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Abstract: In this paper, we prove that convolution of two half-plane harmonic mappings with suitable dilatation is convex in some particular direction. We also prove that convolution of slanted half-plane mappings and slanted strip mapping with fixed dilatation is convex in a particular direction. We give some examples for support of our results. PubDate: 2021-10-07

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Abstract: Let \(K = {\mathbb {Q}} (\alpha )\) be a pure number field generated by a complex root \(\alpha\) of a monic irreducible polynomial \(F(x) = x^{42} -m \in {{\mathbb {Z}}}[x]\) , where \(m \ne \pm 1\) is a square-free rational integer. In this paper, we study the monogenity of K. We prove that if \(m\not \equiv 1\ \mathrm{(mod }{4})\) , \(m\not \equiv \mp 1 \ \mathrm{(mod }{9})\) , and \(\overline{m}\not \in \{\mp 1, 18, 19, 30, 31\} \ \mathrm{(mod }{49})\) , then K is monogenic. But, if \(m \equiv 1\ \mathrm{(mod }{4})\) , or \(m \equiv 1 \ \mathrm{(mod }{9})\) , or \(m \equiv 1 \ \mathrm{(mod }{49})\) , then K is not monogenic. Our results are illustrated by some examples. PubDate: 2021-10-07

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Abstract: In this paper, we study a generalization of Narayana’s numbers and Padovan’s numbers. This generalization also includes a sequence whose elements are Fibonacci numbers repeated three times. We give combinatorial interpretations and a graph interpretation of these numbers. In addition, we examine matrix generators and determine connections with Pascal’s triangle. PubDate: 2021-10-07

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Abstract: Somos conjectured thousands of Dedekind \(\eta \) -function identities of various levels, around 6200 in number. He did so using computational evidence but has not sought to provide any proof for these identities. In this paper, we prove level 6 of Somos’s Dedekind \(\eta \) -function identities containing five terms in two methods. Further, as an application of these identities, we deduce colored partitions for the same. PubDate: 2021-10-07

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Abstract: In this paper, we first give a new proof of the log-Minkowski inequality of general planar convex bodies and then extend the \(L_p\) -Brunn–Minkowski inequality and \(L_p\) -Minkowski inequality of o-symmetric planar convex bodies for \(p\in (0,1)\) to \(\phi \) -Brunn–Minkowski inequality and \(\phi \) -Minkowski inequality of general planar convex bodies. As an application, a family of \(\phi \) -measures of asymmetry for planar convex bodies is introduced. PubDate: 2021-10-07

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Abstract: In this paper, make use of the Horadam polynomials, we introduce a comprehensive subclass of analytic and bi-univalent functions. For functions belonging to this class we derive coefficient inequalities and the Fekete–Szegö inequalities. Also, variety observations of the results presented here are also discussed. PubDate: 2021-10-07

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Abstract: Let \(\{V_n(P,Q)\}\) be the Lucas sequence of the second kind at the nonzero relatively prime parameters P and Q. In this paper, we present techniques for studying the solutions (x, n) with \(x \ge 2, n \ge 0\) of any Diophantine equation of the form $$\begin{aligned} \frac{1}{V_n(P_2,Q_2)}=\sum _{k=1}^{\infty }\frac{V_{k-1}(P_1,Q_1)}{x^k} \end{aligned}$$ in both cases \((P_1,Q_1)=(P_2,Q_2)\) and \((P_1,Q_1)\ne (P_2,Q_2)\) , where \(Q_1, Q_2 \in \{-1,1\}\) . Furthermore, we represent the procedures of these techniques in case of \( -2 \le P_1, P_2 \le 4\) . PubDate: 2021-10-01

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Abstract: In this paper, we prove some common fixed point results of Ćirić-type \(F_M\) -contraction for weakly compatible mappings in metric space. We generalize common fixed point theorems for twelve self-mappings with (CLR)-property or common property (E.A). Also, we present examples and applications of these mappings in metric space and integral equations. PubDate: 2021-09-22

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Abstract: In this paper, we study a schistosomiasis model incorporating the miracidia and cercariae dynamics, discrete-time delays as well as control measures like water treatment. Modelling the dynamics of schistosomiasis infectious disease is quite challenging because of the different larval forms assumed by the parasite and the requirement of two hosts during the life cycle. Our model is generic in the sense that it considers both situations where particle depletion by hosts or snails could have or not a negligible impact on particle dynamics. Precisely, we introduce two parameters u and v such that when \(u=v=0\) , then particle depletion by hosts or snails is not considered; when \(u=v=1\) , then particle depletion by hosts and snails is considered. The model is analyzed to gain insights into the qualitative features of the disease-free equilibrium which allows the determination of the basic reproductive number \({\mathcal {R}}_{u,v}\) . The Center Manifold Theory is used to discuss existence and local stability of an endemic equilibrium. Global sensitivity analysis (SA) of the schistosomiasis model and the basic reproduction number are carried out. SA results of the model point out the leading role of \(\eta \) , the parameter that shapes infection-induced death rate in humans, on the dynamics of humans (susceptible and infected), miracidia, cercariae and infected snails. They also reveal the pervasive role of \(\theta \) , the water treatment-induced death rate of snails, on the dynamics of infected humans, miracidia, snails (susceptible and infected) and cercariae. SA results of the basic reproduction number highlight the role of \(\eta \) , \(\theta \) , \(\lambda \) (the contact rate of transmission of miracidia to susceptible snails), \(\varpi \) (production rate of miracidia from feces of infected humans) and \(\gamma \) (the transmission rate of cercariae to susceptible humans). Therefore, a possible way to control the disease could rely on the intensification of sanitization campaigns that will result in an increase of \(\theta \) together with sensitization about the necessity to have a treatment once you are infected to reduce \(\eta \) . PubDate: 2021-09-19

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Abstract: In this current work, we introduce a subfamily of analytic functions endowed with k-Pell–Lucas numbers. The radius problems, basic geometric properties and general coefficient relations are obtained for the former class. PubDate: 2021-09-16

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Abstract: We find the sharp bound for the third Hankel determinant $$\begin{aligned} H_{3,1}(f):= \left {\begin{array}{*{20}c} {a_{1} } & {a_{2} } & {a_{3} } \\ {a_{2} } & {a_{3} } & {a_{4} } \\ {a_{3} } & {a_{4} } & {a_{5} } \\ \end{array} } \right \end{aligned}$$ for analytic functions f with \(a_n:=f^{(n)}(0)/n!,\ n\in \mathbb N,\ a_1:=1,\) such that $$\begin{aligned} {{\,\mathrm{Re}\,}}f'(z)>0,\quad z\in \mathbb D:=\{z \in \mathbb C: z <1\}. \end{aligned}$$ PubDate: 2021-09-04

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Abstract: In this paper, we determine all repdigits in base b for \(2\le b\le 10,\) which are products of two Pell numbers or Pell–Lucas numbers. It is shown that the largest Pell number which is a base b-repdigit is \(P_{6}=70=(77)_{9} =7+7\cdot 9.\) Also, we give the result that the equations \(P_{m}P_{n}+1=b^{k}\) and \(Q_{m}Q_{n}+1=b^{k}\) have no solutions for \(n\ge 5\) and \(n\ge 1,\) respectively, where \(1\le m\le n\) . PubDate: 2021-09-04

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Abstract: We propose two algorithms to approximate the Hausdorff distance between two sets of the Euclidean space \(\mathbb {R}^{m}\) , whenever such sets can be described as the image of continuous functions defined in suitable domains. Such algorithms are derived from the main results of the paper, which are based on the so called \(\alpha\) -dense curves. These curves allow us to consider a single variable problem to obtain the desired approximations with a lower computational time than other methods. Likewise, upper bounds are provided for the approximation error of our results. PubDate: 2021-09-04

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Abstract: By simple means we prove irrationality results for some series whose terms are recursively defined. In particular, we prove that the roots and the quotient of such series are irrational numbers. PubDate: 2021-09-04

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Abstract: The main purpose of this note is to prove the following: let \(n\ge 3\) be a fixed integer number and let K be a convex body such that for every equiangular circumscribed n-gon, the midpoints of its sides belong to K. Then K is a disc. We also prove that a 3-dimensional convex body K such that the centers of the faces of all its circumscribed regular tetrahedra belong to K is a ball. PubDate: 2021-08-25

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Abstract: This work addresses the strictly convex unconstrained minimization problem via a modified version of the gradient method. The proposal is a line search method that uses a search direction based on the gradient method. This new direction is constructed by a mixture of the negative direction of the gradient with another particular direction that uses second-order information. Unlike Newton-type methods, our algorithm does not need to compute the inverse of the Hessian of the objective function. We analyze the global convergence under an exact line search. A numerical study is carried out, to illustrate the numerical effectiveness of the method by comparing it with some conjugate gradient methods and also with the Barzilai–Borwein gradient method both in quadratic and non-linear problems. PubDate: 2021-08-25

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Abstract: In this paper, we prove a new version of the sub-super solution method for non-local problem and, as application, the existence of a positive solution for a nonlinear elliptic p- Kirchhoff system in different type of nonlinearities. PubDate: 2021-08-07

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Abstract: An intersecting r-uniform straight line system is an intersecting linear system whose lines consist of r points on straight line segments of \(\mathbb {R}^2\) and where any two lines share a point. Recently, the author [A. Vázquez-Ávila, On intersecting straight line systems, J. Discret. Math. Sci. Cryptogr. Accepted] proved that any intersecting r-uniform straight line system \((P,\mathcal {L})\) with \(r\ge \nu _2\) has transversal number at most \(\nu _2-1\) , where \(\nu _2\) is the maximum cardinality of a subset of lines \(R\subseteq \mathcal {L}\) such that every triplet of different elements of R does not have a common point. This paper improves such upper bound if the intersecting r-uniform straight line system satisfies \(r=\nu _2\) . Also, those results have immediate consequences for some questions given by Oliveros et al. in [D. Oliveros, C. O’Neill and S. Zerbib, The geometry and combinatorics of discrete line segment hypergraphs, Discrete Math. 343 (2020), no. 6, 111825]. PubDate: 2021-08-07

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Abstract: We shall present new oscillation criteria for non-canonical higher order nonlinear neutral dynamic equations of the form $$\begin{aligned} \left( (a(t)\left( y^{\varDelta ^{n-1}}(t)\right) ^{\alpha }\right) ^{\varDelta }+q(t)x^{\beta }(\tau (t))=0, \end{aligned}$$ where \(y(t)=x(t)+p(t)x^{\gamma }(\delta (t)).\) We also present some oscillation results for certain second order dynamic equations with sublinear as well as super linear neutral terms. PubDate: 2021-08-07