Supersymmetry, homology with twisted coefficients and n-dimensional knots

Let $n$ be any natural number. Let $K$ be any $n$-dimensional knot in $S^{n+2}$. We define a supersymmetric quantum system for $K$ with the following properties. We firstly construct a set of functional spaces (spaces of fermionic \{resp. bosonic\} states) and a set of operators (supersymmetric infinitesimal transformations) in an explicit way. Thus we obtain a set of the Witten indexes for $K$. Our Witten indexes are topological invariants for $n$-dimensional knots. Our Witten indexes are not zero in general. If $K$ is equivalent to the trivial knot, all of our Witten indexes are zero. Our Witten indexes restrict the Alexander polynomials of $n$-knots. If one of our Witten indexes for an $n$-knot $K$ is nonzero, then one of the Alexander polynomials of $K$ is nontrivial. Our Witten indexes are connected with homology with twisted coefficients. Roughly speaking, our Witten indexes have path integral representation by using a usual manner of supersymmetric theory.Comment: 10pages, no figure

General Sub-packetized Access-Optimal Regenerating Codes

This paper presents a novel construction of $(n,k,d=n-1)$ access-optimal regenerating codes for an arbitrary sub-packetization level $\alpha$ for exact repair of any systematic node. We refer to these codes as general sub-packetized because we provide an algorithm for constructing codes for any $\alpha$ less than or equal to $r^{\lceil \frac{k}{r} \rceil}$ where $\frac{k}{r}$ is not necessarily an integer. This leads to a flexible construction of codes for different code rates compared to existing approaches. We derive the lower and the upper bound of the repair bandwidth. The repair bandwidth depends on the code parameters and $\alpha$. The repair process of a failed systematic node is linear and highly parallelized, which means that a set of $\lceil \frac{\alpha}{r} \rceil$ symbols is independently repaired first and used along with the accessed data from other nodes to recover the remaining symbols

New method for the 3D Ising model

A simple, general and practically exact method is developed for the equilibrium properties of the macroscopic physical systems with translational symmetry. Applied to the Ising model in two and three dimension, a modest calculation gives the spontaneous magnetization and the specific heat to less than 1% error

An Abstract Approach to Stratification in Linear Logic

We study the notion of stratification, as used in subsystems of linear logic with low complexity bounds on the cut-elimination procedure (the so-called light logics), from an abstract point of view, introducing a logical system in which stratification is handled by a separate modality. This modality, which is a generalization of the paragraph modality of Girard's light linear logic, arises from a general categorical construction applicable to all models of linear logic. We thus learn that stratification may be formulated independently of exponential modalities; when it is forced to be connected to exponential modalities, it yields interesting complexity properties. In particular, from our analysis stem three alternative reformulations of Baillot and Mazza's linear logic by levels: one geometric, one interactive, and one semantic

Optical variability of the BL Lacertae object GC 0109+224. Multiband behaviour and time scales from a 7-years monitoring campaign

We present the most continuous data base of optical $BVR_{c}I_{c}$ observations ever published on the BL Lacertae object GC 0109+224, collected mainly by the robotic telescope of the Perugia University Observatory in the period November 1994-February 2002. These observations have been complemented by data from the Torino Observatory, collected in the period July 1995-January 1999, and Mt. Maidanak Observatory (December 2000). GC 0109+224 showed rapid optical variations and six major outbursts were observed at the beginning and end of 1996, in fall 1998, at the beginning and at the end of 2000, and at the beginning of 2002. Fast and large-amplitude drops characterized its flux behaviour. The $R_c$ magnitude ranged from 13.3 (16.16 mJy) to 16.46 (0.8 mJy), with a mean value of 14.9 (3.38 mJy). In the periods where we collected multi-filter observations, we analyzed colour and spectral indexes, and the variability patterns during some flares. The long-term behaviour seems approximatively achromatic, but during some isolated outbursts we found evidence of the typical loop-like hysteresis behaviour, suggesting that rapid optical variability is dominated by non-thermal cooling of a single emitting particle population. We performed also a statistical analysis of the data, through the discrete correlation function (DCF), the structure function (SF), and the Lomb-Scargle periodogram, to identify characteristic times scales, from days to months, in the light curves, and to quantify the mode of variability. We also include the reconstruction of the historical light curve and a photometric calibration of comparison stars, to favour further extensive optical monitoring of this interesting blazar.Comment: 13 pages, 11 PS figures, 1 EPS figure, 3 tables, accepted by Astronomy and Astrophysics. Uses A&A documentclass aa.cls, and the package graphicx.st