This comprehensive volume presents essential mathematical results devoted to topics of mathematical analysis, differential equations and their various applications. It focuses on differential operators, Wardowski maps, low-oscillation functions, Galois and Pataki connections, Hardy-type inequalities, to name just a few.
Effort has been made for this unique title to have an interdisciplinary flavor and features several applications such as in tomography, elastic scattering, fluid mechanics, etc.
This work could serve as a useful reference text to benefit professionals, academics and graduate students working in theoretical computer science, computer mathematics, and general applied mathematics.
Sample Chapter(s)
Preface
Chapter 1: Galois and Pataki Connections on Power Sets
Contents:
Readership: Researchers, professionals, academics and graduate students in theoretical computer science, computer maths, and general applied maths.
https://doi.org/10.1142/9789811267048_fmatter
The following sections are included:
https://doi.org/10.1142/9789811267048_0001
If X is a set and R is a relation on X, then having in mind the abbreviation poset by Birkhoff for a partially ordered set, the ordered pair X(R) = (X, R) will be called a goset instead of a generalized ordered set.
In the sequel, we shall suppose that X(R) and Y (S) are gosets, φ is a function of X to itself, f is a function of X to Y and g is a function of Y to X. Thus, in particular, we may have φ = g ∘ f…
https://doi.org/10.1142/9789811267048_0002
We consider a functional equation related to inner product spaces in probabilistic normed spaces in the sense of Šerstnev. The work presents a relationship between various disciplines: the theory of metric linear spaces, the theory of triangular norms, the theory of probabilistic normed spaces, the theory of fixed points and the theory of functional equations.
https://doi.org/10.1142/9789811267048_0003
In this chapter, we prove the generalized Hyers–Ulam (or Hyers–Ulam–Rassias) stability of the following
The concept of Ulam–Hyers–Rassias stability originated from Rassias’ stability theorem that appeared in his paper (Rassias in Proc. Am. Math. Soc. 72:297–300, 1978).
https://doi.org/10.1142/9789811267048_0004
In this chapter, we introduce two evolutionary algorithmic heuristics which utilize competitive selection, (linear and nonlinear) regression and stochastic gradient descent to evolve generations of new points toward identifying entire solution sets of generalized Nash games with jointly convex constraints. Both algorithms involve the so-called shadow point function, a novel penalty function for generalized Nash equilibrium problems (GNEPs), similar to the Nikaidô–Isoda penalty function in optimization. It is well known that GNEPs have in general entire sets of Nash equilibria, and the question of identification of the entire solution set for a given GNEP is still open, for large classes of GNEPs. Traditional optimization-based methods to solve GNEPs are based on KKT systems, quasi-variational inequalities and projected differential equations/inclusions, which are, with a few exceptions, only designed to find one point in the solution set rather than the set itself. Our algorithms are evaluated on 2- and 3-player games in 2 and 3 dimensions, with both linear and nonlinear shared constraints. The success of these algorithms is discussed and compared, and future work is outlined.
https://doi.org/10.1142/9789811267048_0005
The finite element method (FEM) is a well-established approach for the numerical solution of ordinary differential equations (ODEs) and partial differential equations (PDEs). This method is a powerful tool in the study of various problems and has many applications, such as structural and fluid mechanics. In this review chapter, we mainly focus on applying the method to fluid mechanics problems. Initially, we present the FEM along with the basic theorems and examples. We analyze the error estimates for linear problems and the base functions that help distinguish the problem under consideration. We present the numerical solution of the Duffing equation, using the Galerkin FEM.
Additionally, we concentrate on the two-dimensional Stokes problem. We further introduce novel methods, such as the Discontinuous Galerkin (DG) FEM. The notion of adaptive mesh is also discussed. Lastly, we study the two-dimensional Navier–Stokes equations using the Galerkin FEM. These advanced methods provide reliable numerical results in all studied cases. This is achieved with the application of FEM to “test problems”, such as the backward-facing step.We obtain all the numerical results utilizing the software programs MATLAB and FEniCS.
https://doi.org/10.1142/9789811267048_0006
In this chapter, by combining the alternating direction method with the SQP method, we suggest and analyze two descent SQP alternating direction schemes for the separable constrained convex programming problem. Under certain conditions, the global convergence of both methods is proved. It is proved theoretically that the lower bound of the progress obtained by the second method is greater than that by the first one. Numerical results are given to verify the theoretical assertion.
https://doi.org/10.1142/9789811267048_0007
This chapter deals with a class of frictional unilateral contact problems with damage, for viscoelastic bodies. This model is composed by the momentum equilibrium equation combined with a heat-transfer equation coupled to a flow rule describing the damage evolution. The contact and friction are modeled with contact conditions formulated in displacement velocities and Coulomb’s dry friction law, respectively. The existence and uniqueness results are obtained by using some general results on the evolutionary variational inequalities, monotone operator theory and a fixed-point argument.
https://doi.org/10.1142/9789811267048_0008
Let H be a complex Hilbert space, f be an analytic function on the domain G with 0 ∈ G, A be an operator with Sp (A) ⊂ G and γ be a closed path in G \ {0} with Sp (A) ⊂ ins (γ), then for B, C ∈ B (H) we have
for x, y ∈ H, where
Some natural applications for numerical radius and p-Schatten norm are also provided.
https://doi.org/10.1142/9789811267048_0009
Given an isometric representation of a compact topological group on a Banach space and an invariant locally Lipschitz map, we establish general results including a deformation theorem and a minimax principle based on a notion of equivariant linking. As a byproduct, we generalize Bartsch’s fountain theorem. The main novelty is that we allow the Banach space to contain also nonadmissible representations. Moreover, our results are established under a Cerami-type condition more general than the usual one. Finally, we provide new properties and examples of admissible representations.
https://doi.org/10.1142/9789811267048_0010
The aim of this chapter is to model the interactions in some island environment between a seabird (the prey), a rodent (the mesopredator) and a raptor (the predator). The objective is also to take into account the massive arrival and departure of another seabird S which only frequents the island during some part of the year to reproduce and lay eggs and which thus temporarily affects the interactions between the resident species.
https://doi.org/10.1142/9789811267048_0011
There are numerous unresolved issues and difficulties in the highly active field of quantum calculus (QC). The creation of q-analogs for specific functions, the investigation of q-difference equations, and the use of q-calculus in mathematical physics and quantum information theory are some of the most current study areas. This notion has recently gained traction in the field of geometric function theory. This chapter deals with the QC (Jackson’s calculus) to define the q-differential operator related to the q-Raina function. With this newly created operator, we create a new subclass of analytical functions in conical domains. We examine one of the most well-liked characteristics of the suggested operator geometrically. Our strategy was motivated by the differential subordination hypothesis. We identified some well-known corollaries of our main conclusions as the outcome.
https://doi.org/10.1142/9789811267048_0012
A strong and adaptable mathematical paradigm is provided by dynamic systems with complex variables, which finds use in a number of scientific, engineering and mathematical fields. They promote clearer understanding of intricate phenomena, make mathematical expressions simpler and provide a concise manner to depict oscillatory processes. In this effort, we extend the complex Liu system into K-symbol fractional calculus. Then we examine the stability and stabilization of complex variables’ K-symbol fractional Liu system based on the extended well-known theory of stability. Calculations involving fractions are conceptualized in terms of real and complex K-symbol Caputo derivatives. The approach relies on fractional-order system stability concepts. It forces numerical solutions, as well as the solvability of the extended system (existence and uniqueness). In addition, certain requirements for the solution are developed, such as founding the upper solution of the suggested system. We shall indicate that the upper solution is formulated by the K-symbol Mittag–Leffler function. In numerous mathematical and scientific fields, investigation on the Mittag–Leffler function is underway. It is a flexible tool for characterizing intricate and unusual behavior in a variety of situations. Finally, examples for finding the solutions are illustrated covering the theory on this effort. The solutions are formulated in terms of the K-symbol Mittag–Leffler function.
https://doi.org/10.1142/9789811267048_0013
In this chapter, the direct and inverse scattering problem of time harmonic elastic waves by an inhomogeneous medium with buried objects inside is studied. Initially, the well-posedness of the direct scattering problem by the variational method in a suitable Sobolev space setting is presented and proved. Uniqueness, existence and continuity dependence of solution of the problem from the boundary data of the buried obstacles are established. Further, the corresponding inverse problem is studied and in particular the factorization method for shape reconstruction and location of the support of the inhomogeneous medium are exploited. In addition, an inversion algorithm for shape recovering of the medium is presented and proved as well. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one in elastic media are given.
https://doi.org/10.1142/9789811267048_0014
In this chapter, we first introduced a new class of functions called generalized-m−(((h1○g)p, (h2○g)q); (η1 , η2))-convex defined on (m, g; σ)invex set and also discover two interesting identities regarding Gauss–Jacobi and Hermite–Hadamard-type integral inequalities. By using the first lemma as an auxiliary result, some new bounds with respect to Gauss–Jacobi-type integral inequalities are established. Also, using the second lemma, some new estimates with respect to Hermite–Hadamardtype integral inequalities via k-fractional integrals for generalized-m − (((h1○g)p, (h2○g)q); (η1, η2))-convex mappings are obtained. It is pointed out that some new special cases can be deduced from main results. At the end, some applications to special means for different positive real numbers are provided as well.
https://doi.org/10.1142/9789811267048_0015
In this chapter, we introduce an action update model for two-player continuous action social dilemma games. The model applies to continuous action versions of social dilemma games such as Prisoner’s Dilemma, Hawk and Dove, and Stag Hunt. In our formulation, action updates depend on the current cooperation levels of the two players, independent of specific payoffs. This means that the same update equations can be applied to each of the abovementioned games. We present a preliminary investigation, limited to a particular action update model. As we explain in the sequel, this model admits a large number of modifications and extensions and in fact it belongs to a more general family which is further explored in future publications.
https://doi.org/10.1142/9789811267048_0016
The target of this chapter is to investigate (and try to add more results to) the statistical results based on logarithm Sobolev inequalities. Heisenberg’s uncertainty principle can be considered as such a problem. Moreover, the γ-order generalized normal distribution is emerged from logarithm Sobolev inequality and provides a number of extensions to statistical results, such as measures on information theory and the heat equation.
https://doi.org/10.1142/9789811267048_0017
In order to present known and new identities and relations for Catalantype numbers and polynomials, the method used in this chapter covers the techniques used for p-adic integral equations and special functions, especially generating functions with their functional equations. The following briefly summarizes the results to be given in this chapter. The aim of this chapter is to survey some certain families of Catalan-type numbers and polynomials by generating function approach. In this chapter, in the sequel of reminding the Catalan numbers and some features of these numbers and their relations with some special numbers, we focus on some Catalan-type numbers and analyze some of their fundamental properties. Moreover, by applying p-adic integrals to a function arising from the Catalan–Qi function, we get some p-adic integral formulas involving the Catalan–Qi function and other certain families of special numbers. Finally, with the implementation of our algorithms in Wolfram Mathematica Online (namely, in the Wolfram Cloud (Wolfram Research Inc., 2022)) by using the Wolfram programming language, we demonstrate some numerical values of Catalan-type numbers inside tables.
https://doi.org/10.1142/9789811267048_0018
In this article, we present both the discrete and the integral form of Cauchy–Bunyakovsky–Schwarz (CBS) inequality, some important generalizations in the n-dimensional Euclidean space and in linear subspaces of it, as well as the strengthened CBS. The last CBS inequality plays an important role in elasticity problems. A geometrical interpretation and a collection of the most important proofs of it are, also, presented.
https://doi.org/10.1142/9789811267048_0019
In this chapter, we introduce Lie bracket derivations in complex Banach Lie algebras. Using the direct method and the fixed point method, we prove the Hyers–Ulam stability of Lie bracket derivations in complex Banach Lie algebras.
https://doi.org/10.1142/9789811267048_0020
In this chapter, we establish approximate solutions of linear differential equations of second order with constant coefficients in the sense of Ulam and Hyers by using Aboodh integral transformation. By applying Aboodh integral transform, we prove the different types of Ulam stabilities such as Hyers–Ulam stability, Hyers–Ulam–Rassias stability, Mittag–Leffler–Hyers–Ulam stability and Mittag–Leffler–Hyers–Ulam–Rassias stability of linear differential differential equations of second order with constant coefficients for homogeneous and nonhomogeneous cases. We also obtain the Hyers–Ulam stability constants of these differential equations by using the Aboodh integral transform and give some examples to illustrate our main results.
https://doi.org/10.1142/9789811267048_0021
It is well known that extended general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this chapter, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener–Hopf equations, dynamical systems and auxiliary principle. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the extended general variational inequalities include variational inequalities and implicit complementarity problems as special cases, results presented continue to hold for these problems. Several open problems have been suggested for further research in these areas.
https://doi.org/10.1142/9789811267048_0022
In this chapter, we consider and introduce some new concepts of the biconvex functions and monotone operators involving an arbitrary bifunction. Some new relationships among various concepts of biconvex functions have been established. We have shown that the optimality conditions for the biconvex functions can be characterized by a class of bivariational inequalities. Auxiliary principle technique is used to propose proximal point methods for solving bivariational-like inequalities. We also discussed the conversance criteria for the suggested methods under pseudo-monotonicity. Our method of proof is very simple as compared with methods. Several special cases are discussed as applications of our main concepts and results. Our method of proof of convergence is very simple as compared with techniques. This chapter can be viewed as expository nature. It is a challenging problem to explore the applications of the bivariational inequalities in pure and applied sciences.
https://doi.org/10.1142/9789811267048_0023
We investigate a class of parametric network games which encompasses both the cases of strategic complements and strategic substitutes. In the case of a bounded strategy space, we derive a representation formula for the unique Nash equilibrium. We also prove a comparison result between the Nash equilibrium and the social optimum and then compute the price of anarchy for some simple test problems.
https://doi.org/10.1142/9789811267048_0024
The analytical inversion of the celebrated two-dimensional Radon transform of a function involves a certain integral; the computation of this integral requires the derivative of the Hilbert transform of the Radon transform of the initial function. This inversion provides great insight in the field of medical imaging, especially in positron emission tomography. In this chapter, following our previous works based on third-degree splines, we present a novel numerical implementation of the inversion of the Radon transform based on Chebyshev polynomials and the corresponding reconstruction algorithm. The Chebyshev approximation scheme has the advantage of significantly simplifying the mathematical formulas associated with the inversion of the Radon transform in two dimensions.
https://doi.org/10.1142/9789811267048_0025
This chapter is a survey of the authors’ [49, 50] articles published in recent years, and the sections in these articles have been prepared with the addition of new and applicable results. That is, we survey on the beta-type distribution associated with the Bernstein-type basis functions and the beta function, which were defined by authors [49, 50]. When it comes to the content of this chapter, the keywords that will be covered in this article are the following topics that have vital applications in all applied sciences: beta distribution, Bernstein polynomials, generating function, moment generating function, Stirling numbers, digamma function, beta function, gamma function and expected values for the logarithm of random variables. By blending the topics mentioned above, old and new results are provided. The relationships between these are examined and interpreted. Moreover, the known consequences of Bernstein polynomials are exhibited in the results related to the approximation theory to which they are related.
https://doi.org/10.1142/9789811267048_0026
Providing the right data to a machine learning model is an important step to insure the performance of the model. Noncompliant training data instances may lead to wrong predictions yielding models that cannot be used in production. Instance or prototype selection methods are often used to curate training sets thus leading to more reliable and efficient models. In this work, we investigate if diversity is helpful as a criterion for choosing which instances to remove from a given training set. We test our hypothesis against a random selection method and Mahalanobis outlier selection, using benchmark datasets with different data characteristics. Our computational experiments demonstrate that selection by diversity achieves better classification performance than random selection and can hence be considered as an alternative data selection criterion for effective model training.
https://doi.org/10.1142/9789811267048_0027
All Wardowski-type contractions modeled by low-oscillation functions are, in fact, Matkowski ones. Technical connections with some particular statements in the area obtained by Dung and Hang Vietnam J. Math. 43:743–755, 2015) and Fulga and Proca (Adv. Theory Nonlinear Anal. Appl. 1:57–63, 2017) are also discussed.
https://doi.org/10.1142/9789811267048_0028
Some fixed-point results are given for a class of implicit contractions acting on relational metrical-compatible quasimetric spaces. The connections with a related statement in Turinici (Modern Directions Metrical Fixed Point Theory, 2023) are also being discussed.
https://doi.org/10.1142/9789811267048_0029
In this chapter, the Hyers–Ulam–Rassias stability of the nonlinear fractional differential equations with the ρ-fractional derivative on the continuous function space is investigated by using the weighted space method. Some sufficient conditions are obtained in order that the nonlinear fractional differential equations are stable on the continuous function space. The results improve and extend some recent results.
https://doi.org/10.1142/9789811267048_0030
This chapter deals with the motion of an infinitesimal body near the out-of-plane equilibrium points in the restricted problem of three bodies under the radial component of Poynting–Robertson (P–R) drag and radiation pressure of the primaries as well as their angular velocity. In particular, the out-of-plane equilibria are first determined analytically and it is found that their existence and positions depend on the perturbing forces involved in the equations of motion. Due to the symmetry of the problem, these points appear in pairs and, depending on the parameter values, their number may be zero, two (L6,7) or four (L6,7 and L8,9). Finally, the effects of the parameters are shown on the positions of the out-of-plane equilibrium points for the binary systems Kruger-60 and Achird. An investigation of the stability of the out-of-plane equilibrium points shows that they are unstable for both binary systems.
https://doi.org/10.1142/9789811267048_bmatter
The following section is included:
Sample Chapter(s)
Preface
Chapter 1: Galois and Pataki Connections on Power Sets