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تسجيل مستخدم جديدCircle incidence theorems
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Larry Hoehn discovered a remarkable concurrence theorem about pentagrams. Draw cicles through two consecutive vertices and the intersection points of the sides in between, Then the radical axes of each pair of consecutive circles are concurrent or parallel. In this note we prove a generalisation to n-gons.
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Laczkovich proved that if bounded subsets $A$ and $B$ of $R^k$ have the same non-zero Lebesgue measure and the box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translate
d to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-on
Let $C$ be the unit circle in $mathbb{R}^2$. We can view $C$ as a plane graph whose vertices are all the points on $C$, and the distance between any two points on $C$ is the length of the smaller arc between them. We consider a graph augmentation pro
blem on $C$, where we want to place $kgeq 1$ emph{shortcuts} on $C$ such that the diameter of the resulting graph is minimized. We analyze for each $k$ with $1leq kleq 7$ what the optimal set of shortcuts is. Interestingly, the minimum diameter one can obtain is not a strictly decreasing function of~$k$. For example, with seven shortcuts one cannot obtain a smaller diameter than with six shortcuts. Finally, we prove that the optimal diameter is $2 + Theta(1/k^{frac{2}{3}})$ for any~$k$.
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobjec
t classifier of incidence hypergraphs. Moreover, the injective envelope is calculated and shown to contain the class of uniform hypergraphs -- providing a combinatorial framework for the entries of incidence matrices. A multivariable all-minors characteristic polynomial is obtained for both the determinant and permanent of the oriented hypergraphic Laplacian and adjacency matrices arising from any integer incidence matrix. The coefficients of each polynomial are shown to be submonic maps from the same family into the injective envelope limited by the subobject classifier. These results provide a unifying theorem for oriented hypergraphic matrix-tree-type and Sachs-coefficient-type theorems. Finally, by specializing to bidirected graphs, the trivial subclasses for the degree-$k$ monomials of the Laplacian are shown to be in one-to-one correspondence with $k$-arborescences.
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In 1945, A. W. Goodman and R. E. Goodman proved the following conjecture by P. ErdH{o}s: Given a family of (round) disks of radii $r_1$, $ldots$, $r_n$ in the plane it is always possible to cover them by a disk of radius $R = sum r_i$, provided they
cannot be separated into two subfamilies by a straight line disjoint from the disks. In this note we show that essentially the same idea may work for different analogues and generalizations of their result. In particular, we prove the following: Given a family of positive homothetic copies of a fixed convex body $K subset mathbb{R}^d$ with homothety coefficients $tau_1, ldots, tau_n > 0$ it is always possible to cover them by a translate of $frac{d+1}{2}left(sum tau_iright)K$, provided they cannot be separated into two subfamilies by a hyperplane disjoint from the homothets.
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