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.     

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.     

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.     

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.     

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.     

环F_2+uF_2+u~2F_2上线性码的广义Gray像

摘要:引入了环F_2+uF_2+u~2F_2与F_2之间的广义Gray映射,利用环F_2+uF_2+u~2F_2上线性码的生成矩阵得出了广义Gray像φ(C)的生成矩阵,证明了F_2+uF2+u2F2上线性码自正交码的广义Gray像仍为自正交码和F_2+uF_2+u~2F_2上循环码的广义Gray像是F_2上的准循环码.     

A taxonomy of best-reply multifunctions in 2×2×2 trimatrix games

This paper provides an overview of the various shapes the best-reply multifunctions can take in 2×2×2 trimatrix games. It is shown that, unlike in 2×2 bimatrix games, the best replies to the opponents’ pure strategies do not completely determine the structure of the Nash equilibrium set.      

Generic 2×2 matrices with involution

Procesi has given a linear basis for the ring ofm generic 2×2 matrices. We do the same for the ring ofm generic 2×2 matrices with transpose involution.     

On the determination of solutions of simultaneous Pell equations $$x^{2}-(a^{2}-1)y^{2}=y^{2}-pz^{2}=1$$ x 2 - ( a 2 - 1 ) y 2 = y 2 - p z 2 = 1

Periodica Mathematica Hungarica - In this paper, we consider the simultaneous Pell equations 0.1 $$\begin{aligned} x^{2}-(a^{2}-1)y^{2}= & {} 1, \nonumber \\ y^{2}-pz^{2}= & {} 1,...     

环F2+uF2+vF2上线性码广义Gray像

本文定义了环F2+uF2+vF2到域F2的广义Gray映射φ像,研究了环F2+uF2+vF2上线性码的广义Gray像.利用广义Gray映射φ的线性性,证明了环F2+uF2+vF2上线性码C的广义Gray像φ(C)满足dH(C)=dH(φ(C))且φ(C⊥)φ(C)⊥.同时,给出了F2+uF2+vF2上循环码C的广义Gray像φ(C)为F2上的4-拟循环码.     

2级矩阵的迹

给出了2级矩阵的迹的几个结果,同时给出了若干应用例子.     

2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.     

Characterisations of A p+2 and L 2(2 q )

Let G be a transitive group of degree p+2 with p|G| where p ≧ 5 is a prime number, then (i) G is isomorphic to S _p+2 or A _p+2, if G has an element of order 4, (ii) G is isomorphic to L ₂(2 ^q ) or P Γ L ₂ (2 ^q ), if 2 ^q − 1=p is a Mersenne prime and G has no element of order 4.     

Capable 2-Generator 2-Groups of Class Two

A group is called capable if it is a central factor group. We characterize the capable 2-generator 2-groups of class 2 in terms of a standard presentation.     

The 2-adjugate mod 2 class of designs

In this paper we define a 2-adjugate mod 2 class of designs obtained by application of process of 2-adjugation to the incidence matrix of given design and subsequently reducing to modulo 2. General properties of such designs are discussed and its application to the class of unreduced Balanced Incomplete Block Designs is investigated in detail. It is found that in this case we obtain a class of generalized partially balanced incomplete block designs with 2 associate classes and with triangular association scheme, but with unequal block sizes.     

I_2-CONVERGENCE AND I_2-CAUCHY DOUBLE SEQUENCES

In this article, we prove a decomposition theorem for I2-convergent double sequences and introduce the notions of I2-Cauchy and I*2-Cauchy double sequence, and then study their certain properties. Finally, we introduce the notions of regularly(I2, I)-convergence and(I2, I)-Cauchy double sequence.     

环F_2+uF_2上长为2~e的(1+u)-循环码

最近,环F2+uF2上的线性码引起了编码研究者极大的兴趣.本文证明了R[x]/〈xn+1+u〉是有限链环,其中R=F2+uF2=F2[u]/〈u2〉且n=2e.从而给出了F2+uF2上的所有长为2e的(1+u)-循环码,进而给出了所有(1+u)-循环码的对偶码.证明了F2+uF2上不存在长为2e的非平凡的自对偶的(1+u)-循环码.