TY - JOUR AB - A corner cut in dimension d is a finite subset of N0d that can be separated from its complement in N0d by an affine hyperplane disjoint from N0d. Corner cuts were first investigated by Onn and Sturmfels [Adv. Appl. Math. 23 (1999) 29-48], their original motivation stemmed from computational commutative algebra. Let us write (Nd0k)cut for the set of corner cuts of cardinality k; in the computational geometer's terminology, these are the k-sets of N0d. Among other things, Onn and Sturmfels give an upper bound of O(k2d(d-1)/(d+1)) for the size of (Nd0k)cut when the dimension is fixed. In two dimensions, it is known (see [Corteel et al., Adv. Appl. Math. 23 (1) (1999) 49-53]) that #(Nd0k)cut = Θ(k log k). We will see that in general, for any fixed dimension d, the order of magnitude of #(Nd0k)cut is between kd-1 log k and (k log k)d-1. (It has been communicated to me that the same bounds have been found independently by G. Rémond.) In fact, the elements of (Nd0k)cut correspond to the vertices of a certain polytope, and what our proof shows is that the above upper bound holds for the total number of flags of that polytope. AU - Wagner, Uli ID - 2420 IS - 2 JF - Advances in Applied Mathematics SN - 0196-8858 TI - On the number of corner cuts VL - 29 ER - TY - CHAP AB - Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, ρ, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is 2πℏ2aρ/m where a is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange N7/5 law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy. AU - Lieb, Élliott AU - Solovej, Jan AU - Seiringer, Robert AU - Yngvason, Jakob ID - 2338 SN - 9781571461018 T2 - Current Developments in Mathematics, 2001 TI - The ground state of the Bose gas ER - TY - JOUR AB - A new solvent-free composite polymer electrolyte consisting of high-molecular mass polyethylene oxide (PEO) filled with titanium oxide and containing LiI and I2 was developed. The introduction of the inorganic filler (TiO2 Degussa P25) into the polymer matrix produces dramatic morphological changes to the host polymer structure. Upon addition of the inorganic oxide, the surface roughness increases, with respect to the original polymer and in parallel, the fractal dimension decreases. Both the thermograms and the atomic force microscope (AFM) pictures confirm the amorphicity of the composite electrolyte. The polymer sub-units are held together in a parallel orientation, forming straight long chains of about 500 nm in width, along which TiO2 spherical particles of about 20-25 nm in diameter are distributed. The polymer chains separated by the titania particles are arranged in a three-dimensional, mechanically stable network, that creates free space and voids into which the iodide/triodide anions can easily migrate. All solid-state dye-sensitized solar cells fabricated using this composite electrolyte present high efficiencies (typical maximum incident photon to current efficiency (IPCE) as high as 40% at 520 nm and overall conversion efficiency (η) of 0.96% (Voc = 0.67 V, Jsc = 2.050 mA/cm2, FF = 39%) under direct solar irradiation. Further improvement of the photovoltaic performance is expected by optimization of the electrolyte parameters and of the cell assembly. AU - Katsaros, Georgios AU - Stergiopoulos, Thomas AU - Arabatzis, Iannis AU - Papadokostaki, Kyriaki AU - Falaras, Polycarpos ID - 1737 IS - 1-3 JF - Journal of Photochemistry and Photobiology A: Chemistry SN - 1010-6030 TI - A solvent-free composite polymer/inorganic oxide electrolyte for high efficiency solid-state dye-sensitized solar cells VL - 149 ER - TY - JOUR AB - Using the Pauli-Fierz model of non-relativistic quantum electrodynamics, we calculate the binding energy of an electron in the field of a nucleus of charge Z and in presence of the quantized radiation field. We consider the case of small coupling constant α, but fixed Zα and ultraviolet cut-off Λ. We prove that after renormalizing the mass the binding energy has, to leading order in α, a finite limit as Λ goes to infinity; i.e., the cut-off can be removed. The expression for the ground state energy shift thus obtained agrees with Bethe's formula for small values of Zα, but shows a different behavior for bigger values. AU - Hainzl, Christian AU - Seiringer, Robert ID - 2350 IS - 5 JF - Advances in Theoretical and Mathematical Physics SN - 1095-0761 TI - Mass renormalization and energy level shift in non-relativistic QED VL - 6 ER - TY - JOUR AB - We study fitness landscape in the space of protein sequences by relating sets of human pathogenic missense mutations in 32 proteins to amino acid substitutions that occurred in the course of evolution of these proteins. On average, ≈10% of deviations of a nonhuman protein from its human ortholog are compensated pathogenic deviations (CPDs), i.e., are caused by an amino acid substitution that, at this site, would be pathogenic to humans. Normal functioning of a CPD-containing protein must be caused by other, compensatory deviations of the nonhuman species from humans. Together, a CPD and the corresponding compensatory deviation form a Dobzhansky-Muller incompatibility that can be visualized as the corner on a fitness ridge. Thus, proteins evolve along fitness ridges which contain only ≈10 steps between sucessive corners. The fraction of CPDs among all deviations of a protein from its human ortholog does not increase with the evolutionary distance between the proteins, indicating that subtitutions that carry evolving proteins around these corners occur in rapid succession, driven by positive selection. Data on fitness of interspecies hybrids suggest that the compensatory change that makes a CPD fit usually occurs within the same protein. Data on protein structures and on cooccurrence of amino acids at different sites of multiple orthologous proteins often make it possible to provisionally identify the substitution that compensates a partiCUlar CPD. AU - Kondrashov, Alexey AU - Sunyaev, Shamil AU - Kondrashov, Fyodor ID - 885 IS - 23 JF - PNAS SN - 0027-8424 TI - Dobzhansky-Muller incompatibilities in protein evolution VL - 99 ER -