Related papers: The Only Complex 4-Net Is the Hesse Configuration
In two dimensional digital geometry, two lattice points are 4-connected (resp. 8-connected) if their Euclidean distance is at most one (resp. $\sqrt{2}$). A set $S \subset Z^2$ is 4-connected (resp. 8-connected) if for all pair of points…
We show that the complex projective space is the only projective manifold compactifying $\mathbb{C}^n$ by a smooth connected hypersurface, provided $n$ is even. In the odd dimensional case we give some partial results. The case when $n…
We construct a tensor network that delivers an unnormalized quantum state whose coefficients are the solutions to a given instance of 3SAT, an NP-complete problem. The tensor network contraction that corresponds to the norm of the state…
We determine all affinely homogeneous hypersurfaces S^3 in R^4 whose Hessian is (invariantly) of constant rank 2, including the simply transitive ones. We find 34 inequivalent terminal branches yielding each to a nonempty moduli space of…
In Petri net synthesis we ask whether a given transition system $A$ can be implemented by a Petri net $N$. Depending on the level of accuracy, there are three ways how $N$ can implement $A$: an embedding, the least accurate implementation,…
The representation of complex systems as networks is inappropriate for the study of certain problems. We show several examples of social, biological, ecological and technological systems where the use of complex networks gives very limited…
Multinets are certain configurations of lines and points with multiplicities in the complex projective plane $\mathbb{P}^2$. They appear in the study of resonance and characteristic varieties of complex hyperplane arrangement complements…
In this paper, we propose a subnetwork decomposition/combination approach to investigate the single rate $2$-pair unicast problem. It is shown that the solvability of a $2$-pair unicast problem is completely determined by four specific link…
We show that there exists a link with 2 components which is not smoothly slice in $\mathbb{CP}^2 \# \overline{\mathbb{CP}^2}$. By contrast, it is well-known that every knot (i.e., link with 1 component) is smoothly slice therein. Our proof…
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…
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
We investigate the computational complexity of testing dominance and consistency in CP-nets. Previously, the complexity of dominance has been determined for restricted classes in which the dependency graph of the CP-net is acyclic. However,…
We analyze the symplectic and complex structures on the panelled web 4-manifolds. In particular, we give infinite family of examples of almost complex but not symplectic and not complex 4-manifolds in the non-simply connected case.
We consider geodesic nets (critical points of a length functional on the space of embedded graphs) on doubled polygons (topological 2-spheres endowed with a flat metric away from finitely many cone singularities). We use the theorem of…
The Conditional Preference Network (CP-net) graphically represents user's qualitative and conditional preference statements under the ceteris paribus interpretation. The constrained CP-net is an extension of the CP-net, to a set of…
We prove the uniqueness of crepant resolutions for some quotient singularities and for some nilpotent orbits. The finiteness of non-isomorphic symplectic resolutions for 4-dimenensional symplectic singularities is proved. We also give an…
Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for…
Given a smooth, oriented, simply-connected $4$-manifold $M$, the homological Nielsen realization problem asks: when does a finite group of isometries $G\leq O(H_2(M;\mathbb{Z}))$ preserving the intersection form lift isomorphically to a…
Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We…
A crystallographic arrangement is a set of linear hyperplanes satisfying a certain integrality property and decomposing the space into simplicial cones. Crystallographic arrangements were completely classified in a series of papers by…