on first order ordinary differential equations with nonnegative right hand sides existence of extremal solutions for scalar initial value problems is investigated we shall concentrate upon nonlinearities having constant sign which lead to new existence results

the circle method and bounds for l functions i let f be a hecke maass or holomorphic primitive cusp form of arbitrary level and nebentypus and let chi be a primitive character of conductor m for the twisted l function l s f otimes chi we establish the hybrid subconvex bound begin aligned l left frac 1 2 it f otimes chi right ll m 3 t frac 1 2 frac 1 18 varepsilon end aligned for t in mathbb r the implied constant depends only on the form f and varepsilon

application of computational geometry to multiuser detection in cdma the maximum likelihood multiuser detection problem in code division multiple access is known to be an optimization problem with an objective function that is required to be optimized over a combinatorial decision region conventional suboptimal detectors relax the combinatorial decision region by a convex region without altering the objective function to be optimized we take an approach wherein the objective function is reduced to a form appropriate for the application of a polynomial complexity algorithm in computational geometry while keeping the decision region combinatorial the resulting detector allows a tradeoff between performance and computational complexity the bit error rate performance of the detector has been found to be better than the decorrelator and the linear minimum mean square error detectors for the same level of complexity

on the link between l1 pca and ica principal component analysis pca based on l1 norm maximization is an emerging technique that has drawn growing interest in the signal processing and machine learning research communities especially due to its robustness to outliers the present work proves that l1 norm pca can perform independent component analysis ica under the whitening assumption however when the source probability distributions fulfil certain conditions the l1 norm criterion needs to be minimized rather than maximized which can be accomplished by simple modifications on existing optimal algorithms for l1 pca if the sources have symmetric distributions we show in addition that l1 pca is linked to kurtosis optimization a number of numerical experiments illustrate the theoretical results and analyze the comparative performance of different algorithms for ica via l1 pca although our analysis is asymptotic in the sample size this equivalence opens interesting new perspectives for performing ica using optimal algorithms for l1 pca with guaranteed global convergence while inheriting the increased robustness to outliers of the l1 norm criterion

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

equidistribution of eisenstein series on geodesic segments abstract we show an asymptotic formula for the l 2 norm of the eisenstein series e z 1 2 i t restricted to a segment of a geodesic connecting infinity and an arbitrary real for generic geodesics of this form the asymptotic formula shows that the eisenstein series satisfies restricted que on the other hand for rational geodesics the eisenstein series does not satisfy restricted quantum ergodicity as an application we show that the zero set of the eisenstein series intersects every such geodesic segment provided t is large we also make analogous conjectures for the maass cusp forms in particular we predict that cusp forms do not satisfy restricted quantum ergodicity for rational geodesics

breaking the 12 barrier for the twisted second moment of dirichlet l functions abstract we study the second moment of dirichlet l functions to a large prime modulus q twisted by the square of an arbitrary dirichlet polynomial we break the 1 2 barrier in this problem and obtain an asymptotic formula provided that the length of the dirichlet polynomial is less than q 51 101 q 1 2 1 202 as an application we obtain an upper bound of the correct order of magnitude for the third moment of dirichlet l functions we give further results when the coefficients of the dirichlet polynomial are more specialized

optimal algorithms for l_ 1 subspace signal processing we describe ways to define and calculate l 1 norm signal subspaces that are less sensitive to outlying data than l 2 calculated subspaces we start with the computation of the l 1 maximum projection principal component of a data matrix containing n signal samples of dimension d we show that while the general problem is formally np hard in asymptotically large n d the case of engineering interest of fixed dimension d and asymptotically large sample size n is not in particular for the case where the sample size is less than the fixed dimension we present in explicit form an optimal algorithm of computational cost 2 n for the case n d we present an optimal algorithm of complexity o n d we generalize to multiple l 1 max projection components and present an explicit optimal l 1 subspace calculation algorithm of complexity o n dk k 1 where k is the desired number of l 1 principal components subspace rank we conclude with illustrations of l 1 subspace signal processing in the fields of data dimensionality reduction direction of arrival estimation and image conditioning restoration

a polynomial case of unconstrained zero one quadratic optimization unconstrained zero one quadratic maximization problems can be solved in polynomial time when the symmetric matrix describing the objective function is positive semidefinite of fixed rank with known spectral decomposition

bombieri vinogradov for multiplicative functions and beyond the x1 2 barrier abstract part and parcel of the study of multiplicative number theory is the study of the distribution of multiplicative functions in arithmetic progressions although appropriate analogies to the bombieri vingradov theorem have been proved for particular examples of multiplicative functions there has not previously been headway on a general theory seemingly none of the different proofs of the bombieri vingradov theorem for primes adapt well to this situation in this article we find out why such a result has been so elusive and discover what can be proved along these lines and develop some limitations for a fixed residue class a we extend such averages out to moduli x 20 39

spectral bounds for unconstrained 1 1 quadratic optimization problems abstract given an unconstrained quadratic optimization problem in the following form qp min x t qx x 1 1 n with q r n n we present different methods for computing bounds on its optimal objective value some of the lower bounds introduced are shown to generally improve over the one given by a classical semidefinite relaxation we report on theoretical results on these new bounds and provide preliminary computational experiments on small instances of the maximum cut problem illustrating their performance

fixed rank rayleigh quotient maximization by an mpsk sequence certain optimization problems in communication systems such as limited feedback constant envelope beamforming or noncoherent m ary phase shift keying mpsk sequence detection result in the maximization of a fixed rank positive semidefinite quadratic form over the mpsk alphabet this form is a special case of the rayleigh quotient of a matrix and in general its maximization by an mpsk sequence is mathcal np hard however if the rank of the matrix is not a function of its size then the optimal solution can be computed with polynomial complexity in the matrix size in this work we develop a new technique to efficiently solve this problem by utilizing auxiliary continuous valued angles and partitioning the resulting continuous space of solutions into a polynomial size set of regions each of which corresponds to a distinct mpsk sequence the sequence that maximizes the rayleigh quotient is shown to belong to this polynomial size set of sequences thus efficiently reducing the size of the feasible set from exponential to polynomial based on this analysis we also develop an algorithm that constructs this set in polynomial time and show that it is fully parallelizable memory efficient and rank scalable the proposed algorithm compares favorably with other solvers for this problem that have appeared recently in the literature

the spin of prime ideals fixing a nontrivial automorphism of a number field k we associate to ideals in k an invariant with values in 0 1 which we call the spin and for which the associated l function does not possess euler products we are nevertheless able using the techniques of bilinear forms to handle spin value distribution over primes obtaining stronger results than the analogous ones which follow from the technology of l functions in its current state the initial application of our theorem is to the arithmetic statistics of selmer groups of elliptic curves

efficient computation of the binary vector that maximizes a rank deficient quadratic form the maximization of a full rank quadratic form over the binary alphabet can be performed through exponential complexity exhaustive search however if the rank of the form is not a function of the problem size then it can be maximized in polynomial time by introducing auxiliary spherical coordinates we show that the rank deficient quadratic form maximization problem is converted into a double maximization of a linear form over a multidimensional continuous set the multidimensional set is partitioned into a polynomial size set of regions which are associated with distinct candidate binary vectors and the optimal binary vector belongs to the polynomial size set of candidate vectors thus the size of the candidate set is reduced from exponential to polynomial we also develop an algorithm that constructs the polynomial size candidate set in polynomial time and show that it is fully parallelizable and rank scalable finally we demonstrate the efficiency of the proposed algorithm in the context of adaptive spreading code design

on the gap between the quadratic integer programming problem and its semidefinite relaxation consider the semidefinite relaxation sdr of the quadratic integer program qip where q is a given symmetric matrix and d is diagonal we consider the sdr gap we establish the uniqueness of the sdr solution and prove that if and only if r n 1max xtvvtx x 1 1 n 1 where v is an orthogonal matrix whose columns span the r dimensional null space of d q and where d is the unique sdr solution we also give a test for establishing whether that involves 2r 1 function evaluations in the case that r 1 we derive an upper bound on which is tighter than thus we show that breaching the sdr gap for the qip problem is as difficult as the solution of a qip with the rank of the cost function matrix equal to the dimension of the null space of d q this reduced rank qip problem has been recently shown to be solvable in polynomial time for fixed r