相关论文: The Euclidean distance degree of smooth complex pr…

We study the projective geometry of algebraic neural layers, namely families of maps induced by a polynomial activation function, with particular emphasis on the generic Euclidean Distance degree ($\mathrm{gED}$). This invariant is…

We give a formula for scalar product and distance in generalized barycentric coordinates for Euclidean and Pseudo-Euclidean spaces.

We define the equivariant Chern-Schwartz-MacPherson class of a possibly singular algebraic variety with a group action over the complex number field (or a field of characteristic 0). In fact, we construct a natural transformation from the…

Distance metrics and their nonlinear variant play a crucial role in machine learning based real-world problem solving. We demonstrated how Euclidean and cosine distance measures differ not only theoretically but also in real-world medical…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

We study a formulation of lattice gravity defined via Euclidean dynamical triangulations (EDT). After fine-tuning a non-trivial local measure term we find evidence that four-dimensional, semi-classical geometries are recovered at long…

We study the wall-crossing for moduli spaces of coherent systems of dimension one and order one on a smooth projective variety over the complex numbers. We compute the topological Euler characteristic of the moduli spaces in the particular…

Distance measuring is a very important task in digital geometry and digital image processing. Due to our natural approach to geometry we think of the set of points that are equally far from a given point as a Euclidean circle. Using the…

Constitutive tensors are of common use in mechanics of materials. To determine the relevant symmetry class of an experimental tensor is still a tedious problem. For instance, it requires numerical methods in three-dimensional elasticity. We…

We give explicit formulas for the Chern-Schwartz-MacPherson classes of all Schubert varieties in the Grassmannian of $d$-planes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small…

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most basic intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth…

Let $X$ be a non-singular quasi-projective variety over a field, and let $\mathcal E$ be a vector bundle over $X$. Let $\mathbb G_X({d}, \mathcal E)$ be the Grassmann bundle of $\mathcal E$ over $X$ parametrizing corank $d$ subbundles of…

We give a proof of the degree formula for the Euler characteristic previously obtained by Kirill Zainoulline. The arguments used here are considerably simpler, and allow us to remove all restrictions on the characteristic of the base field.

We study relations of some classes of $k$-convex, $k$-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular,…

The purpose of this paper is to construct a crepant resolution of quotient singularities by trihedral groups ( finite subgroups of SL(3,C) of certain type ), and prove that each Euler number of the minimal model is equal to the number of…

For every positive integer $n\in \mathbb{Z}_+$ we define an `Euler polynomial' $\mathscr{E}_n(t)\in \mathbb{Z}[t]$, and observe that for a fixed $n$ all Chern numbers (as well as other numerical invariants) of all smooth hypersurfaces in…

From a combinatorial point of view, we consider the Earth Mover's Distance (EMD) associated with a metric measure space. The specific case considered is deceptively simple: Let the finite set [n] = {1,...,n} be regarded as a metric space by…

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on…

We propose methods towards a systematic determination of d dimensional curved spaces where Euclidean field theories with rigid supersymmetry can be defined. The analysis is carried out from a group theory as well as from a supergravity…

The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree. The two special loci of the data points where the number of critical points is smaller then the…