Angle between two random segments

The study of "random segments" is a classic issue in geometrical probability, whose complexity depends on how it is defined. But in apparently simple models, the random behavior is not immediate. In the present manuscript the following setting is considered. Let four independent random points that follow a uniform distribution on the unit disk. Two random segments are built with them, which always are inside of the disk. We compute the density function of the angle between these two random segments when they intersect each other. This type of problem tends to be complex due to the high stochastic dependency that exists between the elements that form them. The expression obtained is in terms of integrals, however it allows us to understand the behavior of the distribution of the random angle between the two random segments

Concentration of the roots for the Kac polynomials

In this paper we study how the roots of the so-called Kac polynomial $W_n(z) = \sum_{k=0}^{n-1} \xi_k z^k$ are concentrating to the unit circle when its the coefficients of $W_n$ are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial are concentrating around the unit circle as $n\to\infty$ if and only if $\mathbb{E}\left({\log\left(1+ |\xi_0|\right)}\right)<\infty$. Under the condition of $\mathbb{E}\left({\xi^2_0}\right)<\infty$ we show that there exists an annulus of width $\mathrm{O}(n^{-2}(\log n)^{-3})$ around the unit circle which is \textit{free} of roots with probability $1-\mathrm{O}({(\log n)^{-{1}/{2}}})$. The proof relies on the so-called small ball probability inequalities.Comment: New version. The title of the manuscript was update

Salem–Zygmund inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

In this manuscript we give an extension of the classic Salem-Zygmund inequality for locally sub-Gaussian random variables. As an application, the concentration of the roots of a Kac polynomial is studied, which is the main contribution of this manuscript. More precisely, we assume the existence of the moment generating function for the iid random coefficients for the Kac polynomial and prove that there exists an annulus of width O(n(-2)(log n)(-1/2-gamma)), gamma > 1/2 around the unit circle that does not contain roots with high probability. As an another application, we show that the smallest singular value of a random circulant matrix is at least n(-rho), rho is an element of (0, 1/4) with probability 1 - O(n(-2 rho)).Peer reviewe

Salem–Zygmund inequality for locally sub-Gaussian random variables, random trigonometric polynomials, and random circulant matrices

In this manuscript we give an extension of the classic Salem-Zygmund inequality for locally sub-Gaussian random variables. As an application, the concentration of the roots of a Kac polynomial is studied, which is the main contribution of this manuscript. More precisely, we assume the existence of the moment generating function for the iid random coefficients for the Kac polynomial and prove that there exists an annulus of width O(n(-2)(log n)(-1/2-gamma)), gamma > 1/2 around the unit circle that does not contain roots with high probability. As an another application, we show that the smallest singular value of a random circulant matrix is at least n(-rho), rho is an element of (0, 1/4) with probability 1 - O(n(-2 rho)).Peer reviewe

Zero-free neighborhoods around the unit circle for Kac polynomials

In this paper, we study how the roots of the Kac polynomials concentrate around the unit circle when the coefficients are independent and identically distributed non-degenerate real random variables. It is well-known that the roots of a Kac polynomial concentrate around the unit circle as the dimension growths if and only if some log-moment is finite. Under the finiteness of the second moment, we show that there exists an annulus of small width around the unit circle which is free of roots with high probability. The proof relies on small ball probability inequalities and the least common denominator.Peer reviewe

The asymptotic distribution of the condition number for random circulant matrices

In this manuscript, we study the limiting distribution for the joint law of the largest and the smallest singular values for random circulant matrices with generating sequence given by independent and identically distributed random elements satisfying the so-called Lyapunov condition. Under an appropriated normalization, the joint law of the extremal singular values converges in distribution, as the matrix dimension tends to infinity, to an independent product of Rayleigh and Gumbel laws. The latter implies that a normalized condition number converges in distribution to a Frechet law as the dimension of the matrix increases.Peer reviewe

THE ROLE OF THE XAVANTE INDIGENOUS PEOPLE IN WILDLIFE CONSERVATION

There is an urgent demand to evaluate and document the environmental conditions of the territories of indigenous people. This is basic in the efforts to achieve sustainable development goals adopted by all United Nations Member States in 2015. The Xavante people are hunters/gatherers and depend on natural resources for their physical, spiritual, and cultural survival. Their lands are localized in the state of Mato Grosso, Brazil, in a transitional area between the Cerrado vegetation and the Amazon rainforest. They have been developing environmental projects ~in order to manage their territory correctly for decades, as part of their survival strategy. In recent fieldwork, we stated that some major game species may still be abundant in the territory and we suggest that certain wildlife management measures in the past may be responsible for this. We easily registered most game species handled by the Xavantes, except for some edentates that were rarely detected. We confirm the giant anteater as the most vulnerable species to hunting effects. In this article, we point out the main threats for the territory and present new recommendations that may be fundamental for the conservation of biodiversity in the region and the survival of the Xavante people