We introduce a motivic version of Yamamoto’s integrals, which are integrals attached to labeled posets. For each labeled poset, this motivic version is defined as a sum of motivic iterated integrals. We extend both the infinitesimal coaction and the coaction from motivic iterated integrals to the motivic version of Yamamoto’s integrals. Our main theorem provides an explicit formula for the infinitesimal coaction on the motivic version of Yamamoto’s integrals. The summands in this formula are canonically indexed by irreducible labeled subposets, and, in the totally ordered case, the formula recovers the usual infinitesimal coaction formula for motivic iterated integrals. In addition, we prove an explicit summation formula for the contributions to the coaction expansion associated with a fixed subposet. This gives a method for computing the coaction, and, in the totally ordered case, this summation recovers the corresponding terms in the coaction formula for motivic iterated integrals. As an application, we study Schur multiple zeta values with constant diagonal entries, for which integral expressions are known, and use our formulas to compute the infinitesimal coaction and the coaction for certain classes of motivic Yamamoto integrals.
We construct skew-adjoint operators associated to G-invariant nowhere zero vector fields on manifolds with proper cocompact actions of Lie group G. The mod 2 indices of these operators do not depend on the choice of G-invariant nowhere zero vector fields and provide potentially invariants for such G-manifolds. We adapt the deformation idea of (Witten, Supersymmetry and Morse theory. J. Differ. Geom. 17, 661–692 (1982)) to prove a vanishing property of the above mod 2 indices and use the analytic techniques developed by (Bismut and Lebeau, Complex immersions and Quillen metrics. Publ. Math. IHES. 74, 1–297 (1991)).
Let k be a field of characteristic not 2 and X a smooth intersection of two quadrics in ℙ_k^5 . In this note we show that if the 2-cohomological dimension of k is 1, then X has a rational point.
We consider a homogeneous Bose gas composed of N identical bosons occupying an infinite two-dimensional space. This gas exhibits both an attractive two-body interaction and a repulsive three-body interaction. Via the intermediate Hartree theory, we rigorously derive the homogeneous cubic-quintic nonlinear Schrödinger functional as the mean-field limit of the model. Our investigation focuses on the system’s behavior in relation to the two-body interaction.
