Related papers: Focal values of plane cubic centers

An effective search bound is established for the least non-trivial integer zero of an arbitrary integral cubic form in at least 17 variables.

We show a non-vanishing result for the averages of L-functions associated with the orthogonal basis of the space of cusp forms of vector-valued modular forms on the full group. We also show the existence of at least one basis element whose…

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in…

A recent result shows that a general smooth plane quartic can be recovered from its 24 inflection lines and a single inflection point. Nevertheless, the question whether or not a smooth plane curve of degree at least 4 is determined by its…

In the paper, we first give the least upper bound formula on the number of centers of planar real polynomial Hamiltonian vector fields. This formula reveals that the greater the number of invariant straight lines of the vector field and the…

In this paper, I have proved that for a class of polynomial differential systems of degree n+1 ( where n is an arbitrary positive integer) the composition conjecture is true. I give the sufficient and necessary conditions for these…

In this paper, we proved that there are infinite cube--free numbers of the form $[n^c]$ for any fixed real number $1<c<11/6$.

The cone-volume measure of a polytope with centroid at the origin is proved to satisfy the subspace concentration condition. As a consequence a conjectured (a dozen years ago) fundamental sharp affine isoperimetric inequality for the…

Using essentially only algebra, we give a proof that a cubic rational function over $\mathbb{C}$ with real critical points is equivalent to a real rational function. We also show that the natural generalization to $\mathbb{Q}_p$ fails for…

This paper contains two parts. In the first part, we shall study the Abelian integrals for Zoladek's example [13], in which it is claimed the existence integrals of 11 small-amplitude limit cycles around a singular point in a particular…

A set S of 2n+1 points in the plane is said to be in general position if no three points of S are collinear and no four are concyclic. A circle is called halving with respect to S if it has three points of S on its circumference, n-1 points…

The nonlinear differential system $ \dot{x}=\sum_{i=0}^{\ell}P_{m_i}(x,y),\ \dot{y}=\sum_{i=0}^{\ell}Q_{m_i}(x,y)$ is considered, where $P_{m_i}$ and $Q_{m_i}$ are homogeneous polynomials of degree $m_i\geq 1$ in $x$ and $y$, $m_0=1$. The…

We give a new proof of the main theorem in the theory of C(6) small cancellation complexes. We prove the fundamental theorem of cubical small cancellation theory for C(9) cubical small cancellation complexes.

Although the Unimodality Conjecture holds for some certain classes of cubical polytopes (e.g. cubes, capped cubical polytopes, neighborly cubical polytopes), it fails for cubical polytopes in general. A 12-dimensional cubical polytope with…

We study core stability in non-centroid clustering under the max-loss objective, where each agent's loss is the maximum distance to other members of their cluster. We prove that for all $k\geq 3$ there exist metric instances with $n\ge 9$…

We prove that there are infinitely many Maass--Hecke cuspforms over the field $\mathbb{Q}[\sqrt{-3}]$ such that the corresponding symmetric cube $L$-series does not vanish at the center of the critical strip. This is done by using a result…

We describe here, for the first time, a lower bound on the total number of fundamental constants required for a mathematical description of our physical universe to be complete. The answer is shown to be one. The formal arithmetized…

Given a set of n unit squares in the plane, the goal is to rank them in space in such a way that only few squares see each other vertically. We prove that ranking the squares according to the lexicographic order of their centers results in…

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and…

In this paper we prove that no multiple of the linear system of plane curves of degree $d\geq 4$ with a point of multiplicity $d-m$ (with $2 \leq m \leq d$) and $m(2d-m)$ simple general points is effective.