We offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $\frac{n+1}{n-1}$ provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $\frac{n-1}{2}$. With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $\frac{n-1}{2}$.

We mainly focus on regression estimation in a longitudinal study with nonignorable intermittent nonresponse and dropout. To handle the identifiability issue, we take a time-independent covariate as nonresponse instrument which is independent of nonresponse propensity conditioned on other covariates and responses to ensure the identifiability of nonresponse propensity. The nonresponse propensity is assumed to be a parametric model, and the corresponding parameters are estimated by using the generalized method of moments approach. Then the marginal response means are estimated by inverse probability weighting method. Furthermore, to improve the robustness of estimators, we derive an augmented inverse probability weighting estimator which is shown to be consistent and asymptotically normally distributed. Simulation studies and a real-data analysis show that the proposed approach yields highly efficient estimators.

In this paper, a new 4-parameter exponentiated generalized inverse flexible Weibull distribution is proposed. Some of its statistical properties are studied. The aim of this paper is to estimate the model parameters via several approaches, namely, maximum likelihood, maximum product spacing and Bayesian. According to Bayesian approach, several techniques are used to get the Bayesian estimators, namely, standard error function, Linex loss function and entropy loss function. The estimation herein is based on complete and censored samples. Markov Chain Monte Carlo simulation is used to discuss the behavior of the estimators for each approach. Finally, two real data sets are analyzed to obtain the flexibility of the proposed model.

The present paper examines matrix root properties and embedding conditions for discrete-time Markov chains with three states and a transition matrix having complex eigenvalues. Necessary as well as sufficient conditions for the existence of an m-th stochastic root of the transition matrix are investigated. Matrix roots are expressed in analytical form based on the spectral decomposition of the transition matrix, and properties of these matrix roots are proved.

In this paper, we have characterized the sum of two general measures associated with two distributions with discrete random variables. One of these measures is logarithmic, while others contains the power of variables, named as joint representation of Renyi’s–Tsallis divergence measure. Then, we propose a divergence measure based on Jensen–Renyi’s–Tsallis entropy which is known as a Jensen–Renyi’s–Tsallis divergence measure. It is a generalization of J-divergence information measure. One of the silent features of this measure is that we can allot the equal weight to each probability distribution. This makes it specifically reasonable for the study of decision problems, where the weights could be the prior probabilities. Further, the idea has been generalized from probabilistic to fuzzy similarity/dissimilarity measure. Besides the validation of the proposed measure, some of its key properties are also studied. Further, the performance of the proposed measure is contrasted with some existing measures. At last, some illustrative examples are solved in the context of clustering analysis, financial diagnosis and pattern recognition which demonstrate the practicality and adequacy of the proposed measure between two fuzzy sets (FSs).

In this article, the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss, and discuss their relations with Besov and Triebel–Lizorkin spaces. As an application, the authors establish the difference characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space ${{\mathcal {X}}}$ via using the geometrical property of ${{\mathcal {X}}}$ expressed by its dyadic reference points, dyadic cubes, and the (local) lower bound. Moreover, some results when $p\le 1$ but near to 1 are new even when ${{\mathcal {X}}}$ is an RD-space.

Topology optimization plays an important role in a wide range of engineering applications. In this paper, we propose a novel isogeometric topology optimization algorithm based on deep learning. Unlike the other neural network-based methods, the density distributions in the design domain are represented in the B-spline space. In addition, we use relatively novel technologies, U-Net and DenseNet, to form the neural network structure. The 2D and 3D numerical experiments show that the proposed method has an accuracy rate of over 97% for the final optimization results. After training, the new approach can save time greatly for the new topology optimization compared with traditional solid isotropic material with penalization method and IGA method. The approach can also overcome the checkerboard phenomenon.

Throughout this paper, all groups are finite and G always denotes a finite group; $\sigma $ is some partition of the set of all primes $\mathbb {P}$. A group G is said to be $\sigma $-primary if G is a $\pi $-group for some $\pi \in \sigma $. A $\pi $-semiprojector of G [29] is a subgroup H of G such that HN/N is a maximal $\pi $-subgroup of G/N for all normal subgroups N of G. Let $\Pi \subseteq \sigma $. Then we say that $ {\mathcal {X}} = \{X_ {1}, \ldots , X_ {t} \} $ is a $ \Pi $-covering subgroup system for a subgroup H in G if all members of the set $ {\mathcal {X}} $ are $ \sigma $-primary subgroups of G and for each $ \pi \in \Pi $ with $\pi \cap \pi (H)\ne \emptyset $ there are an index i and a $ \pi $-semiprojector U of H such that $ U \le X_ {i}$. We study the embedding properties of subgroups H of G under the hypothesis that G has a $ \Pi $-covering subgroup system $ {\mathcal {X}}$ such that H permutes with $X^{x}$ for all $X\in {\mathcal {X}}$ and $x\in G$. Some well-known results are generalized.