Related papers: Some cases on Strassen additive conjecture
A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case.…
Let $A=(a^i_j)$ be an orthogonal matrix with no entries zero. Let $B=(b^i_j)$ be the matrix defined by $b^i_j=\frac 1{a^i_j}$. M. Kontsevich conjectured that the rank of $B$ is never equal to three. We interpret this conjecture…
We introduce a class of links whose bracket polynomials admit an expansion over perfect matchings of a plane bipartite graph. This class includes 2-bridge links, pretzel links, and Montesinos links. Our first main result (Theorem A)…
In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is smaller than $p-1$, then the order of any element is the same in the additive and…
In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…
Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and conjectured they did not all vanish.…
Recently, the bunkbed conjecture has been shown to be false, which naturally prompts questions on how to classify the graphs that still satisfy the conjecture. We distinguish between a weak version of the bunkbed conjecture where all the…
The notion of denominator vectors can be extended to all generic basis elements of upper cluster algebras in a natural way. Under a weakened version of generic pairing assumption, we provide a representation-theoretic interpretation for…
Several conjectures on acyclic skew-symmetrizable cluster algebras are proven as direct consequences of their categorification via valued quivers. These include conjectures of Fomin-Zelevinsky, Reading-Speyer, and Reading-Stella related to…
We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the…
We give a short proof for a well-known formula for the rank of a $G$-crossed braided extension of a modular tensor category.
Stanley asked whether a tree is determined up to isomorphism by its chromatic symmetric function. We approach Stanley's problem by studying the relationship between the chromatic symmetric function and other invariants. First, we prove…
Introduced by Goulden and Jackson in their 1996 paper, the matchings-Jack conjecture and the hypermap-Jack conjecture (also known as the $b$-conjecture) are two major open questions relating Jack symmetric functions, the representation…
In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove…
Let $\mathcal{P}$ be a set of points in the plane, and $\mathcal{S}$ a strictly convex set of points. In this note, we show that if $\mathcal{P}$ contains many translates of $\mathcal{S}$, then these translates must come from a generalized…
The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2\pi i c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture…
Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-manifold. We prove the K-theoretic Farrell-Jones Conjecture (up to dimension one) and the L-theoretic Farrell-Jones Conjecture for G, where we…
The combinatorial invariance conjecture (due independently to G. Lusztig and M. Dyer) predicts that if $[x,y]$ and $[x',y']$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan-Lusztig…
Measures of association in contingency tables, such as odds ratios and their generalizations, are often studied under different sampling schemes that either fix or leave random the margins of the table. While classical results show that…
In a recent paper, Hauenstein, Sturmfels, and the second author discovered a conjectural bijection between critical points of the likelihood function on the complex variety of matrices of rank r and critical points on the complex variety of…