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We give the full solution of the following problem: obtain sharp inequalities between the moduli of smoothness The main tool is the new Hardy–Littlewood–Nikol’skii inequalities. More precisely, we obtained the asymptotic behavior of the quantity We also prove the Ulyanov and Kolyada-type inequalities in the Hardy spaces. Finally, we apply the obtained estimates to derive new embedding theorems for the Lipschitz and Besov spaces.

The hierarchical deep-learning neural network (HiDeNN) is systematically developed through the construction of structured deep neural networks (DNNs) in a hierarchical manner, and a special case of HiDeNN for representing Finite Element Method (or HiDeNN-FEM in short) is established. In HiDeNN-FEM, weights and biases are functions of the nodal positions, hence the training process in HiDeNN-FEM includes the optimization of the nodal coordinates. This is the spirit of r-adaptivity, and it increases both the local and global accuracy of the interpolants. By fixing the number of hidden layers and increasing the number of neurons by training the DNNs, rh-adaptivity can be achieved, which leads to further improvement of the accuracy for the solutions. The generalization of rational functions is achieved by the development of three fundamental building blocks of constructing deep hierarchical neural networks. The three building blocks are linear functions, multiplication, and inversion. With these building blocks, the class of deep learning interpolation functions are demonstrated for interpolation theories such as Lagrange polynomials, NURBS, isogeometric, reproducing kernel particle method, and others. In HiDeNN-FEM, enrichment functions through the multiplication of neurons is equivalent to the enrichment in standard finite element methods, that is, generalized, extended, and partition of unity finite element methods. Numerical examples performed by HiDeNN-FEM exhibit reduced approximation error compared with the standard FEM. Finally, an outlook for the generalized HiDeNN to high-order continuity for multiple dimensions and topology optimizations are illustrated through the hierarchy of the proposed DNNs.

The paper aims to establish diverse soliton solutions for the integrable Kuralay equations and to explore the integrable motion of space curves induced by these equations. The solitons arising from the integrable Kuralay equations are examined through qualitative studies. They are considered highly significant for understanding various phenomena in fields such as nonlinear optics, optical fibers, and ferromagnetic materials. This model is analyzed using the new generalized exponential rational function method and the new extended hyperbolic function method. With symbolic computations, the new extended hyperbolic function generates closed-form solutions to the integrable Kuralay equations, expressed in hyperbolic, trigonometric, and exponential forms. In contrast, the new extended hyperbolic function method provides hyperbolic, trigonometric, polynomial, and exponential solutions. The model is found to exhibit soliton solutions like periodic oscillating nonlinear waves, kink-wave profiles, multiple soliton profiles, singular solution, mixed singular solution, mixed hyperbolic solution, periodic pattern with anti-troughs and anti-peaked crests, mixed periodic, mixed complex solitary shock solution, mixed shock singular solution, mixed trigonometric solution, and periodic solution. These solutions are novel and have not been previously reported in the open literature. Using symbolic computation by Mathematica 11.3 or Maple , these newly derived soliton solutions are verified by substituting them back into the associated system.

Let $B$ be a rational function of degree at least two that is neither a Lattès map nor conjugate to $z^{\\pm n}$ or $\\pm T_{n}$. We provide a method for describing the set $C_{B}$ consisting of all rational functions commuting with $B$. Specifically, we define an equivalence relation $\\underset{B}{{\\sim}}$ on $C_{B}$ such that the quotient $C_{B}/\\underset{B}{{\\sim}}$ possesses the structure of a finite group $G_{B}$, and describe generators of $G_{B}$ in terms of the fundamental group of a special graph associated with $B$.

The recently introduced technique, namely the generalized exponential rational function method, is applied to acquire some new exact optical solitons for the generalized Benjamin–Bona–Mahony (GBBM) equation. Appropriately, we obtain many families of solutions for the considered equation. To better understand of the physical features of solutions, some physical interpretations of solutions are also included. We examined the symmetries of obtained solitary waves solutions through figures. It is concluded that our approach is very efficient and powerful for integrating different nonlinear pdes. All symbolic computations are performed in Maple package.

Given a p-coin that lands heads with unknown probability p, we wish to produce an f(p)-coin for a given function f : (0, 1) → (0, 1). This problem is commonly known as the Bernoulli factory and results on its solvability and complexity have been obtained in (ACM Trans. Model. Comput. Simul. 4 (1994) 213–219; Ann. Appl. Probab. 15 (2005) 93–115). Nevertheless, generic ways to design a practical Bernoulli factory for a given function f exist only in a few special cases. We present a constructive way to build an efficient Bernoulli factory when f(p) is a rational function with coefficients in ℝ. Moreover, we extend the Bernoulli factory problem to a more general setting where we have access to an m-sided die and we wish to roll a v-sided one; that is, we consider rational functions between open probability simplices. Our construction consists of rephrasing the original problem as simulating from the stationary distribution of a certain class of Markov chains—a task that we show can be achieved using perfect simulation techniques with the original m-sided die as the only source of randomness. In the Bernoulli factory case, the number of tosses needed by the algorithm has exponential tails and its expected value can be bounded uniformly in p. En route to optimizing the algorithm we show a fact of independent interest: every finite, integer valued, random variable will eventually become log-concave after convolving with enough Bernoulli trials.

We characterize the noncommutative Aleksandrov–Clark measures and the minimal realization formulas of contractive and, in particular, isometric noncommutative rational multipliers of the Fock space. Here, the full Fock space over $\\mathbb {C} ^d$ is defined as the Hilbert space of square-summable power series in several noncommuting (NC) formal variables, and we interpret this space as the noncommutative and multivariable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov–Clark measure theory for noncommutative and contractive rational multipliers. Noncommutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz algebra, the unital $C^*$ -algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz–Toeplitz and Cuntz algebras, and the emerging field of noncommutative rational functions.

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However, the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

In this paper, we prove a Turán-type inequality for rational functions and thereby extend it to a more general class of rational functions of degree mn with prescribed poles, where is a polynomial of degree m . These results not only generalize some Turán-type inequalities for rational functions, but also improve as well as generalize some known polynomial inequalities.

This study examines the optical soliton structure solutions, sensitivity, and modulation stability analysis of the chiral nonlinear Schrödinger equation (NLSE) with Bohm potential, a basic model in nonlinear optics and quantum mechanics. The equation represents wave group dynamics in numerous physical systems, involving quantum Hall and nonlinear optical structures. To attain exact solutions, we utilize the generalized exponential rational function (GERF) method and the Paul‐Painlevé (PP) approach, revealing a various set of optical soliton solutions, containing exponential, hyperbolic, and trigonometric function‐based solitons. By utilizing suitable parametric values, we provide 3D, 2D, and contour visualizations to demonstrate the soliton characteristics. Moreover, a comprehensive sensitivity and modulation stability analysis is carried out, presenting the impact of parametric alteration on soliton behavior. The findings exhibit the emergence of dark, bright, and periodic soliton solutions, which play a key role in understanding nonlinear wave propagation in topological and optical systems. The novelty of this work lies in enhancing the study of chiral soliton within the Bohm potential framework, yielding new observations into their stability, modulation, and structural properties.