Abstract: A new method dealing with recognition of partially occluded and affine distortion objects is presented. The method is designed for objects with smooth curved boundary. It divides an object into affine-invariant parts based on the feature point. And a new approach for matching each part is presented in this paper. Robust Hausdorff distance (RHD) is introduced to measure the similarity between feature points set of model and that of target. In terms of the new RHD, the optimal affine transform can be estimated. And then the sub-curve match pairs are calculated based on the optimal affine transformation. The experimental results show proposed algorithm are capable of coping with partial occlusion and affine transformation.

Abstract: Inspired by a recent work of Haddad, Jimenez and Montenegro, we give a new and simple approach to the recently established general affine Polya-Szego principle. Our approach is based on the general $L_p$ Busemann-Petty centroid inequality and does not rely on the general $L_p$ Petty projection inequality or the solution of the $L_p$ Minkowski problem. A Brothers-Ziemer-type result for the general affine Polya-Szego principle is also established. As applications, we reprove some sharp affine Sobolev-type inequalities and settle their equality conditions. We also prove a stability estimate for the affine Sobolev inequality on functions of bounded variation by using our new approach. As a corollary of this stability result, we deduce a stability estimate for the affine logarithmic--Sobolev inequality.

Abstract: The statistical structure on a manifold \(\mathfrak {M}\) is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection \( abla \) on the \(T\mathfrak {M}\), such that \( abla g\) is totally symmetric, forming, by definition, a “Codazzi pair” \(\{ abla , g\}\). In this paper, we first investigate various transformations of affine connections, including additive translation (by an arbitrary (1,2)-tensor K), multiplicative perturbation (through an arbitrary invertible operator L on \(T\mathfrak {M}\)), and conjugation (through a non-degenerate two-form h). We then study the Codazzi coupling of \( abla \) with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our derivations allow affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. Our systematic approach establishes a general setting for the study of Information Geometry based on transformations and coupling relations of affine connections – in particular, we provide a generalization of conformal-projective transformation.

Abstract: The research of affine scale space is to create a more general approach to the affine invariant image scale representation by modifying the corresponding Gaussian filters in order to cope with the specific change of view point. It has the purpose to retain a linear relationship with the transiting of the view point. With this linear relationship, the affine scale space could be established as a more general approach for the affine invariant image retrieval, including affine feature detection and affine feature descriptor. The scope of this paper is to discuss the accessible to the affine scale space, its performance and a practical implementation to construct it in order to cope with the high complexity brought in by the scale space and the affine adaptation.

Abstract: This paper gives sufficient conditions for controllability of one-input affine control systems evolving on finite-dimensional real (matrix) simple Lie groups. A class of one-input invariant control affine systems evolving on the special linear group SL(n, R) is studied.

Abstract: We provide a new description of the Pieri rule of the homology of the affine Grassmannian and an affine analogue of the charge statistics in terms of bounded partitions. This makes it possible to extend the formulation of the Kostka–Foulkes polynomials in terms of solvable lattice models by Nakayashiki and Yamada to the affine setting.

Abstract: This paper is devoted to a geometric-measure-theoretic study of the brand new affine BV-capacity which is essentially different from the classic BV-capacity in dimension greater than one.

Abstract: In this paper, we develop basic theory for the Orlicz affine surface areas for log-concave and $s$-concave functions. Our definitions were motivated by recently developed 1) Orlicz affine and geominimal surface areas for convex bodies, and 2) $L_p$ affine surface areas for log-concave and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants, and establish related functional affine isoperimetric inequalities as well as generalized functional Blaschke-Santal\'o and inverse Santal\'o inequalities.

Abstract: Affine Schur algebras have several equivalent definitions given by [8, 11, 18, 20, 21] respectively. There are clear correspondences between these definitions on basis elements. Affine Schur algebras play a central role in linking the representations of affine quantum groups and affine Hecke algebras. Affine cellular algebras are introduced by Koenig and Xi in [17]. They extend the framework of cellular algebras due to Graham and Lehrer in [9] to affine cellular algebras which are not necessarily finite dimensional over a field. Many examples such as affine Temperley-Lieb algebras and affine Hecke algebras of type A when the parameter q is not a root of Poincaré polynomial are proved to be affine cellular in [17]. Recently A. S. Kleshchev, J. W. Loubert, V. Miemietz and J. Guilhot prove that KLR algebras of type A and affine Hecke algebras of rank two are affine cellular in [15] and [12] respectively. The aim of this paper is to prove that the affine Schur algebra S (2, 2) in case q = 1 is an affine cellular algebra over Q. We use the equivalent definitions of S (n, r) given by [8, 18, 21] respectively. By using the multiplication formulas given in [3, 18, 22], we investigate the ideal generated by the idempotent corresponding to a particular diagonal matrix and construct a chain of idempotent ideals in S (2, 2). Then we prove that this chain

Abstract: In this article, we put forward the concept of the (i, j)-type L p -mixed affine surface area, such that the notion of L p -affine surface area which be shown by Lutwak is its special cases. Furthermore, applying this concept, the Minkowski inequality for the (i,−p)-type L p -mixed affine surface area and the extensions of the well-known L p -Petty affine projection inequality are established, respectively. Besides, we give an affirmative answer for the generalized L p -Winterniz monotonicity problem.

Abstract: Good parametrisations of affine transformations are essential to interpolation, deformation, and analysis of shape, motion, and animation. It has been one of the central research topics in computer graphics. However, there is no single perfect method and each one has both advantages and disadvantages. In this paper, we propose a novel parametrisation of affine transformations, which is a generalisation to or an improvement of existing methods. Our method adds yet another choice to the existing toolbox and shows better performance in some applications. A C++ implementation is available to make our framework ready to use in various applications.

Abstract: We determine the set of connected components of closed affine Deligne-Lusztig varieties for hyperspecial maximal parahoric subgroups of unramified connected reductive groups. This extends the work by Viehmann for split reductive groups, and the work by Chen-Kisin-Viehmann on minuscule affine Deligne-Lusztig varieties.

Abstract: In this paper, we introduce a new parameter, the affine twist parameter for the affine deformation of a sphere with holes. We show that the affine deformation space can be parametrized by Margulis invariants and affine twist parameters. The affine twist parameter is canonically regarded as a correspondence to the Fenchel-Nielsen twist parameter in Teichmuller theory.

Abstract: Let AS (2v, Fq) be a 2-dimensional affine symplectic space over finite fields Fq. In this paper, we construct a family of error-tolerant pooling designs with the incidence matrix of two types of flats (i.e., (m, s)-flats and (r, 0)-flats) over affine symplectic space AS (2v, Fq). We also discuss the error-tolerant and error-correcting properties of our designs.

Abstract: We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the basepoint of the subspace. These results provide evidence for the conjecture that all affine subspaces of Euclidean space are of Khintchine type for divergence.

Abstract: In this paper we study involutions over a finite field of order 2n We present some classes, several constructions of involutions and we study the set of their fixed points

Abstract: Many quantum systems of interest are initially correlated with their environments and the reduced dynamics of open systems are an interesting while challenging topic. Affine maps, as an extension of completely positive maps, are a useful tool to describe the reduced dynamics of open systems with initial correlations. However, it is unclear what kind of initial state shares an affine map. In this study, we give a sufficient condition of initial states, in which the reduced dynamics can always be described by an affine map. Our result shows that if the initial states of the combined system constitute a convex set, and if the correspondence between the initial states of the open system and those of the combined system, defined by taking the partial trace, is a bijection, then the reduced dynamics of the open system can be described by an affine map.

Abstract: This paper presents a matrix projection method for matching a pair of shapes, and estimating the affine transformation that aligns the two shapes. First, shapes are considered as 2D disordered point sets, and their normalized forms are given. Then, we prove that only a rotational transform exists between the normalized forms of the two shapes under affine distortions. Second, correspondences are found by minimizing the inner product between one matrix and projection of the other. Finally, the affine transformation for shape registration is estimated by the correspondences. Experimental results show that our approach compares favorably to other methods under affine distortions.