In this paper we study the geometric location of periodic points of power series defined over fields of prime characteristic @math . More specifically, we find a lower bound for the absolute value of all periodic points in the open unit disk of minimal period @math of 2-ramified power series. We prove that this bound is optimal for a large class of power series. Our main technical result is a computation of the first significant terms of the @math th iterate of 2-ramified power series. As a by-product we obtain a self-contained proof of the characterization of 2-ramified power series.
In @cite_5 the authors give a corresponding result of Theorem for minimally ramified power series, where @math is expressed in terms of the coefficients of the first two non-linear terms. Provided the information from Theorem we can make a corresponding version of Corollary for minimally ramified power series, where the conditions for optimality are expressed in terms of the four lowest degree non-linear terms.
@cite_5
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We study analytic germs in one variable with a parabolic fixed point at the origin, over an ultrametric ground field of positive characteristic. It is conjectured that for such a germ the origin is isolated as a periodic point. Our main result is an affirmative solution of this conjecture in the case of a generic germ with a prescribed multiplier. The genericity condition is explicit: the power series is minimally ramified, i.e. the degree of the first nonlinear term of each of its iterates is as small as possible. Our main technical result is a computation of the first significant terms of a minimally ramified power series. From this we obtain a lower bound for the norm of nonzero periodic points, from which we deduce our main result. As a by-product we give a new and self-contained proof of a characterization of minimally ramified power series in terms of the iterative residue. Let ( O ) k be the ring of integers of a finite extension k of the field ( Q ) p of p-adic numbers. The endomorphisms of a formal group law defined over ( O ) k provide nontrivial examples of commuting formal series with coefficients in ( O ) k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in ( Z ) p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over ( Q ) p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of ( Z ) p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing Cebysev polynomials The following theorem is proved. Let f(x_1, , x_m) be a binary nonzero polynomial of m variables of degree . H the number of binary m -tuples (a_1, , a_m) with f(a_1, , a_m) = 1 is less than 2^ m- +1 , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. where m + and 3 . This theorem completely characterizes the codewords of the th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results. In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-form formula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an @math -approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of @math , which is better than the ratio of @math for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349. This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension d ≥ 2, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem. LetH=(T,U) be a connected graph,V?Ta set, andca non-negative function on the unordered pairs of elements ofV. In theminimum0-extension problem(*), one is asked to minimize the inner productcmover all metricsmonVsuch that (i)mcoincides with the distance function ofHwithinT; and (ii) eachv?Vis at zero distance from somes?T, i.e.m(v,s)=0. This problem is known to be NP-hard ifH=K3(as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable ifH=K2(the minimum cut problem) orH=K2,r(the minimum (2,r)-metric problem). We study problem (*) for all fixedH. More precisely, we consider the linear programming relaxation (**) of (*) that is obtained by dropping condition (ii) above, and callHminimizableif the minima in (*) and (**) coincide for allVandc. Note that for such anHproblem (*) is solvable in strongly polynomial time. Our main theorem asserts thatHis minimizable if and only ifHis bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense thatHcan be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics. Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in 4, we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphsH. Other results are also presented. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We show that, over Q, if an n-variate polynomial of degree d = nO(1) is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it can also be computed by a depth three circuit (i.e. a ΣΠΣ-circuit) of size exp(O(√(d log n log d log s))) (respectively of size exp(O(√(d log n log s))). In particular this yields a ΣΠΣ circuit of size exp(O(√(d log d))) computing the d × d determinant Detd. It also means that if we can prove a lower bound of exp(omega(√(d log d))) on the size of any ΣΠΣ-circuit computing the d × d permanent Permd then we get super polynomial lower bounds for the size of any arithmetic branching program computing Permd. We then give some further results pertaining to derandomizing polynomial identity testing and circuit lower bounds. The ΣΠΣ circuits that we construct have the property that (some of) the intermediate polynomials have degree much higher than d. Indeed such a counterintuitive construction is unavoidable - it is known that in any ΣΠΣ circuit C computing either Detd or Perm_d, if every multiplication gate has fanin at most d (or any constant multiple thereof) then C must have size at least exp(Ω(d)). In this paper, we propose a method for the approximation of the solution of high- dimensional weakly coercive problems formulated in tensor spaces using low-rank approximation for- mats. The method can be seen as a perturbation of a minimal residual method with a measure of the residual corresponding to the error in a specified solution norm. The residual norm can be de- signed such that the resulting low-rank approximations are optimal with respect to particular norms of interest, thus allowing to take into account a particular objective in the definition of reduced order approximations of high-dimensional problems. We introduce and analyze an iterative algorithm that is able to provide an approximation of the optimal approximation of the solution in a given low-rank subset, without any ap rioriinformation on this solution. We also introduce a weak greedy algorithm which uses this perturbed minimal residual method for the computation of successive greedy correc- tions in small tensor subsets. We prove its convergence under some conditions on the parameters of the algorithm. The proposed numerical method is applied to the solution of a stochastic partial differential equation which is discretized using standard Galerkin methods in tensor product spaces. Mathematics Subject Classification. 15A69, 35J50, 41A63, 65D15, 65N12. We estimate @math phases (angles) from noisy pairwise relative phase measurements. The task is modeled as a nonconvex least-squares optimization problem. It was recently shown that this problem can be solved in polynomial time via convex relaxation, under some conditions on the noise. In this paper, under similar but more restrictive conditions, we show that a modified version of the power method converges to the global optimum. This is simpler and (empirically) faster than convex approaches. Empirically, they both succeed in the same regime. Further analysis shows that, in the same noise regime as previously studied, second-order necessary optimality conditions for this quadratically constrained quadratic program are also sufficient, despite nonconvexity. We develop a framework for proving approximation limits of polynomial size linear programs (LPs) from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any LP as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n1 2-ϵ)-approximations for CLIQUE require LPs of size 2nΩ(ϵ). This lower bound applies to LPs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by LPs. Our main technical ingredient is a quantitative improvement of Razborov’s [38] rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix. The main result of the paper consists of the theorem that under certain, natural assumptions the local conditional maximum @math of the function f on the set [ A = x R^n | _i (x) 0, _j (x) = 0,i = 1, ,k,j = 1, ,l ] is identical with the unconditional maximum of the potential function [ p(x, ) = f(x) + _ i = 1 ^k neg ( _i (x)) - _ j = 1 ^l | (x) | , x R^n , 0, ] for @math sufficiently small. There is also provided a draft of a modified gradient procedure for maximizing the potential @math since it is generally nonsmooth even for differentiable f, @math and @math . We develop power series approximations for a discrete-time queueing system with two parallel queues and one processor. If both queues are nonempty, a customer of queue 1 is served with probability β, and a customer of queue 2 is served with probability 1?β. If one of the queues is empty, a customer of the other queue is served with probability 1. We first describe the generating function U(z 1,z 2) of the stationary queue lengths in terms of a functional equation, and show how to solve this using the theory of boundary value problems. Then, we propose to use the same functional equation to obtain a power series for U(z 1,z 2) in β. The first coefficient of this power series corresponds to the priority case β=0, which allows for an explicit solution. All higher coefficients are expressed in terms of the priority case. Accurate approximations for the mean stationary queue lengths are obtained from combining truncated power series and Pade approximation. We present parallel approximation algorithms for maximization problems expressible by integer linear programs of a restricted syntactic form introduced by [BKT96]. One of our motivations was to show whether the approximation results in the framework of holds in the parallel setting. Our results are a confirmation of this, and thus we have a new common framework for both computational settings. Also, we prove almost tight non-approximability results, thus solving a main open question of We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we show non-approximability results and develop parallel approximation algorithms. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of Probabilistically Checkable Proofs and Multi-Prover One-Round Proof Systems [Raz95, Has97, AS97, RS97]. We study the performance of a family of randomized parallel coordinate descent methods for minimizing the sum of a nonsmooth and separable convex functions. The problem class includes as a special case L1-regularized L1 regression and the minimization of the exponential loss ("AdaBoost problem"). We assume the input data defining the loss function is contained in a sparse @math matrix @math with at most @math nonzeros in each row. Our methods need @math iterations to find an approximate solution with high probability, where @math is the number of processors and @math for the fastest variant. The notation hides dependence on quantities such as the required accuracy and confidence levels and the distance of the starting iterate from an optimal point. Since @math is a decreasing function of @math , the method needs fewer iterations when more processors are used. Certain variants of our algorithms perform on average only @math arithmetic operations during a single iteration per processor and, because @math decreases when @math does, fewer iterations are needed for sparser problems. We investigate recovery of nonnegative vectors from nonadaptive compressive measurements in the presence of noise of unknown power. It is known in the literature that under additional assumptions on the measurement design recovery of such vectors is possible with nonnegative least squares without any regularization. We show that uniqueness results known for the noiseless case carry over to robust guarantees in the noisy setting. We present guarantees which hold instantaneously by connecting the relation to the robust nullspace property. As an important example, we prove that an m × n random iid. 0 1-valued Bernoulli matrix with m = O(s log(n)) rows admits the robust nullspace property with high probability and meets the design requirements for nonnegative least squares recovery. Our analysis is motivated by applications in wireless network activity detection. We consider the problem of finding the minimizer of a convex function @math of the form @math where a low-rank factorization of @math is readily available. We consider the regime where @math . As second-order methods prove to be effective in finding the minimizer to a high-precision, in this work, we propose randomized Newton-type algorithms that exploit sub-sampling of @math , as well as inexact updates, as means to reduce the computational complexity. Two non-uniform sampling distributions based on block norm squares and block partial leverage scores are considered in order to capture important terms among @math . We show that at each iteration non-uniformly sampling at most @math terms from @math is sufficient to achieve a linear-quadratic convergence rate in @math when a suitable initial point is provided. In addition, we show that our algorithms achieve a lower computational complexity and exhibit more robustness and better dependence on problem specific quantities, such as the condition number, compared to similar existing methods, especially the ones based on uniform sampling. Finally, we empirically demonstrate that our methods are at least twice as fast as Newton's methods with ridge logistic regression on several real datasets. This paper introduces a parallel and distributed algorithm for solving the following minimization problem with linear constraints: @math minimizef1(x1)+ź+fN(xN)subject toA1x1+ź+ANxN=c,x1źX1,ź,xNźXN,where @math Nź2, @math fi are convex functions, @math Ai are matrices, and @math Xi are feasible sets for variable @math xi. Our algorithm extends the alternating direction method of multipliers (ADMM) and decomposes the original problem into N smaller subproblems and solves them in parallel at each iteration. This paper shows that the classic ADMM can be extended to the N-block Jacobi fashion and preserve convergence in the following two cases: (i) matrices @math Ai are mutually near-orthogonal and have full column-rank, or (ii) proximal terms are added to the N subproblems (but without any assumption on matrices @math Ai). In the latter case, certain proximal terms can let the subproblem be solved in more flexible and efficient ways. We show that @math źxk+1-xkźM2 converges at a rate of o(1 k) where M is a symmetric positive semi-definte matrix. Since the parameters used in the convergence analysis are conservative, we introduce a strategy for automatically tuning the parameters to substantially accelerate our algorithm in practice. We implemented our algorithm (for the case ii above) on Amazon EC2 and tested it on basis pursuit problems with >300 GB of distributed data. This is the first time that successfully solving a compressive sensing problem of such a large scale is reported. We introduce a generalization of a deterministic relaxation algorithm for edge-preserving regularization in linear inverse problems. This algorithm transforms the original (possibly nonconvex) optimization problem into a sequence of quadratic optimization problems, and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex. We prove that our more general algorithm is globally convergent (i.e., converges to a local minimum from any initialization) under less restrictive conditions, even when the original cost functional is nonconvex. We apply this algorithm to tomographic reconstruction from limited-angle data by formulating the problem as one of regularized least-squares optimization. The results demonstrate that the constraint of piecewise smoothness, applied through the use of edge-preserving regularization, can provide excellent limited-angle tomographic reconstructions. Two edge-preserving regularizers-one convex, the other nonconvex-are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited-angle scenarios, and to explore how factors, such as the choice of error norm, angular sampling rate and amount of noise, affect the reconstruction quality and algorithm performance. These simulation results show that for this application, the nonconvex regularizer produces consistently superior results. We consider the makespan-minimization problem on unrelated machines in the context of algorithmic mechanism design. No truthful mechanisms with non-trivial approximation guarantees are known for this multidimensional domain. We study a well-motivated special case (also a multidimensional domain), where the processing time of a job on each machine is either "low" or "high." We give a general technique to convert any c-approximation algorithm (in a black-box fashion) to a 3c-approximation truthful-in-expectation mechanism. Our construction uses fractional truthful mechanisms as a building block, and builds upon a technique of Lavi and Swamy [Lavi, R., Swamy, C., 2005. Truthful and near-optimal mechanism design via linear programming. In: Proc. 46th FOCS, pp. 595-604]. When all jobs have identical low and high values, we devise a deterministic 2-approximation truthful mechanism. The chief novelty of our results is that we do not utilize explicit price definitions to prove truthfulness. Instead we design algorithms that satisfy cycle monotonicity [Rochet, J., 1987. A necessary and sufficient condition for rationalizability in a quasilinear context. J. Math. Econ. 16, 191-200], a necessary and sufficient condition for truthfulness in multidimensional settings; this is the first work that leverages this characterization.