uniqueness results for the one dimensional p laplacian this paper is concerned with positive solutions of the boundary value problem y p 2y f y 0 y b 0 y b where p 1 b is a positive parameter assume that f is continuous on 0 changes sign from nonpositive to positive and f y yp 1 is nondecreasing in the interval of f 0 the uniqueness results are proved using a time mapping analysis

extrapolation algorithms for discrete signals with application in spectral estimation in this paper we present some new algorithms useful for extrapolation and spectral estimation of band limited sequences in one and two dimensions first we show that many of the existing extrapolation algorithms for noiseless observations are unified under the criterion of minimum norm least squares mnls extrapolation for example the iterative algorithms proposed in 2 and 8 10 are shown to be special cases of a one step gradient algorithm which has linear convergence convergence and other numerical properties are improved by going to a conjugate gradient algorithm for noisy observations these algorithms could be extended by considering a mean square extrapolation criterion which gives rise to a mean square extrapolation filter and also to a recursive extrapolation filter examples and application of these methods are given extension of these algorithms is made for problems where the signal is known to be periodic a new set of functions called the periodic discrete prolate spheroidal sequences p dpss analogous to dpss 21 22 are introduced and their properties are studied finally several of these algorithms are generalized to two dimensions and the relevant equations are given

on a conjecture of voisin on the gonality of very general abelian varieties abstract we study orbits for rational equivalence of zero cycles on very general abelian varieties by adapting a method of voisin to powers of abelian varieties we deduce that for k at least 3 a very general abelian variety of dimension at least 2 k 2 has covering gonality greater than k this settles a conjecture of voisin we also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality in particular we obtain a strengthening of the alzati pirola bound on the degree of irrationality of abelian varieties

exact multiplicity results for a class of two point boundary value problems abstract this paper is concerned with the exact number of positive solutions for boundary value problems y p 2 y f y 0 and y 1 y 1 0 where p 1 and 0 is a positive parameter we consider the case in which the nonlinearity f is positive on 0 and p 1 f u u f u changes sign from negative to positive

energy concentration in band limited extrapolation the authors present a novel bandlimited extrapolation method that incorporates qualitative prior knowledge of the distribution of signal energy in both the time domain spatial domain and the frequency domain the extrapolation is easily computed by simple noniterative algorithm the prior information is incorporated by means of a weighted energy measure and the resulting solution is derived by straightforward hilbert space optimization the algorithm accommodates complex valued sequences nonuniform sampling and arbitrary spectral support and is applicable in one or several dimensions many existing extrapolation algorithms are noted as special cases examples of signal extrapolation and spectrum estimation using the new algorithm are given in one and two dimensions algorithm modification for noisy data is investigated and error bounds are presented

a singular boundary value problem for the one dimensionalp laplacian abstract the singular boundary value problem formula is studied in this paper whereg s s p 2s p 1 the singularity may appear atu 0 and att 0 ort 1 and the functionfmay be discontinuous the authors prove that for anyp 1 and for any positive nonincreasing functionfand nonnegative measurable functionkwith some integrability conditions the above mentioned problem has a unique solution also the properties of the solution are discussed in the paper

uniqueness results for the one dimensional m laplacian considering superlinear nonlinearities abstract we study existence and uniqueness of positive solutions of the boundary value problem p u m 2 u f u in 0 1 u 0 u 1 0 where is a positive parameter m 1 and f 0 r is a continuous function which vanishes at most once in 0 assuming that f is superlinear at we study its behavior near zero to obtain uniqueness results which are proved using the shooting method

some aspects of band limited signal extrapolation models discrete approximations and noise we present some theoretical results on the band limited signal extrapolation problem in section i we describe four basic models for the extrapolation problem these models are useful in understanding the relationship between the continuous extrapolation problem and some discrete algorithms given in 1 and 2 one of these models was shown to approximate the continuous band limited extrapolation problem 3 another model is obtained when the discrete fourier transform dft is used to implement the well known iterative algorithm given in 4 and 5 which was designed for solving the continuous extrapolation problem in section ii this model is related to the continuous model by means of an interesting approximation theorem also an important conjecture is presented section iii shows some approximation results specifically we prove that some discrete discrete and discrete continuous extrapolations of noisy signals converge to solutions of a certain continuous continuous noisy extrapolation problem when the noise is bounded by a known number max eta x leq epsilon this convergence is obtained by using normal families of entire functions in nand some other complex analysis tools we also show that the extrapolation problem is very sensitive to noise even in cases where only small amounts of extrapolation are desired this result indicates that in the presence of noise extrapolation techniques should be used judiciously in order to obtain reasonable results

exact number of positive solutions for semipositone problems this paper is concerned with the exact number of positive solutions for the boundary value problem y p 2y f y 0 and y 1 y 1 0 where p 1 and 0 is a positive parameter we consider the case in which both f u and g u p 1 f u uf u change sign exactly once from negative to positive on 0

on the chow group of zero cycles of a generalized kummer variety abstract for a generalized kummer variety x of dimension 2n we will construct for each 0 i n some co isotropic subvarieties in x foliated by i dimensional constant cycle subvarieties these subvarieties serve to prove that the rational orbit filtration introduced by voisin on the chow group of zero cycles of a generalized kummer variety coincides with the induced beauville decomposition from the chow ring of abelian varieties as a consequence the rational orbit filtration is opposite to the conjectural bloch beilinson filtration for generalized kummer varieties

on the use of singular value decomposition and decimation in discrete time band limited signal extrapolation the problem of extrapolating a band limited signal in discrete time is viewed as one of solving an underdetermined system of linear equations choosing the minimum norm least squares mnls solution is one criterion for singling out an extrapolation from all the possible solutions to the linear system use of the moore penrose inverse yields the mnls solution and singular value decomposition svd provides a means for implementing the moore penrose inverse an expression for the mean square error incurred in solving a linear system via svd is derived this can be used to estimate the number of singular values needed to form the inverse the error expression also indicates that decimation can be applied in the extrapolation problem to reduce the high computational cost of svd without degrading the extrapolation the results developed for the one dimensional case are extended to higher dimensions examples of the svd approach to extrapolation are given along with examples using other extrapolation techniques for comparison the svd approach compares favorably with known mnls extrapolation methods

multiple positive solutions for some p laplacian boundary value problems this paper deals with the existence of multiple positive solutions for the one dimensional p laplacian p x t q t f t x t x t 0 t 0 1 subject to one of the following boundary conditions p x 0 p x 0 0 p x 1 p x 1 0 or x 0 g1 x 0 0 x 1 g2 x 1 0 where p s s p 2s p 1 by means of a fixed point theorem due to avery and peterson sufficient conditions are obtained that guarantee the existence of at least three positive solutions the interesting point is the nonlinear term f is involved with the first order derivative explicitly

optimal estimation for band limited signals including time domain considerations the problem of estimation for band limited signals is investigated the desired estimate is based on a finite number of time samples taken from a measurement interval of length t m with no additional information the estimate must be based on these samples alone when the signal is band limited with normalized bandwidth this provides additional information which is used to improve the estimate by extrapolating the signal an extrapolation based on the measured time samples and the known bandwidth is concentrated in time about the measurement interval the length of this effective extrapolation interval is t l approx t m 2 beta several band limited estimation methods of this type have been described a new estimation method for band limited signals is described which is also based on a time concentrated extrapolation of the measurements this new method improves estimates by specifically incorporating the interval t l into the problem formulation and thereby improving the extrapolation on this relevant interval estimation error bounds and examples are presented in comparing these two types of band limited estimation procedures estimation with imperfect data is considered sensitivity of the band limited methods to errors in the data is discussed estimation with smoothing is utilized in reducing the effects of this noise

rank reduction for modeling stationary signals rank reduction is developed as a general principle for trading off model bias and model variance in the analysis and synthesis of signals the principle is applied to three basic problems stationary time series modeling stationary time series whitening and vector quantization each problem brings its own surprises and insights

multiplicity results for the 1 dimensional generalized p laplacian abstract using energy relations we prove the existence and multiplicity of solutions of the boundary value problem formula

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

finite prolate spheroidal sequences and their applications i generation and properties the finite prolate spheroidal sequences are those finite sequences which have extremal energy concentrations both in space and spatial frequency the first part of the paper is devoted to a study of the eigenvalue problem defining the sequences this reveals those fundamental properties of the sequences which are relevant in image processing applications and shows how they can be generated efficiently a new version of the sampling theorem is also demonstrated

exact number of solutions of a class of two point boundary value problems involving concave and convex nonlinearities in this paper exact number of solutions are obtained for a class of two point boundary value problems involving concave and convex nonlinearities moreover some properties of the solutions are studied in details

an identity for a quasilinear ode and its applications to the uniqueness of solutions of bvps abstract the following boundary value problem 1 1 p u a x f u 0 x 0 x x 1 1 2 u x 0 u x 1 0 is considered where p s s p 2 s p 1 a c 1 x 0 x 1 a x 0 for x x 0 x 1 and f c 1 r an identity for solutions of 1 1 and its linearized equation is derived some applications of the identity to uniqueness of solutions of problem 1 1 1 2 are presented non uniqueness examples for problem 1 1 1 2 are also established moreover the results obtained here are applied to the study of radially symmetric solutions of the dirichlet problem for elliptic equations in annular domains

interpolation and the discrete papoulis gerchberg algorithm analyze the performance of an iterative algorithm similar to the discrete papoulis gerchberg algorithm and which can be used to recover missing samples in finite length records of band limited data no assumptions are made regarding the distribution of the missing samples in contrast with the often studied extrapolation problem in which the known samples are grouped together indeed it is possible to regard the observed signal as a sampled version of the original one and to interpret the reconstruction result studied as a sampling result the authors show that the iterative algorithm converges if the density of the sampling set exceeds a certain minimum value which naturally increases with the bandwidth of the data they give upper and lower bounds for the error as a function of the number of iterations together with the signals for which the bounds are attained also they analyze the effect of a relaxation constant present in the algorithm on the spectral radius of the iteration matrix from this analysis they infer the optimum value of the relaxation constant they also point out among all sampling sets with the same density those for which the convergence rate of the recovery algorithm is maximum or minimum for low pass signals it turns out that the best convergence rates result when the distances among the missing samples are a multiple of a certain integer the worst convergence rates generally occur when the missing samples are contiguous

unbounded solutions of the singular boundary value problems for second order differential equations on the half line this paper presents some results on the existence of multiple unbounded solutions for the boundary value problems for second order differential equations on half line and on the uniqueness of unbounded solution

discrete and continuous band limited signal extrapolation this paper has two main purposes first some new algorithms for the extrapolation of multidimensional band limited sequences are presented these algorithms extend those given in 1 in two ways 1 we do not impose any restrictions on either the shape of the region containing the set of samples of the two dimensional signal to be extrapolated or on the shape of its passband zone 2 we obtain a class of sequences which are band limited extrapolations of the given data including the minimum norm solution of 1 as a special case the second objective of out paper is to relate the discrete extrapolation problem to the continuous signal extrapolation problem 2 specifically we prove that the solution obtained by using our approach for the discrete extrapolation problem tends to the solution of the continuous problem when the sampling rate used in the known part of the continuous signal to be extrapolated approaches infinity

minimax robust discrete time matched filters the problem of designing finite length discrete time matched filters is considered for situations in which exact knowledge of the input signal and or noise characteristics is not available such situations arise in many applications due to channel distortion incoherencies nonlinear effects and other modeling uncertainties in such cases it is often of interest to design a minimax robust matched filter i e a nonadaptive filter with an optimum level of worst case performance for the expected uncertainty class this problem is investigated here for three types of uncertainty models for the input signal namely the mean absolute mean square and maximum absolute distortion classes and for a wide generality of norm deviation models for the noise covariance matrix some numerical examples illustrate the robustness properties of the proposed designs

minimum energy band limited predictor with dynamic subspace selection for time variant flat fading channels in this paper we develop and analyze the basic methodology for minimum energy me band limited prediction of sampled time variant flat fading channels this predictor is based on a subspace spanned by time concentrated and band limited sequences the time concentration of these sequences is matched to the length of the observation interval and the band limitation is determined by the support of the doppler power spectral density of the fading process slepian showed that discrete prolate spheroidal dps sequences can be used to calculate the me band limited continuation of a finite sequence we utilize this property to perform channel prediction we generalize the concept of time concentrated and band limited sequences to a band limiting region consisting of disjoint intervals for a fading process with constant spectrum over its possibly discontiguous support we prove that the me band limited predictor is identical to a reduced rank maximum likelihood predictor which is a close approximation of a wiener predictor in current cellular communication systems the time selective fading process is highly oversampled the essential dimension of the subspace spanned by time concentrated and band limited sequences is in the order of two to five only the prediction error mainly depends on the support of the doppler spectrum we exploit this fact to propose low complexity time variant flat fading channel predictors using dynamically selected predefined subspaces the subspace selection is based on a probabilistic bound on the reconstruction error we compare the performance of the me band limited predictor with a predictor based on complex exponentials for a prediction horizon of one eights of a wavelength the numerical simulation results show that the me band limited predictor with dynamic subspace selection performs better than or similar to a predictor based on complex exponentials with perfectly known frequencies for a prediction horizons of three eights of a wavelength the performance of the me band limited predictor approaches that of a wiener predictor with perfectly known doppler bandwidth

bandlimited extrapolation using time bandwidth dimension the problem of extrapolating discrete index bandlimited signals from a finite number of samples is addressed in this paper the algorithm presented in this paper exploits the fact that the set of bandlimited signals that are also essentially time limited is approximated well by a low dimensional linear subspace this fact which is well known for one dimensional 1 d signals with contiguous passbands and time concentration intervals is established for a more general class of multidimensional m d signals with discontiguous passbands and discontiguous time concentration regions a criterion is presented for determining the dimension of the approximating subspace and the minimax optimal subspace itself based on knowledge of the passband time concentration regions energy concentration factor and bounds on the tolerable extrapolation error the extrapolation is constrained to lie in this subspace and parameters characterizing the extrapolation are obtained from the data by solving a linear system of equations for certain sampling patterns the system is ill conditioned and a second rank reduction is needed to reduce the deleterious effects of observation noise and modeling error a novel criterion for rank selection based on known bounds on noise power and modeling error is presented the effectiveness of the new algorithm and the rank selection criterion are demonstrated by means of computer simulations

multiple positive solutions of nonlinear two point boundary value problems we improve the results obtained by erbe hu and wang in a recent paper we show that there exist at least two positive solutions of two point boundary value problems under conditions weaker than those used by erbe hu and wang

rational orbits on three symmetric products of abelian varieties let a be an n dimensional abelian variety n 2 let ch 0 a be the group of zero cycles of a modulo rational equivalence by regarding an effective degree k zero cycle as a point on s k a the k symmetric product of a and by considering the associated rational equivalence class we get a map s k a ch 0 a whose fibres are called orbits for any n 2 in this paper we determine the maximal dimension of the orbits when k 2 or 3 it is respectively 1 and 2 and the maximal dimension of families of orbits moreover for generic a we get some refinements and in particular we show that if dim a 4 s 3 a does not contain any orbit note that it implies that a generic abelian four fold does not contain any trigonal curve