<abstract><title style='display:none'>Abstract</title> In this paper our aim is to find the solutions of time and space fractional heat differential equations by using new definition of fractional derivative called conformable fractional derivative. Also based on conformable fractional derivative definition conformable Fourier Transform is defined. Fourier sine and Fourier cosine transform definitions are given and space fractional heat equation is solved by conformable Fourier transform.</abstract>

Department of Mathematics, Faculty of Science, Selçuk University, Turkey

Department of Mathematics, Faculty of Science and Art, Mustafa Kemal University, Turkey

The solutions of time and space conformable fractional heat equations with conformable Fourier transform

<abstract><title style='display:none'>Abstract</title> We survey arithmetic and asymptotic properties of the alternating sum-of-divisors function β defined by β(pa) = pa − pa−1 + pa−2 − · · · + (−1)a for every prime power pa (a ≥ 1), and extended by multiplicativity. Certain open problems are also stated.</abstract>

A survey of the alternating sum-of-divisors function

<abstract><title style='display:none'>Abstract</title>For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph ΓE(R) with vertex set Z(RE) \ {[0]} = RE \ {[0], [1]} defined by RE = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph ΓE(R), the relationship of metric dimension between ΓE(R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of ΓE(R). We provide a formula for the number of vertices of the family of graphs given by ΓE(R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of ΓE(R).</abstract>

Computing metric dimension of compressed zero divisor graphs associated to rings

<abstract><title style='display:none'>Abstract</title>This article is devoted to study Elzaki transform and its applications in Free Electron Laser equation involving Hilfer-Prabhakar fractional derivative. We derive formula of Elzaki transform for Hilfer–Prabhakar derivative and its regularized version. The solution of Free Electron Laser equation involving Hilfer-Prabhakar fractional derivative of fractional order is presented in terms of Mittag-Leffler type function. Furthermore, we find the application of the generalized Hilfer-Prabhakar derivative in linear partial differential equation and some problems of Mathematical Physics.</abstract>

On the Elzaki transform and its applications in fractional free electron laser equation

<abstract><title style='display:none'>Abstract</title> Worst, best and average number of messages and running time of leader election algorithms of different distributed systems are analyzed. Among others the known characterizations of the expected number of messages for LCR algorithm and of the worst number of messages of Hirschberg-Sinclair algorithm are improved</abstract>

Leader election in synchronous networks

<abstract><title style='display:none'>Abstract</title>In this paper, we introduce and investigate new subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers. Furthermore, we find estimates of first two coefficients of functions in these classes. Also, we determine Fekete-Szegö inequalities for these function classes.</abstract>

Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers

<abstract>
<title style='display:none'>Abstract</title>
The aim is to introduce some relation theoretic variants of F−contraction in an F−metric space endowed with a binary relation <italic>R</italic> and to prove results for its fixed point. In the sequel, several classes of contractions are sharpened, generalized, and improved. Numerical examples are presented to illustrate the theoretical conclusions. As applications of the main results, we solve a Dirichlet-Neumann initial value problem and two Dirichlet boundary value problems.
</abstract>

On relation-theoretic F−contractions and applications in F−metric spaces

<abstract>
<title style='display:none'>Abstract</title>
Let G = (V, E) be a simple graph with n vertices and m edges. ν(G) and c(G) = m − n + θ be the matching number and cyclomatic number of G, where θ is the number of connected components of G, respectively. Wang and Wong in [<xref ref-type="bibr" rid="j_ausm-2023-0019_ref_018">18</xref>] provided formulae for the upper and the lower bounds of the nullity η(G) of G as η(G) = n − 2ν(G) + 2c(G) and η(G) = n − 2ν(G) − c(G), respectively. In this paper, we restate the upper and the lower bounds of nullity η(G) of G utilizing omega invariant and inherently vertex degrees of G. Also, in the case of the maximal and the minimal nullity conditions are satisfied for G, we present both two main inequalities and many inequalities in terms of Omega invariant, analogously cyclomatic number, number of connected components and vertex degrees of G.
</abstract>

A survey of the maximal and the minimal nullity in terms of omega invariant on graphs

<abstract>
<title style='display:none'>Abstract</title>
Let 𝒢 = 𝒢 (A, M, N, B) be a generalized matrix algebra over a commutative ring with unity. In the present article, we study k-semi-centralizing maps of generalized matrix algebras.
</abstract>

On k-semi-centralizing maps of generalized matrix algebras

<abstract>
<title style='display:none'>Abstract</title>
We show the uniformly boundedness of the L1 norm of general matrix transform kernel functions with respect to the Walsh-Paley system. Special such matrix means are the well-known Cesàro, Riesz, Bohner-Riesz means. Under some conditions, we verify that the kernels <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0013_eq_001.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>K</mml:mtext></mml:mrow><mml:mtext>n</mml:mtext><mml:mtext>T</mml:mtext></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>k</mml:mtext><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mtext>n</mml:mtext></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mtext>t</mml:mtext></mml:mrow><mml:mrow><mml:mtext>k</mml:mtext><mml:mo>,</mml:mo><mml:mtext>n</mml:mtext></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mtext>D</mml:mtext></mml:mrow><mml:mtext>k</mml:mtext></mml:msub></mml:mrow></mml:mrow></mml:math>
<tex-math>{\rm{K}}_{\rm{n}}^{\rm{T}} = \sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{D}}_{\rm{k}}}}</tex-math>
</alternatives>
</inline-formula>, (where Dk is the kth Dirichlet kernel) satisfy
<disp-formula>
<alternatives>
<graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0013_eq_002.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mrow><mml:mo>‖</mml:mo><mml:mrow><mml:msubsup><mml:mrow><mml:mtext>K</mml:mtext></mml:mrow><mml:mtext>n</mml:mtext><mml:mtext>T</mml:mtext></mml:msubsup></mml:mrow><mml:mo>‖</mml:mo></mml:mrow></mml:mrow></mml:mrow><mml:mn>1</mml:mn></mml:msub><mml:mo>≤</mml:mo><mml:mtext>c</mml:mtext><mml:mtext>.</mml:mtext></mml:mrow></mml:math>
<tex-math>{\left\| {{\rm{K}}_{\rm{n}}^{\rm{T}}} \right\|_1} \le {\rm{c}}{\rm{.}}</tex-math>
</alternatives>
</disp-formula>
As a result of this we prove that for any 1 <italic>≤</italic> p &lt; ∞ and f ∈ Lp the Lp-norm convergence <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0013_eq_003.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>k</mml:mtext><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mtext>n</mml:mtext></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mtext>t</mml:mtext></mml:mrow><mml:mrow><mml:mtext>k</mml:mtext><mml:mo>,</mml:mo><mml:mtext>n</mml:mtext></mml:mrow></mml:msub><mml:msub><mml:mrow><mml:mtext>S</mml:mtext></mml:mrow><mml:mtext>k</mml:mtext></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mtext>f</mml:mtext><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>→</mml:mo><mml:mtext>f</mml:mtext></mml:mrow></mml:math>
<tex-math>\sum\nolimits_{{\rm{k = 1}}}^{\rm{n}} {{{\rm{t}}_{{\rm{k}},{\rm{n}}}}{{\rm{S}}_{\rm{k}}}\left( {\rm{f}} \right)} \to {\rm{f}}</tex-math>
</alternatives>
</inline-formula> holds. Besides, for each integrable function f we have that these means converge to f almost everywhere.
</abstract>

Norm and almost everywhere convergence of matrix transform means of Walsh-Fourier series

<abstract>
<title style='display:none'>Abstract</title>
Let <italic>S</italic> denote the class of functions f which are analytic and univalent in the unit disk 𝔻 = {z : |z| &lt; 1} and normalized with <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0017_eq_001.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mtext>f</mml:mtext><mml:mrow><mml:mo>(</mml:mo><mml:mtext>z</mml:mtext><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mtext>z</mml:mtext><mml:mo>+</mml:mo><mml:msubsup><mml:mo>∑</mml:mo><mml:mrow><mml:mtext>n</mml:mtext><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow><mml:mo>∞</mml:mo></mml:msubsup><mml:mrow><mml:msub><mml:mrow><mml:mi>α</mml:mi></mml:mrow><mml:mtext>n</mml:mtext></mml:msub><mml:msup><mml:mrow><mml:mtext>z</mml:mtext></mml:mrow><mml:mtext>n</mml:mtext></mml:msup></mml:mrow></mml:mrow></mml:math>
<tex-math>{\rm{f}}\left( {\rm{z}} \right) = {\rm{z}} + \sum\nolimits_{{\rm{n = 2}}}^\infty {{\alpha _{\rm{n}}}{{\rm{z}}^{\rm{n}}}}</tex-math>
</alternatives>
</inline-formula>. Using a method based on Grusky coefficients we study two problems over the class <italic>S</italic>: estimate of the fourth logarithmic coefficient and upper bound of the coefficient difference |α5| − |α4|.
</abstract>

Two applications of Grunsky coefficients in the theory of univalent functions

<abstract>
<title style='display:none'>Abstract</title>
The aim is to utilize a new metric called an <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0015_eq_002.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mtext>v</mml:mtext><mml:mtext>b</mml:mtext></mml:msubsup></mml:mrow></mml:math>
<tex-math>M_{\rm{v}}^{\rm{b}}</tex-math>
</alternatives>
</inline-formula>–metric which is an improvement and generalization of Mv−metric to revisit the celebrated Banach and Sehgal contractions in <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0015_eq_003.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mtext>v</mml:mtext><mml:mtext>b</mml:mtext></mml:msubsup></mml:mrow></mml:math>
<tex-math>M_{\rm{v}}^{\rm{b}}</tex-math>
</alternatives>
</inline-formula>–metric space. We demonstrate that the collection of open balls forms a basis on <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0015_eq_004.png"/>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msubsup><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mtext>v</mml:mtext><mml:mtext>b</mml:mtext></mml:msubsup></mml:mrow></mml:math>
<tex-math>M_{\rm{v}}^{\rm{b}}</tex-math>
</alternatives></inline-formula>-metric space. Further, we give some examples for the verification of established results. Towards the end, we solve a non-linear matrix equation and an equation of rotation of a hanging cable to substantiate the utility of these extensions.
</abstract>

Fixed point in <inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2023-0015_eq_001.png"/>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>ℳ</mi></mrow><mtext>v</mtext><mtext>b</mtext></msubsup></mrow></math>
<tex-math>M_{\rm{v}}^{\rm{b}}</tex-math>
</alternatives>
</inline-formula>– metric space and applications

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