To Solve Forward and Backward Nonlocal Wave Problems with Pascal Bases Automatically Satisfying the Specified Conditions

Neural oscillations play a crucial role in cognition, and several studies emphasize the importance of considering them as traveling waves that propagate through brain regions. In scalp EEG, waves typically travel along the anterior–posterior axis: forward waves (occipital-to-frontal) predominate during sensory stimulation, while backward waves (frontal-to-occipital) emerge during rest and top-down modulation. Within the predictive coding framework, backward waves are proposed to convey predictive signals, whereas forward waves reflect sensory processes and feedforward propagation of prediction errors. In this study, we investigated traveling wave dynamics during a visual entrainment task in neurotypical (N=25) and autistic (Autism Spectrum Disorder - ASD) adults (N=24). Our results replicated previous findings in neurotypical participants, as we observed an increase in backward waves during rhythmic visual stimulation, consistent with enhanced top-down predictions. Notably, we observed the opposite pattern in the ASD group, characterized by a pronounced increase in forward waves at the entrained frequency during visual stimulation. These results align with predictive coding accounts of autistic perception, which hypothesize an imbalance between predictions and sensory evidence. Specifically, an increase in forward wave may reflect a bias toward bottom-up sensory signaling over predictive feedback, due to atypical hierarchical communication across brain regions in ASD. Together, our findings shed new light on the oscillatory dynamics involved in visual entrainment in neurotypical adults and provide novel evidence in favor of predictive coding accounts of autistic perception, as well as a consequent bias toward bottom-up sensory signaling over predictive feedback in a context of visual entrainment. Significance statement Traveling waves reflect the spatiotemporal propagation of neural oscillations, providing a window into hierarchical brain communication. By examining traveling wave dynamics during visual entrainment, we show that neurotypical adults exhibit increased backward (frontal-to-occipital) waves, consistent with enhanced top-down predictive signaling. In contrast, adults with Autism Spectrum Disorder (ASD) display a marked increase in forward (occipital-to-frontal) waves, indicating stronger bottom-up sensory drive. These findings provide electrophysiological evidence for atypical predictive processing in ASD, supporting predictive coding theories that propose an imbalance between sensory evidence and prior expectations. Our results highlight traveling waves as a sensitive neural marker of hierarchical signaling and predictive dynamics across typical and atypical perceptual systems.

This article primarily aims at introducing a novel operational calculus of Mikusiński's type for the Dzherbashian‐Nersesian operator. Using this calculus, we are able to derive exact solutions for the forward and backward source problems of a differential equation that features Dzherbashian‐Nersesian operator in time and intertwined with nonlocal boundary conditions. The initial condition is expressed in terms of Riemann‐Liouville integral. Solutions are presented using Mittag‐Leffler (ML) type functions. The outcomes related to the existence and uniqueness subject to certain conditions of regularity on the input data are developed.

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameterδcharacterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part asδ → 0, the proposed Neumann-type boundary formulation recovers the local case asO(δ2) in theL∞(Ω) norm, which is optimal considering theO(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges withO(δ2) convergence.

The agent-based model is the principal scientific instrument of generative social science. Typically, we design completed agents-fully endowed with rules and parameters-to grow macroscopic target patterns from the bottom up. Inverse generative science (iGSS) stands this approach on its head: Rather than handcrafting completed agents to grow a target-the forward problem-we start with the macro-target and evolve micro-agents that generate it, stipulating only primitive agent-rule constituents and permissible combinators. Rather than specific agents as designed inputs, we are interested in agents-indeed, families of agents-as evolved outputs. This is the backward problem and tools from Evolutionary Computing can help us solve it. In this overarching essay of the current JASSS Special Section, Part 1 discusses the motivation for iGSS. Part 2 discusses its goals, as distinct from other approaches. Part 3 discusses how to do it concretely, previewing the five iGSS applications that follow. Part 4 discusses several foundational issues for agent-based modeling and economics. Part 5 proposes a central future application of iGSS: to evolve explicit formal alternatives to the Rational Actor, with Agent_Zero as one possible point of evolutionary departure. Conclusions and future research directions are offered in Part 6. Looking 'backward to the future,' I also include, as Appendices, a pair of 1992 memoranda to the then President of the Santa Fe Institute on the forward (growing artificial societies from the bottom up) and backward (iGSS) problems.

A semiclassical model of nonstationary stimulated Raman scattering with account of forward and backward waves of different frequencies is proposed. Equations for the elements of a collective density matrix and electric-field amplitudes are obtained applying the averaging method. It is shown that forward waves are generated by a united polarization component, whereas each of the backward waves is formed by its own component. The description is based on the approximation of small scattering angles and assumes that the quantities considered do not depend on transverse coordinates. The emission dynamics of forward-backward Stokes pulses is investigated both for the linear case and under the strong depletion of pumping and level populations. The domain of applicability of the model of nonstationary forward stimulated Raman scattering is determined.

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

In this article, we investigate a problem with nonlocal boundary conditions which are perturbations of antiperiodical conditions in bounded $m$-dimensional parallelepiped using Fourier method. We describe properties of a transformation operator $R:L_2(G) \to L_2(G),$ which gives us a connection between selfadjoint operator $L_0$ of the problem with antiperiodical conditions and operator $L$ of perturbation of the nonlocal problem $RL_0=LR.$ Also we construct a commutative group of transformation operators $\Gamma(L_0).$ We show that some abstract nonlocal problem corresponds to any transformation operator $R \in \Gamma(L_0):L_2(G) \to L_2(G)$ and vice versa. We construct a system $V(L)$ of root functions of operator $L,$ which consists of infinite number of adjoint functions. Also we define conditions under which the system $V(L)$ is total and minimal in the space $L_{2}(G),$ and conditions under which it is a Riesz basis in the space $L_{2}(G)$. In case if $V(L)$ is a Riesz basis in the space $L_{2}(G),$ we obtain sufficient conditions under which the nonlocal problem has a unique solution in the form of Fourier series by system $V(L).$

This study proposes the Chebyshev Wavelet Colocation method for solving a class of rth-order Boundary-Value Problems (BVPs) with nonlocal boundary conditions. This method is an extension of the Chebyshev wavelet method to the linear and nonlinear BVPs with a class of nonlocal boundary conditions. In this study, the method is tested on second and fourth-order BVPs and approximate solutions are compared with the existing methods in the literature and analytical solutions. The proposed method has promising results in terms of the accuracy.

Pulsatile components of blood pressure may arise from forward (ventricular generated) or backward wave travel in the arterial tree. The objective of this study was to determine the relative contributions of forward and backward waves to pulsatility. We used wave intensity and wave separation analysis to determine pulsatile components of blood pressure during inotropic and vasopressor stimulation by dobutamine and norepinephrine in normotensive subjects and compared pulse pressure components in hypertensive (mean±SD, 48.8±11.3 years; 165±26.6/99±14.2 mm Hg) and normotensive subjects (52.2±12.6 years; 120±14.2/71±8.2 mm Hg). Dobutamine (7.5 μg/kg per minute) increased the forward compression wave generated by the ventricle and increased pulse pressure from 36.8±3.7 to 59.0±3.4 mm Hg (mean±SE) but had no significant effect on mean arterial pressure or the midsystolic backward compression wave. By contrast, norepinephrine (50 ng/kg per minute) had no significant effect on the forward compression wave but increased the midsystolic backward compression wave. Despite this increase in the backward compression wave, and an increase in mean arterial pressure, norepinephrine increased central pulse pressure less than dobutamine (increases of 22.1±3.8 and 7.2±2.8 mm Hg for dobutamine and norepinephrine, respectively; P<0.02). An elevated forward wave component (mean±SE, 50.4±3.4 versus 35.2±1.8 mm Hg, in hypertensive and normotensive subjects, respectively; P<0.001) accounted for approximately two thirds of the total difference in central pulse pressures between hypertensive and normotensive subjects. Increased central pulse pressure during inotropic stimulation and in essential hypertension results primarily from the forward compression wave.

We investigate layered acoustic metamaterials capable of exhibiting a wide variety of wave propagation phenomena, including backward and forward waves with and without negative refraction. The metamaterials are formed by periodically perforated hard plates, which we describe analytically in the limit of small holes compared to both the period and the separation between plates. In particular, we derive expressions for the index of refraction and the transmission and reflection coefficients of finite slabs. We provide illustrative examples of near fields for all four combinations of backward and forward waves with and without negative refraction, along with dispersion diagrams that explain the observed behavior. A comprehensive study of the range of geometrical parameters in which these distinct phenomena are observed is presented as well. These metamaterials hold great promise for achieving full control of sound using simple structures, with potential application to acoustic technologies relying on subwavelength control and imaging of sound.

The paper deals with the determination of the symmetric model and proper boundary conditions for solving nonlocal elastic symmetric structures. The above concepts, in the context of nonlocal integral elasticity, turn out to be different with respect to the standard ones, classically applied when dealing with local elastic symmetric structures. Indeed, when only a symmetric portion of the structure is analyzed, the nonlocal effects induced by the remaining (cut) portions are lost, this necessitates the consideration of an enlarged symmetric model on which appropriate nonlocal boundary conditions have to be imposed. It has to be pointed out how the width of such an enlarged model depends on the nonlocal material parameters, while the correct unknown nonlocal boundary conditions are here obtained and enforced by an iterative procedure. The accuracy of the proposed approach in solving nonlocal structural symmetric problems is tested with the aid of two numerical examples.

Summary form only given. The maximum achievable gain in individual sections of helix type traveling wave amplifiers is limited by the requirement that the device be stable with respect to the excitation of backward waves, Generally, there is a critical length above which the backward wave is absolutely unstable. Whether this length is exceeded in a specific device depends on a number of details that must be computed numerically. We will present a model, based on the TWT simulation code CHRISTINE, in which BWO stability is determined including a number of important effects. These are: the placement of severs, their reflection and transmission coefficients, the profile of attenuation along the interaction length, the presence of a driven signal, and the coupling of forward and backward waves due to asymmetries in the helix support structure. Specifically our model calculates the gain for the coupled forward and backward waves in a specified frequency band. Asymmetries result in a stop band near the pi point formed by the coupling of the forward and backward waves. Here the beam interacts with a composite mode formed by the fundamental spatial harmonic of the forward wave and first spatial harmonic of the backward wave. These modes will have differing coupling coefficients with the beam which we calculate using an improved solution of the tape-helix dispersion equations.

A new algorithm for direct and backward problems of heat conduction equation

We study a nonlinear fractional boundary value problem (BVP) subject to non-local multipoint boundary conditions. By introducing an appropriate parametrization technique we reduce the original problem to an equivalent one with already two-point restrictions. Using a notion of Chebyshev nodes and Lagrange polynomials we construct a successive iteration scheme, that converges to the exact solution of the non-local problem for particular values of the unknown parameters, which are calculated numerically.&#x0D; For mote information see https://ejde.math.txstate.edu/Volumes/2023/58/abstr.html&#x0D;

A novel nonreciprocal bandstop filter (NBF) using metamaterial Thue-Morse multiferroic superlattices (MTMMS) is presented, where the multiferroic domains with polarization and magnetization simultaneous are arranged along x-axis according to the iterative rule of the Thue-Morse sequence. The dispersion relation is obtained by the transfer matrix method and indicates that the band gaps for the forward wave (FW) and the backward wave (BW) are not symmetrical. Transmission spectrum is utilized to illustrate the band gaps clearly. Two band gaps in the frequency ranges of ω = 20.9 – 23.03 GHz and ω = 63.98 – 64.41GHz can be observed in the MTMMS with the sequence of l=2 for the FW. However, the band gaps for the BW appear in the frequency ranges of ω = 19.49 – 21.92 GHz and ω = 63.12 – 63.98 GHz. The maximum bandwidth of 2.1GHz and an obvious nonreciprocity with respect to the FW and BW can be obtained in this case. The S-parameters can verify this property further. In addition, the NBSF is easy to integrate into practical systems since its size of forming stable band gaps is about one centimeter.