arXiv research

Paper defines generalized braids and proves their subgroup status.

Quasitoric spaces were introduced by Davis and Januskiewicz in their 1991 Duke paper. There they extensively studied topological invariants of quasitoric manifolds. These manifolds are generalizations or topological counterparts of nonsingular projective toric varieties. In this article we study structures and invarian…

The paper finds minimal generating sets and abelianizes the quasitoric braid group.

Quasitoric manifolds, introduced by M. Davis and T. Januskiewicz in 1991, are topological generalizations of smooth complex projective spaces. In 1992, Banchoff and Kühnel constructed a 10-vertex equilibrium triangulations of $\CP^2$. We generalize this construction for quasitoric manifolds and construct some equilibri…

Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.

In this note we prove that equivariantly homeomorphic quasitoric manifolds are diffeomorphic. As a consequence we show that up to finite ambiguity the diffeomorphism type of certain quasitoric manifolds $M$ is determined by their cohomology rings and first Pontrjagin classes.

We give three sufficient criteria for two quasitoric manifolds (M,M') to be (weakly) equivariantly homeomorphic. We apply these criteria to count the weakly equivariant homeomorphism types of quasitoric manifolds with a given cohomology ring.

We give hodge structures on quasitoric orbifolds. We define orbifold hodge numbers and show a correspondence of orbifold hodge numbers for crepant resolutions of quasitoric orbifolds. In short we extend hodge structures to a non complex setting .

We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.

We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are non-trivial.

We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The obtained sets of integers are interesting, e. g. those representable as the sum of two…

We prove the existence of invariant almost complex structure on any positively omnioriented quasitoric orbifold. We construct blowdowns. We define Chen-Ruan cohomology ring for any omnioriented quasitoric orbifold. We prove that the Euler characteristic of this cohomology is preserved by a crepant blowdown. We prove th…

We show Mckay correspondence of Betti numbers of Chen-Ruan coho- mology for omnioriented quasitoric orbifolds. In previous articles with M. Poddar [8], [9], we proved the correspondence for four dimension and six dimensions. Here we deal with the general case.

We prove the existence of torus invariant almost complex structure on any positively omnioriented four dimensional primitive quasitoric orbifold. We construct pseudo-holomorphic blowdown maps for such orbifolds. We prove a version of McKay correspondence when the blowdowns are crepant.

We establish a vanishing result for indices of certain twisted Dirac operators on $\text{Spin}^c$-manifolds with non-abelian Lie-group actions. We apply this result to study non-abelian symmetries of quasitoric manifolds. We give upper bounds for the degree of symmetry of these manifolds.

A quasitoric manifold $M$ is a $2n$-dimensional manifold which admits an action of an $n$-dimensional torus which has some nice properties. We determine the isomorphism type of a maximal compact connected Lie-subgroup $G$ of $\text{Homeo}(M)$ which contains the torus. Moreover, we show that this group is unique up to c…

We prove that any quasitoric manifold $M^{2n}$ admits a $T^n$-invariant almost complex structure if and only if $M$ admits a positive omniorientation. In particular, we show that all obstructions to existence of $T^n$-invariant almost complex structure on $M^{2n}$ arise from cohomology of underlying polytope - and henc…

We present some classification results for quasitoric manifolds (M) with (p_1(M)=-\sum a_i^2) for some (a_i\in H^2(M)) which admit an action of a compact connected Lie-group (G) such that (\dim M/G \leq 1). In contrast to Kuroki's work we do not require that the action of (G) extends the torus action on (M).

We investigate small covers and quasitoric over the duals of neighborly simplicial polytopes with small number of vertices in dimensions $4$, $5$, $6$ and $7$. In the most of the considered cases we obtain the complete classification of small covers. The lifting conjecture in all cases is verified to be true. The probl…

In 2006 Masuda and Suh asked if two compact non-singular toric varieties having isomorphic cohomology rings are homeomorphic. In the first part of this paper we discuss this question for topological generalizations of toric varieties, so-called torus manifolds. For example we show that there are homotopy equivalent tor…

New rigidity results for complex and quaternionic moment-angle manifolds.

A Bott tower is the total space of a tower of fibre bundles with base CP^1 and fibres CP^1. Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to fixed points. We show that a (qua…

Paper classifies pseudomanifolds over stratified spaces.

The equivariant cohomology of a space with a group action is not only a ring but also an algebra over the cohomology ring of the classifying space of the acting group. We prove that toric manifolds (i.e. compact smooth toric varieties) are isomorphic as varieties if and only if their equivariant cohomology algebras are…

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial gen…

We show that the Witten genus of a string manifold $M$ vanishes, if there is an effective action of a torus $T$ on $M$ such that $\dim T>b_2(M)$. We apply this result to study group actions on $M\times G/T$, where $G$ is a compact connected Lie group and $T$ a maximal torus of $G$. Moreover, we use the methods which ar…

The aim of this paper is to classify simply connected 6-dimensional torus manifolds with vanishing odd degree cohomology. It is shown that there is a one-to-one correspondence between equivariant diffeomorphism types of these manifolds and 3-valent labelled graphs, called torus graphs introduced by Maeda-Masuda-Panov. …

For any given integer $r \geq 1$ and a quasitoric braid $β_r=(σ_r^{-ε} σ_{r-1}^ε...$ $σ_{1}^{(-1)^{r}ε})^3$ with $ε=\pm 1$, we prove that the maximum degree in $z$ of the HOMFLYPT polynomial $P_{W_2(\hatβ_r)}(v,z)$ of the doubled link $W_2(\hatβ_r)$ of the closure $\hatβ_r$ is equal to $6r-1$. As an application, we gi…

Given an arbitrary non-zero simplicial cycle and a generic vector coloring of its vertices, there is a way to produce a graded Poincare duality algebra associated with these data. The procedure relies on the theory of volume polynomials and multi-fans. This construction includes many important examples, such as cohomol…

A family of closed manifolds is called cohomologically rigid if a cohomology ring isomorphism implies a diffeomorphism for any two manifolds in the family. We establish cohomological rigidity for large families of 3-dimensional and 6-dimensional manifolds defined by 3-dimensional polytopes. We consider the class P of 3…

A new method normalizes activations to match batch normalization without batch dependence.

Online Normalization is a new technique for normalizing the hidden activations of a neural network. Like Batch Normalization, it normalizes the sample dimension. While Online Normalization does not use batches, it is as accurate as Batch Normalization. We resolve a theoretical limitation of Batch Normalization by intro…

This paper analyzes how normalization layers improve neural network training.

Normalization techniques have only recently begun to be exploited in supervised learning tasks. Batch normalization exploits mini-batch statistics to normalize the activations. This was shown to speed up training and result in better models. However its success has been very limited when dealing with recurrent neural n…

This paper presents a normalization mechanism called Instance-Level Meta Normalization (ILM~Norm) to address a learning-to-normalize problem. ILM~Norm learns to predict the normalization parameters via both the feature feed-forward and the gradient back-propagation paths. ILM~Norm provides a meta normalization mechanis…

The paper analyzes various normalization methods in deep learning.

Distance, normals, and double normals for real plane curves with singularities

Normalizes pseudo-Einstein contact forms for easier analysis.

This paper reviews normalization techniques for DNNs.

We consider the Laplace normal vector field of relatively normalized ruled surfaces with non-vanishing Gaussian curvature in the three-dimensional Euclidean space $\mathbb{R}^{3}$. We determine all ruled surfaces and all relative normalizations for which the Laplace normal image degenerates into a point or into a curve…

Monotonicity of normalized implied-volatility coordinates under no-arbitrage

Training state-of-the-art, deep neural networks is computationally expensive. One way to reduce the training time is to normalize the activities of the neurons. A recently introduced technique called batch normalization uses the distribution of the summed input to a neuron over a mini-batch of training cases to compute…

The paper defines normal forms for rational 3-tangles and shows a sequence of moves to transform one form to another.

This paper investigates the impact of normalization on deep neural networks for click-through rate prediction.

We propose a simple but effective multi-source domain generalization technique based on deep neural networks by incorporating optimized normalization layers that are specific to individual domains. Our approach employs multiple normalization methods while learning separate affine parameters per domain. For each domain,…

The study finds abundant normal generators for mapping class groups.

This paper deals with skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$ which are equipped with polar normalizations, that is, relative normalizations such that the relative normal at each point of the ruled surface lies on the corresponding polar plane. We determine the invariants of a such normalized ruled …

New method trains normalizing flows using entropy-regularized transport.