Analytic Continuation Research Articles

The Neocolonial Structural Model Re-Foundation and The Pragmatic Inference of Organizational Culture in Ngugi Wa Thiongo’s Devil on The Cross (1980)

Through the frame frequency of the imported state and the westernization of the African political order, the social construction of the African bureaucracy appears as a structural device of abstraction, exploitation, and brutality. Then, through the frame of reference with the importation of conceptual juridical coordination and perceptual representative consensus, the African public administration enhances a frameshift transformation and a succession of causality. Therefore, in combinatorial and transactional analysis, the framework of Devil On The Cross progresses in a method of empirical processing decomposition. Within this respect, the elemental and pure imaginary of the African administration proceeds as a functional derivative governmentality and a colonial functional inception power. Subsequently, with its interaction design, processing system, and volume technique, the realm of Devil On The Cross inserts a paradigmatic analysis and a value engineering within which this African administration happens as an characteristic structural formula and an inefficient relational frame theory. Moreover, within the space-time continuum, the configuration management and the linear configuration programming of the African management apparatus, Devil On The Cross becomes a holomorphic function and an analytic continuation that reveal the anachronism nature and the asynchronous system of the African post-colonial administration.

Characterizations of Kerr-de Sitter in arbitrary dimension from null infinity.

The Kerr and Kerr-de Sitter metrics share remarkable local geometric properties in four dimensions. Gibbons et al. found a generalization of the Kerr-de Sitter metric to higher dimensions, to which the local characterization above cannot be applied. One viable approach to characterize this family is to understand the behaviour of these metrics at future null infinity. We review Friedrich's and Fefferman-Graham formalisms to discuss the asymptotic initial value problem of ([Formula: see text])-vacuum spacetimes in arbitrary dimensions and study their properties: geometric identification and conformal equivalence of data, Killing initial data and conformal equivalence of boundary conformal Killing vectors (CKV). These results are used to review a recent characterization of Kerr-de Sitter in terms of its asymptotic data, namely conformal flatness at [Formula: see text] together with a canonical TT tensor constructed from specific CKV at [Formula: see text]. Allowing for arbitrary CKV defines the (larger) Kerr-de Sitter-like class. All these metrics can be obtained explicitly as limits or analytic extensions of Kerr-de Sitter. The Kerr-de Sitter-like class is also characterized by the property of being Kerr-Schild and fulfilling a certain falloff condition. In addition, in five dimensions, this class corresponds to all algebraically special metrics with non-degenerate optical matrix. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Boundary signature of singularity in the presence of a shock wave

Matter falling into a Schwarzschild-AdS black hole from the left causes increased focussing of ingoing geodesics from the right, and, as a consequence, they reach the singularity sooner. In a standard Penrose diagram, the singularity “bends down”. We show how to detect this feature of the singularity holographically, using a boundary two-point function. We model the matter with a shock wave, and show that this bending down of the singularity can be read off from a novel analytic continuation of the boundary two-point function. Along the way, we obtain a generalization of the recently proposed thermal product formula for two-point correlators.

The expressivity of classical and quantum neural networks on entanglement entropy

Analytically continuing the von Neumann entropy from Rényi entropies is a challenging task in quantum field theory. While the n-th Rényi entropy can be computed using the replica method in the path integral representation of quantum field theory, the analytic continuation can only be achieved for some simple systems on a case-by-case basis. In this work, we propose a general framework to tackle this problem using classical and quantum neural networks with supervised learning. We begin by studying several examples with known von Neumann entropy, where the input data is generated by representing TrρAn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Tr}}\\rho _A^n$$\\end{document} with a generating function. We adopt KerasTuner to determine the optimal network architecture and hyperparameters with limited data. In addition, we frame a similar problem in terms of quantum machine learning models, where the expressivity of the quantum models for the entanglement entropy as a partial Fourier series is established. Our proposed methods can accurately predict the von Neumann and Rényi entropies numerically, highlighting the potential of deep learning techniques for solving problems in quantum information theory.

Metallic Ratios and Angles of a Real Argument

We extend the concept of metallic ratios to the real argument n considered as a dimension by analytic continuation showing that they are defined by an argument of a normalized complex number, and for rational n ≠ {0, ±2}, they are defined by Pythagorean triples. We further extend the concept of metallic ratios to metallic angles.

Vacuum energy of scalar fields on spherical shells with general matching conditions

In this work we analyze the spectral zeta function for massless scalar fields propagating in a D-dimensional flat space under the influence of a shell potential. The static nature of the potential, and the spherical symmetry, allows us to focus on the spatial part of the field which satisfies a one-dimensional Schrodinger equation endowed with a point potential. The shell potential is defined in terms of the two-interval self-adjoint extensions of the one-dimensional Schrodinger equation that describes the radial part of the scalar field. After performing the necessary analytic continuation, we utilize the spectral zeta function of the system to compute the vacuum energy of the field.

Gravitational form factors of the proton from near-threshold vector meson photoproduction* *Supported by the National Natural Science Foundation of China (12065014, 12247101) and the Natural Science Foundation of Gansu province (22JR5RA266). We acknowledge the West Light Foundation of the Chinese Academy of Sciences (21JR7RA201)

We systematically analyze the quark and gluon gravitational form factors (GFFs) of the proton by connecting the energy-momentum tensor and near-threshold vector meson photoproduction (NTVMP). Specifically, the quark contributions of GFFs are determined by applying global fitting to the cross section of the lightest vector meson photoproduction. Combined with the gluon GFFs obtained from heavy quarkonium photoproduction data, the complete GFFs are obtained and compared with the experimental results and lattice quantum chromodynamics determinations. In addition, we use the resonance via Padé (RVP) method based on the Schlessinger point method to obtain a model-independent quark D-term distribution through direct analytical continuation of deeply virtual Compton scattering experimental data. If errors are considered, the results obtained with RVP are essentially consistent with those obtained by NTVMP. Moreover, the comprehensive information on GFFs helps to uncover the mass distribution and mechanical properties inside the proton. This study is not only an important basis for delving into the enigmatic properties of the proton, but also has significance for theoretically guiding future JLab and EIC experimental measurements.

On a Ramanujan-type series associated with the Heegner number 163

Using the Wolfram NumberTheory package and the Recognize command, together with numerical estimates involving the elliptic lambda and elliptic alpha functions, Bagis and Glasser, in 2013, introduced a conjectural Ramanujan-type series related to the class number h(−d)=1 for a quadratic form with discriminant d=163. This conjectured series is of level one and has positive terms, and recalls the Chudnovsky brothers' alternating series of the same level, given the connection between the Chudnovsky–Chudnovsky formula and the Heegner number d=163 such that Q(−d) has class number one. We prove Bagis and Glasser's conjecture by proving evaluations for λ⁎(163) and α(163), which we derive using the Chudnovsky brothers' formula together with the analytic continuation of a formula due to the Borwein brothers for Ramanujan-type series of level one. As a byproduct of our method, we obtain an infinite family of Ramanujan-type series for 1π generalizing the Chudnovsky algorithm.

Zhang neural networks: an introduction to predictive computations for discretized time-varying matrix problems

This paper wants to increase our understanding and computational know-how for time-varying matrix problems and Zhang Neural Networks. These neural networks were invented for time or single parameter-varying matrix problems around 2001 in China and almost all of their advances have been made in and most still come from its birthplace. Zhang Neural Network methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and other engineering applications in China. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. A typical discretized Zhang Neural Network algorithm needs seven distinct steps in its initial set-up. The construction of discretized Zhang Neural Network algorithms starts from a model equation with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multi-step style solver that predicts the future state of the system reliably from current and earlier state and solution data. Matlab codes of discretized Zhang Neural Network algorithms for time varying matrix problems typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized Zhang Neural network based algorithms highly competitive with ordinary differential equation initial value analytic continuation methods for function given data that are designed to work adaptively. Discretized Zhang Neural Network methods have different characteristics and applicabilities than multi-step ordinary differential equations (ODEs) initial value solvers. These new time-varying matrix methods can solve matrix-given problems from sensor data with constant sampling gaps or from functional equations. To illustrate the adaptability of discretized Zhang Neural Networks and further the understanding of this method, this paper details the seven step set-up process for Zhang Neural Networks and twelve separate time-varying matrix models. It supplies new codes for seven of these. Open problems are mentioned as well as detailed references to recent work on discretized Zhang Neural Networks and time-varying matrix computations. Comparisons are given to standard non-predictive multi-step methods that use initial value problems ODE solvers and analytic continuation methods.

Generalized Maxwell equal area law and black holes in complex free energy

Maxwell equal area law is an important and traditional analytical tool in thermodynamic phase transition, especially in the calculation of gas-liquid phase transition, which reconciles the theoretical calculation with the experimental results. Undoubtedly, its importance is also self-evident for the black hole thermodynamic system. In this study, we construct a generalized Maxwell equal area law, which allows different states of thermodynamic systems to be within the generalized free energy. The black hole thermodynamic characteristics are spontaneously emerged in the free energy landscape. Furthermore, by analytic continuation, we utilize the properties of analytical functions to investigate some universal characteristics of thermodynamic phase transitions in black holes, and preliminarily establish the counterpart of thermodynamic phase transitions in the complex domain.

Total absorption spectrum of benzene aggregates obtained from two different approaches.

The study of molecular aggregation effects on the electronic spectrum is essential for the development of optoelectronic devices. However, investigating the entire valence absorption spectrum of aggregates using quantum mechanical methods is a challenging task. In this work, we perform systematic simulations of the absorption spectrum of benzene molecular clusters up to 35 eV applying two approaches based on time-dependent density functional theory. The results show that depending on the dimer packing, different energy shifts occur for the symmetry allowed [Formula: see text] transition, in comparison to the monomer. The transition intensity increases for the band around 6 eV for larger aggregates from the monomer to dimers and tetramer, indicating the occurrence of the symmetry forbidden (in [Formula: see text] point group) [Formula: see text] [Formula: see text] transition. The benzene crystal exhibits a large redshift following the experimental spectrum. Also, the continuum regions of all spectra show a good agreement with the experiments both in gas and solid phases. Geometry optimization of the monomer was carried out with Gaussian 09 software using the PBE0/def2-TZVP level of theory. We used dimers and tetramer molecular geometries extracted from the experimental crystal structure. The absorption spectra were directly obtained by the Liouville-Lanczos TDDFT approach with plane waves basis set or indirectly by TDDFT pseudo-spectra calculated in a [Formula: see text] basis followed by analytic continuation procedure to obtain complex polarizability. The former is available at Quantum ESPRESSO, and the latter was calculated using Gaussian 09 with the post-processing performed with a code previously developed in our group.

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function FK to Symmetric Domains

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function FK with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks.

Dc conductivity of two-dimensional ionic Hubbard model extracted by analytic continuation

The ionic Hubbard model in a two-dimensional square lattice is revisited using determinant quantum Monte Carlo (DQMC) and numerical analytic continuation. From the dc conductivity extracted by stochastic maximum entropy of the DQMC data, we find that the band insulator induced by a staggered potential [Formula: see text] goes through a metallic phase before entering a Mott insulator as the on-site Hubbard interaction U is increased. Moreover, we map out the phase diagram in the [Formula: see text] plane, and reveal a smaller metallic region than that found in the previous studies. Our results provide a more exact location of the phase boundaries, and are an important complement to those obtained by an approximate approach.

Celestial conformal blocks of massless scalars and analytic continuation of the Appell function F1

In celestial conformal field theory (CCFT), the 4d massless scalars are represented by 2d conformal operators with conformal dimensions h = h¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\overline{h} $$\\end{document} = (1 + iλ)/2. The Mellin transform of 4d massless scalar amplitudes gives the conformal correlators of CCFT. We study the conformal block decomposition of these celestial correlators in CCFT and obtain the explicit blocks. This conformal block decomposition is highly nontrivial, even for the simplest 4d massless scalar amplitude. We use the analytic continuation of the Appell hypergeometric function F1 and the method of monodromy projection of conformal blocks, to achieve this block decomposition. This procedure is consistent with the crossing symmetry, in both the correlator-level and each explicit block-level. We also investigate its behavior in the conformal soft limit and find that the Appell hypergeometric function F1 does not reduce to the Gauss hypergeometric function. This is different from the block decomposition of celestial gluons we studied before, where the Appell hypergeometric function F1 reduces to the Gauss hypergeometric function. This difference comes from the shift of conformal dimensions and is the reason why we adopt the new method here for the block decomposition of celestial massless scalars.

Cosmology from the vacuum

We argue that standard tools of holography can be used to describe fully non-perturbative microscopic models of cosmology in which a period of accelerated expansion may result from the positive potential energy of time-dependent scalar fields evolving towards a region with negative potential. In these models, the fundamental cosmological constant is negative, and the Universe eventually recollapses in a time-reversal symmetric way. The microscopic description naturally selects a special state for the cosmology. In this framework, physics in the cosmological spacetime is dual to the vacuum physics in a static planar asymptotically AdS Lorentzian wormhole spacetime, in the sense that the background spacetimes and observables are related by analytic continuation. The dual spacetime is weakly curved everywhere, so any cosmological observables can be computed in the dual picture via effective field theory without detailed knowledge of the UV completion or the physics near the big bang. In particular, while inflation may explain the origin of perturbations in the cosmology picture, the perturbations can be deduced from the dual picture without any knowledge of the inflationary potential.

Two circular incompressible liquid inclusions

AbstractWe apply Muskhelishvili's complex variable method to solve the plane strain problem associated with two circular incompressible liquid inclusions embedded in an infinite isotropic elastic matrix subjected to uniform remote in‐plane normal stresses. An analytical solution to the problem is derived primarily with the aid of conformal mapping and analytic continuation. To demonstrate our analytical solution, we present detailed numerical results characterizing the internal uniform hydrostatic stresses within the two liquid inclusions, the nonuniform hoop stresses along the two circular liquid‐solid interfaces on the matrix side and the nonuniform rigid body rotations within the two circular liquid inclusions.

Lee-Yang and Langer edge singularities from analytic continuation of scaling functions

Lee-Yang and Langer edge singularities from analytic continuation of scaling functions

Nuclear induction line shape: Non-Markovian diffusion with boundaries.

The dynamics of viscoelastic fluids are governed by a memory function, essential yet challenging to compute, especially when diffusion faces boundary restrictions. We propose a computational method that captures memory effects by analyzing the time-correlation function of the pressure tensor, a viscosity indicator, through the Stokes-Einstein equation's analytic continuation into the Laplace domain. We integrate this equation with molecular dynamics simulations to derive necessary parameters. Our approach computes nuclear magnetic resonance (NMR) line shapes using a generalized diffusion coefficient, accounting for temperature and confinement geometry. This method directly links the memory function with thermal transport parameters, facilitating accurate NMR signal computation for non-Markovian fluids in confined geometries.

The Singularities of Selberg- and Dotsenko–Fateev-Like Integrals

We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of 2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call “DF-symmetric,” we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.

On Complex Orders in Fractional Calculus: Floors, Ceilings, and Analytic Continuation

Many sources on fractional calculus include the assumption that all fractional orders are real. However, the usual definitions of Riemann–Liouville fractional integrals and derivatives can be used without modification in the case that the fractional orders are complex, and allowing complex orders creates a more rich structure in which the immense power of analytic continuation can be brought to bear to make many results on fractional derivatives trivially provable from the corresponding (easier) results on fractional integrals. This short paper summarises, for the benefit of the fractional community, the facts of using complex orders of differintegration, including the care that must be taken over defining the fractional derivative formulae precisely, and the usefulness of analytic continuation in this context.