Related papers: Partition and Analytic Rank are Equivalent over La…
We study the symmetric tensor rank of multiplication over finite field extensions using linearized polynomials. Via field trace, symmetric linearized polynomials are identified with symmetric bilinear forms and symmetric matrices, allowing…
We prove that a conjecture of Fomin, Fulton, Li, and Poon, associated to ordered pairs of partitions, holds for many infinite families of such pairs. We also show that the bounded height case can be reduced to checking that the conjecture…
Given a tract $F$ in the sense of Baker and Bowler and a matrix $A$ with entries in $F$, we define several notions of rank for $A$. In this way, we are able to unify and find conceptually satisfying proofs for various results about ranks of…
By a tensor we mean an element of a tensor product of vector spaces over a field. Up to a choice of bases in factors of tensor products, every tensor may be coordinatized, that is, represented as an array consisting of numbers. This note is…
Learning to rank is a supervised learning problem where the output space is the space of rankings but the supervision space is the space of relevance scores. We make theoretical contributions to the learning to rank problem both in the…
A celebrated result of Gowers states that for every \epsilon > 0 there is a graph G so that every \epsilon-regular partition of G (in the sense of Szemeredi's regularity lemma) has order given by a tower of exponents of height polynomial in…
We present an efficient low-rank approximation algorithm for non-negative tensors. The algorithm is derived from our two findings: First, we show that rank-1 approximation for tensors can be viewed as a mean-field approximation by treating…
The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in…
Notions of rank abound in the literature on tensor decomposition. We prove that strength, recently introduced for homogeneous polynomials by Ananyan-Hochster in their proof of Stillman's conjecture and generalised here to other tensors, is…
If $\mathcal{C}$ is a cocomplete monoidal category in which tensoring from both sides preserves coequalizers, then the category of monoids over $\mathcal{C}$ is cocomplete. The same holds if $\mathcal{C}$ has regular factorizations and…
Journal ranking is becoming more important in assessing the quality of academic research. Several indices have been suggested for this purpose, typically on the basis of a citation graph between the journals. We follow an axiomatic approach…
We show that the augmented base locus coincides with the exceptional locus (i.e. null locus) for any nef $\mathbb{R}$-Cartier divisor on any scheme projective over a field (of any characteristic). Next we prove a semi-ampleness criterion in…
We present novel equivalences in random matrix and tensor models between complex and self-adjoint theories with nontrivial quadratic terms in the action, established through an intermediate field representation. More precisely, we show that…
Inspired by the study of the minimal excludant in integer partitions by G.E. Andrews and D. Newman, we introduce a pair of new partition statistics, sqrank and rerank. They are related to a polynomial bosonic form of statistical…
Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including $spt$-crank, overpartition pairs, and $t$-core partitions. As corollaries, we obtain new proofs of…
We continue the study on sheaves of rings on finite posets. We present examples where the ring of global sections coincide with toric faces rings, quotients of a polynomial ring by a monomial ideal and algebras with straightening laws. We…
In a generalized tournament, players may have an arbitrary number of matches against each other and the outcome of the games is measured on a cardinal scale with a lower and upper bound. An axiomatic approach is applied to the problem of…
Recently, ranking-based semantics is proposed to rank-order arguments from the most acceptable to the weakest one(s), which provides a graded assessment to arguments. In general, the ranking on arguments is derived from the strength values…
We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We…
In this paper, we have considered the dense rank for assigning positions to alternatives in weak orders. If we arrange the alternatives in tiers (i.e., indifference classes), the dense rank assigns position 1 to all the alternatives in the…