Existence of homoclinic solutions for nonlinear second-order coupled systems

This work presents sufficient conditions for the existence of homoclinic solutions for second order coupled discontinuous systems of differential equations on the real line without the usual growth condition in the literature.The arguments apply the fixed point theory, Green's functions technique, $L^1$-Carathéodory functions, lower and upper solutions and Schauder's fixed point theorem.     

A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Panisuffrutin A,a highly degraded seco-triterpene derivative from Paeonia suffruticosa var. papaveracea (Andr.) Kerner

Panisuffrutin A (1), a highly degraded seco-triterpene derivative, together with the known palbinone, has been isolated from the whole plant Paeonia suffruticosa var. papaveracea (Andr.) Kerner. The structure with absolute configuration of 1 was determined via comprehensive NMR and MS analyses, as well as NMR and ECD calculations. A plausible biosynthetic pathway for 1 was proposed. Compound 1 showed weak cytotoxicity against Hela cancer cell line with an IC₅₀ value of 26.2 μM, while palbinone exhibited a moderate inhibition on NO production with an IC₅₀ of 18.3 μM.     

Hyphenone A,the first 3,3-diisoprenylated bicyclic polyprenylated acylphloroglucinols as Cav3.1 T-type calcium channel inhibitor from Hypericum henryi

Hyphenone A (1), a new type of bicyclic polyprenylated acylphloroglucinols (BPAPs) featured with an unprecedented 3,3-diisoprenylated 1,3,5-trione core, were characterized from the roots of Hypericum henryi together with two new congeners (hyphenones B–C, 2–3) and five known biosynthetic related PPAPs. Biogenetically, 1 should be derived from a novel 3,3,5,5-tetraisoprenylated MPAPs precursor. Their structures were elucidated by comprehensive spectroscopic data and X-ray diffraction. Moreover, 1–3 were identified as the first PPAPs type Ca_v3.1 T-type calcium channel (TTCC) inhibitors, with IC₅₀ values of 7.07, 6.19, and 5.47 μM, respectively.     

A practical access to new pyrrolizine carboxylates via KHMDS-catalyzed carbocyclizations

Easy and effective preparation of new 1H-pyrrolizine carboxylates was achieved with high efficiency via KHMDS-induced carbocyclization of N-alkynyl proline carboxylates under substantially mild conditions. Meanwhile, some trans-diiodoallylic N-proline carboxylates were obtained from N-propargyl proline carboxylates using molecular I₂ with or without KHMDS. This method is quite feasible in terms of practical and quick access to the pyrrolizines and their derivatives over the formation of carbanions.     

Dayaolingzhiols A−E,AchE inhibitory meroterpenoids from Ganoderma lucidum

Ganoderma mushrooms are widely used as effective medicines for treating and preventing chronic diseases. In this study, five new meroterpenoids, dayaolingzhiols A (1) and B (2) featuring a 6/6/6 ring system, dayaolingzhiols C?E (3–5) harboring a γ-lactone motif, along with eight known ones (6–13), were purified from G. lucidum. To clarify their chemical structures, spectroscopic and ECD and OR computational methods were used. In addition, the inhibition of all the new meroterpenoids against AChE was evaluated, disclosing that (+)-4 and (±)-5 possess potent AChE inhibitory activities with respective IC₅₀ values of 8.52?±?1.90?μM and 7.37?±?0.52?μM.     

An anisotropic area-preserving flow for convex plane curves

This paper will deal with an anisotropic area-preserving flow which keeps the convexity of the evolving curve and the limiting curve converges to a homothety of a symmetric smooth strictly convex plane curve.     

Nash multiplicities and isolated points of maximum multiplicity

Let X be an algebraic variety defined over a field of characteristic zero, and let $\xi \in \underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$ be a point in the closed subset of maximum multiplicity of X. We provide a criterion, given in terms of arcs, to determine whether ξ is isolated in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$. More precisely, we use invariants of arcs derived from the Nash multiplicity sequence to characterize when ξ is an isolated point in $\underset{\_}{\mathrm{Max}}\mathrm{mult}(X)$.