Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators

The spectral properties for n order differential operators are considered. When given a spectral gap (a,b) of the minimal operator T0 with deficiency index r, arbitrary m points βi  (i=1,2,…,m) in (a,b), and a positive integer function p such that ∑i=1mp(βi)≤r, T0 has a self-adjoint extension T̃ such that each βi  (i=1,2,…,m) is an eigenvalue of T̃ with multiplicity at least p(βi)

Effect of n-3 polyunsaturated fatty acids on dopaminergic neurons in substantia nigra, brain inflammatory response and behavior in mice with Parkinson’s disease

Purpose: To examine the effect of n-3 polyunsaturated fatty acids (PUFAs) on dopaminergic neurons in substantia nigra, intracerebral inflammatory response and ethology in mice with Parkinson’s disease (PD). Methods: Four groups of male C57BL/6 mice (n = 48) were used: normal control, negative control, n3PUFA, and Madopa groups. Except for normal control group, all groups were given 6- hydroxydopamine hydrochloride (6-OHDA) to establish Parkinson’s mice model. The expressions of tyrosine hydroxylase (TH) and calcium-binding protein (CB) in substantia nigra dopaminergic neurons were determined with immunohistochemistry and Western blot. The contents of nitric oxide (NO), tumor necrosis factor (TNF-α) and interferon γ (IFN-γ) (indices of intracerebral inflammatory response) were measured. Tremor paralysis, moving grid number, standing times, swimming ability, and the number of rollers in each group were observed as indices of ethology. Results: The number of TH and CB-positive neurons in the substantia nigra of n-3PUFA-treated mice was significantly increased, relative to those in Madopa-treated mice (p < 0.05). The expressions of TH and CB proteins in substantia nigra in n-3PUFA group were markedly higher than the corresponding expressions in Madopa-treated mice (p < 0.05). Decreased levels of NO, TNF-α and IFN-γ levels were seen in 3PUFA group, when compared to mice in Madopa group, but higher behavioral scores were obtained in n-3PUFA-treated mice, relative to Madopa-treated mice (p < 0.05). Conclusion: The n-3PUFAs protect substantia nigra compact dopaminergic neurons against Parkinson’s disease, alleviate immune inflammation, and improve the coordination of limb movement. Thus, n-3PUFAs have potential therapeutic application in the management of Parkinson’s disease

2,2′-Bis(prop-2-yn­yloxy)-1,1′-binaphth­yl

In the title compound, C26H18O2, the mol­ecule is located on a twofold rotation axis. The two naphthyl ring planes in the mol­ecule are nearly perpendicular to each other [dihedral angle = 82.42 (1)°. No classical hydrogen bonds or aromatic π–π stacking inter­actions were observed

Elastic properties of phases with nonspherical nuclei in dense matter

We consider the elastic constants of phases with nonspherical nuclei, so-called pasta phases, predicted to occur in the inner crust of a neutron star. First, we treat perfectly ordered phases and give numerical estimates for lasagna and spaghetti when the pasta elements are spatially uniform: the results are in order-of-magnitude agreement with the numerical simulations of Caplan, Schneider, and Horowitz, Phys. Rev. Lett. 121, 132701 (2018). We then turn to pasta phases without long-range order and calculate upper (Voigt) and lower (Reuss) bounds on the effective shear modulus and find that the lower bound is zero, but the upper bound is nonzero. To obtain better estimates, we then apply the self-consistent formalism and find that this predicts that the shear modulus of the phases without long-range order is zero if the pasta elements are spatially uniform. In numerical simulations, the pasta elements are found to be modulated spatially and we show that this modulation is crucial to obtaining a nonzero elastic moduli for pasta phases without long-range order. In the self-consistent formalism we find that, for lasagna, the effective shear modulus is linear in the elastic constants that do not vanish when the pasta elements are spatially uniform while, for spaghetti, it varies as the square root of these elastic constants. We also consider the behavior of the elastic constant associated with a homologous strain (hydrostatic compression) of the structure of the pasta phases without long-range order.Comment: 9 pages, 6 figure