functional differential equations with compressed arguments and polynomial coefficients asymptotics of the solutions abstract functional differential equations with linearly compressed arguments and polynomial coefficients are considered we prove under some mild restrictions on the coefficients that each solution y t of such an equation satisfying estimate y t c exp in2 t t where 0

capacity and cutoff rate of noncoherent fsk with nonselective rician fading the capacity and cutoff rate of frequency shift keying fsk modulation and noncoherent reception when the signal is subject to rician fading are calculated both hard and soft decisions with maximum likelihood combining are considered as well as soft decisions with square law combining optimal code rates are found that minimize the required signal to noise ratio for reliable communication

on a class of convolutional codes for the case when k divides n we introduce a special class of n k f ary convolutional codes f gf q a finite field by considering the input to an n k encoder as a sequence over gf q k the output as a sequence over gf q n an idea first used by dym 10 and then considering encoders which correspond to convolving the input with a fixed sequence gamma 0 gamma 1 cdots gamma m over gf q n a means of obtaining an encoder g d from the polynomial gamma d gamma 0 gamma 1 d cdots gamma m d m with respect to a basis for gf q n over gf q is described a criterion on gamma d in order for any g d obtained from it to be noncatastrophic is established which involves computing only the greatest common divisor gcd among s n k polynomials over gf q k this criterion is shown to coincide with that of massey and sain when k 1 it is shown that if gamma d is noncatastrophic i e if encoders obtained from it are noncatastrophic and has zero delay then any encoder g d obtained from it is minimal and has a zero delay feed forward inverse the number of zero delay noncatastrophic polynomials over gf q n of degree m is shown to be q nm q n 1 q n k 1 q n k q k 1 a formula which coincides with that of shusta 11 when k 1 the class of codes just described is shown to form a group under multiplication if the basis is normal the class is shown to be dosed under cyclic shifting when k 1 the class of codes described coincides with the class of all n 1 f ary convolutional codes hence we obtain new proofs of certain well known results about this latter class of codes finally the binary rate 1 2 convolutional codes obtained from the noncatastrophic divisors of d 15 1 over gf 2 2 are studied and optimal codes of constraint lengths 6 8 and 12 found

regular solutions of the shabat equation the shabat equationf t q2f qt f2 t q2f2 qt t r where andqare parameters is the simplest self similar reduction of the so called dressing chain for constructing and analyzing exactly solvable schrodinger equations it is so relevant to the study of theq oscillator algebra in quantum mechanics the main objective of this paper is to investigate whether the shabat equation has a nontrivial global solutionf t c1 r that is normalizable in the sense that the corresponding potentialu t f2 t f t 1 q2 is in the function space l1 r and whether in the case 1 q2 0 it has a global solutionf t c1 r that is regular in the sense thatf t 1 q2 o t 2 andf t o t 3 ast

jordan algebraic approach to convexity theorems for quadratic mappings we describe a jordan algebraic version of results related to convexity of images of quadratic mappings as well as related results on exactness of symmetric relaxations of certain classes of nonconvex optimization problems the exactness of relaxations is proved based on rank estimates our approach provides a unifying viewpoint on a large number of classical results related to cones of hermitian matrices over real and complex numbers we describe apparently new results related to cones of hermitian matrices with quaternion entries and to the exceptional 27 dimensional euclidean jordan algebra

on the design and selection of convolutional codes for an uninterleaved bursty rician channel this article focuses on code design and code selection rules under power and decoding delay constraints for an antipodal bpsk modulated and convolutionally encoded communication system the system operates over a slowly fading awgn channel described by the block fading model we emphasize perfect coherent detection with maximum likelihood decoding assuming ideal channel information the instantaneous fading values the dominant design criterion in this scenario is the code diversity level in terms of blocks while the standard hamming distance plays a secondary role a code design procedure based on maximum distance separable mds cyclic block codes is presented along with a code search algorithm the performance results of selected codes are assessed via simulation and compared to those achieved by reed solomon codes with erasure and error decoding

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

signal design and convolutional coding for noncoherent space time communication on the block rayleigh fading channel we consider the problem of designing signal constellations for the multiple transmit multiple receive antenna rayleigh fading communication channel when neither the transmitter nor the receiver know the fading in particular by employing the asymptotic union bound aub on the probability of error we give a new formulation of the problem of signal design for the noncoherent fading channel since unitary signals are optimal for this channel in the limit of large signal to noise ratios snrs the problem can be posed in terms of packings on the grassmanian manifold a key difference in our approach from that of other authors is that we use a notion of distance on this manifold that is suggested by the union bound as a consequence of our use of this distance measure we obtain signal designs that are guaranteed to achieve the full diversity order of the channel a result that does not hold when the chordal distance is used we introduce a new method of recursively designing signals termed successive updates to approximately optimize this performance measure we then examine the use of our signals with several convolutional codes over the fading channel an upper bound on the bit error probability of the maximum likelihood decoder is presented together with an asymptotic analysis of that bound

a bound on the caratheodory number abstract the caratheodory number k of a pointed closed convex cone k is the minimum among all the for which every element of k can be written as a nonnegative linear combination of at most elements belonging to extreme rays caratheodory s theorem gives the bound k dim k in this work we observe that this bound can be sharpened to k l k 1 where l k is the length of the longest chain of nonempty faces contained in k thus tying the caratheodory number with a key quantity that appears in the analysis of facial reduction algorithms we show that this bound is tight for several families of cones which include symmetric cones and the so called smooth cones we also give a simple example showing that this bound can also fail to be sharp in addition we furnish a new proof of a result by guler and tuncel which states that the caratheodory number of a symmetric cone is equal to its rank finally we connect our discussion to the notion of cp rank for completely positive matrices

recent advances in the numerical analysis of volterra functional differential equations with variable delays the numerical analysis of volterra functional integro differential equations with vanishing delays has to overcome a number of challenges that are not encountered when solving classical delay differential equations with non vanishing delays in this paper i shall describe recent results in the analysis of optimal global and local superconvergence orders in collocation methods for such evolutionary problems following a brief survey of results for equations containing volterra integral operators with non vanishing delays the discussion will focus on pantograph type volterra integro differential equations with linear and nonlinear vanishing delays the paper concludes with a section on open problems these include the asymptotic stability of collocation solutions u h on uniform meshes for pantograph type functional equations and the analysis of collocation methods for pantograph type functional equations with advanced arguments

design and analysis of trellis codes for orthogonal signal sets this paper studies trellis codes to be used with orthogonal signal constellations the intended application is in direct sequence code division multiple access communication systems the problem of designing good codes is addressed showing how to construct trellis codes from block codes with good burst error detecting capabilities several examples are given resulting in codes equivalent to those found previously by computer search ryan and wilson 1991 the procedure presented can be done by hand and is applicable for designing codes with long constraint lengths and large signal sets where a computer search becomes prohibitive performance of such coded systems can be determined through transfer functions of the specific codes used a general transfer function is given which provides a reasonable approximation to the actual transfer function of good codes thus a first order approximation to the performance of a given system can be determined without actually going through the process of designing and analyzing a specific code

stability analysis of theta methods for neutral functional differential equations this paper deals with the subject of numerical stability for the neutral functional differential equation y t ay t by qt cy pt qquad t 0 it is proved that numerical solutions generated by theta methods are convergent if c b if and only if theta geq 1 2 and that the one leg theta method is lambda stable if theta 1 we also find out that inappropriate stepsize causes spurious solution in the marginal case where rm re a 0 and a b

discontinuous galerkin methods for delay differential equations of pantograph type this paper is concerned with the application of the discontinuous galerkin method to delay differential equations with vanishing delay qt 0 q 1 our aim is to establish optimal global and local superconvergence results on uniform meshes and compare these with analogous estimates for collocation methods the theoretical results are illustrated by a broad range of numerical examples

local minima and convergence in low rank semidefinite programming the low rank semidefinite programming problem lrsdpr is a restriction of the semidefinite programming problem sdp in which a bound r is imposed on the rank of x and it is well known that lrsdpr is equivalent to sdp if r is not too small in this paper we classify the local minima of lrsdpr and prove the optimal convergence of a slight variant of the successful yet experimental algorithm of burer and monteiro 5 which handles lrsdpr via the nonconvex change of variables x rrt in addition for particular problem classes we describe a practical technique for obtaining lower bounds on the optimal solution value during the execution of the algorithm computational results are presented on a set of combinatorial optimization relaxations including some of the largest quadratic assignment sdps solved to date

new multilevel codes over gf q set partitioning is applied to multidimensional signal spaces over gf q i e gf sup n1 q n1 or d are presented these codes use reed solomon codes as component codes longer multilevel block codes are also constructed using q ary block codes with block length longer than q 1 as component codes some quaternary multilevel block codes are presented with the same length and number of information symbols as but larger distance than the best previously known quaternary one level block codes it is proved that if all the component block codes are linear the multilevel block code is also linear low rate q ary convolutional codes word error correcting convolutional codes and binary to q ary convolutional codes can also be used to construct multilevel trellis codes over gf q or binary to q ary trellis codes

asymptotic stability of nonlinear functional differential equations in this paper we study asymptotic stability of solutions of the following functional differential equation x t a t x t b t x p t f t x t x p t x o xo 1 here a b 0 oo c f 0 oo c c c are continuous functions x0 e c and p 0 oo 0 oo is a continuously differentiable function such that the following conditions hold i limt oo p t oo ii p t t t e 0 co iii 0 limt oo p t oo the technical condition iii holds for example for the case of proportional delays our results can be generalised for equations with finite number of delays and also for systems to achieve the latter use of logarithmic norm cf 1 of a matrix is essential for the ease of presentation however we consider only the simplest case earlier asymptotic stability of solutions of the neutral functional differential equation

a leray schauder type theorem and applications to boundary value problems for neutral equations consider the following two theorems which motivate the main result of this paper theorem 1 1 fitzpatrick and petryshyn l let x be a frechet space and let d be an open neighbourhood of 0 suppose that f o x is a continuous x condensing map such that a hx f x x e w and h 1 then f has a fixed point here for a subset d of x i3d and d denote the boundary and closure of d respectively further x is the ball measure of noncompactness on x defined as in 2

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

some extended results on the search for good convolutional codes we provide useful results on two classes of convolutional codes binary codes and nonbinary codes the best codes or the best known codes for these two classes of convolutional codes are found by computer search some of them are better than those found in the past we specify these codes by their transfer function matrices distance spectra and information weight spectra furthermore we derive an upper bound on the free distances of binary to m ary codes and q ary to m ary codes numerical values of this bound closely fit the computer searched values

on the attainable order of collocation methods for pantograph integro differential equations for the pantograph integro differential equation pide with nonhomogeneous term y t ay t 01 y q t d q 01 y q t dv q f t t 0 y 0 y0 with proportional delays q t and q t 0 q q 1 0 q 1 we consider the attainable order of m stage implicit collocation based runge kutta methods at the first mesh point t h and give conditions on the collocation polynomials mm t of degree m to u th t 0 1 such that u h y h o h2m 1 where y t is the solution and u t is the collocation solution of pide if m 2 or f t is a polynomial of t whose degree is less than or equal to m then such conditions of mm t are simplified a numerical example is also included

optimal superconvergence results for delay integro differential equations of pantograph type we analyze the optimal global and local orders of superconvergence of collocation solutions u h on uniform meshes i h for delay volterra integro differential equations with proportional delay functions given by theta t qt 0 q 1 t in 0 t in particular we show that if u h is a continuous piecewise polynomial of degree m geq 2 and if collocation is at the gauss legendre points then the optimal order of local superconvergence on i h is p ast m 2 it turns out that the same order p ast holds for nonlinear strictly increasing delay functions vanishing at t 0 however on judiciously chosen geometric meshes collocation at the gauss points yields the order 2m varepsilon n where varepsilon n rightarrow 0 as the number n of mesh points tends to infinity optimal local superconvergence results for the pantograph delay differential equation are obtained as special cases of our general analysis

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

periodic solutions of impulsive systems with a small delay for an impulsive system with delay it is proved that if the corresponding system without delay has an isolated omega periodic solution then in any neighbourhood of this orbit the system considered also has an omega periodic solution if the delay is small enough

on nonlinear delay differential equations we examine qualitative behaviour of delay differential equations of the form y t h y t y qt y o yo where h is a given function and q 0 we commence by investigating existence of periodic solutions in the case of h u v f u p v where f is an analytic function and p a polynomial in that case we prove that unless q is a rational number of a fairly simple form no nonconstant periodic solutions exist in particular in the special case when f is a linear function we rule out periodicity except for the case when q 1 degp if in addition p is a quadratic or a quartic we show that this result is the best possible and that a nonconstant periodic solution exists for q 1 or 1 respectively pro2 4 epcivl po vided that g is a bivariate polynomial we investigate solutions of the delay differential equation by expanding them into dirichlet series coefficients and arguments of these series are derived by means of a recurrence relation and their index set is isomorphic to a subset of planar graphs convergence results for these dirichlet series rely heavily upon the derivation of generating functions of such graphs counted with respect to certain recursively defined functionals we prove existence and convergence of dirichlet series under different general conditions thereby deducing much useful information about global behaviour of the solution

two classes of convolutional codes over gf q for q ary orthogonal signaling two classes of nonbinary convolutional codes for error control in communication systems employing q ary orthogonal signaling are discussed these classes include as special cases the dual k codes and the codes treated by b d trumpis 1975 the types of code extensions of previous bounds on free distance the noncatastrophic condition for these codes and conditions for catastrophic error propagation are described codes obtained by a computer search optimal in that the truncated transfer function bound on the probability of symbol error is minimized are presented it is shown how good convolutional codes can be constructed from cyclic block codes the performance of some of these codes over additive white gaussian noise channels rayleigh channels and partial band interference channels is also discussed

on neutral functional differential equations with proportional delays abstract in this paper we develop a comprehensive theory on the well posedness of the initial value problem for the neutral functional differential equation y t ay t i 1 b i y q i t i 1 c i y p i t t 0 y 0 y 0 and the asymptotic behaviour of its solutions we prove that the existence and uniqueness of solutions depend mainly on the coefficientsci i 1 2 and on the smoothness of functions in the solution space as far as the asymptotic behaviour of analytic solutions is concerned thecihave little effect we prove that if re a 0 then the solutiony t either grows exponentially or is polynomial the most interesting result is that if re a 0 anda 0 then the asymptotic behaviour of the solution depends mainly on the characteristic equation a i 1 b i q i 0 these results can be generalized to systems of equations finally we present some examples to illustrate the change of asymptotic behaviour in response to the variation of some parameters the main idea used in this paper is to express the solution in either dirichlet or dirichlet taylor series form

some convolutional codes whose free distances are maximal the free distance of a convolutional code of rate 1 n is bounded by n times the constant length of its encoder two classes of convolutional codes whose free distances meet this bound are studied a technique of constructing convolutional codes that meet this bound for all sufficiently large n is given

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

finite time stability for a class of fractional order fuzzy neural networks with proportional delay abstract in this paper we consider a class of fuzzy neural networks of fractional order and proportional time delay we derive sufficient conditions for the finite time stability of the considered system using finite time stability theory and fractional calculus and employing differential inequality techniques the results are obtained in terms of simple algebraic inequalities which can be easily verified in the end we give a numerical example to validate the analytical findings