rss_2.0Tatra Mountains Mathematical Publications FeedSciendo RSS Feed for Tatra Mountains Mathematical Publications Mountains Mathematical Publications 's Cover Some Properties of Difference Operator with Some Characterizations<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we introduce the generalized spaces of the form ℋ(<italic>f, g,</italic> Δ<italic><sub>n</sub><sup>m</sup></italic>), where ℋ represents one of the spaces ℓ<sub>∞</sub>, <italic>c</italic> or <italic>c</italic><sub>0</sub>.The köthe duals corresponding to these spaces will be computed and construction of the Schauder bases for ℋ ∈ {<italic>c, c</italic><sub>0</sub>} will be given. Also, some matrix characterizations concerning these spaces will be computed. Moreover, the characterization of some classes of compact operators on the spaces ℓ<sub>∞</sub>(<italic>f, g,</italic> Δ<italic><sub>n</sub><sup>m</sup></italic>)and <italic>c</italic><sub>0</sub>(<italic>f, g,</italic> Δ<italic><sub>n</sub><sup>m</sup></italic>)by employing the Hausdorff measure of non-compactness will be determined.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Numerical Radius of Bounded Operators with ℓ-Norm<abstract> <title style='display:none'>Abstract</title> <p>We study the numerical radius of bounded operators on direct sum of a family of Hilbert spaces with respect to the ℓ<italic><sup>p</sup></italic>-norm, where 1 ≤ <italic>p</italic> ≤∞. We propose a new method which enables us to prove validity of many inequalities on numerical radius of bounded operators on Hilbert spaces when the underling space is a direct sum of Hilbert spaces with ℓ<italic><sup>p</sup></italic>-norm, where 1 ≤ <italic>p</italic> ≤ 2. We also provide an example to show that some known results on numerical radius are not true for a space that is the set of bounded operators on ℓ<italic><sup>p</sup></italic>-sum of Hilbert spaces where 2 <italic>&lt;p &lt;</italic> ∞. We also present some applications of our results.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Global Phase Portraits of Quadratic Polynomial Differential Systems Having as Solution Some Classical Planar Algebraic Curves of Degree 6<abstract> <title style='display:none'>Abstract</title> <p>The main goal of this paper is to classify the global phase portraits of seven quadratic polynomial differential systems, exhibiting as invariant algebraic curves seven well-known algebraic curves of degree six.</p> <p>We prove that these systems have five topologically different phase portraits in the Poincarédisc.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Stabilities and Non-Stabilities of a New Reciprocal Functional Equation<abstract> <title style='display:none'>Abstract</title> <p>The intention of this study is to present some stronger results by investigating Ulam-JRassias product stability and Ulam-JRassias mixed-type sum-product stability of a new reciprocal functional equation. Also, a suitable counter-example is presented to show the failure of stability result for the singular case.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Remark on a Theorem of Tonelli<abstract> <title style='display:none'>Abstract</title> <p>It is well known that if the surface <italic>f</italic> : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections <italic>f<sub>x</sub></italic>(<italic>y</italic>)= <italic>f</italic> (<italic>x, y</italic>)and <italic>f <sup>y</sup></italic>(<italic>x</italic>) = <italic>f</italic> (<italic>x, y</italic>)of <italic>f</italic> are finite almost everywhere in [−1, 1]. In the note it is proved that these variations can be infinite on residual subsets of [−1, 1].</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Some Fractal Properties of Sets Having the Moran Structure<abstract> <title style='display:none'>Abstract</title> <p>This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Modulus of Smoothness and K-Functionals Constructed by Generalized Laguerre-Bessel Operator<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we prove the equivalence between a K-functional and a modulus of smoothness generated by Laguerre-Bessel operator on <disp-formula> <alternatives> <graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2022-0008_eq_001.png"/> <mml:math xmlns:mml="" display="block"><mml:mrow><mml:mi>𝕂</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo>×</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mo>+</mml:mo><mml:mo>∞</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mo>.</mml:mo></mml:mrow></mml:math> <tex-math>\mathbb{K} = [0, + \infty [ \times [0, + \infty [.</tex-math> </alternatives> </disp-formula></p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Some Inequalities Involving Interpolations Between Arithmetic and Geometric Mean<abstract> <title style='display:none'>Abstract</title> <p>In this article, we mainly study the interpolations between arithmetic mean and geometric mean—power mean, Heron mean and Heinz mean. First, we obtain the improvement and reverse improvement of arithmetic-power mean inequalities by the convexity of the function. We show that the proof of Heron mean inequality due to Yang and Ren: [<italic>Some results of Heron mean and Young’s inequalities</italic>, J. Inequal. Appl. <bold>2018</bold> (2018), paper no, 172], is not substantial. In addition, we also obtain Heron-Heinz mean inequalities for <italic>t</italic> ∈ ℝ. Further corresponding operator versions and generalizations are also established.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Remarks on Semi-Menger and Star Semi-Menger Spaces<abstract> <title style='display:none'>Abstract</title> <p>It is proved that for an extremally disconnected <italic>S</italic>-paracompact-<italic>T</italic><sub>2</sub> spaces the properties semi-Menger, Menger, strongly star semi-Menger, strongly star-Menger, star semi-Menger, star-Menger, almost semi-Menger, almost Menger, almost star semi-Menger, almost star-Menger are equivalent. We show that a weakly semi-continuous (strongly <italic>θ</italic>-semi-continuous) image of semi-Menger (almost semi-Menger) spaces is almost semi-Menger (semi-Menger), respectively. The characterizations of semi-Menger and star semi-Menger spaces are also provided.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Selective Version of Star-Semi-Lindelöfness in () Topological Spaces<abstract> <title style='display:none'>Abstract</title> <p>In this paper, we deal with the properties (<italic>a</italic>)<italic>R</italic>-star-semi-Lindelöf and (<italic>a</italic>)<italic>M</italic>-star-semi-Lindelöf in (<italic>a</italic>)topological spaces. These properties are interesting as every (<italic>a</italic>)<italic>R<sup>s</sup></italic>-separable space is (<italic>a</italic>)<italic>R</italic>-star-semi-Lindelöf and every (<italic>a</italic>)<italic><sup>s</sup></italic>-semi-Lindelöf space is (<italic>a</italic>)<italic>R</italic>-star-semi-Lindelöf but not every (<italic>a</italic>)<italic>R</italic>-star-semi-Lindelöf space is (<italic>a</italic>)<italic>R<sup>s</sup></italic>-separable or (<italic>a</italic>)<italic><sup>s</sup></italic>-semi-Lindelöf. It is shown that if an (<italic>a</italic>)topological space <italic>X</italic> is the union of countably many (<italic>a</italic>)-open and (<italic>a</italic>)<italic>R</italic>star-semi-Lindelöf subspaces, then <italic>X</italic> is (<italic>a</italic>)<italic>R</italic>-star-semi-Lindelöf. Similar results are obtained in the context of (<italic>a</italic>)<italic>M</italic>-star-semi-Lindelöf spaces. Further, suitable and required counterexamples are given.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00On the Generalized Inequalities for Co-Ordinated Convex Functions<abstract> <title style='display:none'>Abstract</title> <p>The aim of this paper is to establish some generalized integral inequalities for convex functions of 2-variables on the co-ordinat. Then, we will give a generalized identity and with the help of this integral identity, we will investigate some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities involving Riemann integrals and Riemann-Liouville fractional integrals.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Integrability and Non-Existence of Periodic Orbits for a Class of Kolmogorov Systems<abstract> <title style='display:none'>Abstract</title> <p>In this article, we study the integrability and the non-existence of periodic orbits for the planar Kolmogorov differential systems of the form <disp-formula> <alternatives> <graphic xmlns:xlink="" xlink:href="graphic/j_tmmp-2022-0011_eq_001.png"/> <mml:math xmlns:mml="" display="block"><mml:mrow><mml:mtable columnalign="left"><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mover accent="true"><mml:mi>x</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></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:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>P</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></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:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr><mml:mtr columnalign="left"><mml:mtd columnalign="left"><mml:mrow><mml:mover accent="true"><mml:mi>y</mml:mi><mml:mo>˙</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mi>y</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>R</mml:mi></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:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>Q</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></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:mo>(</mml:mo><mml:mrow><mml:mi>x</mml:mi><mml:mo>,</mml:mo><mml:mi>y</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:mtd></mml:mtr></mml:mtable></mml:mrow></mml:math> <tex-math>\matrix{ {\dot x = x\left( {{R_{n - 1}}\left( {x,y} \right) + {P_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr {\dot y = y\left( {{R_{n - 1}}\left( {x,y} \right) + {Q_n}\left( {x,y} \right) + {S_{n + 1}}\left( {x,y} \right)} \right),} \hfill \cr }</tex-math> </alternatives> </disp-formula> where <italic>n</italic> is a positive integer, <italic>R<sub>n−1</sub></italic>, <italic>P<sub>n</sub></italic>, <italic>Q<sub>n</sub></italic> and <italic>S<sub>n</sub></italic><sub>+1</sub> are homogeneous polynomials of degree <italic>n</italic> − 1, <italic>n</italic>, <italic>n</italic> and <italic>n</italic> + 1, respectively. Applications of Kolmogorov systems can be found particularly in modeling population dynamics in biology and ecology.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00Successive Approximations for Caputo-Fabrizio Fractional Differential Equations<abstract> <title style='display:none'>Abstract</title> <p>In this work we deal with a uniqueness result of solutions for a class of fractional differential equations involving the Caputo-Fabrizio derivative. We provide a result on the global convergence of successive approximations.</p> </abstract>ARTICLE2022-11-29T00:00:00.000+00:00What 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+00:00en-us-1