A complete description of Golay pairs for lengths up to 100

In his 1961 paper, Marcel Golay showed how the search for pairs of binary sequences of length with complementary autocorrelation is at worst a problem. Andres, in his 1977 master's thesis, developed an algorithm which reduced this to a search and investigated lengths up to 58 for existence of pairs. In this paper, we describe refinements to this algorithm, enabling a search at length 82. We find no new pairs at the outstanding lengths 74 and 82. In extending the theory of composition, we are able to obtain a closed formula for the number of pairs of length generated by a primitive pair of length . Combining this with the results of searches at all allowable lengths up to 100, we identify five primitive pairs. All others pairs of lengths less than 100 may be derived using the methods outlined.

     

Varieties of Complete Pairs of Zero-Dimensional Subschemes of Lengths ≥2 and ≥4 in Algebraic Surfaces

We prove that the varieties of complete pairs of zero-dimensional subschemes of lengths d ₁ 2, d ₂ 4 on a smooth irreducible projective algebraic surface are singular.     

Supercuspidal Gelfand pairs

If G is a totally disconnected group and H is a closed subgroup then, according to the Gelfand-Kazhdan Lemma, if the double coset space H?G/H is preserved by an antiautomorphism of G of order two then (G,H) must be a Gelfand pair in the sense that Hom_H(π,1) has dimension at most one for each irreducible, admissible representation π of G. Under certain rather general restrictions, we show that if the symmetry property holds only for almost all double cosets, then (G,H) is a supercuspidal Gelfand pair in the sense that for all irreducible, supercuspidal representations π of G. There exist examples of supercuspidal Gelfand pairs which are not Gelfand pairs.     

Sheaves on Involutive Quantaloids

We extend the well-known notions of a singleton, complete -set, presheaf and sheaf over a complete Heyting algebra or a right-sided idempotent quantale to arbitrary involutive quantaloids. We show that sheaves on and complete -sets come to the same thing. This paper can be considered as a symmetric version of an earlier work of the author.     

Compact weakly symmetric spaces and spherical pairs

Let be a spherical pair and assume that is a connected compact simple Lie group and a closed subgroup of . We prove in this paper that the homogeneous manifold is weakly symmetric with respect to and possibly an additional fixed isometry . It follows that M. Krämer's classification list of such spherical pairs also becomes a classification list of compact weakly symmetric spaces. In fact, our proof involves a case-by-case study of the isotropy representations of all spherical pairs on Krämer's list.

     

激子自旋弛豫对简并量子点发射光子对纠缠度的影响

利用粒子数运动方程和量子回归理论，计算了单个半导体量子点双激子体系脉冲激发下粒子在各能级间辐射跃迁的二阶交叉相关函数以及系统发射光子对的偏振密度矩阵.分析了激子态能级简并量子点体系发射光子对偏振纠缠特性，讨论了纠缠度随激子态间自旋弛豫的变化关系.研究表明，激子自旋弛豫会破坏该系统发射光子对的纠缠度. 关键词： 纠缠光子对 半导体量子点 二阶相关函数     

Quantum secret sharing between multiparty and multiparty with four states

A protocol of quantum secret sharing between multiparty and multiparty with four states was presented. It was shown that this protocol can nullify the Trojan horse attack with a multi-photon signal, the fake-signal attack with Einstein-Podolsky-Rosen pairs, the attack with single photons, and the attack with invisible photons. In addition, the upper bounds of the average success probabilities were given for dishonest agent eavesdropping encryption using the fake-signal attack with any two-particle entangled states. Supported by the National Natural Science Foundation of China (Grant No. 10671054), the Key Project of Science and Technology Research of Education Ministry of China (Grant No. 207011) and the Natural Science Foundation of Hebei Province, China (Grant Nos. A2005000140 and 07M006)     

The factors of spontaneous parametric down conversion process in BiB3O6 crystal with a broadband pump

We investigate the spontaneous parametric down-conversion (SPDC) process in biaxial nonlinear crystal BiB₃O₆ (BIBO) with a broadband pump. Under the type I and type II phase-matching conditions, we numerically calculate the influencing factors on SPDC process in BIBO crystal, such as temporal and spatial walk-off, the acceptance angles, and spectral acceptance bandwidth. Comparing the two types of phase-matching, we could conclude that the type II phase-matching in BIBO crystal is better for the SPDC because of zero dispersion and bigger acceptance parameters. These results can be used to construct a system to generate ultra-fast entangled photon pairs.     

Ramanujan-Slater type identities related to the moduli 18 and 24

We present several new families of Rogers-Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.     

Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers CP¹→CP¹ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.