transmitter optimization for the multi antenna downlink with per antenna power constraints this paper considers the transmitter optimization problem for a multiuser downlink channel with multiple transmit antennas at the base station in contrast to the conventional sum power constraint on the transmit antennas this paper adopts a more realistic per antenna power constraint because in practical implementations each antenna is equipped with its own power amplifier and is limited individually by the linearity of the amplifier assuming perfect channel knowledge at the transmitter this paper investigates two different transmission schemes under the per antenna power constraint a minimum power beamforming design for downlink channels with a single antenna at each remote user and a capacity achieving transmitter design for downlink channels with multiple antennas at each remote user it is shown that in both cases the per antenna downlink transmitter optimization problem may be transformed into a dual uplink problem with an uncertain noise this generalizes previous uplink downlink duality results and transforms the per antenna transmitter optimization problem into an equivalent minimax optimization problem further it is shown that various notions of uplink downlink duality may be unified under a lagrangian duality framework this new interpretation of duality gives rise to efficient numerical optimization techniques for solving the downlink per antenna transmitter optimization problem

a hill series approach to wavefunction nodes the hill series method is shown to have several advantages for treating the schrodinger equation when the wavefunction obeys the dirichlet boundary condition r 0 a novel combination of the virial theorem the hill series method and a finite difference method is used to find the node positions of the wavefunction for several excited states of perturbed oscillator and coulombic systems and to find critical radii for the hydrogen atom

multiple solutions for an eigenvalue problem involving p laplacian type operators abstract in this paper we establish the existence of two nontrivial weak solutions of some eigenvalue problems involving a general elliptic operator in divergence form first we study a one parameter problem under homogeneous dirichlet boundary conditions in bounded domains and then a two parameters problem under inhomogeneous nonlinear robin boundary conditions in regular bounded domains these results complete in several directions some recent papers where a ricceri type critical point theorem was used the main argument of this paper is based on two recent theorems due to arcoya and carmona 2007 in 2 which we use in a slightly modified version see section 2

nonuniformly elliptic equations of p laplacian type abstract this paper deals with the existence of a generalized solution in w 0 1 p to a nonuniformly nonlinear elliptic equation of the form div a x u f x u in a bounded domain of r n here a satisfies a x c 0 h 0 x h 1 x p 1 for any in r n a e x h 0 l p p 1 and h 1 l loc 1

the capacity of gaussian mimo channels under total and per antenna power constraints the capacity of a fixed gaussian multiple input multiple output mimo channel and the optimal transmission strategy under the total power tp constraint and full channel state information are well known this problem remains open in the general case under individual per antenna pa power constraints while some special cases have been solved these include a full rank solution for the mimo channel and a general solution for the multiple input single output miso channel in this paper the fixed gaussian miso channel is considered and its capacity and optimal transmission strategies are determined in a closed form under the joint total and pa power constraints in the general case in particular the optimal strategy is hybrid and includes two parts first is equal gain transmission and second is maximum ratio transmission which are responsible for the pa and tp constraints respectively the optimal beamforming vector is given in a closed form and an accurate yet simple approximation to the capacity is proposed finally the above results are extended to the mimo case by establishing the ergodic capacity of fading mimo channels under the joint power constraints when the fading distribution is right unitary invariant of which i i d and semi correlated rayleigh fading are special cases unlike the fixed miso case the optimal signaling is shown to be isotropic in this case

classification of oscillators in the heisenberg matrix representation for an arbitrary finite and not too singular presumably phenomenological superposition of potentials rdelta with rational exponents we solve the old problem of conversion of the corresponding differential schrodinger bound state problem into its matrix equivalent with the minimalized number l of non zero diagonals the construction with proofs is performed via the standard power series method in its non hermitean matrix sometimes called hill determinant version all the simplest and in this sense algebraizable interactions are then displayed up to l 5 an example of application in perturbation theory is included

eigenvalues of the schrodinger equation with coulomb potentials plus linear and harmonic radial terms analytic approximations for the eigenvalues have been obtained using the two point quasifractional approximants method here we extend and generalize previous work where the method was applied to determine eigenenergies of two dimensional coulomb potentials in the presence of a magnetic field of arbitrary strength we have also shown the advantage of our results with respect to other calculations using the shifted 1 n method or pade approximants

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

multiple solutions for p harmonic type equations abstract in this paper we establish the existence of multiple solutions for an equation involving a p harmonic operator subject to dirichlet boundary conditions in a bounded smooth open domain of r n while the nonlinearity has a p 1 sublinear growth at infinity using a three critical points theorem we prove the existence of at least three distinct weak solutions in w 0 2 p to this problem

on the hill determinant method the conditions of applicability of the hill determinant method to the potentials v r v1r v2r2 and v x x2 lambda x2 1 gx2 are investigated it is shown that the procedure leads to the actual eigenvalues in the former case provided the ansatz is properly chosen but it completely fails to yield the right answer in the latter one

the capacity and optimal signaling for gaussian mimo channels under interference constraints gaussian mimo channel under the joint transmit tx and interference power constraints tpc and ipc is studied a closed form solution for the optimal tx covariance matrix in the general case is obtained using the kkt based approach up to dual variables a number of more explicit closed form solutions are given with optimal dual variables including full rank and rank 1 beamforming cases as well as the case of identical eigenvectors typical for massive mimo settings which differer significantly from the standard water filling solution a whitening filter is shown to be an important part of optimal precoding under interference constraints sufficient and necessary conditions for each constraint to be redundant are given capacity scaling with the snr is shown to be determined by a natural linear algebraic structure of sub spaces induced by channel matrices of multiple users a number of unusual properties of optimal tx covariance matrix under the joint tpc and ipc are pointed out and a bound on its rank is established an interplay between the tpc and ipc is studied including the transition from power limited to interference limited regimes as the tx power increases while closed from solutions for optimal dual variables are given in some special cases an iterative bisection algorithm iba is proposed to find optimal dual variables in the general case and its convergence is proved for some special cases numerical experiments illustrate its efficient performance bounds for the optimal dual variables are given

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

on the hill determinant method explicit examples are given of the false eigenvalues of the hill determinant method the existence of which has been suspected by workers using analytic methods a simple criterion is discovered for distinguishing between false and physical eigenvalues

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

two sided eigenvalue bounds for the spherically symmetric states of the schro dinger equation the eigenvalues of the radial schr6dinger equation are calculated very accurately by obtaining exact upper and lower bounds by truncating the usual unbounded domain 0 cx of the system to a finite interval of the form 0 g two auxiliary eigenvalue problems are defined it is then proved that the eigenvalues of the resulting confined systems provide upper and lower bounds converging monotonically to the true eigenvalues required moreover each auxiliary eigenvalue problem gives rise to an orthonormal set involving bessel functions the matrix representation of the hamiltonian is therefore derived by expanding the wave function into a fourier bessel series numerical results for single and doublewell polynomial oscillators as well as gaussian type non polynomial potentials illustrate that the eigenvalues can be calculated to an arbitrary accuracy whenever the boundary parameter g is in the neighborhood of some critical value denoted by ecr 1998 elsevier science b v all rights reserved

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

comments on a paper concerning the analytic continued fraction technique for bound states the author shows that the treatment of bound and confined states for a class of confinement potentials by application of the analytic continued fraction method as presented in a recent paper dalta et al ibid vol 13 p 3161 1980 is erroneous and may lead to wrong results for the eigenvalues of the relevant schrodinger equation

iterative mode dropping for the sum capacity of mimo mac with per antenna power constraint we propose an iterative mode dropping algorithm that optimizes input signals to achieve the sum capacity of the mimo mac with per antenna power constraint the algorithm successively optimizes each user s input covariance matrix by applying mode dropping to the equivalent single user mimo rate maximization problem both analysis and simulation show fast convergence we then use the algorithm to briefly highlight the difference in mimo mac capacities under sum and per antenna power constraints

on the applicability of the hill determinant and the analytic continued fraction method to anharmonic oscillators the authors finish a line of argument started in an earlier paper and show that the application of the hill determinant and the associated analytic continued fraction method for the calculation of eigenvalues of anharmonic oscillators is of dubious validity and may thus lead to erroneous results in this context some theorems concerning the eigenvalues of the lambda x4 and the doubly anharmonic oscillators are proved

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

on the applicability of the hill determinant method the authors show that contrarily to what flessas and anagnostatos claim 1982 the so called hill determinant method is perfectly suited to the numerical resolution of schrodinger s equation

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

the eigenvalues of the schrodinger equation for spherically symmetric states for various types of potentials in two three and n dimensions by using perturbative and non perturbative methods the energy levels of the schrodinger equation involving various model potentials in two three and n dimensional space are calculated by using perturbative and non perturbative methods the methods have been used effectively to calculate the energy eigenvalues even for high state numbers and large values of the perturbation parameters

the exponential cosine screened potential the concept of forward nested multiplication is used to develop a unified power series hill determinant method for the calculation of schrodinger equation energy levels a numerical application is made to the exponential cosine screened potential

mountain pass solutions to equations of p laplacian type abstract this work is devoted to study the existence of solutions to equations of p laplacian type we prove the existence of at least one solution and under further assumptions the existence of infinitely many solutions in order to apply mountain pass results we introduce a notion of uniformly convex functional that generalizes the notion of uniformly convex norm

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

on the improvement of convergence of hill determinants we propose a simple approach to analytically determine a parameter of a convergence factor exp gx2 4 in the hill determinant method it provides much more rapid convergence of the method than the other analytical approaches currently used for the g determination

false eigenvalues of the hill determinant method it is shown analytically that the false eigenenergies of a potential obtained by the method of hill determinants are the physical energies of a different potential

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

power allocation in multi antenna wireless systems subject to simultaneous power constraints we address the problem of power allocation to maximize ergodic capacity subject to multiple power constraints assuming that perfect causal channel state information csi is available at both the transmitter and the receiver we characterize the optimal power allocation subject to both long term and short term power constraints which depends upon the ratio of the two power levels additionally we find a suboptimal power allocation if the input power is subject to long term and per antenna power constraints we characterize the conditions for which one power constraint dominates and the other can be ignored numerical results suggest that for the rayleigh fading case a short term power constraint that is larger than a long term power constraint does not significantly impact the ergodic capacity of the channel the effect of per antenna power constraints is also explored for the case of rayleigh fading through our numerical results