rss_2.0Mathematics FeedSciendo RSS Feed for Mathematics Feed was the River Ister in the Time of Strabo? A Mathematical Approach<abstract><title style='display:none'>Abstract</title><p>We introduce a novel method for map registration and apply it to transformation of the river Ister from <italic>Strabo’s map of the World</italic> to the current map in the World Geodetic System. This transformation leads to the surprising but convincing result that Strabo’s river Ister best coincides with the nowadays Tauernbach-Isel-Drava-Danube course and not with the Danube river what is commonly assumed. Such a result is supported by carefully designed mathematical measurements and it resolves all related controversies otherwise appearing in understanding and translation of Strabo’s original text. Based on this result, we also show that <italic>Strabo’s Suevi in the Hercynian Forest</italic> corresponds to the Slavic people in the Carpathian-Alpine basin and thus that the compact Slavic settlement was there already at the beginning of the first millennium AD.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillation Results for Third-Order Quasi-Linear Emden-Fowler Differential Equations with Unbounded Neutral Coefficients<abstract><title style='display:none'>Abstract</title><p>Some new oscillation criteria are obtained for a class of thirdorder quasi-linear Emden-Fowler differential equations with unbounded neutral coefficients of the form <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0028_eq_001.png"/><mml:math xmlns:mml="" display="inline"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>z</mml:mi><mml:mo>″</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mi>α</mml:mi></mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mo>′</mml:mo><mml:mo>+</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi>y</mml:mi><mml:mi>λ</mml:mi></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>t</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>\[(a(t){(z(t))^\alpha })' + f(t){y^\lambda }(g(t)) = 0,\]</tex-math></alternatives></inline-formula> where <italic>z</italic>(<italic>t</italic>) = <italic>y</italic>(<italic>t</italic>) + <italic>p</italic>(<italic>t</italic>)<italic>y</italic>(<italic>σ</italic>(<italic>t</italic>)) and <italic>α, λ</italic> are ratios of odd positive integers. The established results generalize, improve and complement to known results.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00A Fractional Order Delay Differential Model for Survival of Red Blood Cells in an Animal: Stability Analysis<abstract><title style='display:none'>Abstract</title><p>In this paper, we analyse stability of survival of red blood cells in animal fractional order model with time delay. Results have been illustrated by numerical simulations.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Finite Volume Schemes for the Affine Morphological Scale Space (Amss) Model<abstract><title style='display:none'>Abstract</title><p>Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter <italic>θ</italic>, 0 ≤ <italic>θ</italic> ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (<italic>θ</italic> = 0), semi-implicit, fully-implicit (<italic>θ</italic> = 1) and Crank-Nicolson (<italic>θ</italic> = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Improvement and Handling of the Segmentation Model with an Inflation Term<abstract><title style='display:none'>Abstract</title><p>The use of balloon models to address the problems of “snakes” based models was introduced by Laurent D. Cohen. This paper presents a geodesic active contours model with a modified external force term that includes a balloon model. This balloon model makes the segmentation surface to behave like a balloon inflated by the external forces. In this paper, we show an automatic way to control the behaviour of the external force with respect to the segmentation evolution. The external forces, comprised of edge and inflation terms, push the segmentation surface to edges, while curvature regularizes the evolution. As segmentation evolves, the influence of the applied inflation force is determined by how close we are to the edges. With this setup, the initial segmentation does not need to be close to the object’s edges, instead it is inflated by the balloon model towards the edges. Closer to the edges, the influence of the inflation force is adjusted accordingly. The force’s influence is completely turned off when the evolution is stable (reached the edges), then only the curvature and edge information is used to evolve the segmentation.</p><p>This approach solves the issues associated with inclusion of balloon model. These issues are that the inflation force can overpower forces from weak edges, or they can cause the contour to be slightly larger than the actual minima. We present examples of the improved model for segmentation of human bladder images. Weak edges are more prevalent in medical images, and the automated handling of the inflation forces gives promising results for this kind of images.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problems<abstract><title style='display:none'>Abstract</title><p>In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (<italic>θ</italic>-method, <italic>θ</italic> ∈ [1/2, 1] (<italic>θ</italic> = 1 corresponds to the backward Euler method and <italic>θ</italic> = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders <italic>O</italic>(Δ<italic>t</italic> + <italic>h</italic><sup>3</sup>) for the backward Euler method and <italic>O</italic>(Δ<italic>t</italic><sup>2</sup> + <italic>h</italic><sup>3</sup>) for the Crank-Nicolson method, where Δ<italic>t</italic> and <italic>h</italic> are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Solving Nonlinear Volterra-Fredholm Integral Equations using an Accurate Spectral Collocation Method<abstract><title style='display:none'>Abstract</title><p>In this paper, we present a Jacobi spectral collocation method to solve nonlinear Volterra-Fredholm integral equations with smooth kernels. The main idea in this approach is to convert the original problem into an equivalent one through appropriate variable transformations so that the resulting equation can be accurately solved using spectral collocation at the Jacobi-Gauss points. The convergence and error analysis are discussed for both <italic>L</italic><sup>∞</sup> and weighted <italic>L</italic><sup>2</sup> norms. We confirm the theoretical prediction of the exponential rate of convergence by the numerical results which are compared with well-known methods.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Application of Logical Regression Function Model in Credit Business of Commercial Banks<abstract> <title style='display:none'>Abstract</title> <p>This paper takes the credit risk management of commercial banks in China as the mainline, and puts forward a quantitative model that is suitable for the credit risk management of commercial banks in China at present – Logistic regression model, and takes a commercial bank as an example, using the regression model to conduct empirical research on the credit risk of enterprises. The estimated Logistic model was tested with confirmation samples. The results show that when the cut-off point is set to 0.5, the overall correct rate of the model for the credit risk measurement of natural persons and for enterprises reaches 84.9% and 88%, respectively. When the cut-off point is set at 0.7, the overall accuracy is 89.2%. In general, the results of credit risk measurement of bank customers by the Logistic model are quite satisfactory. The Logistic Regression model is easy to understand and efficient, so it is worth popularising and putting into practice in commercial banks in China.</p> </abstract>ARTICLE2021-12-27T00:00:00.000+00:00Extreme Values of Euler-Kronecker Constants<abstract> <title style='display:none'>Abstract</title> <p>In a family of <italic>S<sub>n</sub></italic>-fields (<italic>n ≤</italic> 5), we show that except for a density zero set, the lower and upper bounds of the Euler-Kronecker constants are <italic>−</italic>(<italic>n −</italic> 1) log log <italic>d<sub>K</sub></italic>+ <italic>O</italic>(log log log <italic>d<sub>K</sub></italic>) and loglog <italic>d<sub>K</sub></italic> + <italic>O</italic>(log log log <italic>d<sub>K</sub></italic>), resp., where <italic>d<sub>K</sub></italic> is the absolute value of the discriminant of a number field <italic>K</italic>.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00On the Classification of Solutions of Quantum Functional Equations with Cyclic and Semi-Cyclic Supports<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we classify all solutions with cyclic and semi-cyclic semigroup supports of the functional equations arising from multiplication of quantum integers with fields of coefficients of characteristic zero. This also solves completely the classification problem proposed by Melvyn Nathanson and Yang Wang concerning the solutions, with semigroup supports which are not prime subsemigroups of ℕ, to these functional equations for the case of rational field of coefficients. As a consequence, we obtain some results for other problems raised by Nathanson concerning maximal solutions and extension of supports of solutions to these functional equations in the case where the semigroup supports are not prime subsemigroups of ℕ.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00Uniform Distribution of the Weighted Sum-of-Digits Functions<abstract> <title style='display:none'>Abstract</title> <p>The higher-dimensional generalization of the weighted <italic>q</italic>-adic sum-of-digits functions <italic>s<sub>q,γ</sub></italic>(<italic>n</italic>), <italic>n</italic> =0, 1, 2,..., covers several important cases of sequences investigated in the theory of uniformly distributed sequences, e.g., <italic>d</italic>-dimensional van der Corput-Halton or <italic>d</italic>-dimensional Kronecker sequences. We prove a necessary and sufficient condition for the higher-dimensional weighted <italic>q</italic>-adic sum-of-digits functions to be uniformly distributed modulo one in terms of a trigonometric product. As applications of our condition we prove some upper estimates of the extreme discrepancies of such sequences, and that the existence of distribution function <italic>g</italic>(<italic>x</italic>)= <italic>x</italic> implies the uniform distribution modulo one of the weighted <italic>q</italic>-adic sum-of-digits function <italic>s<sub>q,γ</sub></italic> (<italic>n</italic>), <italic>n</italic> = 0, 1, 2,... We also prove the uniform distribution modulo one of related sequences <italic>h</italic><sub>1</sub><italic>s<sub>q, γ</sub></italic> (<italic>n</italic>)+<italic>h</italic><sub>2</sub><italic>s<sub>q,γ</sub></italic> (<italic>n</italic> +1), where <italic>h</italic><sub>1</sub> and <italic>h</italic><sub>2</sub> are integers such that <italic>h</italic><sub>1</sub> + <italic>h</italic><sub>2</sub> ≠ 0 and that the akin two-dimensional sequence <italic>s<sub>q,γ</sub></italic> (<italic>n</italic>), <italic>s<sub>q,γ</sub></italic> (<italic>n</italic> +1) cannot be uniformly distributed modulo one if <italic>q ≥</italic> 3. The properties of the two-dimensional sequence <italic>s<sub>q,γ</sub></italic> (<italic>n</italic>),s<italic><sub>q,γ</sub></italic> (<italic>n</italic> +1), <italic>n</italic> =0, 1, 2,..., will be instrumental in the proofs of the final section, where we show how the growth properties of the sequence of weights influence the distribution of values of the weighted sum-of-digits function which in turn imply a new property of the van der Corput sequence.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00The Inequality of Erdős-Turán-Koksma in the Terms of the Functions of the System Γ<abstract> <title style='display:none'>Abstract</title> <p>In the present paper the author uses the function system Γ<sub><italic>ℬ</italic><sub><italic>s</italic></sub></sub>constructed in Cantor bases to show upper bounds of the extreme and star discrepancy of an arbitrary net in the terms of the trigonometric sum of this net with respect to the functions of this system. The obtained estimations are inequalities of the type of Erdős-Turán-Koksma. These inequalities are very suitable for studying of nets constructed in the same Cantor system.</p> </abstract>ARTICLE2021-10-30T00:00:00.000+00:00Application of the Extended Fan Sub-Equation Method to Time Fractional Burgers-Fisher Equation<abstract><title style='display:none'>Abstract</title><p>In this paper, the extended Fan sub-equation method to obtain the exact solutions of the generalized time fractional Burgers-Fisher equation is applied. By applying this method, we obtain different solutions that are benefit to further comprise the concepts of complex nonlinear physical phenomena. This method is simple and can be applied to several nonlinear equations. Fractional derivatives are taken in the sense of Jumarie’s modified Riemann-Liouville derivative. A comparative study with the other methods approves the validity and effectiveness of the technique, and on the other hand, for suitable parameter values, we plot 2D and 3D graphics of the exact solutions by using the extended Fan sub-equation method. In this work, we use Mathematica for computations and programming.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments<abstract><title style='display:none'>Abstract</title><p>This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0021_eq_001.png"/><mml:math xmlns:mml=""><mml:mrow><mml:mi>Δ</mml:mi><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>ℕ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mtext> </mml:mtext><mml:mo stretchy="false">[</mml:mo><mml:mo>∇</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>−</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>ℕ</mml:mi><mml:mo stretchy="false">]</mml:mo><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>\[\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\</tex-math></alternatives></inline-formula>, where <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0021_eq_002.png"/><mml:math xmlns:mml=""><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math/></alternatives></inline-formula> and <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0021_eq_003.png"/><mml:math xmlns:mml=""><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math/></alternatives></inline-formula> are sequences of nonnegative real numbers and <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0021_eq_004.png"/><mml:math xmlns:mml=""><mml:mrow><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>τ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math><tex-math>\[{(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}\]</tex-math></alternatives></inline-formula> are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Explicit Non Algebraic Limit Cycle for a Discontinuous Piecewise Differential Systems Separated by One Straight Line and Formed by Linear Center and Linear System Without Equilibria<abstract><title style='display:none'>Abstract</title><p>In this paper, we deal with the discontinuous piecewise differential linear systems formed by two differential systems separated by a straight line when one of these two differential systems is a linear without equilibria and the other is a linear center. We are going to show that the maximum number of crossing limit cycles is one, and if exists, it is non algebraic and analytically given.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillation Behaviour of Solutions for a Class of a Discrete Nonlinear Fractional-Order Derivatives<abstract><title style='display:none'>Abstract</title><p>Based on the generalized Riccati transformation technique and some inequality, we study some oscillation behaviour of solutions for a class of a discrete nonlinear fractional-order derivative equation <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0022_eq_001.png"/><mml:math xmlns:mml=""><mml:mrow><mml:mi>Δ</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>γ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mrow><mml:mo stretchy="false">[</mml:mo><mml:mi>α</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msup><mml:mi>Δ</mml:mi><mml:mi>μ</mml:mi></mml:msup><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:mrow><mml:mi>η</mml:mi></mml:msup><mml:mo stretchy="false">]</mml:mo><mml:mo>+</mml:mo><mml:mi>ϕ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mi>f</mml:mi><mml:mo stretchy="false">[</mml:mo><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo stretchy="false">]</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>N</mml:mi><mml:mrow><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo></mml:mrow></mml:math><tex-math>\[\Delta [\gamma (\ell ){[\alpha (\ell ) + \beta (\ell ){\Delta ^\mu }u(\ell )]^\eta }] + \phi (\ell )f[G(\ell )] = 0,\ell \in {N_{{\ell _0} + 1 - \mu }},\] </tex-math></alternatives></inline-formula> where <inline-formula><alternatives><inline-graphic xmlns:xlink="" xlink:href="graphic/tmmp-2021-0022_eq_002.png"/><mml:math xmlns:mml=""><mml:mrow><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>G</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mstyle displaystyle="true"><mml:munderover><mml:mo>∑</mml:mo><mml:mrow><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mi>ℓ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow><mml:mrow><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>μ</mml:mi></mml:mrow></mml:munderover><mml:mrow><mml:msup><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>ℓ</mml:mi><mml:mo>−</mml:mo><mml:mi>j</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mo>−</mml:mo><mml:mi>μ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>j</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:mstyle></mml:mrow></mml:math><tex-math>\[{\ell _0} &gt; 0,\quad G(\ell ) = \sum\limits_{j = {\ell _0}}^{\ell - 1 + \mu } {{{(\ell - j - 1)}^{( - \mu )}}u(j)} \]</tex-math></alternatives></inline-formula> and Δ<sup><italic>μ</italic></sup> is the Riemann-Liouville (R-L) difference operator of the derivative of order <italic>μ</italic>, 0 &lt; <italic>μ</italic> ≤ 1 and <italic>η</italic> is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Terms<abstract><title style='display:none'>Abstract</title><p>The authors examine the oscillation of second-order nonlinear differential equations with mixed nonlinear neutral terms. They present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated by some examples.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Some Results Involving the Airy Functions and Airy Transforms<abstract><title style='display:none'>Abstract</title><p>In the present work, the author studied some properties of the modified Bessel’s functions and Airy functions. It is worth mentioning that the Airy functions are used in many fields of physics. They are applied in many branches of classical and quantum physics. The author also studied certain properties of the Airy transform and derived some new integral relations involving the Airy functions. Non-trivial illustrative examples are provided as well. All the results are presented in lucid and comprehensible language.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Properties of the Katugampola Fractional Operators<abstract><title style='display:none'>Abstract</title><p>In this work, there are considered higher order fractional operators defined in the sense of Katugampola. There are proved some fundamental properties of the Katugampola fractional operators of any arbitrary real order. Moreover, there are given conditions ensuring existence of the higher order Katugampola fractional derivative in space of the absolutely continuous functions.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00Certain Singular Distributions and Fractals<abstract><title style='display:none'>Abstract</title><p>In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.</p></abstract>ARTICLE2022-01-01T00:00:00.000+00:00en-us-1