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We establish a formula which decomposes the cohomologies of various sheaves on a simple normal crossings divisor (SNC) $D$ in terms of the simplicial cohomologies of the dual complex $\Delta(D)$ with coefficients in a presheaf of vector…
In this paper, two parallel notions of convexity of sets are introduced in the abelian semigroup setting. The connection of these notions to algebraic and to set-theoretic operations is investigated. A formula for the computation of the…
In general a contractible complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one "gets stuck", that is one can choose the collapses in such a way that one…
The fundamental geometry of self-similar sets becomes significantly more complex when the generating contractive maps include non-trivial rotational components. A well-known family exemplifying this complexity is that of the dragon curves…
We consider the set of points chosen randomly, independently and uniformly in the $d$-dimensional spherical layer. A set of points is called $1$-convex if all its points are vertices of the convex hull of this set. In \cite{3} an estimate…
We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary…
Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive…
The associahedron is a polytope whose graph is the graph of flips on triangulations of a convex polygon. Pseudotriangulations and multitriangulations generalize triangulations in two different ways, which have been unified by Pilaud and…
We describe a general framework for studying duality between different phase spaces which share the same symmetry group $\mathrm{H}$. Solutions corresponding to collective dynamics become dual in the sense that they are generated by the…
Diagram-chasing arguments frequently lead to "magical" relations between distant points of diagrams: exactness implications, connecting morphisms, etc.. These long connections are usually composites of short "unmagical" connections, but the…
The stable Andrews-Curtis conjecture in combinatorial group theory is the statement that every balanced presentation of the trivial group can be simplified to the trivial form by elementary moves corresponding to "handle-slides" together…
A set in the Euclidean plane is said to be biconvex if, for some angle $\theta\in[0,\pi/2)$, all its sections along straight lines with inclination angles $\theta$ and $\theta+\pi/2$ are convex sets (i.e, empty sets or segments).…
We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an…
For a finite simplicial complex K and a CW-pair (X,A), there is an associated CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse theory to a particular CW-structure on the n-sphere moment-angle complexes Z_K(D^{n},…
We study connection points on the double regular $n$-gon translation surface, for $n \geq 7$ odd and its staircase model. For $n \neq 9$, we provide a large family of points with coordinates in the trace field that are not connection…
Let $f:X\to S$ be an extremal contraction from a threefolds with terminal singularities onto a surface (so called Mori conic bundle). We study some particular cases of such contractions: quotients of usual conic bundles and index two…
We study Poisson structures over singular varieties. In this purpose, we consider the Koszul complex associated to the equations of a complete intersection. This complex forms a differential graded algebra which is equivalent to the algebra…
We provide the combinatorial proofs of the log-convexity for the derangement numbers in the symmetric group $\mathfrak{S}_n$, hyperoctahedral group $\mathfrak{B}_n$, and the demihyperoctahedral group $\mathfrak{D}_n$. We also show that the…
Deciding whether the union of two convex polyhedra is itself a convex polyhedron is a basic problem in polyhedral computations; having important applications in the field of constrained control and in the synthesis, analysis, verification…
The role of discrete (or point-group) symmetries is discussed in the framework of the Cluster Shell Model which describes the splitting of single-particle levels in the deformed field of cluster potentials. We discuss the classification of…