Finite automata can be useful for proving the congruences for combinatorial numbers. We offer several examples as applications: First we consider the congruence of central Delannoy numbers module 3, then we find the congruences of the number of graphs and the number of total edges for noncrossing connected graphs module 3. One of these results answers a conjecture of Deutsch and Sagan in affirm...

Inspired by Bárány’s colourful Carathéodory theorem [Bár82], we introduce a colourful generalization of Liu’s simplicial depth [Liu90]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any ddimensional configuration is d + 1 and that the maximum is d + 1. We exhibit configurations attaining each of these depths, and apply our results to ...

Inspired by Bárány’s Colourful Carathéodory Theorem [4], we introduce a colourful generalization of Liu’s simplicial depth [13]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 +1 and that the maximum is dd+1 +1. We exhibit configurations attaining each of these depths, and apply our results to the ...

We provide matching upper and lower bounds on the mutual information in noisy reconstruction of parity check codes and thereby prove a long-standing conjecture by Montanari [IEEE Transactions on Information Theory 2005]. Besides extending a prior concentration result of Abbe and Montanari [Theory of Computing 2015] to the case of odd check degrees, we precisely determine the conjectured formula...

We use Cayley graphs to construct several dual-containing codes, all of which have sparse graphs. These codes’ properties are promising compared to other quantum error-correcting codes. This paper builds on the ideas of the earlier paper Sparse-Graph Codes for Quantum ErrorCorrection (quant-ph/0304161), which the reader is encouraged to refer to. To recap: Our aim is to make classical error-cor...

we investigate the classical h.~zassenhaus conjecture‎ ‎for integral group rings of alternating groups $a_9$ and $a_{10}$ of degree‎ ‎$9$ and $10$‎, ‎respectively‎. ‎as a consequence of our‎ ‎previous results we confirm the prime graph‎ ‎conjecture for integral group rings of $a_n$ for all $n leq 10$‎.

In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times Four Color Conjecture. He asked whether 4-coloring complex planar triangulation could have two components in which all colorings same parity. this note we answer Tutte’s contrary to speculations by showing that there are triangulations plane whose coloring complexes arbitrarily many even an...

For a global field K and an elliptic curve Eη over K(T), Silverman's specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the exampl...

For a global field K and an elliptic curve Eη over K(T), Silverman's specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the exampl...

For a global fieldK and an elliptic curve Eη overK(T ), Silverman’s specialization theorem implies rank(Eη(K(T ))) ≤ rank(Et(K)) for all but finitely many t ∈ P(K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the examples ...