Search Results (218)

In this article, the mechanism of relaxation polarization currents occurring at a constant temperature (isothermal process) in crystals with ionic-molecular chemical bonds (CIMBs) in an alternating electric field was investigated. Methods of the quasi-classical kinetic theory of dielectric relaxation, based on solutions of the nonlinear system of Fokker–Planck and Poisson equations (for the blocking electrode model) and perturbation theory (by expanding into an infinite series in powers of a dimensionless small parameter) were used. Generalized nonlinear mathematical expressions for calculating the complex amplitudes of relaxation modes of the volume-charge distribution of the main charge carriers (ions, protons, water molecules, etc.) were obtained. On this basis, formulas for the current density of relaxation polarization (for transient processes in a dielectric) in the k-th approximation of perturbation theory were constructed. The isothermal polarization currents are investigated in detail in the first four approximations (k = 1, 2, 3, 4) of perturbation theory. These expressions will be applied in the future to compare the results of theory and experiment, in analytical studies of the kinetics of isothermal ion-relaxation (in crystals with hydrogen bonds (HBC), proton-relaxation) polarization and in calculating the parameters of relaxers (molecular characteristics of charge carriers and crystal lattice parameters) in a wide range of field parameters (0.1–1000 MV/m) and temperatures (1–1550 K). Asymptotic (far from transient processes) recurrent formulas are constructed for complex amplitudes of relaxation modes and for the polarization current density in an arbitrary approximation k of perturbation theory with a multiplicity r by the polarizing field (a multiple of the fundamental frequency of the field). The high degree of reliability of the theoretical results obtained is justified by the complete agreement of the equations of the mathematical model for transient and stationary processes in the system with a harmonic external disturbance. This work is of a theoretical nature and is focused on the construction and analysis of nonlinear properties of a physical and mathematical model of isothermal ion-relaxation polarization in CIMB crystals under various parameters of electrical and temperature effects. The theoretical foundations for research (construction of equations and working formulas, algorithms, and computer programs for numerical calculations) of nonlinear kinetic phenomena during thermally stimulated relaxation polarization have been laid. This allows, with a higher degree of resolution of measuring instruments, to reveal the physical mechanisms of dielectric relaxation and conductivity and to calculate the parameters of a wide class of relaxators in dielectrics in a wide experimental temperature range (25–550 K). Full article

This paper investigates the stability of continued fractions with complex partial denominators and numerators equal to one. Such fractions are an important tool for function approximation and have wide application in physics, engineering, and mathematics. A formula is derived for the relative error of the approximant of a continued fraction, which depends on both the relative errors of the fraction’s elements and the elements themselves. Based on this formula, using the methodology of element sets and their corresponding value sets, conditions are established under which the approximants of continued fractions are stable to perturbations of their elements. Stability sets are constructed, which are sets of admissible values for the fraction’s elements that guarantee bounded errors in the approximants. For each of these sets, an estimate of the relative error that arises from the perturbation of the continued fraction’s elements is obtained. The results are illustrated with an example of a continued fraction that is an expansion of the ratio of Bessel functions of the first kind. A numerical experiment is conducted, comparing two methods for calculating the approximants of a continued fraction: the backward and forward algorithms. The computational stability of the backward algorithm is demonstrated, which corresponds to the theoretical research results. The errors in calculating approximants with this algorithm are close to the unit round-off, regardless of the order of approximation, which demonstrates the advantages of continued fractions in high-precision computation tasks. Another example is a comparative analysis of the accuracy and stability to perturbations of second-order polynomial model and so-called second-order continued fraction model in the problem of wood drying modeling. Experimental studies have shown that the continued fraction model shows better results both in terms of approximation accuracy and stability to perturbations, which makes it more suitable for modeling processes with pronounced asymptotic behavior. Full article

This paper studies $(p,q)$-harmonic maps by unified geometric analytic methods. First, we deduce variation formulas of the $(p,q)$-energy functional. Second, we analyze weakly conformal and horizontally conformal $(p,q)$-harmonic maps and prove Liouville results for $(p,q)$-harmonic maps under Hessian and asymptotic conditions on complete Riemannian manifolds. Finally, we define the $(p,q)$-$SSU$ manifold and prove that non-constant stable $(p,q)$-harmonic maps do not exist. Full article

The continuous-wave (CW) illuminator, whose fundamentals are related to the theoretical understanding of loop antennas loaded with ferrite materials, is a device which plays an important role in electromagnetic pulse (EMP) susceptibility assessment. However, existing theoretical formulas do not consider cases where ferrite materials are loaded into the loop antenna. This paper provides a new explicit theoretical formula for the impedance of a circular loop antenna loaded with ferrite materials for CW illuminator design, and explores the variation regularity of its input impedance. Loading ferrite materials affects the internal impedance of the loop antenna and forces some modifications to the classical calculation procedure, resulting in an asymptotic numerical calculation method and a closed-form solution. The full-wave simulation results from CST Studio Suite show a maximum error of less than 0.99%, compared to the classical theory. With ferrite material loaded, the input impedance of the loop antenna is significantly reduced and smoothed in a wide range of normalized radii. For a loop antenna with a fixed circumference, the input impedance indicates that the Q-factor decreases as the thickness of the ferrite material increases. Conversely, for a ferrite-loaded loop antenna with a constant material thickness, a larger loop circumference results in a higher Q-factor. In summary, this study provides a fast and accurate computational method for the input impedance design of CW illuminators, while also offering an effective tool for further research on the performance of ferrite-loaded loop antennas. Full article

The associations between sialylated anti-cyclic citrullinated peptide (anti-CCP) antibodies bearing α-2,6-sialic acid (SIA), ST6Gal1 and Neu1 enzymes, and clinical disease activity measures such as disease activity score 28 (DAS28), the Simplified Disease Activity Index (SDAI), and Clinical Disease Activity Index (CDAI) are unknown in rheumatoid arthritis (RA). To address this gap, this study included 97 patients with RA evaluated at baseline (month 0) and at 6 and 12 months. At each visit, blood cells were analyzed for B-cell ST6Gal1 and Neu1 expressions, and plasma samples were assessed for ST6Gal1 and Neu1 levels. The erythrocyte sedimentation rate (ESR), C-reactive protein (CRP), monocyte chemotactic protein-1 (MCP-1), and IgG anti-CCP with its α-2,6-SIA modification were measured. Disease activity measures, namely DAS28-ESR, DAS28-CRP, DAS28-MCP-1, SDAI, and CDAI, were calculated. Correlations and Receiver Operating Characteristics among ST6Gal, Neu1, SIA/anti-CCP ratios, and disease activity measures were assessed. Multivariate regression analyses were performed to reveal confounding factors in such correlations. The total SIA content of anti-CCP antibodies was inversely correlated with B-cell Neu1 levels (ρ = −0.317 with p = 0.013. Plasma (free-form) Neu1 levels were inversely correlated with SIA/IgG anti-CCP ratios (ρ = −0.361, p = 0.001) in the DAS28-MCP-1 < 2.2 (remission) subgroup. No such correlation was observed for the DAS28-ESR, DAS28-CRP, SDAI, or CDAI subgroups. B-cell ST6Gal1 levels correlated inversely with SDAI ≤ 11 and DAS28-MCP-1 ≤ 3.6 combined remission and low-disease-activity subgroups (ρ = −0.315 with p = 0.001 and ρ = −0.237 with p = 0.008, respectively). The same was observed for B-cell ST6Gal1/Neu1 ratios correlating with the SDAI ≤ 11 subgroup (ρ = −0.261, p = 0.009). Nevertheless, B-cell ST6Gal1/Neu1 ratios against SDAI ≤ 11 and DAS28-MCP-1 ≤ 3.6 subgroups produced significant area-under-curve (AUC) values of 0.616 and 0.600, respectively (asymptotic p-Values 0.004 and 0.018, respectively). Through multivariate regression analyses, we found that biologics (a confounding factor) interfered with p-Values related to the B-cell ST6Gal1 enzyme but did not interfere with p-Values related to the pure B-cell Neu1 enzyme. In addition, disease duration interfered with p-Values related to the pure Neu1 enzyme on B-cells or in plasma. Moreover, plasma ST6Gal1/Neu1 ratios against the DAS28-MCP-1 < 2.2 remission subgroup produced an AUC of 0.628 and asymptotic p = 0.003. Therefore, it is suggested that B-cell ST6Gal1/Neu1 ratios can be used as clinical indicators for the combined remission and low-disease-activity subgroup of SDAI and DAS28-MCP-1 formulae. Plasma ST6Gal1/Neu1 ratios are also good indicators of DAS28-MCP-1 remission. Full article

Denote by ${\Delta }_1(x;\phi )$ the error term in the classical Rankin–Selberg problem. Denote by $\zeta \left(s\right)$ the Riemann zeta-function. We establish an upper bound for this integral ${\int }_0^T{\Delta }_1(t;\phi ){\left|\zeta \left({\displaystyle \frac{1}{2}}+it\right)\right|}^{2}dt.$ In addition, when $2\le k\le 4$ is a fixed integer, we will derive an asymptotic formula for the integral ${\int }_1^T{\Delta }_1^k(t;\phi ){\left|\zeta \left({\displaystyle \frac{1}{2}}+it\right)\right|}^{2}dt.$ The results rely on the power moments of ${\Delta }_1(t;\phi )$ and $E\left(t\right)$, the classical error term in the asymptotic formula for the mean square of $\left|\zeta \left({\displaystyle \frac{1}{2}}+it\right)\right|$. Full article

For income distributions divided into middle, lower, and higher regions based on scalar median cut-offs, this paper establishes the asymptotic distribution properties—including explicit empirically applicable variance formulas and hence standard errors—of sample estimates of the proportion of the population within the group, their share of total income, and the groups’ mean incomes. It then applies these results for relative mean income ratios, various polarization measures, and decile-mean income ratios. Since the derived formulas are not distribution-free, the study advises using a density estimation technique proposed by Comte and Genon-Catalot. A shrinking middle-income group with declining relative incomes and marked upper-tail polarization among men’s incomes are all found to be highly statistically significant. Full article

This paper investigates a class of nonlinear rational difference equations with delayed terms, which often arise in various mathematical models. We analyze the iterative behavior of these rational functions and show how their iterations can be represented through second-order linear recurrence relations. By establishing a connection with generalized Balancing sequences, we derive explicit formulas that describe the system’s asymptotic behavior. Our main contribution is proving the existence of a unique globally asymptotically stable equilibrium point for all trajectories, regardless of initial conditions. We also provide analytical expressions for the solutions and support our findings with numerical examples. These results offer valuable insights into the dynamics of nonlinear rational systems and form a theoretical basis for further exploration in this area. Full article

In this paper, the remainder term of anti-Gaussian quadrature rules for analytic integrands with respect to Jacobi weight functions ${\omega }_{a,b}\left(x\right)={(1-x)}^a{(1+x)}^b$, where $a,b>-1$, is analyzed, and sharp estimates of the error are provided. These kinds of quadrature formulas were introduced by D.P. Laurie and have been recently studied by M.M. Spalević for the case of Jacobi-type weight functions $\omega$. Full article

Setting an expandable polystyrene (EPS) board on box culverts can reduce the vertical earth pressure (VEP) acting on the culvert roof. However, long-term backfill load will induce creep in both the EPS board and the surrounding soil, resulting in a change in the stress state of the culvert–soil system. A mechanical model for the long-term interaction of “backfill–EPS board–box culvert” was established, and theoretical formulas were derived for calculating the earth pressure around the culvert. Numerical simulation was employed to validate the accuracy of the proposed theoretical approach. Research indicates that, with EPS board, the VEP decreases rapidly then slightly increases with time and eventually approaches an asymptotic value, ultimately decreasing by 33%. However, the horizontal earth pressure (HEP) shows the opposite pattern and ultimately increases by 15%. The foundation contact pressure (FCP) increases nonlinearly and reaches a stable value, ultimately increasing by 10.2%. Without the EPS board, the VEP and HEP are significantly different from those with the EPS board. Although EPS boards can reduce the VEP on the culvert, attention should be paid to the variation of HEP caused by the creep of the EPS board and backfill. Full article

In this paper, we introduce a general fractional master equation involving regularized general fractional derivatives with Sonin kernels, and we discuss its physical characteristics and mathematical properties. First, we show that this master equation can be embedded into the framework of continuous time random walks, and we derive an explicit formula for the waiting time probability density function of the continuous time random walk model in form of a convolution series generated by the Sonin kernel associated with the kernel of the regularized general fractional derivative. Next, we derive a fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels from the continuous time random walk model in the asymptotical sense of long times and large distances. Another important result presented in this paper is a concise formula for the mean squared displacement of the particles governed by this fractional diffusion equation. Finally, we discuss several mathematical aspects of the fractional diffusion equation involving regularized general fractional derivatives with Sonin kernels, including the non-negativity of its fundamental solution and the validity of an appropriately formulated maximum principle for its solutions on the bounded domains. Full article

In the context of defining $\mathcal{NP}$-completeness, a tableau represents a hypothetical accepting computation path p of a nondeterministic polynomial time Turing machine N on an input w. The tableau is encoded by the propositional logic formula $\psi$, defined as $\psi ={\psi }_{cell}\wedge {\psi }_{rest}$. The component ${\psi }_{cell}$ enforces the constraint that each cell in the tableau contains exactly one symbol, while ${\psi }_{rest}$ incorporates constraints governing the step-by-step behavior of N on w. Intuitively, ${\psi }_{rest}$ appears to pose a much greater challenge for satisfiability. This raises the question of whether the distinction between ${\psi }_{cell}$ being a 3cnf formula, rather than a cheap 2cnf formula, actually matters. We show that if, hypothetically, ${\psi }_{rest}$ can be succinctly represented as a Horn formula, then satisfying $\psi$ can be achieved efficiently in $K^{f(n,k)}$ steps, where N operates within $O\left(n^k\right)$ steps and both k and K are constants. Asymptotically, $f(n,k)\approx n^{{\displaystyle \frac{2}{3}}k}$. Our method has the potential for iterative application. Technically, we trim ${\psi }_{cell}$ down to a 2cnf–Horn formula, whose satisfiability allows for empty cells, or “holes”, in the tableau. This modified tableau represents exponentially many paths of N on w, rather than a single accepting path p. While a tableau with holes conceptualizes the satisfiability of ${\psi }_{trim}$—a trimmed-down version of $\psi$—it does not directly address the satisfiability of $\psi$. Therefore, we introduce an external user who efficiently employs backtracking to fill in specific holes, ultimately verifying the satisfiability of the original $\psi$. Full article

In this article, a new higher-order convergence Laplace–Fourier method is developed to obtain the solutions of linear neutral delay differential equations. The proposed method provides more accurate solutions than the ones provided by the pure Laplace method and the original Laplace–Fourier method. We develop and show the crucial modifications of the Laplace–Fourier method. As with the original Laplace–Fourier method, the new method combines the Laplace transform method with Fourier series theory. All of the beneficial features from the original Laplace–Fourier method are retained. The solution still includes a component that accounts for the terms in the tail of the infinite series, allowing one to obtain more accurate solutions. The Laplace–Fourier method requires us to approximate the formula for the residues with an asymptotic expansion. This is essential to enable us to use the Fourier series results that enable us to account for the tail. The improvement is achieved by deriving a new asymptotic expansion which minimizes the error between the actual residues and those which are obtained from this asymptotic expansion. With both the pure Laplace and improved Laplace–Fourier methods, increasing the number of terms in the truncated series obviously increases the accuracy. However, with the pure Laplace method, this improvement is small. As we shall show, with the improved higher-order convergence Laplace–Fourier method, the improvement is significantly larger. We show that the convergence rate of the new Laplace–Fourier solution has a remarkable order of convergence. The validity of the new technique is corroborated by means of illustrative examples. Comparisons of the solutions of the new method with those generated by the pure Laplace method and the unmodified Laplace–Fourier approach are presented. Full article

Neural networks, mimicking the structural and functional aspects of the human brain, have found widespread applications in diverse fields such as pattern recognition, control systems, and information processing. A critical phenomenon in these systems is synchronization, where multiple neurons or neural networks harmonize their dynamic behaviors to a common rhythm, contributing significantly to their efficient operation. However, the inherent complexity and nonlinearity of neural networks pose significant challenges in understanding and controlling this synchronization process. In this paper, we focus on the synchronization of a class of fractional-order, delayed, and non-autonomous neural networks. Fractional-order dynamics, characterized by their ability to capture memory effects and non-local interactions, introduce additional layers of complexity to the synchronization problem. Time delays, which are ubiquitous in real-world systems, further complicate the analysis by introducing temporal asynchrony among the neurons. To address these challenges, we propose a straightforward yet powerful global synchronization framework. Our approach leverages novel state feedback control to derive an analytical formula for the synchronization controller. This controller is designed to adjust the states of the neural networks in such a way that they converge to a common trajectory, achieving synchronization. To establish the asymptotic stability of the error system, which measures the deviation between the states of the neural networks, we construct a Lyapunov function. This function provides a scalar measure of the system’s energy, and by showing that this measure decreases over time, we demonstrate the stability of the synchronized state. Our analysis yields sufficient conditions that guarantee global synchronization in fractional-order neural networks with time delays and Caputo derivatives. These conditions provide a clear roadmap for designing neural networks that exhibit robust and stable synchronization properties. To validate our theoretical findings, we present numerical simulations that demonstrate the effectiveness of our proposed approach. The simulations show that, under the derived conditions, the neural networks successfully synchronize, confirming the practical applicability of our framework. Full article

In this work, we analyze the vanishing cycles of Feynman loop integrals by the means of the Mayer–Vietoris spectral sequence. A complete classification of possible vanishing geometries is obtained. We use this result for establishing an asymptotic expansion for the loop integrals near their singularity locus and then give explicit formulas for the coefficients of such an expansion. Further development of this framework may potentially lead to exact calculations of one- and two-loop Feynman diagrams, as well as other next-to-leading and higher-order diagrams, in studies of radiative corrections for upcoming lepton–hadron scattering experiments. Full article