Let E be an elliptic curve defined over the rational numbers and r a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of E and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to x which have Frobenius trace equal to r. However, when one sums the main term in their asymptotic over ...

The striking results on noncrossed products were their existence (Amitsur) and the determination of Q(t) and Q((t)) as their smallest possible centres (Brussel). This paper gives the first fully explicit noncrossed product example over Q((t)). As a consequence, the use of deep number theoretic theorems (local-global principles such as the Hasse norm theorem and density theorems) in order to pro...

Let B be an m × n (m ≥ n) complex (or real) matrix. It is known that there is a unique polar decomposition B = QH, where Q * Q = I, the n × n identity matrix, and H is positive definite, provided B has full column rank. If B is perturbed to B, how do the polar factors Q and H changes? This question has been investigated quite extensively, but most work so far was on how the perturbation changed...

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries chain, and $q=e^{\frac{i\pi}{p}}$ a $2p$-th root unity $ - its $p$-dimensional analogue. Both cases are parametrised by continuous "spin" $\alpha\in\mathbb{C}$. To motivate our cons...

We discuss some growth rates of composite entire functions on the basis of the definition of relative (p, q)th order (relative (p, q)th lower order) with respect to another entire function which improve some earlier results of Roy (2010) where p and q are any two positive integers.

Recently we explained that the classical $Q$ Schur functions stand behind various well-known properties of cubic Kontsevich model, and next step is to ask what happens in this approach generalized model (GKM) with monomial potential $X^{n+1}$. We propose use Hall-Littlewood polynomials at parameter equal $n$-th root unity as a generalization from $n=2$ arbitrary $n>2$. They are associated $n$-s...

Using functional equations, we derive noncommutative extensions of Ramanujan's 1 ψ 1 summation. 1. Introduction. Hypergeometric series with noncommutative parameters and argument, in the special case involving square matrices, have been the subject of recent study, see e.g. the papers by Duval and Ovsienko [DO], Grünbaum [G], Tirao [T], and some of the references mentioned therein. Of course, t...

Recent studies reveal an intimate connection between the quantum knot invariant and the “nearly modular form” especially with the half integral weight. In Ref. 8 Lawrence and Zagier studied an asymptotic expansion of the Witten–Reshetikhin–Turaev invariant of the Poincaré homology sphere, and they showed that the invariant can be regarded as the Eichler integral of the modular form of weight 3/...

It is known that Q-derived univariate polynomials (polynomials defined over Q, with the property that they and all their derivatives have all their roots in Q) can be completely classified subject to two conjectures: that no quartic with four distinct roots is Q-derived, and that no quintic with a triple root and two other distinct roots is Q-derived. We prove the second of these conjectures. A...