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In this study, geometrical and mechanical identification of the talus and cochlea tibiae in horse and ox is presented. The shape of the expressed bones of these animals can be considered as rotational surfaces of planar curves. The model is established based on the bolt-nut mechanism while interpreting the relationship between talus and tibiae, and describing geometrical differences between ox and horse in functional anatomy. The results obtained from the geometric and mechanical data have led to the conclusion that lateral and friction forces applied on talus and cochlea tibiae of the horse are higher than that in ox. For this reason, it can be concluded that horses can stand longer than cattle, consistent with the way of life.

Bu çalışmada, dördüncü mertebeden lineer olmayan, Elektro-Elastik (EE) çubuktaki yalnız gezen dalgalara karşılık gelen EE kısmi diferensiyel denklemi ele alındı. Denklemin gezici dalga çözümlerini araştırmak için, ( 𝐺′/𝐺 ) açılım yöntemi kullanıldı. Yöntemin içerdiği farklı durumlar için tam çözümler oluşturuldu. Trigonometrik, hiperbolik ve rasyonel fonksiyonları içeren çözümler elde edildi. Elde edilen çözümlerden biri için kararlılık analizi yapıldı. Son olarak çözümlerin farklı parametrelerdeki bazı özel değerleri için grafikleri, Maple programı ile çizdirilerek, parametreler değiştiğinde grafiklerin değişimi karşılaştırmalı olarak incelenmeye sunulmuştur. In this paper, the Elektro-Elastic (EE) partial differential equation corresponding to traveling alone waves in a fourth order nonlinear (EE) rod is discussed. The expansion method ( 𝐺′/𝐺 ) was used for investigate the traveling wave solutions of the equation. Exact solutions have been created for the different situations involved in the method. Solutions including trigonometric, hyperbolic and rational functions were obtained. Stability analysis was performed for one of the obtained solutions. Finally, the graphs for some special values of the solutions with different parameters were drawn with Maple program, and the change of the graphics when the parameters changed was presented to be examined comparatively.

There is a very wide application of mathematics in communication theory, signal processing and networking. A network is formed by nodes communicating with each other. Graph theory plays a vital role in the area of networking research. A topological index is a numeric quantity obtained from a graph structure which is invariant under graph isomorphism. In this study, we obtained a generalized degree-based topological index, called -Zagreb index of some derived networks such as Sierpinski network, butterfly network, Benes network and mesh-derived network and hence consider some particular cases. Council of Scientific & Industrial Research (CSIR) - India 09/973(0016)/2017-EMR-I

Optical soliton pulses or molecules with the Sasa-Satsuma equation have been worked by modified simple equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

The Kundu–Mukherjee–Naskar model in birefringent fibers is examined to uncover quite important optical solitons in (2+1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Triki-Biswas equation that is a new model for soliton dynamics of sub pico-second pulses in a monomodal optical fiber with non-Kerr dispersion is considered to get optical solitos of its in this paper. Optical bright and singular solitons are given by trial equation approach. Also, singular periodic solutions are located.

The Chen–Lee–Liu equation in birefringent fibers is examined to uncover quite important optical solitons of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Biswas–Arshed model which is a new model for soliton transmission through optical fibers is employed to obtain optical solitons of its in this work. The trial equation technique is used to catch out both highly important solitons and singular periodic solutions.

The Biswas-Arshed model in birefringent fibers which is first written in two component form for vector solitons is employed to obtain optical solitons of its in this work. The trial equation technique is used to catch out both highly important solitons and singular periodic solutions.

The inverse problem for integer-valued topological indices is about the existence of a graph having its index value equal to a given integer. We solve this problem for the first and second Zagreb indices, and present analogous results also for the forgotten and hyper-Zagreb index. The first Zagreb index of connected graphs can take any even positive integer value, except 4 and 8. The same is true if one restricts to trees or to molecular graphs. The second Zagreb index of connected graphs can take any positive integer value, except 2, 3, 5, 6, 7, 10, 11, 13, 15 and 17. The same is true if one restricts to trees or to molecular graphs.

This paper examines soliton dynamics of sub pico-second pulses modeled by Kaup–Newell equation. The modified simple equation scheme and trial equation approach retrieves dark, bright as well as singular solitons to the model. The constraint relations guarantee the existence of such solitons.

This paper studies differential group delay in optical fibers with coupled Fokas-Lenells equation. Two integration schemes are adopted. They are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are retrieved together with their respective existence criteria that are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains bright, dark and singular dispersive optical soliton solutions to Schrodinger-Hirota equation by the aid of trial equation method. Both Kerr and power laws of nonlinearity are studied. Singular periodic solutions are also obtained as a byproduct of this scheme. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

Apart from its applications in Chemistry, Biology, Physics, Social Sciences, Anthropology, etc., there are close relations between graph theory and other areas of Mathematics. Fibonacci numbers are of utmost interest due to their relation with the golden ratio and also due to many applications in different areas from Biology, Architecture, Anatomy to Finance. In this paper, we define Fibonacci graphs as graphs having degree sequence consisting of n consecutive Fibonacci numbers and use the invariant &Omega, to obtain some more information on these graphs. We give the necessary and sufficient conditions for the realizability of a set D of n successive Fibonacci numbers for every n and also list all possible realizations called Fibonacci graphs for 1&le, n&le, 4.

Special number sequences play important role in many areas of science. One of them named as Fibonacci sequence dates back to 820 years ago. There is a lot of research on Fibonacci numbers due to their relation with the golden ratio and also due to many applications in Chemistry, Physics, Biology, Anthropology, Social Sciences, Architecture, Anatomy, Finance, etc. A slight variant of the Fibonacci sequence was obtained in the eighteenth century by Lucas and therefore named as Lucas sequence. There are very natural close relations between graph theory and other areas of Mathematics including number theory. Recently Fibonacci graphs have been introduced as graphs having consecutive Fibonacci numbers as vertex degrees. In that paper, graph theory was connected with number theory by means of a new graph invariant called Omega(D) for a realizable degree sequence D defined recently. Omega(D) gives information on the realizability, number of components, chords, loops, pendant edges, faces, bridges, connectedness, cyclicness, etc. of the realizations of D and is shown to have several applications in graph theory. In this paper, we define Lucas graphs as graphs having degree sequence consisting of n consecutive Lucas numbers and by using Sl and its properties, we obtain a characterization of these graphs. We state the necessary and sufficient conditions for the realizability of a given set D consisting of n successive Lucas numbers for every n and also list all possible realizations called Lucas graphs for 1 = 5.

Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\ n\geq 2,\ xyz\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers., Comment: 12 pages, to appear, Colloquium Mathematicum (2020)

We want to remark explicitly that, by using the L-n (x) functions (essentially linked to Lucas polynomials of the second kind), our methodology builds a bridge, to our knowledge not previously well known, between the Theory of Geometric Functions and that of Special Functions, which are usually considered as very different fields. Thus, also making use of the differential operator I-p,q(k), we introduce a new class of analytic bi-univalent functions. Coefficient estimates, Fekete-Szego inequalities and several special consequences of the results are obtained.

In the acquaint article, we scrutinize some fundamental attribute of a subclass of harmonic univalent functions defined by a new alteration. Like these, coefficient disparities, distortion bounds, convolutions, convex combinations and extreme points.

The modified simple equation scheme is employed to secure dark and singular optical soliton solution to the perturbed Gerdjikov-Ivanov equation that models pulse dynamics in optical fibers and PCF. The solvability conditions for this algorithm is also presented. National Science Foundation for Young Scientists of Wuhan Donghu University (2017dhzk001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains optical soliton solution to perturbed Gerdjikov-Ivanov equation by trial equation approach. Bright, dark and singular soliton solutions are derived. Additional solutions such as singular periodic solutions also emerge of this integration scheme. National Science Foundation for Young Scientists of Wuhan Donghu University (2017dhzk001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper obtains optical soliton solutions to the Kundu Eckhaus equation with general coefficients. The Riccati Bernoulli's sub-ODE method as well as Kudryashov's scheme are employed to obtain soliton solutions to the model. This generalized earlier reported result of the model with specific coefficients. Subsequently, the multiplier approach was utilized to secure a conserved quantity with the bright soliton solution that was reported earlier. Young Foundation of Wuhan Donghu University (2017DHKZ001) Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.

In this manuscript, generating functions are constructed for the new special families of polynomials and numbers using the p-adic q-integral technique. Partial derivative equations, functional equations and other properties of these generating functions are given. With the help of these equations, many interesting and useful identities, relations, and formulas are derived. We also give p-adic q-integral representations of these numbers and polynomials. The results we have obtained for these special numbers and polynomials are closely related to well-known families of polynomials and numbers including the Bernoulli numbers, the Apostol-type Bernoulli numbers and polynomials and the Frobenius-Euler numbers, the Stirling numbers, and the Daehee numbers. We give some remarks and observations on the results of this paper. Akdeniz Üniversitesi

In this study, we consider three model equations of shallow water waves. Shallow water equations model the propagation of strongly nonlinear waves up to breaking and run-up in nearshore zones. We perform multiple exp-function method which is known as a generalization of Hirota's perturbation scheme. We yield one-, two-, and three-wave solutions. The obtained solutions can be used as benchmarks for numerical solutions of the underlying equations. Faculty Research Committee of FAST, North-West University, Mafikeng Campus, SouthAfrica

In this paper, by considering a Frenet curve lying on an oriented hypersurface, we extend the Darboux frame field into Euclidean 4 -space E-4. Depending on the linear independency of the curvature vector with the hypersurface's normal, we obtain two cases for this extension. For each case, we obtain some geometrical meanings of new invariants along the curve on the hypersurface. We also give the relationships between the Frenet frame curvatures and Darboux frame curvatures in E-4. Finally, we compute the expressions of the new invariants of a Frenet curve lying on an implicit hypersurface. Yıldız Teknik Üniversitesi- 2013-01-03-KAP01

Three main tools to study graphs mathematically are to make use of the vertex degrees, distances and matrices. The classical graph energy was defined by means of the adjacency matrix in 1978 by Gutman and has a large number of applications in chemistry, physics and related areas. As a result of its importance and numerous applications, several modifications of the notion of energy have been introduced since then. Most of them are defined by means of graph matrices constructed by vertex degrees. In this paper we define another type of energy called q-distance energy by means of distances and matrices. We study some fundamental properties and also establish some upper and lower bounds for this new energy type.

Optical soliton pulses or molecules with the Sasa–Satsuma equation have been worked by trial equation methodology in this paper. Soliton type solutions have been obtained as a result of this study. These quite important optical solitons are known as dark optical soliton, bright optical soliton in additionally singular optical soliton.

The Kundu–Mukherjee–Naskar model is examined to uncover quite important optical solitons in (2 + 1)-dimensions of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by modified simple equation technique with parameter restrictions.

The Chen–Lee–Liu equation is examined to uncover quite important optical solitons of its. Dark, bright and also singular solitons in additionally singular periodic solutions are yield by trial equation technique with parameter restrictions.

The Biswas-Arshed equation in birefringent fibers is considered to get optical solitos of its in this paper. Optical dark, bright and singular solitons are given by modified simple equation approach. Also, singular periodic solutions are located.

The authors recently defined a new graph invariant denoted by Ω(G) only in terms of a given degree sequence which is also related to the Euler characteristic. It has many important combinatorial applications in graph theory and gives direct information compared to the better known Euler characteristic on the realizability, connectedness, cyclicness, components, chords, loops etc. Many similar classification problems can be solved by means of Ω. All graphs G so that $$\Omega(G)\leq-4$$ are shown to be disconnected, and if $$\Omega(G)\geq-2$$ , then the graph is potentially connected. It is also shown that if the realization is a connected graph and $$\Omega(G)\geq-2$$ , then certainly the graph should be a tree. Similarly, it is shown that if the realization is a connected graph G and $$\Omega(G)\geq0$$ , then certainly the graph should be cyclic. Also, when $$\Omega(G)\geq-4$$ , the components of the disconnected graph could not all be cyclic and if all the components of G are cyclic, then $$\Omega(G)\geq0$$ . In this paper, we study an extremal problem regarding graphs. We find the maximum number of loops for three possible classes of graphs. We also state a result giving the maximum number of components amongst all possible realizations of a given degree sequence.

The Biswas-Arshed model which is a new model for soliton transmission through optical fibers is taken into account for the purpose of acquiring very valuable optical solitons of its in this paper. The modified simple equation methodology is utilized for obtaining both highly important solitons and singular periodic solutions.

This paper implements two efficient integration algorithms to retrieve soliton solutions that appear with weak nonlocal nonlinearity. These are trial equation method and modified simple equation scheme. Bright, dark and singular soliton solutions are recovered and their existence criteria are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this work, we derive the complexiton solutions for Date-Jimbo-Kashiwara-Miwa (DJKM) equation using the extended transformed rational function algorithm that relies on the Hirota bilinear form of the considered equation. Additional solutions such as complex-valued solutions also fall out of this integration scheme. Multisoliton-type solutions, in other words one-soliton, two-soliton and three-soliton solutions, which comprise both wave frequencies and generic phase shifts are presented through the medium of the multiple exp-function methodology which falls out as a result of generalisation of Hirota's perturbation technique.

In this work, we consider the perturbed Radhakrishnan-Kundu-Lakshmanan equation (RKL) which is the higher order nonlinear Schrodinger equation with cubic nonlinear terms in Kerr law. This equation is used to model propagation of solitons through an optical fiber. The multi-waves, interactions and Peregrine-like rational solutions of this equation are investigated with the aid of logarithmic transformation and symbolic computation without using Hirota bilinear forms By selecting appropriate values of the parameter, three dimensional plots are drawn to obtained solitons. Moreover, physical meanings of the solutions obtained were also presented.

WOS: 000374768100008, For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as M-1(G) = Sigma v(i is an element of V(G))d(C)(v(i))(2), where d(G) (v(i)) is the degree of vertex v(i), in G. Recently Xu et al. introduced two graphical invariants (Pi) over bar (1) (G) = Pi v(i)v(j is an element of E(G)) (dG (v(i))+dG (v(j))) and (Pi) over bar (2)(G) = Pi(vivj is an element of E(G)) (dG (v(i))+dG (v(j))) named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = Pi(n)(i=1) d(G) (v(i)). The irregularity index t(G) of G is defined as the num=1 ber of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M-1(G) of graphs and trees in terms of number of vertices, irregularity index, maximum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and NarumiKatayama index, and (first and second) multiplicative Zagreb index and coindices of graphs., National Research Foundation - Korean governmentKorean Government [2013R1A1A2009341]; Research Project Office of N. Erbakan University; Research Project Office of Uludag UniversityUludag University; Research Project Office of Selcuk UniversitySelcuk University; TUBITAK 2221-Programme, The first author is supported by the National Research Foundation funded by the Korean government with the grant no. 2013R1A1A2009341. The other authors are partially supported by Research Project Offices of N. Erbakan, Uludag and Selcuk Universities. This paper has been prepared during the Kinkar Ch. Das's visit in Turkey that was partially funded by TUBITAK 2221-Programme.

In this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

The trial equation method is applied to obtain soliton solutions to Lakshmanan–Porsezian–Daniel model in optical fibers, PCF and metamaterials. This integration procedure is implemented into the model with three forms of nonlinearity. Bright, dark and singular soliton solutions are recovered with conditions that guarantee their existence.

This paper retrieves soliton solutions to the perturbed resonant nonlinear Schrodinger's equation with time-dependent coefficients using the modified simple equation method. The four types of exotic non-Kerr laws studied in this paper are quadratic-cubic law, anti cubic law, cubic-quintic-septic law and triple-power law. These soliton solutions appear with constraint conditions that guarantee their existence. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper retrieves optical soliton solution to the perturbed complex Ginzburg-Landau equation that is studied with nine different forms of nonlinearity. The trial solutions approach is the integration algorithm adopted in this paper. The perturbation terms appear with full nonlinearity to get a taste of generalized setting. Bright, dark and singular soliton solutions are obtained. The existence criteria of such solitons are also presented. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

This paper secures bright, dark and singualr soliton solutions in birefringent fibers. Both Kerr law and parabolic law nonlinearity are studied. The trial equation method successfully retrieves these solitons with constraint relations that assure their existence. Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

In this paper, we discuss the various properties of a newly-constructed subclass of the class of normalized bi-univalent functions in the open unit disk, which is defined here by using a symmetric basic (or q-) derivative operator. Moreover, for functions belonging to this new basic (or q-) class of normalized biunivalent functions, we investigate the estimates and inequalities involving the second Hankel determinant.

The modified simple equation method applied to perturbed complex Ginzburg-Landau equation gives dark and singular solutions to the model. The perturbation terms appear with full nonlinearity. There are eight nonlinear forms studied in this paper. These solitons appear with constraint conditions that gurantee its existence and these are also presented. Chutian Scholar Program of Hubei Government in China Department of Mathematics and Statistics at Tshwane University of Technology South African National Foundation (92052 IRF1202210126) National Research Foundation of Korea Qatar National Research Fund (QNRF) (NPRP 8-028-1-001)

Let Σ g denote a closed orientable surface of genus g ≥ 2 . We consider a certain infinite family of Σ g -bundles over circle whose monodromies are taken from some collection of pseudo-Anosov diffeomorphisms. We show the existence of tight contact structure on every closed 3-manifold obtained via rational r -surgery along a section of any member of the family whenever r ≠ 2 g − 1 . Combining with Thurston's hyperbolic Dehn surgery theorem, we obtain infinitely many hyperbolic closed 3-manifolds admitting tight contact structures.

In this study, we perform the extended Kudryashov method to nonlinear Schrdinger equation (NLSE) with spatio-temporal dispersion that arises in a propagation of light in nonlinear optical fibers, planar waveguides, Bose-Einstein condensate theory. Four types of nonlinearity - Kerr law, power law, parabolic law and dual-power law - are being considered for the model. By using this scheme, the topological, singular soliton and rational solutions are obtained. In addition, some graphical simulations of solutions are provided. It is demonstrated that the proposed algorithm is effective and can be handled for many other nonlinear complex differential equations.

In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submersions according to characteristic vector field is vertical or horizontal., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1309.2487

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized tractrices in Euclidean (n + 1)-space En+1. Further, we introduce some kind of generalized rotational surfaces in Euclidean spaces E-3 and E-4, respectively. We have also obtained some basic properties of generalized rotational surfaces in E-4 and some results of their curvatures. Finally, we give some examples of generalized Beltrami surfaces in E-3 and E-4, respectively.

In this paper, all the solutions of the Diophantine equations x2 + 5a · pb = yn (for p = 29, 41) are given for nonnegative integers a, b, x, y, n ≥ 3 with x and y coprime. © 2017, University of Nis. All rights reserved.

Triangular numbers denoted by T-n are the numbers of the form T-n = n(2+1)/2 for n >= 0. There are infinitely many triangular numbers that are also square numbers. These numbers are called square triangular numbers and denoted by S-n. One can write Sn as S-n = s(n)(2) =t(n)(t(n)+1)/2 where s(n) and to denote the sides of the corresponding square and triangle. In this work, we derive some algebraic identities on triangular, square triangular numbers and also squares and triangles. We construct a connection between triangular and square triangular numbers. We determine when the equality T-n. = S-n, holds by using s(n), and t(n). We also deduce some formulas on perfect squares, sums of s(n), t(n), S-n,T-n, divisibility properties and integer solutions of Pell equations.

In the present paper we consider a special class of Lorentz surfaces in the four-dimensional pseudo-Euclidean space with neutral metric which are one-parameter systems of meridians of rotational hypersurfaces with timelike or spacelike axis and call them meridian surfaces. We give the complete classification of minimal and quasi-minimal meridian surfaces. We also classify the meridian surfaces with non-zero constant mean curvature., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1606.00047