a distributed newton method for network utility maximization i algorithm most existing works use dual decomposition and first order methods to solve network utility maximization num problems in a distributed manner which suffer from slow rate of convergence properties this paper develops an alternative distributed newton type fast converging algorithm for solving num problems by using novel matrix splitting techniques both primal and dual updates for the newton step can be computed using iterative schemes in a decentralized manner we propose a stepsize rule and provide a distributed procedure to compute it in finitely many iterations the key feature of our direction and stepsize computation schemes is that both are implemented using the same distributed information exchange mechanism employed by first order methods we describe the details of the inexact algorithm here and in part ii of this paper we show that under some assumptions even when the newton direction and the stepsize in our method are computed within some error due to finite truncation of the iterative schemes the resulting objective function value still converges superlinearly in terms of primal iterations to an explicitly characterized error neighborhood simulation results demonstrate significant convergence rate improvement of our algorithm relative to the existing first order methods based on dual decomposition

on the dirichlet problem for pseudodifferential operators generating feller semigroups abstract for pseudo differential operators generating symmetric feller semigroups we discuss several approaches to the dirichlet problem and show that under suitable regularity assumptions the solutions obtained by different methods do all coincide in particular we give a reasonable analytic interpretation to a probabilistic approach to the dirichlet problem

existence theorems for the one dimensional singular p laplacian equation with a nonlinear boundary condition in this paper general existence theorems are presented for the singular equation f p y q t f t y 0

approximate primal solutions and rate analysis for dual subgradient methods in this paper we study methods for generating approximate primal solutions as a byproduct of subgradient methods applied to the lagrangian dual of a primal convex possibly nondifferentiable constrained optimization problem our work is motivated by constrained primal problems with a favorable dual problem structure that leads to efficient implementation of dual subgradient methods such as the recent resource allocation problems in large scale networks for such problems we propose and analyze dual subgradient methods that use averaging schemes to generate approximate primal optimal solutions these algorithms use a constant stepsize in view of its simplicity and practical significance we provide estimates on the primal infeasibility and primal suboptimality of the generated approximate primal solutions these estimates are given per iteration thus providing a basis for analyzing the trade offs between the desired level of error and the selection of the stepsize value our analysis relies on the slater condition and the inherited boundedness properties of the dual problem under this condition it also relies on the boundedness of subgradients which is ensured by assuming the compactness of the constraint set

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

sobolev inequalities the poisson semigroup and analysis on the sphere sn abstract hypercontractive estimates are obtained for the poisson semigroup on lp sn such estimates are determined by a sharp logarithmic sobolev inequality that relates the entropy of a function on sn to its smoothness

rearrangement inequalities for functionals with monotone integrands the inequalities of hardy littlewood and riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement it is known that these inequalities extend to integrands of the form f u1 um where f is supermodular in particular they hold when f has nonnegative mixed second derivatives i jf for all i j this paper concerns the regularity assumptions on f and the equality cases it is shown here that extended hardy littlewood and riesz inequalities are valid for supermodular integrands that are just borel measurable under some nondegeneracy conditions all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant i e measure preserving maps for the hardy littlewood inequality translations for the riesz inequality the proofs rely on monotone changes of variables in the spirit of sklar s theorem

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

multiplicity of positive solutions for singular three point boundary value problems at resonance abstract in this paper we study the existence and multiplicity of positive solutions for the three point boundary value problems with singular nonlinear perturbations at resonance the existence results are applicable to both the case of a strong singularity and the case of a weak singularity the proof is based on a nonlinear alternative leray schauder principle and a fixed point theorem in cones for completely continuous operators

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

convergence of limited communication gradient methods distributed optimization increasingly plays a central role in economical and sustainable operation of cyber physical systems nevertheless the complete potential of the technology has not yet been fully exploited in practice due to communication limitations posed by the real world infrastructures this work investigates fundamental properties of distributed optimization based on gradient methods where gradient information is communicated using a limited number of bits in particular a general class of quantized gradient methods are studied where the gradient direction is approximated by a finite quantization set sufficient and necessary conditions are provided on such a quantization set to guarantee that the methods minimize any convex objective function with lipschitz continuous gradient and a nonempty and bounded set of optimizers a lower bound on the cardinality of the quantization set is provided along with specific examples of minimal quantizations convergence rate results are established that connect the fineness of the quantization and the number of iterations needed to reach a predefined solution accuracy generalizations of the results to a relevant class of constrained problems using projections are considered finally the results are illustrated by simulations of practical systems

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

positive solutions of singular p laplacian dynamic equations with sign changing nonlinearity abstract let t be a time scale such that 0 t t by the schauder fixed point theorem and the upper and lower solution method we present some existence criteria of the positive solution of m point singular p laplacian dynamic equation p u t q t f t u t 0 t 0 t t with boundary conditions u 0 0 i 1 m 1 i u i u t 0 m 2 where p s s p 2 s with p 1 i r r is continuous for i 1 2 m 1 and nonincreasing if m 3 0 1 2 m 2 m 1 t the nonlinear term may be singular in its dependent variable and is allowed to change sign our results are new even for the corresponding differential t r and difference equations t z as an application an example is given to illustrate our result

upper and lower solutions for the singular p laplacian with sign changing nonlinearities and nonlinear boundary data this paper presents existence results for singular p laplacian equations with nonlinear boundary data our nonlinearity f t u in the differential equation is allowed to change sign and the singularity may occur at u 0 t 0 and t 1

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

existence theorems for the one dimensional singular p laplacian equation with sign changing nonlinearities in this paper general existence theorems are presented for the singular equation f p y f t y 0 y 0 y 1 0 throughout our nonlinearity is allowed to change sign

an uncertainty principle for fermions with generalized kinetic energy we derive semiclassical upper bounds for the number of bound states and the sum of negative eigenvalues of the one particle hamiltoniansh f i v x acting onl2 n these bounds are then used to derive a lower bound on the kinetic energy sum limits j 1 n left langle psi f i nabla j psi right rangle for ann fermion wavefunction we discuss two examples in more detail f p p andf p p2 m2 1 2 m both in three dimensions

extremals of functionals with competing symmetries abstract we present a new method of producing optimizing sequences for highly symmetric functionals the sequences have good convergence properties built in we apply the method in different settings to give elementary proofs of some classical inequalities such as the hardy littlewood sobolev and the logarithmic sobolev inequality in their sharp form

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

extremal metrics for spectral functions of dirac operators in even and odd dimensions abstract let m n g be a closed smooth riemannian spin manifold and denote by its atiyah singer dirac operator we study the variation of riemannian metrics for the zeta function and functional determinant of 2 and prove finiteness of the morse index at stationary metrics and local extremality at such metrics under general i e not only conformal change of metrics in even dimensions which is also a new case for the conformal laplacian the relevant stability operator is of log polyhomogeneous pseudodifferential type and we prove new results of independent interest on the spectrum for such operators we use this to prove local extremality under variation of the riemannian metric which in the important example when m n g is the round n sphere gives a partial verification of branson s conjecture on the pattern of extremals thus det 2 has a local max max min min when the dimension is 4 k 4 k 1 4 k 2 4 k 3 respectively

compactness via symmetrization consider two types of translation invariant functionals i and j on rm and a sequence of functions fn whose corresponding symmetric rearrangements f n are convergent we show that fn themselves converge up to translations if either limn i fn limn i fn or limn j fn limn j fn these compactness results lead to applications in variational problems and stability problems in stellar dynamics

inequalities for integral means over symmetric sets abstract we prove that the integral of n functions over a symmetric set l in r n with additional properties increases when the functions are replaced by their symmetric decreasing rearrangements the result is known when l is a centrally symmetric convex set and our result extends it to nonconvex sets we deduce as consequences inequalities for the average of a function whose level sets are of the same type as l over measurable sets in r n the average of such a function on e is maximized by the average over the symmetric set e

best constants for sobolev inequalities for higher order fractional derivatives we obtain sharp constants for sobolev inequalities for higher order fractional derivatives as an application we give a new proof of a theorem of w beckner concerning conformally invariant higher order differential operators on the sphere

spherical dirac gjms operator determinants motivated by ads cft the extension is made to spin half of a scalar calculation of the conformal anomalies and functional determinants of gjms operators on spheres the formal aspects are heuristic but sufficient a barnes zeta function representation again proves effective the determinants are calculated for the two factorizations of the general function intertwiner form of the gjms operator and shown to be equal even including any multiplicative anomaly a comment is made on the general eigenvalue problem and a few numerical results are presented an alternative approach is detailed for odd dimensions and it is shown that the scalar determinants are expressed in terms of the spinor ones and vice versa an explicit general form is given

critical metrics for the determinant of the laplacian in odd dimensions let m be a closed compact n dimensional manifold with n odd we calculate the first and second variations of the zeta regularized determinants det a and det l as the metric on m varies where a denotes the laplacian on functions and l denotes the conformal laplacian we see that the behavior of these functionals depends on the dimension indeed every critical metric for 1 n 1 2det a or 1 n 1 21 det l has finite index consequently there are no local maxima if n 4rm 1 and no local minima if n 4m 3 we show that the standard 3 sphere is a local maximum for det a while the standard 4m 3 sphere with m 1 2 is a saddle point by contrast for all odd n the standard n sphere is a local extremal for det l an important tool in our work is the canonical trace on odd class operators in odd dimensions this trace is related to the determinant by the formula det q tr log q and we prove some basic results on how to calculate this trace

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

determinants and conformal anomalies of gjms operators on spheres the conformal anomalies and functional determinants of the branson gjms operators p2k on the d dimensional sphere are evaluated in explicit terms for any d and k such that k d 2 if d is even the determinants are given in terms of multiple gamma functions and a rational multiplicative anomaly which vanishes for odd d taking the mode system on the sphere as the union of neumann and dirichlet ones on the hemisphere is a basic part of the method and leads to a heuristic explanation of the non existence of super critical operators 2k d for even d significant use is made of the barnes zeta function the results are given in terms of ratios of determinants of operators on a d 1 dimensional bulk dual sphere for odd dimensions the log determinant is written in terms of multiple sine functions and agreement is found with holographic computations yielding an integral over a plancherel measure the n d determinant ratio is also found explicitly for even dimensions ehrhart polynomials are encountered