Earticle

Topology provides a new approach to understanding contemporary architecture from a perspective different from Euclidean geometry. For this purpose, concepts such as homeomorphism, topological space, continuous deformation, and the Möbius band are particularly important. This allows for a topological intervention in architecture through the application of the concept of 'folding' to explain the relationships and positions of spaces. Until now, topology has been applied without distinction between formal and informal architecture. However, the spatial structure of three-dimensional curved surfaces allows for more effective analysis when approached topologically. Therefore, topological concepts become more suitable tools for grasping the structures of informal architecture. This paper explores the potential of appropriating topological geometry architecturally by analyzing the Möbius House as a case of 'homeomorphism' and the Kunsthal as a case of the topological operation 'piercing.' Subsequently, it conducts a comprehensive analysis of the topological spaces of the Yokohama International Terminal and the Mercedes-Benz Museum, presenting topological geometry as a medium for research in informal architecture. Additionally, it emphasizes the necessity for topological architectural research to connect with the horizons of the mobility paradigm, which has recently gained more attention.

In this paper, the homotopy analysis method, whose original concept comes from algebraic topology, is applied to connect the Black-Scholes option price (the good initial guess) to the option price under general stochastic volatility environment in a recursive manner. We obtain the homotopy solutions for the European vanilla and barrier options as well as the relevant convergence conditions.

We apply the proposed method (NHPSTM) which is the combination of new homotopy perturbation methodand Sumudu transform to solve analytical linear and nonlinear Klein-Gordon equations. The proposed method finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The fact that the proposed technique solves nonlinear problems without using Adomian’s polynomials can be considered as a clear advantage of this new method over the decomposition method. Obtained results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems arising in mathematical physics and engineering.

An analytical technique, namely the homotopy analysis method (HAM), is applied to solve periodic solutions for superharmonic resonances of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter h. In this paper, periodic analytic approximations for superharmonic resonances of nonlinear oscillations with parametric excitation are obtained by using the HAM, which agree well with numerical results. This article shows that the HAM is a powerful and effective technique for nonlinear dynamical systems.

The reconstruction of a volatility based on a Black-Scholes option pricing model is ill-posed. In order to overcome the ill-posedness, a homotopy perturbation inversion method is designed to solve the inverse problem. The proposed method is a modified version of the Landweber method. The reconstruction of a volatility is a nonlinear problem which is needed to be linearized. Hence, numerical experiments consist of the reconstruction of a policy parameter based on a Todaro model which is a linear inverse problem and the reconstruction of a volatility based on a Black-Scholes option pricing model in order to test the performance of the proposed method. Numerical examples show that the proposed method is more accurate and faster than the Landweber method.

An improved and flexible matrix algorithm for solving interlinked separation problems which is based on the homotopy continuation method has been developed. A flexible model of the interlinked stream in standardized matrix form and JACOBIAN generation I algorithm for homotopy continuation are suggested. Also DOF analysis is performed for easy-understanding of equation based simulation of complex column systems. The Algorithm is tested on several problems of interlinked separation processes and some of results are documented.

기존의 도함수에 기초한 수치적 최적화 기법들(derivative-based optimization)은 비선형 최적화 문제를 풀기 위해 목적식의 1차 도함수의 정보를 이용하여 정류점(stable point)인 최적해를 찾아 나가는 방식을 취하고 있다. 그러나 이런 방법들은 목적식의 국부 최적해(local minimum)을 찾는 것은 보장하나, 전역 최적해(global minimum)를 찾는 데에는 실패할 경우가 많다. 국부 최적해와 전역 최적해는 모두 목적식의 1차 도함수가 '0'인 값을 가지는 특징이 있으므로, 국부 또는 전역 최적해를 구하는 구하는 과정은 목적식의 1차 도함수가 '0'인 해를 찾는 방정식 문제로 변환될 수 있다. 따라서 본 논문에서는 비선형 방정식의 해를 찾는데 좋은 성능을 보이는 Homotopy 방법을 이용하여 목적식의 1차 도함수에 관한 비선형 방정식을 풀고, 이를 통해 비선형 최적화 문제의 모든 국부 최적해를 찾아냄으로써 전역 최적화 문제를 해결하는 방법을 제안하고자 한다. 제안된 방법론을 다양한 전역 최적화 문제에 적용한 결과, 기존의 방법들에 비해 더 좋은 성능을 보임을 알 수 있었다.

Let X be a p-compace groyp, Y -> X bd a p-compact-subgroup of X and G -> X be a p-compact toral subgroup of X with $(X/Y)^{hG} \neq 0$. In this paper we show that the homotopy fixed point set of the homotopy fibre $(X/Y)^{hG}$ is $F_p$-finite.

A map 𝜂_n : Sⁿ⁺¹ → Sⁿ is the generator of π_n+1(Sⁿ) and there exists a mapping cone C_𝜂n = Sⁿ ∪_𝜂n eⁿ⁺². The focus of this study is to clearly identify the generators of the homotopy groups and cohomotopy groups of C_𝜂n for n ≥ 3.

First, we generalize homotopy results of O'Regan [6] for Mönch type multimaps to Chandrabhan type multimaps. Second, we show that the better admissible Chandrabhan type multimaps have fixed point properties whenever their ranges are Klee approximable. Finally, we give examples of essential maps for various class of multimaps including 𝚽-condensing multimaps.

For given spaces X and Y, let map(X, Y) and map_*(X, Y) be the unbased and based mapping spaces from X to Y, equipped with compact-open topology respectively. Then let map(X, Y ; f) and map_*(X, Y ; g) be the path component of map(X, Y) containing f and map_*(X, Y) containing g, respectively. In this paper, we compute cohomotopy groups of suspended complex plane π^n+m(ΣⁿℂP²) for m = 6, 7. Using these results, we classify path components of the spaces map(ΣⁿℂP², S^m) up to homotopy equivalence. We also determine the generalized Gottlieb groups G_n(ℂP², S^m). Finally, we compute homotopy groups of mapping spaces map(ΣⁿℂP², S^m; f) for all generators [f] of [ΣⁿℂP², S^m], and Gottlieb groups of mapping components containing constant map map(ΣⁿℂP², S^m; *).

We determine the sets of homotopy minimal periods of all self-maps on the Klein bottle by using a single formula for homotopy minimal periods of maps on the infra-solvmanifolds of type (R). This provides an alternate but an easy proof for the main results of [12].

The classical group completion theorem states that under a certain condition the homology of ${\Omega}BM$ is computed by inverting ${\pi}_0M$ in the homology of M. McDuff and Segal extended this theorem in terms of homology fibration. Recently, more general group completion theorem for simplicial spaces was developed. In this paper, we construct a symmetric monoidal 2-category ${\mathcal{A}}$. The 1-morphisms of ${\mathcal{A}}$ are generated by three atomic 2-dimensional CW-complexes and the set of 2-morphisms is given by the group of path components of the space of homotopy equivalences of 1-morphisms. The main part of the paper is to compute the homotopy type of the group completion of the classifying space of ${\mathcal{A}}$, which is shown to be homotopy equivalent to ${\mathbb{Z}}{\times}BAut^+_{\infty}$.

We generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups.

First, we show the finiteness property of the homotopy fixed point set of p-discrete toral group. Let $G_\infty$ be a p-discrete toral group and X be a finite complex with an action of $G_\infty such that X^K$ is nilpotent for each finit p-subgroup K of $G_\infty$. Assume X is $F_\rho-complete$. Then X(sup)hG$\infty$ is F(sub)p-finite. Using this result, we give the condition so that X$^{hG}$ is $F_\rho-finite for \rho-compact$ toral group G.

We show that if a simple $C^*$-algebra A satisfies certain $K_1$-group conditions, then two unitarily equivalent projections are homotopic. Also we show that the equivalence of projections determined by a dimension function is a homotopy.

We use L_∞ models to compute the rational homotopy type of the mapping space of the component of the natural inclusion i_n,k : ℂPⁿ ↪ ℂP^n+k between complex projective spaces and show that it has the rational homotopy type of a product of odd dimensional spheres and a complex projective space. We also characterize the mapping aut₁ ℂPⁿ → map(ℂPⁿ, ℂP^n+k; i_n,k) and the resulting G-sequence.

Given a topological space X and a non-negative integer k, ${\varepsilon}^{\sharp}_k(X)$ is the set of all self-homotopy equivalences of X that do not change maps from X to an t-sphere $S^t$ homotopically by the composition for all $t{\geq}k$. This set is a subgroup of the self-homotopy equivalence group ${\varepsilon}(X)$. We find certain homotopic tools for computations of ${\varepsilon}^{\sharp}_k(X)$. Using these results, we determine ${\varepsilon}^{\sharp}_k(M(G,n))$ for $k{\geq}n$, where M(G, n) is a Moore space type of (G, n) for a finitely generated abelian group G.