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https://www.selleckchem.com/MEK.html We study the spectrum of generalized Wishart matrices, defined as F=(XY^⊤+YX^⊤)/2T, where X and Y are N×T matrices with zero mean, unit variance independent and identically distributed entries and such that E[X_itY_jt]=cδ_i,j. The limit c=1 corresponds to the Marčenko-Pastur problem. For a general c, we show that the Stieltjes transform of F is the solution of a cubic equation. In the limit c=0, T≫N, the density of eigenvalues converges to the Wigner semicircle.Key aspects of glasses are controlled by the presence of excitations in which a group of particles can rearrange. Surprisingly, recent observations indicate that their density is dramatically reduced and their size decreases as the temperature of the supercooled liquid is lowered. Some theories predict these excitations to cause a gap in the spectrum of quasilocalized modes of the Hessian that grows upon cooling, while others predict a pseudogap D_L(ω)∼ω^α. To unify these views and observations, we generate glassy configurations of controlled gap magnitude ω_c at temperature T=0, using so-called breathing particles, and study how such gapped states respond to thermal fluctuations. We find that (i) the gap always fills up at finite T with D_L(ω)≈A_4(T)ω^4 and A_4∼exp(-E_a/T) at low T, (ii) E_a rapidly grows with ω_c, in reasonable agreement with a simple scaling prediction E_a∼ω_c^4 and (iii) at larger ω_c excitations involve fewer particles, as we rationalize, and eventually become stringlike. We propose an interpretation of mean-field theories of the glass transition, in which the modes beyond the gap act as an excitation reservoir, from which a pseudogap distribution is populated with its magnitude rapidly decreasing at lower T. We discuss how this picture unifies the rarefaction as well as the decreasing size of excitations upon cooling, together with a stringlike relaxation occurring near the glass transition.We study the large scale behavior of a collection of hard co

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