Perturbation Theory around Non-Nested Fermi Surfaces I. Keeping the Fermi Surface Fixed

The perturbation expansion for a general class of many-fermion systems with a non-nested, non-spherical Fermi surface is renormalized to all orders. In the limit as the infrared cutoff is removed, the counterterms converge to a finite limit which is differentiable in the band structure. The map from the renormalized to the bare band structure is shown to be locally injective. A new classification of graphs as overlapping or non-overlapping is given, and improved power counting bounds are derived from it. They imply that the only subgraphs that can generate $r$ factorials in the $r^{\rm th}$ order of the renormalized perturbation series are indeed the ladder graphs and thus give a precise sense to the statement that `ladders are the most divergent diagrams'. Our results apply directly to the Hubbard model at any filling except for half-filling. The half-filled Hubbard model is treated in another place.Comment: plain TeX with postscript figures in a uuencoded gz-compressed tar file. Put it on a separate directory before unpacking, since it contains about 40 files. If you have problems, requests or comments, send e-mail to [email protected]

Exact Half-BPS Flux Solutions in M-theory with D(2,1;c′;0)2D(2, 1; c'; 0)^2 Symmetry: Local Solutions

We construct local solutions to 11-dimensional supergravity (or M-theory), which are invariant under the superalgebra $D(2, 1; c'; 0)\oplus D(2, 1; c'; 0)$ for all values of the parameter $c'$. The BPS constraints are reduced to a single linear PDE on a complex function $G$. The PDE was solved in 0806.0605 modulo application of boundary and regularity conditions. The physical fields of the solutions are determined by $c'$, a harmonic function $h$, and the complex function $G$. $h$ and $G$ are both functions on a 2-dimensional compact Riemannian manifold. The harmonic function $h$ is freely chosen. We obtain the expressions for the metric and the field strength in terms of $G$, $h$, and $c'$ and show that these are indeed valid solutions of the Einstein, Maxwell, and Bianchi equations. Finally we give a construction of one parameter deformations of $AdS_7 \times S^4$ and $AdS_4 \times S^7$ as a function of $c'$

Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces

The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean functions, but until recently (O'Donnell and Servedio) nothing was known about efficient algorithms for \emph{reconstructing} $f$ (exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the \emph{Chow Parameters Problem.} Our main result is a new algorithm for the Chow Parameters Problem which, given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function $f$, runs in time \tilde{O}(n^2)\cdot (1/\eps)^{O(\log^2(1/\eps))} and with high probability outputs a representation of an LTF $f'$ that is \eps-close to $f$. The only previous algorithm (O'Donnell and Servedio) had running time \poly(n) \cdot 2^{2^{\tilde{O}(1/\eps^2)}}. As a byproduct of our approach, we show that for any linear threshold function $f$ over $\{-1,1\}^n$, there is a linear threshold function $f'$ which is \eps-close to $f$ and has all weights that are integers at most \sqrt{n} \cdot (1/\eps)^{O(\log^2(1/\eps))}. This significantly improves the best previous result of Diakonikolas and Servedio which gave a \poly(n) \cdot 2^{\tilde{O}(1/\eps^{2/3})} weight bound, and is close to the known lower bound of $\max\{\sqrt{n},$ (1/\eps)^{\Omega(\log \log (1/\eps))}\} (Goldberg, Servedio). Our techniques also yield improved algorithms for related problems in learning theory

Evolution of cracks in selvedge of the coal bed during its stationary working

Purpose. To study rupturing of the coal/rock seam selvedge by natural gas-filled cracks as phenomena that prepare and initiate sudden outbursts of coal, rock, and gas during the steady movement of the face. Methods. The work is based on theoretical studies, including methods of thermodynamics, statistical physics, and asymptotic analysis. Findings. A generalized Griffith’s criterion for the material rupture by a crack is applied to the selvedge part of a gas-saturated coal/rock seam during its stationary unloading. Originality. The kinetic theory describing processes of gas-containing materials destruction is developed using the example of rupturing the selvedge part of a coal/rock seam by natural gas-filled cracks. Practical implications. The criterion for changing the control parameters (reservoir gas pressure, crack dimensions, rock pressure, surface coal/rock energy, elastic modulus) is found at which spontaneous failure of the seam becomes possible. This allows to discuss the possibility of predicting sudden outbursts of coal, rock and gas.Цель. Исследование процессов разрыва краевой части угольного/породного пласта природными газонаполненными трещинами как явлений, подготавливающих и инициирующих внезапные выбросы угля, породы и газа при стационарном подвигании забоя. Методика. Работа выполнена на основе теоретических исследований, включающих методы термодинамики, статистической физики, асимптотического анализа. Результаты. Получен обобщенный критерий Гриффитса разрыва материала трещиной применительно к краевой части газонасыщенного угольного/породного пласта при его стационарной разгрузке. Научная новизна. Развита кинетическая теория процессов разрушения газосодержащих материалов на примере разрыва краевой части угольного/породного пласта природными газонаполненными трещинами. Практическая значимость. Найден критерий изменения управляющих параметров (пластового давления газа, размеров трещин, горного давления, поверхностной энергии угля/породы, модуля упругости), при котором спонтанное разрушение пласта становится возможным. Это позволяет обсуждать возможность прогноза внезапных выбросов угля, породы и газа.Мета. Дослідження процесів розриву крайової частини вугільного/породного пласта природними газонаповненими тріщинами як явищ, що підготовлюють та ініціюють раптові викиди вугілля, породи й газу при стаціонарному посуванні вибою. Методика. Робота виконана на основі теоретичних досліджень, що включають методи термодинаміки, статистичної фізики, асимптотичного аналізу. Результати. Отримано узагальнений критерій Гріффітса розриву матеріалу тріщиною стосовно крайової частини газонасиченого вугільного/породного пласта при його стаціонарному розвантаженні. Наукова новизна. Розвинено кінетичну теорію процесів руйнування матеріалів, що містять газ, на прикладі розриву крайової частини вугільного/породного пласта природними газонаповненими тріщинами. Практична значимість. Знайдено критерій зміни керуючих параметрів (пластового тиску газу, розмірів тріщин, гірського тиску, поверхневої енергії вугілля/породи, модуля пружності), при якому спонтанне руйнування пласта стає можливим. Це дозволяє обговорювати можливість прогнозу раптових викидів вугілля, породи і газу.Работа выполнена в рамках научно-исследовательс-кой темы НАН Украины “Массо-, теплоперенос и физика предвыбросных явлений в газонасыщенном трещиновато-пористом слоистом углепородном массиве”

Rectification in Luttinger liquids

We investigate the rectification of an ac bias in Luttinger liquids in the presence of an asymmetric potential (the ratchet effect). We show that strong repulsive electron interaction enhances the ratchet current in comparison with Fermi liquid systems, and the I-V curve is strongly asymmetric in the low-voltage regime even for a weak asymmetric potential. At higher voltages the ratchet current exhibits an oscillatory voltage dependence.Comment: 5 pages, Revte

Packing Returning Secretaries

We study online secretary problems with returns in combinatorial packing domains with $n$ candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All $2n$ arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio at least $0.5721 - o(1)$ for growing $n$, and an algorithm with ratio at least $0.5459$ for all $n \ge 1$. We extend all algorithms and ratios to $k \ge 2$ arrivals per candidate. In the second variant, there is a pool of undecided candidates. In each round, a random candidate from the pool arrives. Upon arrival a candidate can be either decided (accept/reject) or postponed (returned into the pool). We mainly focus on minimizing the expected number of postponements when computing an optimal solution. An expected number of $\Theta(n \log n)$ is always sufficient. For matroids, we show that the expected number can be reduced to $O(r \log (n/r))$, where $r \le n/2$ is the minimum of the ranks of matroid and dual matroid. For bipartite matching, we show a bound of $O(r \log n)$, where $r$ is the size of the optimum matching. For general packing, we show a lower bound of $\Omega(n \log \log n)$, even when the size of the optimum is $r = \Theta(\log n)$.Comment: 23 pages, 5 figure

Clustering of fermionic truncated expectation values via functional integration

I give a simple proof that the correlation functions of many-fermion systems have a convergent functional Grassmann integral representation, and use this representation to show that the cumulants of fermionic quantum statistical mechanics satisfy l^1-clustering estimates

Phyllosilicate Transitions in Ferromagnesian Soils: Short-Range Order Materials and Smectites Dominate Secondary Phases

Analyses of X-ray diffraction (XRD) patterns taken by the CheMin instrument on the Curiosity Rover in Gale crater have documented the presence of clay minerals interpreted as smectites and a suite of amorphous to short-range order materials termed X-ray amorphous materials. These X-ray amorphous materials are commonly ironrich and aluminum poor and likely some of them are weathering products rather than primary glasses due to the presence of volatiles. Outstanding questions remain regarding the chemical composition and mineral structure of these X-ray amorphous materials and the smectites present at Gale crater and what they indicate about environmental conditions during their formation. To gain a better understanding of the mineral transitions that occur within ferromagnesian parent materials, we have investigated the development of secondary clay minerals and shortrange order materials in two soil chronosequences with varying climates developing on ultramafic bedrock. Field Sites: We investigated soil weathering within two field locations, the Klamath Mountains of Northern California, and the Tablelands of Newfoundland, Canada. Both sites possess age dated or correlated recently deglaciated soils and undated but substantially older soils. In the Klamath mountains the Trinity Ultramafic Body was deglaciated roughly 15,000 years bp while in the Tablelands a moraine was dated to about 17,600 years bp. The Klamath Mountains feature a seasonally wet and dry climate while the Tablelands are wet year-round with saturated soil conditions observed during sampling and standing water observed within 3 of 4 soil pit sampling locations