Mathematica Moravica, Vol. 27, No. 2 (2023)
Nonlinear contractions and Caputo tempered impulsive implicit fractional differential equations in $b$-metric spaces
Mathematica Moravica, Vol. 27, No. 2 (2023), 1–24.
doi: https://doi.org/10.5937/MatMor2302001K
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Abstract and keywords
Abstract. This paper deals with some existence and uniqueness
results for a class of problems for nonlinear Caputo tempered implicit fractional
differential equations in $b$-Metric spaces with initial nonlocal conditions and instantaneous impulses.
The results are based on the ${\omega}-{\delta}$-Geraghty type contraction, the $F$-contraction and the fixed point theory.
Furthermore, some illustrations are presented to demonstrate the plausibility of our results.
Keywords. Fixed point, implicit differential equations, impulses,
tempered fractional derivative, ${\omega}-{\delta}$-Geraghty contraction, $F$-contraction, nonlocal condition.
Blow-up phenomena for a $p(x)$-biharmonic heat equation with variable exponent
Mathematica Moravica, Vol. 27, No. 2 (2023), 25–32.
doi: https://doi.org/10.5937/MatMor2302025P
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Abstract and keywords
Abstract. In this paper, we deal with a $p(x)$-biharmonic heat equation with variable exponent under Dirichlet
boundary and initial condition. We prove the blow up of solutions under
suitable conditions.
Keywords. Blow up, heat equation, variable exponent.
Fixed and coincidence point theorems on partial metric spaces with an application
Mathematica Moravica, Vol. 27, No. 2 (2023), 33–53.
doi: https://doi.org/10.5937/MatMor2302033KK
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Abstract and keywords
Abstract. The aim of this paper is to investigate some fixed
and coincidence point theorems in complete, orbitally complete
and $(T, f)$-orbitally complete partial metric spaces under the generalized contractive type conditions of mappings.
Moreover, our results generalize and extend the several obtained results in the literature.
Additionally some non-trivial examples are demonstrated, and an application has discussed to integral equations.
Keywords. Fixed point, coincidence point, orbital continuity, orbital completeness, partial metric and Hausdorff metric.
A Study on lung cancer using nabla discrete fractional-order model
Mathematica Moravica, Vol. 27, No. 2 (2023), 55–76.
doi: https://doi.org/10.5937/MatMor2302055A
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Abstract and keywords
Abstract. This study proposes a nabla discrete fractional-order system of differential
equations to model lung cancer and its interactions with lung epithelial cells, mutated cells, oncogenes, tumor suppressor genes,
immune cells, cytokines, growth factors, angiogenic factors, and extracellular matrix.
The proposed model can help predict cancer growth, metastasis, and response to treatment.
Analytical results show the system is stable with a unique solution, and the model predicts that
the immune system responds to cancer cells but eventually becomes overpowered.
The numerical analysis employed the forward and backward Euler method and demonstrated that
changes in parameter values have significant effects on the steady-state solution.
The findings show that the growth of lung epithelial cells or their interaction with immune cells can cause
an increase in the number of lung cancer cells. Conversely, an increase in cell death or
a reduction in the interaction between lung epithelial cells and immune cells can decrease the number of lung cancer cells.
The study highlights the usefulness of the nabla discrete fractional model in studying lung cancer dynamics.
Keywords. Mathematical modeling, Discrete fractional-order, Nable difference operator, Lung cancer.
Suspension bridge model with laminated beam
Mathematica Moravica, Vol. 27, No. 2 (2023), 77–90.
doi: https://doi.org/10.5937/MatMor2302077R
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Abstract and keywords
Abstract. This manuscript introduces a suspension bridge system where laminated beams model the deck.
The action of frictional damping is considered.
Well-posedness is proved using the Lumer-Phillips theorem, and the exponential stability is
obtained by applying the Gearhart-Huang-Prüss theorem.
Keywords. Suspension bridge, laminated beam, Timoshenko system.
Graph polynomials associated with Dyson--Schwinger equations
Mathematica Moravica, Vol. 27, No. 2 (2023), 91–114.
doi: https://doi.org/10.5937/MatMor2302091S
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Abstract and keywords
Abstract. Quantum motions are encoded by a particular family of
recursive Hochschild equations in the renormalization Hopf algebra which represent Dyson--Schwinger equations, combinatorially.
Feynman graphons, which topologically complete the space of Feynman diagrams of a gauge field theory,
are considered to formulate some random graph representations for solutions of quantum motions.
This framework leads us to explain the structures of Tutte and Kirchhoff--Symanzik polynomials associated with solutions of Dyson--Schwinger equations.
These new graph polynomials are applied to formulate a new parametric representation of large Feynman diagrams and their corresponding Feynman rules.
Keywords. Combinatorial Dyson--Schwinger equations, Feynman graphons, Graph polynomials, Parametric representations.
On the bi-periodic Padovan sequences
Mathematica Moravica, Vol. 27, No. 2 (2023), 115–126.
doi: https://doi.org/10.5937/MatMor2302115D
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Abstract and keywords
Abstract. In this study, we define a new generalization of
the Padovan numbers, which shall also be called the bi-periodic Padovan sequence.
Also, we consider a generalized bi-periodic Padovan matrix sequence.
Finally, we investigate the Binet formulas, generating functions,
series and partial sum formulas for these sequences.
Keywords. Padovan sequence, Binet-like formula, generating function,
biperiodic Padovan sequence, matrix.
$k-$regular decomposable incidence structure of maximum degree
Mathematica Moravica, Vol. 27, No. 2 (2023), 127–136.
doi: https://doi.org/10.5937/MatMor2302127S
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Abstract and keywords
Abstract. This paper discusses incidence structures and their rank.
The aim of this paper is to prove that there exists a regular decomposable incidence structure $ \mathcal{J}=\left(\mathbb{P},\mathcal{B} \right) $
of maximum degree depending on the size of the set and a predetermined rank.
Furthermore, an algorithm for construction of this structures is given.
In the proof of the main result, the points of the set $\mathbb{P}$ are shown by Euler’s formula of complex number.
Two examples of construction the described incidence structures of maximum degree 6 and maximum degree 30 are given.
Keywords. Regular incidence structure, partition, Euler’s formula of complex number.