# Volume 15 (2023), Number 1-2

## On a Reduction Method Using Max-Plus Algebra for a Initial Value Problem in Classic Algebra and the Solution of the Problem

### Author(s): ZELIHA AYDOĞMUŞ and AHMET ÏPEK

**Abstract:** In this paper, we first will develop a reduction method in max-plus
algebra for the initial value problem given by $$\{\begin{array}{l}x\left(t+n\right)=\mathrm{max}\left\{{a}_{n-1}\left(t\right)+x\left(t+n-1\right),\dots ,{a}_{1}\left(t\right)+x\left(t+1\right),{a}_{0}\left(t\right)+x\left(t\right),f\left(t\right)\right\}\\ x\left({t}_{0}\right)={c}_{1},\text{\hspace{0.17em}}x\left({t}_{0}+1\right)={c}_{2},\dots ,x\left({t}_{0}+n-1\right)={c}_{n}\end{array}$$ and then we obtain the solutions to the
this equation.

## A New Type Random Iteration Scheme for Random Common Fixed Point of Three Operators

### Author(s): MUHAMMED EMIN BATUHAN and ISA YILDIRIM

**Abstract:** In this paper, we introduce a random iteration scheme for three asymptotically
nonexpansive random operators defined on a uniformly convex separable Banach space and prove its
convergence to a common fixed point of three random operators.

## Reduced-Order Modelling Based on Koopman Operator Theory

### Author(s): DIANA A. BISTRIAN, GABRIEL DIMITRIU and IONEL M. NAVON

**Abstract:** The present study focuses on a subject of significant interest in fluid dynamics:
the identification of a model with decreased computational complexity from numerical code output
using Koopman operator theory. A reduced-order modelling method that incorporates a novel strategy
for identifying the most impactful Koopman modes was used to numerically approximate the Koopman
composition operator.

## Singularities, Torsion, Cauchy Integrals and Their Spectra on Space-Time

### Author(s): DR. FRANCISCO BULNES

**Abstract:** All field sources are identified as fields φAB , which can be identified too as
poles or singularities in the complex Riemannian manifold model of the space-time including field
sources, such that their integrals can calculate their value through the Cauchy type integrals as
the Conway integrals to any loop generated in the local causal structure of the space-time around
of these fields. The integrals are solutions of the spinor equation associated to the corresponding
twistor field equation. A theorem is mentioned on the evidence of field torsion as field invariant
and geometrical invariant in poles of Cauchy type integrals in spinor-twistor frame. Then an
immediate result is that torsion existence in the space-time induces gravitational waves in a
projective bundle. Sources are evidence at least locally of torsion existence. Then exists curvature
here. Some conjectures and technical lemmas are mentioned as references of other works and is included
a new application conjecture too.

## Analytic Parametrization of the Algebraic Points of Given Degree on the Curve of Affine Equation ${y}^{2}=157\left({x}^{2}-2\right)\left({x}^{2}+x\right)\left({x}^{2}+1\right)$

### Author(s): MOHAMADOU MOR DIOGOU DIALLO

**Abstract:** We give an explicit parametrization of the set of algebraic points of given degree on
$\mathbb{Q}$ over the affine equation curve: ${y}^{2}=157\left({x}^{2}-2\right)\left({x}^{2}+x\right)\left({x}^{2}+1\right)$. This note treat aspecial case of the curves
described by Anna ARNTH-JENSEN and Victor FLYNN in [1], where the generators of the Mordell-Weil group explained.

## Some Ostrowski Type Inequalities for Two Sin-Integral Transforms of Absolutely Continuous Functions

### Author(s): SILVESTRU SEVER DRAGOMIR AND GABRIELE SORRENTINO

**Abstract:** For a Lebesgue integrable function $f:\left[a,b\right]\subset \left[-\pi /\mathrm{4,}\pi /4\right]\to \u2102$ we consider the sin-integral transforms
$${S}_{f}\left(x\right):={\displaystyle {\int}_{a}^{b}f\left(t\right)\mathrm{sin}\left(x-t\right)dt},\text{\hspace{0.17em}}x\in \left[a,b\right]$$
and
$${\tilde{S}}_{f}\left(x\right):={\displaystyle {\int}_{a}^{x}f\left(t\right)\mathrm{sin}\left(t-a\right)dt}+{\displaystyle {\int}_{x}^{b}f\left(t\right)\mathrm{sin}\left(b-t\right)dt},\text{\hspace{0.17em}}x\in \left[a,b\right]$$
We provide in this paper some upper bounds for the quantities
$$\left|f\left(b\right)\mathrm{cos}\left(b-x\right)-f\left(a\right)\mathrm{cos}\left(x-a\right)-{S}_{f}\left(x\right)\right|$$
and
$$\left|{\tilde{S}}_{f}\left(x\right)-\left[f\left(a\right)+f\left(b\right)-2\mathrm{cos}\left(\frac{b-a}{2}\right)\mathrm{cos}\left(x-\frac{a+b}{2}\right)f\left(x\right)\right]\right|$$
for $x\in \left[a,b\right]$, in terms of the $p$-norms of the derivative
${f}^{\prime}$ for absolutely continuous functions
$f:\left[a,b\right]\subset \left[-\pi /\mathrm{4,}\pi /4\right]\to \u2102$.

## Trigonometrically $tgs$-Convexity

### Author(s): HURIYE KADAKAL and MAHIR KADAKAL

**Abstract:** In this manuscript, we introduce and study the concept of trigonometrically $tgs$-convex function and prove two Hermite-Hadamard type integral inequalities
for the newly introduced class of functions. Also, some applications to special means of real numbers are also given.

## Generalized Fresnel Integrals and the Dirac Representative Sequences Generated by Them

### Author(s): WILHELM W. KECS

**Abstract:** The Fresnel cosine and sine integrals are generalized in the Euclidean space ${\mathbb{R}}^{n}$, $n\ge 2$. Two families of functions are associated with them and it is shown that they converge in the sense
of distributions towards Dirac’s distribution. The properties of these Dirac representative sequences are established, and
the obtained results are exemplified in cases $n=\mathrm{1,2,3}$.

## Generalized Biconvex Functions and Directional Bivariational Inequalities

### Author(s): MUHAMMAD ASLAM NOOR and KHALIDA INAYAT NOOR

**Abstract:** Some new classes of generalized biconvex sets and biconvex functions with respect to an arbitrary function
$k$ and the bifunction $\beta \left(.-.\right)$ are introduced and studied. These new biconvex functions
are nonconvex functions and include the convex functions and $k$-convex as special cases.
We study some basic properties of generalized biconvex functions. It is shown that the minimum of generalized biconvex functions on
the generalized biconvex sets can be characterized by a class of variational inequalities, which is called the directional bivariational
inequalities. Using the auxiliary technique, several new inertial type methods for solving the bivariational inequalities are proposed
and analyzed. Convergence analysis of the proposed methods is considered under suitable pseudomonotonicity, which are weaker conditions
than monotonicity. Some open problems are also suggested for future research.

## On the Solution of a Linear Matrix Difference Equation

### Author(s): AHMET ÖZDAĞ and AHMET ÏPEK

**Abstract:** In this paper we present the closed-form expression for the solution of the linear matrix difference equation
$${X}_{n+1}=A{X}_{n}+{X}_{n}B,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}X\left(0\right)=C,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=\mathrm{0,1,2,}\dots $$
which we obtained with the aid of the Kronecker sum, Kronecker product and $Vec$ operator.

## Application of the Least Squares Method to Study the Correlation Between Climatic Factors in Romania

### Author(s): ANAMARIA POPESCU

**Abstract:** The paper presents linear and non-linear regression mathematical models that estimate the evolution of processes or
phenomena based on some parameters that define the processes and phenomena in order to perform calculations and approximations of
experimental data. Having a series of data on climatic factors, the analysis of the results from a period from 1901 to the present,
consisting of decades, is presented, approaching the approximation by the least squares method and verifying the existence of a correlation
and identifying the model.