Related papers: The Gaussian Double-Bubble Conjecture
The Galois Alperin weight (GAW) conjecture has been reduced to the inductive GAW condition for simple groups. We proceed in two steps to refine this reduction. First, we propose the blockwise Galois Alperin weight (BGAW) conjecture and…
The iterative absorption method has recently led to major progress in the area of (hyper-)graph decompositions. Amongst other results, a new proof of the Existence conjecture for combinatorial designs, and some generalizations, was…
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to…
We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $\mu(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non…
We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an…
Biclustering is a problem in machine learning and data mining that seeks to group together rows and columns of a dataset according to certain criteria. In this work, we highlight the natural relation that quantum computing models like boson…
For compact subsets $E$ of the unit disk $ \mathbb{D}$ we study the capacity of the condenser ${\rm cap}( \mathbb{D},E)$ by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of…
We present an orbifold GUT model in which the NMSSM Higgs trilinear couplings are unified with the three Standard Model gauge couplings. The model is constructed as an N=2 supersymmetric SU(8) gauge theory in six dimensions, which is…
We study the minimum mean-squared error for 2-means clustering when the outcomes of the vector-valued random variable to be clustered are on two touching spheres of unit radius in $n$-dimensional Euclidean space and the underlying…
We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this…
We consider the isoperimetric problem for the sum of two Gaussian densities in the line and the plane. We prove that the double Gaussian isoperimetric regions in the line are rays and that if the double Gaussian isoperimetric regions in the…
Mutually unbiased bases correspond to highly useful pairs of measurements in quantum information theory. In the smallest composite dimension, six, it is known that between three and seven mutually unbiased bases exist, with a decades-old…
We investigate warped compactification with an abelian gauge theory in six dimensions. The vanishing cosmological constant in four dimensions can generically be realized with a regular metric even in a 3-brane background without fine tuning…
We prove the double bubble conjecture in the three-sphere $S^3$ and hyperbolic three-space $H^3$ in the cases where we can apply Hutchings theory: 1) in $S^3$, each enclosed volume and the complement occupy at least 10% of the volume of…
We extend Gleason's theorem to the two-dimensional Hilbert space of a qubit by invoking the standard axiom that describes composite quantum systems. The tensor-product structure allows us to derive density matrices and Born's rule for $d=2$…
In this paper we propose a special class of 3-algebras, called double-symplectic 3-algebras. We further show that a consistent contraction of the double-symplectic 3-algebra gives a new 3-algebra, called an N=4 three-algebra, which is then…
Some of the peculiar electrodynamical effects associated with gauged ``dimension bubbles'' are presented. Such bubbles, which effectively enclose a region of 5d spacetime, can arise from a 5d theory with a compact extra dimension. Bubbles…
We introduce a dimension reduction method for visualizing the clustering structure obtained from a finite mixture of Gaussian densities. Information on the dimension reduction subspace is obtained from the variation on group means and,…
The analysis of the extremal structure of the scalar potentials of gauged maximally extended supergravity models in five, four, and three dimensions, and hence the determination of possible vacuum states of these models is a computationally…
Gaussian cluster states are ideal infinitely squeezed states. In practice it is possible to construct only approximated version of them with finite squeezing. Here we show how to determine the specific multi-mode squeezing transformation,…