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Gravitational lenses on galaxy scales are plausibly modelled as having ellipsoidal symmetry and a universal dark matter density profile, with a Sersic profile to describe the distribution of baryonic matter. Predicting all lensing effects…
In this paper we develop a global correspondence between immersed horospherically convex hypersurfaces in hyperbolic space and complete conformal metrics on domains in the sphere. We establish results on when the hyperbolic Gauss map is…
It has recently been proved that, for constant density stars, there is a critical value $\Lambda^{*}=1$ for the dimensionless density parameter $\Lambda\equiv 4\pi R^2\rho_{\text{max}}$ of the star above which the asymptotically measured…
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion.…
In this paper, motivated by the work of Bonk, Heinonen, and Koskela (Asterisque, 2001), we consider the problem of the equivalence of the Gromov boundary and Euclidean boundary. Our strategy to study this problem comes from the recent work…
The Poisson boundary of a group G with a probability measure \mu is the space of ergodic components of the time shift in the path space of the associated random walk. Via a generalization of the classical Poisson formula it gives an…
We study the random planar maps obtained from supercritical Galton--Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and…
Observations indicate that our universe is characterized by a late-time accelerating phase, possibly driven by a cosmological constant $\Lambda$, with the dimensionless parameter $\Lambda L_P^2 \simeq 10^{-122}$, where $L_P = (G \hbar…
We consider percolation on the Voronoi tessellation generated by a homogeneous Poisson point process on the hyperbolic plane. We show that the critical probability for the existence of an infinite cluster is asymptotically equal to $\pi…
It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length…
Let $P$ be a $\delta$-separated $(\delta, s, C_P)$-set of points in $B(0, 1)\subset \mathbb{R}^d$ and $\Pi$ be a $\delta$-separated $(\delta, t, C_\Pi)$-set of hyperplanes intersecting $B(0, 1)$ in $\mathbb{R}^d$. Define \[I_{C\delta}(P,…
We present results on thermodynamic quantities, resistivity and optical conductivity for the Hubbard model on a simple hypercubic lattice in infinite dimensions. Our results for the paramagnetic phase display the features expected from an…
Many algorithms require discriminative boundaries, such as separating hyperplanes or hyperballs, or are specifically designed to work on spherical data. By applying inversive geometry, we show that the two discriminative boundaries can be…
We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces $\Sigma^n \subset \mathbb R^{n+1}$ that are second order stable critical points of compact…
The space ML(F) of measured geodesic laminations on a given closed hyperbolic surface F has a canonical linear structure arising in fact from different sources in 2-dimensional hyperbolic (earthquake theory) or complex projective (grafting)…
The cosmological particle horizon is the maximum measurable length in the Universe. The existence of such a maximum observable length scale implies a modification of the quantum uncertainty principle. Thus due to non-locality of quantum…
It is well known that the similar universal behavior of infinite-size (bulk) systems of different nature requires the same basic conditions: space dimensionality; number components of order parameter; the type (short- or long-range) of the…
The guiding center plasma model (also known as kinetic MHD) is a rigorous sub-cyclotron-frequency closure of the Vlasov-Maxwell system. While the model has been known for decades, and it plays a fundamental role in describing the physics of…
The chase of universal bounds on diffusivities in strongly coupled systems and holographic models has a long track record. The identification of a universal velocity scale, independent of the presence of well-defined quasiparticle…
We investigate strong gravitational lensing in the concordance $\Lambda$CDM cosmology by carrying out ray-tracing along past light cones through the Millennium Simulation, the largest simulation of cosmic structure formation ever carried…