Basis Of Operators Research Articles

Operator bases, S-matrices, and their partition functions

Relativistic quantum systems that admit scattering experiments are quantitatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary descriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) momentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal. These frameworks systematically handle redundancies associated with equations of motion (on-shell) and integration by parts (momentum conservation).We introduce a partition function, termed the Hilbert series, to enumerate the operator basis — correspondingly, the S-matrix — and derive a matrix integral expression to compute the Hilbert series. The expression is general, easily applied in any spacetime dimension, with arbitrary field content and (linearly realized) symmetries.In addition to counting, we discuss construction of the basis. Simple algorithms follow from the algebraic formulation in momentum space. We explicitly compute the basis for operators involving up to n = 5 scalar fields. This construction universally applies to fields with spin, since the operator basis for scalars encodes the momentum dependence of n-point amplitudes.We discuss in detail the operator basis for non-linearly realized symmetries. In the presence of massless particles, there is freedom to impose additional structure on the S- matrix in the form of soft limits. The most na¨ıve implementation for massless scalars leads to the operator basis for pions, which we confirm using the standard CCWZ formulation for non-linear realizations.Although primarily discussed in the language of EFT, some of our results — conceptual and quantitative — may be of broader use in studying conformal field theories as well as the AdS/CFT correspondence.

Nonautonomous Hamiltonian quantum systems, operator equations, and representations of the Bender–Dunne Weyl-ordered basis under time-dependent canonical transformations

We solve the problem of integrating operator equations for the dynamics of nonautonomous quantum systems by using time-dependent canonical transformations. The studied operator equations essentially reproduce the classical integrability conditions at the quantum level in the basic cases of one-dimensional nonautonomous dynamical systems. We seek solutions in the form of operator series in the Bender–Dunne basis of pseudodifferential operators. Together with this problem, we consider quantum canonical transformations. The minimal solution of the operator equation in the representation of the basis at a fixed time corresponds to the lowest-order contribution of the solution obtained as a result of applying a canonical linear transformation to the basis elements.

Entanglement classification in the noninteracting Fermi gas

In this paper, entanglement classification shared among the spins of localized fermions in the noninteracting Fermi gas is studied. It is proven that the Fermi gas density matrix is block diagonal on the basis of the projection operators to the irreducible representations of symmetric group [Formula: see text]. Every block of density matrix is in the form of the direct product of a matrix and identity matrix. Then it is useful to study entanglement in every block of density matrix separately. The basis of corresponding Hilbert space are identified from the Schur–Weyl duality theorem. Also, it can be shown that the symmetric part of the density matrix is fully separable. Then it has been shown that the entanglement measure which is introduced in Eltschka et al. [New J. Phys. 10, 043104 (2008)] and Guhne et al. [New J. Phys. 7, 229 (2005)], is zero for the even [Formula: see text] qubit Fermi gas density matrix. Then by focusing on three spin reduced density matrix, the entanglement classes have been investigated. In three qubit states there is an entanglement measure which is called 3-tangle. It can be shown that 3-tangle is zero for three qubit density matrix, but the density matrix is not biseparable for all possible values of its parameters and its eigenvectors are in the form of W-states. Then an entanglement witness for detecting non-separable state and an entanglement witness for detecting nonbiseparable states, have been introduced for three qubit density matrix by using convex optimization problem. Finally, the four spin reduced density matrix has been investigated by restricting the density matrix to the irreducible representations of [Formula: see text]. The restricted density matrix to the subspaces of the irreducible representations: [Formula: see text], [Formula: see text] and [Formula: see text] are denoted by [Formula: see text], [Formula: see text] and [Formula: see text], respectively. It has been shown that some highly entangled classes (by using the results of Miyake [Phys. Rev. A 67, 012108 (2003)] for entanglement classification) do not exist in the blocks of density matrix [Formula: see text] and [Formula: see text], so these classes do not exist in the total Fermi gas density matrix.

The leptonic future of the Higgs

Precision study of electroweak symmetry breaking strongly motivates the construction of a lepton collider with center-of-mass energy of at least 240 GeV. Besides Higgsstrahlung (e+e− → hZ), such a collider would measure weak boson pair production (e+e− → WW) with an astonishing precision. The weak-boson-fusion production process ( {e}^{+}{e}^{-}to nu overline{nu}h ) provides an increasingly powerful handle at higher center-of-mass energies. High energies also benefit the associated top-Higgs production ( {e}^{+}{e}^{-}to toverline{t}h ) that is crucial to constrain directly the top Yukawa coupling. The impact and complementarity of differential measurements, at different center-of-mass energies and for several beam polarization configurations, are studied in a global effective-field-theory framework. We define a global determinant parameter (GDP) which characterizes the overall strengthening of constraints independently of the choice of operator basis. The reach of the CEPC, CLIC, FCC-ee, and ILC designs is assessed.

Application of the phase difference correct method in rolling bearing fault diagnosis

On the basis of empirical mode decomposition and teager energy operator, the new approach for rolling bearing fault diagnosis based on phase difference correction method is presented. It focuses on...

Multi-attribute group decision making under probabilistic hesitant fuzzy environment with application to evaluate the transformation efficiency

Hesitant fuzzy sets were introduced by Torra to efficiently address situations in which the membership degree of an element in a set is expressed by several different values. However, there is a large shortcoming in hesitant fuzzy sets—the serious loss of information. Therefore, in this paper, we improve upon hesitant fuzzy sets by implementing a probabilistic hesitant fuzzy set (PHFS). Then, we introduce some new basic operations on the probabilistic hesitant fuzzy elements (PHFEs) using the Frank t-conorm and t-norm. Based on these proposed operations, we further develop probabilistic hesitant fuzzy weighted arithmetic and geometric aggregation operators. The desired properties and the relationships among them are investigated in detail. In addition, an approach to multi-attribute group decision making (MAGDM) is investigated on the basis of the new operators. Finally, a numerical example of public company efficiency evaluation is provided to illustrate the application and validity of the proposed approach.

The space of vacua of 3d mathcal{N}=3 abelian theories

We use brane techniques to study the space of vacua of abelian 3d mathcal{N}=3 gauge theories. The coordinates on these spaces are the vevs of chiral monopole and meson operators, which are realized in the type IIB brane configuration of the theory by adding semi-infinite (1, k) strings or F1 strings. The study of various brane setups allows us to determine a basis of chiral operators and chiral ring relations relevant to each branch of vacua, leading to the algebraic description of these branches. The method is mostly graphical and does not require actual computations. We apply it and provide explicit results in various examples. For linear quivers we find that the space of vacua has in general a collection of Coulomb-like branches, a Higgs branch and mixed branches. For circular quivers we find an extra branch, the geometric branch, parametrized by monopoles with equal magnetic charges in all U(1) nodes and meson operators. We explain how to include FI and mass deformations. We also study mathcal{N}=3 theories realized with (p, q) 5-branes.

2, 84, 30, 993, 560, 15456, 11962, 261485, . . .: higher dimension operators in the SM EFT

In a companion paper [1], we show that operator bases for general effective field theories are controlled by the conformal algebra. Equations of motion and integration by parts identities can be systematically treated by organizing operators into irreducible representations of the conformal group. In the present work, we use this result to study the standard model effective field theory (SM EFT), determining the content and number of higher dimension operators up to dimension 12, for an arbitrary number of fermion generations. We find additional operators to those that have appeared in the literature at dimension 7 (specifically in the case of more than one fermion generation) and at dimension 8. (The title sequence is the total number of independent operators in the SM EFT with one fermion generation, including hermitian conjugates, ordered in mass dimension, starting at dimension 5.)

Exact many-body wave function and properties of trapped bosons in the infinite-particle limit

The emphasis of this work is on the computation of physical properties as well as of the wave function of interacting bosons in a trap potential. Many-body perturbation theory is employed to study the leading term of these quantities for finite numbers of bosons, and exact solutions are aimed at in the infinite-particle limit. As discussed before, a suitable starting point is the second-quantized Hamiltonian represented in the basis of destruction and creation operators of its own mean-field potential. This choice leads to expressions for the perturbation terms of all quantities which exhibit a very weak dependence on the particle number. Importantly, when applying ideas similar to Bogoliubov's, the Hamiltonian can be reduced in the infinite-particle limit to a much simplified form which is a priori particle-number conserving. The resulting phonon Hamiltonian is diagonalizable by a linear transformation for which an explicit eigenvalue equation is given. Physical properties can be expressed explicitly by elements of this transformation, and of particular relevance is that the particle-number-conserving wave functions of the original many-boson system can be reconstructed using recursion relations. The reconstruction of the particle-conserving wave function from the phonon Hamiltonian can also be used to assess when the infinite-particle limit is reached in practice for finite trapped condensates. Two applications are discussed in detail. For one of them, an exact solution is known which is found, in the infinite-particle limit, to exactly coincide with that of the phonon Hamiltonian. In both examples expressions for the properties are given in closed form. The physics behind the phonon Hamiltonian and its physical properties is discussed.

Operators up to dimension seven in standard model effective field theory extended with sterile neutrinos

We revisit the effective field theory of the standard model that is extended with sterile neutrinos, $N$. We examine the basis of complete and independent effective operators involving $N$ up to mass dimension seven (dim-7). By employing equations of motion, integration by parts, and Fierz and group identities, we construct relations among operators that were considered independent in the previous literature, and find seven redundant operators at dim-6, sixteen redundant operators and two new operators at dim-7. The correct numbers of operators involving $N$ are, without counting Hermitian conjugates, $16~(L\cap B)+1~(\slashed{L}\cap B)+2~(\slashed{L}\cap\slashed{B})$ at dim-6, and $47~(\slashed{L}\cap B)+5~(\slashed{L}\cap\slashed{B})$ at dim-7. Here $L/B~(\slashed L/\slashed B)$ stands for lepton/baryon number conservation (violation). We verify our counting by the Hilbert series approach for $n_f$ generations of the standard model fermions and sterile neutrinos. When operators involving different flavors of fermions are counted separately and their Hermitian conjugates are included, we find there are $29~(1614)$ and $80~(4206)$ operators involving sterile neutrinos at dim-6 and dim-7 respectively for $n_f=1~(3)$.

A subleading operator basis and matching for gg \u2192 H

The Soft Collinear Effective Theory (SCET) is a powerful framework for studying factorization of amplitudes and cross sections in QCD. While factorization at leading power has been well studied, much less is known at subleading powers in the λ ≪ 1 expansion. In SCET subleading soft and collinear corrections to a hard scattering process are described by power suppressed operators, which must be fixed case by case, and by well established power suppressed Lagrangians, which correct the leading power dynamics of soft and collinear radiation. Here we present a complete basis of power suppressed operators for gg → H, classifying all operators which contribute to the cross section at mathcal{O}left({lambda}^2right) , and showing how helicity selection rules significantly simplify the construction of the operator basis. We perform matching calculations to determine the tree level Wilson coefficients of our operators. These results are useful for studies of power corrections in both resummed and fixed order perturbation theory, and for understanding the factorization properties of gauge theory amplitudes and cross sections at subleading power. As one example, our basis of operators can be used to analytically compute power corrections for N -jettiness subtractions for gg induced color singlet production at the LHC.

Hilbert series and operator basis for NRQED and NRQCD/HQET

We use a Hilbert series to construct an operator basis in the 1/m expansion of a theory with a nonrelativistic heavy fermion in an electromagnetic (NRQED) or color gauge field (NRQCD/HQET). We present a list of effective operators with mass dimension d≤8. Comparing to the current literature, our results for NRQED agree for d≤8, but there are some discrepancies in NRQCD/HQET at d=7 and 8.

Feynman rules for the Standard Model Effective Field Theory in R\u03be-gauges

We assume that New Physics effects are parametrized within the Standard Model Effective Field Theory (SMEFT) written in a complete basis of gauge invariant operators up to dimension 6, commonly referred to as “Warsaw basis”. We discuss all steps necessary to obtain a consistent transition to the spontaneously broken theory and several other important aspects, including the BRST-invariance of the SMEFT action for linear Rξ-gauges. The final theory is expressed in a basis characterized by SM-like propagators for all physical and unphysical fields. The effect of the non-renormalizable operators appears explicitly in triple or higher multiplicity vertices. In this mass basis we derive the complete set of Feynman rules, without resorting to any simplifying assumptions such as baryon-, lepton-number or CP conservation. As it turns out, for most SMEFT vertices the expressions are reasonably short, with a noticeable exception of those involving 4, 5 and 6 gluons. We have also supplemented our set of Feynman rules, given in an appendix here, with a publicly available Mathematica code working with the FeynRules package and producing output which can be integrated with other symbolic algebra or numerical codes for automatic SMEFT amplitude calculations.

A Method for Multi-Criteria Group Decision Making with 2-Tuple Linguistic Information Based on Cloud Model

This paper presents a new approach to solve the multi-criteria group decision making (MCGDM) problem where criteria values take the form of 2-tuple linguistic information. Firstly, a 2-tuple hybrid ordered weighted geometric (THOWG) operator is proposed, which synthetically considers the importance of both individual and the ordered position so as to overcome the defects of existing operators. Secondly, combining the advantages of the cloud model and 2-tuple linguistic variable, a new generating cloud method is proposed to transform 2-tuple linguistic variables into clouds. Thirdly, we further define some new cloud algorithms, such as cloud possibility degree and cloud support degree which can be respectively used to compare clouds and determine the criteria weights. Furthermore, a new approach for 2-tuple linguistic group decision making is presented on the basis of the THOWG operator, the improved generating cloud method as well as the new cloud algorithms. Finally, an example of assessing the social effects of biomass power plants (BPPS) is illustrated to verify the application and feasible of the developed approach, and a comparative analysis is also conducted to validate the effectiveness of the proposed method.

Abstract 233: Practice Process Variability and Outcomes, Relation to Interventionalist Volume

Background: For the hospital, the cardiac catheterization lab is a high risk, high volume, high cost center. With the evolving imperative to provide value over volume, cardiac catheterization lab utilization, processes, and outcomes are an increasing focus of both internal and external scrutiny on an individual operator and aggregate basis. A hospital initiated multidisciplinary task force reviews real time and quarterly data from the NCDR and New York State to provide operator and cath lab managerial feedback, modify practice, and to assess progress. Ongoing interventions include; worksheets, EMR and Cath Lab templates, timely data review and feedback to managerial staff and operators. Methods: Stony Brook University has an open Cardiac Cath Lab with services provided by 5 full time faculty and 12 voluntary (private practice) interventional cardiologists. Using internal, NYS [2009-2015], and 2015 NCDR data, utilization, processes, and outcomes of high volume [HV] operators [>100 PCI/yr] and those with a disproportionate rate of cath to PCI conversions [>50%] were compared to lower volume [LV] and low conversion rate operators. Results: High volume operators [6] performed an average of 171 interventions compared to an average of 36 for LV operators [11]. Cases rated appropriate increased from 26% (102 of 391) in 2010 to 96.4% (1448 of 1502) in 2015. Cases rated inappropriate decreased from 24.8% (97 of 391) in 2010 to 1.0% (15 of 1502) in 2015; HV operators 0.5% versus 2.2% LV operators in 2015. The cases rated uncertain also decreased from 49% (192 of 391) in 2010 to 2.6% (39 of 1502) in 2015; HV operators 1.6% versus 5.1% for LV operators in 2015. This is statistically significant with p < 0.05 by chi-square. Process indicators of stent utilization/case [1.8 vs 1.6] and contrast use [179 vs 166 cc/case] were similar in HV and LV operators with an increase in fluoroscopy times [20.2 vs 18.3 min], Total Area Dose [11,017 vs 9,154], Skin Dose 1,759 vs 1,536] in the LV operators. Bleeding complications were higher [5.8% vs 2.1%] in HV operators and mortalities were confined to the HV operators and those performing STEMI interventions. Conclusions: A multidisciplinary task force is an effective method of targeting multiple variables impacting on cath lab value metrics. Marked progress has been made towards the consistent and appropriate use of cardiac catheterization and angioplasty. The highest rates of cath-PCI conversion, bleeding complications and mortality were for HV full time cardiologists, providing 7x24 coverage for STEMI interventions and caring for a disproportionate number of ACS patients. Initiated processes, oversight and feedback changes have effectively minimized process and potential outcome differences between high and low volume operators.

The OWA distance operator and its application in business failure

PurposeThe purpose of this paper is to develop a goodness index based on Hamming distance and ordered weighted averaging distance (OWAD), which is useful to make decisions. These alternative measures enrich the results of diagnostic fuzzy models and facilitate the experts’ task in decision-making. An application to a set of firms to verify the results is also presented.Design/methodology/approachThe paper follows the basis of OWA operators to design a methodology to reduce the map of causes of business failure into monitoring key areas.FindingsThe present paper introduces two alternative measures to test the proposal of grouping. In the empirical application, the superiority of the minimum T-norm over other decision rules is verified. The ordered weighted averaging distance (OWAD) goodness index predicts a better adjustment over the index built using OWA and Hamming distance measures.Practical implicationsA useful mechanism to reduce the map of causes or diseases detected in key areas is added through this analysis. At the same time, these key areas can be disaggregated once some alert indicator is identified; this allows knowing the causes that require special attention. This application of OWA can encourage the development of suitable computer systems for monitoring the firm’s problems, alerting regarding failures and easing decision-making.Originality/valueA comparison of grouping causes into key areas through a goodness index based on Hamming distance and OWAD is proposed. These contributions enrich the Vigier and Terceño (2008) model and could be applied to any model of fuzzy diagnosis to test the results.

REGULATION OF THE RELATIONSHIP BETWEEN ADDITIVE REDUCTION AND METRICS METHODS

Objectives. The aim of the work is to determine the relationship between generalised criterion and target programming methods. Methods. The paper considers the aggregation operation that underlies many decision-making procedures used in input-output models, in neural network technologies and in the study of multi-purpose systems. The use of certain metrics within the framework of target programming can lead to solutions that are not Pareto-optimal. Therefore, in targeted programming, a significant place is given to finding the conditions under which the use of one or another metric obviously leads to Pareto-optimal solutions. The necessary (Carlin's theorem) and sufficient conditions of Pareto-optimality are known to perform the additive reduction. For a generalised criterion on the basis of order operators of weighted aggregation, two theorems proven by the author (the theorem on the inclusion of the set of Pareto-optimal solutions into a set of effective solutions and the Pareto optimality theorem for the solution obtained) are presented. Results. The proof of the Pareto optimality theorem of the solution is given, maximising the generalised criterion obtained on the basis of the order operations of weighted aggregation, which justifies the use of operations of this type for solving the problems of vector optimisation or multicriteria choice. The theorem on the existence of an additive reduction for a metric is true only in the particular case and is based on Carlin's theorem, according to which a subset of Pareto-set points maximises some additive reduction. Conclusion. In the paper a relationship between the additive reduction and metrics methods is established. An assertion concerning the relationship between the parameters of the distance function in the target programming method and the weighting coefficients of the additive reduction is formulated and proved, which ensures the equivalence of the optimal Pareto solutions.

A Case Study of IPv6 Network Performance: Packet Delay, Loss, and Reordering

Internet Protocol (IP) is used to identify and locate computers on the Internet. Currently, IPv4 still routes most Internet traffic. However, with the exhausting of IPv4 addresses, the transition to IPv6 is imminent, because, as the successor of IPv4, IPv6 can provide a larger available address space. Existing studies have addressed the notion that IPv6‐centric next generation networks are widely deployed and applied. In order to gain a deep understanding of IPv6, this paper revisits several critical IPv6 performance metrics. Our extensive measurement shows that packet delay and loss rate of IPv6 are similar to IPv4 when the AS‐level paths are roughly the same. Specifically, when the link utilization exceeds a threshold, for example, 0.83 in our study, variation of packet delay presents a similar pattern with the variation of link utilization. If packet delay of a path is large, packet‐loss rate of that path is more likely to fluctuate. In addition, we conduct a first‐ever analysis of packet reordering in IPv6 world. Few IPv6 probe packets are out‐of‐order and the reordering rate is , which is much lower than that of 0.79% in IPv4 world. Our analysis consolidates an experimental basis for operators and researchers of IPv6 networks.

An effective modified migrating birds optimization for hybrid flowshop scheduling problem with lot streaming

In this paper, the problem of hybrid flowshop hybridizing with lot streaming (HLFS) with the objective of minimizing the total flow time is addressed. We propose a mathematical model and an effective modified migrating birds optimization (EMBO) to solve this problem within an acceptable computational time. A so-called shortest waiting time rule (SWT) is introduced to schedule the jobs concurrently arriving at stages more reasonably. A combined neighborhood search strategy is developed that unites two different neighborhood operators during evolution, not only taking full advantage of their specializations but also promoting their joint efforts. Two competitive mechanisms are respectively used to increase the probability of locating better solutions at the front of the flock and enhance the interaction between two lines. The scout phase on the basis of the Glover operator and a well-designed local search is applied to the individuals trapped into local optimums and helps the algorithm explore potential promising domains. The dynamic solution acceptance criteria is developed to strike a compromise between intensification and diversification mechanisms. The performance of our proposed algorithm is evaluated by comparisons with seven other efficient algorithms in the literature. And the extensive numerical illustrations demonstrate that the proposed algorithm performs much more effectively for the addressed problem.

Lorentz symmetry violation in the fermion number anomaly with the chiral overlap operator

Recently, Grabowska and Kaplan proposed a four-dimensional lattice formulation of chiral gauge theories on the basis of a chiral overlap operator. We compute the classical continuum limit of the fermion number anomaly in this formulation. Unexpectedly, we find that the continuum limit contains a term which is not Lorentz invariant. The term is, however, proportional to the gauge anomaly coefficient, and thus the fermion number anomaly in this lattice formulation automatically restores the Lorentz-invariant form when and only when the anomaly cancellation condition is met.