2022 | Buch

# Approximation and Computation in Science and Engineering

herausgegeben von: Nicholas J. Daras, Themistocles M. Rassias

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

2022 | Buch

herausgegeben von: Nicholas J. Daras, Themistocles M. Rassias

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

In recent years, extensive research has been conducted by eminent mathematicians and engineers whose results and proposed problems are presented in this new volume. It is addressed to graduate students, research mathematicians, physicists, and engineers. Individual contributions are devoted to topics of approximation theory, functional equations and inequalities, fixed point theory, numerical analysis, theory of wavelets, convex analysis, topology, operator theory, differential operators, fractional integral operators, integro-differential equations, ternary algebras, super and hyper relators, variational analysis, discrete mathematics, cryptography, and a variety of applications in interdisciplinary topics. Several of these domains have a strong connection with both theories and problems of linear and nonlinear optimization. The combination of results from various domains provides the reader with a solid, state-of-the-art interdisciplinary reference to theory and problems. Some of the works provide guidelines for further research and proposals for new directions and open problems with relevant discussions.

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We deal here with the minimum and the maximum of ∑ i = 1 n F a 2 i − 1 , a 2 i , a ∈ ℝ 2 n $$\displaystyle \sum _{i=1}^{n}F\left ( a_{2i-1},a_{2i}\right ) ,\left ( \mathbf {a}\right ) \in \mathbb {R} ^{2n} $$ and of ∑ i = 1 n F a i , a i + 1 , a n + 1 = a 1 , a ∈ ℝ n $$\displaystyle \sum _{i=1}^{n}F\left ( a_{i},a_{i+1}\right ) ,\ \ a_{n+1}=a_{1},\left ( \mathbf {a}\right ) \in \mathbb {R} ^{n} $$ obtained by using rearrangement techniques. The results depend on the arrangement of a $$\left ( \mathbf {a}\right ) $$ and are used in proving Jensen-type inequalities.

Our aim is to investigate the generalized Hyers-Ulam-Rassias stability for the following general Jensen functional equation: ∑ k = 0 n − 1 f ( x + b k y ) = n f ( x ) , $$\displaystyle \sum _{k=0}^{n-1} f(x+ b_{k}y)=nf(x), $$ where n ∈ ℕ 2 $$n \in \mathbb {N}_{2}$$ , b k = exp ( 2 i π k n ) $$b_{k}=\exp (\frac {2i\pi k}{n})$$ for 0 ≤ k ≤ n − 1, in 2-Banach spaces by using a new version of Brzdȩk’s fixed point theorem. In addition, we prove some hyperstability results for the considered equation and the general inhomogeneous Jensen equation ∑ k = 0 n − 1 f ( x + b k y ) = n f ( x ) + G ( x , y ) . $$\displaystyle \sum _{k=0}^{n-1} f(x+ b_{k}y)=nf(x)+G(x,y). $$

We obtain the complete asymptotic expansion of the sequence defined by ∫ 0 1 f ( x ) g ( x n ) d x $$\int _0^1f(x)g(x^n)dx$$ , where the functions f and g satisfy various conditions. The main result is applied in Sect. 4 to find the complete asymptotic expansion of some classical sequences.

Pseudoprimes are composite integers which share properties of the prime numbers, and they have applications in many areas, as, for example, in public-key cryptography. Numerous types of pseudoprimes are known to exist, many of them defined by linear recurrent sequences. In this material, we present some novel classes of pseudoprimes related to the generalized Lucas sequences. First, we present some arithmetic properties of the generalized Lucas and Pell–Lucas sequences and review some classical pseudoprimality notions defined for Fibonacci, Lucas, Pell, and Pell–Lucas sequences and their generalizations. Then we define new notions of pseudoprimality which do not involve the use of the Jacobi symbol and include many classical pseudoprimes. For these, we present associated integer sequences recently added to the Online Encyclopedia of Integer Sequences, identify some key properties, and propose a few conjectures.

We discuss approximations of square integrable periodic functions by their projections in finite shift-invariant subspaces and highlight the role of principal shift invariance. We also show how we may produce a variety of sampling representations based on finite frame theory and we discuss some applications.

In this chapter, we study the Jaynes–Cummings model under multiphoton excitation and in the general case of intensity-dependent coupling strength given by an arbitrary function f. The Jaynes–Cummings theoretical model is of great interest to atomic physics, quantum optics, solid-state physics, and quantum information theory with several applications in coherent control and quantum information processing. As the initial state of the radiation mode, we consider a squeezed state, which is the most general Gaussian pure state. The time evolution of the mean photon number and the dispersions of the two quadrature components of the electromagnetic field are calculated for an arbitrary function f. The mean value of the inversion operator of the atom is also calculated for some simple forms of the function f.

In this chapter, a new version of multi-quadratic mappings are characterized. By this characterization, every multi-additive-quadratic-cubic mapping which is defined as system of functional equations can be unified as a single equation. In addition, by applying two fixed point theorems, the generalized Hyers-Ulam stability of multi-additive-quadratic-cubic mappings in normed and non-Archimedean normed spaces are studied. A few corollaries corresponding to some known stability and hyperstability outcomes for multi-additive, multi-quadratic, multi-cubic, and multi-additive-quadratic-cubic mappings (functional equations) are presented.

In the present chapter, we have generalized the truncated M-fractional derivative. This new differential operator denoted by i , p D M , k , α , β σ , γ , q , $${ }_{i,p}\mathscr {D}_{M, k, \alpha , \beta }^{\sigma , \gamma ,q},$$ where the parameter σ associated with the order of the derivative is such that 0 < σ < 1 and M is the notation to designate that the function to be derived involves the truncated (p, k)-Mittag-Leffler function. The operator i , p D M , k , α , β σ , γ , q $${ }_{i,p}\mathscr {D}_{M, k, \alpha , \beta }^{\sigma , \gamma ,q}$$ satisfies the properties of the integer-order calculus. We also present the respective fractional integral from which emerges, as a natural consequence, the result, which can be interpreted as an inverse property. Finally, we obtain the analytical solution of the M-fractional heat equation, linear fractional differential equation, and present a graphical analysis.

The aim of this paper is to study the Hyers-Ulam-Rassias stability for a Volterra-Hammerstein functional integral equation in three variables via Picard operators.

EEG recordings give extremely noisy signals that do not allow classical methods to clearly display such as the existence of power laws or even more so the critical state that is a signature of the normal operation of biological tissues (Contoyiannis et al., Phys Rev Lett 93:098101, 2004; Contoyiannis et al., Nat Hazards Earth Syst Sci 13:125–139, 2013; Kosmidis et al., Eur J Neurosci, 2018. https://doi.org/10.1111/ejn.14117 ). We have recently introduced a method, based on Haar wavelet transformation (Contoyiannis et al. Phys. Rev. E 101:052104, 2020), that completely ignores noise and thus can reveal the information of the power law in EEGs. It calculates the exponent of the power law and thus gives us the ability to determine whether the brain is in critical state in terms of physics, i.e., in a state of normal biological function. Pathological conditions, such as epilepsy, are quantified through this method so we can observe their evolution.

Let [ ⋅ ] be the floor function. In this paper, we show that every sufficiently large positive integer N can be represented in the form N = [ p 1 log p 1 ] + [ p 2 log p 2 ] + [ p 3 log p 3 ] , $$\displaystyle N=[p_1\log p_1]+[p_2\log p_2]+[p_3\log p_3], $$ where p1, p2, andp3 are prime numbers. We also establish an asymptotic formula for the number of such representations, when p1, p2, andp3 do not exceed given sufficiently large positive number.

In this paper, we introduce the concepts of Hermite-Hadamard trapezoid and mid-point divergences that are closely related to the Jensen divergence considered by Burbea and Rao in 1982. The joint convexity of these divergences and several inequalities involving these measures are established. Various examples concerning the Csiszár, Lin-Wong, and HH f-divergence measures are also given.

Let f : 0 , ∞ → ℝ $$f:\left ( 0,\infty \right ) \rightarrow \mathbb {R}$$ be a convex function on 0 , ∞ $$\left ( 0,\infty \right ) $$ . The associated two variables perspective function P f : 0 , ∞ × 0 , ∞ → ℝ $$P_{f}:\left ( 0,\infty \right ) \times \left ( 0,\infty \right ) \rightarrow \mathbb {R}$$ is defined by P f x , y : = x f y x . $$\displaystyle P_{f}\left ( x,y\right ) :=xf\left ( \frac {y}{x}\right ) . $$ In this paper, we establish some basic and double integral inequalities for the perspective function Pf defined above. Some double integral inequalities in the case of rectangles, squares, and circular sectors are also given.

In this paper, we find the maximum value of a multi-variable function, related by an optimization problem. Our method of maximizing this function is geometric, without applying the partial derivatives tests and the concept of Hessian matrix.

The study of the functional characteristics of the brain plays a crucial role in modern medical imaging. An important and effective nuclear medicine technique is positron emission tomography (PET), whose utility is based upon the noninvasive measure of the in vivo distribution of imaging agents, which are labeled with positron-emitting radionuclides. The main mathematical problem of PET involves the inverse Radon transform, leading to the development of several methods toward this direction. Herein, we present an improved formulation based on Chebyshev polynomials, according to which a novel numerical algorithm is employed in order to interpolate exact simulated values of the Randon transform via an analytical Shepp–Logan phantom representation. This approach appears to be efficient in calculating the Hilbert transform and its derivative, being incorporated within the final analytical formulae. The numerical tests are validated by comparing the presented methodology to the well-known spline reconstruction technique.

The main aim of this chapter is to introduce several mixed operators between Choquet integral operators and max-product operators and to study their approximation, shape preserving, and localization properties. Section 2 contains some preliminaries on the Choquet integral. In Sect. 3, we obtain quantitative estimates in uniform and pointwise approximation for the following mixed type operators: max-product Bernstein–Kantorovich–Choquet operator, max-product Szász–Mirakjan–Kantorovich–Choquet operators, nontruncated and truncated cases, and max-product Baskakov–Kantorovich–Choquet operators, nontruncated and truncated cases. We show that for large classes of functions, the max-product Bernstein–Kantorovich–Choquet operators approximate better than their classical correspondents, and we construct new max-product Szász–Mirakjan–Kantorovich–Choquet and max-product Baskakov–Kantorovich–Choquet operators, which approximate uniformly f in each compact subinterval of [0, +∞) with the order ω 1 ( f ; λ n ) $$\omega _{1}(f; \sqrt {\lambda _{n}})$$ , where arbitrary fast. Also, shape preserving and localization results for max-product Bernstein–Kantorovich–Choquet operators are obtained. Section 4 contains quantitative approximation results for discrete max-product Picard–Kantorovich–Choquet, discrete max-product Gauss–Weierstrass–Kantorovich–Choquet operators, and discrete max-product Poisson–Cauchy–Kantorovich–Choquet operators. Section 5 deals with the approximation properties of the max-product Kantorovich–Choquet operators based on (φ, ψ)-kernels. It is worth to mention that with respect to their max-product correspondents, while they keep their good properties, the mixed max-product Choquet operators present, in addition, the advantage of a great flexibility by the many possible choices for the families of set functions used in their definitions. The results obtained present potential applications in sampling theory, neural networks, and learning theory.

We investigate suitable expressions for the mean time to extinction of the corresponding “adjoint” circuit chains describing uniquely the discrete-time birth–death model in random environments.

By combining the two versions of Brzdȩk’s fixed point theorem in non-Archimedean Banach spaces Brzdȩk and Ciepliński (Nonlinear Analy 74:6861–6867, 2011) and that in 2-Banach spaces Brzdȩk and Ciepliński (Acta Math Sci 38(2):377–390, 2018), we will investigate the hyperstability of the following σ-Jensen functional equation: f ( x + y ) + f ( x + σ ( y ) ) = 2 f ( x ) , $$\displaystyle f(x+y)+f(x+\sigma (y))=2f(x), $$ where f : X → Y such that X is a normed space, Y is a non-Archimedean 2-Banach space, and σ is a homomorphism of X. In addition, we prove some interesting corollaries corresponding to some inhomogeneous outcomes and particular cases of our main results in C∗-algebras.

We consider the d-dimensional Hermite–Hadamard inequality 1 1 S ∫ S f ( x ) d x ≤ Q tra ( f ) : = 1 ∂ S ∫ ∂ S f ( x ) d γ . $$\displaystyle {} \frac {1}{\left |S\right |} \int _{S}f({\boldsymbol x}) \, d{\boldsymbol x} \leq Q^{\text{ tra}}(f):= \frac {1}{\left |\partial S\right |} \int \limits _{\partial S}f({\boldsymbol x})d\gamma . $$ Here f is a convex function defined on a simplex S ⊂ ℝ d , ( d ∈ ℕ ) . $$S\subset \mathbb {R}^d, (d\in \mathbb {N}).$$ We give necessary and sufficient conditions on S for the validity of (1). More specifically, we establish that (1) holds if and only if S is an equiareal simplex. We will give two proofs of this result: • The first proof is based on Green’s identity. Here, in addition to the convexity requirement, the C1-regularity assumption is necessary. • In the second proof, the convexity is only required. A series of equivalent criteria for validity of (1) is simply reformulated in terms of coincidences of certain simplex centers.

In the framework of the planar circular restricted three-body problem (R3BP), we explore the effects of oblateness of the infinitesimal mass body as well as radiation pressure and triaxiality of the two primaries on the position and stability of the triangular equilibrium points (TEPs). It is found that all the involved parameters affect the positions and stability of these points. Specifically, it has been shown that TEPs are stable for 0 < μ < μc and unstable for μ c ⩽ μ ⩽ 1 ∕ 2 $$\mu _c \leqslant \mu \leqslant 1/2$$ , where μc denotes the critical mass parameter which depends on system’s parameters. In addition, all the parameters of the bigger primary, except that of triaxiality, have destabilizing tendencies resulting in a decrease in the size of the region of stability. Finally, we justify the relevance of the model in astronomy by applying it to the binary Lalande 21258 system for which the equilibrium points have been seen to be unstable.

The symmetric differential operator SDO is a simplification functioning of the recognized ordinary derivative. The purpose of this effort is to provide a study of SDO connected with the geometric function theory. These differential operators indicate a generalization of well known differential operator including the Sàlàgean differential operator. Our contribution is to deliver two classes of symmetric differential operators in the open unit disk and to describe the further development of these operators by introducing convex linear symmetric operators. In addition, by acting these SDOs on the class of univalent functions, we display a set of sub-classes of analytic functions having geometric representation, such as starlikeness and convexity properties. Investigations in this direction lead to some applications in the univalent function theory of well known formulas, by defining and studying some sub-classes of analytic functions type Janowski function, bounded turning function subclass and convolution structures. Consequently, we define a linear combination differential operator involving the Sàlàgean differential operator and the Ruscheweyh derivative. The new operator is a generalization of the Lupus differential operator. Moreover, we aim to solve some special complex boundary problems for differential equations, spatially the class of Briot-Bouquet differential equations. All solutions are symmetric under the suggested SDOs. Additionally, by using the SDOs, we introduce a generalized class of Briot-Bouquet differential equations to deliver, what is called the symmetric Briot-Bouquet differential equations. We shall show that the upper solution is symmetric in the open unit disk by considering a set of examples of univalent functions.

In this article, we first introduced a new class of generalized ((p1, p2);(ψ1, ψ2))–convex mappings and an interesting lemma regarding Hermite–Hadamard type conformable fractional integral inequalities. By using the notion of generalized ((p1, p2);(ψ1, ψ2))–convexity and lemma as an auxiliary result, some new estimates with respect to Hermite–Hadamard type integral inequalities associated with twice differentiable generalized ((p1, p2);(ψ1, ψ2))–convex mappings via conformable fractional integrals are established. It is pointed out that some new special cases can be deduced from main results of the article. At the end, some applications to special means are also given.

In the present paper, the authors establish a new version of the Hermite–Hadamard and Ostrowski type fractional integral inequalities for a class of n-polynomial s-type convex functions. Using our generalizations we are able to also deduce some already known results. We present two different techniques, for functions whose first and second derivatives in absolute value at certain powers are n-polynomial s-type convex by employing k-fractional integral operators. These techniques have yielded some interesting results. In the form of corollaries, some estimates of k-fractional integrals are obtained which contain bounds of RL-fractional integrals. We also obtain a refined bound of the Midpoint, Trapezoidal, and Simpson type inequalities for twice differentiable n-polynomial s-type convex functions.

In this chapter, by using the orthogonally fixed point method, we prove the Hyers–Ulam stability and the hyperstability of orthogonally 3-Lie homomorphisms for additive ρ-functional equation in 3-Lie algebras. Indeed, we investigate the stability and the hyperstability of the system of functional equations f ( x + y ) − f ( x ) − f ( y ) = ρ 2 f x + y 2 + f ( x ) + f ( y ) , f ( [ [ u , v ] , w ] ) = [ [ f ( u ) , f ( v ) ] , f ( w ) ] $$\displaystyle \begin{array}{@{}rcl@{}} \left \{ \begin {array}{ll} f(x+y)-f(x)-f(y)= \rho \left (2f\left (\frac {x+y}{2}\right )+ f(x)+ f(y)\right ),\\ f([[u,v],w])=[[f(u),f(v)],f(w)] \end {array} \right . \end{array} $$ in 3-Lie algebras where ρ≠1 is a fixed real number.

In this chapter, we introduce some new fractional integral operators and fractional area balance operators. The corresponding integral operator inequalities are established.

In this chapter, we introduce some very general new notions of Kg strongly convex functional in normed linear spaces. As their applications, new generalized Ostrowski type and perturbed Simpson type inequalities are established. We apply these inequalities to provide approximations for the integral of a real valued function.

In (Park et al., Rocky Mountain J Math 49:593–607, 2019), Park introduced the following bi-additive s-functional inequality 1 ∥ f ( x + y , z − w ) + f ( x − y , z + w ) − 2 f ( x , z ) + 2 f ( y , w ) ∥ ≤ s 2 f x + y 2 , z − w + 2 f x − y 2 , z + w − 2 f ( x , z ) + 2 f ( y , w ) , $$\displaystyle \begin{aligned} & \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \\ & \quad \le \left \|s \left (2f\left (\frac {x+y}{2}, z-w\right ) + 2f\left (\frac {x-y}{2}, z+w\right ) - 2f(x,z )+ 2 f(y, w)\right )\right \|,{} \end{aligned} $$ where s is a fixed nonzero complex number with |s| < 1. Using the fixed point method, we prove the Hyers–Ulam stability of ternary biderivations and ternary bihomomorphism in C∗-ternary algebras, associated with the bi-additive s-functional inequality (1).

Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of Lie biderivations and Lie bihomomorphisms in Lie Banach algebras, associated with the bi-additive functional inequality 1 ∥ f ( x + y , z + w ) + f ( x + y , z − w ) + f ( x − y , z + w ) + f ( x − y , z − w ) − 4 f ( x , z ) ∥ ≤ s 2 f x + y , z − w + 2 f x − y , z + w − 4 f ( x , z ) + 4 f ( y , w ) , $$\displaystyle \begin{aligned} & \| f(x+y, z+w) + f(x+y, z-w) + f(x-y, z+w) \\ &\qquad + f(x-y, z-w) -4f(x,z)\| \\ & \quad \le \left \|s \left (2f\left (x\kern -0.7pt+\kern -0.7pt y, z\kern -0.7pt-\kern -0.7pt w\right ) \kern -0.7pt+\kern -0.7pt 2f\left (x-y, z\kern -0.7pt+\kern -0.7pt w\right ) \kern -0.7pt-\kern -0.7pt 4f(x,z )\kern -0.7pt+\kern -0.7pt 4 f(y, w)\right )\right \|, \end{aligned} $$ where s is a fixed nonzero complex number with |s| < 1.

Let λ j j = 1 ∞ $$\left \{ \lambda _{j}\right \} _{j=1}^{\infty }$$ be a sequence of distinct positive numbers. Let w be a non-negative function, integrable on the real line. One can form orthogonal Dirichlet polynomials ϕ n $$\left \{ \phi _{n}\right \} $$ from linear combinations of λ j − i t j = 1 n $$\left \{ \lambda _{j}^{-it}\right \} _{j=1}^{n}$$ , satisfying the orthogonality relation ∫ − ∞ ∞ ϕ n t ϕ m t ¯ w t d t = δ m n . $$\displaystyle \int _{-\infty }^{\infty }\phi _{n}\left ( t\right ) \overline {\phi _{m}\left ( t\right ) }w\left ( t\right ) dt=\delta _{mn}. $$ Weights that have been considered include the arctan density w t = 1 π 1 + t 2 $$w\left ( t\right ) =\frac {1}{\pi \left ( 1+t^{2}\right ) }$$ ; rational function choices of w; w t = e − t $$w\left ( t\right ) =e^{-t}$$ ; and w t $$w\left ( t\right ) $$ constant on an interval symmetric about 0. We survey these results and discuss possible future directions.

In this paper, we present a survey of some recent results concerning generalizations and improvements of approximations of some analytic functions including trigonometric, inverse trigonometric, polynomial, and irrational functions.

In the present paper, we prove that Z $$\mathcal {Z}$$ -contractions, weakly type contractions, and some type of F-contractions are actually Meir–Keeler contractions.

Zech’s logarithm is a function closely related to the Discrete Logarithm. It has applications in communications, cryptography, and computing. In this paper, we provide polynomial and exponential formulas for Zech’s logarithm over prime fields.

We present the theoretical framework and the approximations needed to numerically simulate the response of alkali metal atoms under multi-photon excitation. By applying the semi-classical approximation, we obtain a system of coupled ordinary and partial differential equations accounting both for the nonlinear dynamics of the atomic medium and the spatiotemporal evolution of the emitted fields. The case of two-photon excitation by a laser field with an additional one-photon coupling field is investigated by numerically solving the set of differential equations employing a self-consistent computational scheme. The computation of the emission intensities and atomic level populations and coherences is then possible.

In this paper, we define and introduce some new concepts of the higher order strongly general preinvex functions and higher order strongly monotone operators involving the arbitrary bifunction. Some new relationships among various concepts of higher order strongly general preinvex functions have been established. It is shown that the new parallelogram laws for Banach spaces can be obtained as applications of higher order strongly affine general preinvex functions, which is itself a novel application. It is proved that the optimality conditions of the higher order strongly general preinvex functions are characterized by a class of variational inequalities, which are called the higher order strongly general variational-like inequalities. An auxiliary principle technique is used to suggest an implicit method for solving strongly general variational-like inequalities. Convergence analysis of the proposed method is investigated using the pseudo-monotonicity of the operator. Some special cases are also discussed. Results obtained in this paper can be viewed as a refinement and improvement of previously known results.

We consider a class of games played on networks in which the utility functions consist of both deterministic and random terms. In order to find the Nash equilibrium of the game we formulate the problem as a variational inequality in a probabilistic Lebesgue space which is solved numerically to provide approximations for the mean value of the random equilibrium. We also numerically compare the solution thus obtained, with the solution computed by solving the deterministic variational inequality derived by taking the expectation of the pseudo-gradient of the game with respect to the random parameters.

Let (X, d) be a metric space, G be a graph associated with X and f : X → X be an operator which satisfies two main assumptions: (1) f is generalized G-monotone; (2) f is a G-contraction with respect to d. In the above framework, we will present sufficient conditions under which: (i) f is a Picard operator; (ii) the fixed point problem x = f(x), x ∈ X is well-posed in the sense of Reich and Zaslavski; (iii) the fixed point problem x = f(x), x ∈ X has the Ulam-Hyers stability property; (iv) f has the Ostrowski stability property; (v) f satisfies to some Gronwall type inequalities. Some open questions are presented.

This chapter deals with the approximate solution of Fredholm integral equations and a type of integro-differential equations having non-separable kernels, as they appear in many applications. The procedure proposed consists of firstly approximating the non-separable kernel by a finite partial sum of a power series and then constructing the solution of the degenerate equation explicitly by a direct matrix method. The method, which is easily programmable in a computer algebra system, is explained and tested by solving several examples from the literature.

If R $$\mathscr {R}$$ is a family of relations on X to Y , U $$\hskip 0.2 mm \mathscr {U}$$ is a family of relations on P ( X ) $$\mathscr {P}\hskip 0.2 mm(X)$$ to Y , and V $$\mathscr {V}$$ is a family of relations on P ( X ) $$\mathscr {P}\hskip 0.2 mm(X)$$ to P ( Y ) $$\mathscr {P}\hskip 0.2 mm(Y)$$ , then we say that R $$\mathscr {R}$$ is an ordinary relator, U $$\mathscr {U}$$ is a super relator, and V $$\mathscr {V}$$ is a hyper relator on X to Y .We show that the X = Y , U = { U } $$\mathscr {U}=\{\hskip 0.2 mm U\hskip 0.2 mm\}$$ and V = { V } $$\mathscr {V}=\{\hskip 0.2 mm V\hskip 0.2 mm\}$$ particular case of the non-conventional three relator space ( X , Y ) ( R , U , V ) $$\hskip 0.2 mm(\hskip 0.2 mm X\hskip 0.2 mm, \,Y\hskip 0.2 mm)(\hskip 0.2 mm\mathscr {R}\hskip 0.2 mm, \,\mathscr {U}\hskip 0.2 mm, \,\mathscr {V}\hskip 1.2 mm)$$ can be used to treat, in a unified way, the various generalized open sets studied by a great number of topologists.

The purpose of this chapter is to survey and make a compilation that covers many families of the special numbers and polynomials including the Apostol-Bernoulli numbers and polynomials, the Apostol-Euler numbers and polynomials, the Apostol-Genocchi numbers and polynomials, the Fubini numbers, the Stirling numbers, the Frobenius-Euler polynomials, and the others, blending new results for of the polynomials Wn(x;λ), which were given in: Y. Simsek, Computation methods for combinatorial sums and Euler-type numbers related to new families of numbers, Math. Meth. Appl. Sci., 40 (2017), 2347–2361. Many well-known results of these polynomials are given in this chapter. Using these known and new results, a large number of new formulas and new relations are created. Some well-known relations among the polynomials Wn(x;λ), the Bernoulli and Euler polynomials of higher order, Apostol-type polynomials (Apostol-Bernoulli polynomials, Apostol-Euler polynomials, Apostol-Genocchi polynomials, etc.) are given. It has been presented in new relations related to these polynomials. Some open problems are raised from the results for the polynomials Wn(x;λ). Behaviors of the polynomials Wn(x;λ) under integral transforms are also examined in this chapter. Firstly, Laplace transform of the polynomials Wn(x;λ) is given. With the help of this transformation, new infinite series representations are found. Then, the behavior of the polynomials Wn(x;λ) under the Melin transform is also given with help of the works Kucukoglu et al. (Quaest Math 42(4):465–478, 2019) and Simsek (AIP Conf Proc 1978:040012-1–040012-4, 2018). With the aid of this transformation, some relationships with the family of zeta functions are also blended in detail with the previously well-known results using values from negative integers. Since these results are known to be used frequently in both approximation theory, number theory, analysis of functions, and mathematical physics, these results can potentially be used in these scientific areas. In addition, it has been tried to give a detailed perspective on the applications of the polynomials Wn(x;λ) with their generating functions in approximation theory. Firstly, with the help of computational algorithms, basic known information about numerical values and graphics of these polynomials are introduced. With the help of the algorithm given for these numerical values, the approach steps are tried to be given in detail. These details are then illustrated on graphics and shapes, so that the visual approach steps are made clearer. As a result, this chapter is compiled by blending, interpreting, and comparing the fundamental properties of the polynomials Wn(x;λ) and the numbers Wn(λ) with their generating functions and other special numbers and polynomials.

A signed graph is a graph that has a sign assigned to each of its edges. Signed graphs were introduced by Harary in 1953 in relation to certain problems in social psychology, and the matroids of signed graphs were first introduced by Zaslavsky in 1982. The investigation of the spectra of signed graphs has gained much attention in recent years by various authors. In this chapter, we focus on some of the most important results related to the eigenvalues of the adjacency and the Laplacian matrices of signed graphs.

A geometric extension is given for the perturbed contraction principle in Aydi et al. [Abstr. Appl. Anal., Volume 2013, Article ID 312479].

The aim of this chapter is to introduce the notion of G(σ, h)-convex functions a generalized exponentially (σ, h)-convex functions. We show that for suitable choices of real function h(.), the class of G(σ, h)-convex functions reduces to some other new classes of Gσ-convex functions. We also show that for G = exp $${\mathrm {G}}=\exp $$ , we have another new class which is called as G(σ, h)-convex function. For the applications of this class we derive some new variants of Hermite-Hadamard’s inequality using the class of G(σ, h)-convex functions. In the last section, we define the class of strongly G(σ, h)-convexity. We also derive a new Hermite-Hadamard like inequality involving strongly G(σ, h)-convexity. Several new special cases which can be deduced from the main results of the chapter are also discussed.