• 충청수학회지
• /
• 제21권3호
• /
• pp.371-376
• /
• 2008

Let $P(M,G,{\pi})=:P$ be a principal fibre bundle with structure Lie group G over a base manifold M. In this paper we get the following facts: 1. The tangent bundle TG of the structure Lie group G in $P(M,G,{\pi})=:P$ is a Lie group. 2. The Lie algebra ${\mathcal{g}}=T_eG$ is a normal subgroup of the Lie group TG. 3. $TP(TM,TG,{\pi}_*)=:TP$ is a principal fibre bundle with structure Lie group TG and projection ${\pi}_*$ over base manifold TM, where ${\pi}_*$ is the differential map of the projection ${\pi}$ of P onto M. 4. for a Lie group $H,\;TH=H{\circ}T_eH=T_eH{\circ}H=TH$ and $H{\cap}T_eH=\{e\}$, but H is not a normal subgroup of the group TH in general.

• 대한수학회지
• /
• 제46권3호
• /
• pp.551-560
• /
• 2009

A theorem of Robert Blumenthal is used here in order to obtain a sufficient condition for a function between two Finsler manifolds to be a fibre bundle map. Our study is connected with two possible constructions: 1) a Finslerian generalization of usually Kaluza-Klein theories which use Riemannian metrics, the well-known particular case of Finsler metrics, 2) a Finslerian version of reduction process from geometric mechanics. Due to a condition in the Blumenthal's result the completeness of Euler-Lagrange vector fields of Finslerian type is discussed in detail and two situations yielding completeness are given: one concerning the energy and a second related to Finslerian fundamental function. The connection of our last framework, namely a regular Lagrangian having the energy as a proper (in topological sense) function, with the celebrated $Poincar{\acute{e}}$ Recurrence Theorem is pointed out.

• Fibers and Polymers
• /
• 제7권3호
• /
• pp.310-316
• /
• 2006

In this article, an attempt has been made to explain the failure mechanism of spun yams. The mechanism includes the aspects of generation and distribution of forces on a fibre under the tensile loading of a yam, the free body diagram of forces, the conditions for gripping and slipping of a fibre, and the initiation, propagation, and ultimate yam rupture in its weakest link. A simple mathematical model for the tenacity of spun yams has been proposed. The model is based on the translation of fibre bundle tenacity into the yam tenacity.

• Kyungpook Mathematical Journal
• /
• 제13권2호
• /
• pp.235-240
• /
• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• 대한수학회논문집
• /
• 제11권4호
• /
• pp.1159-1174
• /
• 1996

The spectrum of the Laplacian matrix of a graph gives an information of the structure of the graph. For example, the product of non-zero eigenvalues of the characteristic polynomial of the Laplacian matrix of a graph with n vertices is n times of the number of spanning trees of that graph. The characteristic polynomial of the Laplacian matrix of a graph tells us the number of spanning trees and the connectivity of given graph. in this paper, we compute the characteristic polynomial of the Laplacian matrix of a graph bundle when its voltage lie in an abelian subgroup of the full automorphism group of the fibre; in particular, the automorphism group of the fibre is abelian. Also we study a relation between the characteristic polynomial of the Laplacian matrix of a graph G and that of the Laplacian matrix of a graph bundle over G. Some applications are also discussed.

• 한국식품과학회지
• /
• 제9권4호
• /
• pp.271-276
• /
• 1977

한우(韓牛)의 round muscke에 papain을 농도별(濃度別)로 증가(增加)시켜서 longitudinal section과 cross section하여 결체조직(結締組織)의 collagenous fibre와 elastic fibre를 각각(各各) 염색(染色)하여 현미경으로 관찰한 결과(結果)는 다음과 같다. 1. 효소(酵素) 농도(濃度)의 증가(增加)에 따라 비정형 상태로 뭉쳐져 있는 collagenous fibre는 점점 풀어지는 상태(狀態)를 지나 후에는 붕괴되며 또한 염색성(染色性)이 현저하게 약해졌다. 2. elastic fibre의 방향(方向)은 muscle fibre에 평행(平行)하게 존재(存在)하며 효소(酵素) 농도(濃度)의 증가(增加)에 따라 점차 탄성(彈性)을 잃고 loose헤지며 직선적(直線的)으로 보이고 fragment되는 것도 현저하다. 3. collagenous fibre와 elastic fibre 모두 효소(酵素) 처리구(處理區)가 대조구(對照區)에 비하여 조직변화(組織變化)가 현저하다.

• 대한수학회보
• /
• 제38권4호
• /
• pp.741-751
• /
• 2001

In this paper when $(M, \omega)$ is a compact weakly exact symplectic manifold with nonempty boundary satisfying $c_1｜{\pi}_2(M)$ = 0, we construct an action spectrum bundle over the group of Hamil-tonian diffeomorphisms of the manifold M generated by the time-dependent Hamiltonian vector fields, whose fibre is nowhere dense and invariant under symplectic conjugation.

• 충청수학회지
• /
• 제26권4호
• /
• pp.793-798
• /
• 2013

Let ${\phi}$ be a map of a manifold M into another manifold N, L(N) the bundle of all linear frames over N, and ${\phi}^{-1}$(L(N)) the bundle over M which is induced from ${\phi}$ and L(N). Then, we construct a structure equation for the torsion form in ${\phi}^{-1}$(L(N)) which is induced from a torsion form in L(N).

• Journal of applied mathematics & informatics
• /
• 제22권1_2호
• /
• pp.39-48
• /
• 2006

Let ${\gamma}$(G) be the domination number of a graph G. It is shown that for any $k {\ge} 0$ there exists a Cartesian graph bundle $B{\Box}_{\varphi}F$ such that ${\gamma}(B{\Box}_{\varphi}F) ={\gamma}(B){\gamma}(F)-2k$. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality ${\gamma}(B{\bigotimes}_{\varphi}F){\le}{\gamma}(B){\gamma}(F)$ for strong graph bundles. Examples of graphs Band F with ${\gamma}(B{\bigotimes}_{\varphi}F) < {\gamma}(B){\gamma}(F)$ are given.

• 대한수학회지
• /
• 제43권6호
• /
• pp.1269-1287
• /
• 2006

As a continuation of computing the zeta function of a regular covering graph by Mizuno and Sato in [9], we derive in this paper computational formulae for the zeta functions of a graph bundle and of any (regular or irregular) covering of a graph. If the voltages to derive them lie in an abelian or dihedral group and its fibre is a regular graph, those formulae can be simplified. As a by-product, the zeta function of the cartesian product of a graph and a regular graph is obtained. The same work is also done for a discrete torus and for a discrete Klein bottle.