On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E₈ lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f∈R (where R=1+xZ?x?) can be written as f=gn for g∈R, n?2. Let Pn:={gn|g∈R} and let . We show among other things that (i) for f∈R, f∈Pn⇔f (mod μn)∈Pn, and (ii) if f∈Pn, there is a unique g∈Pn with coefficients mod μn/n such that f≡gn (mod μn). In particular, if f≡1 (mod μn) then f∈Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E₈ in R⁸) is in Pn if n is of the form i²j³k⁵ (i?3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length m² is in Pr² (and similarly that the theta series of the Barnes-Wall lattice BWm² is in Pm²). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f∈Pn (n?2) with coefficients restricted to the set {1,2,…,n}.     

Membrane separation of nitrogen from natural gas: A case study from membrane synthesis to commercial deployment

Fourteen percent of U.S. natural gas contains excess nitrogen, and cannot be sent to the national pipeline without treatment. Nitrogen is difficult to economically separate from methane, by any technology. Currently, the only process used on a large scale is cryogenic liquefaction and fractionation, but this technology requires economies of scale to be practical. Many owners of small gas fields cannot produce their gas for lack of suitable nitrogen separation technology. This paper describes the development of selective membranes to treat natural gas containing high concentrations of nitrogen. Membranes selectively permeate either nitrogen or methane, the principal constituent of natural gas. Our work has shown that methane-selective membranes are generally preferable and membranes with high permeances and methane/nitrogen selectivities of approximately 3–3.5 were developed. This selectivity is modest, so commercial systems often require multi-stage or multi-step process designs. Despite the design complexity and compression requirements, multi-step/multi-stage membrane systems are the lowest cost nitrogen removal technology in many applications. To date, 12 membrane-based systems for nitrogen removal for natural gas processing have been installed.     

Integer valued polynomials and Lubin-Tate formal groups

If R is an integral domain and K is its field of fractions, we let Int(R) stand for the subring of K[x] which maps R into itself. We show that if R is the ring of integers of a p-adic field, then Int(R) is generated, as an R-algebra, by the coefficients of the endomorphisms of any Lubin-Tate group attached to R.     

A formal framework for quantifying voter-controlled privacy

Privacy is a necessary requirement for voting. Without privacy, voters can be forced to vote in specific ways, and the forcing party can check their compliance. But offering privacy does not suffice: if a voter can reduce her privacy, an attacker can force her to do so. In this paper, we distinguish various ways that a voter can communicate with the intruder to reduce her privacy and classify them according to their ability to reduce the privacy of a voter. We develop a framework combining knowledge reasoning and trace equivalences to formally model voting protocols and define voter-controlled privacy. Our framework is quantitative, in the sense that it defines a measure for the privacy of a voter. Therefore, the framework can precisely measure the level of privacy for a voter for each of the identified privacy-reduction classes. The quantification allows our framework to capture receipts that reduce, but not nullify, the privacy of the voter.     

两线阵CCD相机几何指标差异对立体测绘的影响

为了准确、全面地揭示相机几何指标在立体测绘中的误差传播规律,研究了两线阵CCD相机几何指标差异与定位准确度之间的关系.针对两线阵立体测绘特点,分析了影响测绘定位的几何指标差异参量.在前方空间交会地面点计算公式的基础上,推导并建立了两线阵相机定位准确度模型,据此研究了内方位元素标定误差间的几何指标差异对定位准确度的影响规律.结果表明:主点位置x方向误差间的差异对定位准确度影响显著,当其值从1 μm增至2.5 μm时,定位准确度明显下降了60％,焦距f误差间的差异对定位准确度的影响几乎可忽略,本文所建立的模型与相关研究结论对两线阵CCD相机指标的量化与优化设计具有重要指导意义.     

强正则剩余格值逻辑系统L~N及其完备性

正则剩余格是一类重要的模糊逻辑代数系统,而常见的模糊逻辑形式系统大多数带有非联接词,并且相应的Lindenbaum代数都是正则剩余格.本文以强正则剩余格为语义,建立了一个一般的命题演算形式系统LN,并且证明了这个系统的完备性.几种常见的带有非联接词的模糊逻辑形式系统都是系统LN的扩张.     

The product of lattice covolume and discrete series formal dimension: p-adic GL(2)

Let $F$ be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.     

A simple algorithm for expanding a power series as a continued fraction

I present and discuss an extremely simple algorithm for expanding a formal power series as a continued fraction. This algorithm, which goes back to Euler (1746) and Viscovatov (1805), deserves to be better known. I also discuss the connection of this algorithm with the work of Gauss (1812), Stieltjes (1889), Rogers (1907) and Ramanujan, and a combinatorial interpretation based on the work of Flajolet (1980).