Proceedings of the Royal Society of Edinburgh Section A-Mathematics最新文献

Wigner's theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn's generalization of Wigner's theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${mathfrak {D}}$, and simple smooth modules over the twisted Heisenberg–Virasoro algebra $bar {mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${mathfrak {D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${mathfrak {D}}$-modules and $bar {mathfrak {D}}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $mathfrak {D}$ and $bar {mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg m

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$. Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$, based on the singular theorem.

We present a unified approach to the processes of inversion and duality for quasilinear and $1$-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.

In this paper, we look for minimizers of the energy functional for isotropic compressible elasticity taking into consideration the effect of a gravitational field induced by the body itself. We consider two types of problems: the displacement problem in which the outer boundary of the body is subjected to a Dirichlet-type boundary condition, and the one with zero traction on the boundary but with an internal pressure function. For a spherically symmetric body occupying the unit ball $mathcal {B}in mathbb {R}^3$, the minimization is done within the class of radially symmetric deformations. We give conditions for the existence of such minimizers, for satisfaction of the Euler–Lagrange equations, and show that for large displacements or large internal pressures, the minimizer must develop a cavity at the centre. We discuss a numerical scheme for approximating the minimizers for the displacement problem, together with some simulations that show the dependence of the cavity radius and minimum energy on the displacement and mass density of the body.

We generalize the one-dimensional population model of Anguige & Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller & Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.