A new class for large sets of almost Hamilton cycle decompositions

A k-cycle system of order v with index λ, denoted by CS(v, k, λ), is a collection A of k-cycles (blocks) of K _v such that each edge in K _v appears in exactly λ blocks of A. A large set of CS(v, k, λ)s is a partition of the set of all k-cycles of K _v into CS(v, k, λ)_s, and is denoted by LCS(v, k, λ). A (v ?1)-cycle in K _v is called almost Hamilton. The completion of the existence problem for LCS(v, v?1, λ) depends only on one case: all v ≥ 4 for λ = 2. In this paper, it is shown that there exists an LCS(v, v ? 1, 2) for all v ≡ 2 (mod 4), v ≥ 6.     

一类四进制U-正交函数系

本文给出了一类四进制的分段多项式正交系(简称QU-系统)的构造方法,讨论了该构造算法的可行性,并给出了1至3次QU-系统的一组显式表达式.同时,研究了QU-系统的性质以及与二进制U-系统之间的关系,并推导出了QU-系统的基函数值与Fourier-QU系数的计算公式.由本文的方法可以构造一类L2[0,1]中的完备的正交系...     

Three new classes of optimal frequency-hopping sequence sets

The study of frequency-hopping sequences (FHSs) has been focused on the establishment of theoretical bounds for the parameters of FHSs as well as on the construction of optimal FHSs with respect to the bounds. Peng and Fan (IEEE Trans Inf Theory 50(9):2149–2154, 2004) derived two lower bounds on the maximum nontrivial Hamming correlation of an FHS set, which is an important indicator in measuring the performance of an FHS set employed in practice. In this paper, we obtain two main results. We study the construction of new optimal frequency-hopping sequence sets by using cyclic codes over finite fields. Let \(\mathcal {C}\) be a cyclic code of length n over a finite field \(\mathbb {F}_q\) such that \(\mathcal {C}\) contains the one-dimensional subcode \( \mathcal {C}_0=\{(\alpha ,\alpha ,\ldots ,\alpha )\in \mathbb {F}_q^n\,|\,\alpha \in \mathbb {F}_q\}. \) Two codewords of \(\mathcal {C}\) are said to be equivalent if one can be obtained from the other through applying the cyclic shift a certain number of times. We present a necessary and sufficient condition under which the equivalence class of any codeword in \(\mathcal {C}\setminus \mathcal {C}_0\) has size n. This result addresses an open question raised by Ding et al. (IEEE Trans Inf Theory 55(7):3297–3304, 2009). As a consequence, three new classes of optimal FHS sets with respect to the Singleton bound are obtained, some of which are also optimal with respect to the Peng–Fan bound at the same time. We also show that the two Peng–Fan bounds are, in fact, identical.     

A class of primitive substitutions and scrambled sets

Consider the subshifts induced by constant-length primitive substitutions on two symbols. By investigating the equivalent version for the existence of Li-Yorke scrambled sets and by proving the non-existence of Schweizer-Smítal scrambled sets, we completely reveal for this class of subshifts the chaotic behaviors possibly occurring in the sense of Li-Yorke and Schweizer-Smítal.     

A class of uniqueness sets for double Walsh series

A problem of uniqueness for multiple Walsh series is considered. A class of uniqueness sets for double Walsh series is obtained, this class contains known classes of uniqueness sets.     

A special class of Williamson matrices and difference sets

A construction is given of a very special class of Hadamard matrices. This yields Hadamard matrices of the Williamson kind and difference sets of order 4 · 3^2m.     

One class of partial sets

We shall establish that any semirecursive -hyperhypersimple set has partial Turing degree.Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 473–478, October, 1976.     

A class of sets of uniqueness for multiple Walsh series

The uniqueness problem for multiple series over orthogonal systems of functions is considered. Classes of sets of uniqueness for multiple Walsh series and for multiple series over a mixed system of functions are obtained, which extends known classes of sets of uniqueness. A many-dimensional analogue of Privalov’s theorem is established.     

A class of functions whose sublevel sets are absolute retracts

In this paper, we prove a result of which the following is a corollary: If X is a Banach space and J:X→R is a contraction, then the nonempty sublevel sets of the function x→‖x‖+J(x) are absolute retracts.     

A class of spectral self-affine measures with four-element digit sets

The self-affine measure _μM,D${\mu }_{M,D}$ associated with an expanding matrix M∈_Mn(Z)$M\in M_n(\mathbb{Z})$ and a finite digit set D⊂Zⁿ$D\subset {\mathbb{Z}}^n$ is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures _μM,D${\mu }_{M,D}$ with four-element digit sets in the higher dimensions (n≥3$n\ge 3$) such that the Hilbert space L²(_μM,D)$L^2({\mu }_{M,D})$ possesses an orthogonal exponential basis. That is, _μM,D${\mu }_{M,D}$ is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.