We wish to characterize when a Lévy process Xt crosses boundaries like t�, �>0, in a oneor two-sided sense, for small times t; thus, we inquire when lim sup t↓0|Xt|/t�, lim sup t↓0Xt/t� and/or lim inf t↓0Xt/t� are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of �>0. This completes and extends a line of research goi...

1. Prove the results about the smoothness of the members of the Matèrn family. Proof. From the class we know that for small values of τ , we have K ν (τ) ≈ Γ(ν)2 ν−1 τ −ν. Also, for the derivatives we have that d dτ (τ ν K ν (τ)) = −τ ν K ν−1 (τ). If the Matèrn family has the correlation function ρ(τ) = 1 Γ(ν)2 ν−1 τ ν K ν (τ), then ρ (τ) = −1 Γ(ν)2 ν−1 τ ν K ν−1 (τ) Note that K ν (τ) = K −ν (τ...

The long standing Lech’s conjecture in commutative algebra states that for a flat local extension $$(R,\mathfrak {m})\rightarrow (S,\mathfrak {n})$$ of Noetherian rings, we have an inequality on the Hilbert–Samuel multiplicities: $$e(R)\le e(S)$$ . In general is wide open when $$\dim R>3$$ , even equal characteristic. this paper, prove all dimensions, provided {m})$$ standard graded ring over p...

In this study we evaluate, at full atomic detail, the folding processes of two small helical proteins, the B domain of protein A and the Villin headpiece. Folding kinetics are studied by performing a large number of ab initio Monte Carlo folding simulations using a single transferable all-atom potential. Using these trajectories, we examine the relaxation behavior, secondary structure formation...