Related papers: A Caporaso-Harris type Formula for relative refine…
Refined algebraic domains are regions in the plane surrounded by finitely many non-singular real algebraic curves which may intersect with normal crossing. We are interested in shapes of such regions with surrounding real algebraic curves.…
We consider a coefficient inverse problem for the dielectric permittivity in Maxwell's equations, with data consisting of boundary measurements of one or two backscattered or transmitted waves. The problem is treated using a Lagrangian…
We give a new proof of Mikhalkin's Theorem on the topological classification of simple Harnack curves, which in particular extends Mikhalkin's result to real pseudoholomorphic curves.
The Fast Reciprocal Square Root Algorithm is a well-established approximation technique consisting of two stages: first, a coarse approximation is obtained by manipulating the bit pattern of the floating point argument using integer…
Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate…
We obtain a recursive formula for the number of rational degree $d$ curves in $\mathbb{CP}^2$ that pass through $3d+1-m$ generic points and that have an $m$-fold singular point. The special case of counting curves with a triple point was…
Kontsevich introduced certain ribbon graphs as cell decompositions for combinatorial models of moduli spaces of complex curves with boundaries in his proof of Witten's conjecture. In this work, we define four types of generalised Kontsevich…
While the exterior Helmholtz problem with Dirichlet boundary conditions is always well-posed, the associated standard boundary integral equations are not if the squared wavenumber agrees with an eigenvalue of the interior Dirichlet problem.…
We refine the bit complexity analysis of an algorithm for the computation of at least one point per connected component of a smooth real algebraic set, yielding exponential speedup (with respect to the number of variables) compared to prior…
In this paper, we obtain the invariant curves of quasi-periodic reversible mappings with finite smoothness. Since the reversible property is difficult to maintain in the process of approximating smooth functions by analytical ones,…
We derive recursive equations for the characteristic numbers of rational nodal plane curves with at most one cusp, subject to point conditions, tangent conditions and flag conditions, developing techniques akin to quantum cohomology on a…
In this paper we prove the algebraic-tropical correspondence for stable maps of rational curves with marked points to toric varieties such that the marked points are mapped to given orbits in the big torus and in the boundary divisor, the…
In mathematics there is a wide class of knot invariants that may be expressed in the form of multiple line integrals computed along the trajectory C describing the spatial conformation of the knot. In this work it is addressed the problem…
In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices…
We derive a combinatorial formula for quantized Donaldson-Thomas invariants of the m-loop quiver. Our main tools are the combinatorics of noncommutative Hilbert schemes and a degenerate version of the Cohomological Hall algebra of this…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…
A simply laced Dynkin diagram gives rise to a family of curves over $\mathbb{Q}$ and a coregular representation, using deformations of simple singularities and Vinberg theory respectively. Thorne has conjectured and partially proven a…
This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This theory finds applications in multi-wave imaging, greedy methods to…
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by…
In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation…