Brinkmann's theorem in general relativity

In the early 1920s Brinkmann made the first significant contribution to the study ofpp waves in general relativity. Reference to his original work and to the later work of Ehlers and Kundt is made in order to rectify and clarify statements made recently by McInnes.     

New Spin Generalisation for Long Range Interaction Models

We study new interactions between degrees of freedom for Calogero, Sutherland and confined Calogero spin models. These interactions are encoded by the generators of the Lie algebra so(N) or sp(N). We find the symmetry algebras of these new models: the half-loop algebra based on so(N) or sp(N) for the Calogero models and the Yangian of so(N) or sp(N) for the two types of other models. Surprisingly, these symmetry occur only for a specific value of the coupling constant.Dedicated to my PhD supervisor and friend D. Arnaudon.     

A Multi-Dimensional Lieb-Schultz-Mattis Theorem

For a large class of finite-range quantum spin models with half-integer spins, we prove that uniqueness of the ground state implies the existence of a low-lying excited state. For systems of linear size L, with arbitrary finite dimension, we obtain an upper bound on the excitation energy (i.e., the gap above the ground state) of the form (C log L)/L. This result can be regarded as a multi-dimensional Lieb-Schultz-Mattis theorem [14] and provides a rigorous proof of the main result in [8].     

A Symplectic Slice Theorem

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.     

A General No-Cloning Theorem

A simple theorem is proved which has most of the known no-cloning and no-broadcasting results as corollaries. It also implies the standard restrictions on measuring non-commuting observables.     

A Theorem of Ludwig Revisited

Using a recent result of Busch and Gudder, we reconsider a theorem of Ludwig which allows one to identify a class of effect automorphisms as the symmetry transformations in quantum mechanics.     

Generalisation of the modified tight-binding approximation in transition metals

The modified tight-binding approach (MTBA) used in calculations of the electron-phonon coupling in d-band metals was criticized before because the non-orthogonality of the wavefunctions was not taken into account. It is shown that the wavefunctions can be made orthogonal to first order in the overlap integrals. The modifications thus introduced in the electron-phonon matrix element are derived, and their impact on former calculations done in the MTBA is discussed.     

A Theorem About Uniform Distribution

Many problems of combinatorial number theory can be formulated in terms of behavior of orbits of certain transformations acting on the spaces of integers or their subsets. Their analysis can be reduced to problems embracing number theory, probability theory and dynamical systems. In this paper we consider one such question originated from the famous (3x+1)-problem, which illustrates the difficulties which sometimes arise. The main theorem gives the limiting uniform distribution of certain functionals of independent random variables.Dedicated to F. DysonAcknowledgement Financial support from NSF, grant DMS-0070698 and RFFI, grant 99-01-00314 are highly appreciated. I thank S. Konyagin and an unknown referee for their very careful reading of the text and many useful remarks.     

A Toroidal Magnetic Field Theorem

In the framework of magnetohydrodynamics the kinematic dynamo problem in a spherical fluid volume as well as in a plane layer is considered. On the premises of a purely toroidal magnetic field a nonlinear evolution equation for the toroidal scalar is derived. In this equation the flow field is constrained in such a way that no poloidal magnetic field can arise, but is otherwise arbitrary; the magnetic diffusivity is assumed to be spherically (horizontally, resp.) symmetric. Solutions of this problem are of particular interest since the magnetic field is confined to the fluid volume and therefore invisible to an external observer. It is proved in this paper that the maximum norm of smooth solutions of this equation decays exponentially fast to zero. Thus, dynamo solutions, i.e. nondecaying solutions, of this type do not exist.     

A Quantum Version of Sanov's Theorem

We present a quantum version of Sanov's theorem focussing on a hypothesis testing aspect of the theorem: There exists a sequence of typical subspaces for a given set Ψ of stationary quantum product states asymptotically separating them from another fixed stationary product state. Analogously to the classical case, the separating rate on a logarithmic scale is equal to the infimum of the quantum relative entropy with respect to the quantum reference state over the set Ψ. While in the classical case the separating subsets can be chosen universally, in the sense that they depend only on the chosen set of i.i.d. processes, in the quantum case the choice of the separating subspaces depends additionally on the reference state.     

A Nonlinear Transfer Operator Theorem

In recent papers, Kenyon et al. (Ergod Theory Dyn Syst 32:1567–1584 2012), and Fan et al. (C R Math Acad Sci Paris 349:961–964 2011, Adv Math 295:271–333 2016) introduced a form of non-linear thermodynamic formalism based on solutions to a non-linear equation using matrices. In this note we consider the more general setting of Hölder continuous functions.     

A conventional proof of Kerr's Theorem

A proof of Kerr's Theorem for generating geodesic and shearfree null congruences in Minkowski space is given in the Newman-Penrose null tetrad formalism.Supported in part by NSF Grant GP-43844-X.     

PAC-Bayes Unleashed: Generalisation Bounds with Unbounded Losses

We present new PAC-Bayesian generalisation bounds for learning problems with unbounded loss functions. This extends the relevance and applicability of the PAC-Bayes learning framework, where most of the existing literature focuses on supervised learning problems with a bounded loss function (typically assumed to take values in the interval [0;1]). In order to relax this classical assumption, we propose to allow the range of the loss to depend on each predictor. This relaxation is captured by our new notion of HYPothesis-dependent rangE (HYPE). Based on this, we derive a novel PAC-Bayesian generalisation bound for unbounded loss functions, and we instantiate it on a linear regression problem. To make our theory usable by the largest audience possible, we include discussions on actual computation, practicality and limitations of our assumptions.     

Generalisation of the Concept of symmetry and its classification in physics

Symmetry played a central role in the works of E.P. Wigner. He made a great contribution to the extension and reinterpretation of the concept of symmetry. He proposed a new classification. The paper investigates in the light of Wigner’s works how do we see now the classification of physical symmetries.     

A Decoupling Theorem for the BPHZL-Scheme

Conditions are stated, which are sufficient for the heavy-mass-suppression of BPHZL- subtracted Feynman-integrals containing propagators of “heavy fields”. This result generalizes the Decoupling Theorems of Ambjørn, Manoukian and Landsman to cases, where massless fields (e.g., gauge fields) are present.     

A Meinardus Theorem with Multiple Singularities

We solve the quantum version of the A ₁ T-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum A ₁ Q-system and generalize it to the fully non-commutative case. We give the relation between the quantum T-system and the quantum lattice Liouville equation, which is the quantized Y-system.     

A generalization of the Proximity Force Theorem

We generalize the Proximity Force Theorem of J. Blocki et al. (Ann. Phys. (N.Y.)105 (1977), 427) (valid for gently curved surfaces) to include surfaces that may have large curvatures (but are still characterized by small angles between relevant portions of the interacting surfaces). A general proof is given for the approximate continuity of the proximity force when a gap configuration goes over into a crevice after contact. Simple and some-what improved formulae are given for the universal proximity potential functions Φ and $\text{φ}$ for gaps and crevices.     

A Stochastic Version of the Noether Theorem

A stochastic version of the Noether theorem is derived for systems under the action of external random forces. The concept of moment generating functional is employed to describe the symmetry of the stochastic forces. The theorem is applied to two kinds of random covariant forces. One of them generated in an electrodynamic way and the other is defined in the rest frame of the particle as a function of the proper time. For both of them, it is shown the conservation of the mean value of a random drift momentum. The validity of the theorem makes clear that random systems can produce causal stochastic correlations between two faraway separated systems, that had interacted in the past. In addition possible connections of the discussion with the Ives Couder’s experimental results are remarked.