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...

The following question is studied: For which (nonintegrable) kernels A does lim,.^ / Óv4(/ — s)x(s) ds = 0 imply that lim,^^ x(f) — 0 when x is bounded and satisfies a Tauberian condition.

Phytochemicals may reduce chronic inflammation and cancer risk in part by modulating T-cell nuclear factor-kappaB (NF-kappaB) activation. Therefore, we examined the effects of curcumin (Cur) and limonin (Lim) feeding on NF-kappaB-dependent CD4(+) T-cell proliferation. DO11.10 transgenic mice (n = 5-7) were fed diets containing 1% Cur or 0.02% Lim combined with either (n-6) PUFA [5% corn oil (CO...

LIM-homeodomain (LIM-HD) transcription factors form a combinatorial 'LIM code' that contributes to the specification of cell types. In the ventral spinal cord, the binary LIM homeobox protein 3 (Lhx3)/LIM domain-binding protein 1 (Ldb1) complex specifies the formation of V2 interneurons. The additional expression of islet-1 (Isl1) in adjacent cells instead specifies the formation of motor neuro...