The strong thirteen spheres problem

The thirteen spheres problem is asking if 13 equal size nonoverlapping spheres in three dimensions can touch another sphere of the same size. This problem was the subject of the famous discussion between Isaac Newton and David Gregory in 1694. The problem was solved by Schutte and van der Waerden only in 1953. A natural extension of this problem is the strong thirteen spheres problem (or the Tammes problem for 13 points) which asks to find an arrangement and the maximum radius of 13 equal size nonoverlapping spheres touching the unit sphere. In the paper we give a solution of this long-standing open problem in geometry. Our computer-assisted proof is based on a enumeration of the so-called irreducible graphs.Comment: Modified lemma 2, 16 pages, 12 figures. Uploaded program packag

Sign refinement for combinatorial link Floer homology

Link Floer homology is an invariant for links which has recently been described entirely in a combinatorial way. Originally constructed with mod 2 coefficients, it was generalized to integer coefficients thanks to a sign refinement. In this paper, thanks to the spin extension of the permutation group we give an alternative construction of the combinatorial link Floer chain complex associated to a grid diagram with integer coefficients. We prove that the filtered homology of this complex is an invariant for the link and that it gives the previous sign refinement by means of a 2-cohomological class corresponding to the spin extension of the permutation group.Comment: 17 pages, 10 figures. correction of the Alexander grading and of the formula of lemma 5.2 of the sign refinemen

Empty pentagons in point sets with collinearities

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328k^2 points in the plane contains an empty pentagon or k collinear points. This is optimal up to a constant factor since the (k-1)x(k-1) grid contains no empty pentagon and no k collinear points. The previous best known bound was doubly exponential.Comment: 15 pages, 11 figure

Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.Comment: 40 Pages + 25 Pages of Appendices. 38 figures. Typos and bibliographic amendments and minor correction

Electronic and magnetic properties of carbon-based and boron-based nano materials, 2017

The structural and electronic properties of covalently and non-covalently functionalized graphene are investigated by means of first-principles density-functional-theory. The electronic characteristics of non-covalently functionalized graphene by a planar covalent organic framework (COF) are investigated. The aromatic central molecule of the COF acts as an electron donor while the linker of the COF acts as an electron acceptor. The concerted interaction of donor acceptor promotes the formation of planar COF networks on graphene. The distinctive electronic properties of covalently functionalized fluorinated epitaxial graphene are attributed to the polar covalent CF bond. The partial ionic character of the CF bond results in the hyperconjugation of CF ?-bonds with an sp2 network of graphene. The implications of resonant-orbital-induced doping for the electronic and magnetic properties of fluorinated epitaxial graphene are discussed. Isolation of single-walled carbon nanotubes (SWNTs) with specific chirality and diameters is critical. Water-soluble poly [(m- phenyleneethynylene)- alt- (p- phenyleneethynylene)], 3, is found to exhibit high selectivity in dispersing SWNT (6,5). The polymers ability to sort out SWNT (6,5) appears to be related to the carboncarbon triple bond, whose free rotation allows a unique assembly. We have also demonstrated the important role of dispersion forces on the structural and electronic stability of parallel displaced and Y-shaped benzene dimer conformations. Long-range dispersive forces play a significant role in determining the relative stability of benzene dimer. The effective dispersion of SWNT depends on the helical pitch length associated with the conformations of linkages as well as ?-? stacking configurations. We have revisited the constructing schemes for a large family of stable hollow boron fullerenes with 80 + 8n (n = 0,2,3,...) atoms. In contrast to the hollow pentagon boron fullerenes the stable structures constitute 12 filled pentagons and 12 additional hollow hexagons. Based on results from density-functional calculations, an empirical rule for filled pentagons is proposed along with a revised electron counting scheme. We have also studied the relative stability of various boron fullerene structures and structural and electronic properties of B80 bucky ball and boron nanotubes. Our results reveal that the energy order of fullerenes strongly depends on the exchange-correlation functional employed in the calculation. A systematic study elucidates the importance of incorporating dispersion forces to account for the intricate interplay of two and three centered bonding in boron nanostructures. KEY TERMS: Density Functional Theory, Dispersion-Correction, Nano Materials, Condensed Matter Physics, Materials Chemistry, Organic Chemistry, Physical Chemistr