Several parameters of generalized Mycielskians

The generalized Mycielskians (also known as cones over graphs) are the natural generalization of the Mycielski graphs (which were first introduced by Mycielski in 1955). Given a graph G and any integer m?0, one can transform G into a new graph μ_m(G), the generalized Mycielskian of G. This paper investigates circular clique number, total domination number, open packing number, fractional open packing number, vertex cover number, determinant, spectrum, and biclique partition number of μ_m(G).     

跳汰机的性能及统计模型

在煤用重选设备评定的国际标准（ＩＳＯ９２３）和以之为蓝本的国家标准（ＧＢ／Ｔ１５７１５）中的第一项评定指标“可能偏差”及和其有关的“不完善度”的确定是由以下步骤决定：１．由仅仅一次浮沉试验取得一组数据（一个观察值）并由此计算两段分离所得产品的分配率;２．用手工凭想象中的“Ｓ型曲线”把分配率中的两组６ 至８ 个点联起来得到两条“分配曲线”;３．由这两条粗糙的曲线“量出”各自的２５％ 和７５％ 分位点,以此得到能反映重选设备分离能力好坏的两个指标：“可能偏差”Ｅ和“不完善度”Ｉ,很难想象,这样由一个观察值得到的指标并通过没有模型的手工绘图而得到的结果会有任何实际意义,本文建议利用尽可能多的观察值来拟合ｌｏｇｉｓｔｉｃ回归模型,并依此得到分配曲线和计算出参数Ｅ和Ｉ,本文还利用一组实际数据来说明我们方法的合理性     

The Gold Partition Conjecture for 6-Thin Posets

We consider the Gold Partition Conjecture (GPC) that implies the 1/3–2/3 Conjecture. We prove the GPC in the case where every element of the poset is incomparable with at most six others. The proof involves the extensive use of computers. This paper contains results obtained using computer resources of the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM), University of Warsaw.     

A condition-based algorithm for solving polyhedral feasibility problems

It is known that Polyhedral Feasibility Problems can be solved via interior-point methods whose real number complexity is polynomial in the dimension of the problem and the logarithm of a condition number of the problem instance. A limitation of these results is that they do not apply to ill-posed instances, for which the condition number is infinite. We propose an algorithm for solving polyhedral feasibility problems in homogeneous form that is applicable to all problem instances, and whose real number complexity is polynomial in the dimension of the problem instance and in the logarithm of an “extended condition number” that is always finite.     

Enumerative properties of shifted-Dyck partitions

We study the enumerative properties of a new class of (skew) shifted partitions. This class arises in the computation of certain parabolic Kazhdan-Lusztig polynomials and is closely related to ballot sequences. As consequences of our results, we obtain new identities for the parabolic Kazhdan-Lusztig polynomials of Hermitian symmetric pairs and for the ordinary Kazhdan-Lusztig polynomials of certain Weyl groups.     

Ramanujan-type congruences for broken 2-diamond partitions modulo 3

The notion of broken k-diamond partitions was introduced by Andrews and Paule.Let△k(n)denote the number of broken k-diamond partitions of n.Andrews and Paule also posed three conjectures on the congruences of△2(n)modulo 2,5 and 25.Hirschhorn and Sellers proved the conjectures for modulo 2,and Chan proved the two cases of modulo 5.For the case of modulo 3,Radu and Sellers obtained an infinite family of congruences for△2(n).In this paper,we obtain two infinite families of congruences for△2(n)modulo 3 based on a formula of Radu and Sellers,a 3-dissection formula of the generating function of triangular number due to Berndt,and the properties of the U-operator,the V-operator,the Hecke operator and the Hecke eigenform.For example,we find that△2(243n+142)≡△2(243n+223)≡0(mod 3).The infinite family of Radu and Sellers and the two infinite families derived in this paper have two congruences in common,namely,△2(27n+16)≡△2(27n+25)≡0(mod 3).     

Partitioned Tensor Products and Their Spectra

A pleasant family of graphs defined by Godsil and McKay is shown to have easily computed eigenvalues in many cases.     

Rule-based models of multivariable functions

This study is devoted to fuzzy rule based modelling of multiple-input single-output nonlinear numerical relationships. The model under investigation is viewed as a collection of conditional statements “if state Ω_i then y = g_i(x, a_i)”, i = 1, 2,…, N with Ω_i being a fuzzy relation defined in the space of the input variables. In contrast to the commonly encountered identification approach that is dwelled upon discrete experimental data, the one proposed in this study is concerned with explicitly articulated nonlinear input-output relationship. The main thrust is in the development of a fuzzy partition of the input variables completed through a sequence of fuzzy relations rather than Cartesian products of fuzzy sets. This approach allows us to maintain the number of necessary rules under control and avoid a combinatorial explosion otherwise inevitable in situations of highly multivariable functions. Introduced are criteria of separability and function variability whose objective is to guide a distribution and granularity of the linguistic labels forming the condition part of the rules.