Quantum cohomology of flag manifolds and Toda lattices

We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.Comment: 35 page

Soliton solutions of Calogero model in harmonic potential

A classical Calogero model in an external harmonic potential is known to be integrable for any number of particles. We consider here reductions which play a role of "soliton" solutions of the model. We obtain these solutions both for the model with finite number of particles and in a hydrodynamic limit. In the latter limit the model is described by hydrodynamic equations on continuous density and velocity fields. Soliton solutions in this case are finite dimensional reductions of the hydrodynamic model and describe the propagation of lumps of density and velocity in the nontrivial background.Comment: 25 pages, 2 figure

Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds

Using 4-dimensional arithmetic hyperbolic manifolds, we construct some new homological quantum error correcting codes. They are LDPC codes with linear rate and distance $n^\epsilon$. Their rate is evaluated via Euler characteristic arguments and their distance using $\mathbb{Z}_2$-systolic geometry. This construction answers a queston of Z\'emor, who asked whether homological codes with such parameters could exist at all.Comment: 21 page

On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of \Co^{N}. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in \Co^{2}.Comment: minor changes in the formulation and the proof of Lemma 8.

On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local perturbation of the operator might embed an eigenvalue into the continuous spectrum (a feature uncommon for periodic elliptic operators of second order). In all known constructions of such examples, the corresponding eigenfunction is compactly supported. One wonders whether this must always be the case. The paper answers this question affirmatively. What is more surprising, one can estimate that the eigenmode must be localized not far away from the perturbation (in a neighborhood of the perturbation's support, the width of the neighborhood determined by the unperturbed operator only). The validity of this result requires the condition of irreducibility of the Fermi (Floquet) surface of the periodic operator, which is expected to be satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio

Methane pyrolysis on sponge iron powder for sustainable hydrogen production

Methane pyrolysis is one of the possible methods to produce low-carbon hydrogen. One of the most promising catalysts for methane pyrolysis is Fe due to its availability, relatively low cost and high working temperature. In the presented paper, the methane pyrolysis on unsupported (without a carrier) sponge iron in the form of powder was studied in the temperature range of 700–1100 ◦ C. Methane pyrolysis was carried out in a stainless-steel tube reactor with an inner diameter of 10 mm. The reactor was heated locally by propane burner, the length of the heated zone was about 8 cm along the reactor tube. Methane feed rates were about 50, 100, and 200 ml/min, and the residence time of methane in the 8 cm long reaction zone was about 4, 2 and 1 s, respectively. The hydrogen yield increased with an increase in the temperature and a decrease in methane feed rate. At 700–800 ◦C, the hydrogen yield did not exceed 20%. At 900 ◦C, the yield reached 28.6% at a residence time of about 4 s. At 1000 ◦C, hydrogen yield was about 40 and 66.5% at a residence time of about 1 and 4 s, respectively. At 1100 ◦C, hydrogen yield varied in the range of 70–85%. The use of a catalyst increased the hydrogen yield by 81% compared to the experiment without a catalyst at 1100 ◦C. The catalytic effect of sponge iron powder can be used in the development of methane pyrolysis plants

The Alexander-Orbach conjecture holds in high dimensions

We examine the incipient infinite cluster (IIC) of critical percolation in regimes where mean-field behavior has been established, namely when the dimension d is large enough or when d>6 and the lattice is sufficiently spread out. We find that random walk on the IIC exhibits anomalous diffusion with the spectral dimension d_s=4/3, that is, p_t(x,x)= t^{-2/3+o(1)}. This establishes a conjecture of Alexander and Orbach. En route we calculate the one-arm exponent with respect to the intrinsic distance.Comment: 25 pages, 2 figures. To appear in Inventiones Mathematica

Results from the first use of low radioactivity argon in a dark matter search

Liquid argon is a bright scintillator with potent particle identification properties, making it an attractive target for direct-detection dark matter searches. The DarkSide-50 dark matter search here reports the first WIMP search results obtained using a target of low-radioactivity argon. DarkSide-50 is a dark matter detector, using two-phase liquid argon time projection chamber, located at the Laboratori Nazionali del Gran Sasso. The underground argon is shown to contain Ar-39 at a level reduced by a factor (1.4 +- 0.2) x 10^3 relative to atmospheric argon. We report a background-free null result from (2616 +- 43) kg d of data, accumulated over 70.9 live-days. When combined with our previous search using an atmospheric argon, the 90 % C.L. upper limit on the WIMP-nucleon spin-independent cross section based on zero events found in the WIMP search regions, is 2.0 x 10^-44 cm^2 (8.6 x 10^-44 cm^2, 8.0 x 10^-43 cm^2) for a WIMP mass of 100 GeV/c^2 (1 TeV/c^2 , 10 TeV/c^2).Comment: Accepted by Phys. Rev.

Regularity of harmonic discs in spaces with quadratic isoperimetric inequality

We study harmonic and quasi-harmonic discs in metric spaces admitting a uniformly local quadratic isoperimetric inequality for curves. The class of such metric spaces includes compact Lipschitz manifolds, metric spaces with upper or lower curvature bounds in the sense of Alexandrov, some sub-Riemannian manifolds, and many more. In this setting, we prove local Hölder continuity and continuity up to the boundary of harmonic and quasi-harmonic discs

Area minimizing discs in metric spaces

We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Radò and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces