Some Comparison Inequalities for Generalized Muirhead and Identric Means

For x,y>0,  a,b∈ℝ, with a+b≠0, the generalized Muirhead mean M(a,b;x,y) with parameters a and b and the identric mean I(x,y) are defined by M(a,b;x,y)=((xayb+xbya)/2)1/(a+b) and I(x,y)=(1/e)(yy/xx)1/(y−x), x≠y, I(x,y)=x, x=y, respectively. In this paper, the following results are established: (1) M(a,b;x,y)>I(x,y) for all x,y>0 with x≠y and (a,b)∈{(a,b)∈ℝ2:a+b>0, ab≤0, 2(a−b)2−3(a+b)+1≥0, 3(a−b)2−2(a+b)≥0}; (2) M(a,b;x,y)<I(x,y) for all x,y>0 with x≠y and (a,b)∈{(a,b)∈ℝ2:a≥0, b≥0, 3(a−b)2−2(a+b)≤0}∪{(a,b)∈ℝ2:a+b<0}; (3) if (a,b)∈{(a,b)∈ℝ2:a>0, b>0, 3(a−b)2−2(a+b)>0}∪{(a,b)∈ℝ2:ab<0, 3(a−b)2−2(a+b)<0}, then there exist x1,y1,x2,y2>0 such that M(a,b;x1,y1)>I(x1,y1) and M(a,b;x2,y2)<I(x2,y2)

Dramatic Increases of Soil Microbial Functional Gene Diversity at the Treeline Ecotone of Changbai Mountain.

The elevational and latitudinal diversity patterns of microbial taxa have attracted great attention in the past decade. Recently, the distribution of functional attributes has been in the spotlight. Here, we report a study profiling soil microbial communities along an elevation gradient (500-2200 m) on Changbai Mountain. Using a comprehensive functional gene microarray (GeoChip 5.0), we found that microbial functional gene richness exhibited a dramatic increase at the treeline ecotone, but the bacterial taxonomic and phylogenetic diversity based on 16S rRNA gene sequencing did not exhibit such a similar trend. However, the β-diversity (compositional dissimilarity among sites) pattern for both bacterial taxa and functional genes was similar, showing significant elevational distance-decay patterns which presented increased dissimilarity with elevation. The bacterial taxonomic diversity/structure was strongly influenced by soil pH, while the functional gene diversity/structure was significantly correlated with soil dissolved organic carbon (DOC). This finding highlights that soil DOC may be a good predictor in determining the elevational distribution of microbial functional genes. The finding of significant shifts in functional gene diversity at the treeline ecotone could also provide valuable information for predicting the responses of microbial functions to climate change

Federated Learning with Intermediate Representation Regularization

In contrast to centralized model training that involves data collection, federated learning (FL) enables remote clients to collaboratively train a model without exposing their private data. However, model performance usually degrades in FL due to the heterogeneous data generated by clients of diverse characteristics. One promising strategy to maintain good performance is by limiting the local training from drifting far away from the global model. Previous studies accomplish this by regularizing the distance between the representations learned by the local and global models. However, they only consider representations from the early layers of a model or the layer preceding the output layer. In this study, we introduce FedIntR, which provides a more fine-grained regularization by integrating the representations of intermediate layers into the local training process. Specifically, FedIntR computes a regularization term that encourages the closeness between the intermediate layer representations of the local and global models. Additionally, FedIntR automatically determines the contribution of each layer's representation to the regularization term based on the similarity between local and global representations. We conduct extensive experiments on various datasets to show that FedIntR can achieve equivalent or higher performance compared to the state-of-the-art approaches. Our code is available at https://github.com/YLTun/FedIntR.Comment: IEEE BigComp 202

Sharp Power Mean Bounds for the Combination of Seiffert and Geometric Means

We answer the question: for α∈(0,1), what are the greatest value p and the least value q such that the double inequality Mp(a,b)0 with a≠b. Here, Mp(a,b), P(a,b), and G(a,b) denote the power of order p, Seiffert, and geometric means of two positive numbers a and b, respectively

An optimal double inequality among the one-parameter, arithmetic and harmonic means

For $p\in\mathbb{R}$, the one-parameter mean $J_{p}(a,b)$, arithmetic mean $A(a,b)$, and harmonic mean $H(a,b)$ of two positive real numbers $a$ and $b$ are defined by\begin{equation*}J_{p}(a,b)=\begin{cases}\tfrac{p(a^{p+1}-b^{p+1})}{(p+1)(a^p-b^p)}, & a\neq b,p\neq 0,-1,\\\tfrac{a-b}{\log{a}-\log{b}}, & a\neq b,p=0,\\\tfrac{ab(\log{a}-\log{b})}{a-b}, & a\neq b,p=-1,\\a, & a=b,\end{cases}\end{equation*}$A(a,b)=\tfrac{a+b}{2}$, and $H(a,b)=\tfrac{2ab}{a+b}$,respectively. In this paper, we answer the question: For $\alpha\in(0,1)$, what are the greatest value $r_{1}$ and the least value $r_{2}$ such that the double inequality $J_{r_{1}}(a,b)0$ with $a\neq b$