Discrete Integration Research Articles

The discrete fractional Karamata theorem and its applications

Abstract We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

Integrable Discretization and Multi-soliton Solutions of Negative Order AKNS Equation

Integrable Discretization and Multi-soliton Solutions of Negative Order AKNS Equation

Inverse scattering transform for continuous and discrete space‐time‐shifted integrable equations

AbstractNonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space‐time‐shifted nonlinear Schrödinger (NLS) and space‐time‐shifted modified Korteweg‐de Vries equations are studied. Additionally, the semidiscrete IST is developed for the time, space, and space‐time‐shifted variants of the Ablowitz–Ladik integrable discretization of the NLS. One‐soliton solutions are constructed for all continuous and discrete cases.

Decomposing Imaginary-Time Feynman Diagrams Using Separable Basis Functions: Anderson Impurity Model Strong-Coupling Expansion

We present a deterministic algorithm for the efficient evaluation of imaginary-time diagrams based on the recently introduced discrete Lehmann representation (DLR) of imaginary-time Green’s functions. In addition to the efficient discretization of diagrammatic integrals afforded by its approximation properties, the DLR basis is separable in imaginary-time, allowing us to decompose diagrams into linear combinations of nested sequences of one-dimensional products and convolutions. Focusing on the strong-coupling bold-line expansion of generalized Anderson impurity models, we show that our strategy reduces the computational complexity of evaluating an Mth-order diagram at inverse temperature β and spectral width ωmax from O((βωmax)2M−1) for a direct quadrature to O(M(log(βωmax))M+1), with controllable high-order accuracy. We benchmark our algorithm using third-order expansions for multiband impurity problems with off-diagonal hybridization and spin-orbit coupling, presenting comparisons with exact diagonalization and quantum Monte Carlo approaches. In particular, we perform a self-consistent dynamical mean-field theory calculation for a three-band Hubbard model with strong spin-orbit coupling representing a minimal model of Ca2RuO4, demonstrating the promise of the method for modeling realistic strongly correlated multiband materials. For both strong and weak coupling expansions of low and intermediate order, in which diagrams can be enumerated, our method provides an efficient, straightforward, and robust blackbox evaluation procedure. In this sense, it fills a gap between diagrammatic approximations of the lowest order, which are simple and inexpensive but inaccurate, and those based on Monte Carlo sampling of high-order diagrams. Published by the American Physical Society 2024

Exact solutions to SIR epidemic models via integrable discretization

An integrable discretization of the SIR model with vaccination is proposed. Through the discretization, the conserved quantities of the continuous model are inherited to the discrete model, since the discretization is based on the intersection structure of the non-algebraic invariant curve defined by the conserved quantities. Uniqueness of the forward/backward evolution of the discrete model is demonstrated in terms of the single-valuedness of the Lambert W function on the positive real axis. Furthermore, the exact solution to the continuous SIR model with vaccination is constructed via the integrable discretization. When applied to the original SIR model, the discretization procedure leads to two kinds of integrable discretization, and the exact solution to the continuous SIR model is also deduced. It is furthermore shown that the discrete SIR model geometrically linearizes the time evolution by using the non-autonomous parallel translation of the line intersecting the invariant curve.

BPS Spectra and Algebraic Solutions of Discrete Integrable Systems

This paper extends the correspondence between discrete Cluster Integrable Systems and BPS spectra of five-dimensional N=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {N}}=1$$\\end{document} QFTs on R4×S1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^4\ imes S^1$$\\end{document} by proving that algebraic solutions of the integrable systems are exact solutions for the system of TBA equations arising from the BPS spectral problem. This statement is exemplified in the case of M-theory compactifications on local del Pezzo Calabi–Yau threefolds, corresponding to q-Painlevé equations and SU(2) gauge theories with matter. A degeneration scheme is introduced, allowing to obtain closed-form expression for the BPS spectrum also in systems without algebraic solutions. By studying the example of local del Pezzo 3, it is shown that when the region in moduli space associated to an algebraic solution is a “wall of marginal stability”, the BPS spectrum contains states of arbitrarily high spin, and corresponds to a 5d uplift of a four-dimensional nonlagrangian theory.

A new computational framework for log-concave density estimation

In statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order 1/T up to logarithmic factors on the objective function scale, where T denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository LogConcComp (Log-Concave Computation).

A relation algebraic approach to universal integrals

In a series of papers Klement et al. investigated discrete integrals such as the Choquet and Sugeno integral and their axiomatization. As part of their study they showed that universal integrals are based on semicopulas, and they provided lower and upper bounds of the integral operations based on a given semicopula. These real-valued resp. unit interval valued integrals can be considered as proper aggregation tool in the context of fuzzy sets. The aim of the current paper is to generalize this approach to so-called L-fuzzy sets and relations, i.e., fuzzy sets and relations that use an arbitrary Heyting algebra L as membership degree instead of the unit interval. Furthermore, we present the theory within arrow categories, i.e., we abstract from concrete sets and relations and work within a suitable algebraic framework. The current paper also shows that the results of the previous work can be proven without referring to the real numbers and specific measures such as the Lebesque measure and the induced measurable spaces.

Analysis of the behavior of a mobile robot with precise and fuzzy pid controller in an uncertain environment

The aim of this study was to develop methods for building intelligent systems that ensure safe movement on a planned and specified trajectory in an environment with unknown obstacles for planning the movement of a mobile robot. To achieve this goal, modern concepts and methods of planning systems development were analyzed. By moving a mobile robot in an unknown environment, it is proposed to develop an intelligent system for planning its modeling in an uncertain static environment, to realize an intelligent system for planning the movement of a mobile robot in an unknown dynamic environment and modeling the resulting system. As a result, analytical studies were conducted on a computer using mathematical modeling, analytical geometry, kinematic and dynamic analysis, fuzzy logic, neural networks, robotics, mechatronics, discrete integration and applied programming methods, and experimental studies were conducted using a mobile robot model.

A Generalized Sine-Gordon Equation: Reductions and Integrable Discretizations

A Generalized Sine-Gordon Equation: Reductions and Integrable Discretizations

Gas mass transfer characteristics in shales: Insights from multiple flow mechanisms, effective viscosity, and poromechanics

Accurate characterization of gas transport behavior is indispensable for successful shale gas reservoir development. In this study, a fractal-theory-based apparent permeability (AP) model is developed to better characterize real gas mass transfer in shales. The proposed model involves multi-scale organic matter (OM) and inorganic matter (IOM) flow channels and incorporates poromechanics-related dynamic pore size, multiple gas transport mechanisms, and dynamic viscosity. The discrete integration method (DIM) is employed to address viscosity changes during fractal scaling up. Experimental data, molecular dynamics simulations, and published theoretical models were used to verify the proposed model and reasonable agreements have been achieved. Results indicate that as the pore pressure and size increase, dominant mechanisms for gas transport change from surface diffusion at low pressure (lower than switch pressure) to viscous flow at higher pressure and larger pore sizes. Furthermore, higher organic matter content makes the permeability curve more similar to OM type. Ignoring effective viscosity results in underestimation of permeability, which is more noticeable when pressure rises and pore shrinks. Poromechanical properties primarily affect pore size, larger elastic modulus and smaller Poisson's ratio leads to significantly lower permeability. The real gas effect on AP reduction cannot be ignored, especially at high pressure. This work provides insights into the gas transport characteristics in shales, which have practical implications for shale gas reservoir simulation and development.

An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions

An integral discretization scheme on a graded mesh for a fractional differential equation with integral boundary conditions

Weak noise theory of the O'Connell-Yor polymer as an integrable discretization of the nonlinear Schrödinger equation.

We investigate and solve the weak noise theory for the semidiscrete O'Connell-Yor directed polymer. In the large deviation regime, the most probable evolution of the partition function obeys a classical nonlinear system which is a nonstandard discretization of the nonlinear Schrödinger equationwith mixed initial-final conditions. We show that this system is integrable and find its general solution through an inverse scattering method and a non-standard Fredholm determinant framework that we develop. This allows us to obtain the large deviation rate function of the free energy of the polymer model from its conserved quantities and to study its convergence to the large deviations of the Kardar-Parisi-Zhang equation. Our model also degenerates to the classical Toda chain, which further substantiates the applicability of our Fredholm framework.

Product placement in cinematography as a tool of marketing communication

Product placement in cinematography is an effective marketing tactic, consisting of the discrete integration of products or services into films. This strategy gives brands significant exposure to the general public without seeming intrusive. By subtly placing products in scenes, movies can influence consumer preferences and opinions. This marketing communication tool has grown in popularity over the decades, and brands are willing to invest significant sums to see their products featured in impactful films. However, the success of product placement depends on the context of the film and the authenticity of the integration of the products into the story.

Topological error correcting processes from fixed-point path integrals

We propose a unifying paradigm for analyzing and constructing topological quantum error correcting codes as dynamical circuits of geometrically local channels and measurements. To this end, we relate such circuits to discrete fixed-point path integrals in Euclidean spacetime, which describe the underlying topological order: If we fix a history of measurement outcomes, we obtain a fixed-point path integral carrying a pattern of topological defects. As an example, we show that the stabilizer toric code, subsystem toric code, and CSS Floquet code can be viewed as one and the same code on different spacetime lattices, and the honeycomb Floquet code is equivalent to the CSS Floquet code under a change of basis. We also use our formalism to derive two new error-correcting codes, namely a Floquet version of the 3+1-dimensional toric code using only 2-body measurements, as well as a dynamic code based on the double-semion string-net path integral.

A Memory-Efficient Neural Ordinary Differential Equation Framework Based on High-Level Adjoint Differentiation

Neural ordinary differential equations (neural ODEs) have emerged as a novel network architecture that bridges dynamical systems and deep learning. However, the gradient obtained with the continuous adjoint method in the vanilla neural ODE is not reverse-accurate. Other approaches suffer either from an excessive memory requirement due to deep computational graphs or from limited choices for the time integration scheme, hampering their application to large-scale complex dynamical systems. To achieve accurate gradients without compromising memory efficiency and flexibility, we present a new neural ODE framework, PNODE, based on high-level discrete adjoint algorithmic differentiation. By leveraging discrete adjoint time integrators and advanced checkpointing strategies tailored for these integrators, PNODE can provide a balance between memory and computational costs, while computing the gradients consistently and accurately. We provide an open-source implementation based on PyTorch and PETSc, one of the most commonly used portable, scalable scientific computing libraries. We demonstrate the performance through extensive numerical experiments on image classification and continuous normalizing flow problems. We show that PNODE achieves the highest memory efficiency when compared with other reverse-accurate methods. On the image classification problems, PNODE is up to two times faster than the vanilla neural ODE and up to 2.3 times faster than the best existing reverse-accurate method. We also show that PNODE enables the use of the implicit time integration methods that are needed for stiff dynamical systems.

Design and simulation of measuring model for horizontal flight distance of projectile based on discrete acceleration

The ability of the projectile to strike accurately in the air depends on the accurate measurement of its flight distance. Due to the influence of the external environment, it is very difficult to measure the distance of high-speed projectiles in real time with ranging equipment such as radar and laser. Based on this, a real-time measurement model of the horizontal flight distance of the projectile was designed, which combines trajectory equation and discrete acceleration. First, the angle of inclination of the projectile at each position was calculated according to the six-degree-of-freedom rigid outer trajectory model. Then, two kinds of distance measurement models based on discrete acceleration integration were introduced, and the accuracy simulation was carried out. Simulation results show that the Simpson integral method is more accurate. The flight arc length of the projectile was calculated by the Simpson integral method. Finally, the horizontal flight distance of the projectile was obtained by combining the flight arc length and inclination angle of the projectile. Finally, the flight distance measurement model was simulated, and the simulation results show that the model has high accuracy. The simulation results show that the horizontal flight distance of the projectile can be measured by the trajectory equation and discrete acceleration according to the set conditions. This method can be used to measure the distance of the projectile by carrying the sensor and is less affected by the external environment. It can provide a theoretical basis and reference for the measurement of the horizontal distance of the projectile.

B\"{a}cklund transformations as integrable discretization. The geometric approach

We present interpretation of known results in the theory of discrete asymptotic and discrete conjugate nets from the "discretization by B\"{a}cklund transformations" point of view. We collect both classical formulas of XIXth century differential geometry of surfaces and their transformations, and more recent results from geometric theory of integrable discrete equations. We first present transformations of hyperbolic surfaces within the context of the Moutard equation and Weingarten congruences. The permutability property of the transformations provides a way to construct integrable discrete analogs of the asymptotic nets for such surfaces. Then after presenting the theory of conjugate nets and their transformations we apply the principle that B\"{a}cklund transformations provide integrable discretization to obtain known results on the discrete conjugate nets. The same approach gives, via the Ribaucour transformations, discrete integrable analogs of orthogonal conjugate nets.

Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

In this paper, we give a procedure for discretizing recursion operators by utilizing unified bilinear forms within integrable hierarchies. To illustrate this approach, we present unified bilinear forms for both the AKNS hierarchy and the KdV hierarchy, derived from their respective recursion operators. Leveraging the inherent connection between soliton equations and their auto-B\"acklund transformations, we discretize the bilinear integrable hierarchies and derive discrete recursion operators. These discrete recursion operators exhibit convergence towards the original continuous forms when subjected to a standard limiting process.

The effect of loss/gain and Hamiltonian perturbations of the Ablowitz—Ladik lattice on the recurrence of periodic anomalous waves

Using the finite gap method, in this paper we extend the recently developed perturbation theory for anomalous waves (AWs) of the periodic nonlinear Schrödinger (NLS) type equations to lattice equations, using as basic model the Ablowitz–Ladik (AL) lattices, integrable discretizations of the focusing and defocusing NLS equations. We study the effect of physically relevant perturbations of the AL equations, like linear loss, gain, and/or Hamiltonian corrections, on the AW recurrence, in the simplest case of one unstable mode. We show that these small perturbations induce O(1) effects on the periodic AW dynamics, generating three distinguished asymptotic patterns. Since dissipation and higher order Hamiltonian corrections can hardly be avoided in natural phenomena involving AWs, we expect that the asymptotic states described analytically in this paper will play a basic role in the theory of periodic AWs in natural phenomena described by discrete systems. The quantitative agreement between the analytic formulas of this paper and numerical experiments is excellent.