Abstract: We propose an extension to the non-rigid factorisation method to solve the affine structure and motion of a deformable object, where the shape basis is selected automatically. In contrast to earlier approaches, we assume a general uncalibrated, affine camera model whereas most of the previous approaches assume a special case such as an orthographic, weak-perspective or paraperspective camera model. In general, there is a global affine ambiguity for the shape bases. It turns out that a natural way of selecting the shape bases is to pick up the bases that are statistically as independent as possible. The independent bases can be found by independent subspace analysis (ISA) which leads to the minimisation of mutual information between the basis shapes. After selecting the shape basis by ISA, only the in-the-subspace affine ambiguities remain from the general affine ambiguity. To solve the remaining unknowns of the general affine transformation, we propose an iterative method that recovers the block structure of the factored motion matrix. The experiments are provided with synthetic structure and real face expression data in 2D and 3D, which show promising results.

Abstract: We first describe how the Kashiwara involution on crystals of affine type $A$ is encoded by the combinatorics of aperiodic multisegments. This yields a simple relation between this involution and the Zelevinsky involution on the set of simple modules for the affine Hecke algebras. We then give efficient procedures for computing these involutions. Remarkably, these procedures do not use the underlying crystal structure. They also permit to match explicitly the Ginzburg and Ariki parametrizations of the simple modules associated to affine and cyclotomic Hecke algebras, respectively .

Abstract: In the present paper the generalized Propeller theorem from planar Euclidean geometry is extended to all planar affine Cayley–Klein geometries. Since there are no equilateral triangles in affine Cayley–Klein planes (except for the Euclidean case), there is no direct extension of the Propeller theorem. In order to find the respective non-Euclidean analogues of it, we introduce the notion of Ωk-equilateral triangle. Some properties of such triangles are given, too. Finally, we prove a Propeller theorem related to isocentric triangles in all affine Cayley–Klein planes.

Abstract: In order to characterize the (a)symmetries of cut-and-project sets, we prove the following: any cut-and-project set with the two projections being injective on the lattice is fixed by an affine transformation if and only if (1) the window restricted on the projection of the lattice is fixed by another affine transformation, and (2) both affine transformations induce via the two projections the same transformation on the lattice. By this theorem, we prove that any Pisot tilings are asymmetric with respect to any affine transformations.

Abstract: In this paper we obtain exact bounds for dimensions of the Lie algebras of infinitesimal affine transformations of the direct product of two nonprojective Euclidean affinely connected spaces without torsion.

Abstract: Given a bounded convex domain Ω with C ∞ boundary and a function ϕ ∈ C ∞ (∂Ω), Li-Simon-Chen can construct an Euclidean complete and W -complete convex hypersurface M with constant affine Gauss-Kronecker curvature, and they guess the M is also affine complete. In this paper, we give a confirmation answer.

Abstract: In this paper we consider a differentiable mapping of a p-dimensional affine space into the differentiable manifold \( \mathfrak{M} \)N of all centered m-planes in an n-dimensional Euclidean space. We pay special attention to describing geometric images defined by a fundamental geometric object of the mentioned mapping.

Abstract: A complete description of the bijective affine map on C ( X, I ) is given. This provides an answer to a question of [2] on the affine bijections on C ( X, I ). Keywords: Affine map; Riesz isomorphism Quaestiones Mathematicae 32(2009), 115–117

Abstract: In this paper, we present the convergence analysis of the method for solving the monotone affine variational inequalities on supply chain equilibrium model which were proposed by Ferris and Mangasarian, and the convergence rate was also given under same conditions. Some numerical experiments of the method are also reported in this paper.

Abstract: The aim of the paper is to establish a natural affine frame for affine Lagrangians and Hamiltonians, generalizing the well-known classical field theory. Scalar and volume-valued Lagrangians and Hamiltonians can be lifted to the new classes. Using the Hamilton–Jacobi principle, we analyze variational problems corresponding to actions defined by the affine Lagrangians and Hamiltonians. The extremals verify generalizations of the Euler–Lagrange and De Donder–Weyl PDEs. They improve the information about the dynamical solutions of the classical variational problems and refresh the Lagrange–Hamilton theories.