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This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on…
We provide a resultant-based formula for the total degree w.r.t. the spatial variables of the generic offset to a parametric surface. The parametrization of the surface is not assumed to be proper.
A dominant rational self-map on a projective variety is called $p$-cohomologically hyperbolic if the $p$-th dynamical degree is strictly larger than other dynamical degrees. For such a map defined over $\overline{\mathbb{Q}}$, we study…
This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \cite{K2}, for \emph{traversally generic…
For a target variety $X$ and a nodal curve $C$, we introduce a one-parameter stability condition, termed $\epsilon$-admissibility, for maps from nodal curves to $X\times C$. If $X$ is a point, $\epsilon$-admissibility interpolates between…
We examine maps between noncommutative projective spaces. A surjection of graded rings A-->A/J induces a closed immersion Proj(A/J)-->Proj(A). A homomorphism f:A-->B between graded rings induces an affine map U --> Proj(A) from a non-empty…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
For Lie algebras whose Poisson semi-center is a polynomial ring we give a bound for the sum of the degrees of the generating semi-invariants. This bound was previously known in many special cases.
We consider metric graph Gaussian free field (GFF) defined on polygons of $\delta\mathbb{Z}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary…
The paper studies a general inverse eigenvalue problem which contains as special cases many well studied pole placement and matrix extension problems. It is shown that the studied problem corresponds on the geometric side to a central…
We provide criteria for deciding whether a given planar curve is an image of a given spatial curve, obtained by a central or a parallel projection with unknown parameters. These criteria reduce the projection problem to a certain…
In this paper we present several formulae for computing the partial degrees of the defining polynomial of the offset curve to an irreducible affine plane curve given implicitly, and we see how these formulae particularize to the case of…
We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…
Network analysis has emerged as a key technique in communication studies, economics, geography, history and sociology, among others. A fundamental issue is how to identify key nodes, for which purpose a number of centrality measures have…
Projective structures on compact real manifolds are classical objects in real differential geometry. Complex manifolds with a holomorphic projective structure on the other hand form a special class as soon as the dimension is greater than…
We study the intersection theory on the moduli spaces of maps of $n$-pointed curves $f:(C,s_1,... s_n)\to V$ which are stable with respect to a weight data $(a_1,..., a_n)$, $0\le a_i\le 1$. After describing the structure of these moduli…
Given a real algebraic curve, embedded in projective space, we study the computational problem of deciding whether there exists a hyperplane meeting the curve in real points only. More generally, given any divisor on such a curve, we may…
We present a generalization of multiview varieties as closures of images obtained by projecting subspaces of a given dimension onto several views, from the photographic and geometric points of view. Motivated by applications in Computer…
The "dancing metric" is a pseudo-riemannian metric $\pmb{g}$ of signature $(2,2)$ on the space $M^4$ of non-incident point-line pairs in the real projective plane $\mathbb{RP}^2$. The null-curves of $(M^4,\pmb{g})$ are given by the "dancing…
The purpose of this paper is to analyze the degree index and clustering index in random graphs. The degree index in our setup is a certain measure of degree irregularity whose basic properties are well studied in the literature, and the…