Combinatorial principles between RRT 2 2 and RT 2 2

We study the strength of some combinatorial principles weaker than Ramsey theorem for pairs over RCA0. First, we prove that Rainbow Ramsey theorem for pairs does not imply Thin Set theorem for pairs. Furthermore, we get some other related results on reverse mathematics using the same method. For instance, Rainbow Ramsey theorem for pairs is strictly weaker than ErdSs- Moser theorem under RCA0.     

Infra-red spectral studies of CHF2-CHCl2 and CBrCl2-CBrCl2

The infra-red spectra of two polyhalogenated ethanes, 1, 1-difluoro-2, 2-dichloro ethane and 1, 2-dibromo-1, 1, 2, 2-tetrachloro ethane were obtained in the region of 2·5–38 microns. The spectra of CHF₂-CHCl₂ were obtained in gas, liquid and solid states and also in solvents of widely varying dielectric constants. These spectral studies showed that the molecule exists in the form of two rotational isomers. The fundamental modes of the two isomers were identified on the basis of intensity changes from gas to liquid to solid and in solvents of different dielectric constants. The observed infra-red absorption bands have been assigned in terms of the fundamental absorption bands. Spectra of CBrCl₂-CBrCl₂ were obtained in the solid and vapour phase and in various solvents. These spectra showed that CBrCl₂-CBrCl₂ exists only in the trans form. The infra-red spectra, combined with Raman data available from literature was used for complete assignment of the observed absorption bands.     

P2Q2Iso2D=2D Isoparametric FEM in Matlab

A short Matlab implementation realizes a flexible isoparametric finite element method up to quadratic order for the approximation of elliptic problems in two-dimensional domains with curved boundaries. Triangles and quadrilaterals equipped with varying quadrature rules allow for mesh refinement. Numerical examples for the Laplace equation with mixed boundary conditions indicate the flexibility of isoparametric finite elements.     

Non-symmetric 2-designs modulo 2

Necessary conditions are obtained for the existence of a 2 – (v, k, ) design, for which the block intersection sizess ₁,s ₂, ...,s _n satisfys ₁ s ₂ ... s _n s (mod 2 ^e ), wheree is odd. These conditions are obtained by combining restrictions on the Smith Normal Form of the incidence matrix of the design with some well known properties of self-orthogonal binary codes with all weights divisible by 4.Research done at AT&T Bell Laboratories.     

Descent on 2-Fibrations and Strongly 2-Regular 2-Categories

We consider pseudo-descent in the context of 2-fibrations. A 2-category of descent data is associated to a 3-truncated simplicial object in the base 2-category. A morphism q in the base induces (via comma-objects and pullbacks) an internal category whose truncated simplicial nerve induces in turn the 2-category of descent data for q. When the 2-fibration admits direct images, we provide the analogous of the Beck–Bénabou–Roubaud theorem, identifying the 2-category of descent data with that of pseudo-algebras for the pseudo-monad q ^*Σ _q . We introduce a notion of strong 2-regularity for a 2-category R, so that its basic 2-fibration of internal fibrations c od:F ib(R)→R admits direct images. In this context, we show that essentially-surjective-on-objects morphisms, defined by a certain lax colimit, are of effective descent by means of a Beck-style pseudo-monadicity theorem.     

环$F_2 uF_2 u^2F_2$上的循环码及自对偶码

We give the structures of a cyclic code over ring
R = F2 ＋ uF2 ＋ u^2F2 = {0, 1,u, u^2,v, v^2,uv, v^3},
where u^3 = 0, of odd length and its dual code. For the cyclic code, necessary and sufficient conditions for the existence of self-dual code are provided.     

Comparison of relative risk,attributable risk and logistic response procedures for 2×2×2 andc 2×2×2 contingency tables

Summary A numerical evaluation was made on three different χ² statistics in order to compare mutagenic risk frequencies between two experimental groups in a 2×2×2 contingency table andc 2×2×2 contingency tables. The three methods involve 1) a relative risk approach, 2) an attributable risk approach and 3) a logistic response approach. A relatively large difference was observed among χ² statistics in the three approaches using actual data under the hypothesis that there is no difference between the two groups under scrutiny. With regard to approximate power, it appears that approaches 2) and 3) give fairly stable results. Approach 1) has greater power if there is a small difference in the control groups. It was confirmed that the approximate power of approach 1) is extremely large when the difference in the relative risk frequencies between the two groups under contrast is constant and each relative risk frequency is small. This paper was presented at the 48th annual meeting of Japan Statistical Association in Tokyo, Japan, 1980.     

Generic 2-surfaces and 2-2 foliations in general relativity

Summary Special points of spacelike and timelike 2-surfaces are defined by means of algebraic relations between the second order invariants of their immersion. Generic immersions are then defined and by means of direction fields constructed over the surface, the least number of such points is related to global properties of the surface. Moreover it is proved that every compact spacelike 2-surface has at least one point where the normal curvature is zero. Consequences are drawn for generic 2-spheres and trapped surfaces. The theorems are generalised to 2-2 foliations and fibrations. Entrata in Redazione il 9 ottobre 1973.     

Galois 2-extensions unramified outside 2

We classify quadratic, biquadratic and degree 4 cyclic 2-rational number fields. We also classify those quadratic number fields which are not 2-rational, but have a degree 2 extension, which is Galois over Q and is 2-rational. In this case we explicitly describe the Galois group of their maximal pro-2 extension unramified outside 2 and infinity using a result of Herfort-Ribes-Zalesskii on virtually free pro-p groups.     

环F_2+uF_2+u~2F_2上循环码的Gray距离

定义了环R=F2+uF2+u2 F2(u3=0)到F32的一个新的Gray映射.首先介绍环R上奇长度的循环码的挠码,给出了各阶挠码的生成多项式.利用一阶挠码与二阶挠码确立了R上奇长度的循环码的Gray距离.     

On A_2^2 -Polyhedra

本文提出A_2^2同调可环，证明它的实现性，即在任给的抽象的同调可环的正则同构类中，必有某一A_2^2多面体的同调可环。又证明同调可环间的正则同态，必为连续映像所实现，于是获得一个函子使A_2^2多面体的伦型与同调可环的正则同构类一一对应。这里提供一个实例，说明这样的函子是代数与几何的桥梁。     

R^2上带奇异性的非线性椭圆型方程的正整解

以Ｓｃｈａｕｄｅｒ－Ｔｙｃｈｏｎｏｆｆ不动点定理为工具,建立了一类在Ｒ＾２上带奇异性的非线性椭圆型方程正整解的存在性定理,并给出了解在无穷远的性质。     

方程ǖ+bur+g(u(t-τ))=p(t)的2π-周期解

本文对一类不具耗散项的时滞微分方程进行了分析,利用重合度理论,得到了方程至少存在一个2π-周期解的充分条件.     

A note on the simultaneous Pell equations $$x^2-(a^2-1)y^2=1$$ x 2 - ( a 2 - 1 ) y 2 = 1 and $$y^2-bz^2=1$$ y 2 - b z 2 = 1

Periodica Mathematica Hungarica - Let $$a>1,b$$ be two positive integers where the square-free part of b is 2pq with p, q two distinct odd primes. Recently, Cipu (Proc Am Math Soc...     

环F_2+uF_2上长度为2n(n为奇数)的循环码

研究了环F2+uF2上长度为2n(n为奇数)的循环码,给出了循环码及其对偶码的生成多项式,以及循环码为自对偶码的充要条件,最后进一步给出了循环码极小Lee重量的一些相关结论