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Let b¯2,3(n), which enumerates the number of (2, 3)-regular overcubic bipartition of n. The purpose of the paper is to describe some congruences modulo 8 for b¯2,3(n). For example, for each α ≥ 0 and n ≥ 1, b¯2,3(8n+5)≡0(mod8), b¯2,3(2⋅3α+3n+4⋅3α+2)≡0(mod8).

H.C. Chan has studied the congruence properties of cubic partition function a(n), which is defined by ∑n=0∞a(n)qn=1(q;q)∞(q2;q2)∞.

To establish several congruence modulo 8 for b¯2,3(n), here the author keeps to the classical spirit of q-series techniques in the proofs.

The results established in the work are extension to those proved in ℓ-regular cubic partition pairs.

Complex technology not only provides potential economic benefits but also increases the difficulty of application. Whether and how upstream technological complexity affects downstream manufacturers' innovation through vertical separation structure is worth discussing, but it has not been effectively discussed.

Through theoretical analysis and empirical testing, this article discusses the cost effect and market competition effect caused by upstream technological complexity on downstream manufacturers and further elucidates the impact of upstream technological complexity on downstream manufacturers' innovation.

Research has found that the impact of upstream technological complexity on the downstream manufacturers' innovation depends on the cost effect and market competition effect. The cost effect caused by the complexity of upstream technology inhibits the innovation of downstream manufacturers. In contrast, the market competition effect promotes the innovation of downstream manufacturers. There are differences in the cost effect and market competition effect of upstream technological complexity on different types of downstream manufacturers, so there is also significant heterogeneity in the impact of upstream technological complexity on innovation of different types of downstream manufacturers.

The conclusions of this article improve the understanding of the relationship between upstream technological complexity and downstream innovation and provide helpful implications for industrial chain innovation.

The discrete Fourier transform (dft) of a fractional process is studied. An exact representation of the dft is given in terms of the component data, leading to the frequency domain form of the model for a fractional process. This representation is particularly useful in analyzing the asymptotic behavior of the dft and periodogram in the nonstationary case when the memory parameter d≥12. Various asymptotic approximations are established including some new hypergeometric function representations that are of independent interest. It is shown that smoothed periodogram spectral estimates remain consistent for frequencies away from the origin in the nonstationary case provided the memory parameter d < 1. When d = 1, the spectral estimates are inconsistent and converge weakly to random variates. Applications of the theory to log periodogram regression and local Whittle estimation of the memory parameter are discussed and some modified versions of these procedures are suggested for nonstationary cases.

This chapter extends the work of Baltagi, Bresson, Chaturvedi, and Lacroix (2018) to the popular dynamic panel data model. The authors investigate the robustness of Bayesian panel data models to possible misspecification of the prior distribution. The proposed robust Bayesian approach departs from the standard Bayesian framework in two ways. First, the authors consider the ε-contamination class of prior distributions for the model parameters as well as for the individual effects. Second, both the base elicited priors and the ε-contamination priors use Zellner’s (1986) g-priors for the variance–covariance matrices. The authors propose a general “toolbox” for a wide range of specifications which includes the dynamic panel model with random effects, with cross-correlated effects à la Chamberlain, for the Hausman–Taylor world and for dynamic panel data models with homogeneous/heterogeneous slopes and cross-sectional dependence. Using a Monte Carlo simulation study, the authors compare the finite sample properties of the proposed estimator to those of standard classical estimators. The chapter contributes to the dynamic panel data literature by proposing a general robust Bayesian framework which encompasses the conventional frequentist specifications and their associated estimation methods as special cases.

This chapter develops a structural framework for the analysis of scoring procurement auctions where bidder’s quality and bid are taken into account. With exogenous quality, the authors characterize the optimal mechanism whether the buyer is private or public and show that the optimal scoring rule need not be linear in the bid. The model primitives include the buyer benefit function, the bidders’ cost inefficiencies distribution and cost function, and potentially the cost of public funds. We show that the model primitives are nonparametrically identified under mild functional assumptions from the buyer’s choice, firms’ bids and qualities. The authors then develop a multistep kernel-based procedure to estimate the model primitives and provide their convergence rates. Our identification and estimation results are general as they apply to other scoring rules including quasi-linear ones.

Let p_[1,r;t] be defined by ∑n=0∞p[1,r;t](n)qn=(E1Er)t, where t is a non-zero rational number, r ≥ 1 is an integer and Er=∏n=0∞(1−qr(n+1)) for |q| < 1. The function p_[1,r;t](n) is the generalisation of the two-colour partition function p_[1,r;−1](n). In this paper, the authors prove some new congruences modulo odd prime ℓ by taking r = 5, 7, 11 and 13, and non-integral rational values of t.

Using q-series expansion/identities, the authors established general congruence modulo prime number for two-colour partition function.

In the paper, the authors study congruence properties of two-colour partition function for fractional values. The authors also give some particular cases as examples.

The partition functions for fractional value is studied in 2019 by Chan and Wang for Ramanujan's general partition function and then extended by Xia and Zhu in 2020. In 2021, Baruah and Das also proved some congruences related to fractional partition functions previously investigated by Chan and Wang. In this sequel, some congruences are proved for two-colour partitions in this paper. The results presented in the paper are original.

In order to make full use of the generalized greyness of interval grey number, this paper analyzes the properties and its applications of generalized greyness.

Firstly, the static properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain are analyzed. Then, this paper gives the dynamic properties of generalized greyness in bounded background domain, infinite background domain and infinitesimal background domain and explains the dialectical principle contained in it. Finally, the generalized greyness is used to judge the effectiveness of interval grey number transformation.

The results show that the generalized greyness of interval grey number has relativity, normativity, unity, eternity and conservation. The static and dynamic properties of generalized greyness are the same in the infinite and infinitesimal background domain, while they are different in the bounded background domain. The generalized greyness can be used as an index to judge whether the grey number transformation is greyness or information preservation.

The research shows that the generalized greyness can be used as an index to judge the validity of the grey number transformation and also can be applied in grey evaluation, grey decision-making and grey prediction and so on.

The paper succeeds in realizing the mathematical principle of “white is black”, the “greyness clock-slow effect” of the value domain of interval grey number and the generalized greyness conservation principle, which provides a theoretical basis for the rational use of generalized greyness of interval grey number.