Journal of the London Mathematical Society-Second Series最新文献

We introduce the symplectic density property and the Hamiltonian density property together with the corresponding versions of Andersén–Lempert theory. We establish these properties for the Calogero–Moser space $�C_n$ of $�n$ particles and describe its group of holomorphic symplectic automorphisms.

We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with permutations of the points.

In this paper, we conduct a comprehensive study of the global well-posedness of solution for a class of nonlocal wave equations with variable-order fractional Laplacian and variable exponent nonlinearity by constructing a suitable framework of the variational theory. We first prove the local-in-time existence of the weak solution via the Galerkin approximation technique and fixed point theory. Then by constructing the potential well theory, we classify the initial data leading to the global existence and finite time blowup of the solution for three different initial energy cases, that is, subcritical initial energy case, critical initial energy case, and supercritical initial energy case. For the subcritical and critical initial energy cases, we show that the solution exists globally in time when the initial data belong to the stable manifold and blows up in finite time when the initial data belong to the unstable manifold. For the supercritical initial energy case, we observe some initial conditions that enable the finite time blow-up solution by an adapted concavity method, and the issue of global existence still remains unsolved. As a further study of finite time blowup, we estimate the upper and lower bounds of blow-up time by using different strategies, that is, applying some first-order differential inequality regardless of the different initial energy levels, to give a unified expression for the lower bound estimation for three initial energy levels. For the upper bound estimation, we utilize two second-order differential inequalities influenced by the different energy levels to give the upper bound estimations of the blow-up time at each initial energy level.

We show that smooth del Pezzo varieties in positive characteristic are quasi-$�F$-split. To this end, we introduce weak quasi-$�F$-splitting and we prove that general ladders of smooth del Pezzo varieties are normal.

We consider a one-dimensional hydrodynamic model featuring nonlocal attraction–repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch–Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of solutions.

We consider a free energy functional on Cartan–Hadamard manifolds and investigate the existence of its global minimizers. The energy functional consists of two components: an entropy (or internal energy) and an interaction energy modelled by an attractive potential. The two components have competing effects, as they favour spreading by linear diffusion and blow-up by non-local attractive interactions, respectively. We find necessary and sufficient conditions for ground states to exist, in terms of the behaviours of the attractive potential at infinity and at zero. In particular, for general Cartan–Hadamard manifolds, superlinear growth at infinity of the attractive potential prevents the spreading. The behaviour can be relaxed for homogeneous manifolds, for which only linear growth of the potential is sufficient for this purpose. As a key tool in our analysis, we develop a new logarithmic Hardy–Littlewood–Sobolev inequality on Cartan–Hadamard manifolds.

We complete the determination of the $�\ell$-block distribution of characters for quasi-simple exceptional groups of Lie type up to some minor ambiguities relating to non-uniqueness of Jordan decomposition. For this, we first determine the $�\ell$-block distribution for finite reductive groups whose ambient algebraic group defined in characteristic different from $�\ell$ has connected centre. As a consequence, we derive a compatibility between $�\ell$-blocks, $�e$-Harish-Chandra series and Jordan decomposition. Further, we apply our results to complete the proof of Robinson's conjecture on defects of characters.

We prove an analogue of Miller's stable splitting of the unitary group $��U�\left(�m�\right)�$ for spaces of commuting elements in $��U�\left(�m�\right)�$. After inverting $��m�!�$, the space $��Hom�(�Z^n�,�U��\left(�m�\right)��)�$ splits stably as a wedge of Thom-like spaces of bundles of commuting varieties over certain partial flag manifolds. Using Steenrod operations, we prove that our splitting does not hold integrally. Analogous decompositions for symplectic and orthogonal groups as well as homological results for the one-point compactification of the commuting variety in a Lie algebra are also provided.

We consider a type of Hardy–Sobolev inequality, whose weight function is singular on the whole domain boundary. We are concerned with the attainability of the best constant of such inequality. In dimension two, we link the inequality to a conformally invariant one using the conformal radius of the domain. The best constant of such inequality on a smooth bounded domain is achieved if and only if the domain is non-convex. In higher dimensions, the best constant is achieved if the domain has negative mean curvature somewhere. If the mean curvature vanishes but is non-umbilic somewhere, we also establish the attainability for some special cases. In the other direction, we also show that the best constant is not achieved if the domain is sufficiently close to a ball in $�C^2$ sense.

Semi-arithmetic Fuchsian groups is a wide class of discrete groups of isometries of the hyperbolic plane which includes arithmetic Fuchsian groups, hyperbolic triangle groups, groups admitting a modular embedding, and others. We introduce a new geometric invariant of a semi-arithmetic group called stretch. Its definition is based on the notion of the Riemannian center of mass developed by Karcher and collaborators. We show that there exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch and coarea. The proof of this result uses the arithmetic Margulis lemma. We also show that when stretch is not bounded there exist infinite sequences of such groups.