Small ball probability for the condition number of random matrices

Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max}(A)/s_{\min}(A)$ satisfies the small ball probability estimate $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, ${\mathbb P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq k\leq n,$ obtained (under some additional assumptions) by Nguyen.Comment: Some changes according to the Referee's comment

Random polytopes obtained by matrices with heavy tailed entries

Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $\Gamma^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $\Gamma$). In particular, we show that $$\Gamma B_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right).$$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion is equivalent to so-called $\ell_1$-quotient property and plays an important role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic

Random polytopes obtained by matrices with heavy tailed entries

Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly. We investigate properties of the corresponding random polytope $\Gamma^* B_1^N$ in $\mathbb{R}$ (the absolute convex hull of rows of $\Gamma$). In particular, we show that $$\Gamma B_1^N \supset b^{-1} \left( B_{\infty}^n \cap \sqrt{\ln (N/n)}\, B_2^n \right).$$ where $b$ depends only on parameters in small ball inequality. This extends results of \cite{LPRT} and recent results of \cite{KKR}. This inclusion is equivalent to so-called $\ell_1$-quotient property and plays an important role in compressive sensing (see \cite{KKR} and references therein).Comment: Last version, to appear in Communications in Contemporary Mathematic

Predicting soil wind erosion potential under different corn residue management scenarios in the central Great Plains

Various models and simplified equations are available to predict wind erosion potential. However, their performance can be often site-specific, depending on soil characteristics and agronomic practices, warranting sitespecific model validations. Thus, in this study, we 1) validated the wind erodible fraction (WEF) predictive equations by Fryrear et al. (1994) and López et al. (2007) and 2) estimated the total soil loss with the Singleevent Wind Erosion Evaluation Program (SWEEP) using 3-yr measured data from six experiments located across a precipitation gradient in the central Great Plains. Each site had three corn (Zea mays L.) residue removal treatments: control (no removal), grazed, and baled. The measured and predicted WEF were significantly correlated. While the Fryrear et al. (1994) equation performed better than the López et al. (2007) equation, it underestimated WEF with 59% uncertainty across site-years. To reduce this underestimation and uncertainty, we developed a new statistical equation (WEF%=84.3+2.64×% silt-0.30×% clay-7.43×% organic matter- 0.15×% residue cover; r2=0.56). The predictive ability of the new equation was, however, no better than that of the existing predictive equations, suggesting the need for further refinement of WEF equations for the region. Simulated total soil loss by wind using the SWEEP model indicated that corn residue baling may increase soil loss if residue cover drops below 20% in the study region. Overall, the existing WEF equations could under- or overestimate WEF based on site-specific residue management, warranting further model refinement and site-specific validation, whereas the SWEEP estimated soil loss corroborates the critical importance of maintaining sufficient residue cover (\u3e 20%) to reduce wind erosion

U.S. Billion-ton Update: Biomass Supply for a Bioenergy and Bioproducts Industry

The Report, Biomass as Feedstock for a Bioenergy and Bioproducts Industry: The Technical Feasibility of a Billion-Ton Annual Supply (generally referred to as the Billion-Ton Study or 2005 BTS), was an estimate of “potential” biomass within the contiguous United States based on numerous assumptions about current and future inventory and production capacity, availability, and technology. In the 2005 BTS, a strategic analysis was undertaken to determine if U.S. agriculture and forest resources have the capability to potentially produce at least one billion dry tons of biomass annually, in a sustainable manner—enough to displace approximately 30% of the country’s present petroleum consumption. To ensure reasonable confidence in the study results, an effort was made to use relatively conservative assumptions. However, for both agriculture and forestry, the resource potential was not restricted by price. That is, all identified biomass was potentially available, even though some potential feedstock would more than likely be too expensive to actually be economically available. In addition to updating the 2005 study, this report attempts to address a number of its shortcoming

What makes young Russians happy and satisfied with their lives?

Participants (N = 10,672 with the mean age of 20.7 years) of the Russian Character and Personality Survey (RCPS), involving 40 universities or colleges from across the Russian Federation, rated their happiness and satisfaction with life; the ratings were combined into an index of subjective well-being (SWB). Using the National Character Survey (NCS), participants also rated their own personality characteristics as well as those of an ideal person and a typical Russian living in their own region. Only two personality (test) subscales—N3: Depression and E6: Positive Emotions—were correlated with SWB on the between-individual level of analysis. Although spiritual values associated with a negative attitude toward money are typically regarded as an essential part of the Russian national character, our results demonstrated that only satisfaction with one’s own financial situation was a reliable predictor of SWB. In those regions where people had, on average, a higher life expectancy, better education, and a higher level of wealth, individuals also tended to be happier and more satisfied with their lives

Random polytopes obtained by matrices with heavy-tailed entries

International audienc

Age stereotypes of the employees and their specific manifestations

Охарактеризованы мотивационные факторы возрастной стереотипизации. Выявлена содержательная составляющая возрастных стереотипов сотрудников. The motivational factors of age stereotyping are described. The content component of age stereotypes of employees is revealed