Finite Interval Research Articles

Schmidt’s game and nondense orbits of a class of interval maps

Let T be a piecewise C 1 + δ expanding interval map with finite branches on a finite interval X. For any ξ ∈ X , we consider the set BAD ( ξ ) of x whose forward orbits are bounded away from ξ, i.e. BAD ( ξ ) = { x ∈ X : inf i ≥ 0 | T i ( x ) − ξ | > 0 } . We prove that the set BAD ( ξ ) is 1/2-strong winning for any ξ ∈ X in the sense of Schmidt’s game. Our result extends that of Färm et al. [Dimension of countable intersections of some sets arising in non-integer bases, Fund. Math. 209(2) (2010), pp. 157–176.] for beta-transformations with simpler proofs.

Monotonic Switching Iterative Learning Control Method for a Class of Discrete Time Switched System

This paper deal with the problem of monotonic tracking convergence error (MC) of discrete time switched system. We begin our review by proposing the considered switched systems are operated during a finite time interval repetitively, and then the iterative learning control (ILC) scheme can be introduced between subsystems. Namely, the tracking error converges with non-zero constant initial error. After the switched system is transformed into a 2D switched system, sufficient conditions in terms of linear matrix inequalities (LMIs) are derived by using the 2 l-norm. It is shown that the tracking error converges monotonically to zero in the sense of 2 l-norm. A numerical simulation example is established shown the effectiveness of the proposed method.

ON THE APPROXIMATION OF A BOUNDARY VALUE PROBLEM FOR DELAY INTEGRO-DIFFERENTIAL EQUATIONS

A linear two-point boundary value problem for a system of integro-differential equations with a constant delay argument is investigated on a finite interval. By dividing the interval by parts, the integral term of the integrodifferential equation with constant delay argument is replaced by the quadrature formula. With this replacement, the linear two-point boundary value problem for a system of integro-differential equations with a constant delay argument is approximated by the linear boundary value problem for a system of loaded differential equations with a constant delay argument. Definitions of correct solvability of boundary value problem for system of integro-differential equations with delay argument and constructed boundary value problem for system of loaded differential equations with constant delay argument are introduced. Conditions of correct solvability of linear boundary value problem for system of integro-differential equations with delay argument and linear boundary value problem for system of loaded differential equations with delay argument are established. Relationship between correct solvabilities of linear two-point boundary value problem for system of integro-differential equations with constant delay argument and approximating linear two-point boundary value problem for system of loaded differential equations with constant delay argument is shown.

Time-dependent accelerated Unruh-DeWitt detector without event horizon

Abstract We investigate unitarily inequivalent representations of the algebra of operators in quantum field theory in the cases where there is a Fock representation of the commutation relations. We examine more closely the operational definition of a measurement device, discussing the Unruh-DeWitt and Glauber models of quantum detectors. The transition probability per unit of proper time of both detectors in different non-inertial frames of reference in Minkowski spacetime is evaluated. We first study detectors traveling in a stationary worldline with a constant proper acceleration, i.e., a hyperbolic motion, interacting with the field prepared in the Poincaré invariant Fock vacuum state. Next, we study the Unruh-DeWitt detector at rest in a non-uniformly accelerated frame, with a time dependent acceleration interacting with the field in the Poincar'e invariant Fock vacuum state. We evaluate the positive frequency Wightman function for the non-uniformly accelerated frame in a finite time interval and find that it is similar to the two-point correlation function of a system in equilibrium with a thermal bath. Calculating the transition rate, the non-uniformly accelerated Unruh-DeWitt detector can be excited even if the scalar field is prepared in the vacuum state.

A fast convolution method for the fractional Laplacian in R

In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on R, as well as some more general singular integrals. After mapping R into a finite interval, we discretize the integral operator using a modified midpoint rule. The result of this procedure can be cast as a discrete convolution, which can be evaluated efficiently using the Fast-Fourier Transform (FFT). The method provides an efficient, second-order accurate approximation to the fractional Laplacian, without the need to truncate the domain. We first prove that the method gives a second-order approximation for the fractional Laplacian and other related singular integrals; then, we detail the implementation of the method using a fast convolution algorithm, and give numerical examples that support its efficacy and efficiency; finally, as an example of its applicability to an evolution problem, we employ the method for the discretization of the nonlocal part of the one-dimensional cubic fractional Schrödinger equation in the focusing case.

Sparse Representation of Random Field Via Probability Chaos

ABSTRACTThe main purpose of the article is to develop adaptive sparse representation theory by functions of a fixed random variable. The results are divided into two parts. One is a generalization of completeness of orthogonal polynomials in all cases where the set of monomials functions, , is contained in for a PDF supported in a compact interval . What we offer is unified treatment for all PDFs having compact support that include, and extend the classical finite interval case to its widest possibility. The second part introduces a new sparse representation methodology for random variables of second order contained in the Bochner type spaces with systems of functions of any fixed random variable of second moment.

Diffusion-mediated adsorption versus absorption at partially reactive targets: a renewal approach

Abstract Renewal theory is finding increasing applications in non-equilibrium statistical physics. One example relates the probability density and survival probability of a Brownian particle or an active run-and-tumble particle with stochastic resetting to the corresponding quantities without resetting. A second example is so-called snapping out Brownian motion, which sews together diffusions on either side of an impermeable interface to obtain the corresponding stochastic dynamics across a semi-permeable interface. A third example relates diffusion-mediated surface adsorption–desorption (reversible adsorption) to the case of irreversible adsorption. In this paper we apply renewal theory to diffusion-mediated adsorption processes in which an adsorbed particle may be permanently removed (absorbed) prior to desorption. We construct a pair of renewal equations that relate the probability density and first passage time (FPT) density for absorption to the corresponding quantities for irreversible adsorption. We first consider the example of diffusion in a finite interval with a partially reactive target at one end. We use the renewal equations together with an encounter-based formalism to explore the effects of non-Markovian adsorption/desorption on the moments and long-time behaviour of the FPT density for absorption. We then analyse the corresponding renewal equations for a partially reactive semi-infinite trap and show how the solutions can be expressed in terms of a Neumann series expansion. Finally, we construct higher-dimensional versions of the renewal equations and derive general expression for the FPT density using spectral decompositions.

Integral Representation of Solutions to Initial‐Boundary Value Problems on a Finite Interval in the Framework of the Hyperbolic Heat Equation

ABSTRACTWe consider initial‐boundary value problems on a finite interval for the system of the energy balance equation and modified Fourier's law (constitutive equation, commonly called the Cattaneo equation) describing temperature (or internal energy) and heat flux. This hyperbolic system of first‐order partial differential equations is the simplest model of non‐Fourier heat conduction. Boundary conditions include various models of behavior of a physical system at the boundaries, including boundary conditions describing Newton's law, which states that the heat flux at the boundary is directly proportional to the difference in the temperature of the physical system and ambient temperature. In this case, the boundary conditions express the relationship of unknown functions (temperature or internal energy and heat flux) with each other. These problems cannot be solved by the Fourier method and the Laplace transform. To solve the problems, we apply the Fokas unified transform method. To illustrate the final formula, we consider numerical examples for special cases in which heat exchange at one of the boundaries obeys Newton's law.

Inverse random scattering for the one-dimensional Helmholtz equation

Abstract This paper concerns the inverse random potential scattering problem for the Helmholtz equation in one dimension. The random potential is assumed to be a generalized microlocally isotropic Gaussian random field, whose covariance is a classical pseudodifferential operator. To investigate the inverse problem, as an effective mathematical tool, the scattering theory is employed to obtain an analytic domain and estimates for the resolvent of the elliptic operator with rough potentials. For the inverse problem, based on the resolvent estimates, we show that the principal symbol of the covariance operator can be reconstructed by a single realization of the boundary data averaged over the high-frequency band almost surely. Consequently, by analyticity of the resolvent, the uniqueness of the inverse problem can be achieved by data at multiple frequencies only in a finite interval. The method developed here is unified and can be applied to other stochastic inverse scattering problems in higher dimensions for various wave equations by boundary measurements.

Pinching Sudakov

In this paper, we discuss the factorization of the Sudakov form factor on the Coulomb branch of maximally supersymmetric Yang-Mills theory in the near mass-shell limit. We unravel all pinch singularities of this observable making use of the Method of Regions. We find their operator content in terms of matrix elements of Wilson lines on semi-infinite and finite intervals for the jet and ultrasoft functions, respectively. However, naive factorization into these incoherent momentum components is broken at two-loop order by effects subleading in the parameter of dimensional regularization. To save the day, we perform an appropriate twisting of the functions involved as well as simultaneous finite scheme transformation of the ’t Hooft coupling. Infrared physics of twisted jet and ultrasoft functions is governed by the octagon anomalous dimension, while the untwisted ultrasoft function possesses infrared evolution driven by an anomalous dimension different from the ubiquitous cusp.

Uncertainty of Tipping-Bucket Data May Hamper Detection and Analysis of Secular Changes in Short-Term Rainfall Rates

Exploring the secular tendency to intensification of short-interval rainfall intensities, such as those associated with convective storms, requires rainfall data having sufficient accuracy and temporal resolution. Light rainfalls also exhibit secular change, and documenting these imposes considerable demands on data quality. Tipping-bucket rain gauges are the most widely deployed globally for data collection, but they cannot record rainfall amount or rainfall rate instantaneously. Both require data to be collected over some finite time interval, the accumulation time (AT), during which one or more buckets must fill and tip. Relatively short ATs, such as when analysing 15 min rainfall amounts and rates, are associated with increased uncertainty in TBRG data. Quantifying the resulting uncertainty forms the subject of the present work. Worst-case rainfall depth and rainfall rate errors that would arise in TBRG data for constant rainfall rates of 5, 10, 20, 30, 40, and 50 mm h−1 are determined for ATs from 5 min to 50 min. Errors frequently considerably exceed the 1–2% accuracy levels claimed by many manufacturers of TBRGs. The errors found pose challenges for the detection of secular change in rainfall rates. The present results point to the need for fuller analysis of errors in TBRG data for short-duration rainfalls and for gauge specifications to specify uncertainty separately for rainfall depth and rainfall rate.

Eigenvalue bounds for perturbed periodic Dirac operators

We characterise regions in the complex plane that contain all non-embedded eigenvalues of a perturbed periodic Dirac operator on the real line with real-valued periodic potential and a generally non-symmetric matrix-valued perturbation V . We show that the eigenvalues are located close to the end-points of the spectral bands for small V\in L^{1}(\mathbb{R})^{2\times 2} , but only close to the spectral bands as a whole for small V\in L^{p}(\mathbb{R})^{2\times 2} , p > 1 . As auxiliary results, we prove the relative compactness of matrix multiplication operators in L^{2p}(\mathbb{R})^{2\times 2} with respect to the periodic operator under minimal hypotheses, and find the asymptotic solution of the Dirac equation on a finite interval for spectral parameters with large imaginary part.

A model for cancer chemo-immunotherapy treatment schedules

This work constructs a generic model for chemo-immunotherapy of cancer in a healthy tissue. The main interests are the modeling and computer study of chemotherapy, immunotherapy and the advantages in their combined applications. It describes the dynamics of Healthy, Immune, and Cancer cells when chemotherapy and immunotherapy are applied, either separately or combined. The analysis of the model shows that its solutions exist, are bounded and nonnegative on each finite time interval, and thus are biologically feasible. The model simulations describe the development of the disease without intervention, when only chemotherapy or immunotherapy are administered periodically, when the two modalities are combined. In this manner, once validated, the model can be used to design treatment schedules for improved outcomes.

Impact of magnetic field and radiation on Casson fluid flow and heat transfer over a 3D exponentially expanding sheet using the Bernoulli wavelet method

This study examines the impact of a magnetic field and thermal radiation on the flow, heat transfer and mass transfer of an incompressible Casson fluid over a three-dimensional exponentially stretching sheet. The governing equations, including the continuity, momentum, energy and concentration equations, are formulated while incorporating the effects of the applied magnetic field, radiative heat transfer and the Casson fluid model. To facilitate numerical analysis, similarity transformations are employed to non-dimensionalize the system. A coordinate transformation is introduced to convert the semi-infinite domain [ 0 , ∞ ) into a finite domain [0, 1] , aligning with the applicability of the numerical wavelet method in a finite interval. The transformed equations are then solved numerically using the Bernoulli wavelet method, offering an efficient and accurate approach to addressing complex boundary layer problems. The validity of the results is confirmed through comparison with existing literature, demonstrating strong agreement. The effects of key parameters, including the Casson fluid parameter, magnetic field strength, radiation parameter and stretching rate, are analysed in terms of velocity, temperature and concentration distributions. The findings reveal that an increasing magnetic field suppresses velocity due to the Lorentz force, while thermal radiation enhances temperature distribution. Moreover, mass diffusion plays a crucial role in shaping concentration profiles. This study provides valuable insights into non-Newtonian fluid behaviour and highlights the effectiveness of wavelet-based numerical techniques in solving boundary layer problems.

Antiferromagnetic Covariance Structure of Coulomb Chain

We consider a system of particles lined up on a finite interval in a 3-dimensional space with Coulomb interactions between the nearest and next to the nearest neighbours. This model was introduced by Malyshev (Probl Inf Transm 51(1):31–36, 2015) to study the flow of charged particles. The distribution of spacings between the consecutive particles is of interest. Notably, even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here, assuming zero external force, we show that interactions beyond the nearest ones lead to qualitatively new features of the system. In particular, the order of decay (in terms of the total number of particles) of covariances between the spacings is changed when compared with the former nearest-neighbours case. Furthermore, we discover that the covariances between spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of the number of spacings between them, while their amplitude decays. In the course of the proof of these results, a conditional Central Limit Theorem for dependent random variables is established.

Estimation of anisotropic viscosities for the stochastic primitive equations

Abstract The viscosity parameters play a fundamental role in applications involving stochastic primitive equations (SPE), such as accurate weather predictions, climate modeling, and ocean current simulations. In this paper, we develop several novel estimators for the anisotropic viscosities in the SPE, using a finite number of Fourier modes of a single sample path observed within a finite time interval. The focus is on analyzing the consistency and asymptotic normality of these estimators. We consider a torus domain and treat strong, pathwise solutions in the presence of additive white noise (in time). Notably, the analysis for estimating horizontal and vertical viscosities differs due to the unique structure of the SPE and the fact that both parameters of interest are adjacent to the highest-order derivative. To the best of our knowledge, this is the first work addressing the estimation of anisotropic viscosities, with the potential applicability of the developed methodology to other models.

Regional fractional zone moving control for linear hyperbolic systems: Numerical approximation and optimization

SummaryThis work focuses on the exact fractional controllability of linear hyperbolic systems over a finite time interval. The control is applied to the internal moving zone of a predefined actuator location in the spatial domain of the system. We use the Hilbert uniqueness method (HUM) to characterize the minimum‐energy control that enables us to achieve the desired fractional state at time . We therefore consider a numerical approach that allows us to determine the explicit formula for optimal control. We also verify the results' applicability using computer simulations for the one‐dimensional wave equation with a moving zone actuator.

Toward quantum tunneling from excited states: Recovering imaginary-time instantons from a real-time analysis

We revisit the path integral description of quantum tunneling and its generalization to excited states. For clarity, we focus on the simple toy model of a point particle in a double-well potential, for which we perform all steps explicitly. Instead of performing the familiar Wick rotation from physical to imaginary time—which is inconsistent with the requisite boundary conditions when treating tunneling from excited states—we regularize the path integral by adding an infinitesimal complex contribution to the Hamiltonian, while keeping time strictly real. We find that this gives rise to a complex stationary-phase solution, in agreement with recent insights from Picard-Lefshetz theory. We then show that there exists a class of analytic solutions for the corresponding equations of motion, which can be made to match the appropriate boundary conditions in the physically relevant limits of a vanishing regulator and an infinite physical time. We provide a detailed discussion of this nontrivial limit. We find that, for systems without an explicit time-dependence, our approach reproduces the picture of an instantonlike solution defined on a finite Euclidean-time interval. Lastly, we discuss the generalization of our approach to broader classes of systems, for which it serves as a reliable framework for high-precision calculations. Published by the American Physical Society 2025

Relative Controllability and Finite Interval Stability of Impulsive Fractional Order Switched Delay Differential System with Nonlinear Perturbation

Relative Controllability and Finite Interval Stability of Impulsive Fractional Order Switched Delay Differential System with Nonlinear Perturbation

Diffuse interface model for two-phase flows on evolving surfaces with different densities: global well-posedness

We show global in time existence and uniqueness on any finite time interval of strong solutions to a Navier–Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.