Recursion Relations Research Articles

Out-of-time-order asymptotic observables are quasi-isomorphic to time-ordered amplitudes

Asymptotic observables in quantum field theory beyond the familiar S-matrix have recently attracted much interest, for instance in the context of gravity waveforms. Such observables can be understood in terms of Schwinger-Keldysh-type ‘amplitudes’ computed by a set of modified Feynman rules involving cut internal legs and external legs labelled by time-folds.In parallel, a homotopy-algebraic understanding of perturbative quantum field theory has emerged in recent years. In particular, passing through homotopy transfer, the S-matrix of a perturbative quantum field theory can be understood as the minimal model of an associated (quantum) L∞-algebra.Here we bring these two developments together. In particular, we show that Schwinger-Keldysh amplitudes are naturally encoded in an L∞-algebra, similar to ordinary scattering amplitudes. As before, they are computed via homotopy transfer, but using deformation-retract data that are not canonical (in contrast to the conventional S-matrix). We further show that the L∞-algebras encoding Schwinger-Keldysh amplitudes and ordinary amplitudes are quasi-isomorphic (meaning, in a suitable sense, equivalent). This entails a set of recursion relations that enable one to compute Schwinger-Keldysh amplitudes in terms of ordinary amplitudes or vice versa.

Neural network-based hybrid modeling approach incorporating Bayesian optimization with industrial soft sensor application

Hybrid modeling combines physical and data-driven models to improve the performance of industrial soft sensors. However, simplified physical assumptions and extensive parameter optimization increase the complexity of the model and decrease its robustness and accuracy. In this study, a novel neural network-based approach for hybrid modeling was developed, based on Runge-Kutta (RK) and Bayesian hyperparameter optimization. First, the recursive solution equation for the quality variable was derived using the master equation, which included the black-box term and RK method. Second, the numerical solution of the black-box term was inverted using the derived equation as a label for the training samples, and hyperparameters were optimized using Bayesian optimization. Finally, during the prediction phase, quality variables at subsequent times were derived by solving a recursive equation containing neural network approximations. The superiority of the proposed soft sensing method was assessed through studies on a continuous stirred tank reactor (CSTR) case and a simulated penicillin case. The experimental findings demonstrate that the proposed method is better than existing methods in terms of robustness and accuracy.

APPLICATION OF DYNAMIC PROGRAMMING IN DETERMINING THE SHORTEST ROUTE PT JNE USING BACKWARD RECURSIVE EQUATION

Technological progress speeds up human movement, the distribution of goods, and service provision. PT JNE, a prominent firm in Indonesia, concentrates on delivering goods swiftly and with comprehensive services.. On the other hand, problems include daily changes in delivery, delays, less than optimal routes, and expensive gasoline. The shortest ideal route for the transportation of products is found using dynamic programming and a backward recursive equation technique in this study. The Medan Belawan District as well as the JNE Medan Representative Office provided the data that was used. The findings show that a route of a → f → d → c → g → b → e → h → a is the ideal total distance for goods delivery using a dynamic programming graph is 12,2 km, with an average courier covering 16,5 km. Based on data analysis, this shows a 26% increase in efficiency over the current routes. The layout of the routes makes it easier for couriers to choose the fastest route.

Stable envelopes for slices of the affine Grassmannian

The affine Grassmannian associated to a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}$$\\end{document} is an affine analogue of the usual flag varieties. It is a rich source of Poisson varieties and their symplectic resolutions. These spaces are examples of conical symplectic resolutions dual to the Nakajima quiver varieties. We study the cohomological stable envelopes of Maulik and Okounkov (Astérisque 408:ix+209, 2019) in this family. We construct an explicit recursive relation for the stable envelopes in the G=PSL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{G}}= \ extbf{PSL}_{2}$$\\end{document} case and compute the first-order correction in the general case. This allows us to write an exact formula for multiplication by a divisor.

Color-kinematic numerators for fermion Compton amplitudes

We introduce a novel approach to compute Compton amplitudes involving a fermion pair inspired by Hopf algebra amplitude constructions. This approach features a recursive relation employing quasi-shuffle sets, directly verifiable by massive factorization properties. We derive results for minimal gauge invariant color-kinematic numerators with physical massive poles using this method. We have also deduced a graphical method for deriving numerators that simplifies the numerator generation and eliminates redundancies, thus providing several computational advantages.

Integrable structure of higher spin CFT and the ODE/IM correspondence

We study two dimensional systems with extended conformal symmetry generated by the W\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{W} $$\\end{document}3 algebra. These are expected to have an infinite number of commuting conserved charges, which we refer to as the quantum Boussinesq charges. We compute the eigenvalues of the quantum Boussinesq charges in both the vacuum and first excited states of the higher spin module through the ODE/IM correspondence. By studying the higher spin conformal field theory on the torus, we also calculate thermal correlators involving the energy-momentum tensor and the spin-3 current by making use of the Zhu recursion relations. By combining these results, we show that it is possible to derive the current densities, whose integrals are the quantum Boussinesq charges. We also evaluate the thermal expectation values of the conserved charges, and show that these are quasi-modular differential operators acting on the character of the higher spin module.

A novel skewed generalized normal distribution: properties, statistical inference, and its applications

In this paper, a novel skewed generalized normal distribution (NSGN), which encompasses several well-known distributions, is a two-component mixture model based on the generalized normal distribution. The NSGN distribution offers flexibility in modeling skewed data characterized by high kurtosis and strong skewness, making it highly applicable in finance, medicine, and engineering. The primary characteristics and properties of this new distribution are introduced, and the explicit expressions for the moments of order statistics are provided using recursive relations. The maximum likelihood estimation based on a profile likelihood approach, L-moments estimation, and a two-step estimation method combining Bayesian posterior maximum estimation with moment estimates are presented to estimate the four parameters of NSGN distribution. A simulation study compares the performance of these methods across different sample sizes and different parameters. The appropriateness of the proposed distribution has been tested by comparing it with several skewed distributions. Additionally, applications to real datasets showcase the beneficial properties of the NSGN for applied statistical research.

Asymptotic expansions of truncated hypergeometric series for 1/π

In this paper, we consider rational hypergeometric series of the formpπ=∑k=0∞ukwithuk=(12)k(q)k(1−q)k(k!)3(r+sk)tk, where (a)k denotes the Pochhammer symbol and p,q,r,s,t are algebraic coefficients. Using only the first n+1 terms of this series, we define the remainderRn=pπ−∑k=0nuk=∑k=n+1∞uk. We consider an asymptotic expansion of Rn. More precisely, we provide a recursive relation for determining the coefficients cj such thatRn=(12)n(q)n(1−q)nn!3ntn(∑j=0J−1cjnj+O(n−J)),n→∞. Here we need J<∞ to approximate Rn, because (like the Stirling series) this series diverges if J→∞. By applying our recursive relation to the Chudnovsky formula, we solve an open problem posed by Han and Chen.

Machine Learning-Enhanced Probabilistic Validation Techniques with Minimal Data Exposure in Distributed Systems

This article studies the distributed state estimation issue for complex networks with nonlinear uncertainty. The extended state approach is used to deal with the nonlinear uncertainty. The distributed state predictor is designed based on the extended state system model, and the distributed state estimator is designed by using the measurement of the corresponding node. The prediction error and the estimation error are derived. The prediction error covariance (PEC) is obtained in terms of the recursive Riccati equation, and the upper bound of the PEC is minimized by designing an optimal estimator gain. With the vectorization approach, a sufficient condition concerning stability of the upper bound is developed. Finally, a numerical example is presented to illustrate the effectiveness of the designed extended state estimator.

The in-out formalism for in-in correlators

Cosmological correlators, the natural observables of the primordial universe, have been extensively studied in the past two decades using the in-in formalism pioneered by Schwinger and Keldysh for the study of dissipative open systems. Ironically, most applications in cosmology have focused on non-dissipative closed systems. We show that, for non-dissipative systems, correlators can be equivalently computed using the in-out formalism with the familiar Feynman rules. In particular, the myriad of in-in propagators is reduced to a single (Feynman) time-ordered propagator and no sum over the labelling of vertices is required. In de Sitter spacetime, this requires extending the expanding Poincaré patch with a contracting patch, which prepares the bra from the future. Our results are valid for fields of any mass and spin but assuming the absence of infrared divergences.We present three applications of the in-out formalism: a representation of correlators in terms of a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation that computes Minkowski correlators in terms of lower order ones; and the derivation of cutting rules from Veltman’s largest time equation, which we explicitly develop and exemplify for two-vertex diagrams to all loop orders.The in-out formalism leads to a natural definition of a de Sitter scattering matrix, which we discuss in simple examples. Remarkably, we show that our scattering matrix satisfies the standard optical theorem and the positivity that follows from it in the forward limit.

Conceptualizing Records: Community Archives Describing Themselves

ABSTRACT The concept of records is foundational to archival studies, yet empirical research on how members and volunteers of community and grassroots archives conceptualize records remains limited. To understand how members and volunteers of community archives conceptualize records, this study asks how members and volunteers of the Black Bottom Archives (BBA), the Detroit Sound Conservancy (DSC), the Faulkner Morgan Archive (FMA), the Hula Preservation Society (HPS), and The History Project (THP) conceptualize records and how these conceptualizations inform their programs and practices. Based on ten semistructured interviews with five archives in the United States, this research reveals that members and volunteers view records as multifaceted and contingent on ongoing negotiation, borrowing, and intervention for larger goals rather than strictly being tied to abstract, institutional, or professional notions of records. Such a view also points to a recursive and generative relationship between records, programs, and practices and keeping track of power and legitimacy.

Synthesis of Itˆo Equations for a Shaping Filter with a Given Spectrum

The analytical method for the synthesis of a generator of a random process with a given spectrum in the form of a linear system of Ito’s equations is proposed. The stationarity of a random process is assumed, the spectral and corresponding transfer functions of which are defined in the form of rational fractions. The coefficients of the system of Ito’s equations of the generator are found from recurrent algebraic relations. The method is focused on working with mathematical models of nature random processes, such as the Dryden’s wind model. The transformation of the spectra of the wind gust model in three directions is presented in detail and the corresponding stochastic equations are given.

Half-exceeded Symmetric Permutations and the Inverse Pairs

In this paper, we introduce a class of restricted symmetric permutations, called half-exceeded symmetric permutations. We deduce the enumerative formula of the permutations of { 1 , 2 , … , 2 n } and give it a refinement according to the distribution of the inverse pairs. As a consequence, we obtain new combinatorial interpretations of some well-known sequences, such as Stirling numbers of the second kind and ordered Bell numbers. Moreover, we introduce the ordered Stirling number of the second kind and establish a combinatorial proof of the recursive relation of the sequence.

Joint moments of higher order derivatives of CUE characteristic polynomials II: structures, recursive relations, and applications

In a companion paper (Keating and Wei 2023 Int. Math. Res. Not. 2024 9607–32), we established asymptotic formulae for the joint moments of higher order derivatives of the characteristic polynomials of CUE random matrices. The leading order coefficients of these asymptotic formulae are expressed as partition sums of derivatives of determinants of Hankel matrices involving I-Bessel functions, with column indices shifted by Young diagrams. In this paper, we continue the study of these joint moments and establish more properties for their leading order coefficients, including structure theorems and recursive relations. We also build a connection to a solution of the σ-Painlevé III ′ equation. In the process, we give recursive formulae for the Taylor coefficients of the Hankel determinants formed from I-Bessel functions that appear at zero and find some differential equations these determinants satisfy. The approach we establish is applicable to determinants of general Hankel matrices whose columns are shifted by Young diagrams.

Exact solution of the [formula omitted] quantum spin chain with open boundary condition

In this paper, we studied the exact solution of the C2(1) invariant quantum spin chain with off-diagonal open boundary condition. We obtain a solution of the reflection equation where the all matrix element of reflection matrix are nonzeros. By using the technique of fusion, we construct the fused transfer matrix and find the closed recursive relations among the transfer matrices. Based on the algebraic analysis, we obtain the eigenvalue of the system and express it as the inhomogeneous T−Q relation.

On amplitudes and field redefinitions

We derive an off-shell recursion relation for correlators that holds at all loop orders. This allows us to prove how generalized amplitudes transform under generic field redefinitions, starting from an assumed behavior of the one-particle-irreducible effective action. The form of the recursion relation resembles the operation of raising the rank of a tensor by acting with a covariant derivative. This inspires a geometric interpretation, whose features and flaws we investigate.

Strebel Differentials and String Field Theory

Abstract A closed string worldsheet of genus g with n punctures can be presented as a contact interaction in which n semi-infinite cylinders are glued together in a specific way via the Strebel differential on it, if $n\ge 1,\ 2g-2+n\gt 0$. We construct a string field theory of closed strings such that all the Feynman diagrams are represented by such contact interactions. In order to do so, we define off-shell amplitudes in the underlying string theory using the combinatorial Fenchel–Nielsen coordinates to describe the moduli space and derive a recursion relation satisfied by them. Utilizing the Fokker–Planck formalism, we construct a string field theory from which the recursion relation can be deduced through the Schwinger–Dyson equation. The Fokker–Planck Hamiltonian consists of kinetic terms and three-string interaction terms.

Establishing a Reciprocal and Recursive Relationship Between Sociopolitical Development and Wellbeing for Early Emerging Adult College Students When “There Was a lot Happening in Both the World. . . and Within My Own Personal World”

Establishing a Reciprocal and Recursive Relationship Between Sociopolitical Development and Wellbeing for Early Emerging Adult College Students When “There Was a lot Happening in Both the World. . . and Within My Own Personal World”

Modularity in Argyres-Douglas theories with a = c

We consider a family of Argyres-Douglas theories, which are 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 strongly coupled superconformal field theories (SCFTs) but share many features with 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theories. In particular, the two central charges of these theories are the same, namely a = c. We derive a simple and illuminating formula for the Schur index of these theories, which factorizes into the product of a Casimir term and a term referred to as the Schur partition function. While the former is controlled by the anomaly, the latter is identified with the vacuum character of the corresponding chiral algebra and is expected to satisfy the modular linear differential equation. Our simple expression for the Schur partition function, which can be regarded as the generalization of MacMahon’s generalized sum-of-divisor function, allows one to numerically compute the series expansions efficiently, and furthermore find the corresponding modular linear differential equation. In a special case where the chiral algebra is known, we are able to derive the corresponding modular linear differential equation using Zhu’s recursion relation. We further study the solutions to the modular linear differential equations and discuss their modular transformations. As an application, we study the high temperature limit or the Cardy-like limit of the Schur index using its simple expression and modular properties, thus shedding light on the 1/4-BPS microstates of genuine N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 SCFTs with a = c and their dual quantum gravity via the AdS/CFT correspondence.

Cascading failures in bipartite networks with directional support links.

We study the cascading failures in a system of two interdependent networks whose internetwork supply links are directional. We will show that, by utilizing generating function formalism, the cascading process can be modeled by a set of recursive relations. Most importantly, the functions involved in these relations are solely dependent upon the choice of the degree distribution of ingoing links. Simulation results in the limit of very large networks based on different choices of degree distributions for outgoing links, e.g., Kronecker delta, Poisson and Pareto, are indeed identical and are in excellent agreement with the theory. However, for Pareto distribution with the shape parameter 1<α<2, the convergence is slow. In general, directional networks can be more vulnerable or less vulnerable than their bidirectional counterparts. For three special settings of interdependent networks, we analytically compare their vulnerability. For practical applications it is important to predict if a system responds to the size of the initial attack continuously or if there is catastrophic collapse of the system if the attack exceeds a specific transition size. We analytically show that systems with lower average degrees are more resilient against this abrupt transition. We also establish an equivalence of this transition with the liquid-gas transition in statistical mechanics. In the last section, we derive the set of recursive relation to describe the cascading process where the initial attack is not restricted to a single network.