Related papers: New Results on Doubly Adjacent Pattern-Replacement…
In our earlier posting "Matrix Pencils and Entanglement Classification", arXiv:0911.1803, we gave a polynomial-time algorithm for deciding if two states in a space of dimension $2\otimes m\otimes n$ are SLOCC equivalent. In this note, we…
We put forward an alternative approach to the SLOCC classification of entanglement states of three-qubit and four-qubit systems. By directly solving matrix equations, we obtain the relations satisfied by the amplitudes of states. The…
A classical enumerative result states that, given a graph $G$ and a vertex $u$, the number of connected subgraphs of $G$ is equal to the number of orientations of $G$ such that every vertex can reach $u$ by a directed path. We show that…
In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or…
Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of…
We considerably improve Ono's and Ahlgren-Ono's work on the frequent occurrence of Ramanujan-type congruences for the partition function, and demonstrate that Ramanujan-type congruences occur in families that are governed by square-classes.…
We study a curious class of partitions, the parts of which obey an exceedingly strict congruence condition we refer to as "sequential congruence": the $m$th part is congruent to the $(m+1)$th part modulo $m$, with the smallest part…
We show the existence of equivalence classes for large deviations. Stochastic dynamics within an equivalence class share the same large deviation properties.
The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive…
Given k similarity classes of invertible matrices, the Deligne-Simpson problem asks to determine whether or not one can find matrices in these classes whose product is the identity and with no common invariant subspace. The first author…
Prior work of Gavryushkin, Khoussainov, Jain and Stephan investigated what algebraic structures can be realised in worlds given by a positive (= recursively enumerable) equivalence relation which partitions the natural numbers into…
We leverage probabilistic models of neural representations to investigate how residual networks fit classes. To this end, we estimate class-conditional density models for representations learned by deep ResNets. We then use these models to…
Let $ped(n)$ denote the number of partitions of $n$ wherein even parts are distinct (and odd parts are unrestricted). We show infinite families of congruences for $ped(n)$ modulo $8$. We also examine the behavior of $ped_{-2}(n)$ modulo $8$…
We consider two families of equivalence classes in the Weyl groups of type $B_n$ which are suggested by the study of left cells in unequal parameter Iwahori-Hecke algebras. Both families are indexed by a non-negative integer $r$. For $r=0$,…
A frequent topic in the study of pattern avoidance is identifying when two sets of patterns $\Pi, \Pi'$ are Wilf equivalent, that is, when $|\text{Av}_n(\Pi)| = |\text{Av}_n(\Pi')|$ for all $n$. In recent work of Dokos et al. the notion of…
In this paper we explore the noncommutative analogues, $\mathrm{VP}_{nc}$ and $\mathrm{VNP}_{nc}$, of Valiant's algebraic complexity classes and show some striking connections to classical formal language theory. Our main results are the…
We explicitly calculate the triangle inequalities for the group PSO(8). Therefore we explicitly solve the eigenvalues of sum problem for this group (equivalently describing the side-lengths of geodesic triangles in the corresponding…
The consistency problem for a class of algebraic structures asks for an algorithm to decide for any given conjunction of equations whether it admits a non-trivial satisfying assignment within some member of the class. By Adyan (1955) and…
In 2012, Peter Paule and Cristian-Silviu Radu proved an infinite family of Ramanujan type congruences for $2$-colored Frobenius partitions $c\phi_2$ introduced by George Andrews. Recently, Frank Garvan, James Sellers and Nicolas Smoot…
We approach multipartite entanglement classification in the symmetric subspace in terms of algebraic geometry, its natural language. We show that the class of symmetric separable states has the structure of a Veronese variety and that its…