We prove the existence of a large class of dynamical solutions to the Einstein–Euler equations for which the fluid density and spatial three-velocity converge to a solution of the Poisson–Euler equations of Newtonian gravity. The results presented here generalize those of The Newtonian limit for perfect fluids to allow for a larger class of initial data. As in The Newtonian limit for perfect fluids, the proof is based on a nonlocal symmetric hyperbolic formulation of the Einstein–Euler equations, which contain a singular parameter $\epsilon = v_T /c$ with $v_T$ a characteristic speed associated to the fluid and $c$ the speed of light. Energy and dispersive estimates on weighted Sobolev spaces are the main technical tools used to analyze the solutions in the singular limit $\epsilon \searrow 0$.

Starting from the classic Friedmann–Robertson–Walker theory with big bang it is shown that the solutions of the field equations can be extended to negative times. Choosing a new cosmic time scale instead of proper time one achieves complete differentiability of the scale factor and of suitable thermodynamic quantities equivalent to pressure and energy density. Then, the singularity of big bang manifests itself only by the vanishing of the scale factor at time zero. Moreover, all solutions of the field equations are defined for all times from $-\infty$ to $+\infty$. In a separate section, the horizon structure of the extended theory is studied. Some weak assumptions guarantee that there are no horizons. Hence, the horizon problem in a strict sense disappears. An intensive discussion of the results is given at the end of the paper.

We systematically study the factorization and recursion relations in higher genus correlation functions of Berenstein–Maldacena–Nastase (BMN) operators in free $\mathcal{N} = 4$ super Yang–Mills theory. These properties were found in a previous paper by the author, and were conjectured to result from the correspondence with type IIB string theory on the infinitely curved pp-wave background, where the strings become effectively infinitely long. Here we push the calculations to higher genus, provide more clarifications and verifications of the factorization and recursion relations. Our calculations provide conjectural indirect tests of the AdS/ CFT correspondence for multi-loop superstring amplitudes of stringy modes.

We derive generalized TKNN-equations via bundle representations of the noncommutative torus with rational deformation parameter, the bundle coming from spectral projections in the torus algebra. These equations relate Chern numbers of dual bundles, which we interpret as Hall conductances for Dirac-like Hamiltonians describing magnetic Bloch electrons in a strong magnetic field. We also present their generalizations for irrational values of the deformation parameter.

We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool is fractional calculus, which is cast in a way convenient for the definition of the differential structure, distances, volumes, and symmetries. By an extensive use of concepts and techniques of fractal geometry, we clarify the relation between fractional calculus and fractals, showing that fractional spaces can be regarded as fractals when the ratio of their Hausdorff and spectral dimension is greater than one. All the results are analytic and constitute the foundation for field theories living on multi-fractal spacetimes, which are presented in a companion paper.

We provide a model independent construction of a net of $C*$-algebras satisfying the Haag–Kastler axioms over any spacetime manifold. Such a net, called the net of causal loops, is constructed by selecting a suitable base $K$ encoding causal and symmetry properties of the spacetime. Considering $K$ as a partially ordered set (poset) with respect to the inclusion order relation, we define groups of closed paths (loops) formed by the elements of $K$. These groups come equipped with a causal disjointness relation and an action of the symmetry group of the spacetime. In this way, the local algebras of the net are the group $C*$-algebras of the groups of loops, quotiented by the causal disjointness relation. We also provide a geometric interpretation of a class of representations of this net in terms of causal and covariant connections of the poset K. In the case of the Minkowski spacetime, we prove the existence of Poincaré covariant representations satisfying the spectrum condition. This is obtained by virtue of a remarkable feature of our construction: any Hermitian scalar quantum field defines causal and covariant connections of $K$. Similar results hold for the chiral spacetime $S^1$ with conformal symmetry.

It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper’s operator) has a gap in its spectrum with that labelling number. This answers the strong version of the so-called "Ten Martini Problem". When specialized to the particular case where the coupling constant is equal to one, it follows that the "Hofstadter butterfly" has for any quantum Hall conductance the exact number of components prescribed by the recursive scheme to build this fractal structure.

It is known that given a stable holomorphic pair $(E,\phi)$, where $E$ is a holomorphic vector bundle on a compact Kähler manifold $X$ and $\phi$ is a holomorphic section of $E$, the vector bundle $E$ admits a Hermitian metric solving the vortex equation. We generalize this to pairs $(E,\phi)$, where $E$ is a reflexive sheaf on $X$.