Related papers: Sard's conjecture for degenerate Engel

This work studies the problem of estimating a two-dimensional superposition of point sources or spikes from samples of their convolution with a Gaussian kernel. Our results show that minimizing a continuous counterpart of the $\ell_1$ norm…

We study subcritical two-dimensional oriented percolation seen from its rightmost point on the set of infinite configurations which are bounded above. This a Feller process whose state space is not compact and has no invariant measures. We…

Consider a manifold, of dimension greater than 3, equipped with a bracket-generating distribution. In this article we prove complete h-principles for embedded regular horizontal curves and for embedded transverse curves. These results…

We introduce a dense counterpart of graph degeneracy, which extends the recently-proposed invariant symmetric difference. We say that a graph has sd-degeneracy (for symmetric-difference degeneracy) at most $d$ if it admits an elimination…

In a series of three papers we develop an end space theory for directed graphs. As for undirected graphs, the ends of a digraph are points at infinity to which its rays converge. Unlike for undirected graphs, some ends are joined by limit…

Menger's Edge Theorem asserts that there exist $k$ pairwise edge-disjoint paths between two vertices in an undirected graph if and only if a deletion of any $k-1$ or less edges does not disconnect these two vertices. Alternatively, there…

Let K be a field and let L/K be a finite extension. Let X/K be a scheme of finite type. A point of X(L) is said to be new if it does not belong to the union of X(F), when F runs over all proper subextensions of L. Fix now an integer g>0 and…

The zero locus of a function f on a graph G is defined as the graph with vertex set consisting of all complete subgraphs of G, on which f changes sign and where x,y are connected if one is contained in the other. For d-graphs, finite simple…

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart…

F. Escalante and T. Gallai studied in the seventies the structure of different kind of separations and cuts between a vertex pair in a (possibly infinite) graph. One of their results is that if there is a finite separation, then the optimal…

An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erd\H{o}s and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of…

We consider a surface that admits a $\mathbb{Q}$-Gorenstein degeneration to a cyclic quotient singularity $\frac{1}{dn^2}(1,dna-1)$. Under several technical assumptions, we construct $d$ exceptional vector bundles of rank $n$ which are…

A symplectic bundle over an algebraic curve has a natural invariant $\sLag$ determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give…

A graph $G$ is said to be a $k$-degenerate graph if any subgraph of $G$ contains a vertex of degree at most $k$. Let $\kappa$ be any non-negative function on the vertex set of $G$. We first define a $\kappa$-degenerate graph. Next we give…

A dichotomous ordinal graph consists of an undirected graph with a partition of the edges into short and long edges. A geometric realization of a dichotomous ordinal graph $G$ in a metric space $X$ is a drawing of $G$ in $X$ in which every…

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $\mathbb R^5$. We solve the analogous problem for germs of generic rank 2 distributions in ${\mathbb R}^n$…

The Tree Decomposition Conjecture by Bar\'at and Thomassen states that for every tree $T$ there exists a natural number $k(T)$ such that the following holds: If $G$ is a $k(T)$-edge-connected simple graph with size divisible by the size of…

We reduce a question of Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich, which asks for a classification of all $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich-Zorich spectrum, to a…

In this paper we investigate the Erd\"os/Falconer distance conjecture for a natural class of sets statistically, though not necessarily arithmetically, similar to a lattice. We prove a good upper bound for spherical means that have been…

A graph $G$ is $k$-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree $k$ or less. We prove that every connected planar graph with average degree $d \ge 2$ has a 4-degenerate induced…