singularity evaluation of the straight wire mixed potential integral equation in the method of moments procedure a rigorous treatment on the computation of the various integrals that arise in the method of moments mom formulation of the straight wire electric field integral equation is provided for triangle basis functions along with delta function pulse or triangle weights particular attention is given to integrals whose integrands are weakly singular a singularity extraction technique is employed that splits the integral under question into two parts one that is numerically integrable and one that is analytically integrable closed form approximations based on taylor series techniques are also provided for the former these approximations are very robust resulting in errors less than 0 1 when 2 1 here is the length of a single mom segment and is the wire radius results are compared with data from the literature to demonstrate the robustness of the presented approach for fat wires

on bar frameworks stress matrices and semidefinite programming a bar framework g p in r dimensional euclidean space is a graph g on the vertices 1 2 n where each vertex i is located at point p i in mathbb r r given a framework g p in mathbb r r a problem of great interest is that of determining whether or not there exists another framework g q not obtained from g p by a rigid motion such that q i q j 2 p i p j 2 for each edge i j of g this problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework g q is restricted to be in the same r dimensional space or not the stress matrix s of a bar framework g p plays a key role in these and other related problems in this paper semidefinite programming sdp theory is used to address in a unified manner several problems concerning universal rigidity new results are presented as well as new proofs of previously known theorems in particular we use the notion of sdp non degeneracy to obtain a sufficient condition for universal rigidity and we show that this condition yields the previously known sufficient condition for generic universal rigidity we present new results concerning positive semidefinite stress matrices and we use a semidefinite version of farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix s

surface wave and superdirectivity aspects of effective current for linear antennas previous papers have defined the concept of effective current for linear antennas as a method of postprocessing the unphysical solutions obtained by applying discretization galerkin methods to the usual integral equations that employ the approximate or reduced kernel such an unphysical solution can be thought of as an auxiliary rapidly oscillating current located on the axis of the linear antenna while the effective current is the magnetic field due to the oscillating current at a distance equal to the antenna radius multiplied by a factor equal to the antenna circumference in the present paper we focus on the rapid decrease of the aforementioned magnetic field and point out similarities to surface waves to the well known phenomenon of superdirectivity and to the method of auxiliary sources mas our study sheds light on the unusual nature of electromagnetic waves generated by currents with very rapid spatial variations subject to the condition that the discretization is fine we treat th

interior point trajectories in semidefinite programming in this paper we study interior point trajectories in semidefinite programming sdp including the central path of an sdp this work was inspired by the seminal work of megiddo on linear programming trajectories progress in math programming interior point algorithms and related methods n megiddo ed springer verlag berlin 1989 pp 131 158 under an assumption of primal and dual strict feasibility we show that the primal and dual central paths exist and converge to the analytic centers of the optimal faces of respectively the primal and the dual problems we consider a class of trajectories that are similar to the central path but can be constructed to pass through any given interior feasible or infeasible point and study their convergence finally we study the derivatives of these trajectories and their convergence

positive semidefinite matrix completions on chordal graphs and constraint nondegeneracy in semidefinite programming let g v e be a graph in matrix completion theory it is known that the following two conditions are equivalent i g is a chordal graph ii every g partial positive semidefinite matrix has a positive semidefinite matrix completion in this paper we relate these two conditions to constraint nondegeneracy condition in semidefinite programming and prove that they are each equivalent to iii for any g partial positive definite matrix that has a positive semidefinite completion constraint nondegeneracy is satisfied at each of its positive semidefinite matrix completions

a family of simple non parametric kernel learning algorithms previous studies of non parametric kernel learning npkl usually formulate the learning task as a semi definite programming sdp problem that is often solved by some general purpose sdp solvers however for n data examples the time complexity of npkl using a standard interior point sdp solver could be as high as o n6 5 which prohibits npkl methods applicable to real applications even for data sets of moderate size in this paper we present a family of efficient npkl algorithms termed simplenpkl which can learn non parametric kernels from a large set of pairwise constraints efficiently in particular we propose two efficient simplenpkl algorithms one is simplenpkl algorithm with linear loss which enjoys a closed form solution that can be efficiently computed by the lanczos sparse eigen decomposition technique another one is simplenpkl algorithm with other loss functions including square hinge loss hinge loss square loss that can be re formulated as a saddle point optimization problem which can be further resolved by a fast iterative algorithm in contrast to the previous npkl approaches our empirical results show that the proposed new technique maintaining the same accuracy is significantly more efficient and scalable finally we also demonstrate that the proposed new technique is also applicable to speed up many kernel learning tasks including colored maximum variance unfolding minimum volume embedding and structure preserving embedding

a squared smoothing newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems we study a smoothing newton method for solving a nonsmooth matrix equation that includes semidefinite programming and the semidefinite complementarity problem as special cases this method if specialized for solving semidefinite programs needs to solve only one linear system per iteration and achieves quadratic convergence under strict complementarity and nondegeneracy we also establish quadratic convergence of this method applied to the semidefinite complementarity problem under the assumption that the jacobian of the problem is positive definite on the affine hull of the critical cone at the solution these results are based on the strong semismoothness and complete characterization of the b subdifferential of a corresponding squared smoothing matrix function which are of general theoretical interest

entire domain mom analysis of an array of arbitrarily oriented circular loop antennas a general formulation this paper presents a general formulation for the entire domain method of moments analysis of an n element array of arbitrarily oriented coupled circular loop antennas the orientations of the loops with respect to each other are defined through the use of eulerian angles the positions of the loops in space are defined by translation vectors from the center of a loop to another by making use of coordinate transformations and vector algebra a concise set of linear equations for solving the currents on the loops is obtained the formulation is sufficiently general to be used for all the previously reported coupled loop orientations it is also ideal for the study of skewed circular loops which may be oriented in innumerable ways and in turn demand a general formulation for their analysis the use of the general formulation is illustrated by analyzing the coupling currents between collocated skewed loops the formulation is also applied to study the effects on the transfer impedance between a pair of noncollocated skewed loops as one of the loops is rotated about various axes

solving some large scale semidefinite programs via the conjugate residual method most current implementations of interior point methods for semidefinite programming use a direct method to solve the schur complement equation sce m delta y h in computing the search direction when the number of constraints is large the problem of having insufficient memory to store m can be avoided if an iterative method is used instead numerical experiments have shown that the conjugate residual cr method typically takes a huge number of steps to generate a high accuracy solution on the other hand it is difficult to incorporate traditional preconditioners into the sce except for block diagonal preconditioners we decompose the sce into a 2 2 block system by decomposing delta y similarly for h into two orthogonal components with one lying in a certain subspace that is determined from the structure of m numerical experiments on semidefinite programming problems arising from the lovasz theta function of graphs and maxcut problems show that high accuracy solutions can be obtained with a moderate number of cr steps using the proposed equation

on the use of nonsingular kernels in certain integral equations for thin wire circular loop antennas fundamental properties of certain popular integral equations for thin wire circular loop antennas with nonsingular kernels are studied the cornerstone of our study is a large asymptotic formula for the n th fourier coefficient of the nonsingular kernel four different methods of driving the circular loop are considered namely the delta function generator the finite gap generator a certain type of frill generator and the case of plane wave incidence the excitation model is crucial to our discussions since it determines the behavior of the solution convergence also discussed are associated difficulties that may arise when moment methods are applied to the aforementioned equations as well as a simple improvement to the fourier series method for determining the current in the case of the frill generator the main results herein closely parallel recent results for the case of the straight antenna and similarities and differences between the straight and circular cases are discussed

mismatched decoding revisited general alphabets channels with memory and the wide band limit the mismatch capacity of a channel is the highest rate at which reliable communication is possible over the channel with a given possibly suboptimal decoding rule this quantity has been studied extensively for single letter decoding rules over discrete memoryless channels dmcs here we extend the study to memoryless channels with general alphabets and to channels with memory with possibly non single letter decoding rules we also study the wide band limit and in particular the mismatch capacity per unit cost and the achievable rates on an additive noise spread spectrum system with single letter decoding and binary signaling

the exact kernel for cylindrical antenna in the numerical calculation for a cylindrical antenna the thin wire kernel is widely used as a convenient approximation to the exact kernel of the electric field equation however this thin wire approximation is valid only if the radius of wire is much less than its length and the wavelength furthermore arbitrarily using this approximation may not obtain the convergent outcomes the expression of the exact kernel for the cylindrical antenna which would be useful in the numerical work is presented

efficient modeling of radiation and scattering for a large array of loops a computationally efficient technique based on the method of moments mom formulation is invoked in the characterization of radiation and scattering properties of an array of coaxial circular non identical loops a set of pocklington type integral equations for the loop currents is formulated and subsequently discretized by a standard procedure thanks to a suitable choice of the basis functions the resulting matrix corresponding to the pertinent linear system is forced to consist of circulant blocks this type of system is solvable by an innovative recursive algorithm featuring several important advantages such as lower memory and execution time consumption over standard purely numerical inversion the overall procedure is simpler in implementation than already existing methods based on fourier analysis the procedure invokes almost exclusively elementary functions and is applicable to large arrays with respect to diameter or number of loops data for such configurations are presented for the first time in literature

a new formulation of pocklington s equation for thin wires using the exact kernel pocklington s integro differential equation for thin wires with the exact kernel is reformulated using a second derivative formula for improper integrals this allows for analytical evaluation of the second derivatives resulting in a pure integral equation in a similar manner to what is done when using the approximate kernel however as opposed to using the approximate kernel the resulting integral equation developed here is numerically stable even with a simple pulse function point matching solution good convergence for the current is obtained using pulse functions and the severe unphysical oscillations of the current that are encountered when using pulse functions with the approximate kernel are avoided

comments on exact solutions of electromagnetic fields in both near and far zones radiated by thin circular loop antennas a general representation with reply this paper presents an alternative vector analysis of the electromagnetic em fields radiated from thin circular loop antennas of arbitrary radius a this method which employs the dyadic green s function in the derivation of the em radiated fields makes the analysis more general compact and straightforward than those two methods published recently by werner 1996 and overfelt 1996 both near and far zones are considered so that the em radiated fields are expressed in terms of the vector wave eigenfunctions not only the exact solution of the em fields in the near and far zones outside the region where r a is derived by the use of the spherical hankel function of the first kind but also the closed series form of the em fields radiated in the near zone inside the region 0 spl les r a is obtained in series of the spherical bessel functions of the first kind as an example a fourier cosine series is used to expand an arbitrary current distribution along the loop and the exact representations of the em radiated fields due to the loop everywhere are obtained in closed form the closed form reduces to those for the sinusoidal current loop and further for the uniform current loop validity of the approximate formulas is discussed and clarified error analysis based on numerical computations of the radiated fields is also given to show the accuracy of the limiting cases

on the spectral efficiency of noncoherent doubly selective block fading channels in this paper we consider noncoherent single antenna communication over doubly selective block fading channels with discrete block fading interval n in our noncoherent setup neither the transmitter nor the receiver know the channel fading coefficients though both know the channel statistics in particular we consider discrete time channels whose impulse response trajectories obey a complex exponential basis expansion model with uncorrelated coefficients and we show that such a model holds in the limit n for pulse shaped transmission reception over certain wide sense stationary uncorrelated scattering channels first we show that when the inputs are chosen from continuous distributions the channel s multiplexing gain i e capacity pre log factor equals max 0 1 n delay n dopp n for discrete delay spread n delay and discrete doppler spread n dopp next for the case of strictly doubly selective fading i e n dopp 1 and n delay 1 we establish that for cyclic prefixed affine pilot aided transmission pat schemes designed to minimize the mean squared error mse attained by pilot aided minimum mse channel estimation the pre log factor of the achievable rate is less than the channel s multiplexing gain we then provide guidelines for the design of pat schemes whose achievable rate pre log factor equals the channel s multiplexing gain and construct an example

efficient kernel calculation of cylindrical antennas this paper presents a technique for evaluating analytically and without limitation the singular part of the kernel integral of cylindrical wires due to uniform current distribution this approach uses the static green s function expression in cylindrical coordinates the formula of the singular part converges rapidly and illustrates its usefulness for kernel calculations without loss of accuracy

electromagnetic characteristics of superquadric wire loop antennas the analysis of antenna configurations in the form of a generalized superquadric loop which includes circular elliptical and rectangular loop geometries is presented in this paper use of a galerkin form of the moment method with piecewise sinusoidal subsectional basis and testing functions provides rapid numerical convergence and accurate representation of the antenna current a convenient parametric representation for the superquadric curve is developed to allow a subsectional formulation using curved wire segments rather than the commonly employed piecewise linear segments to construct the geometry both magnetic frill and delta gap source models are implemented to allow a detailed study of input impedance directivity radiation pattern and current distribution as a function of various geometrical parameters the results are shown to compare well with previous results for the special case of a circular loop antenna some useful curves are presented to aid in the design of practical superquadric loop antennas

fading channels how perfect need perfect side information be the analysis of flat fading channels is often performed under the assumption that the additive noise is white and gaussian and that the receiver has precise knowledge of the realization of the fading process these assumptions imply the optimality of gaussian codebooks and of scaled nearest neighbor decoding here we study the robustness of this communication scheme with respect to errors in the estimation of the fading process we quantify the degradation in performance that results from such estimation errors and demonstrate the lack of robustness of this scheme for some situations we suggest the rule of thumb that in order to avoid degradation the estimation error should be negligible compared to the reciprocal of the signal to noise ratio snr

unitary space time modulation for multiple antenna communications in rayleigh flat fading motivated by information theoretic considerations we propose a signaling scheme unitary space time modulation for multiple antenna communication links this modulation is ideally suited for rayleigh fast fading environments since it does not require the receiver to know or learn the propagation coefficients unitary space time modulation uses constellations of t spl times m space time signals spl phi sub i l 1 l where t represents the coherence interval during which the fading is approximately constant and m t is the number of transmitter antennas the columns of each spl phi sub i are orthonormal when the receiver does not know the propagation coefficients which between pairs of transmitter and receiver antennas are modeled as statistically independent this modulation performs very well either when the signal to noise ratio snr is high or when t spl gt m we design some multiple antenna signal constellations and simulate their effectiveness as measured by bit error probability with maximum likelihood decoding we demonstrate that two antennas have a 6 db diversity gain over one antenna at 15 db snr

on the application of numerical methods to hallen s equation the so called hallen integral equation for the current on a finite linear antenna center driven by a delta function generator takes two forms depending on the choice of kernel the two kernels are usually referred to as the exact and the approximate or reduced kernel with the approximate kernel the integral equation has no solution nevertheless the same numerical method is often applied to both forms of the integral equation in this paper the behavior of the numerical solutions thus obtained is investigated and the similarities and differences between the two numerical solutions are discussed the numerical method is galerkin s method with pulse functions we first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length in this case the method yields an infinite toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter the infinite system is solved exactly for nonzero pulse width the exact solution is then developed asymptotically for the case where the pulse width is small when the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna the latter problem is greatly facilitated for the approximate kernel the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the pocklington type

the use of the frill generator in thin wire integral equations for a frill generator feed certain fundamental mathematical properties of hallen s and pocklington s equations with the approximate kernel are stated and derived for a particular moment method procedure galerkin s method with pulse functions applied to hallen s equation the consequences to the numerical solutions are carefully examined generalizations to other numerical methods with subsectional basis functions are discussed many of our results come from studying the simpler problem of the infinite dipole analytically and applying the understanding thus obtained to the case of the finite dipole

general interior point maps and existence of weighted paths for nonlinear semidefinite complementarity problems extending the previous work of monteiro and pang 1998 this paper studies properties of fundamental maps that can be used to describe the central path of the monotone nonlinear com plementarity problems over the cone of symmetric positive semidefinite matrices instead of focusing our attention on a specific map as was done in the approach of monteiro and pang 1998 this paper considers a general form of a fundamental map and introduces conditions on the map that allow us to extend the main results of monteiro and pang 1998 to this general map each fundamental map leads to a family of weighted continuous trajectories which include the central trajectory as a special case hence for complementarity problems over the cone of symmetric positive semidefinite matrices the notion of weighted central path depends on the fundamental map used to represent the central path

matrices with high completely positive semidefinite rank a real symmetric matrix m is completely positive semidefinite if it admits a gram representation by hermitian positive semidefinite matrices of any size d the smallest such d is called the complex completely positive semidefinite rank of m and it is an open question whether there exists an upper bound on this number as a function of the matrix size we construct completely positive semidefinite matrices of size 4k2 2k 2 with complex completely positive semidefinite rank 2k for any positive integer k this shows that if such an upper bound exists it has to be at least exponential in the matrix size for this we exploit connections to quantum information theory and we construct extremal bipartite correlation matrices of large rank we also exhibit a class of completely positive matrices with quadratic in terms of the matrix size completely positive rank but with linear completely positive semidefinite rank and we make a connection to the existence of hadamard matrices

the exact analytical solution for large circular loops radiating around a dielectric coated conducting sphere in this paper the electromagnetic field of electrically large circular loop antennas with different radii and different nonuniform current values around a dielectric coated conducting sphere is considered one or more loop antennas are located on the outer surface of a spherical dielectric shell covering a conducting sphere eigenfunction series solutions for the field are assumed in two regions the current distribution on the wire loop driven by a voltage source is determined by fourier series expansion and all necessary harmonics are taken into account exact analytical field expressions in closed forms are derived and field patterns are plotted the antenna model and formulation presented in this paper offer exact analytical solutions to several loop antenna problems

solving log determinant optimization problems by a newton cg primal proximal point algorithm we propose a newton cg primal proximal point algorithm ppa for solving large scale log determinant optimization problems our algorithm employs the essential ideas of ppa the newton method and the preconditioned cg solver when applying the newton method to solve the inner subproblem we find that the log determinant term plays the role of a smoothing term as in the traditional smoothing newton technique focusing on the problem of maximum likelihood sparse estimation of a gaussian graphical model we demonstrate that our algorithm performs favorably compared to existing state of the art algorithms and is much preferred when a high quality solution is required for problems with many equality constraints

solution of hallen s integral equations by means of nystrom method to solve hallen s integral equations accurately nystrom method is applied in this paper to obtain an exact and finite solution both the exact kernel and a finite gap width must be taken into consideration in the case of a gap feed a modification function that converts the approximate kernel to the exact kernel is derived as the modification function is singular a one point correction method is proposed as a singularity subtraction method using a step function source we can use various quadrature rules to solve the problem the results show small difference between them the validity of this theory is confirmed by comparing the measured results and the calculated results the effect of the feed system is shown to be strong

nearest neighbor decoding for additive non gaussian noise channels we study the performance of a transmission scheme employing random gaussian codebooks and nearest neighbor decoding over a power limited additive non gaussian noise channel we show that the achievable rates depend on the noise distribution only via its power and thus coincide with the capacity region of a white gaussian noise channel with signal and noise power equal to those of the original channel the results are presented for single user channels as well as multiple access channels and are extended to fading channels with side information at the receiver

multiple antenna signaling over fading channels with estimated channel state information performance analysis multiple antenna concepts for wireless communication systems promise high spectral efficiency and low error rates by proper exploitation of the randomness in multipath propagation in this paper we investigate the impact of channel uncertainty caused by channel estimation errors on the error rate performance we consider a training based multiple antenna system that reserves a portion of time to sound the channel training symbols are used to estimate the channel by means of an arbitrary linear filter at the receiver no channel state information csi is assumed at the transmitter we present a new framework to analyze training based multiple antenna systems by introducing an equivalent system model that specifies the channel by the estimated and hence known channel coefficients and an uncorrelated data dependent multiplicative noise we derive the maximum likelihood ml detector and highlight its behavior in the limiting cases of perfect csi and no csi and its relation to several mismatched detectors we deduce new exact expressions and chernoff bounds of the pairwise error probability pep used to assess word error and bit error rate bounds for ml and mismatched detection finally we review the code design guidelines in terms of the deleterious effect of channel uncertainty for coherent and noncoherent signaling schemes and present numerical results

comment on analytical results for a bessel function times legendre polynomials class integrals a result is obtained stemming from gegenbauer where the products of certain bessel functions and exponentials are expressed in terms of an infinite series of spherical bessel functions and products of associated legendre functions closed form solutions for integrals involving bessel functions times associated legendre functions times exponentials recently elucidated by neves et al j phys a math gen 39 l293 are then shown to result directly from the orthogonality properties of the associated legendre functions this result offers greater flexibility in the treatment of classical heisenberg chains and may do so in other problems such as occur in electromagnetic diffraction theory

on the outage capacity of a practical decoder accounting for channel estimation inaccuracies the optimal decoder achieving the outage capacity under imperfect channel estimation is investigated first by searching into the family of nearest neighbor decoders which can be easily implemented on most practical coded modulation systems we derive a decoding metric that minimizes the average of the transmission error probability over all channel estimation errors next we specialize our general expression to obtain the corresponding decoding metric for fading mimo channels according to the notion of estimation induced outage eio capacity introduced in our previous work and assuming a block rayleigh fading channel we characterize the maximal achievable information rates using gaussian codebooks associated to the proposed decoder these achievable rates are compared to the rates achieved by the classical mismatched maximum likelihood ml decoder and the ultimate limits given by the eio capacity numerical results show that the derived metric provides significant gains in terms of achievable eio rates and bit error rate ber in a bit interleaved coded modulation bicm framework without introducing any additional decoding complexity however the achievable rates of such metric are still far from the eio capacity

gaussian codes and weighted nearest neighbor decoding in fading multiple antenna channels we investigate the fading multiple antenna channel the decoder is assumed to possess imperfect channel fading information a modified nearest neighbor decoder with an innovative weighting factor is introduced and an expression for the generalized mutual information gmi the achievable rate is obtained we show that under certain conditions the achievable rate is equivalent to that of a fading multiple antenna gaussian channel where fading is known to the receiver and is equal to the channel estimation and where noise is due to both the channel noise and the channel estimation error we show that for our communication scheme the minimum mean square error mmse channel estimator is optimal in the sense that it achieves the highest value of gmi and hence the highest communication rate additionally a training based multiple input multiple output mimo scheme in a block fading channel is investigated and it is shown that the number of degrees of freedom depends on the signal to noise ratio snr

evaluation of potential and impedance integrals in analysis of axially symmetric metallic structures to prescribed accuracy up to machine precision this paper presents a novel method for evaluating potential and impedance integrals appearing in the method of moment analysis of arbitrary axially symmetric metallic structures based on exact wire kernel and higher order bases due to new variable transforms proposed for singularity cancellation and smoothing the integrands high accuracy up to machine precision is achieved using relatively small number of integration points simple formulas are determined for predicting a number of integration points needed for prescribed accuracy benefits of high precision evaluation of impedance integrals are illustrated on a number of numerical examples

fast simple and accurate computation of the currents on an arbitrarily large circular loop antenna this paper is presenting a very efficient method for computing the currents on a circular loop antenna with arbitrarily large size pocklington s integral equation is formulated for the circular loop geometry and the method of moments with point matching is applied to cast the equation s discrete counterpart the basis functions are chosen in such a way that the relevant square matrix is circulant and therefore amenable to exact eigenvalue analysis subsequently the matrix is diagonalized and inverted analytically yielding simple analytical expressions for the weights of the basis functions and hence the current and the input admittance of the antenna the algorithm utilizes mainly elementary mathematical functions as opposed to already known expressions in the literature and yields results in the form of a rapidly convergent single series applicable to extremely large loops

capacity bounds via duality with applications to multiple antenna systems on flat fading channels a technique is proposed for the derivation of upper bounds on channel capacity it is based on a dual expression for channel capacity where the maximization of mutual information over distributions on the channel input alphabet is replaced with a minimization of average relative entropy over distributions on the channel output alphabet we also propose a technique for the analysis of the asymptotic capacity of cost constrained channels the technique is based on the observation that under fairly mild conditions capacity achieving input distributions escape to infinity the above techniques are applied to multiple antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown or only partially known at the receiver it is demonstrated that for high signal to noise ratio snr the capacity of such channels typically grows only double logarithmically in the snr to better understand this phenomenon and the rates at which it occurs we introduce the fading number as the second order term in the high snr asymptotic expansion of capacity and derive estimates on its value for various systems it is suggested that at rates that are significantly higher than the fading number communication becomes extremely power inefficient thus posing a practical limit on practically achievable rates upper and lower bounds on the fading number are also presented for single input single output siso systems the bounds coincide thus yielding a complete characterization of the fading number for general stationary and ergodic fading processes we also demonstrate that for memoryless multiple input single output miso channels the fading number is achievable using beam forming and we derive an expression for the optimal beam direction this direction depends on the fading law and is in general not the direction that maximizes the snr on the induced siso channel using a new closed form expression for the expectation of the logarithm of a noncentral chi square distributed random variable we provide some closed form expressions for the fading number of some systems with gaussian fading including siso systems with circularly symmetric stationary and ergodic gaussian fading the fading number of the latter is determined by the fading mean fading variance and the mean squared error in predicting the present fading from its past it is not directly related to the doppler spread for the rayleigh ricean and multiple antenna rayleigh fading channels we also present firm nonasymptotic upper and lower bounds on channel capacity these bounds are asymptotically tight in the sense that their difference from capacity approaches zero at high snr and their ratio to capacity approaches one at low snr

vector coloring the categorical product of graphs a vector t coloring of a graph is an assignment of real vectors p 1 ldots p n to its vertices such that p i tp i t 1 for all i 1 ldots n and p i tp j le 1 whenever i and j are adjacent the vector chromatic number of g is the smallest number t ge 1 for which a vector t coloring of g exists for a graph h and a vector t coloring p 1 ldots p n of g the map taking i ell in v g times v h to p i is a vector t coloring of the categorical product g times h it follows that the vector chromatic number of g times h is at most the minimum of the vector chromatic numbers of the factors we prove that equality always holds constituting a vector coloring analog of the famous hedetniemi conjecture from graph coloring furthermore we prove necessary and sufficient conditions under which all optimal vector colorings of g times h are induced by optimal vector colorings of the factors our proofs rely on various semidefinite programming formulations of the vector chromatic number and a theory of optimal vector colorings we develop along the way which is of independent interest

a new graph parameter related to bounded rank positive semidefinite matrix completions the gram dimension mathrm gd g of a graph g is the smallest integer k ge 1 such that any partial real symmetric matrix whose entries are specified on the diagonal and at the off diagonal positions corresponding to edges of g can be completed to a positive semidefinite matrix of rank at most k assuming a positive semidefinite completion exists for any fixed k the class of graphs satisfying mathrm gd g le k is minor closed hence it can be characterized by a finite list of forbidden minors we show that the only minimal forbidden minor is k k 1 for k le 3 and that there are two minimal forbidden minors k 5 and k 2 2 2 for k 4 we also show some close connections to euclidean realizations of graphs and to the graph parameter nu g of van der holst combinatorica 23 4 633 651 2003 in particular our characterization of the graphs with mathrm gd g le 4 implies the forbidden minor characterization of the 3 realizable graphs of belk discret comput geom 37 139 162 2007 and belk and connelly discret comput geom 37 125 137 2007 and of the graphs with nu g le 4 of van der holst combinatorica 23 4 633 651 2003

a vector potential for a partly penetrating well and flux in an approximate method of images a vector potential is formulated for three dimensional steady groundwater flow within a framework that is consistent with the lagrange stream function for twodimensional flow analytical expressions are developed for vector potentials of a point sink and a line sink of both finite and infinite length a previously unpublished vector potential for a partly penetrating well in a horizontal confined aquifer is obtained using the method of images where an infinite series of image line sinks is approximated using a finite number of images and two semi infinite elements singularities and discontinuities within these vector potentials are interpreted as producing a virtual discharge connecting vortex lines into continuous loops over infinity the vector potential is used to accurately and efficiently compute the net flux through surfaces containing singularities in the specific discharge this net flux quantifies errors in the elevation of streamlines associated with a finite number of images and provides a measure to ensure a relatively seamless link between local three dimensional flow and farfield two dimensional flow

exact closed form expression of the electromagnetic field excited by pulse shaped and triangular line currents the exact closed form expression for the electromagnetic field excited by pulse shaped and triangular line currents is presented the analytical formulation based on the incomplete hankel functions shows that the field is composed of cylindrical waves excited near the current axis and spherical waves arising from the source critical points spherical waves related with pulse shaped current basis functions are shown to have stronger field singularities than for triangular basis functions because of the impulsive charges present at the current truncations for triangular line currents it is further shown that an additional spherical wave arises from the critical point featuring a charge jump discontinuity near and high frequency asymptotic field expressions show explicitly the nature of the field singularities using the analytical properties of the incomplete hankel functions the galerkin s impedance matrix coefficients useful to solve radiation and scattering problems in truncated cylindrical structures using the method of moments are finally derived in a closed form numerical examples show the accuracy of the proposed field representation

integral evaluation using the mellin transform and generalized hypergeometric functions tutorial and applications to antenna problems this is a tutorial presentation of the mellin transform mt method for the exact calculation of one dimensional definite integrals and an illustration of the application of this method to antenna electromagnetics problems once the basics have been mastered one quickly realizes that the mt method is extremely powerful often yielding closed form expressions very difficult to come up with other methods or to deduce from the usual tables of integrals yet as opposed to other methods the mt method is very straightforward to apply it usually requires laborious calculations but little ingenuity in fact the mt method is used by mathematica to symbolically calculate definite integrals the first part of this paper is a step by step tutorial proceeding from first principles it includes basic information on mellin barnes integrals and generalized hypergeometric functions and summarizes the key ideas of the mt method in the remaining parts the mt method is applied to three examples from the antenna area the results here are believed to be new at least in the antenna electromagnetics literature in our first example we obtain a closed form expression as a generalized hypergeometric function for the power radiated by a constant current circular loop antenna this quantity has been extensively discussed recently our second example concerns the admittance of a 2 d slot antenna in both these examples the exact closed form expressions are applied to improve upon existing formulas in standard antenna textbooks in our third example finally we obtain a very simple expression for an integral arising in recent unpublished studies of unbounded biaxially anisotropic media

an exact solution of the generalized exponential integral and its application to moment method formulations the generalized exponential integral is one of the most fundamental integrals in antenna theory and for many years exact solutions to this integral have been sought this paper considers an exact solution to the generalized exponential integral which is completely general and independent of the usual restrictions involving the wavelength field point distance and dipole length is considered the exact series representation presented converges rapidly in the induction and near field regions of the antenna and therefore provides an alternative to numerical integration two method of moments formulations are considered they use the exact expression for the generalized exponential integral in the computation of the impedance matrix elements it is demonstrated that for very thin straight wire antennas an asymptotic expansion can be used to obtain a numerically convenient form of the generalized exponential integral

local duality of nonlinear semidefinite programming recently chan and sun chan z x d sun constraint nondegeneracy strong regularity and nonsingularity in semidefinite programming siam j optim 19 370 376 reported for semidefinite programming sdp that the primal dual constraint nondegeneracy is equivalent to the dual primal strong second order sufficient condition ssosc this result is responsible for a number of important results in stability analysis of sdp in this paper we study duality of this type in nonlinear semidefinite programming nsdp we introduce the dual ssosc at a karush kuhn tucker kkt triple of nsdp and study its various characterizations and relationships to the primal nondegeneracy although the dual ssosc is nothing but the ssosc for the wolfe dual of the nsdp it suggests new information for the primal nsdp for example it ensures that the inverse of the hessian of the lagrangian function exists at the kkt triple and the inverse is positive definite on some normal space it also ensures the primal nondegeneracy some of our results generalize the corresponding classical duality results in nonlinear programming studied by fujiwara et al fujiwara o s p han o l mangasarian 1984 local duality of nonlinear programs siam j control optim 22 162 169 for the convex quadratic sdp qsdp we have complete characterizations for the primal and dual ssosc our results reveal that the nearest correlation matrix problem satisfies not only the primal and dual ssosc but also the primal and dual nondegeneracy at its solution suggesting that it is a well conditioned qsdp

rigidity and energy suppose one holds two sticks in the form of a cross in one hand and places a rubber band in tension around the four ends when it is released it comes to rest in the shape of a convex quadrilateral in a plane it always returns to the same shape no matter how it is distorted as long as the ends do not slip and the rubber band does not break this is a very simple example of a rigid framework of the type we discuss here energy explains why a framework such as the one above is rigid the rubber band deforms in such a way as to minimize the total energy in the framework only in the final deformed shape will the framework have a minimum energy this idea of introducing energy functions is very useful it can be used to prove a key lemma that was used by cauchy 5 in 1813 to show that convex polyhedral surfaces are rigid it can explain why some and perhaps all of r buckminster fuller s tensegrity structures 10 stay up when applied to spider webs it shows why they can only take on certain geometric shapes it also can be used to prove conjecture 6 of branko gri inbaum and g c shephard in their lectures on lost mathematics 12 this conjecture says that if a framework in the shape of a convex polygon with rods sticks on the boundary and cables inside is rigid in the plane so is the framework obtained by reversing the roles of rods and cables it is interesting to compare some of these results with the opening a rm theorem of axel schur 18 see chern 6 also which is a very close smooth analogue to cauchy s lemma schur s theorem says that if a convex planar smooth arc the arm is opened that is it is moved to another position with the same length but with corresponding points having smaller curvature then the two ends are moved apart another amusing application is to show that a regular pentagon in 4 space a pentagon with all 5 sides equal and all 5 angles between the sides equal has its angles bounded between 36 and 108 a comment of o bottema in 4

primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

global completability with applications to self consistent quantum tomography let p 1 ldots p n in mathbb r d be unknown vectors and let omega subseteq 1 ldots n 2 assume that the inner products p i t p j are fixed for all i j in omega do these inner product constraints up to simultaneous rotation of all vectors determine p 1 ldots p n uniquely here we derive a necessary and sufficient condition for the uniqueness of p 1 ldots p n i e global completability which is applicable to a large class of practically relevant sets omega moreover given omega we show that the condition for global completability is universal in the sense that for almost all vectors p 1 ldots p n in mathbb r d the completability of p 1 ldots p n only depends on omega and not on the specific values of p i t p j for i j in omega this work was motivated by practical considerations namely matrix factorization techniques and self consistent quantum tomography

exact field expressions for circular loop antennas using spherical functions expansion a new technique is introduced for evaluating the vector potential of circular loop antennas clas with arbitrary current distribution and dimensions which can be used to obtain the exact electromagnetic fields in near and far regions of the antenna the technique is based on the spherical bessel functions sbfs and associated legendre polynomials alps expansion this expansion is used to decouple the azimuth variables of the integrand of the vector potential integrals from the other variables in a separate functions that depend exponentially on azimuth variables this technique reduces greatly the mathematical operations for evaluation of the vector potential to only one simple integration the technique is used to obtain exact electromagnetic fields expressions for a thin wire cla of arbitrary current distribution and dimension which is valid everywhere except on the loop the well known far fields for the cla of arbitrary current distribution as well as near and far fields for uniform current loop and small loop are obtained from the general field expressions presented in this paper as an special cases

a method of moments approach for the efficient and accurate modeling of moderately thick cylindrical wire antennas this paper introduces a moment method formulation which is capable of accurately modeling moderately thick cylindrical wire antennas new algorithms are presented for the efficient computation of the cylindrical wire kernel and related impedance matrix integrals these algorithms make use of exact series representations as well as efficient numerical procedures and lead to a significant reduction in overall computation time for thicker wires another major advantage of this moment method technique is that it is no longer restricted by the segment length to radius ratio limitations inherent in past formulations thereby making it possible to achieve solution convergence for a much wider class of wire antenna structures several examples illustrating the superior convergence properties of this new moment method formulation are presented and discussed

new exact solution procedure for the near fields of the general thin circular loop antenna an analytical method is presented which gives the various fields including the scalar potential of a general circular loop antenna in cylindrical coordinates in terms of six separate series three of the series are elementary and the other three involve legendre functions of the second kind and half integral order the results obtained are compared with direct numerical integration and with two far field formulae the standard textbook expression and what appears to be a new formula which gives more accurate results the series solution is more robust than numerical integration for the higher fourier terms of the general solution close to the loop the solution has also been compared with the solution of werner and it agrees exactly the solution presented here gives better convergence close to the loop whereas werner s solution is better far from the loop sample calculations are given for the axisymmetric case

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

design and analysis of mmse pilot aided cyclic prefixed block transmissions for doubly selective channels this paper considers affine cyclic prefixed block based pilot aided transmission pat over the single antenna doubly selective channel where the channel is assumed to obey a complex exponential basis expansion model first a tight lower bound on the mean squared error mse of pilot aided channel estimates is derived along with necessary and sufficient conditions on the pilot data pattern that achieves this bound from these conditions novel minimum mse mmse pat schemes are proposed and upper lower bounds on their ergodic achievable rates are derived a pilot data power allocation technique is also developed a high snr asymptotic analysis of the ergodic achievable rate of affine mmse pat is then performed which suggests that the channel s spreading parameters should be taken into account when choosing among affine mmse pat schemes specifically we establish that multicarrier mmse pat achieves higher rates than single carrier mmse pat when the channel s delay spread dominates its doppler spread and vice versa

on eigenvalue optimization in this paper we study optimization problems involving eigenvalues of symmetric matrices one of the difficulties with numerical analysis of such problems is that the eigenvalues considered as functions of a symmetric matrix are not differentiable at those points where they coalesce we present a general framework for a smooth differentiable approach to such problems it is based on the concept of transversality borrowed from differential geometry in that framework we discuss first and second order optimality conditions and rates of convergence of the corresponding second order algorithms finally we present some results on the sensitivity analysis of such problems

the real positive semidefinite completion problem for series parallel graphs abstract we consider the partial real symmetric matrices x whose diagonal entries are equal to 1 and whose off diagonal entries are specified only on a subset of the positions the question is to determine whether x can be completed to a positive semidefinite matrix extending a result of barrett et al 3 we give a set of necessary conditions for x to be completable and show that these conditions are also sufficient if and only if the graph corresponding to the positions of the specified entries is series parallel i e has no k 4 minor

on communication over unknown sparse frequency selective block fading channels this paper considers the problem of reliable communication over discrete time channels whose impulse responses have length l and exactly s l non zero coefficients and whose support and coefficients remain fixed over blocks of n l channel uses but change independently from block to block here it is assumed that the channel s support and coefficient realizations are both unknown although their statistics are known assuming gaussian non zero coefficients and noise and focusing on the high snr regime it is first shown that the ergodic noncoherent channel capacity has pre log factor 1 s n for any l it is then shown that to communicate with arbitrarily small error probability at rates in accordance with the capacity pre log factor it suffices to use pilot aided orthogonal frequency division multiplexing ofdm with s pilots per fading block in conjunction with an appropriate noncoherent decoder since the achievability result is proven using a noncoherent decoder whose complexity grows exponentially in the number of fading blocks k a simpler decoder based on s 1 pilots is also proposed its e achievable rate is shown to have pre log factor equal to 1 s 1 n with the previously considered channel while its achievable rate is shown to have pre log factor 1 s 1 n when the support of the block fading channel remains fixed over time

near fields of the constant current thin circular loop antenna of arbitrary radius assuming a known constant current distribution on the thin circular loop antenna of arbitrary radius in free space an exact integration of the vector potential is performed without recourse to approximations the only restrictions on the solution variables are that the observation point distance must be greater than the loop radius and that the polar angle must run between 0 and spl pi the resulting vector potential infinite series solution possesses a real part composed of linear combinations of complete elliptic integrals of the first and second kind and an imaginary part composed of elementary functions thus it is possible to obtain an exact solution which is valid everywhere that r a and 0 spl les spl theta spl les spl pi the electromagnetic field components of the constant current circular loop antenna are then determined by direct series differentiation these solutions are valid in the near and induction fields converging rapidly there and are also valid in the far field although many terms of the series are needed for convergence

an exact integration procedure for vector potentials of thin circular loop antennas a direct integration procedure for far zone vector potentials of thin circular loop antennas has been known for many years this method is general in the sense that it leads to simple integrals which have closed form solutions for most commonly assumed loop current distributions however a comparable integration technique has not been available for evaluating the more complicated near zone vector potentials this paper introduces a systematic approach for the exact integration of general near zone vector potentials associated with current carrying circular loop antennas a particular example is considered where this new integration technique is used to find exact solutions to the vector potential and electromagnetic field integrals for loops with a fourier cosine series expansion of the current the observation is made that degenerate forms of these exact representations lead to simplified expressions for the important special cases of a uniform and cosinusoidal current loop two equivalent forms of exact series expansions are derived for the uniform current vector potential and field integrals it is shown that the familiar small loop approximations as well as the classical far field expressions may be obtained as limiting cases of the more general exact series representations for the uniform current loop obtained in this paper convenient asymptotic far field expansions are derived for the loop with a cosinusoidal current distribution finally the far field analysis for the cosinusoidal loop is generalized to loops having an arbitrary current represented by a fourier cosine series

an exact duality theory for semidefinite programming and its complexity implications in this paper we present a new and more complete duality for semidefinite programming sdp with the following features begin itemize item this dual is an explicit semidefinite program whose number of variables and the coefficient bitlengths are polynomial in those of the primal item if the primal is feasible then it is bounded if and only if the dual is feasible item the duality gap ie the difference between the primal and the dual objective function values is zero whenever the primal is feasible and bounded also in this case the dual attains its optimum item it yields a precise farkas lemma for semidefinite feasibility systems ie characterization of the it infeasibility of a semidefinite inequality in terms of the it feasibility of another polynomial size semidefinite inequality end itemize note that the standard duality for linear programming satisfies all of the above features but no such duality theory was previously known for sdp without slater like conditions being assumed then we apply the dual to derive certain complexity results for semidefinite programming problems the decision problem of semidefinite feasibility sdfp ie that of determining if a given semidefinite inequality system is feasible is the central problem of interest the complexity of sdfp is unknown but we show the following 1 in the turing machine model sdfp is not np complete unless np co np 2 in the real number model of blum shub and smale cite bss sdfp is in np cap co np we then give polynomial reductions from the following problems to sdfp 1 checking whether an sdp is bounded 2 checking whether a feasible and bounded sdp attains the optimum 3 checking the optimality of a feasible solution

synthesis of wire antennas using the multipole expansion method a rigorous mathematical solution to the problem of synthesizing wire antennas using the multipole expansion method mem is presented the input current and required size of the antenna are estimated using the knowledge of the required field pattern this solution is based on three steps first obtaining the expansion s weighting coefficients that carry both the signature of the current and size of the antenna in essence the analysis of the direct modeling needs to be addressed first in order to define these coefficients second extracting the coefficients from the required field pattern using the natural orthogonality property of the spherical harmonic functions finally performing a numerical search for the argument of the spherical bessel functions to satisfy the ratio between two successive moments once the size of the antenna is found a recursive expression is used to obtain the input current for the current distributions assumed herein the odd order coefficients of the mem are shown to be the only necessary coefficients to construct the radiation far field also it is found that the estimated model parameters are in excellent agreement with the analytical ones

optimal placement of training for frequency selective block fading channels the problem of placing training symbols optimally for orthogonal frequency division multiplexing ofdm and single carrier systems is considered the channel is assumed to be quasi static with a finite impulse response of length l 1 samples under the assumptions that neither the transmitter nor the receiver knows the channel and that the receiver forms a minimum mean square error mmse channel estimate based on training symbols only training is optimized by maximizing a tight lower bound on the ergodic training based independent and identically distributed i i d capacity for ofdm systems it is shown that the lower bound is maximized by placing the known symbols periodically in frequency for single carrier systems under the assumption that the training symbols are placed in clusters of length spl alpha spl ges 2l 1 it is shown that the lower bound is maximized by a family of placement schemes called qpp spl alpha where qpp stands for quasi periodic placement these placement schemes are formed by grouping the known symbols into as many clusters as possible and then placing these clusters periodically in the packet for both ofdm and single carrier systems the optimum energy tradeoff between training and data is also obtained

response to comments on exact field expressions for circular loop antennas using spherical functions expansion the author would like to thank the commentators dr zheng and dr li for reviewing and commenting on the work reported in the author disagrees with their conclusion that the author in is criticizing the merits of their published papers the author is merely explaining the differences between the author s work in and the commentators work reported in this should not detract from the contribution they made in as indicated in their comments

positive semidefinite matrix completion universal rigidity and the strong arnold property this paper addresses the following three topics positive semidenite psd matrix completions universal rigidity of frameworks and the strong arnold property sap we show some strong connections among these topics using semidenite programming as unifying theme our main contribution is a sufcient condition for constructing partial psd matrices which admit a unique completion to a full psd matrix such partial matrices are an essential tool in the study of the gram dimension gd g of a graph g a recently studied graph parameter related to the low psd matrix completion problem additionally we derive an elementary proof of connelly s sucient condition for universal rigidity of tensegrity frameworks and we investigate the links between these two sucient conditions we also give a geometric characterization of psd matrices satisfying the strong arnold property in terms of nondegeneracy of an associated semidenite program which we use to establish some links between the gram dimension gd and the colin de verdi ere type graph parameter

analysis and regularization of the thin wire integral equation with reduced kernel for the straight wire modeled as a hollow tube we establish a conditional equivalence relation between the integral equations with exact and reduced kernel this relation allows us to examine the existence and uniqueness conditions for the integral equation with reduced kernel based on a local argument further we characterize the ill posedness of integral equation with reduced kernel and we propose a regularization and filtering procedure to extend the range of this integral equation

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

an exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius an exact formulation is given for the vector potential and corresponding electric fields associated with a uniform current cylindrical dipole antenna of arbitrary radius the exact expressions converge rapidly in the induction and near field regions of the antenna they are completely general and independent of the usual restrictions involving the wavelength field point distance dipole radius and length the properties of these expressions allow them to be readily used for the efficient as well as accurate computational modeling of cylindrical wire antennas an exact expression for the cylindrical wire kernel which is analytically as well as numerically convenient to work with is presented it is demonstrated that the integral of the kernel can be expressed in terms of a series which involves a generalized exponential integral and higher order associated integrals several methods for evaluating the generalized exponential integral are discussed including some useful asymptotic expansions a numerically stable forward recurrence relation for the higher order associated integrals is also presented

the real positive definite completion problem for a simple cycle abstract we consider the question of whether a real partial positive definite matrix in which the specified off diagonal entries consist of a full n cycle has a positive definite completion this lies in contrast to the previously studied chordal case we give two solutions in one we describe about n 2 independent conditions on angles associated with a normalization of the data that are necessary and sufficient the second is more computational and allows presentation of all positive definite completions as well as answering the existence question

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

first and second order analysis of nonlinear semidefinite programs in this paper we study nonlinear semidefinite programming problems convexity duality and first order optimality conditions for such problems are presented a second order analysis is also given second order necessary and sufficient optimality conditions are derived finally sensitivity analysis of such programs is discussed

complementarity and nondegeneracy in semidefinite programming primal and dual nondegeneracy conditions are defined for semidefinite programming given the existence of primal and dual solutions it is shown that primal nondegeneracy implies a unique dual solution and that dual nondegeneracy implies a unique primal solution the converses hold if strict complementarity is assumed primal and dual nondegeneracy assumptions do not imply strict complementarity as they do in lp the primal and dual nondegeneracy assumptions imply a range of possible ranks for primal and dual solutions x and z this is in contrast with lp where nondegeneracy assumptions exactly determine the number of variables which are zero it is shown that primal and dual nondegeneracy and strict complementarity all hold generically numerical experiments suggest probability distributions for the ranks of x and z which are consistent with the nondegeneracy conditions

method of moments analysis of resonant circular arrays of cylindrical dipoles we perform a method of moments mom analysis of a circular array of cylindrical dipoles the array is known from earlier theoretical and experimental studies to possess very narrow resonances the earlier theoretical studies were carried out using the two term theory the present paper is a direct continuation of a recent work showing that the problem possesses unique and particular difficulties the main difficulties are overcome herein using a set of improved kernels in the usual hallen type integral equations these kernels had been developed in previous works and were successfully incorporated into the aforementioned two term theory analyses we make a detailed comparison of our mom results to two term theory results and also to the earlier experimental results

on affine motions and universal rigidity of tensegrity frameworks abstract recently alfakih and ye 2013 4 proved that if an r dimensional bar framework g p on n r 2 nodes in general position in r r admits a positive semidefinite stress matrix with rank n r 1 then g p is universally rigid in this paper we generalize this result in two directions first we extend this result to tensegrity frameworks second we replace the general position assumption by the weaker assumption that in configuration p each point and its neighbors in g affinely span r r

eliminating unphysical oscillations arising in galerkin solutions to classical integral equations of antenna theory an asymptotic study previous works have discussed in detail the difficulties occurring when one applies numerical methods to hallen s and pocklington s integral equations for the current distribution along a linear antenna when the so called approximate kernel is used the main difficulty is the appearance of unphysical oscillations near the driving point and or near the ends of the antenna another work has proposed an easy to apply possible remedy to overcome these unnatural oscillations the basic idea is to define a new current from the near magnetic field produced by the original oscillating current in the present paper for the case of an antenna center driven by a delta function generator we place this remedy on a much firmer basis by means of an asymptotic study for the case of the linear antenna of infinite length we provide specific connections of our study to actual finite length antennas

evaluation and integration of the thin wire kernel new approaches for numerically computing the thin wire kernel and wire potential integrals are presented the singular behavior of the kernel integral is removed by transforming the integration variable to produce a smooth integrand subsequent integration of the kernel to obtain potential integrals uses quadrature schemes catering to its behavior this technique allows standard algorithms for numerical quadrature to be used with updated integration weights that account for the transformed behavior obviating the need for singularity subtraction techniques the result is a procedure for evaluating the potential integrals that is independent of the basis functions

nearest neighbor decoding in mimo block fading channels with imperfect csir this paper studies communication outages in multiple input multiple output mimo block fading channels with imperfect channel state information at the receiver csir using mismatched decoding error exponents we prove the achievability of the generalized outage probability the probability that the generalized mutual information gmi is less than the data rate and show that this probability is the fundamental limit for independent and identically distributed i i d codebooks then using nearest neighbor decoding we study the generalized outage probability in the high signal to noise ratio snr regime for random codes with gaussian and discrete signal constellations in particular we study the snr exponent which is defined as the high snr slope of the error probability curve on a logarithmic logarithmic scale we show that the maximum achievable snr exponent of the imperfect csir case is given by the snr exponent of the perfect csir case times the minimum of one and the channel estimation error diversity random codes with gaussian constellations achieve the optimal snr exponent with finite block length as long as the block length is larger than a threshold on the other hand random codes with discrete constellations achieve the optimal snr exponent with block length growing with the logarithm of the snr the results hold for many fading distributions including rayleigh rician nakagami nakagami and weibull as well as for optical wireless scintillation distributions such as lognormal rice and gamma gamma

techniques for evaluating the uniform current vector potential at the isolated singularity of the cylindrical wire kernel the cylindrical wire kernel possesses a singularity which must be properly treated in order to evaluate the uniform current vector potential traditionally the singular part of the kernel is extracted resulting in a slowly varying function which is convenient for numerical integration this paper provides some new accurate and computationally efficient methods for evaluating the remaining singular integral it is shown that this double integral may be converted to a single integral which no longer possesses a singular integrand and consequently may be efficiently evaluated numerically this form of the integral is independent of the restrictions involving wire length and radius which are inherent in various approximations also presented is a highly convergent exact series representation of the integral which is valid except in the immediate vicinity of the singularity finally a new approximation is derived which is found to be an improvement over the classical thin wire approximation it is demonstrated that each of these methods provides extremely accurate as well as efficient results for a wide range of wire radii and field point locations

on two types of convergence in the method of auxiliary sources the method of auxiliary sources mas is often applied to problems involving an externally illuminated smooth perfect electric conductor pec one seeks to approximately satisfy the boundary condition on the pec surface using auxiliary sources located inside the pec surface usually the underlying auxiliary surface is also smooth and closed the currents on the auxiliary sources mas currents are the initial unknowns once they have been found one can easily determine the field due to them mas field at all points external to the auxiliary surface and in particular at all points external to the pec scatterer we show that in the limit n it is possible to have a convergent mas field together with divergent mas currents and that this phenomenon is accompanied by an abrupt behavior of the limiting value of the mas field we show this possibility through an analytical study of a two dimensional scattering problem involving a circular cylinder in which mas fields and currents can be determined explicitly the analytical study proceeds from first principles it involves fundamental electromagnetics and relatively simple mathematical manipulations numerical results supplement the analytical study and demonstrate the nature of the aforementioned divergence our study sheds light on other aspects of mas in particular it establishes interesting similarities and differences between mas and its continuous version and reveals many similarities between mas currents and numerical solutions of hallen s integral equation with the approximate kernel

on the phenomenon of oscillations in the method of auxiliary sources when one applies the method of auxiliary sources to scattering problems involving perfect conductors one first seeks fictitious auxiliary currents located inside the conductor and then determines the field from these currents for a simple two dimensional problem involving an infinite circular cylinder illuminated by an electric current filament it has recently been shown analytically that it is possible to have divergent auxiliary currents to make this statement precise one must properly normalize the currents together with a convergent field it was also shown through numerical investigations that the aforementioned divergence appears as abnormal rapid oscillations in the present paper we investigate such phenomena in more detail with particular emphasis on oscillations for a perfectly conducting ground plane illuminated by an electric current filament we once again demonstrate the possibility of having divergent oscillating currents producing a convergent field we develop an asymptotic formula for the oscillating current values which sheds light on the nature of the oscillations we revisit the circular cylinder problem to develop a similar asymptotic formula we also discuss roundoff errors and possible generalizations to scatterers of other shapes the present study is to a great extent analytical with the analytical predictions confirmed and supplemented by numerical results

comments on exact field expressions for circular loop antennas using spherical functions expansion the author would like to thank the commentators dr zheng and dr li for reviewing and commenting on the work reported in 1 the author disagrees with their conclusion that the author in 1 is criticizing the merits of their published papers the author is merely explaining the differences between the author s work in 1 and the commentators work reported in 2 this should not detract from the contribution they made in 2 as indicated in their comments

on a new cylindrical harmonic representation for spherical waves an exact series representation is presented for integrals whose integrands are products of cosine and spherical wave functions where the argument of the cosine term can be any integral multiple n of the azimuth angle spl phi this series expansion is shown to have the following form i n e sup jkr0 r sub 0 spl delta sub no jk spl sigma sub m 1 sup spl infin c m n k sup 2 spl rho spl rho 0 m h sub m sup 2 kr sub 0 kr sub 0 sup m it is demonstrated that in the special cases n 0 and n 1 this series representation corresponds to existing expressions for the cylindrical wire kernel and the uniform current circular loop vector potential respectively a new series representation for spherical waves in terms of cylindrical harmonics is then derived using this general series representation finally a closed form far field approximation is developed and is shown to reduce to existing expressions for the cylindrical wire kernel and the uniform current loop vector potential as special cases

communication on the grassmann manifold a geometric approach to the noncoherent multiple antenna channel we study the capacity of multiple antenna fading channels we focus on the scenario where the fading coefficients vary quickly thus an accurate estimation of the coefficients is generally not available to either the transmitter or the receiver we use a noncoherent block fading model proposed by marzetta and hochwald see ibid vol 45 p 139 57 1999 the model does not assume any channel side information at the receiver or at the transmitter but assumes that the coefficients remain constant for a coherence interval of length t symbol periods we compute the asymptotic capacity of this channel at high signal to noise ratio snr in terms of the coherence time t the number of transmit antennas m and the number of receive antennas n while the capacity gain of the coherent multiple antenna channel is min m n bits per second per hertz for every 3 db increase in snr the corresponding gain for the noncoherent channel turns out to be m 1 m t bits per second per hertz where m min m n t 2 the capacity expression has a geometric interpretation as sphere packing in the grassmann manifold