Showing papers on "Operator (computer programming) published in 2013"

Proximal Algorithms

Abstract: This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.

The Effective Field Theory of Dark Matter Direct Detection

Abstract: We extend and explore the general non-relativistic effective theory of dark matter (DM) direct detection. We describe the basic non-relativistic building blocks of operators and discuss their symmetry properties, writing down all Galilean-invariant operators up to quadratic order in momentum transfer arising from exchange of particles of spin 1 or less. Any DM particle theory can be translated into the coefficients of an effective operator and any effective operator can be simply related to most general description of the nuclear response. We find several operators which lead to novel nuclear responses. These responses differ significantly from the standard minimal WIMP cases in their relative coupling strengths to various elements, changing how the results from different experiments should be compared against each other. Response functions are evaluated for common DM targets — F, Na, Ge, I, and Xe — using standard shell model techniques. We point out that each of the nuclear responses is familiar from past studies of semi-leptonic electroweak interactions, and thus potentially testable in weak interaction studies. We provide tables of the full set of required matrix elements at finite momentum transfer for a range of common elements, making a careful and fully model-independent analysis possible. Finally, we discuss embedding non-relativistic effective theory operators into UV models of dark matter.

Renormalization Group Evolution of the Standard Model Dimension Six Operators I: Formalism and lambda Dependence

Abstract: We calculate the order \lambda, \lambda^2 and \lambda y^2 terms of the 59 x 59 one-loop anomalous dimension matrix of dimension-six operators, where \lambda and y are the Standard Model Higgs self-coupling and a generic Yukawa coupling, respectively. The dimension-six operators modify the running of the Standard Model parameters themselves, and we compute the complete one-loop result for this. We discuss how there is mixing between operators for which no direct one-particle-irreducible diagram exists, due to operator replacements by the equations of motion.

Renormalization group evolution of the standard model dimension six operators. I: formalism and λ dependence

Abstract: We calculate the order λ, λ2 and λy 2 terms of the 59 × 59 one-loop anomalous dimension matrix of dimension-six operators, where λ and y are the Standard Model Higgs self-coupling and a generic Yukawa coupling, respectively. The dimension-six operators modify the running of the Standard Model parameters themselves, and we compute the complete one-loop result for this. We discuss how there is mixing between operators for which no direct one-particle-irreducible diagram exists, due to operator replacements by the equations of motion.

Framed BPS states

Abstract: We consider a class of line operators in $d = 4, \mathcal{N} = 2$ supersymmetric field theories, which leave four supersymmetries unbroken. Such line operators support a new class of BPS states which we call "framed BPS states." These include halo bound states similar to those of $d = 4, \mathcal{N} = 2$ supergravity, where (ordinary) BPS particles are loosely bound to the line operator. Using this construction, we give a new proof of the Kontsevich-Soibelman wall-crossing formula (WCF) for the ordinary BPS particles, by reducing it to the semiprimitive WCF. After reducing on $S^1$, the expansion of the vevs of the line operators in the IR provides a new physical interpretation of the "Darboux coordinates" on the moduli space M of the theory. Moreover, we introduce a "protected spin character" (PSC) that keeps track of the spin degrees of freedom of the framed BPS states. We show that the generating functions of PSCs admit a multiplication, which defines a deformation of the algebra of holomorphic functions on $\mathcal{M}$. As an illustration of these ideas, we consider the sixdimensional (2, 0) field theory of $A_1$ type compactified on a Riemann surface $\mathcal{C}$. Here, we show (extending previous results) that line operators are classified by certain laminations on a suitably decorated version of $\mathcal{C}$, and we compute the spectrum of framed BPS states in several explicit examples. Finally, we indicate some interesting connections to the theory of cluster algebras.

The bootstrap program for boundary CFT d

Abstract: We study the constraints of crossing symmetry and unitarity for conformal field theories in the presence of a boundary, with a focus on the Ising model in various dimensions. We show that an analytic approach to the bootstrap is feasible for free-field theory and at one loop in the epsilon expansion, but more generally one has to resort to numerical methods. Using the recently developed linear programming techniques we find several interesting bounds for operator dimensions and OPE coefficients and comment on their physical relevance. We also show that the “boundary bootstrap” can be easily applied to correlation functions of tensorial operators and study the stress tensor as an example. In the appendices we present conformal block decompositions of a variety of physically interesting correlation functions.

Integrating scale out and fault tolerance in stream processing using operator state management

Abstract: As users of "big data" applications expect fresh results, we witness a new breed of stream processing systems (SPS) that are designed to scale to large numbers of cloud-hosted machines. Such systems face new challenges: (i) to benefit from the "pay-as-you-go" model of cloud computing, they must scale out on demand, acquiring additional virtual machines (VMs) and parallelising operators when the workload increases; (ii) failures are common with deployments on hundreds of VMs-systems must be fault-tolerant with fast recovery times, yet low per-machine overheads. An open question is how to achieve these two goals when stream queries include stateful operators, which must be scaled out and recovered without affecting query results.Our key idea is to expose internal operator state explicitly to the SPS through a set of state management primitives. Based on them, we describe an integrated approach for dynamic scale out and recovery of stateful operators. Externalised operator state is checkpointed periodically by the SPS and backed up to upstream VMs. The SPS identifies individual operator bottlenecks and automatically scales them out by allocating new VMs and partitioning the checkpointed state. At any point, failed operators are recovered by restoring checkpointed state on a new VM and replaying unprocessed tuples. We evaluate this approach with the Linear Road Benchmark on the Amazon EC2 cloud platform and show that it can scale automatically to a load factor of L=350 with 50 VMs, while recovering quickly from failures.

Differential Evolution With Ranking-Based Mutation Operators

Abstract: Differential evolution (DE) has been proven to be one of the most powerful global numerical optimization algorithms in the evolutionary algorithm family. The core operator of DE is the differential mutation operator. Generally, the parents in the mutation operator are randomly chosen from the current population. In nature, good species always contain good information, and hence, they have more chance to be utilized to guide other species. Inspired by this phenomenon, in this paper, we propose the ranking-based mutation operators for the DE algorithm, where some of the parents in the mutation operators are proportionally selected according to their rankings in the current population. The higher ranking a parent obtains, the more opportunity it will be selected. In order to evaluate the influence of our proposed ranking-based mutation operators on DE, our approach is compared with the jDE algorithm, which is a highly competitive DE variant with self-adaptive parameters, with different mutation operators. In addition, the proposed ranking-based mutation operators are also integrated into other advanced DE variants to verify the effect on them. Experimental results indicate that our proposed ranking-based mutation operators are able to enhance the performance of the original DE algorithm and the advanced DE algorithms.

Neutrinos with Lorentz-violating operators of arbitrary dimension

Abstract: The theoretical description of fermions in the presence of Lorentz and $CPT$ violation is developed. We classify all Lorentz- and $CPT$-violating and invariant terms in the quadratic Lagrange density for a Dirac fermion, including operators of arbitrary mass dimension. The exact dispersion relation is obtained in closed and compact form, and projection operators for the spinors are derived. The Pauli Hamiltonians for particles and antiparticles are extracted, and observable combinations of operators are identified. We characterize and enumerate the coefficients for Lorentz violation for any operator mass dimension via a decomposition using spin-weighted spherical harmonics. The restriction of the general theory to various special cases is presented, including isotropic models, the nonrelativistic and ultrarelativistic limits, and the minimal Standard-Model Extension. Expressions are derived in several limits for the fermion dispersion relation, the associated fermion group velocity, and the fermion spin-precession frequency. We connect the analysis to some other formalisms and use the results to extract constraints from astrophysical observations on isotropic ultrarelativistic spherical coefficients for Lorentz violation.

N=4 superconformal bootstrap.

Abstract: We implement the conformal bootstrap for N=4 superconformal field theories in four dimensions. The consistency of the four-point function of the stress-energy tensor multiplet imposes significant upper bounds for the scaling dimensions of unprotected local operators as functions of the central charge of the theory. At the threshold of exclusion, a particular operator spectrum appears to be singled out by the bootstrap constraints. We conjecture that this extremal spectrum is that of N=4 supersymmetric Yang-Mills theory at an S-duality invariant value of the complexified gauge coupling.

Some intuitionistic fuzzy Einstein hybrid aggregation operators and their application to multiple attribute decision making

Abstract: Intuitionistic fuzzy information aggregation plays an important part in intuitionistic fuzzy set theory, which has emerged to be a new research direction receiving more and more attention in recent years. In this paper, we investigate the multiple attribute decision making (MADM) problems with intuitionistic fuzzy numbers. Then, we first introduce some operations on intuitionistic fuzzy sets, such as Einstein sum, Einstein product, and Einstein exponentiation, and further develop some new Einstein hybrid aggregation operators, such as the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator, which extend the hybrid averaging (HA) operator and the hybrid geometric (HG) operator to accommodate the environment in which the given arguments are intuitionistic fuzzy values. Then, we apply the intuitionistic fuzzy Einstein hybrid averaging (IFEHA) operator and intuitionistic fuzzy Einstein hybrid geometric (IFEHG) operator to deal with multiple attribute decision making under intuitionistic fuzzy environments. Finally, some illustrative examples are given to verify the developed approach and to demonstrate its practicality and effectiveness.

A critical Kirchhoff type problem involving a non-local operator

Abstract: Abstract In this paper we show the existence of non-negative solutions for a Kirchhoff type problem driven by a nonlocal integrodifferential operator, that is − M ( ‖ u ‖ Z 2 ) L K u = λ f ( x , u ) + | u | 2 ∗ − 2 u in Ω , u = 0 in R n ∖ Ω where L K is an integrodifferential operator with kernel K , Ω is a bounded subset of R n , M and f are continuous functions, ‖ ⋅ ‖ Z is a functional norm and 2 ∗ is a fractional Sobolev exponent.

Constraints on conformal field theories in diverse dimensions from the bootstrap mechanism.

Abstract: Recently an efficient numerical method has been developed to implement the constraints of crossing symmetry and unitarity on the operator dimensions and operator product expansion coefficients of conformal field theories in diverse space-time dimensions. It appears that the calculations can be done only for theories lying at the boundary of the allowed parameter space. Here it is pointed out that a similar method can be applied to a larger class of conformal field theories, whether unitary or not, and no free parameter remains, provided we know the fusion algebra of the low lying primary operators. As an example we calculate using first principles, with no phenomenological input, the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity in three and four space dimensions. The edge exponents compare favorably with the latest numerical estimates. A consistency check of this approach on the 3D critical Ising model is also made.

A primal?dual fixed point algorithm for convex separable minimization with applications to image restoration

Abstract: Recently, the minimization of a sum of two convex functions has received considerable interest in a variational image restoration model. In this paper, we propose a general algorithmic framework for solving a separable convex minimization problem from the point of view of fixed point algorithms based on proximity operators (Moreau 1962 C. R. Acad. Sci., Paris I 255 2897–99). Motivated by proximal forward–backward splitting proposed in Combettes and Wajs (2005 Multiscale Model. Simul. 4 1168–200) and fixed point algorithms based on the proximity operator (FP2O) for image denoising (Micchelli et al 2011 Inverse Problems 27 45009–38), we design a primal–dual fixed point algorithm based on the proximity operator (PDFP2Oκ for κ ∈ [0, 1)) and obtain a scheme with a closed-form solution for each iteration. Using the firmly nonexpansive properties of the proximity operator and with the help of a special norm over a product space, we achieve the convergence of the proposed PDFP2Oκ algorithm. Moreover, under some stronger assumptions, we can prove the global linear convergence of the proposed algorithm. We also give the connection of the proposed algorithm with other existing first-order methods. Finally, we illustrate the efficiency of PDFP2Oκ through some numerical examples on image supper-resolution, computerized tomographic reconstruction and parallel magnetic resonance imaging. Generally speaking, our method PDFP2O (κ = 0) is comparable with other state-of-the-art methods in numerical performance, while it has some advantages on parameter selection in real applications.

Intuitionistic fuzzy geometric Heronian mean aggregation operators

Abstract: In this paper, the multi-criteria decision making problem with the assumption that the criteria are correlative is studied under intuitionistic fuzzy environment. Some new aggregation operators for intuitionistic fuzzy information are proposed, including the intuitionistic fuzzy geometric Heronian mean (IFGHM) operator and the intuitionistic fuzzy geometric weighed Heronian mean (IFGWHM) operator. We investigate the properties of the proposed operators, such as idempotency, monotonicity, permutation and boundary. Moreover, an approach is proposed for multi-criteria decision making based on IFGWHM operator. An example about talent introduction is given to illustrate the proposed method.

Renormalization group scaling of Higgs operators and h → γγ decay

Abstract: We compute the renormalization of dimension six Higgs-gauge boson operators that can modify the h → γγ rate at tree-level. Operator mixing is shown to lead to an important modification of new physics effects which has been neglected in past calculations. We also find that the usual formula for the S oblique parameter contribution of these Higgs- gauge boson operators needs additional terms to be consistent with renormalization group evolution. We study the implications of our results for Higgs phenomenology and for new physics models which attempt to explain a deviation in the h → γγ rate. We derive a new relation between the S parameter and the h → γγ and h → Zγ decay rates.

The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator

Abstract: We analyze a nonlocal diffusion operator having as special cases the fractional Laplacian and fractional differential operators that arise in several applications. In our analysis, a nonlocal vector calculus is exploited to define a weak formulation of the nonlocal problem. We demonstrate that, when sufficient conditions on certain kernel functions hold, the solution of the nonlocal equation converges to the solution of the fractional Laplacian equation on bounded domains as the nonlocal interactions become infinite. We also introduce a continuous Galerkin finite element discretization of the nonlocal weak formulation and we derive a priori error estimates. Through several numerical examples we illustrate the theoretical results and we show that by solving the nonlocal problem it is possible to obtain accurate approximations of the solutions of fractional differential equations circumventing the problem of treating infinite-volume constraints.

Distributed Pareto Optimization via Diffusion Strategies

Abstract: We consider solving multi-objective optimization problems in a distributed manner by a network of cooperating and learning agents. The problem is equivalent to optimizing a global cost that is the sum of individual components. The optimizers of the individual components do not necessarily coincide and the network therefore needs to seek Pareto optimal solutions. We develop a distributed solution that relies on a general class of adaptive diffusion strategies. We show how the diffusion process can be represented as the cascade composition of three operators: two combination operators and a gradient descent operator. Using the Banach fixed-point theorem, we establish the existence of a unique fixed point for the composite cascade. We then study how close each agent converges towards this fixed point, and also examine how close the Pareto solution is to the fixed point. We perform a detailed mean-square error analysis and establish that all agents are able to converge to the same Pareto optimal solution within a sufficiently small mean-square-error (MSE) bound even for constant step-sizes. We illustrate one application of the theory to collaborative decision making in finance by a network of agents.

Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators

Abstract: The purpose of this paper is to derive some Lewy-Stampacchia estimates in some cases of interest, such as the ones driven by non-local operators. Since we will perform an abstract approach to the problem, this will provide, as a byproduct, Lewy-Stampacchia estimates in more classi- cal cases as well. In particular, we can recover the known estimates for the standard Laplacian, the p-Laplacian, and the Laplacian in the Heisenberg group. In the non-local framework we prove a Lewy-Stampacchia estimate for a general integrodifferential operator and, as a particular case, for the fractional Laplacian. As far as we know, the abstract framework and the results in the non-local setting are new.

Analysis Operator Learning and its Application to Image Reconstruction

Abstract: Exploiting a priori known structural information lies at the core of many image reconstruction methods that can be stated as inverse problems. The synthesis model, which assumes that images can be decomposed into a linear combination of very few atoms of some dictionary, is now a well established tool for the design of image reconstruction algorithms. An interesting alternative is the analysis model, where the signal is multiplied by an analysis operator and the outcome is assumed to be sparse. This approach has only recently gained increasing interest. The quality of reconstruction methods based on an analysis model severely depends on the right choice of the suitable operator. In this paper, we present an algorithm for learning an analysis operator from training images. Our method is based on lp-norm minimization on the set of full rank matrices with normalized columns. We carefully introduce the employed conjugate gradient method on manifolds, and explain the underlying geometry of the constraints. Moreover, we compare our approach to state-of-the-art methods for image denoising, inpainting, and single image super-resolution. Our numerical results show competitive performance of our general approach in all presented applications compared to the specialized state-of-the-art techniques.

Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number

Abstract: We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovasz' famous ϑ function on general operator systems, as the norm-completion (or stabilization) of a “naive” generalization of ϑ. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lovasz' original ϑ in the classical case, give several applications, and propose to study the operator systems associated with channels as “noncommutative graphs,” using the language of Hilbert modules.

Autonomous vehicle control for impaired driver

Abstract: A condition of an operator of a vehicle is detected. It is determined that the condition is an impaired condition. At least one autonomous operation is performed based on the impaired condition.

Renormalization of dimension-six operators relevant for the Higgs decays h → γγ, γz

Abstract: The discovery of the Higgs boson has opened a new window to test the SM through the measurements of its couplings. Of particular interest is the measured Higgs coupling to photons which arises in the SM at the one-loop level, and can then be significantly affected by new physics. We calculate the one-loop renormalization of the dimension- six operators relevant for h → γγ, γZ, which can be potentially important since it could, in principle, give log-enhanced contributions from operator mixing. We find however that there is no mixing from any current-current operator that could lead to this log-enhanced effect. We show how the right choice of operator basis can make this calculation simple. We then conclude that h → γγ, γZ can only be affected by RG mixing from operators whose Wilson coefficients are expected to be of one-loop size, among them fermion dipole-moment operators which we have also included.

Rapidity evolution of Wilson lines at the next-to-leading order

Abstract: At high energies particles move very fast so the proper degrees of freedom for the fast gluons moving along the straight lines are Wilson-line operators - infinite gauge factors ordered along the line. In the framework of operator expansion in Wilson lines the energy dependence of the amplitudes is determined by the rapidity evolution of Wilson lines. We present the next-to-leading order hierarchy of the evolution equations for Wilson-line operators.

Induced hesitant interval-valued fuzzy Einstein aggregation operators and their application to multiple attribute decision making

Abstract: In this paper, we investigate the multiple attribute decision making MADM problems with hesitant interval-valued fuzzy information. We first introduce some operations on hesitant fuzzy sets, such as Einstein sum, Einstein product, Einstein exponentiation, etc., and propose the hesitant interval-valued fuzzy sets. Then, we further develop some new Einstein aggregation operators with hesitant interval-valued fuzzy information, such as the hesitant interval-valued fuzzy Einstein weighted average HIVFEWA operator, hesitant interval-valued fuzzy Einstein weighted geometric HIVFEWG operator, hesitant interval-valued fuzzy Einstein ordered weighted average HIVFEOWA operator, hesitant interval-valued fuzzy Einstein ordered weighted geometric HIVFEOWG operator, induced hesitant interval-valued fuzzy Einstein ordered weighted averaging I-HIVFEOWA operator and induced hesitant interval-valued fuzzy Einstein ordered weighted geometric I-HIVFEOWG operator. Then, we apply the induced hesitant interval-valued fuzzy Einstein ordered weighted averaging I-HIVFEOWA operator and induced hesitant interval-valued fuzzy Einstein ordered weighted geometric I-HIVFEOWG operator to deal with multiple attribute decision making under the hesitant interval-valued fuzzy environments. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Dynamical Systems Method for Solving Operator Equations

Abstract: Preface Contents 1. Introduction 2. Ill-posed problems 3. DSM for well-posed problems 4. DSM and linear ill-posed problems 5. Some inequalities 6. DSM for monotone operators 7. DSM for general nonlinear operator equations 8 DSM for operators satisfying a spectral assumption 9. DSM in Banach spaces 10. DSM and Newton-type methods without inversion of the derivative 11. DSM and unbounded operators 12. DSM and nonsmooth operators 13. DSM as a theoretical tool 14. DSM and iterative methods 15. Numerical problems arising in applications 16. Auxiliary results from analysis Bibliographical notes Bibliography Index