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DYNAMICS OF THE PREDATOR-PREY MODELS ON THE TWO-PATCH FRAGMENTED HABITAT WITH DISPERSAL
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109374
title: DYNAMICS OF THE PREDATOR-PREY MODELS ON THE TWO-PATCH FRAGMENTED HABITAT WITH DISPERSAL abstract: In this work, we consider the population-dispersal dynamics for predator-prey interactions in a two-patch environment. On each fragmented patch, there is a two-species predator-prey ecological system. It is assumed that the predator species are mobile. The existence and local dynamics of boundary equilibria and interior equilibria with respect to parameters are completely classified. Moreover, global extinction results are established analytically. In particular, the phenomenon of over-exploitation is also found in these discrete patches models. Finally, some biological interpretations are discussed.
<br>DYNAMICS OF THE PREDATOR-PREY MODELS ON THE TWO-PATCH FRAGMENTED HABITAT WITH DISPERSAL
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109373
title: DYNAMICS OF THE PREDATOR-PREY MODELS ON THE TWO-PATCH FRAGMENTED HABITAT WITH DISPERSAL abstract: In this work, we consider the population-dispersal dynamics for predator-prey interactions in a two-patch environment. On each fragmented patch, there is a two-species predator-prey ecological system. It is assumed that the predator species are mobile. The existence and local dynamics of boundary equilibria and interior equilibria with respect to parameters are completely classified. Moreover, global extinction results are established analytically. In particular, the phenomenon of over-exploitation is also found in these discrete patches models. Finally, some biological interpretations are discussed.
<br>REGIME SHIFT IN A PHYTOPLANKTON–PHOSPHORUS MODEL WITH VERTICAL STRUCTURE AND SEASONALITY
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109372
title: REGIME SHIFT IN A PHYTOPLANKTON–PHOSPHORUS MODEL WITH VERTICAL STRUCTURE AND SEASONALITY abstract: Many ecological systems are influenced by positive feedbacks between organisms and abiotic environments, which generates multiple stable equilibria of a mathematical model with a hysteresis structure. In addition, discontinuous shifts of system at equilibrium is predicted, which is often called regime shift in ecosystem sciences. This hysteresis structure is unfavorable from environmental management point of view, because the reconstruction of original equilibrium state requests much lower levels of external forcing. Mathematical models proposed in previous works are simple and mathematically tractable ([7],[2]).However, it is difficult to extrapolate from such simple models the occurrence likelihood of regime shift in natural environments since temporally dynamic features in ecology and physico-chemical environments, and spatial dimension are less explored in those models. In this study, we construct a realistic but mathematically tractable model of interaction between phytoplankton and phosphorus, which incorporates (1) 1-dimensional vertical structure of lake ecosystem and (2) seasonal periodic cycle of water mixing. We aim to understand the impact of changes in seasonality in various types of lakes on the occurrence of multiple attractors (periodic solution) and hysteresis structure.
<br>ON THE EVOLUTIONARY STABILITY OF MALE HARASSMENT IN A COERCIVE MATING GAME
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109371
title: ON THE EVOLUTIONARY STABILITY OF MALE HARASSMENT IN A COERCIVE MATING GAME abstract: In many animals, males employ coercive mating strategies to help them maximize their expected number of offspring. In such systems, selection will favor behavioral adaptations in females that help them mitigate harassment costs and maximize their reproductive fitness. Previously, Bokides et al. [1] presented a model showing how male harassment strategies can co-evolve with female habitat preferences in a mating game. Their results indicated that if females dispersed freely between habitats where males were present and where males were excluded, selection could favor males who strategically harassed at high (or low) levels, depending on the proximity of their phenotype to a threshold level h∗h∗. This article is a continuation of that work addressing the questions of stability at equilibria where males harass at the threshold level (i.e., h∗h∗). We show these states are both locally and globally asymptotically stable. Further, we argue based on these results that h∗h∗ is an evolutionary stable male harassment level at which females will be ideally distributed to match the resource quality and social environments of their alternative habitats.
<br>A REACTION-DIFFUSION SYSTEM AND ITS SHADOW SYSTEM DESCRIBING HARMFUL ALGAL BLOOMS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109370
title: A REACTION-DIFFUSION SYSTEM AND ITS SHADOW SYSTEM DESCRIBING HARMFUL ALGAL BLOOMS abstract: The occurrence of harmful algal blooms (HAB) in ecosystems is a worldwide environmental issue that currently needs to be addressed. An attempt to theoretically understand the mechanism behind the formation of HAB has led to the proposal of a reaction-diffusion model of the Lotka--Volterra type. In particular, a shadow system, as a limiting system of the model in which the diffusion rate tends to infinity, has been proposed to study whether or not stable nonconstant equilibrium solutions of the system exist, because these solutions are mathematically associated with HAB. In this paper, we discuss the convergence property between solutions of the full system and its shadow system from the point of view of an evolutional problem.
<br>LACK OF SYMMETRY IN LINEAR DETERMINACY DUE TO CONVECTIVE EFFECTS IN REACTION-DIFFUSION-CONVECTION PROBLEMS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109369
title: LACK OF SYMMETRY IN LINEAR DETERMINACY DUE TO CONVECTIVE EFFECTS IN REACTION-DIFFUSION-CONVECTION PROBLEMS abstract: This paper is concerned with linear determinacy in monostable reaction- diffusion-convection equations and co-operative systems. We present sufficient conditions for minimal travelling-wave speeds (equivalent to spreading speeds) to equal values obtained from linearisations of the travelling-wave problem about the unstable equilibrium. These conditions involve both reaction and convection terms. We present separate conditions for non-increasing and non-decreasing travelling waves, called `right' and `left' conditions respectively, because of the asymmetry in propagation caused by the convection terms. We also give a necessary condition on the reaction term for the existence of convection terms such that both the right and left conditions can be satisfied simultaneously. Examples show that our sufficient conditions for linear determinacy are not necessary and compare these conditions in the scalar case with alternative conditions observed in Malaguti-Marcelli [15] and Benguria-Depassier-Mendez [3]. We also illustrate, for both an equation and a system, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied. An example is given of an equation which is right but not left linearly determinate.
<br>ON THE EXISTENCE OF TWO STATIONARY SOLUTIONS FOR A FREE BOUNDARY PROBLEM DESCRIBING CELL MOTILITY
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109368
title: ON THE EXISTENCE OF TWO STATIONARY SOLUTIONS FOR A FREE BOUNDARY PROBLEM DESCRIBING CELL MOTILITY abstract: This paper is concerned with the existence of stationary solutions for a free boundary problem related to cell motility. In recent years, the author and Ninomiya \cite{monobe_ninomiya} showed that there exist at least two stationary solutions with disk-shaped domains in isotropic boundary conditions. In this paper, it will be shown that there exist exactly two stationary solutions for the free boundary problem under the same boundary conditions. The proof is based on the weak maximum principle and the mean-valued theorem.
<br>THE OPTIMAL DISPERSAL STRATEGY: A TWO-PATCH MODEL WITH TRAVEL LOSS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109367
title: THE OPTIMAL DISPERSAL STRATEGY: A TWO-PATCH MODEL WITH TRAVEL LOSS abstract: The dispersal of organisms plays an important role in determining the dynamics of ecological models. Ecologically, it is of interest in understanding how dispersal strategy influences the distribution of populations. An ideal free distribution (IFD) of populations has been used to predict the distribution of organisms among patches, where a key assumption is to assume that species can move freely between patches without paying any cost. If instead one assumes that there are losses when species moves from one patch to another, then ideal free distributions may not appear. In this note, we examine a two-patch resident-mutant model with travel loss and predict the optimal dispersal strategy for resident and mutant. Moreover, such strategy which produces a non-IFD is evolutionarily stable. Some same and different features of patch models with travel loss are discussed.
<br>ASYMPTOTIC ANALYSIS OF A MONOSTABLE EQUATION IN PERIODIC MEDIA
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109366
title: ASYMPTOTIC ANALYSIS OF A MONOSTABLE EQUATION IN PERIODIC MEDIA abstract: We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} %[4]of the well-known spreading properties \cite{Wein02}, %[32], \cite{Ber-Ham-02}, %[9],and the solution of a Hamilton-Jacobi equation.
<br>LORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109365
title: LORENTZ SPACE ESTIMATES FOR VECTOR FIELDS WITH DIVERGENCE AND CURL IN HARDY SPACES abstract: In this note, we establish the estimate on the Lorentz space L(3/2,1)L(3/2,1) for vector fields in bounded domains under the assumption that the normal or the tangential component of the vector fields on the boundary vanishes. We prove that the L(3/2,1)L(3/2,1) norm of the vector field can be controlled by the norms of its divergence and curl in the atomic Hardy spaces and the L1L1 norm of the vector field itself.
<br>GROWTH OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH EXTREMAL FUNCTIONS FOR DENJOY'S CONJECTURE AS COEFFICIENTS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109364
title: GROWTH OF SOLUTIONS OF SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH EXTREMAL FUNCTIONS FOR DENJOY'S CONJECTURE AS COEFFICIENTS abstract: The classical problem of finding conditions on the entire coefficients A(z)A(z) and B(z)B(z) guaranteeing that all nontrivial solutions of f′′+A(z)f′+B(z)f=0f″+A(z)f′+B(z)f=0 are of infinite order is discussed. Some such conditions which involve deficient value, Borel exceptional value and extremal functions for Denjoy's conjecture are obtained.
<br>INEQUALITIES FOR POWER SERIES WITH NONNEGATIVE COEFFICIENTS VIA A REVERSE OF JENSEN INEQUALITY
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109363
title: INEQUALITIES FOR POWER SERIES WITH NONNEGATIVE COEFFICIENTS VIA A REVERSE OF JENSEN INEQUALITY abstract: Some inequalities for power series with nonnegative coefficients via a reverse of Jensen inequality obtained by Dragomir & Ionescu in 1994 are given. Applications for some fundamental functions defined by power series are also provided.
<br>ON SOME CLASSES OF INVARIANT SUBMANIFOLDS OF LORENTZIAN PARA-SASAKIAN MANIFOLDS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109362
title: ON SOME CLASSES OF INVARIANT SUBMANIFOLDS OF LORENTZIAN PARA-SASAKIAN MANIFOLDS abstract: The object of the present paper is to study invariant submanifolds of Lorenzian Para-Sasakian manifolds. We consider the recurrent and bi-recurrent invariant submanifolds of Lorentzian para-Sasakian manifolds and pseudo-parallel and generalized Ricci pseudo-parallel invariant submanifolds of Lorentzian para-Sasakian manifolds. Also we search for the conditions Z(X,Y)⋅α=fQ(g,α)Z(X,Y)⋅α=fQ(g,α) and Z(X,Y)⋅α=fQ(S,α)Z(X,Y)⋅α=fQ(S,α) on invariant submanifolds of Lorentzian para-Sasakian manifolds, where ZZ is the concircular curvature tensor. Finally, we construct an example of an invariant submanifold of Lorentzian para Sasakian manifold.
<br>UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS OF ENTIRE FUNCTIONS SHARING ONE VALUE
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109361
title: UNIQUENESS OF DIFFERENCE-DIFFERENTIAL POLYNOMIALS OF ENTIRE FUNCTIONS SHARING ONE VALUE abstract: In this paper, we study the uniqueness of difference-differential polynomials of entire functions ff and gg sharing one value with counting multiplicity. In this paper we extend and generalize the results of X. Y. Zhang, J. F. Chen and W. C. Lin [17] L. Kai, L. Xin-ling and C. Ting-bin [7] and many others [2, 16].
<br>SECOND DEGREE GENERALIZED JACOBI ITERATION METHOD FOR SOLVING SYSTEM OF LINEAR EQUATIONS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109360
title: SECOND DEGREE GENERALIZED JACOBI ITERATION METHOD FOR SOLVING SYSTEM OF LINEAR EQUATIONS abstract: In this paper, a Second degree generalized Jacobi Iteration method for solving system of linear equations, Ax=bAx=b and discuss about the optimal values a1a1 and b1b1 in terms of spectral radius about for the convergence of SDGJ method of x(n+1)=b1[D−1m(Lm+Um)x(n)+k1m]−a1x(n−1).x(n+1)=b1[Dm−1(Lm+Um)x(n)+k1m]−a1x(n−1). Few numerical examples are considered to show that the effective of the Second degree Generalized Jacobi Iteration method (SDGJ) in comparison with FDJ, FDGJ, SDJ.
<br>JOINS, CORONAS AND THEIR VERTEX-EDGE WIENER POLYNOMIALS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109359
title: JOINS, CORONAS AND THEIR VERTEX-EDGE WIENER POLYNOMIALS abstract: The vertex-edge Wiener index of a simple connected graph GG is defined as the sum of distances between vertices and edges of GG. The vertex-edge Wiener polynomial of GG is a generating function whose first derivative is a q−q−analog of the vertex-edge Wiener index. Two possible distances D1(u,e|G)D1(u,e|G) and D2(u,e|G)D2(u,e|G) between a vertex uu and an edge ee of GG can be considered and corresponding to them, the first and second vertex-edge Wiener indices of GG, and the first and second vertex-edge Wiener polynomials of GG are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.
<br>A USEFUL ORTHONORMAL BASIS ON BI-SLANT SUBMANIFOLDS OF ALMOST HERMITIAN MANIFOLDS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109358
title: A USEFUL ORTHONORMAL BASIS ON BI-SLANT SUBMANIFOLDS OF ALMOST HERMITIAN MANIFOLDS abstract: In this paper, we study bi-slant submanifolds of an almost Hermitian manifold for different cases. We introduce a new orthonormal basis on bi-slant submanifold, semi-slant submanifold and hemi-slant submanifold of an almost Hermitian manifold to compute Chen's main inequalities. We investigate these inequalities for semi-slant submanifolds, hemi-slant submanifolds and slant submanifolds of a generalized complex space form. We obtain some characterizations on such submanifolds of a complex space form.
<br>TWIN SIGNED ROMAN DOMINATION NUMBERS IN DIRECTED GRAPHS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109357
title: TWIN SIGNED ROMAN DOMINATION NUMBERS IN DIRECTED GRAPHS abstract: Let DD be a finite simple digraph with vertex set V(D)V(D) and arc set A(D)A(D). A twin signed Roman dominating function (TSRDF) on the digraph DD is a function f:V(D)→{−1,1,2}f:V(D)→{−1,1,2} satisfying the conditions that (i) ∑x∈N−[v]f(x)≥1∑x∈N−[v]f(x)≥1 and ∑x∈N+[v]f(x)≥1∑x∈N+[v]f(x)≥1 for each v∈V(D)v∈V(D), where N−[v]N−[v] (resp. N+[v]N+[v]) consists of vv and all in-neighbors (resp. out-neighbors) of vv, and (ii) every vertex uu for which f(u)=−1f(u)=−1 has an in-neighbor vv and an out-neighbor ww for which f(v)=f(w)=2f(v)=f(w)=2. The weight of an TSRDF ff is ω(f)=∑v∈V(D)f(v)ω(f)=∑v∈V(D)f(v). The twin signed Roman domination number γ∗sR(D)γsR∗(D) of DD is the minimum weight of an TSRDF on DD. In this paper, we initiate the study of twin signed Roman domination in digraphs and we present some sharp bounds on γ∗sR(D)γsR∗(D). In addition, we determine the twin signed Roman domination number of some classes of digraphs.
<br>INEQUALITIES FOR SOME CLASSICAL INTEGRAL TRANSFORMS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109356
title: INEQUALITIES FOR SOME CLASSICAL INTEGRAL TRANSFORMS abstract: By using a form of the Cauchy-Bunyakovsky-Schwarz inequality, we establish new inequalities for some classical integral transforms such as Laplace transform,Fourier transform, Fourier cosine transform, Fourier sine transform, Mellin transform and Hankel transform.
<br>ON CERTAIN INTEGRAL FORMULAS INVOLVING THE PRODUCT OF BESSEL FUNCTION AND JACOBI POLYNOMIAL
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109355
title: ON CERTAIN INTEGRAL FORMULAS INVOLVING THE PRODUCT OF BESSEL FUNCTION AND JACOBI POLYNOMIAL abstract: In the present paper, we establish some interesting integrals involving the product of Bessel function of the first kind with Jacobi polynomial, which are expressed in terms of Kampe de Feriet and Srivastava and Daoust functions. Some other integrals involving the product of Bessel (sine and cosine) function with ultraspherical polynomial, Gegenbauer polynomial, Tchebicheff polynomial, and Legendre polynomial are also established as special cases of our main results. Further, we derive an interesting connection between Kampe de Feriet and Srivastava and Daoust functions.
<br>QUENCHING PROBLEMS FOR A SEMILINEAR REACTION-DIFFUSION SYSTEM WITH SINGULAR BOUNDARY OUTFLUX
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109354
title: QUENCHING PROBLEMS FOR A SEMILINEAR REACTION-DIFFUSION SYSTEM WITH SINGULAR BOUNDARY OUTFLUX abstract: In this paper, we study two quenching problems for the following semilinear reaction-diffusion system:
ut=uxx+(1−v)−p1,0<x<1, 0<t<T,ut=uxx+(1−v)−p1,0<x<1, 0<t<T,
vt=vxx+(1−u)−p2,0<x<1, 0<t<T,vt=vxx+(1−u)−p2,0<x<1, 0<t<T,
ux(0,t)=0, ux(1,t)=−v−q1(1,t), 0<t<T,ux(0,t)=0, ux(1,t)=−v−q1(1,t), 0<t<T,
vx(0,t)=0, vx(1,t)=−u−q2(1,t), 0<t<T,vx(0,t)=0, vx(1,t)=−u−q2(1,t), 0<t<T,
u(x,0)=u0(x)<1,v(x,0)=v0(x)<1, 0≤x≤1,u(x,0)=u0(x)<1,v(x,0)=v0(x)<1, 0≤x≤1,
where p1,p2,q1,q2p1,p2,q1,q2 are positive constants and u0(x),v0(x)u0(x),v0(x) are positive smooth functions. We firstly get a local exisence result for this system. In the first problem, we show that quenching occurs in finite time, the only quenching point is x=0x=0 and (ut,vt)(ut,vt) blows up at the quenching time under the certain conditions. In the second problem, we show that quenching occurs in finite time, the only quenching point is x=1x=1 and (ut,vt)(ut,vt)blows up at the quenching time under the certain conditions.
<br>HERMITE-HADAMARD TYPE INEQUALITIES FOR (P1,H1)-(P2,H2)-CONVEX FUNCTIONS ON THE CO-ORDINATES
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109353
title: HERMITE-HADAMARD TYPE INEQUALITIES FOR (P1,H1)-(P2,H2)-CONVEX FUNCTIONS ON THE CO-ORDINATES abstract: In this paper, we establish some Hermite-Hadamard type inequalities for (p1,h1)(p1,h1)-(p2,h2)(p2,h2)-convex function on the co-ordinates. Furthermore, some inequalities of Hermite-Hadamard type involving product of two convex functions on the co-ordinates are also considered. The results presented here would provide extensions of those given in earlier works.
<br>ON SOLVABILITY OF COUPLED HYBRID SYSTEM OF QUADRATIC FRACTIONAL INTEGRAL EQUATIONS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109352
title: ON SOLVABILITY OF COUPLED HYBRID SYSTEM OF QUADRATIC FRACTIONAL INTEGRAL EQUATIONS abstract: Of concern is studying solvability of the hybrid systems of quadratic fractional integral equations. To this aim applying hybrid fixed point theory due to DhageDhage, existence of at least one positive solution for mentioned systems via so called D-Lipschitzian mappings will be concluded . We illustrate the obtained results by presenting an example.
<br>A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109351
title: A NOTE ON THE LEAST (NORMALIZED) LAPLACIAN EIGHVA;UE OF SIGNED GRAPHS abstract: Let Γ=(G,σ)Γ=(G,σ) be a connected signed graph, and L(Γ)L(Γ) be its Laplacian and L(Γ)L(Γ) its normalized Laplacian with eigenvalues λ1≥λ2≥⋯≥λnλ1≥λ2≥⋯≥λn and μ1≥μ2≥⋯≥μnμ1≥μ2≥⋯≥μn, respectively. It is known that a signed graph ΓΓ is balanced if and only if λn=0λn=0 (or μn=0μn=0). We show that λnλn and μnμn measure how much ΓΓ is far from being balanced by proving that
μn(Γ)λn(Γ)≤min{2ϵ(Γ)m,ν(Γ)ν(Γ)+ν1(Γ)},≤min{λ1(Γ′):Γ−Γ′isbalanced},
μn(Γ)≤min{2ϵ(Γ)m,ν(Γ)ν(Γ)+ν1(Γ)},λn(Γ)≤min{λ1(Γ′):Γ−Γ′isbalanced},
where ν(Γ)ν(Γ) (resp. ϵ(Γ)ϵ(Γ)) denotes the frustration number (resp. the frustration index) of ΓΓ, that is the minimum number of vertices (resp. edges) to be deleted such that the signed graph is balanced.
<br>A GCD AND LCM-LIKE INEQUALITY FOR MULTIPLICATIVE LATTICES
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109350
title: A GCD AND LCM-LIKE INEQUALITY FOR MULTIPLICATIVE LATTICES abstract: Let A1,…,AnA1,…,An (n≥2)(n≥2) be elements of an commutative multiplicative lattice. Let G(k)G(k) (resp., L(k)L(k)) denote the product of all the joins (resp., meets) of kk of the elements. Then we show that
L(n)G(2)G(4)⋯G(2⌊n/2⌋)≤G(1)G(3)⋯G(2⌈n/2⌉−1).
L(n)G(2)G(4)⋯G(2⌊n/2⌋)≤G(1)G(3)⋯G(2⌈n/2⌉−1).
In particular this holds for the lattice of ideals of a commutative ring. We also consider the relationship between
G(n)L(2)L(4)⋯L(2⌊n/2⌋) and L(1)L(3)⋯L(2⌈n/2⌉−1)
G(n)L(2)L(4)⋯L(2⌊n/2⌋) and L(1)L(3)⋯L(2⌈n/2⌉−1)
and show that any inequality relationships are possible.
<br>EXISTENCE THEOREMS FOR GENERALIZED VECTOR EQUILIBRIA WITH VARIABLE ORDERING RELATION
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109349
title: EXISTENCE THEOREMS FOR GENERALIZED VECTOR EQUILIBRIA WITH VARIABLE ORDERING RELATION abstract: In this paper we study the solvability of the generalized vector equilibrium problem (for short, GVEP) with a variable ordering relation in reflexive Banach spaces. The existence results of strong solutions of GVEPs for monotone multifunctions are established with the use of the KKM-Fan theorem. We also investigate the GVEPs without monotonicity assumptions and obtain the corresponding results of weak solutions by applying the Brouwer fixed point theorem. These results are also the extension and improvement of some recent results in the literature.
<br>A NEW GENERAL IDEA FOR STARLIKE AND CONVEX FUNCTIONS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109348
title: A NEW GENERAL IDEA FOR STARLIKE AND CONVEX FUNCTIONS abstract: Let AA be the class of functions f(z)f(z) which are analytic in the open unit disk UU with f(0)=0f(0)=0 and f′(0)=1f′(0)=1. For the class AA, a new general class AkAk is defined. With this general class AkAk, two interesting classes S∗k(α)Sk∗(α) and Kk(α)Kk(α) concerning classes of starlike of order αα in UU and convex of order αα in UU are considered.
<br>ISOTROPIC GEOMETRY OF GRAPH SURFACES ASSOCIATED WITH PRODUCT PRODUCTION FUNCTIONS IN ECONOMICS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109347
title: ISOTROPIC GEOMETRY OF GRAPH SURFACES ASSOCIATED WITH PRODUCT PRODUCTION FUNCTIONS IN ECONOMICS abstract: A production function is a mathematical formalization in economics which denotes the relations between the output generated by a firm, an industry or an economy and the inputs that have been used in obtaining it. In this paper, we study the product production functions of 2 variables in terms of the geometry of their associated graph surfaces in the isotropic 3−3−space I3I3. In particular, we derive several classification results for the graph surfaces of product production functions in I3I3 with constant curvature.
<br>THE ROMAN BONDAGE NUMBER OF A DIGRAPH
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109346
title: THE ROMAN BONDAGE NUMBER OF A DIGRAPH abstract: Let D=(V,A)D=(V,A) be a finite and simple digraph. A Roman dominating function on DD is a labeling f:V(D)→{0,1,2}f:V(D)→{0,1,2} such that every vertex with label 0 has an in-neighbor with label 2. The weight of an RDF ff is the value ω(f)=∑v∈Vf(v)ω(f)=∑v∈Vf(v). The minimum weight of a Roman dominating function on a digraph DD is called the Roman domination number, denoted by γR(D)γR(D). The Roman bondage number bR(D)bR(D) of a digraph DD with maximum out-degree at least two is the minimum cardinality of all sets A′⊆AA′⊆A for which γR(D−A′)>γR(D)γR(D−A′)>γR(D). In this paper, we initiate the study of the Roman bondage number of a digraph. We determine the Roman bondage number in several classes of digraphs and give some sharp bounds.
<br>EULER-CES`ARO DIFFERENCE SPACES OF BOUNDED, CONVERGENT AND NULL SEQUENCES
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109345
title: EULER-CES`ARO DIFFERENCE SPACES OF BOUNDED, CONVERGENT AND NULL SEQUENCES abstract: In this paper, we introduce the spaces ℓ˘∞ℓ˘∞, c˘c˘ and c˘0c˘0 of Euler-Ces`aro bounded, convergent and null difference sequences and prove that the inclusions ℓ∞⊂ℓ˘∞ℓ∞⊂ℓ˘∞, c⊂c˘c⊂c˘ and c0⊂c˘0c0⊂c˘0 strictly hold. We show that the spaces c˘0c˘0 and c˘c˘ turn out to be the separable BK spaces such that c˘c˘ does not possess any of the following: AK property and monotonicity. We determine the alpha-, beta- and gamma-duals of the new spaces and characterize the matrix classes (c˘:ℓ∞)(c˘:ℓ∞), (c˘:c)(c˘:c) and (c˘:c0)(c˘:c0).
<br>EXTREME MONOPHONIC GRAPHS AND EXTREME GEODESIC GRAPHS
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109344
title: EXTREME MONOPHONIC GRAPHS AND EXTREME GEODESIC GRAPHS abstract: For a connected graph G=(V,E)G=(V,E) of order at least two, a chord of a path PP is an edge joining two non-adjacent vertices of PP. A path PP is called a monophonic path if it is a chordless path. A monophonic set of GG is a set SS of vertices such that every vertex of GG lies on a monophonic path joining some pair of vertices in SS. The monophonic number of GG is the minimum cardinality of its monophonic sets and is denoted by m(G)m(G). A geodetic set of GG is a set SS of vertices such that every vertex of GG lies on a geodesic joining some pair of vertices in SS. The geodetic number of GG is the minimum cardinality of its geodetic sets and is denoted by g(G)g(G). The number of extreme vertices in GG is its extreme order ex(G)ex(G). A graph GG is an extreme monophonic graph if m(G)=ex(G)m(G)=ex(G) and an extreme geodesic graph if g(G)=ex(G)g(G)=ex(G). Extreme monophonic graphs of order pp with monophonic number pp and p−1p−1 are characterized. It is shown that every pair a,ba,b of integers with 0≤a≤b0≤a≤b is realized as the extreme order and monophonic number, respectively, of some graph. For positive integers r,dr,d and k≥3k≥3 with r<dr<d, it is shown that there exists an extreme monophonic graph GG of monophonic radius rr, monophonic diameter dd, and monophonic number kk. Also, we give a characterization result for a graph GG which is both extreme geodesic and extreme monophonic.
<br>EXTENDED CONSTANT PARTS OF BECKER-STARK'S AND SHAFER-FINK'S INEQUALITIES
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109343
title: EXTENDED CONSTANT PARTS OF BECKER-STARK'S AND SHAFER-FINK'S INEQUALITIES abstract: In this paper, we give some inequalities which are extended constant parts of Becker-Stark's and Shafer-Fink's inequality.
<br>ON SEMI-SYMMETRIC METRIC CONNECTION IN SUB-RIEMANNIAN MANIFOLD
https://tkuir.lib.tku.edu.tw/dspace/handle/987654321/109342
title: ON SEMI-SYMMETRIC METRIC CONNECTION IN SUB-RIEMANNIAN MANIFOLD abstract: The authors firstly in this paper define a semi-symmetric metric non-holonomic connection (in briefly, SS-connection) on sub-Riemannian manifolds. An invariant under a SS-connection transformation is obtained. The authors then further give a result that a sub-Riemannian manifold (M,V0,g,∇¯)(M,V0,g,∇¯) is locally horizontally flat if and only if MM is horizontally conformally flat and horizontally Ricci flat.
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