The global well-posedness of the Boltzmann equation with initial data of large amplitude has remained a long-standing open problem. In this paper, by developing a new L x L v 1 L x , v approach, we prove the global existence and uniqueness of mild solutions to the Boltzmann equation in the whole space or torus for a class of initial data with bounded velocity-weighted L x L v 1 L x , v norm under some smallness condition on the L x L v 1 L x , v norm as well as defect mass, energy and entropy so that the initial data allow large amplitude oscillations. Both the hard and soft potentials with angular cut-off are considered, and the large time behavior of solutions in the L x L v 1 L x , v norm with explicit rates of convergence are also studied.

Let $E$ be an elliptic curve over $\dQ$ and $A$ another elliptic curve over a real quadratic number field. We construct a $\dQ$-motive of rank $8$, together with a distinguished class in the associated Bloch--Kato Selmer group, using Hirzebruch--Zagier cycles, that is, graphs of Hirzebruch--Zagier morphisms. We show that, under certain assumptions on $E$ and $A$, the non-vanishing of the central critical value of the (twisted) triple product $L$-function attached to $(E,A)$ implies that the dimension of the associated Bloch--Kato Selmer group of the motive is $0$; and the non-vanishing of the distinguished class implies that the dimension of the associated Bloch--Kato Selmer group of the motive is $1$. This can be viewed as the triple product version of Kolyvagin's work on bounding Selmer groups of a single elliptic curve using Heegner points.

We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [..] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [..] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

This paper introduces a new concept, the quasi-range-preserving operator, and gives necessary and sufficient conditions for a linear operator to be quasi-range-preserving. As a special case Glicksberg's problem is affirmatively answered.

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