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Showing papers on "Operator (computer programming) published in 2016"


Journal ArticleDOI
26 Feb 2016-PLOS ONE
TL;DR: This work presents a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space and demonstrates the usefulness of nonlinear observable subspaces in the design of Koop man operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.
Abstract: In this wIn this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by l1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.ork, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by l1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.

461 citations


Journal ArticleDOI
TL;DR: An interval‐valued Pythagorean fuzzy ELECTRE method is proposed to solve uncertainty MAGDM problem and an illustrative example for evaluating the software developments is given to verify the developed approach and to demonstrate its practicality and effectiveness.
Abstract: In this paper, we investigate the multiple attribute group decision making MAGDM problems with interval-valued Pythagorean fuzzy sets IVPFSs. First, the concept, operational laws, score function, and accuracy function of IVPFSs are defined. Then, based on the operational laws, two interval-valued Pythagorean fuzzy aggregation operators are developed for aggregating the interval-valued Pythagorean fuzzy information, such as interval-valued Pythagorean fuzzy weighted average IVPFWA operator and interval-valued Pythagorean fuzzy weighted geometric IVPFWG operator. A series of inequalities of aggregation operators are studied. Later, we develop some interval-valued Pythagorean fuzzy point operators. Moreover, combining the interval-valued Pythagorean fuzzy point operators with IVPFWA operator, we present some interval-valued Pythagorean fuzzy point weighted averaging IVPFPWA operators, which can adjust the degree of the aggregated arguments with some parameters. Then, we propose an interval-valued Pythagorean fuzzy ELECTRE method to solve uncertainty MAGDM problem. Finally, an illustrative example for evaluating the software developments is given to verify the developed approach and to demonstrate its practicality and effectiveness.

337 citations


Journal ArticleDOI
TL;DR: In this article, a nonintrusive projection-based model reduction approach for full models based on time-dependent partial differential equations is presented, which is applicable to full models that are linear in the state or have a low-order polynomial nonlinear term.

307 citations


Book
23 Aug 2016
TL;DR: Ergodic Theory: Topological Dynamical Systems, Minimality and Recurrence, Measure-Preserving Systems, Recurrence and Ergodicity, Mean Ergodic Theorem, Mixing Dynamical System, Markov Operators on C*-algebra C(K) and the Koopman Operator.
Abstract: What is Ergodic Theory?.- Topological Dynamical Systems.- Minimality and Recurrence.- The C*-algebra C(K) and the Koopman Operator.- Measure-Preserving Systems.- Recurrence and Ergodicity.- The Banach Lattice Lp and the Koopman Operator.- The Mean Ergodic Theorem.- Mixing Dynamical Systems.- Mean Ergodic Operators on C(K).- The Pointwise Ergodic Theorem.- Isomorphisms and Topological Models.- Markov Operators.- Compact Semigroups and Groups.- Topological Dynamics Revisited.- The Jacobs-de Leeuw-Glicksberg Decomposition.- Dynamical Systems with Discrete Spectrum.- A Glimpse at Arithmetic Progressions.- Joinings.- The Host-Kra-Tao Theorem.- More Ergodic Theorems.- Appendix A: Topology.- Appendix B: Measure and Integration Theory.- Appendix C: Functional Analysis.- Appendix D: The Riesz Representation Theorem.- Appendix E: Theorems of Eberlein, Grothendieck, and Ellis.

283 citations


Journal ArticleDOI
TL;DR: The Choquet integral operator for Pythagorean fuzzy aggregation operators, such as Pythagorian fuzzy Choquet Integral average (PFCIA), is defined and two approaches to multiple attribute group decision making with attributes involving dependent and independent by the PFCIA operator and multi‐attributive border approximation area comparison (MABAC) in Pythagian fuzzy environment are proposed.
Abstract: In this paper, we define the Choquet integral operator for Pythagorean fuzzy aggregation operators, such as Pythagorean fuzzy Choquet integral average PFCIA operator and Pythagorean fuzzy Choquet integral geometric PFCIG operator. The operators not only consider the importance of the elements or their ordered positions but also can reflect the correlations among the elements or their ordered positions. It is worth pointing out that most of the existing Pythagorean fuzzy aggregation operators are special cases of our operators. Meanwhile, some basic properties are discussed in detail. Later, we propose two approaches to multiple attribute group decision making with attributes involving dependent and independent by the PFCIA operator and multi-attributive border approximation area comparison MABAC in Pythagorean fuzzy environment. Finally, two illustrative examples have also been taken in the present study to verify the developed approaches and to demonstrate their practicality and effectiveness.

272 citations


Journal ArticleDOI
TL;DR: A new method for Pythagorean fuzzy multiple-criteria decision-making (MCDM) problems with aggregation operators and distance measures with the main advantage that it uses distance measures in a unified framework between the ordered weighted averaging (OWA) operator and weighted average (WA) that considers the degree of importance of each concept in the aggregation.
Abstract: As a generalization of intuitionistic fuzzy set, the Pythagorean fuzzy set is interesting and very useful in modeling uncertain information in real-world decision-making problems. In this paper, we develop a new method for Pythagorean fuzzy multiple-criteria decision-making (MCDM) problems with aggregation operators and distance measures. First, we present the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator. The main advantage of the PFOWAWAD operator is that it uses distance measures in a unified framework between the ordered weighted averaging (OWA) operator and weighted average (WA) that considers the degree of importance of each concept in the aggregation. Some of its main properties and special cases are studied. Then, based on the proposed operator, a hybrid TOPSIS method, called PFOWAWAD-TOPSIS is introduced for Pythagorean fuzzy MCDM problem. Finally, a numerical example is provided to illustrate the practicality and feasibility of the developed method.

250 citations


Journal ArticleDOI
TL;DR: In this paper, a novel operator-theoretic framework is proposed to study global stability of nonlinear systems based on the spectral properties of the so-called Koopman operator.
Abstract: We propose a novel operator-theoretic framework to study global stability of nonlinear systems. Based on the spectral properties of the so-called Koopman operator, our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems. The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles. These results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space.

247 citations


Journal ArticleDOI
TL;DR: Baroni et al. as discussed by the authors considered a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and proved some regularity results for related minimisers.
Abstract: We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p, q)-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima. 1. Almost fifty years of degenerate operators in Russia In 1967 a seminal paper [60] of Ural’tseva appeared, featuring the proof of the C-nature of energy solutions to the degenerate equation (1.1) − div(|Du|p−2Du) = 0 . The one on the left hand side is nowadays very well known as the p-Laplacean operator. This operator is relevant in a large number of situations as for instance in the Calculus of Variations, in Geometric Analysis, in the theory of quasiconformal mappings, in the modelling of non-Newtonian fluids. Ural’tseva herself, by exhibiting a counterexample, showed that the regularity of solutions does not go beyond Holder continuity of the gradient, for some exponent β ∈ (0, 1). The proof the Holder gradient continuity result also appears in the second, yet untranslated, edition of the classical book [41]. Ural’tseva’s fundamental result is at the origin of a huge literature, up to the point that it is nowadays hard to find another single nonlinear operator that has attracted so much attention as long as the elliptic regularity theory is concerned. We quote here the important paper of Uhlenbeck [59], where Ural’tseva’s result has been extended to the vectorial case, and the papers [23, 29, 42, 47], where different proofs and extensions to equations with coefficients have been given. The equation appearing in (1.1) is the Euler-Lagrange equation of the functional (1.2) w 7→ ∫ |Dw| dx , p > 1 and, in fact, several of the results and techniques coming from the analysis of (1.1) have been eventually found to be useful in the Calculus of Variations. Starting from the eighties, new models and functionals related to the one in (1.2) were developed by Zhikov [62, 63, 64, 65, 66], together with a group of Russian mathematicians, in order to describe the behaviour of strongly anisotropic materials in the context of homogenisation and nonlinear elasticity. These functionals revealed to be important 1 2 BARONI, COLOMBO, AND MINGIONE also in the study of duality theory and in the context of the Lavrentiev phenomenon. They are non-autonomous functionals of the form

247 citations


Journal ArticleDOI
TL;DR: In this article, the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas are surveyed and a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases.
Abstract: The article surveys the main techniques and results of the spectral theory of periodic operators arising in mathematical physics and other areas. Close attention is paid to studying analytic properties of Bloch and Fermi varieties, which influence significantly most properties of such operators. The approaches described are applicable not only to the standard model example of Schrodinger operator with periodic electric potential −∆ + V (x), but to a wide variety of elliptic periodic equations and systems, equations on graphs, ∂-operator, and other operators on abelian coverings of compact bases. Many important applications are mentioned. However, due to the size restrictions, they are not dealt with in details.

206 citations


Journal ArticleDOI
TL;DR: The problem of finding a fixed point to a nonexpansive operator (i.e., $x^*=Tx^*), where x is the number of points in a non-convex operator, has been studied in numerical linear algebra, optimization, and other areas of data science as discussed by the authors.
Abstract: Finding a fixed point to a nonexpansive operator, i.e., $x^*=Tx^*$, abstracts many problems in numerical linear algebra, optimization, and other areas of data science. To solve fixed-point problems...

193 citations


Journal ArticleDOI
TL;DR: In this article, the authors present the design and implementation of an augmented reality (AR) tool in aid of operators being in a hybrid, human and robot collaborative industrial environment, which aims to provide production and process related information as well as to enhance the operators' immersion in the safety mechanisms, dictated by the collaborative workspace.

Journal ArticleDOI
TL;DR: This work investigates the uniqueness of solutions for a class of nonlinear boundary value problems for fractional differential equations with Lipschitz constant related to the first eigenvalues corresponding to the relevant operators.

Journal ArticleDOI
TL;DR: In this article, the existence and multiplicity of non-negative entire (weak) solutions of a stationary Kirchhoff eigenvalue problem, involving a general nonlocal integro-differential operator, were established.
Abstract: In this paper we establish existence and multiplicity of nontrivial non-negative entire (weak) solutions of a stationary Kirchhoff eigenvalue problem, involving a general nonlocal integro-differential operator. The model under consideration depends on a real parameter Λ and involves two superlinear nonlinearities, one of which could be critical or even supercritical.

Journal ArticleDOI
TL;DR: In this paper, a matrix product operator/matrix product state algebra within a pure renormalized operator-based matrix renormalization group (DMRG) code is described.
Abstract: Current descriptions of the ab initio density matrix renormalization group (DMRG) algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab initio DMRG sweep using a matrix product operator based code, and the equivalence to the original renormalized operator implementation. Next we describe how to implement the general matrix product operator/matrix product state algebra within a pure renormalized operator-based DMRG code. Finally, we discuss two improvements of the ab initio DMRG sweep algorithm motivated by matrix product operator language: Hamiltonian compression, and a sum over operators representation that allows for perfect computational parallelism. The connections and correspondences described here serve to link the future developments with the past and are important in the efficient implementation of continuing advances in ab initio DMRG and related algorithms.

Journal ArticleDOI
TL;DR: In this paper, the authors review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators, e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD).
Abstract: Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.

Journal ArticleDOI
TL;DR: In this paper, it was shown that any operator with nonzero Poincar\'e charges, and in particular any compactly supported operator, in flat-spacetime quantum field theory must be gravitationally dressed once coupled to gravity, i.e., it must depend on the metric at arbitrarily long distances, and lower bounds on this nonlocal dependence.
Abstract: Quantum field theory---our basic framework for describing all nongravitational physics---conflicts with general relativity: the latter precludes the standard definition of the former's essential principle of locality, in terms of commuting local observables. We examine this conflict more carefully, by investigating implications of gauge (diffeomorphism) invariance for observables in gravity. We prove a dressing theorem, showing that any operator with nonzero Poincar\'e charges, and in particular any compactly supported operator, in flat-spacetime quantum field theory must be gravitationally dressed once coupled to gravity, i.e., it must depend on the metric at arbitrarily long distances, and we put lower bounds on this nonlocal dependence. This departure from standard locality occurs in the most severe way possible: in perturbation theory about flat spacetime, at leading order in Newton's constant. The physical observables in a gravitational theory therefore do not organize themselves into local commuting subalgebras: the principle of locality must apparently be reformulated or abandoned, and in fact we lack a clear definition of the coarser and more basic notion of a quantum subsystem of the Universe. We discuss relational approaches to locality based on diffeomorphism-invariant nonlocal operators, and reinforce arguments that any such locality is state-dependent and approximate. We also find limitations to the utility of bilocal diffeomorphism-invariant operators that are considered in cosmological contexts. An appendix provides a concise review of the canonical covariant formalism for gravity, instrumental in the discussion of Poincar\'e charges and their associated long-range fields.

Journal ArticleDOI
TL;DR: In this paper, the authors explore an extension of Hilbert series techniques to count operators that include derivatives, and conjecture an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts.
Abstract: In this work, we explore an extension of Hilbert series techniques to count operators that include derivatives. For sufficiently low-derivative operators, we conjecture an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts. Specifically, the conjectured technique can be applied whenever there is only one Lorentz invariant for a given partitioning of derivatives among the fields. At higher numbers of derivatives, equation of motion redundancies can be removed, but the increased number of Lorentz contractions spoils the subtraction of integration by parts redundancies. While restricted, this technique is sufficient to automatically recreate the complete set of invariant operators of the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary numbers of flavors). At dimension 8, the algorithm does not automatically generate the complete operator set; however, it suffices for all but five classes of operators. For these remaining classes, there is a well defined procedure to manually determine the number of invariants. Assuming our method is correct, we derive a set of 535 dimension-8 N f = 1 operators.

Journal ArticleDOI
TL;DR: Based on the INPOWA operator and the comparative formula of the INNs, an approach to decision making with INNs is established and an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.
Abstract: On the basis of prioritized aggregated operator and prioritized ordered weighted average (POWA) operator, in this paper, the authors further present interval neutrosophic prioritized ordered weighted aggregation (INPOWA) operator with respect to interval neutrosophic numbers (INNs). Firstly, the definition, operational laws, characteristics, expectation and comparative method of INNs are introduced. Then, the INPOWA operator is developed, and some properties of the operator are analyzed. Furthermore, based on the INPOWA operator and the comparative formula of the INNs, an approach to decision making with INNs is established. Finally, an illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.

Journal ArticleDOI
TL;DR: In this paper, a planar tree-level four-point function in N = 4 SYM is considered, where one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half BPS operators a co-moving frame is chosen in flavour space.
Abstract: We consider a class of planar tree-level four-point functions in N=4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half-BPS operators a co-moving frame is chosen in flavour space. In configuration space, the four-punctured sphere is naturally triangulated by tree-level planar diagrams. We demonstrate on a number of examples that each tile can be associated with a modified hexagon form-factor in such a way as to efficiently reproduce the tree-level four-point function. Our tessellation is not of the OPE type, fostering the hope of finding an independent, integrability-based approach to the computation of planar four-point functions.

Journal ArticleDOI
TL;DR: The variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives is presented, which is illustrated by numerical experiments in image processing.
Abstract: We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators are accessed separately either via forward or via backward evaluations. We present also the variational case when one is interested in the solving of a primal-dual pair of convex optimization problems with complexly structured objectives, which we also illustrate by numerical experiments in image processing.

Journal ArticleDOI
TL;DR: In this paper, a general structure theorem for the singular part of A-free Radon measures, where A is a linear PDE operator, was established and applied to suitably chosen differential operators.
Abstract: We establish a general structure theorem for the singular part of A-free Radon measures, where A is a linear PDE operator. By applying the theorem to suitably chosen differential operators A, we obtain a simple proof of Alberti’s rank-one theorem and, for the first time, its extensions to functions of bounded deformation (BD). We also prove a structure theorem for the singular part of a finite family of normal currents. The latter result implies that the Rademacher theorem on the differentiability of Lipschitz functions can hold only for absolutely continuous measures and that every top-dimensional Ambrosio–Kirchheim metric current in Rd is a Federer–Fleming flat chain.

Proceedings Article
12 Feb 2016
TL;DR: An operator for tabular representations is described, the consistent Bellman operator, which incorporates a notion of local policy consistency, which leads to an increase in the action gap at each state; increasing this gap mitigates the undesirable effects of approximation and estimation errors on the induced greedy policies.
Abstract: This paper introduces new optimality-preserving operators on Q-functions. We first describe an operator for tabular representations, the consistent Bellman operator, which incorporates a notion of local policy consistency. We show that this local consistency leads to an increase in the action gap at each state; increasing this gap, we argue, mitigates the undesirable effects of approximation and estimation errors on the induced greedy policies. This operator can also be applied to discretized continuous space and time problems, and we provide empirical results evidencing superior performance in this context. Extending the idea of a locally consistent operator, we then derive sufficient conditions for an operator to preserve optimality, leading to a family of operators which includes our consistent Bellman operator. As corollaries we provide a proof of optimality for Baird's advantage learning algorithm and derive other gap-increasing operators with interesting properties. We conclude with an empirical study on 60 Atari 2600 games illustrating the strong potential of these new operators.

Journal ArticleDOI
TL;DR: This paper demonstrates a form of interaction without any explicit input by the operator, enabling computer systems to become neuroadaptive, that is, to automatically adapt to specific aspects of their operator’s mindset.
Abstract: The effectiveness of today’s human–machine interaction is limited by a communication bottleneck as operators are required to translate high-level concepts into a machine-mandated sequence of instructions. In contrast, we demonstrate effective, goal-oriented control of a computer system without any form of explicit communication from the human operator. Instead, the system generated the necessary input itself, based on real-time analysis of brain activity. Specific brain responses were evoked by violating the operators’ expectations to varying degrees. The evoked brain activity demonstrated detectable differences reflecting congruency with or deviations from the operators’ expectations. Real-time analysis of this activity was used to build a user model of those expectations, thus representing the optimal (expected) state as perceived by the operator. Based on this model, which was continuously updated, the computer automatically adapted itself to the expectations of its operator. Further analyses showed this evoked activity to originate from the medial prefrontal cortex and to exhibit a linear correspondence to the degree of expectation violation. These findings extend our understanding of human predictive coding and provide evidence that the information used to generate the user model is task-specific and reflects goal congruency. This paper demonstrates a form of interaction without any explicit input by the operator, enabling computer systems to become neuroadaptive, that is, to automatically adapt to specific aspects of their operator’s mindset. Neuroadaptive technology significantly widens the communication bottleneck and has the potential to fundamentally change the way we interact with technology.

Journal ArticleDOI
TL;DR: A new numerical method for solving the distributed fractional differential equations is presented based upon hybrid functions approximation and the Riemann-Liouville fractional integral operator for hybrid functions is introduced.

Journal ArticleDOI
TL;DR: In this article, the extremal correlators of Coulomb branch operators in four-dimensional N = 2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator are derived.
Abstract: We consider the correlation functions of Coulomb branch operators in four-dimensional N=2 Superconformal Field Theories (SCFTs) involving exactly one anti-chiral operator. These extremal correlators are the "minimal" non-holomorphic local observables in the theory. We show that they can be expressed in terms of certain determinants of derivatives of the four-sphere partition function of an appropriate deformation of the SCFT. This relation between the extremal correlators and the deformed four-sphere partition function is non-trivial due to the presence of conformal anomalies, which lead to operator mixing on the sphere. Evaluating the deformed four-sphere partition function using supersymmetric localization, we compute the extremal correlators explicitly in many interesting examples. Additionally, the representation of the extremal correlators mentioned above leads to a system of integrable differential equations. We compare our exact results with previous perturbative computations and with the four-dimensional tt^* equations. We also use our results to study some of the asymptotic properties of the perturbative series expansions we obtain in N=2 SQCD.

Journal ArticleDOI
TL;DR: This paper has presented some new operational laws for SVNNs based on ATT, and proposed a single-valued neutrosophic number-weighted averaging (SVNNWA) operator and a multiple attribute group decision-making problems based on these operators.
Abstract: The single-valued neutrosophic set (SVNS) can be easier to describe the incomplete, indeterminate and inconsistent information, and Archimedean t-conorm and t-norm (ATT) can generalize most of the existing t-conorms and t-norms, including Algebraic, Einstein, and Hamacher Frank t-conorms and t-norms. In this paper, we extended ATT to the single-valued neutrosophic numbers (SVNNs), and proposed a single-valued neutrosophic number-weighted averaging (SVNNWA) operator and a single-valued neutrosophic number-weighted geometric (SVNNWG) operator based on ATT. First, we presented some new operational laws for SVNNs based on ATT, and discussed some special cases and properties of them. Then we proposed the SVNNWA and SVNNWG operators, and studied some properties and special cases of them. Further, we gave the decision-making methods for multiple attribute decision-making (MADM) and multiple attribute group decision-making (MAGDM) problems based on these operators. Finally, two examples are given to verify the developed approaches.

Journal ArticleDOI
TL;DR: In this paper, the existence and stability of quasi-periodic, small amplitude solutions of quasilinear (i.e., strongly nonlinear) autonomous Hamiltonian differentiable perturbations of KdV was proved.

Journal ArticleDOI
TL;DR: In this article, the authors introduce a systematic framework for counting and finding independent operators in effective field theories, taking into account the redundancies associated with use of the classical equations of motion and integration by parts.
Abstract: We introduce a systematic framework for counting and finding independent operators in effective field theories, taking into account the redundancies associated with use of the classical equations of motion and integration by parts. By working in momentum space, we show that the enumeration problem can be mapped onto that of understanding a polynomial ring in the field momenta. All-order information about the number of independent operators in an effective field theory is encoded in a geometrical object of the ring known as the Hilbert series. We obtain the Hilbert series for the theory of N real scalar fields in (0+1) dimensions—an example, free of space-time and internal symmetries, where aspects of our framework are most transparent. Although this is as simple a theory involving derivatives as one could imagine, it provides fruitful lessons to be carried into studies of more complicated theories: we find surprising and rich structure from an interplay between integration by parts and equations of motion and a connection with SL(2, $${{\mathbb{C}}}$$ ) representation theory, which controls the structure of the operator basis.

Journal ArticleDOI
TL;DR: For interacting theories in more than two dimensions, it was shown in this article that the averaged null energy condition must be positive for all spin operators other than the stress tensor, leading to new inequalities for the coupling constants of spinning operators in conformal field theory.
Abstract: Unitary, Lorentz-invariant quantum field theories in flat spacetime obey microcausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $\int du T_{uu}$, must be positive. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to $n$-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress tensor, generating an infinite family of new constraints of the form $\int du X_{uuu\cdots u} \geq 0$. These lead to new inequalities for the coupling constants of spinning operators in conformal field theory, which include as special cases (but are generally stronger than) the existing constraints from the lightcone bootstrap, deep inelastic scattering, conformal collider methods, and relative entropy. We also comment on the relation to the recent derivation of the averaged null energy condition from relative entropy, and suggest a more general connection between causality and information-theoretic inequalities in QFT.

Journal ArticleDOI
TL;DR: In this paper, the authors performed extensive three-loop tests of the hexagon bootstrap approach for structure constants in planar N = 4 SYM theory and found that the first type of correction coincides exactly with the leading wrapping correction for the spectrum.