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It is very meaningful and challenging to efficiently seek common solutions to operator systems (CSOSs), which are widespread in pure and applied mathematics, as well as some closely related optimization problems. The purpose of this paper is to introduce a novel class of multistep implicit iterative algorithms (MSIIAs) for solving general CSOSs. By using Xu's lemma and Maingé's fundamental and important results, we first obtain strong convergence theorems for both one-step and multistep implicit iterative schemes for CSOSs, involving asymptotically demicontractive operators. Finally, for the applications and profits of the main results presented in this paper, we give two numerical examples and present an iterative approximation to solve the general common solution to the variational inequalities and operator equations. [ABSTRACT FROM AUTHOR]

Block matrices play an important role in all branches of pure and applied mathematics. In this paper, we study the two fundamental concepts: inverses and determinants of general n × n block matrices. In the first part, the inverses of 2 × 2 block matrices are given, where one of the blocks is a non-singular matrix, a result which can be generalised to a block matrix of any size, by splitting it into four blocks. The second part focuses on the determinants, which is covered in two different methods. In the first approach, we revise a formula for the determinant of a block matrix A, with blocks elements of R; a commutative subring of M n × n (F) . The determinants of tensor products of two matrices are also given in this part. In the second method for computing the determinant, we give the general formula, which would work for any block matrix, regardless of the ring or the field under consideration. The individual formulas for determinants of 2 × 2 and 3 × 3 block matrices are also produced here. [ABSTRACT FROM AUTHOR]

In this paper, we propose two efficient methods for solving the fractional-order Schrödinger–KdV system. The first method is the Laplace residual power series method (LRPSM), which involves expressing the solution as a power series and using residual correction to improve the accuracy of the solution. The second method is a new iterative method (NIM) that simplifies the problem and obtains a recursive formula for the solution. Both methods are applied to the Schrödinger–KdV system with fractional derivatives, which arises in many physical applications. Numerical experiments are performed to compare the accuracy and efficiency of the two methods. The results show that both methods can produce highly accurate solutions for the fractional Schrödinger–KdV system. However, the new iterative method is more efficient in terms of computational time and memory usage. Overall, our study demonstrates the effectiveness of the residual power series method and the new iterative method in solving fractional-order Schrödinger–KdV systems and provides a valuable tool for researchers and practitioners in applied mathematics and physics. [ABSTRACT FROM AUTHOR]

We study the problem of prediction of binary sequences with expert advice in the online setting, which is a classic example of online machine learning. We interpret the binary sequence as the price history of a stock, and view the predictor as an investor, which converts the problem into a stock prediction problem. In this framework, an investor, who predicts the daily movements of a stock, and an adversarial market, who controls the stock, play against each other over N turns. The investor combines the predictions of n≥2 experts in order to make a decision about how much to invest at each turn, and aims to minimize their regret with respect to the best‐performing expert at the end of the game. We consider the problem with history‐dependent experts, in which each expert uses the previous d days of history of the market in making their predictions. We prove that the value function for this game, rescaled appropriately, converges as N→∞ at a rate of ON−1/6 to the viscosity solution of a nonlinear degenerate elliptic PDE, which can be understood as the Hamilton‐Jacobi‐Issacs equation for the two‐person game. As a result, we are able to deduce asymptotically optimal strategies for the investor. Our results extend those established by the first author and R.V. Kohn [14] for n=2 experts and d≤4 days of history. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

In this paper, solitons solutions of higher-order dispersive cubic–quintic Schrödinger equationincluding third-order as well as fourth-order derivatives with respect to time, that describes the dynamics of ultrashort pulses in optical fibers are investigated in detail. In this respect,a solution procedure in the locality of applied mathematics called the hyperbolic function method is appliedusing multi-linear variable separation approach (MLVSA). As an outcome, a bunch of soliton solutions isderived in conjunction with plotting dark and periodic wave solutions. The credibility of the results is examined by setting each solution back into its governing equation. Through portraits, different forms of wave solutions are depicted. Moreover, the restrictions on the parameters are also given for the existence of the obtained solutions. [ABSTRACT FROM AUTHOR]

Defining efficient families of recurrent neural networks (RNN) models for solving time-varying nonlinear equations is an interesting research topic in applied mathematics. Accordingly, one of the underlying elements in designing RNN is the use of efficient nonlinear activation functions. The role of the activation function is to bring out an output from a set of input values that are supplied into a node. Our goal is to define new family of activation functions consisting of a fixed gain parameter and a functional part. Corresponding zeroing neural networks (ZNN) is defined, termed as varying-parameter improved zeroing neural network (VPIZNN), and applied to solving time-varying nonlinear equations. Compared with previous ZNN models, the new VPIZNN models reach an accelerated finite-time convergence due to the new time-varying activation function which is embedded into the VPIZNN design. Theoretical results and numerical experiments are presented to demonstrate the superiority of the novel VPIZNN formula. The capability of the proposed VPIZNN models are demonstrated in studying and solving the Van der Pol equation and finding the root a (t) m . [ABSTRACT FROM AUTHOR]

The paper presents a method for processing positional observations of near-Earth objects using a numerical model of satellite motion developed at the Research Institute of Applied Mathematics and Mechanics of Tomsk State University (NII PMM TSU). The root-mean-square error of orbit improvement without rejection of observations for such objects does not exceed 0.3″ over a seven-day time interval. The results of the presentation of observations for the next occurrence of the object are obtained, which makes it possible to find the object in a time interval of five months. The orbit has been refined in the joint processing of measurements on several occurrences of the object over a six-month time interval. All results were obtained from observations made on the equipment of the Terskol Observatory Shared Use Center. [ABSTRACT FROM AUTHOR]

This paper addresses discontinuities in the solutions of mathematical physics that describe actual processes and are not observed in experiments. The appearance of discontinuities is associated in this paper with the classical differential calculus based on the analysis of infinitesimal quantities. Nonlocal functions and nonlocal derivatives, which are not specified, in contrast to the traditional approach to a point, but are the results of averaging over small but finite intervals of the independent variable are introduced. Classical equations of mathematical physics preserve the traditional form but include nonlocal functions. These equations are supplemented with additional equations that link nonlocal and traditional functions. The proposed approach results in continuous solutions of the classical singular problems of mathematical physics. The problems of a string and a circular membrane loaded with concentrated forces are used to demonstrate the procedure. Analytical results are supported with experimental data. [ABSTRACT FROM AUTHOR]

This is the second part of a two‐paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the S2‐valued equivariant energy critical wave maps equation on ℝ1+2, with equivariance class k≥4. It is known that every topologically trivial wave map with energy less than twice that of the unique k‐equivariant harmonic map Q→k scatters in both time directions. We study maps with precisely the threshold energy E=2EQ→k. In the first part of the series [15] we gave a refined construction of a threshold wave map that asymptotically decouples into a superposition of two harmonic maps (bubbles), one of which is concentrating in scale. In this paper, we show that this solution is the unique (up to the natural invariances of the equation) two‐bubble wave map. Combined with our earlier work [14] we obtain an exact description of every threshold wave map. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

For a given finite subset S of a compact Riemannian manifold (M,g) whose Schouten curvature tensor belongs to a given cone, we establish a necessary and sufficient condition for the existence and uniqueness of a conformal metric on M\S such that each point of S corresponds to an asymptotically flat end and that the Schouten tensor of the conformal metric belongs to the boundary of the given cone. As a by‐product, we define a purely local notion of Ricci lower bounds for continuous metrics that are conformal to smooth metrics and prove a corresponding volume comparison theorem. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

The article proposes formulating and codifying a set of applied numerical methods, coined as Deep Learning Discrete Calculus (DLDC), that uses the knowledge from discrete numerical methods to interpret the deep learning algorithms through the lens of applied mathematics. The DLDC methods aim to leverage the flexibility and ever-increasing resources of deep learning and rich literature on numerical analysis to formulate a general class of numerical methods that can directly use data with uncertainty to predict the behavior of an unknown system as well as elevate the speed and accuracy of numerical solution of the governing equations for known systems. The article is structured into two major sections. In the first section, the building blocks of the DLDC methods are presented and deep learning structures analogous to traditional numerical methods such as finite difference and finite element methods are constructed with a view to incorporate these techniques in Science, Technology, Engineering, Mathematics syllabus for K-12 students. The second section builds upon the building blocks of the previous discussion and proposes new solution schemes for differential and integral equations pertinent to multiscale mechanics. Each section is accompanied by a mathematical formulation of the numerical methods, analogous DLDC formulation, and suitable examples. [ABSTRACT FROM AUTHOR]

The concept of "useful adaptive result", a key concept in the theory of functional systems by P. K. Anokhin, is discussed. A large body of works devoted to this problem, in the historical aspect and on the basis of results of own researches, allows concluding that useful adaptive results of the system organization of physiological functions include not only the event arising as a result of activity of the system, but also the physiological cost of this activity. The ratio of the physiological cost and performance indicators gives an idea of the effectiveness of system activity and is its integral characteristic. The main approaches to estimation of physiological cost and efficiency of activity are discussed, as well as possibilities of using methods of applied mathematics (correlation and cluster analysis, artificial neural network technology) to describe system organization of physiological functions in healthy people. The question of the use of the proposed regulations to describe the system organization of physiological functions in patients with Parkinson's disease and epilepsy is considered. [ABSTRACT FROM AUTHOR]

Let C k + 1 be a cycle of length k + 1 and C t + 1 be a cycle of length t + 1. A polygon flower with two centers, denoted by F = F (C k + 1 ; P 1 , ... , P k ; C t + 1 ; P k + 1 , ... , P k + t) is obtained by identifying the i th edge of C k + 1 with an edge ei that belongs to an end-polygon of Pi for i = 1 , ... , k , and identifying the j th edge of C t + 1 with an edge ej that belongs to an end-polygon of Pj for j = k + 1 , ... , k + t , where C k + 1 and C t + 1 have a common edge h. In this paper, we determine the order of sandpile group S(F) of F, which can be viewed as generalized of results in paper (Haiyan Chen, Bojan Mohar. The sandpile group of a polygon flower. Discrete Applied Mathematics, 2019). Moreover, the formula and structure for sandpile group of polygon flower can be obtained. Finally, as application of our result, we also present the sandpile group of cata-condensed system with two branched hexagons. [ABSTRACT FROM AUTHOR]

In this paper, the extended simplest equation technique is considered to construct various exact optical solutions to the time-fractional nonlinear Schrodinger equation with second-order spatiotemporal and group velocity dispersion coefficients. The acquired novel optical soliton solutions are illustrated by the hyperbolic functions, the rational functions, and the trigonometric functions. The singular, dark, bright, mixed bright, dark–bright, and wave soliton solutions of the proposed model are successfully constructed. Further, to clarify the magnitude of the present nonlinear time-fractional Schrodinger model several solutions of the new exact optical solutions are plotted via two-dimensional and three-dimensional graphs using suitable values of physical parameters. The results acquired illustrate that the utilized technique is simple and quite efficient for exploring exact soliton solutions for different differential equations of fractional and integer orders arising in optics and applied mathematics. The novel optical solutions can assist researchers with an interest in plasma physics to unravel the mystery of numerous nonlinear phenomena that arise in various plasma models. [ABSTRACT FROM AUTHOR]

Interpolative Kannan contractions are a refinement of Kannan contraction, which is considered as one of the significant notions in fixed point theory. Gb-metric spaces is considered as a generalized concept of both concepts b-metric and G-metric spaces therefore, the significant fixed and common fixed point results of the contraction based on this concept is generalized results for both concepts. The purpose of this manuscript, is to take advantage to interpolative Kannan contraction together with the notion of Ωb which equipped with Gb-metric spaces and H simulation functions to formulate two new interpolative contractions namely, (H, Ωb)-interpolative contraction for self mapping f and generalized (H, Ωb)-interpolative contraction for pair of self mappings (f1, f2). We discuss new fixed and common fixed point theorems. Moreover, to demonstrate the applicability and novelty of our theorems, we formulate numerical examples and applications to illustrate the importance of fixed point theory in applied mathematics and other sciences. [ABSTRACT FROM AUTHOR]

The present paradigm associates the dawn of modern applied mathematics with the first decades of the 19th century. In an investigation of these historical premises, we search for themes investigated today through methods pertaining to applied mathematics in the works of a medieval scholar whose singular vision helped him reach several conclusions that were definitely ahead of his time. Nicole Oresme’s work, Tractatus de configurationibus qualitatum et motuum, written approximately between 1351 and 1355, showcases early mathe- matical applications that would now be classified as works in applied mathematics. [ABSTRACT FROM AUTHOR]

In this paper, methods of second-order and higher-order reverse mathematics are applied to versions of a theorem of Banach that extends the Schröder–Bernstein theorem. Some additional results address statements in higher-order arithmetic formalizing the uncountability of the power set of the natural numbers. In general, the formalizations of higher-order principles here have a Skolemized form asserting the existence of functionals that solve problems uniformly. This facilitates proofs of reversals in axiom systems with restricted choice. [ABSTRACT FROM AUTHOR]

The Reynolds Transport Theorem, colloquially known as "differentiation under the integral sign", is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and statistical physics. In this work, we state and prove generalizations thereof to submanifolds with corners evolving in a manifold via the flow of a smooth time-independent or time-dependent vector field. Thereby we close a practically important gap in the mathematical literature, as related works require various "boundedness conditions" on domain or integrand that are cumbersome to satisfy in common modeling situations. By considering manifolds with corners, a generalization of manifolds and manifolds with boundary, this work constitutes a step towards a unified treatment of classical integral theorems for the "unbounded case" for which the boundary of the evolving set can exhibit some irregularity. [ABSTRACT FROM AUTHOR]

Aneurysm morphology has been correlated with rupture. Previous reports identified several morphologic indices that predict rupture status, but they measure only specific qualities of the morphology of an aneurysm in a semiquantitative fashion. Fractal analysis is a geometric technique whereby the overall complexity of a shape is quantified through the calculation of a fractal dimension (FD). By progressively altering the scale of measurement of a shape and determining the number of segments required to incorporate the entire shape, a noninteger value for the dimension of the shape is derived. We present a proof-of-concept study to calculate the FD of an aneurysm for a small cohort of patients with aneurysms in 2 specific locations to determine whether FD is associated with aneurysm rupture status. Twenty-nine aneurysms of the posterior communicating and middle cerebral arteries were segmented from computed tomography angiograms in 29 patients. FD was calculated using a standard box-counting algorithm extended for use with three-dimensional shapes. Nonsphericity index and undulation index (UI) were used to validate the data against previously reported parameters associated with rupture status. Nineteen ruptured and 10 unruptured aneurysms were analyzed. Through logistic regression analysis, lower FD was found to be significantly associated with rupture status (P = 0.035; odds ratio, 0.64; 95% confidence interval, 0.42–0.97 per FD increment of 0.05). In this proof-of-concept study, we present a novel approach to quantify the geometric complexity of intracranial aneurysms through FD. These data suggest an association between FD and patient-specific aneurysm rupture status. [ABSTRACT FROM AUTHOR]

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains Ω ⊂ ℝ d with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by H 0 s (Ω ¯) , and Sobolev–Slobodeckij̆ spaces, denoted by H s (Ω) , where s > d 2 . Our goal is accomplished by obtaining the Green's solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green's kernels satisfying these problems are symmetric and positive definite reproducing kernels of H 0 s (Ω ¯) and H s (Ω) , respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains. [ABSTRACT FROM AUTHOR]

Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time. [ABSTRACT FROM AUTHOR]

We consider the gravity water waves system with a periodic one‐dimensional interface in infinite depth and give a rigorous proof of a conjecture of Dyachenko‐Zakharov [16] concerning the approximate integrability of these equations. More precisely, we prove a rigorous reduction of the water waves equations to its integrable Birkhoff normal form up to order 4. As a consequence, we also obtain a long‐time stability result: periodic perturbations of a flat interface that are initially of size ε remain regular and small up to times of order ε−3. This time scale is expected to be optimal. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

Previous work treated the problem of a mass sliding over a rough spherical surface in broad generality, providing both analytic and numerical solutions. This paper examines special cases of 2D motion along a surface meridian when the initial speed is precisely chosen so that the sliding mass nearly stops before speeding up and subsequently leaving the surface. Carrying the solution for these critical cases into the time domain via both an analytical method and numerical integration adds richness that might otherwise be missed. The numerical method raises practical mathematical issues that must be handled carefully to obtain accurate results. Although conceptually simple, this classical mechanics problem is an excellent vehicle for students to gain proficiency with mathematical analysis tools and further their appreciation for how applied mathematics can bring new insight into situations where intuition may fall short. [ABSTRACT FROM AUTHOR]

This article is a memorial tribute to Larisa Grigor'evna Afanasyeva, a prominent mathematician and professor at Lomonosov Moscow State University. Afanasyeva specialized in queuing theory, particularly the ergodicity of various queuing systems. She made significant contributions to the field and published over 130 research papers and two books. Afanasyeva's work was recognized internationally, and she was an active scientist until her passing in August 2022. [Extracted from the article]

We propose a mathematical model of vibration of elastic systems formed by conjugated shells of revolution with different geometry in the field of combined static axisymmetric loads. The model is based on the concepts of geometrically nonlinear theory of mean bending and realized within the framework of the classical Kirchhoff–Love theory with the use of contemporary methods of applied mathematics and numerical analysis. The spectral picture of a shell structure with elements of positive, zero, and negative Gaussian curvature is constructed. This picture enables us to detect resonance situations under specific dynamic actions and determine dangerous combinations of static loads in the analysis of stability of the equilibrium states of the structure. [ABSTRACT FROM AUTHOR]

The generalized Kuramoto–Sivashinsky equation arises frequently in engineering, physics, biology, chemistry, and applied mathematics, and because of its extensive applications, this important model has received much attention regarding obtaining numerical solutions. This article introduces a new hybrid technique based on nonpolynomial splines and finite differences for solving the Kuramoto–Sivashinsky equation approximately. Specifically, the truncation error is studied to examine the convergence order of the proposed scheme, some problems are given to show its viability and effectiveness, and the norm errors are determined to compare the current method with the analytic solution and some other methods from the literature. [ABSTRACT FROM AUTHOR]

This article addresses the issue of multicollinearity for the distributed lag model via Bayesian approach. We introduce the Bayes Almon ridge estimator under this approach by employing prior information. Specifically, the resulting estimator is a new Bayesian estimator that is the mean of posterior density function for the coefficient of the Almon model. Also, a restricted version of this estimator is proposed by using Lagrange multipliers to optimize the coefficients under the additional exact linear constraints. In mathematical sense, theoretical comparisons are carried out by employing mean squared error utilizing matrix theory. Additionally, we investigate a selection method for the biasing parameter of the new estimators with the help of the mean squared error comparisons. To test the theoretical suggestions, we perform a real-life data analysis as a means of applied mathematics. Moreover, we benefit from Monte Carlo simulation techniques where we use several levels of different parameters. The outcomes of numerical example and simulation study favor the new estimators and, hence we succeed in eliminating the multicollinearity. [ABSTRACT FROM AUTHOR]

This paper presents the relevance of artificial intelligence in the teaching of mathematics by using an Intelligent Support System made in Wolfram Mathematica. This system allows the design of an unlimited number of personalized items necessary for the training of students passionate about mathematics. Neuroscience research in collaboration with artificial intelligence can lead to deeper and faster personalized math learning, increasing the potential of every student. [ABSTRACT FROM AUTHOR]

These two observations show that while the finger metric is nearly impossible to quantify experimentally, it possesses a significant group of symmetries; in fact, these symmetries act transitively on the viewport space HT ht . A length space is called HT ht -hyperbolic if for any three points I A i , I B i , I C i , the geodesic path connecting I A i and I B i lies within the union of HT ht -tubes around the geodesic trajectories connecting I AC i and I BC i . Of course, such a circular geodesic looks curved to us who reside in the Euclidean space used to describe hyperbolic space: for the native dwellers of HT ht , these semicircles are perfectly straight. [Extracted from the article]

Accurately determining system order plays a vital role in system identification directly related to the accuracy of identification results, especially for nonlinear system identification. Due to the need for human subjective judgment, the traditional sequence determination method easily causes uncertainty in the results; and the phenomenon of the virtual mode or omission occurs. An automatic nonlinear subspace identification method is proposed to address the aforementioned problems. When the eigenvalue decomposition of the constructed Hankel matrix is performed, the calculation range of the modal order of the system is estimated. The similarity coefficient and distance function are introduced to cluster the identified modal results, the poles of the false modes are removed to obtain the cluster stabilization diagram, and the best order of the system is received. Then, the modal parameters and nonlinear coefficients are obtained. Simulation examples are carried out to verify the effectiveness and robustness of the proposed method. An experimental study is carried out on a multilayer building with nonlinear characteristics. Compared with the traditional stabilization graph, the accuracy of the automatic order determination proposed in this paper is proven. [ABSTRACT FROM AUTHOR]