Related papers: Dimensional dual hyperovals in classical polar spa…

In this paper, we study the problem of decomposability of bilinear spaces of dimension four without symmetry, as well as the problem of decomposability of split central simple algebras of degree four with an anti-automorphism. In…

Motivated by the results in {\tt hep-th/0508228}, we perform a careful analysis of the allowed linear constraints on $N=(2,2)$ scalar superfields. We show that only chiral, twisted-chiral and semi-chiral superfields are possible. Various…

Solutions to scalar theories with derivative self-couplings often have regions where non-linearities are important. Given a classical source, there is usually a region, demarcated by the Vainshtein radius, inside of which the classical…

We establish a number of foundational results on Poincar\'e spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there…

We investigate a duality relation between floating and illumination bodies. The definitions of these two bodies suggest that the polar of the floating body should be similar to the illumination body of the polar. We consider this question…

In this survey paper, we take the viewpoint of polar invariants to the local and global study of non-dicritical holomorphic foliations in dimension two and their invariant curves. It appears a characterization of second type foliations and…

We establish a canonical isomorphism between two bigraded cohomology theories for polyhedral spaces: Dolbeault cohomology of superforms and tropical cohomology. Furthermore, we prove Poincar\'e duality for cohomology of tropical manifolds,…

We establish existence and uniqueness results for the singular initial value problem associated with a class of quasilinear, symmetric hyperbolic, partial differential equations of Fuchsian type in several space dimensions. This is an…

We consider various regular graphs defined on the set of elements of given rank of a finite polar space. It is likely that no two such graphs, of the same kind but defined for different ranks, can have the same degree. We shall prove this…

We prove a precise version of a general conjecture on the polar degree stated by June Huh. We confirm Huh's conjectural list of all projective hypersurfaces with isolated singularities and polar degree equal to 2.

There is a theorem known as a Virial theorem that restricts the possible existence of non-trivial static solitary waves with scalar fields in a flat space-time with 3 or more spatial dimensions. This raises the following question: Does the…

Polar spaces over finite fields are fundamental in combinatorial geometry. The concept of polar space was firstly introduced by F. Veldkamp who gave a system of 10 axioms in the spirit of Universal Algebra. Later the axioms were simplified…

In our previous works we have classified real non-singular cubic hypersurfaces in the 5-dimensional projective space up to equivalence that includes both real projective transformations and continuous variations of coefficients preserving…

We have developed in the past several algorithms with intrinsic complexity bounds for the problem of point finding in real algebraic varieties. Our aim here is to give a comprehensive presentation of the geometrical tools which are…

We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic H\"older spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.

Let $\cal P$ be a finite classical polar space of rank $d$. An $m$-regular system with respect to $(k - 1)$-dimensional projective spaces of $\cal P$, $1 \le k \le d - 1$, is a set $\cal R$ of generators of $\cal P$ with the property that…

According to [1] an $n$-dimensional $\mathcal{N}$--set is a compact subset $A$ of $\mathbb{R}^n$ such that for every $x$ in $\mathbb{R}^n$ there is $y$ in $A$ with $y-x$ in $\mathbb{Z}^n$. We prove that every two dimensional…

We show first that a generic hypersurface $V$ of degree $d\geq 3$ in the complex projective space $ \mathbb{P}^n$ of dimension $n \geq 3$ has at least one hyperplane section $V \cap H$ containing exactly $n$ ordinary double points, alias…

We construct higher-spin N=1 super algebras as extensions of the super Virasoro algebra containing generators for all spins $s\ge 3/2$. We find two distinct classical (Poisson) algebras on the phase super space. Our results indicate that…

A polytope in the hyperbolic space $\H^n$ is called an {\it ideal polytope} if all its vertices belong to the boundary of $\H^n$. We prove that no simple ideal Coxeter polytope exist in $\H^n$ for $n>8$.