Related papers: Quadratic isoperimetric inequality for 7-located s…
We consider a $q$-analogue of abstract simplicial complexes, called $q$-complexes, and discuss the notion of shellability for such complexes. It is shown that $q$-complexes formed by independent subspaces of a $q$-matroid are shellable.…
We determine the cohomology groups of the space of seven points in general linear position as representations of the symmetric group on seven elements by making equivariant point counts over finite fields. We also comment on the case of…
Two known $q$-deformed (or `quantum') $7$-spheres, both denoted $\mathbb{S}^7_q$ in the literature, may be distinguished by the presence or absence of symmetry under $\mathrm{SU}_q(2)$. The quaternionic version of $\mathbb{S}^7_q$ has been…
Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power…
In this paper we prove that a generic rational equation of degree $7$ is solvable by 2-fold origami. In particular we show how to septisect an arbitrary angle. This extends the work of Alperin & Lang and Nishimura on 2-fold origami.…
We prove an inequality for polynomials applied in a symmetric way to non-commuting operators.
The classification, up to a center-affinity, of the homogeneous quadratic differential systems defined on $\mathbb{R}^{3}$ that have at least a semisimple nonsingular derivation, is achieved. It is proved that there exist four…
In this paper, we present the quadratic associative symmetry algebra of the 3D nondegenerate maximally quantum superintegrable system. This is the complete symmetry algebra of the system. It is demonstrated that the symmetry algebra…
We study the homology of simplicial and cubical sets with symmetries. These are simplicial and cubical sets with additional maps expressing the symmetries of simplices and cubes. We consider the chain complex computing the homology groups…
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either…
We show that nonlinear optimization techniques can successfully be applied to realize and to inscribe matroid polytopes and simplicial spheres. Thus we obtain a complete classification of neighborly polytopes of dimension $4$, $6$ and $7$…
In this paper we obtain four new parametric ideal solutions of the Tarry-Escott problem of degree 7, that is, of the simultaneous diophantine equations, $\sum_{i=1}^8x_i^r=\sum_{i=1}^8y_i^r,\;r=1,\,2,\,\dots,\,7$. While all the known…
We prove that there does not exist a non-square quadratic polynomial with integer coefficients and an axis of symmetry which takes square values for N consecutive integers for N=7 or N >= 9. At the opposite, if N <= 6 or N=8 there are…
In this paper, we deals with isoperimetric-type inequalities for closed convex curves in the Euclidean plane R^2. We derive a family of parametric inequalities involving the following geometric functionals associated to a given convex curve…
We deal with seven dimensional compact Riemannian manifolds of positive curvature which admit a cohomogeneity one action by a compact Lie group G. We prove that the manifold is diffeomorphic to a sphere if the dimension of the semisimple…
We prove the sharp quantitative isoperimetric inequality in the case of the barycentric asymmetry, for bounded sets. This generalizes the $2$-D case recently proved in~\cite{BCH}.
In this paper, we prove a total curvature estimate of closed hypersurfaces in simply-connected non-positively curved symmetric spaces, and as a corollary, we obtain an isoperimetric inequality for such manifolds.
In this paper we investigate the problem of linearizability for a family of cubic complex planar systems of ordinary differential equations. We give a classification of linearizable systems in the family obtaining conditions for…
We provide an explicit classification of the following four families of surfaces in any homogeneous 3-manifold with 4-dimensional isometry group: isoparametric surfaces, surfaces with constant principal curvatures, homogeneous surfaces, and…
We study structural and enumerative aspects of pure simplicial complexes and clique complexes. We prove a necessary and sufficient condition for any simplicial complex to be a clique complex that depends only on the list of facets. We also…