Mathematical Physics

Quantum r -Airy structures can be constructed as modules of W( gl r ) -algebras via restriction of twisted modules for the underlying Heisenberg algebra. In this paper we classify all such higher quantum Airy structures that arise from modules twisted by automorphisms of the Cartan subalgebra that have repeated cycles of the same length. An interesting feature of these higher quantum Airy structures is that the dilaton shifts must be chosen carefully to satisfy a matrix invertibility condition, with a natural choice being roots of unity. We explore how these higher quantum Airy structures may provide a definition of the Chekhov, Eynard, and Orantin topological recursion for reducible algebraic spectral curves. We also study under which conditions quantum r -Airy structures that come from modules twisted by arbitrary automorphisms can be extended to new quantum (r+1) -Airy structures by appending a trivial one-cycle to the twist without changing the dilaton shifts.

We prove a Gannon-Lee theorem for strongly causal and null pseudoconvex Lorentzian metrics of regularity C 1 . Thereby we generalize earlier results to the non-globally hyperbolic setting in a natural way, while at the same time lowering the regularity to the most general class currently available in the context of the classical singularity theorems. Along the way we also prove that any maximizing causal curve in a C 1 -spacetime is a geodesic and hence of C 2 -regularity.

Kinetic theory describes a dilute monatomic gas using a distribution function f(q,p,t) , the expected phase-space density of particles. The distribution function evolves according to the collisionless Boltzmann equation in the high Knudsen number limit. Fluid dynamics is an alternative description of the gas using hydrodynamic variables that are functions of position and time only. These hydrodynamic variables evolve according to the compressible Euler equations in the inviscid limit. Both systems are noncanonical Hamiltonian systems. Each configuration space is an infinite-dimensional Poisson manifold, and the dynamics is the flow generated by a Hamiltonian functional via a Poisson bracket. We construct a map J 1 from the space of distribution functions to the space of hydrodynamic variables that respects the Poisson brackets on the two spaces i.e. a Poisson map. It maps the p -integral of the Boltzmann entropy flogf to the hydrodynamic entropy density. This map belongs to a family of Poisson maps to spaces that include generalised entropy densities as additional hydrodynamic variables. The whole family can be generated from the Taylor expansion of a further Poisson map that depends on a formal parameter. If the kinetic-theory Hamiltonian factors through the Poisson map J 1 , an exact reduction of kinetic theory to fluid dynamics is possible. This is not the case, but by ignoring the relative entropy of f to its local Maxwellian, we can construct an approximate Hamiltonian that factors through the map. The resulting reduced Hamiltonian generates the compressible Euler equations. We can thus derive the compressible Euler equations as a Hamiltonian approximation to kinetic theory. We also give an analogous Hamiltonian derivation of the compressible Euler--Poisson equations with non-constant entropy, starting from the Vlasov--Poisson equation.

We prove the equivalence between two explicit expressions for two-point Witten-Kontsevich correlators obtained by M. Bertola, B. Dubrovin, D. Yang and by P. Zograf.

We give the asymptotic behavior of the ground state energy of Engel's and Dreizler's relativistic Thomas-Fermi-Weizsäcker-Dirac functional for heavy atoms for fixed ratio of the atomic number and the velocity of light. Using a variation of the lower bound, we show stability of matter.

In this article, we construct a class of explicit, smooth and spherically symmetric solutions to the asymptotically flat vacuum constraint equations which have ADM mass of arbitrary sign ( ?��? , negative, zero, positive). As a direct consequence, there exist asymptotically flat vacuum initial data sets whose metrics are exactly negative mass Schwarzschild outside a given ball. We emphasize that our result does not contradict the spacetime positive energy theorem proven by Eichmair, instead it shows that the decay rate at infinity of the symmetric (0,2) -tensor k stated in the theorem is sharp. The key argument we use in the article is classical, based on the conformal method, in which the conformal equations are equivalently transformed into a single nonlinear equation of functions of one variable.

While investigating the generalization of the Chandrasekhar (1943) dynamical friction to the case of field stars with a power-law mass spectrum and equipartition Maxwell-Boltzmann velocity distribution, a pair of 2-dimensional integrals involving the Error function occurred, with closed form solution in terms of Exponential Integrals (Ciotti 2010). Here we show that both the integrals are very special cases of the family of (real) functions I(\lambda,\mu,\nu; z) :=\int_0^zx^{\lambda}\,\Enu(x^{\mu})\,dx= {\gamma\left({1+\lambda\over\mu},z^{\mu}\right) + z^{1+\lambda}\Enu(z^{\mu})\over 1+\lambda + \mu (\nu -1)}, \quad \mu>0,\quad z\geq 0, \eqno (1) where $\Enu$ is the Exponential Integral, γ is the incomplete Euler gamma function, and for existence λ>max{−1,−1−μ(ν−1)} . Only in one of the consulted tables a related integral appears, that with some work can be reduced to eq.~(1), while computer algebra systems seem to be able to evaluate the integral in closed (and more complicated) form only provided numerical values for some of the parameters are assigned. Here we show how eq.~(1) can in fact be established by elementary methods.

We construct a family of additive entanglement measures for pure multipartite states. The family is parametrised by a simplex and interpolates between the Rényi entropies of the one-particle reduced states and the recently-found universal spectral points (Christandl, Vrana, and Zuiddam, STOC 2018) that serve as monotones for tensor degeneration.

We discuss the Hu-Washizu (HW) variational principle from a geometric standpoint. The mainstay of the present approach is to treat quantities defined on the co-tangent bundles of reference and deformed configurations as primal. Such a treatment invites compatibility equations so that the base space (configurations of the solid body) could be realised as a subset of an Euclidean space. Cartan's method of moving frames and the associated structure equations establish this compatibility. Moreover, they permit us to write the metric and connection using 1-forms. With the mathematical machinery provided by differentiable manifolds, we rewrite the deformation gradient and Cauchy-Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We also show that for a hyperelastic solid, an equation similar to the Doyle-Erciksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite the HW functional in terms of frames and differential forms. Finally, we show that the compatibility of deformation, constitutive rules and equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functional when varied with respect to traction 1-forms, deformation 1-forms and the deformation. This new perspective that involves the notion of kinematic closure precisely explicates the necessary geometrical restrictions on the variational principle, without which the deformed body may not be realized as a subset of the Euclidean space. It also provides a pointer to how these restrictions could be adjusted within a non-Euclidean setting.

The notion of quantum algebras is merged with that of Lie systems in order to establish a new formalism called Poisson-Hopf algebra deformations of Lie systems. The procedure can be naturally applied to Lie systems endowed with a symplectic structure, the so-called Lie-Hamilton systems. This is quite a general approach, as it can be applied to any quantum deformation and any underlying manifold. One of its main features is that, Lie systems are extended to generalized systems described by involutive distributions. In this way, we obtain their new generalized (deformed) counterparts that cover, in particular, a new oscillator system with a time-dependent frequency and a position-dependent mass. Based on a recently developed procedure to construct Poisson-Hopf deformations of Lie-Hamilton systems \cite{Ballesteros5}, a novel unified approach to deformations of Lie-Hamilton systems on the real plane with a Vessiot-Guldberg Lie algebra isomorphic to sl(2) is proposed. In addition, we study the deformed systems obtained from Lie-Hamilton systems associated to the oscillator algebra h 4 , seen as a subalgebra of the 2-photon algebra h 6 . As a particular application, we propose an epidemiological model of SISf type that uses the solvable Lie algebra b 2 as subalgebra of sl(2) , by restriction of the corresponding quantum deformed systems.