Infinite Transitivity on Affine Varieties

In this note we survey recent results on automorphisms of affine algebraic varieties, infinitely transitive group actions and flexibility. We present related constructions and examples, and discuss geometric applications and open problems.

Fermat's Equation in Matrices

The Fermat equation is solved in integral two by two matrices of determinant one as well as in finite order integral three by three matrices.

Holomorphic families of nonequivalent embeddings and of holomorphic group actions on affine space

We construct holomorphic families of proper holomorphic embeddings of \mathbb {C}^{k} into \mathbb {C}^{n} (0\textless k\textless n-1), so that for any two different parameters in the family, no holomorphic automorphism of \mathbb {C}^{n} can map the image of the corresponding two embeddings onto each other. As an application to the study of the group of holomorphic automorphisms of \mathbb {C}^{n}, we derive the existence of families of holomorphic \mathbb {C}^{*}-actions on \mathbb {C}^{n} (n\ge5) so that different actions in the family are not conjugate. This result is surprising in view of the long-standing holomorphic linearization problem, which, in particular, asked whether there would be more than one conjugacy class of \mathbb {C}^{*}-actions on \mathbb {C}^{n} (with prescribed linear part at a fixed point).

Slow-fast systems on algebraic varieties bordering piecewise-smooth dynamical systems

This article extends results contained in Buzzi et al. (2006) [4], Llibre et al. (2007, 2008) [12,13] concerning the dynamics of non-smooth systems. In those papers a piecewise C-k discontinuous vector field Z on R-n is considered when the discontinuities are concentrated on a codimension one submanifold. In this paper our aim is to study the dynamics of a discontinuous system when its discontinuity set belongs to a general class of algebraic sets. In order to do this we first consider F :U -> R a polynomial function defined on the open subset U subset of R-n. The set F-1 (0) divides U into subdomains U-1, U-2,...,U-k, with border F-1(0). These subdomains provide a Whitney stratification on U. We consider Z(i) :U-i -> R-n smooth vector fields and we get Z = (Z(1),...., Z(k)) a discontinuous vector field with discontinuities in F-1(0). Our approach combines several techniques such as epsilon-regularization process, blowing-up method and singular perturbation theory. Recall that an approximation of a discontinuous vector field Z by a one parameter family of continuous vector fields is called an epsilon-regularization of Z (see Sotomayor and Teixeira, 1996 [18]; Llibre and Teixeira, 1997 [15]). Systems as discussed in this paper turn out to be relevant for problems in control theory (Minorsky, 1969 [16]), in systems with hysteresis (Seidman, 2006 [17]) and in mechanical systems with impacts (di Bernardo et al., 2008 [5]). (C) 2011 Elsevier Masson SAS. All rights reserved.

Impact damage on selected varieties of apples, pears and other fruits.

During the past years, different laboratory impact response studies have been carrj.ec out in following fruits: apples (2 varieties), pears (4 varieties), Asian or Nashi pears (4 varieties), melons (2 varieties), peaches (2 varieties) and avocados. The methodology of the tests is described, as well as the results and observations obtained in each group of tests. Impact response is compared to bruising susceptibility, bruise characteristics (appearance and structural features) and varietal and ripeness differences.

A dissertation on the varieties of mankind : being an address read before the Cuvierian and Physiological societies of Wesleyan University, August 2, 1843 /

Mode of access: Internet.

The graphic arts; a treatise on the varieties of drawing, painting, and engraving, in comparison with each other and with nature,

First published in 1882.

On Affine Connections in a Riemannian Manifold with a Circulant Metric and two Circulant Affinor Structures

Ива Р. Докузова, Димитър Р. Разпопов - В настоящата статия е разгледан клас V оттримерни риманови многообразия M с метрика g и два афинорни тензора q и S. Дефинирана е и друга метрика ¯g в M. Локалните координати на всички тези тензори са циркулантни матрици. Намерени са: 1) зависимост между тензора на кривина R породен от g и тензора на кривина ¯R породен от ¯g; 2) тъждество за тензора на кривина R в случая, когато тензорът на кривина ¯R се анулира; 3) зависимост между секционната кривина на прозволна двумерна q-площадка {x, qx} и скаларната кривина на M.

The algebraic density property for affine toric varieties

In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén–Lempert theory. We show that an affine toric variety X satisfies this algebraic density property relative to a closed T-invariant subvariety Y if and only if X∖Y≠TX∖Y≠T. For toric surfaces we are able to classify those which possess a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella's famous classification of complete algebraic vector fields in the affine plane.

Synchrony of planting and proportions of susceptible varieties affect rice tungro disease epidemics in the Philippines.

Surveys were conducted in the Philippines from 1995 to 1997 to examine relationships between production environment variables (agroecosystem, synchrony of planting, and varieties planted) and the occurrence of rice tungro disease epidemics using correspondence analyses. The sites covered were Isabela, Nueva Ecija, North Cotabato, and Bohol provinces as well as Bicol region. Tungro disease incidence in farmers’ fields was assessed visually based on typical symptoms. In addition, leaf samples were collected from each field and indexed serologically by enzyme-linked immunosorbent assay for the presence of Rice tungro bacilliform (RTBV) and Rice tungro spherical (RTSV) viruses. Thus, relationships between the production environment variables and four disease variables — visual incidence and double RTBV and RTSV, single RTSV, and single RTBV infections — were examined. A higher association was observed between site and varieties planted as well as site and synchrony of planting than between site and agroecosystem or site and disease variables (visual incidence, double RTBV and RTSV and single RTSV infections). Disease variables depended on both varieties planted and synchrony of planting and correspondence analysis revealed that the low disease incidence in Nueva Ecija was associated with synchronous planting while the high disease incidence in Isabela was associated with the planting of susceptible varieties and asynchronous planting. Such findings suggest that the relationship between the last two factors at a given site is critical to predicting tungro occurrence. Moreover, correspondence analysis of the relationship among disease variables revealed that tungro incidence is associated with not only double RTBV and RTSV infections but also single RTSV infections. Implications of these results on tungro epidemiology and management are discussed.

End point estimates for Radon transform of radial functions on non-Euclidean spaces

We prove end point estimate for Radon transform of radial functions on affine Grasamannian and real hyperbolic space. We also discuss analogs of these results on the sphere.

Geometric Distortion Insensitive Image Watermarking in Affine Covariant Regions

Feature-based image watermarking schemes, which aim to survive various geometric distortions, have attracted great attention in recent years. Existing schemes have shown robustness against rotation, scaling, and translation, but few are resistant to cropping, nonisotropic scaling, random bending attacks (RBAs), and affine transformations. Seo and Yoo present a geometrically invariant image watermarking based on affine covariant regions (ACRs) that provide a certain degree of robustness. To further enhance the robustness, we propose a new image watermarking scheme on the basis of Seo's work, which is insensitive to geometric distortions as well as common image processing operations. Our scheme is mainly composed of three components: 1) feature selection procedure based on graph theoretical clustering algorithm is applied to obtain a set of stable and nonoverlapped ACRs; 2) for each chosen ACR, local normalization, and orientation alignment are performed to generate a geometrically invariant region, which can obviously improve the robustness of the proposed watermarking scheme; and 3) in order to prevent the degradation in image quality caused by the normalization and inverse normalization, indirect inverse normalization is adopted to achieve a good compromise between the imperceptibility and robustness. Experiments are carried out on an image set of 100 images collected from Internet, and the preliminary results demonstrate that the developed method improves the performance over some representative image watermarking approaches in terms of robustness.

Spodoptera frugiperda (J. E. Smith) (Lepidoptera: Noctuidae) in cotton: vertical distribution of egg masses, effects of adult density and plant age on oviposition behavior

O trabalho teve por objetivo determinar, em casa-de-vegetação, a densidade adequada de adultos, a idade ideal das plantas e a distribuição vertical de ovos nas diferentes partes da planta, visando à realização de estudos de resistência e à melhoria das práticas de manejo de Spodoptera frugiperda em algodoeiro. Os experimentos de oviposição de S. frugiperda em relação à densidade de adultos, relação entre plantas de algodoeiro e oviposição de S. frugiperda e não preferência para oviposição de S. frugiperda em variedades de algodoeiro foram realizados com plantas da variedade de algodoeiro BRS Ita 90. A não preferência (antixenose) para oviposição foi acompanhada nas variedades FiberMax 966, FiberMax 977, DeltaOpal, DeltaPenta, Acala 90, Coodetec 408, Coodetec 409, Coodetec 410, BRS Cedro, BRS Ipê, BRS Aroeira, IPR 96, IPR 20, BRS Araçá, IAC 24 e BRS Ita 90. Concluiu-se que S. frugiperda prefere ovipositar em plantas com cerca de 60 dias de idade, na superfície inferior das folhas situadas no terço superior dos testes de plantas sob condições de estufa. Uma densidade de, pelo menos, três pares de adultos S. frugiperda por planta foi suficiente para realizar testes de não preferência para oviposição em casa de vegetação. As variedades Coodetec 408, BRS Aroeira, BRS Araçá, BRS Ita 90 e DeltaPenta apresentaram resistência do tipo não preferência para oviposição de S. frugiperda. Em regiões com altas infestações de S. frugiperda, seria prudente para o cultivo utilizar a variedade de algodão BRS Ita 90.

Probabilistic Atlas Based Segmentation Using Affine Moment Descriptors and Graph-Cuts

We show a procedure for constructing a probabilistic atlas based on affine moment descriptors. It uses a normalization procedure over the labeled atlas. The proposed linear registration is defined by closed-form expressions involving only geometric moments. This procedure applies both to atlas construction as atlas-based segmentation. We model the likelihood term for each voxel and each label using parametric or nonparametric distributions and the prior term is determined by applying the vote-rule. The probabilistic atlas is built with the variability of our linear registration. We have two segmentation strategy: a) it applies the proposed affine registration to bring the target image into the coordinate frame of the atlas or b) the probabilistic atlas is non-rigidly aligning with the target image, where the probabilistic atlas is previously aligned to the target image with our affine registration. Finally, we adopt a graph cut - Bayesian framework for implementing the atlas-based segmentation.