We introduce a class of Poisson varieties whose symplectic leaves may be thought of as additive analogues of twisted wild character varieties. Some examples parametrize what we call twisted orthogonal meromorphic connections and twisted symplectic meromorphic connections. We show that the Harnad duality gives Poisson isomorphisms between certain moduli spaces of twisted orthogonal (resp. symplectic) connections and those of symplectic (resp. orthogonal) connections on the Riemann sphere.
Let gcd(n1,…,nk) denote the greatest common divisor of the positive integers n1,…,nk, and let ϕ denote the Euler totient function. We consider the asymptotic behaviour of the logarithmic weight of the summatory function S~k(x)≔∑n1⋯nk≤xϕ(gcd(n1,…,nk))logxn1⋯nk for any real number x>3 and any integer k (2≤k≤7). Moreover, under the Riemann Hypothesis, we derive asymptotic formulas for S~k(x) with any integer k≥2.
We study the global in time existence of small solutions to the Cauchy problem for the critical derivative fractional nonlinear Schrödinger equation { iut−1α|∂x|αu=i∂x(|u|2u), t>0, x∈R,u(0,x)=u0(x), x∈R, where α∈(1,2). We prove a local existence of solutions in a suitable function space by using the parabolic regularization method which is not given in our previous paper and presnt a more simpler proof than that of [15].
We construct a 4-dimensional phase space using the orbit space by S1-action in hyperbolic 3-space and the Lagrangian by the Hsiang-Lawson metric. We then consider certain integrable systems as Hamiltonian systems on the phase space, and we give a Haantjes operator on the 4-dimensional phase space and construct a concrete symplectic-Haantjes manifold. In particular, we construct this manifold using a rewritten conditional formulation of the definition formula given by Tempesta and Tondo.
