Determining the number of factors with potentially strong within-block correlations in error terms

<p>We develop methods to estimate the number of factors when error terms have potentially strong correlations in the cross-sectional dimension. The information criteria proposed by Bai and Ng (<a href="#CIT0005" target="_blank">2002</a>) require the cross-sectional correlations between the error terms to be weak. Violation of this weak correlation assumption may lead to inconsistent estimates of the number of factors. We establish two data-dependent estimators that are consistent whether the error terms are weakly or strongly correlated in the cross-sectional dimension. To handle potentially strong cross-sectional correlations between the error terms, we use a block structure in which the within-block correlation may either be weak or strong, but the between-block correlation is limited. Our estimators allow imperfect knowledge and a moderate misspecification of the block structure. Monte-Carlo simulation results show that our estimators perform similarly to existing methods for cases in which the conventional weak correlation assumption is satisfied. When the error terms have a strong cross-sectional correlation, our estimators outperform the existing methods.</p

Effects of different concentrations of decomposed garlic stalk on soil enzyme activities.

<p>Effects of different concentrations of decomposed garlic stalk on soil sucrase activity (A), soil urease activity (B) and soil alkaline phosphatase activity (C). Error bars represent the standard error of the mean. Different letters above the error bars indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

Effects of different garlic stalk decomposition temperatures on soil pH (A) and EC (B).

<p>Error bars represent the standard error of the mean. Different letters above the histograms indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

Effects of different garlic stalk decomposition temperatures on soil enzyme activities.

<p>Effect of different decomposition temperature of garlic stalk on soil sucrase activity (A), soil urease activity (B) and soil alkaline phosphatase activity (C). Error bars represent the standard error of the mean. Different letters above the histograms indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

Effects of different garlic stalk decomposition durations on dynamic changes in soil nutrients.

<p>The data are presented as soil nutrient dynamics. Different letters in the same column indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

Effects of different garlic stalk decomposition temperatures on dynamic changes in soil nutrients.

<p>The data are presented as soil nutrient dynamics. Different letters in the same column indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

Basic characteristics of original soil and garlic stalk.

<p>Basic characteristics of original soil and garlic stalk.</p

Effects of the duration of garlic stalk decomposition on soil pH (A) and EC (B).

<p>Error bars represent as the standard error of the mean. Different letters above the error bars indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

Effects of different concentrations of decomposed garlic stalk on dynamic changes in soil nutrients.

<p>The data are presented as the soil nutrient dynamics. Different letters in the same column indicate significant differences at the 0.05 level (ANOVA and Duncan’s multiple range test), n = 3.</p

<i>N</i>‑Heterocyclic Carbene Catalyzed Enantioselective [3 + 2] Dearomatizing Annulation of Saturated Carboxylic Esters with <i>N</i>‑Iminoisoquinolinium Ylides

The dearomatizing annulation reaction is a significant challenge in organic chemistry. The direct activation of α-carbons of simple saturated esters, as nucleophiles, is an important synthesis strategy. In the present study, we disclose [3 + 2] dearomatizing annulation reactions with direct activating α-carbons of saturated carboxylic esters and <i>N</i>-iminoisoquinolinium ylides, which possess highly enantioselective characteristics, catalyzed by <i>N</i>-heterocyclic carbenes (NHCs). The protocol achieves isoquinoline dearomatization and the construction of tricyclic chiral products under mild conditions with good yield, substrate tolerance, and diastereoselectivity as well as excellent enantioselectivity