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I am currently working in the private sector as the CTO for Lendino. My background is both in mathematics (homological algebra) and computer science (algorithms and data structures).
We consider distance labeling schemes for trees: given a tree with $n$ nodes, label the nodes with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the distance in the tree between the two nodes. A lower bound by Gavoille et. al. (J. Alg. 2004) and an upper bound by Peleg (J. Graph Theor...
We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize th...
We consider NCA labeling schemes: given a rooted tree $T$, label the nodes of $T$ with binary strings such that, given the labels of any two nodes, one can determine, by looking only at the labels, the label of their nearest common ancestor. For trees with $n$ nodes we present upper and lower bounds establishing that labels of size $(2\pm \epsilon)...
The new intersection theorem states that, over a Noetherian local ring R , for any non-exact complex concentrated in degrees n ,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R . One of the results in this paper is that the Grothendieck group of P(len...
This paper explores the interplay between the Frobenius functor and Serre’s vanishing conjecture over a Noetherian local ring R of prime characteristic p. We show that the Frobenius functor induces a diagonalizable map on certain Q-vector spaces, which are tensor products of Q with quotients of Grothendieck groups. This allows us to decompose an el...