We define in the setting of homotopy type theory an H-space structure on S3. Hence we obtain a description of the quaternionic Hopf fibration S3 ↪→ S7 S4, using only homotopy invariant tools.

In this paper, we describe the spaces of stability conditions for the triangulated categories associated to three dimensional Calabi-Yau fibrations. We deal with two cases, the flat elliptic fibrations and smooth K3 (Abelian) fibrations. In the first case, we will see there exist chamber structures similar to those of the movable cone used in birational geometry. In the second case, we will com...

We show “free theorems” in the style of Wadler on homotopy type theory as consequences of relational parametricity. It follows that, for each space, its higher inductive definition and impredicative definition have the same homotopy groups. An impredicative definition of a space can help us to find generators of homotopy groups of the space.

We obtain some explicit calculations of crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups.

We give an alternative to Postnikov’s homotopy classification of maps from 3-dimensional CW-complexes to homogeneous spaces G/H of Lie groups. It describes homotopy classes in terms of lifts to the group G and is suitable for extending the notion of homotopy to Sobolev maps. This is required in applications to variational problems of mathematical physics.

In this paper, we present an efficient numerical algorithm, namely block homotopy perturbation method, for solving fuzzy linear systems based on homotopy perturbation method. Some numerical examples are given to show the efficiency of the algorithm. Keywords—Homotopy perturbation method, Fuzzy linear systems, Block linear system, Fuzzy solution, Embedding parameter.

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for br...

Necessary and sufficient conditions for the existence of a scalar homotopy in the Vinnicombe metric between two given transfer functions are presented. Previous results show that for multivariable transfer functions, the existence of a homotopy is related to a winding number condition on the homotopy end-points. Here, it is shown that for scalar transfer functions the existence of a homotopy is...

We investigate the steady two-dimensional flow of a viscous incompressible fluid in a rectangular domain that is bounded by two permeable surfaces. The governing fourth-order nonlinear differential equation is solved by applying the spectral-homotopy analysis method and a novel successive linearisation method. Semianalytical results are obtained and the convergence rate of the solution series w...

In this paper, we apply the homotopy perturbation method for solving the sixth-order boundary value problems by reformulating them as an equivalent system of integral equations. This equivalent formulation is obtained by using a suitable transformation. The analytical results of the integral equations have been obtained in terms of convergent series with easily computable components. Several ex...