address the question of the linear and spectral stability properties of relative equilibria which are special solutions of the equations of motion our main result is a sufficient condition to detect spectral hence linear instability namely we prove that if the morse index of an equilibrium point with even nullity is odd then the associated relative equilibrium is spectrally unstable the proof is based on some refined formulae for computing the spectral flow as a notable application of our theorem we examine two important classes of singular potentials the homogeneous one with 0 2 which includes the gravitational case and the logarithmic one we also establish for the homogeneous potential an inequality which is useful to test the spectral instability of the associated relative equilibrium

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

adaptive detection and estimation in the presence of useful signal and interference mismatches this paper considers adaptive detection and estimation in the presence of useful signal and interference mismatches we assume a homogeneous environment where the random disturbance components from the primary and secondary data share the same covariance matrix moreover the data under test contains a deterministic interference vector in addition to the possible useful signal we focus on the situation where an energy fraction of both the useful signal and the deterministic interference may lie outside their nominal subspaces conical uncertainty model under these conditions we devise a procedure for the computation of the joint maximum likelihood ml estimators of the useful signal and interference vectors resorting to a suitable rank one decomposition of a semidefinite program sdp problem optimal solution hence we use the aforementioned estimators for the synthesis of adaptive receivers based on different generalized likelihood ratio test glrt criteria at the analysis stage we assess the performance of the new detectors in comparison with some decision rules available in open literature

doubly constrained robust capon beamformer the standard capon beamformer scb is known to have better resolution and much better interference rejection capability than the standard data independent beamformer when the array steering vector is accurately known however the major problem of the scb is that it lacks robustness in the presence of array steering vector errors in this paper we will first provide a complete analysis of a norm constrained capon beamforming nccb approach which uses a norm constraint on the weight vector to improve the robustness against array steering vector errors and noise our analysis of nccb is thorough and sheds more light on the choice of the norm constraint than what was commonly known we also provide a natural extension of the scb which has been obtained via covariance matrix fitting to the case of uncertain steering vectors by enforcing a double constraint on the array steering vector viz a constant norm constraint and a spherical uncertainty set constraint which we refer to as the doubly constrained robust capon beamformer dcrcb nccb and dcrcb can both be efficiently computed at a comparable cost with that of the scb performance comparisons of nccb dcrcb and several other adaptive beamformers via a number of numerical examples are also presented

consensus admm for general quadratically constrained quadratic programming nonconvex quadratically constrained quadratic programming qcqp problems have numerous applications in signal processing machine learning and wireless communications albeit the general qcqp is np hard and several interesting special cases are np hard as well this paper proposes a new algorithm for general qcqp the problem is first reformulated in consensus optimization form to which the alternating direction method of multipliers can be applied the reformulation is done in such a way that each of the subproblems is a qcqp with only one constraint qcqp 1 which is efficiently solvable irrespective of non convexity the core components are carefully designed to make the overall algorithm more scalable including efficient methods for solving qcqp 1 memory efficient implementation parallel distributed implementation and smart initialization the proposed algorithm is then tested in two applications multicast beamforming and phase retrieval the results indicate superior performance over prior state of the art methods

coordinated beamforming for multiuser miso interference channel under rate outage constraints this paper studies the coordinated beamforming design problem for the multiple input single output miso interference channel assuming only channel distribution information cdi known to the transmitters under a given requirement on the rate outage probability for receivers we aim to maximize the system utility e g the weighted sum rate weighted geometric mean rate and the weighed harmonic mean rate subject to the rate outage constraints and individual power constraints the outage constraints however lead to a complicated nonconvex structure for the considered beamforming design problem and render the optimization problem difficult to handle although this nonconvex optimization problem can be solved in an exhaustive search manner this brute force approach is only feasible when the number of transmitter receiver pairs is small for a system with a large number of transmitter receiver pairs computationally efficient alternatives are necessary hence the focus of this paper is the design of such efficient approximation methods in particular by employing semidefinite relaxation sdr and first order approximation techniques we propose an efficient successive convex approximation sca algorithm that provides high quality approximate beamforming solutions via solving a sequence of convex approximation problems the solution thus obtained is further shown to be a stationary point for the sdr of the original outage constrained beamforming design problem furthermore we propose a distributed sca algorithm where each transmitter optimizes its own beamformer using local cdi and information obtained from limited message exchange with the other transmitters our simulation results demonstrate that the proposed sca algorithm and its distributed counterpart indeed converge and promising performance can be achieved for all the considered system utilities

on a subproblem of trust region algorithms for constrained optimization we study a subproblem that arises in some trust region algorithms for equality constrained optimization it is the minimization of a general quadratic function with two special quadratic constraints properties of such subproblems are given it is proved that the hessian of the lagrangian has at most one negative eigenvalue and an example is presented to show that the hessian may have a negative eigenvalue when one constraint is inactive at the solution

on the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues we derive some basic results on the geometry of semidefinite programming sdp and eigenvalue optimization i e the minimization of the sum of the k largest eigenvalues of a smooth matrix valued function we provide upper bounds on the rank of extreme matrices in sdps and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue optimization in the spectrum of an optimal matrix the kth and k 1 st largest eigenvalues tend to be equal and frequently have multiplicity greater than two this clustering is intuitively plausible and has been observed as early as 1975 when the matrix valued function is affine we prove that clustering must occur at extreme points of the set of optimal solutions if the number of variables is sufficiently large we also give a lower bound on the multiplicity of the critical eigenvalue these results generalize to the case of a general matrix valued function under appropriate conditions

on secret sharing systems a secret sharing system permits a secret to be shared among n trustees in such a way that any k of them can recover the secret but any k 1 have complete uncertainty about it a linear coding scheme for secret sharing is exhibited which subsumes the polynomial interpolation method proposed by shamir and can also be viewed as a deterministic version of blakley s probabilistic method bounds on the maximum value of n for a given k and secret size are derived for any system linear or nonlinear the proposed scheme achieves the lower bound which for practical purposes differs insignificantly from the upper bound the scheme may be extended to protect several secrets methods to protect against deliberate tampering by any of the trustees are also presented

noncollision periodic solutions to some singular dynamical systems with very weak forces abstract we prove a double variational characterization of the set of all the periodic solutions of the kepler problem we provide a sufficient condition for the existence of a periodic solution with prescribed period or energy for autonomous second order differential systems with keplerian type potentials the proofs are based on linking arguments

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

strong duality for the cdt subproblem a necessary and sufficient condition in this paper we consider the problem of minimizing a nonconvex quadratic function subject to two quadratic inequality constraints as an application such a quadratic program plays an important role in the trust region method for nonlinear optimization such a problem is known as the celis dennis and tapia cdt subproblem in the literature the lagrangian dual of the cdt subproblem is a semidefinite program sdp hence convex and solvable however a positive duality gap may exist between the cdt subproblem and its lagrangian dual because the cdt subproblem itself is nonconvex in this paper we present a necessary and sufficient condition to characterize when the cdt subproblem and its lagrangian dual admits no duality gap i e the strong duality holds this necessary and sufficient condition is easy verifiable and involves only one any optimal solution of the sdp relaxation for the cdt subproblem moreover the condition reveals that it is actually rare to render a positive duality gap for the cdt subproblems in general moreover if the strong duality holds then an optimal solution for the cdt problem can be retrieved from an optimal solution of the sdp relaxation by means of a matrix rank one decomposition procedure the same analysis is extended to the framework where the necessary and sufficient condition is presented in terms of the lagrangian multipliers at a kkt point furthermore we show that the condition is numerically easy to work with approximatively

trace formula for linear hamiltonian systems with its applications to elliptic lagrangian solutions in the present paper we build up trace formulas for both the linear hamiltonian systems and sturm liouville systems the formula connects the monodromy matrix of a symmetric periodic orbit with the infinite sum of eigenvalues of the hessian of the action functional a natural application is to study the non degeneracy of linear hamiltonian systems precisely by the trace formula we can give an estimation for the upper bound such that the non degeneracy preserves moreover we could estimate the relative morse index by the trace formula consequently a series of new stability criteria for the symmetric periodic orbits is given as a concrete application the trace formula is used to study the linear stability of elliptic lagrangian solutions of the classical planar three body problem which depends on the mass parameter beta in 0 9 and the eccentricity e in 0 1 based on the trace formula we estimate the stable region and hyperbolic region of the elliptic lagrangian solutions

far field multicast beamforming for uniform linear antenna arrays the problem of transmit beamforming to multiple cochannel multicast groups is considered for the important special case when the channel vectors are vandermonde this arises when a uniform linear antenna antenna ula array is used at the transmitter under far field line of sight propagation conditions as provisioned in 802 16e and related wireless backhaul scenarios two design approaches are pursued i minimizing the total transmitted power subject to providing at least a prescribed received signal to interference plus noise ratio sinr to each intended receiver and ii maximizing the minimum received sinr under a total transmit power budget whereas these problems have been recently shown to be np hard in general it is proven here that for vandermonde channel vectors it is possible to recast the optimization in terms of the autocorrelation sequences of the sought beam vectors yielding an equivalent convex reformulation this affords efficient optimal solution using modern interior point methods the optimal beam vectors can then be recovered using spectral factorization robust extensions for the case of partial channel state information where the direction of each receiver is known to lie in an interval are also developed interestingly these also admit convex reformulation the various optimal designs are illustrated and contrasted in a suite of pertinent numerical experiments

secure transmission with multiple antennas part ii the mimome wiretap channel the capacity of the gaussian wiretap channel model is analyzed when there are multiple antennas at the sender intended receiver and eavesdropper the associated channel matrices are fixed and known to all the terminals a computable characterization of the secrecy capacity is established as the saddle point solution to a minimax problem the converse is based on a sato type argument used in other broadcast settings and the coding theorem is based on gaussian wiretap codebooks at high signal to noise ratio snr the secrecy capacity is shown to be attained by simultaneously diagonalizing the channel matrices via the generalized singular value decomposition and independently coding across the resulting parallel channels the associated capacity is expressed in terms of the corresponding generalized singular values it is shown that a semi blind masked multi input multi output mimo transmission strategy that sends information along directions in which there is gain to the intended receiver and synthetic noise along directions in which there is not can be arbitrarily far from capacity in this regime necessary and sufficient conditions for the secrecy capacity to be zero are provided which simplify in the limit of many antennas when the entries of the channel matrices are independent and identically distributed the resulting scaling laws establish that to prevent secure communication the eavesdropper needs three times as many antennas as the sender and intended receiver have jointly and that the optimum division of antennas between sender and intended receiver is in the ratio of 2 1

on the essential spectrum of schrodinger operators abstract an extension of the theorem of hunziker van winter zhislin on the location of the essential spectrum of many body hamiltonians

morse index and stability of elliptic lagrangian solutions in the planar three body problem abstract we illustrate a new way to study the stability problem in celestial mechanics in this paper using the variational nature of elliptic lagrangian solutions in the planar three body problem we study the relation between morse index and its stability via maslov type index theory of periodic solutions of hamiltonian system for elliptic lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the morse index which is the key to understand the stability problem as a special case we provide a criterion to spectral stability of relative equilibrium

spectral flow brouwer degree and hill s determinant formula abstract in 2005 a new topological invariant defined in terms of the brouwer degree of a determinant map was introduced by musso pejsachowicz and the first name author for counting the conjugate points along a semi riemannian geodesic this invariant was defined in terms of a suspension of a complexified family of linear second order dirichlet boundary value problems in this paper starting from this result we generalize this invariant to a general self adjoint morse sturm system and we prove a new spectral flow formula finally we discuss the relation between this spectral flow formula and the hill s determinant formula and we apply this invariant for detecting instability of periodic orbits of a hamiltonian system

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

a special class of fractional qcqp and its applications on cognitive collaborative beamforming in this paper we investigate a special class of the fractional quadratically constrained quadratic problem qcqp with more than two quadratic constraints we propose to effectively solve this special fractional qcqp by one or two convex semidefinite programmings sdps for the two sdps one is equivalent to the original fractional qcqp with rank one relaxation from the charnes cooper transformation and the other exploiting the optimal value of the former sdp always has rank one solution which is optimal to the former sdp theoretical analysis shows that our proposed non iterative sdp based algorithm achieves the global optimal solution to the special fractional qcqp in specific scenarios our proposed non iterative sdp based algorithm has lower computational complexity compared to the second order cone programming socp based and constrained concave convex procedure cccp based iterative algorithms we apply the proposed non iterative sdp based algorithm on two collaborative beamforming problems in cognitive relay networks specifically one is the achievable rate region for two way non regenerative cognitive relay networks and the other is the achievable secrecy rate for one way regenerative cognitive relay networks simulation results have shown that our proposed non iterative sdp based algorithm achieves the same performance as the socp based iterative algorithm our proposed algorithm achieves the better performance than the cccp based iterative algorithm

algorithms for secrecy guarantee with null space beamforming in two way relay networks in this paper we consider a two way relay network with two sources and multiple cooperative relays in the presence of an eavesdropper to guarantee the secrecy a null space beamforming scheme is applied where the relay beamforming vector lies in the nullspace of the equivalent channel of relay link from two sources to the eavesdropper our goal is to obtain the optimal beamforming vector as well as two sources transmit power subject to various criteria we propose three different approaches and solve them in an alternating iterative way where subproblems for solving beamforming and sources power are formulated in each iteration respectively first we minimize the total transmit power under secrecy rate constraint at two sources for beamforming vector subproblem two different methods semi definite programming sdp and sequential quadratic programming sqp are proposed where we analyze and verify sqp has lower complexity than sdp second we maximize the secrecy sum rate subject to total transmit power constraint the beamforming vector subproblem is equivalent to a generalized rayleigh quotient problem with rank constraint third the problem of minimum per user secrecy rate maximization under the total power constraint is investigated for user fairness an iterative procedure utilizing the sdp with bisection search method is proposed to solve beamforming subproblem in each approach the subproblem with two sources power is formulated as a single variable problem and solved by newton s method with line search simulation results demonstrate the validity of proposed approaches and algorithms for both symmetric and asymmetric scenarios

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

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

morse index and bifurcation for figure eight choreographies of the equal mass three body problem we report on the morse index and periodic solutions bifurcating from the figure eight choreography for the equal mass three body problem under homogeneous potential 1 r a for and under lennard jones lj type potential where r is a distance between bodies it is shown that the morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the morse index inversely we observed bifurcation occurs at every point where the morse index changes for the figure eight choreography under 1 r a and for solution under the lj type potential where solution is a figure eight choreography tending to that under 1 r 6 for an infinitely large period thus to our numerical studies change of the morse index is not only necessary but also a sufficient condition for bifurcation for these choreographies further we observed that the change of the morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

outage constrained robust transmit optimization for multiuser miso downlinks tractable approximations by conic optimization in this paper we study a probabilistically robust transmit optimization problem under imperfect channel state information csi at the transmitter and under the multiuser multiple input single output miso downlink scenario the main issue is to keep the probability of each user s achievable rate outage as caused by csi uncertainties below a given threshold as is well known such rate outage constraints present a significant analytical and computational challenge indeed they do not admit simple closed form expressions and are unlikely to be efficiently computable in general assuming gaussian csi uncertainties we first review a traditional robust optimization based method for approximating the rate outage constraints and then develop two novel approximation methods using probabilistic techniques interestingly these three methods can be viewed as implementing different tractable analytic upper bounds on the tail probability of a complex gaussian quadratic form and they provide convex restrictions or safe tractable approximations of the original rate outage constraints in particular a feasible solution from any one of these methods will automatically satisfy the rate outage constraints and all three methods involve convex conic programs that can be solved efficiently using off the shelf solvers we then proceed to study the performance complexity tradeoffs of these methods through computational complexity and comparative approximation performance analyses finally simulation results are provided to benchmark the three convex restriction methods against the state of the art in the literature the results show that all three methods offer significantly improved solution quality and much lower complexity

morse theory and relative equilibria in the planar n vortex problem morse theoretical ideas are applied to the study of relative equilibria in the planar n vortex problem for the case of positive circulations we prove that the morse index of a critical point of the hamiltonian restricted to a level surface of the angular impulse is equal to the number of pairs of real eigenvalues of the corresponding relative equilibrium periodic solution the morse inequalities are then used to prove the instability of some families of relative equilibria in the four vortex problem with two pairs of equal vorticities we also show that for positive circulations relative equilibria cannot accumulate on the collision set

reel bf design achieving the sdp bound for downlink beamforming with arbitrary shaping constraints this paper considers the beamforming design for a multiuser multiple input single output downlink with an arbitrary number of context specific shaping constraints in this setup the state of the art beamforming schemes cannot attain the well known performance bound promised by the semidefinite program sdp relaxation technique to close the gap we propose a redundant signal embedded linear beamforming reel bf scheme where each user is assigned with one information beamformer and several shaping beamformers it is shown that the proposed reel bf scheme can perform general rank k beamforming for user symbols in a low complexity and structured manner in addition sufficient conditions are derived to guarantee that the reel bf scheme always achieves the sdp bound for linear beamforming schemes based on such conditions an efficient algorithm is then developed to obtain the optimal reel bf solution in polynomial time numerical results demonstrate that the proposed scheme enjoys substantial performance gains over the existing alternatives

fredholm operators and essential s spectrum in the quaternionic setting for bounded right linear operators in a right quaternionic hilbert space with a left multiplication defined on it we study the approximate s point spectrum in the same hilbert space then we study the fredholm operators and the fredholm index in particular we prove the invariance of the fredholm index under small norm operator and compact operator perturbations finally in association with the fredholm operators we develop the theory of essential s spectrum we also characterize the s spectrum in terms of the essential s spectrum and fredholm operators in the sequel we study left and right s spectra as needed for the development of the theory presented in this note for bounded right linear operators in a right quaternionic hilbert space with a left multiplication defined on it we study the approximate s point spectrum in the same hilbert space then we study the fredholm operators and the fredholm index in particular we prove the invariance of the fredholm index under small norm operator and compact operator perturbations finally in association with the fredholm operators we develop the theory of essential s spectrum we also characterize the s spectrum in terms of the essential s spectrum and fredholm operators in the sequel we study left and right s spectra as needed for the development of the theory presented in this note

code design to optimize radar detection performance under accuracy and similarity constraints this paper deals with the design of coded waveforms which optimize radar performances in the presence of colored gaussian disturbance we focus on the class of linearly coded pulse trains and determine the optimum radar code according to the following criterion maximization of the detection performance under a control on the region of achievable doppler estimation accuracies and imposing a similarity constraint with a prefixed radar code this last constraint is tantamount to requiring a similarity between the ambiguity functions of the devised waveform and of the pulse train encoded with the prefixed sequence the resulting optimization problem is nonconvex and quadratic in order to solve it we propose a technique with polynomial computational complexity based on the relaxation of the original problem into a semidefinite program thus the best code is determined through a rank one decomposition of an optimal solution of the relaxed problem at the analysis stage we assess the performance of the new encoding technique in terms of detection performance region of achievable doppler estimation accuracies and ambiguity function

on the low rank solutions for linear matrix inequalities in this paper we present a polynomial time procedure to find a low rank solution for a system of linear matrix inequalities lmi the existence of such a low rank solution was shown in the work of au yeung and poon and the work of barvinok in the approach of au yeung and poon an earlier unpublished manuscript of bohnenblust played an essential role both proofs in the work of au yeung and poon and that of barvinok are nonconstructive in nature the aim of this paper is to provide a polynomial time constructive procedure to find such a low rank solution approximatively extensions of our new results and their relations to some of the known results in the literature are discussed

a coding theorem for secret sharing communication systems with two gaussian wiretap channels a coding theorem is proved for the secret sharing communication system sscs with two gaussian wiretap channels this communication system is an extension of both the sscs with two noiseless channels and the gaussian wiretap channel gwc the admissible region of rates and security levels for the sscs with two gwcs is described by the capacities and secrecy capacities of two gwcs the following three cases are considered two wiretappers cannot cooperate with each other they can cooperate to decipher the transmitted information and it is not known whether they can cooperate or not

a note on quadratic forms we extend an interesting theorem of yuan 12 for two quadratic forms to three matrices let c 1 c 2 c 3 be three symmetric matrices in n n if max x t c 1 x x t c 2 x x t c 3 x 0 for all x n it is proved that there exist t i 0 i 1 2 3 such that i 1 3 t i 1 and i 1 3 t i c i has at most one negative eigenvalue

doubly constrained robust capon beamformer with ellipsoidal uncertainty sets the doubly constrained robust dcr capon beamformer with a spherical uncertainty set was introduced and studied by stoica and wang here we consider the generalized dcr problem gdcr in which the uncertainty set is an ellipsoid rather than a sphere although as noted previously by stoica and wang this problem is nonconvex and appears to be intractable we show that it can be solved efficiently in fact we prove that the gdcr beamformer can be obtained as a solution to a convex optimization problem to this end we first derive a strong duality result for nonconvex quadratic programs with two quadratic constraints over the complex domain specializing the results to our context leads to a semidefinite programming formulation of the gdcr beamformer

code design for radar stap via optimization theory in this paper we deal with the problem of constrained code optimization for radar space time adaptive processing stap in the presence of colored gaussian disturbance at the design stage we devise a code design algorithm complying with the following optimality criterion maximization of the detection performance under a control on the regions of achievable values for the temporal and spatial doppler estimation accuracy and on the degree of similarity with a pre fixed radar code the resulting quadratic optimization problem is solved resorting to a convex relaxation that belongs to the semidefinite program sdp class an optimal solution of the initial problem is then constructed through a suitable rank one decomposition of an optimal solution of the relaxed one at the analysis stage we assess the performance of the new algorithm both on simulated data and on the standard challenging the knowledge aided sensor signal processing and expert reasoning kassper datacube

morse index and linear stability of the lagrangian circular orbit in a three body type problem via index theory it is well known that the linear stability of the lagrangian elliptic solutions in the classical planar three body problem depends on a mass parameter and on the eccentricity e of the orbit we consider only the circular case e 0 but under the action of a broader family of singular potentials homogeneous potentials for alpha in 0 2 and the logarithmic one it turns out indeed that the lagrangian circular orbit persists also in this more general setting we discover a region of linear stability expressed in terms of the homogeneity parameter and the mass parameter then we compute the morse index of this orbit and of its iterates and we find that the boundary of the stability region is the envelope of a family of curves on which the morse indices of the iterates jump in order to conduct our analysis we rely on a maslov type index theory devised and developed by y long x hu and s sun a key role is played by an appropriate index theorem and by some precise computations of suitable maslov type indices

a survey of the s lemma in this survey we review the many faces of the s lemma a result about the correctness of the s procedure the basic idea of this widely used method came from control theory but it has important consequences in quadratic and semidefinite optimization convex geometry and linear algebra as well these were all active research areas but as there was little interaction between researchers in these different areas their results remained mainly isolated here we give a unified analysis of the theory by providing three different proofs for the s lemma and revealing hidden connections with various areas of mathematics we prove some new duality results and present applications from control theory error estimation and computational geometry

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

approximation bounds for semidefinite relaxation of max min fair multicast transmit beamforming problem consider a downlink multicast scenario where a base station equipped with multiple antennas wishes to simultaneously broadcast a number of signals to some given groups of users over a common bandwidth the goal of the base station is to select appropriate beamforming vectors so as to maximize the minimum signal to interference plus noise ratio sinr among all users under a power budget constraint since this max min fair transmit beamforming problem is np hard in general a randomized polynomial time approximation approach based on semidefinite relaxation sdr has been proposed recently where excellent performance in both simulated and measured wireless channels has been reported this paper shows that the sdr based approach can provide at least an o 1 m approximation to the optimum solution where m is the total number of users this estimate implies that the sdr solution achieves an sinr that is at most logm o 1 db away from the highest possible value the existence of such a data independent bound certifies the worst case approximation quality of the sdr algorithm for any problem instance and any number of transmit antennas for real valued problems the corresponding approximation ratio is shown to be o 1 m2 while the sinr loss due to sdr approximation is at most 2logm o 1 db

a new approach to robust beamforming in the presence of steering vector errors this correspondence presents a new approach to array beamforming in the presence of steering vector errors such as steering angle and carrier frequency errors the basic idea is based on the principle that the output power of an optimized beamformer achieves a local maximum if the steering vector coincides with that of the desired signal the proposed approach iteratively searches for the correct steering vector by maximizing the array mean output power using a first order taylor series approximation of the steering vector in terms of the steering errors this method has the advantages of easy implementation and suffers no loss in the degrees of freedom for interference rejection furthermore the method is independent of the choice of the phase center coordinate origin contrary to derivative constraint methods conditions for the existence of a solution using the proposed method are also established

pareto boundary of the rate region for single stream mimo interference channels linear transceiver design we consider a multiple input multiple output mimo interference channel ic where a single data stream per user is transmitted and each receiver treats interference as noise the paper focuses on the open problem of computing the outermost boundary so called pareto boundary pb of the achievable rate region under linear transceiver design the pareto boundary consists of the strict pb and non strict pb for the two user case we compute the non strict pb and the two ending points of the strict pb exactly for the strict pb we formulate the problem to maximize one rate while the other rate is fixed such that a strict pb point is reached to solve this non convex optimization problem which results from the hard coupled two transmit beamformers we propose an alternating optimization algorithm furthermore we extend the algorithm to the multi user scenario and show convergence numerical simulations illustrate that the proposed algorithm computes a sequence of well distributed operating points that serve as a reasonable and complete inner bound of the strict pb compared with existing methods

transmitter optimization for per antenna power constrained multi antenna downlinks an slnr maximization methodology in this paper we consider signal to leakage plus noise ratio slnr maximized precoding for multi antenna downlinks under per antenna power constraints papcs in contrast to the conventional sum power constraint the per antenna power constrained slnr precoder design problem no longer admits a simple closed form solution and achieving its optimal solution turns out to be much more challenging to conquer the difficulty we first propose to perform the charnes cooper transformation on the problem of interest which yields a solvable simpler second order cone program socp moving beyond this approach we are then able to obtain a semi closed form solution to the optimal slnr precoder the power allocation among users is also optimized for further performance enhancement for which we devise two numerical methods based on gradient projection and a low complexity approximation finally focusing on the imperfect channel state information csi scenario we develop a robust slnr based precoder to combat the performance degradation caused by random csi errors the superiority of all proposed precoder designs in this work is confirmed by numerical simulations

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

on maximization of quadratic form over intersection of ellipsoids with common center we demonstrate that if a1 a m are symmetric positive semidefinite n n matrices with positive definite sum and a is an arbitrary symmetric n n matrix then the relative accuracy in terms of the optimal value of the semidefinite relaxation max x tr ax mid tr a ix le1 i 1 m x succeq0 eqno hbox sdp of the optimization program x tax to max mid x ta ix le 1 i 1 m eqno hbox p is not worse than 1 frac 1 2 ln 2m 2 it is shown that this bound is sharp in order as far as the dependence on m is concerned and that a feasible solution x to p with x tax ge frac opt hbox rm sdp 2 ln 2m 2 eqno can be found efficiently this somehow improves one of the results of nesterov 4 where bound similar to is established for the case when all ai are of rank 1

quadratic matrix programming we introduce and study a special class of nonconvex quadratic problems in which the objective and constraint functions have the form f boldmath x tr boldmath x t boldmath a boldmath x 2 tr boldmath b t boldmath x c boldmath x in real r n times r the latter formulation is termed quadratic matrix programming qmp of order r we construct a specially devised semidefinite relaxation sdr and dual for the qmp problem and show that under some mild conditions strong duality holds for qmp problems with at most r constraints using a result on the equivalence of two characterizations of the nonnegativity property of quadratic functions of the above form we are able to compare the constructed sdr and dual problems to other known sdrs and dual formulations of the problem an application to robust least squares problems is discussed

rank constrained separable semidefinite programming with applications to optimal beamforming consider a downlink communication system where multiantenna base stations transmit independent data streams to decentralized single antenna users over a common frequency band the goal of the base stations is to jointly adjust the beamforming vectors to minimize the transmission powers while ensuring the signal to interference noise ratio requirement of each user within the system at the same time it may be necessary to keep the interference generated on other coexisting systems under a certain tolerable level in addition one may want to include general individual shaping constraints on the beamforming vectors this beamforming problem is a separable homogeneous quadratically constrained quadratic program and it is difficult to solve in general in this paper we give conditions under which strong duality holds and propose efficient algorithms for the optimal beamforming problem first we study rank constrained solutions of general separable semidefinite programs sdps and propose rank reduction procedures to achieve a lower rank solution then we show that the sdp relaxation of three classes of optimal beamforming problem always has a rank one solution which can be obtained by invoking the rank reduction procedures

on design of collaborative beamforming for two way relay networks this paper studies the achievable rate region for an amplify and forward af based two way relay network with collaborative beamforming with different assumptions of channel reciprocity between the source relay and relay source channels the achievable rate region is characterized under two setups first with reciprocal channels we investigate the achievable rate regions when the relay cluster is subject to a sum power constraint or individual power constraints we show that the optimal beamforming vectors obtained from solving the weighted sum inverse snr minimization wsismin problems are sufficient to characterize the corresponding achievable rate region furthermore we derive the closed form solutions for those optimal beamforming vectors and consequently propose the partially distributed algorithms to implement the optimal beamforming where each relay node only needs the local channel information and one global parameter second with the nonreciprocal channels the achievable rate regions are also characterized for both the sum power constraint case and the individual power constraint case although no closed form solutions are available under this setup we present efficient numerical algorithms by solving a sequence of semi definite programming sdp problems after semi definite relaxation sdr where the optimal beamformer can be obtained under the sum power constraint and approximate optimal solutions can be obtained under the individual power constraints

elliptic relative equilibria in the n body problem a planar central configuration of the n body problem gives rise to a solution where each particle moves on a specific keplerian orbit while the totality of the particles move on a homothety motion if the keplerian orbit is elliptic then the solution is an equilibrium in pulsating coordinates so we call this solution an elliptic relative equilibrium the totality of such solutions forms a four dimensional symplectic subspace and we give a symplectic coordinate system which is adapted to this subspace and its symplectic complement in our coordinate system the linear variational equations of such a solution decouple into three subsystems one subsystem simply gives the motion of the center of mass another is kepler s problem and the third determines the nontrivial characteristic multipliers using these coordinates we study the linear stability of the elliptic relative equilibrium defined by the equilateral triangular central configuration of the three body problem we reproduce the analytic studies of g roberts we also study the linear stability of the four and five body problem where three or four bodies of unit mass are at the vertices of a equilateral triangle or

robust adaptive beamforming for general rank signal model with positive semi definite constraint via potdc the robust adaptive beamforming rab problem for general rank signal model with an additional positive semi definite constraint is considered using the principle of the worst case performance optimization such rab problem leads to a difference of convex functions dc optimization problem the existing approaches for solving the resulted non convex dc problem are based on approximations and find only suboptimal solutions here we aim at finding the globally optimal solution for the non convex dc problem and clarify the conditions under which the solution is guaranteed to be globally optimal particularly we rewrite the problem as the minimization of a one dimensional optimal value function ovf then the ovf is replaced with another equivalent one for which the corresponding optimization problem is convex the new one dimensional ovf is minimized iteratively via polynomial time dc potdc algorithm we show that the potdc converges to a point that satisfies karush kuhn tucker kkt optimality conditions and such point is the global optimum under certain conditions towards this conclusion we prove that the proposed algorithm finds the globally optimal solution if the presumed norm of the mismatch matrix that corresponds to the desired signal covariance matrix is sufficiently small the new rab method shows superior performance compared to the other state of the art general rank rab methods

a morse smale index theorem for indefinite elliptic systems and bifurcation abstract we generalise the semi riemannian morse index theorem to non degenerate elliptic systems of partial differential equations on star shaped domains moreover we apply our theorem to bifurcation from a branch of trivial solutions of semilinear systems where the bifurcation parameter is introduced by shrinking the domain to a point this extends recent results of the authors for scalar equations

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

the spectral theorem for quaternionic unbounded normal operators based on the s spectrum in this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of s spectrum the proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator with this paper we complete the foundation of spectral analysis of quaternionic operators the s spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators the lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators a prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics