Representing a point and the diagonal as zero loci in flag manifolds

The zero locus of a generic section of a vector bundle over a manifold defines a submanifold. A classical problem in geometry asks to realise a specified submanifold in this way. We study two cases; a point in a generalised flag manifold and the diagonal in the direct product of two copies of a generalised flag manifold. These cases are particularly interesting since they are related to ordinary and equivariant Schubert polynomials respectively.Comment: to appear in Algebraic & Geometric Topolog

Three presentations of torus equivariant cohomology of flag manifolds

Let $G$ be a compact connected Lie group and $T$ be its maximal torus. The homogeneous space $G/T$ is called the (complete) flag manifold. One of the main goals of the {\em equivariant Schubert calculus} is to study the $T$-equivariant cohomology $H^*_T(G/T)$ with regard to the $T$-action on $G/T$ by multiplication. There are three presentations known for $H^*_T(G/T)$; (1) the free $H^*(BT)$-module generated by the Schubert varieties (2) (with the rational coefficients) the {\em double coinvariant ring} of the Weyl group (3) the {\em GKM ring} associated to the Hasse graph of the Weyl group. Each presentation has both advantages and disadvantages. In this paper, we describe how to convert an element in one presentation to another by giving an explicit algorithm, which can then be used to compute the equivariant structure constants for the product of Schubert classes. The algorithm is implemented in Maple.Comment: fixed minor errors. added Rem 2.

Mod 2 cohomology of 2-local finite groups of low rank

We determine the mod 2 cohomology over the Steenrod algebra of the classifying space of a free loop group LG for G=Spin(7), Spin(8), Spin(9), F_4, and DI(4). Then we show that it is isomorphic as algebras over the Steenrod algebra to the mod 2 cohomology of the classifying space of a certain 2-local finite group of type G

Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C^2

Investigating the link between postcritical behaviors and the relations of saddle basic sets for Axiom A polynomial skew products on C^2, we characterize various properties concerning the three kinds of accumulation sets defined by DeMarco and Hruska in terms of the saddle basic sets. We also give a new example of higher degree.Comment: 26 pages, 5 figure

Overview of image-to-image translation by use of deep neural networks: denoising, super-resolution, modality conversion, and reconstruction in medical imaging

Since the advent of deep convolutional neural networks (DNNs), computer vision has seen an extremely rapid progress that has led to huge advances in medical imaging. This article does not aim to cover all aspects of the field but focuses on a particular topic, image-to-image translation. Although the topic may not sound familiar, it turns out that many seemingly irrelevant applications can be understood as instances of image-to-image translation. Such applications include (1) noise reduction, (2) super-resolution, (3) image synthesis, and (4) reconstruction. The same underlying principles and algorithms work for various tasks. Our aim is to introduce some of the key ideas on this topic from a uniform point of view. We introduce core ideas and jargon that are specific to image processing by use of DNNs. Having an intuitive grasp of the core ideas of and a knowledge of technical terms would be of great help to the reader for understanding the existing and future applications. Most of the recent applications which build on image-to-image translation are based on one of two fundamental architectures, called pix2pix and CycleGAN, depending on whether the available training data are paired or unpaired. We provide computer codes which implement these two architectures with various enhancements. Our codes are available online with use of the very permissive MIT license. We provide a hands-on tutorial for training a model for denoising based on our codes. We hope that this article, together with the codes, will provide both an overview and the details of the key algorithms, and that it will serve as a basis for the development of new applications.Comment: many typos are fixed. to appear in Radiological Physics and Technolog

A circuit-preserving mapping from multilevel to Boolean dynamics

Many discrete models of biological networks rely exclusively on Boolean variables and many tools and theorems are available for analysis of strictly Boolean models. However, multilevel variables are often required to account for threshold effects, in which knowledge of the Boolean case does not generalise straightforwardly. This motivated the development of conversion methods for multilevel to Boolean models. In particular, Van Ham's method has been shown to yield a one-to-one, neighbour and regulation preserving dynamics, making it the de facto standard approach to the problem. However, Van Ham's method has several drawbacks: most notably, it introduces vast regions of "non-admissible" states that have no counterpart in the multilevel, original model. This raises special difficulties for the analysis of interaction between variables and circuit functionality, which is believed to be central to the understanding of dynamic properties of logical models. Here, we propose a new multilevel to Boolean conversion method, with software implementation. Contrary to Van Ham's, our method doesn't yield a one-to-one transposition of multilevel trajectories, however, it maps each and every Boolean state to a specific multilevel state, thus getting rid of the non-admissible regions and, at the expense of (apparently) more complicated, "parallel" trajectories. One of the prominent features of our method is that it preserves dynamics and interaction of variables in a certain manner. As a demonstration of the usability of our method, we apply it to construct a new Boolean counter-example to the well-known conjecture that a local negative circuit is necessary to generate sustained oscillations. This result illustrates the general relevance of our method for the study of multilevel logical models

The Chow rings of the algebraic groups E_6, E_7, and E_8

We determine the Chow rings of the complex algebraic groups of the exceptional type E_6, E_7, and E_8, giving the explicit generators represented by the pull-back images of Schubert varieties of the corresponding flag varieties. This is a continuation of the work of R. Marlin on the computation of the Chow rings of SO_n, Spin_n, G_2, and F_4. Our method is based on Schubert calculus of the corresponding flag varieties, which has its own interest.Comment: 23 pages, AMS-LaTeX; Result for E_8 added; Title changed from "The Chow rings of the algebraic groups E_6 and E_7"

Homotopy nilpotency in p-compact groups

A p-compact group is a mod p homotopy theoretical analogue of a compact Lie group. It is determined the homotopy nilpotency class of a p-compact group having the homotopy type of the $p$-completion of the direct product of spheres.Comment: 17 page

Products in Equivariant Homology

We refine the intersection product in homology to an equivariant setting, which unifies several known constructions. As an application, we give a common generalisation of the Chas-Sullivan string product on a manifold and the Chataur-Menichi string product on the classifying space by defining a string product on the Borel construction of a manifold. We prove a vanishing result which enables us to define a secondary product. The secondary product is then used to construct secondary versions of the Chataur-Menichi string product, and the equivariant intersection product in the Borel equivariant homology of a manifold with an action of a compact Lie group. The latter reduces to the product in homology of the classifying space defined by Kreck, which coincides with the cup product in negative Tate cohomology if the group is finite

Diagonals of flag bundles

We express the diagonals of projective, Grassmann and, more generally, flag bundles of type (A) using the zero schemes of some vector bundle sections, and do the same for their single point subschemes. We discuss diagonal and point properties of these flag bundles. We study when the complex manifolds G/B for other groups have the point and diagonal properties. We discuss explicit formulas for the classes of diagonals of the varieties G/B.Comment: 18 pages; added Section 7 and Theorem 9; the exposition of the paper has been improve