We study the secular coefficients of $N \times N$ random unitary matrices $U_{N}$ drawn from Circular $\beta$-Ensemble, which are defined as $\{z^n\}$ in characteristic polynomial $\det(1-zU_{N}^{*})$. When $\beta > 4$ we obtain a new class limiting distributions that arise when both $n$ and $N$ tend to infinity simultaneously. solve an open problem Diaconis Gamburd by showing for $\beta=2$, mi...

We construct a map ζ from K0(P) to (Z[x]/xd+1)× × Z, where (Z[x]/xd+1)× is a multiplicative Abelian group with identity 1, and show that ζ induces an isomorphism between K0(P) and its image. This is inspired by a correspondence between Chern and Hilbert polynomials stated in Eisenbud [1, Exercise 19.18]. The equivalence of these two polynomials over Pd is discussed in this paper.

Diversity partitioning has become a popular method for analyzing patterns of alpha and beta diversity. A recent evaluation of the method emphasized a distinction between additive and multiplicative partitioning and further advocated the use of multiplicative partitioning based on a presumed independence between alpha and beta. Concurrently, additive partitioning was criticized for producing dep...

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementary equivalent to the reals with exponentiation and that each model of Peano arithmetic is an integer part of a real clo...

We establish a generalization of Breen’s theory of cubic structures on line bundles over group schemes. We study such “n-cubic structures” inductively using multiextensions. As a result we obtain information on the set of isomorphism classes of line bundles with n-cubic structures over finite multiplicative group schemes over Spec (Z) by relating this set to certain corresponding eigenspaces of...

We show that the operation of taking generic extensions provides the set of isomorphism classes of representations of a quiver of Dynkin type with a monoid structure. Its monoid ring is isomorphic to the specialization at q = 0 of Ringel’s Hall algebra. This provides the latter algebra with a multiplicatively closed basis. Using a crystal-type basis for a two-parameter quantum group, this multi...

We show that the imaginary multiplicative chaos $\exp(i\beta \Gamma)$ determines gradient of underlying field $\Gamma$ for all log-correlated Gaussian fields with covariance form $-\log |x-y| + g(x,y)$ mild regularity conditions on $g$, $d \geq 2$ and $\beta \in (0,\sqrt{d})$. In particular, we 2D continuum zero boundary free is measurable w.r.t. its chaos.

LetM be the multiplicative group of integral weight meromorphic modular forms for some character of SL2(Z) with integer coefficients, leading coefficient 1, and whose zeros and poles are supported at cusps and imaginary quadratic irrationals. Denote by Mk,h ⊂ M the subset which consists of modular forms of weight k and for which order of the pole at infinity is h. We have M = ∪k∈Z,h∈ 1 12 ZMk,h...

Division algebras D generated by some finitely generated nilpotent subgroup G of the multiplicative group D* of D are studied and the question to what extent G is determined by D is considered. Trivial examples show that D does not determine G up to isomorphism. However, it is proved that if F denotes the center of D, then the F-subalgebra of D generated by G is in fact determined up to isomorp...