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Articles 65851 - 65880 of 294560

Full-Text Articles in Physical Sciences and Mathematics

Initial Value Problem For Elastic System In Transversely Isotropic Inhomogeneous Media, Meltem Altunkaynak Jan 2022

Initial Value Problem For Elastic System In Transversely Isotropic Inhomogeneous Media, Meltem Altunkaynak

Turkish Journal of Mathematics

In this paper, we consider an initial value problem (IVP) for three dimensional elasticity system in a transversely isotropic inhomogeneous media. We will rewrite the problem in the form of Fourier images by means of Fourier transform method. After some arrangements, the problem is reduced to integral equations in the vector form. Using the properties of the vector integral equation and successive approximations method, an explicit formula for the solution of the IVP in transversely isotropic inhomogeneous media is constructed, and existence and uniqueness of the solution is stated. By a computational example, we illustrate the robustness of the method.

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A Note On The Transfinite Diameter Of Bernstein Sets, Özcan Yazici Jan 2022

A Note On The Transfinite Diameter Of Bernstein Sets, Özcan Yazici

Turkish Journal of Mathematics

A compact set $K\subset \mathbb C^n$ is called Bernstein set if, for some constant $M>0$, the following inequality $$ D^{\alpha}P _K\leq M^{ \alpha }(\deg P)^{ \alpha } P _K $$ is satisfied for every multiindex $\alpha\in \mathbb N^n$ and for every polynomial $P$. We provide here a lower bound for the transfinite diameter of Bernstein sets by using generalized extremal Leja points.

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On Finite Nonsolvable Groups Whose Cyclic $P$-Subgroups Of Equal Order Are Conjugate, Robert Van Der Waall, Sezgi̇n Sezer Jan 2022

On Finite Nonsolvable Groups Whose Cyclic $P$-Subgroups Of Equal Order Are Conjugate, Robert Van Der Waall, Sezgi̇n Sezer

Turkish Journal of Mathematics

The structure of the nonsolvable (P)-groups is completely described in this article. By definition, a finite group $G$ is called a (P)-group if any two cyclic $p$-subgroups of the same order are conjugate in $G$, whenever $p$ is a prime number dividing the order of $G$.

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Some Convergence, Stability, And Data Dependence Results For $K^{\Ast }$ Iterative Method Of Quasi-Strictly Contractive Mappings, Ruken Çeli̇k, Neci̇p Şi̇mşek Jan 2022

Some Convergence, Stability, And Data Dependence Results For $K^{\Ast }$ Iterative Method Of Quasi-Strictly Contractive Mappings, Ruken Çeli̇k, Neci̇p Şi̇mşek

Turkish Journal of Mathematics

In a recent paper, Yu et al. obtained convergence and stability results of the $K^{\ast }$ iterative method for quasi-strictly contractive mappings [An iteration process for a general class of contractive-like operators: Convergence, stability and polynomiography. AIMS Mathematics 2021; 6 (7): 6699-6714.]. To guarantee these convergence and stability results, the authors imposed some strong conditions on parametric control sequences which are used in the $K^{\ast }$ iterative method. The aim of the presented work is twofold: (a) to recapture the aforementioned results without any restrictions imposed on the mentioned parametric control sequences (b) to complete the work of Yu et …

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On The Convergence And Stability Analysis Of Finite-Difference Methods For The Fractional Newell-Whitehead-Segel Equations, İnci̇ Çi̇li̇ngi̇r Süngü, Emre Aydin Jan 2022

On The Convergence And Stability Analysis Of Finite-Difference Methods For The Fractional Newell-Whitehead-Segel Equations, İnci̇ Çi̇li̇ngi̇r Süngü, Emre Aydin

Turkish Journal of Mathematics

In this study, standard and non-standard finite-difference methods are proposed for numerical solutions of the time-spatial fractional generalized Newell-Whitehead-Segel equations describing the dynamical behavior near the bifurcation point of the Rayleigh-Benard convection of binary fluid mixtures. The numerical solutions have been found for high values of $p$ which shows the degree of nonlinear terms in the equations. The stability and convergence conditions of the obtained difference schemes are determined for each value of $p$. Errors of methods for various values of $p$ are given in tables. The compatibility of exact solutions and numerical solutions and the effectiveness of the methods …

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Adjunction Identity To Hypersemigroups, Niovi Kehayopulu Jan 2022

Adjunction Identity To Hypersemigroups, Niovi Kehayopulu

Turkish Journal of Mathematics

It is shown that some embedding problems on hypersemigroups are actually problems of adjunction. According to the theorem of this paper, for every hypersemigroup $S$ which does not have identity element, an hypersemigroup $T$ having identity element can be constructed in such a way that $S$ is an ideal of $T$. Moreover, if $S$ is regular, intra-regular, right (left) regular, right (left) quasi-regular or semisimple, then so is $T$. If $A$ is an ideal, subidempotent bi-ideal or quasi-ideal of $S$, then it is an ideal, bi-ideal, quasi-ideal of $T$ as well. Illustrative examples are given.

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Discrete Fractional Integrals, Lattice Points On Short Arcs, And Diophantine Approximation, Faruk Temur Jan 2022

Discrete Fractional Integrals, Lattice Points On Short Arcs, And Diophantine Approximation, Faruk Temur

Turkish Journal of Mathematics

Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate quadratic polynomials. We achieve this in part by establishing connections to problems on concentration of lattice points on short arcs of conics, whence we study discrete fractional integrals and lattice point concentration from a unified perspective via tools of sieving and diophantine approximation, and prove theorems that are of interest to researchers in both subjects.

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Nonic B-Spline Algorithms For Numerical Solution Of The Kawahara Equation, Meli̇s Zorşahi̇n Görgülü Jan 2022

Nonic B-Spline Algorithms For Numerical Solution Of The Kawahara Equation, Meli̇s Zorşahi̇n Görgülü

Turkish Journal of Mathematics

In this paper, the nonic (9th order) B-spline functions which have not been used before for the numerical solutions of the partial differential equations by finite element methods are used to solve numerically the Kawahara equation. These approaches involve the collocation and Galerkin finite element methods based on the nonic B-spline functions in space discretization and second order scheme (Crank-Nicolson method) in time discretization. To see the accuracy of the proposed methods three test problems are demonstrated and the obtained numerical results for both of the methods are compared with the exact solution of the Kawahara equation.

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On The Bmo Spaces Associated With The Laplace-Bessel Differential Operator, Si̇nem Sezer, Si̇mten Bayrakçi, Güldane Yildiz, Recep Kahraman Jan 2022

On The Bmo Spaces Associated With The Laplace-Bessel Differential Operator, Si̇nem Sezer, Si̇mten Bayrakçi, Güldane Yildiz, Recep Kahraman

Turkish Journal of Mathematics

In this paper, the characteristic properties of the space of functions of bounded mean oscillation called the $B$-$BMO$ associated with the Laplace-Bessel differential operator are obtained. The John-Nirenberg type inequality on the $B$-$BMO$ space and a relation between the $B$-Poisson integral and the $B$-$BMO$ functions are proved.

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Ulam's Type Stability Analysis Of Fractional Difference Equation With Impulse: Gronwall Inequality Approach, Rabia Ilyas, Mujeeb Ur Rehman Jan 2022

Ulam's Type Stability Analysis Of Fractional Difference Equation With Impulse: Gronwall Inequality Approach, Rabia Ilyas, Mujeeb Ur Rehman

Turkish Journal of Mathematics

In this paper, we present a new Gronwall inequality with an impulsive effect. The existence and uniqueness of the solution is investigated through fixed point theorems. Moreover, with the help of newly developed inequality Ulam's type stability criterion is developed for impulsive fractional difference equation. At last, a model is given to help the hypothetical outcome.

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Nonunique Best Proximity Point Results With An Application To Nonlinear Fractional Differential Equations, Mustafa Aslantaş Jan 2022

Nonunique Best Proximity Point Results With An Application To Nonlinear Fractional Differential Equations, Mustafa Aslantaş

Turkish Journal of Mathematics

In this paper, we point out an error in proving famous Achari type nonunique fixed point results. Also, we prove some best proximity point results in $b$ -metric spaces by introducing new concepts. Hence, we develop some results existing in the literature. Finally, we give a result for the existence of the solution of nonlinear fractional differential equations.

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Existence And Uniqueness Of Mild Solutions For Mixed Caputo And Riemann-Liouville Semilinear Fractional Integrodifferential Equations With Nonlocal Conditions, Ashraf H. A. Radwan Jan 2022

Existence And Uniqueness Of Mild Solutions For Mixed Caputo And Riemann-Liouville Semilinear Fractional Integrodifferential Equations With Nonlocal Conditions, Ashraf H. A. Radwan

Turkish Journal of Mathematics

The purpose of this paper is to investigate the existence and uniqueness of the mild solution to a class of semilinear fractional integrodifferential equations with state-dependent nonlocal fractional conditions. Our problem includes both Caputo and Riemann-Liouville fractional derivatives. Continuous dependence of solutions on initial conditions and $\epsilon$-approximate mild solutions of the considered problem will be discussed.

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Classical Solutions For 1-Dimensional And 2-Dimensional Boussinesq Equations, Svetlin Georgiev, Aissa Boukarou, Khaled Zennir Jan 2022

Classical Solutions For 1-Dimensional And 2-Dimensional Boussinesq Equations, Svetlin Georgiev, Aissa Boukarou, Khaled Zennir

Turkish Journal of Mathematics

In this article we investigate the IVPs for 1-dimensional and 2-dimensional Boussinesq equations. A new topological approach is applied to prove the existence of at least one classical solution and at least two nonnegative classical solutions for the considered IVPs. The arguments are based upon recent theoretical results.

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New Form Of Laguerre Fractional Differential Equation And Applications, Zahra Kavooci, Kazem Ghanbari, Hanif Mirzaei Jan 2022

New Form Of Laguerre Fractional Differential Equation And Applications, Zahra Kavooci, Kazem Ghanbari, Hanif Mirzaei

Turkish Journal of Mathematics

Laguerre differential equation is a well known equation that appears in the quantum mechanical description of the hydrogen atom. In this paper, we aim to develop a new form of Laguerre Fractional Differential Equation (LFDE) of order $2\alpha$ and we investigate the solutions and their properties. For a positive real number $\alpha$, we prove that the equation has solutions of the form $L_{n,\alpha}(x)=\sum_{k=0}^na_kx^k$, where the coefficients of the polynomials are computed explicitly. For integer case $\alpha=1$ we show that these polynomials are identical to classical Laguerre polynomials. Finally, we solve some fractional differential equations by defining a suitable integral transform.

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Solving A Class Of Ordinary Differential Equations And Fractional Differential Equations With Conformable Derivative By Fractional Laplace Transform, Mohammad Molaei, Farhad Dastmalchi Saei, Mohammad Javidi, Yaghoub Mahmoudi Jan 2022

Solving A Class Of Ordinary Differential Equations And Fractional Differential Equations With Conformable Derivative By Fractional Laplace Transform, Mohammad Molaei, Farhad Dastmalchi Saei, Mohammad Javidi, Yaghoub Mahmoudi

Turkish Journal of Mathematics

In this paper, we use the fractional Laplace transform to solve a class of second-order ordinary differential equations (ODEs), as well as some conformable fractional differential equations (CFDEs), including the Laguerre conformable fractional differential equation. Specifically, we apply the transform to convert the differential equations into first-order, linear differential equations. This is done by using the fractional Laplace transform of order $\alpha+\beta$ or $\alpha+\beta+\gamma$. Also, we investigate some more results on the fractional Laplace transform, obtained by Abdeljawad.

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Oscillation Of Second Order Mixed Functional Differential Equations With Sublinear And Superlinear Neutral Terms, Shan Shi, Zhenlai Han Jan 2022

Oscillation Of Second Order Mixed Functional Differential Equations With Sublinear And Superlinear Neutral Terms, Shan Shi, Zhenlai Han

Turkish Journal of Mathematics

In this paper, we shall establish some new oscillation theorems for the functional differential equations with sublinear and superlinear neutral terms of the form $$ (r(t)(z'(t))^\alpha)'=q(t)x^\alpha(\tau(t)), $$ where $z(t)=x(t)+p_1(t)x^\beta(\sigma(t))-p_2(t)x^\gamma(\sigma(t))$ with $0

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Some Recent Results In Plastic Structure On Riemannian Manifold, Akbar Dehghan Nezhad, Zohreh Aral Jan 2022

Some Recent Results In Plastic Structure On Riemannian Manifold, Akbar Dehghan Nezhad, Zohreh Aral

Turkish Journal of Mathematics

The plastic ratio is a fascinating topic that continually generates new ideas. The purpose of this paper is to point out and find some applications of the plastic ratio in the differential manifold. Precisely, we say that an $(1,1)$-tensor field $P$ on a $m$-dimensional Riemannian manifold $(M, g)$ is a plastic structure if it satisfies the equation $ P^3 = P + I $, where $ I $ is the identity. We establish several properties of the plastic structure. Then we show that a plastic structure induces on every invariant submanifold a plastic structure, too.

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Oscillation Of Third-Order Neutral Differential Equations With Oscillatory Operator, Miroslav Bartusek Jan 2022

Oscillation Of Third-Order Neutral Differential Equations With Oscillatory Operator, Miroslav Bartusek

Turkish Journal of Mathematics

A third-order damped neutral sublinear differential equation for which its differential operator is oscillatory is studied. Sufficient conditions are given under which every solution is either oscillatory or the derivative of its neutral term is oscillatory (or it tends to zero).

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On A Subclass Of The Analytic And Bi-Univalent Functions Satisfying Subordinate Condition Defined By $Q$-Derivativ, Ni̇zami̇ Mustafa, Semra Korkmaz Jan 2022

On A Subclass Of The Analytic And Bi-Univalent Functions Satisfying Subordinate Condition Defined By $Q$-Derivativ, Ni̇zami̇ Mustafa, Semra Korkmaz

Turkish Journal of Mathematics

In this paper, we introduce and examine certain subclass $\ M_{q,\Sigma }\left( \varphi ,\beta \right) $ of analytic and bi-univalent functions on the open unit disk in the complex plane. Here, we give coefficient bound estimates, upper bound estimate for the second Hankel determinant and Fekete-Szegö inequality for the function belonging to this class. Some interesting special cases of the results obtained here are also discussed.

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A Galerkin-Type Approach To Solve Systems Of Linear Volterra-Fredholm Integro-Differential Equations, Murat Karaçayir, Şuayi̇p Yüzbaşi Jan 2022

A Galerkin-Type Approach To Solve Systems Of Linear Volterra-Fredholm Integro-Differential Equations, Murat Karaçayir, Şuayi̇p Yüzbaşi

Turkish Journal of Mathematics

The main interest of this paper is to propose a numerical scheme in order to solve linear systems of Volterra-Fredholm integro-differential equations given with mixed conditions. The proposed method is a weighted residual scheme which uses monomials up to a prescribed degree $N$ as the basis functions. By taking inner product of the equation system with the elements of this basis set in a Galerkin-like fashion, the original problem is transformed into a linear algebraic equation system. After a suitable incorporation of the mixed conditions, a final algebraic system is obtained, from which the approximate solutions of the problem are …

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Continuous Wavelet Transform On Triebel-Lizorkin Spaces, Antonio L. Baisón Olmo, Víctor A. Cruz Barriguete, Jaime Navarro Jan 2022

Continuous Wavelet Transform On Triebel-Lizorkin Spaces, Antonio L. Baisón Olmo, Víctor A. Cruz Barriguete, Jaime Navarro

Turkish Journal of Mathematics

The continuous wavelet transform in higher dimensions is used to prove the regularity of weak solutions $u \in L^p(\mathbb R^n)$ under $Qu = f$ where $f$ belongs to the Triebel-Lizorkin space $F^{r,q}_p(\mathbb R^n)$ where $1 < p,q < \infty$, $0< r 0$ with positive constant coefficients $c_{\beta}$.

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On The Behaviour Of Solutions To A Kind Of Third Order Nonlinear Neutral Differential Equation With Delay, Adeleke Ademola, Peter Arawomo, Olufemi Adesina, Samuel Okoya Jan 2022

On The Behaviour Of Solutions To A Kind Of Third Order Nonlinear Neutral Differential Equation With Delay, Adeleke Ademola, Peter Arawomo, Olufemi Adesina, Samuel Okoya

Turkish Journal of Mathematics

This paper presents a novel class of third order nonlinear nonautonomous neutral differential equation with delay. The third order neutral differential equation is cut down to a system of first order, a suitable complete Lyapunov-Krasovskii's functional is constructed and used, to obtain standard conditions on the nonlinear functions to ensure stability and uniform asymptotic stability of the trivial solutions, the existence of a unique periodic solution, uniform boundedness and uniform ultimate boundedness of solutions when the forcing term is nonzero. The obtained results are new and include many prominent results on neutral and nonneutral delay differential equations in literature. Finally, …

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Constant Angle Surfaces In The Lorentzian Warped Product Manifold $I \Times_{F} \Mathbb E^2_1$, Uğur Dursun Jan 2022

Constant Angle Surfaces In The Lorentzian Warped Product Manifold $I \Times_{F} \Mathbb E^2_1$, Uğur Dursun

Turkish Journal of Mathematics

Let $I \times_{f} \mathbb E^2_1$ be a 3-dimensional Lorentzian warped product manifold with the metric $\tilde g = dt^2 + f^2(t) (dx^2 - dy^2)$, where $I$ is an open interval, $f$ is a strictly positive smooth function on $I,$ and $\mathbb E^2_1$ is the Minkowski 2-plane. In this work, we give a classification of all space-like and time-like constant angle surfaces in $I \times_{f} \mathbb E^2_1$ with nonnull principal direction when the surface is time-like. In this classification, we obtain space-like and time-like surfaces with zero mean curvature, rotational surfaces, and surfaces with constant Gaussian curvature. Also, we have some …

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Novel Exact Solutions To Navier-Stokes Momentum Equations Describing An Incompressible Fluid, Yahya Öz Jan 2022

Novel Exact Solutions To Navier-Stokes Momentum Equations Describing An Incompressible Fluid, Yahya Öz

Turkish Journal of Mathematics

An analytical solution to the incompressible Navier-Stokes momentum equations for a divergence-free flow $\boldsymbol{\nabla}\cdot \vec u\left(\vec x,t\right)=0$ with time-dependent dynamic viscosity $\mu\left(t\right)$ is presented. The demonstrated methodology holds for the physically relevent three dimensions. The constructed flow velocities $\vec u\left(\vec x,t\right)$ are eigenvectors of the vector operator curl. Moreover, vortex $\vec \omega\left(\vec x,t\right)$, helicity $H\left(\vec x,t\right)$, enstrophy $\mathcal{E}\left(t\right)$ and enstrophy evolution $\frac{\mathrm{d}\mathcal{E}\left(t\right)}{\mathrm{d}t}$ are explicitly determined.

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$3d$-Flows Generated By The Curl Of A Vector Potential & Maurer-Cartan Equations, Oğul Esen, Partha Guha, Hasan Gümral Jan 2022

$3d$-Flows Generated By The Curl Of A Vector Potential & Maurer-Cartan Equations, Oğul Esen, Partha Guha, Hasan Gümral

Turkish Journal of Mathematics

We examine $3D$ flows $\mathbf{\dot{x}}=\mathbf{v}({\bf x})$ admitting vector identity $M\mathbf{v} = \nabla \times \mathbf{A}$ for a multiplier $M$ and a potential field $\mathbf{A}$. It is established that, for those systems, one can complete the vector field $\mathbf{v}$ into a basis fitting an $\mathfrak{sl}(2)$-algebra. Accordingly, in terms of covariant quantities, the structure equations determine a set of equations in Maurer-Cartan form. This realization permits one to obtain the potential field as well as to investigate the (bi-)Hamiltonian character of the system. The latter occurs if the system has a time-independent first integral. In order to exhibit the theoretical results on some …

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Bessel Equation And Bessel Function On $\Mathbb{T}_{(Q,H)}$, Ahmet Yantir, Burcu Si̇li̇ndi̇r Yantir, Zehra Tuncer Jan 2022

Bessel Equation And Bessel Function On $\Mathbb{T}_{(Q,H)}$, Ahmet Yantir, Burcu Si̇li̇ndi̇r Yantir, Zehra Tuncer

Turkish Journal of Mathematics

This article is devoted to present nabla $(q, h)$-analogues of Bessel equation and Bessel function. In order to construct series solution of nabla $(q, h)$-Bessel equation, we present nabla $(q, h)$-analysis regarding nabla generalized quantum binomial, nabla $(q,h)$-analogues of Taylor's formula, Gauss's binomial formula, Taylor series, analytic functions, analytic exponential function with its fundamental properties, analytic trigonometric and hyperbolic functions. We emphasize that nabla $(q, h)$-Bessel equation recovers classical, $h$- and $q$-discrete Bessel equations. In addition, we establish nabla $(q, h)$-Bessel function which is expressed in terms of an absolutely convergent series in nabla generalized quantum binomials and intimately demonstrate …

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The Interior-Boundary Strichartz Estimate For The Schrödinger Equation On The Half-Line Revisited, Bi̇lge Köksal, Türker Özsari Jan 2022

The Interior-Boundary Strichartz Estimate For The Schrödinger Equation On The Half-Line Revisited, Bi̇lge Köksal, Türker Özsari

Turkish Journal of Mathematics

In recent papers, it was shown for the biharmonic Schrödinger equation and 2D Schrödinger equation that Fokas method-based formulas are capable of defining weak solutions of associated nonlinear initial boundary value problems (ibvps) below the Banach algebra threshold. In view of these results, we revisit the theory of interior-boundary Strichartz estimates for the Schrödinger equation posed on the right half line, considering both Dirichlet and Neumann cases. Finally, we apply these estimates to obtain low regularity solutions for the nonlinear Schrödinger equation (NLS) with Neumann boundary condition and a coupled system of NLS equations defined on the half line with …

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Maximising The Number Of Connected Induced Subgraphs Of Unicyclic Graphs, Audace A V Dossou Olory Jan 2022

Maximising The Number Of Connected Induced Subgraphs Of Unicyclic Graphs, Audace A V Dossou Olory

Turkish Journal of Mathematics

Denote by $\mathcal{G}(n,c,g,k)$ the set of all connected graphs of order $n$, having $c$ cycles, girth $g$, and $k$ pendant vertices. In this paper, we give a partial characterisation of the structure of those graphs in $\mathcal{G}(n,c,g,k)$ maximising the number of connected induced subgraphs. For the special case where $c=1$, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the corresponding maximum number given the following parameters: (1) order, girth, and number of pendant vertices; (2) order and girth; (3) order.

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Geochemical And Petrographic Approach For The Origin Of The Limestone Blocks Of Thewalls Of The Great Temple Of Hattusha, Çorum (N-Turkey), İsmai̇l Ömer Yilmaz, Andreas Schachner, İbrahi̇m Tonguç Uysal, Ezgi̇ Ünal İmer Jan 2022

Geochemical And Petrographic Approach For The Origin Of The Limestone Blocks Of Thewalls Of The Great Temple Of Hattusha, Çorum (N-Turkey), İsmai̇l Ömer Yilmaz, Andreas Schachner, İbrahi̇m Tonguç Uysal, Ezgi̇ Ünal İmer

Turkish Journal of Earth Sciences

The socle stones of the Great Temple in the Lower City of the Hittite capital city, Hattusha (Çorum Province) have been studied for the first time by petrographic and geochemical analysis (Sr isotope, stable isotope, and trace element geochemistry). Study objectives were to determine the carbonate facies of the stone samples used in the Great Temple and investigate their possible sources. Petrographic analysis of 10 limestone samples presents five clusters of different microfacies. Stable isotope (δ13C and δ18O) analysis displayed the presence of 5 groups in parallel with petrographic analysis. Trace element analysis of the same groups displayed considerable difference …

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Stratigraphic Correlation Of The Givetian (Late Middle Devonian) Deposits In Qujing,Yunnan, Southwestern China And The Paleogeographic Implications, Jianhua Wang, Yunchuan Mi, Xiaorou Ma, Yukun Bi Jan 2022

Stratigraphic Correlation Of The Givetian (Late Middle Devonian) Deposits In Qujing,Yunnan, Southwestern China And The Paleogeographic Implications, Jianhua Wang, Yunchuan Mi, Xiaorou Ma, Yukun Bi

Turkish Journal of Earth Sciences

Detailed description of spatio-temporal framework of fossil-bearing strata is of fundamental significance for understanding the evolution of organisms and regional tectonics. The Qujing and Haikou formations are two lithological units exposed in eastern Yunnan, southwestern China, and both bear rich fossils indicative of a Middle Devonian age, while their relationship or lateral correlation remains unclear. Based on detailed observations of outcrops of these two formations in Qujing, eastern Yunnan, and as well as their typical rock characteristics and fossil contents, we propose that the two formations are basically contemporaneous heterotopic facies. From north to south of the Qujing area, the …

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