# Volume 14 (2022), Number 2

## Mathematical Considerations on Randomized Orthogonal Decomposition Method for Developing Twin Data Models

### Author(s): DIANA A. BISTRIAN

**Abstract:** This paper introduces the approach of Randomized Orthogonal Decomposition (ROD) for producing twin data
models in order to overcome the drawbacks of existing reduced order modelling techniques. When compared to Fourier empirical
decomposition, ROD provides orthonormal shape modes that maximize their projection on the data space, which is a significant benefit.
A shock wave event described by the viscous Burgers equation model is used to illustrate and evaluate the novel method. The
new twin data model is thoroughly evaluated using certain criteria of numerical accuracy and computational performance.

## Bounds for the Normalized Determinant of Hadamard Product of Two Positive Operators in Hilbert Spaces

### Author(s): SILVESTRU SEVER DRAGOMIR

**Abstract:** For positive invertible operators $A$ on a Hilbert space
$H$ and a fixed unit vector $x\in H$, define the normalized determinant by ${\Delta}_{x}\left(A\right):=\mathrm{exp}\left(\mathrm{ln}Ax,x\right)$. In this paper we obtain upper and lower bounds for the determinant
${\Delta}_{x}\left(A\circ B\right)$ of the Hadamard product of two operators under some natural assumptions such as
$0<{m}_{1}\le A\le {M}_{1}$ and $0<{m}_{2}\le B\le {M}_{2}$, where ${m}_{i}$, ${M}_{i}$
($i=\mathrm{1,2}$) are constants.

## ${G}_{g}$ - Convex Functions

### Author(s): SERCAN GÜLSU, MAHIR KADAKAL and İMDAT İŞCAN

**Abstract:** In this paper, the concept of ${G}_{g}$-convex
function is given the first time in the literature. Some inequalities of Hadamard’s type for ${G}_{g}$-convex functions are given. Some algebraic properties of ${G}_{g}$-convex functions and special cases are discussed. In addition, we establish some new integral
inequalities for ${G}_{g}$-convex functions by using an integral
identity.

## On Physical and Mathematical Wave Fronts in Temperature Waves

### Author(s): NASSAR H.S. HAIDAR

**Abstract:** A rather ”tenuous” existence of mathematical wavefronts in parabolic temperature waves is revealed to accompany a
certain hyperbolicity dormant in these waves. The revelation is based on a proof that temperature waves do satisfying a certain new
telegrapher’s equation, equivalent to Fourier’s heat conduction equation. This parabolic-equivalent hyperbolic heat equation happens
to be similar to the famous Cattaneo-Vernotte non-Fourier heat conduction equation. A basic result of this work is that temperature
waves can mathematically support proper wavefronts of infinite span. Physically, however they can support wavefronts only in ”shortened”
form. The paper reports also on an associated shrinkage of a triangle for detectable wavefronts of such waves, and on an unknown
frequency dependence of the inclination of wavefronts in classical (parabolic) temperature waves. This, added to the strong spatial
damping and significant dispersion of these waves, has been forming a pathological obstacle in the experimental verification of their
support to conventional wavefronts.

## Impulsive Fractional Differential Equations Involving the Caputo-Hadamard Fractional Derivative in a Banach Space

### Author(s): AMOURIA HAMMOU and SAMIRA HAMANI

**Abstract:** In this paper we establish existence results for a class of initial value problems for impulsive fractional
differential equations involving the Caputo-Hadamard fractional derivative of order $1<r\le 2$.

## Convergence Results for Sequential Henstock Stieltjes Integral in Real Valued Space

### Author(s): ILUEBE V.O. and MOGBADEMU A.A.

**Abstract:** In this paper, we prove the convergence theorems for the Sequential Henstock Stieltjes integral of the real
valued functions and give an example to show its applicability.

## New 3D Thermoelastic Influence Functions, Caused by a Unitary Point Heat Source, Applied in a Quarter of Layer

### Author(s): VICTOR ȘEREMET and ION CREȚU

**Abstract:** The aim of this paper consist in the constructing of the main thermoelastic displacements Green’s functions
(MTDGFs) for a generalized 3D BVP of uncoupled thermoelasticity for a quarter of layer. To reach this aim are derived structural
formulas for MTDGFs, expressed via respective Green’s functions for Poisson’s equation (GFPE) by using harmonic integral
representations method (HIRM). These structural formulas are validated by the checking the equations of thermoelasticity with respect
to point of response in which the thermoelastic displacements appeared and with respect to point of application the heat source and
the nonhomogeneous Poisson’s equation. In addition, they satisfy the homogeneous mechanical boundary conditions for MTDGFs with respect
to point application the displacements and to mechanical boundary conditions and temperature Green’s function with respect to point of
application the heat source. The thermoelastic volume dilatation (TVD) derived separately from respective integral representations has
been equal to the TVD derived by using structural formulas for MTDGFs. The final analytical expressions for MTDGFs obtained on the base
of mentioned above structural formulas for sixteen new 3D BVPs of thermoelasticity within quarter of layer contain Bessel functions of
the zero-order of the second type. These results are presented graphically.

## Some Sum Formulas and New Identities of Bi-Periodic Jacobsthal and Bi-Periodic Jacobsthal Lucas Sequence

### Author(s): SUKRAN UYGUN

**Abstract:** In this article, it is considered that the new generalizations of the Jacobsthal sequence and Jacobsthal-Lucas
sequence. These sequences can arise in the study of continued fractions of quadratic irrationals. Some well-known sequences are special
cases of this generalization. Jacobsthal and Jacobsthal-Lucas sequence is a special case with $a=b=1$. Some new identities and properties of these generalized sequences are
investigated with the aid of its Binet formula, recurrence relation etc. We study specially some different sum formulas for these
sequences. Then, by these formulas we get properties for Jacobsthal sequence, Jacobsthal-Lucas sequence.