<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>In this paper, we consider the performance evaluation of two retrial queueing system. Customers arrive to the system, if upon arrival, the queue is full, the new arriving customers either move into one of the orbits, from which they make a new attempts to reach the primary queue, until they find the server idle or balk and leave the system, these later, and after getting a service may comeback to the system requiring another service. So, we derive for this system, the joint distribution of the server state and retrial queue lengths. Then, we give some numerical results that clarify the relationship between the retrials, arrivals, balking rates, and the retrial queue length.</abstract>

Performance evaluation of two Markovian retrial queueing model with balking and feedback

<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 the paper, the authors find several identities, including a new recurrence relation for the Stirling numbers of the first kind, involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers.</abstract>

Department of Mathematics, College of Science, Tianjin Polytechnic University,

Several identities involving the falling and rising factorials and the Cauchy, Lah, and Stirling numbers

<abstract>
<title style='display:none'>Abstract</title>
In this paper, we study two generalizations of dual-complex Lucas-balancing numbers: dual-complex k-Lucas balancing numbers and dual-complex k-Lucas-balancing numbers. We give some of their properties, among others the Binet formula, Catalan, Cassini, d’Ocagne identities.
</abstract>

Two generalizations of dual-complex Lucas-balancing numbers

<abstract><title style='display:none'>Abstract</title>In this paper, we propose a new approximate method, namely fractional natural decomposition method (FNDM) in order to solve a certain class of nonlinear time-fractional wave-like equations with variable coefficients. The fractional natural decomposition method is a combined form of the natural transform method and the Adomian decomposition method. The nonlinear term can easily be handled with the help of Adomian polynomials which is considered to be a clear advantage of this technique over the decomposition method. Some examples are given to illustrate the applicability and the easiness of this approach.</abstract>

Fractional natural decomposition method for solving a certain class of nonlinear time-fractional wave-like equations with variable coefficients

<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

<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>
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 = 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>
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>
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>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> 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> In this article, the time fractional order Burgers equation has been solved by quadratic B-spline Galerkin method. This method has been applied to three model problems. The obtained numerical solutions and error norms L2 and L∞ have been presented in tables. Absolute error graphics as well as those of exact and numerical solutions have been given.</abstract>

Department of Mathematics, Faculty of Science and Art, Inönü University, Turkey

Numerical solution of time fractional Burgers equation

<abstract><title style='display:none'>Abstract</title>In this note we point out interesting connections among Lah numbers, Laguerre polynomials of order negative one, and exponential polynomials. We also discuss several different expressions for the nth derivative of exp(1/<italic>x</italic>). A new representation of this derivative is given in terms of exponential polynomials.</abstract>

Lah numbers, Laguerre polynomials of order negative one, and the nth derivative of exp(1/<italic>x</italic>)

<abstract><title style='display:none'>Abstract</title>In the paper, the authors find necessary and sufficient conditions such that a function related to the Catalan-Qi function, which is an alternative generalization of the Catalan numbers, is logarithmically complete monotonic.</abstract>

Institute of Mathematics, Henan Polytechnic University, China; College of Mathematics, Inner Mongolia University for Nationalities, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, China

School of Mathematics and Informatics, Henan Polytechnic University, China

Logarithmically complete monotonicity of a function related to the Catalan-Qi function

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