We consider the problem of subharmonic solutions and unbounded solutions for a class of indefinite sublinear equations of the form x ''+f(t,x)=p(t). We overcome the difficulty due to the indefiniteness of the nonlinearity and establish a new approach to study the spiral property of solutions through precise estimations of increments and geometric analysis on the phase plane. Focusing on a typical model equation x ''+a(t)x(1/3)=0 with a(t) being sign-changing and 2 pi-periodic, we can show that it has infinitely many subharmonic solutions when integral(2 pi)(0)a(t)dt>0, with the amplitude becoming larger and larger, and its large amplitude solutions are unbounded when integral(2 pi)(0)a(t)dt<0. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper we are concerned with the existence of invariant curves of planar twist mappings which are almost periodic in a spatial variable. As an application of this result to differential equations we will prove the existence of almost periodic solutions and the boundedness of all solutions for superlinear Duffing’s equation with an almost periodic external force.

Multiple lines of evidence have indicated that pyruvate kinase M2 (PKM2) is upregulated in most cancer cells and it is increasingly recognized as a potential therapeutic target in oncology. In a continuation of our discovery of lead compound 5 and SAR study, the 7-azaindole moiety in compound 5 was systematically optimized. The results showed that compound 6f, which has a difluoroethyl substitution on the 7-azaindole ring, exhibited high PKM2 activation potency and anti-proliferation activities on A375 cell lines. In a xenograft mouse model, oral administration of compound 6f led to significant tumor regression without obvious toxicity. Further mechanistic studies revealed that 6f could influence the translocation of PKM2 into nucleus, as well as induction of apoptosis and autophagy of A375 cells. More importantly, compound 6f significantly inhibited migration of A375 cells in a concentration-dependent manner. Collectively, 6f may serve as a lead compound in the development of potent PKM2 activators for cancer therapy.

We study the persistence of lower-dimensional invariant tori for a nearly integrable completely degenerate Hamiltonian system. It is shown that the majority of unperturbed invariant tori can survive from the perturbations which are only assumed the smallness and smoothness.

In this paper, we are concerned with the existence of lower dimensional invariant tori in nearly integrable reversible systems. By KAM method, we prove that under some reasonable assumptions, there are many so-called degenerate lower dimensional invariant tori, that is one of normal frequencies is zero.

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.

The distal head of the natural orifice transluminal endoscopic surgery (NOTES) platform commonly uses the structure of a snake bone, which cannot rotate, and the manufacturing is often time-consuming. A novel rotatable, one-element snake bone for NOTES is proposed. This paper first describes the movement mechanism and actuation. The new structure, which is composed of hinge pairs for bending and track-sled rings for rotation, was designed to reach a 90 deg bending angle and 62 deg rotational angle. The workspace of the snake bone was derived using screw theory and was simulated on matlab. The relationship between the angle and wire displacement was analyzed in detail. The new snake bone system bent and rotated by manipulating control wires that were actuated by DC motors, and its angular movements were measured by motion sensors with an angle error within ±2.6 deg. The snake bone was mounted on a flexible tube, inserted into a colonoscopy model, and navigated by motor actuation to eventually reach the cecum. The experimental results demonstrate the new snake bone's ability to travel through a natural orifice by rotating and bending, which satisfies the mobility requirement for NOTES.

In this paper we first prove the so-called large twist theorem, then using it to prove the boundedness of all solutions and the existence of quasi-periodic solutions for Duffing's equation $$ \ddot{x}+x^{2n+1}+\dsum_{i=0}^{2n}p_i(t)x^i=0, $$ where $p_i(t)\in C^1(\mathbb{S}) (n+1\leq i\leq 2n)$ and $p_i(t)\in C^0(\mathbb{S}) (0\leq i\leq n)$ with $\mathbb{S}=\mathbb{R}/\mathbb{Z}$.

In this paper we study the dynamical behavior of the differential equationx″+ax+−bx−=f(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, a and b are two different positive constants, f(t) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11]). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation.

A wide range of Rh(iii)-catalyzed ortho-olefinated 7-azaindole derivatives as well as novel tetracyclic heterorings were achieved, which could served as useful starting materials for the construction of biological molecules.

A novel and efficient one-pot procedure for the synthesis of S-vinyl dithiocarbamates from electron deficient allenes, amines and CS2 was presented. The reactions proceed at room temperature for 10-30 min without any catalyst to afford the products in high yields, excellent regioselectivity and stereoselectivity. (C) 2016 Elsevier Ltd. All rights reserved.

Under suitable conditions on the weight functions b, this paper shows the exact asymptotic behavior of the unique solution l(t) at zero to the singular boundary value problem u^'' (t) = b(t)f(u(t)), u(t) > 0, t > 0, u(0) = ∞, u(∞) = 0, where b ∈ C^1(0,∞) which is positive and non-decreasing on (0,∞) (may vanish at zero). We assume that f ( u ) does not grow like u p with p > 1 or faster at infinity, but behaves like u ln α u as u→∞ for some α > 1 .

In this paper, the authors are concerned with the forced isochronous oscillators with a repulsive singularity and a bounded nonlinearityx '' + V'(x) + g( x) = e(t, x, x'),where the assumptions on V, g and e are regular, described precisely in the introduction. Using a variant of Moser's twist theorem of invariant curves, the authors show the existence of quasi-periodic solutions and boundedness of all solutions. This extends the result of Liu to the case of the above system where e depends on the velocity.

Herein we reported a Pd-catalyzed ortho-acetoxylation of arenes through a six-membered system using an amide-oxazoline as the directing group. A wide range of aryls and heteroaryls are tolerated. The approach provides general and straightforward access to phenol esters without the need for extra oxidants.

In this paper, we analyze the second order expansion for the unique solution near the boundary to the singular Dirichlet problem −▵u=b(x)g(u),u>0,x∈Ω,u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, g ∈ C1((0, ∞), (0, ∞)), g is decreasing on (0, ∞) with lims→0+g(s)=∞ and g is normalized regularly varying at zero with index −γ (γ > 1), b∈Clocα(Ω) (0 < α < 1), is positive in Ω, may be vanishing or singular on the boundary and belongs to the Kato class K(Ω). Our analysis is based on the sub-supersolution method and Karamata regular variation theory.

In this paper, we mainly study the asymptotic behavior of solutions to the following problems \({\triangle u \pm a(x)| \nabla u|^{q} = b(x)f(u), x \in \Omega, \ u|_{\partial \Omega} = + \infty}\), where Ω is a bounded domain with a smooth boundary in \({\mathbb{R}^{N} (N \geq 2)}\), q >  0, \({a \in C^{\alpha}(\bar{\Omega})}\) is positive in Ω, and \({b \in C^{\alpha}(\bar{\Omega})}\) is nonnegative in Ω and may be vanishing on the boundary. We assume that f is Γ-varying at ∞, whose variation at ∞ is not regular. Our analysis is based on the sub-supersolution method and Karamata regular variation theory.

We study the second order estimate for the unique solution near the boundary to the singular Dirichlet problem -?u = b(x)g(u); u > 0; x ? ?, u|?? = 0, where ? is a bounded domain with smooth boundary in RN, g ? C1((0,?),(0?)), g is decreasing on (0,?) with lim s?0+ g(s) = 1 and g is normalized regularly varying at zero with index ? (? > 1), b ? C?(??) (0 < ? < 1), is positive in ?, may be vanishing on the boundary. Our analysis is based on Karamata regular variation theory.

In this paper, we mainly study the second order expansion of solutions to the elliptic problems △u=b(x)f(u),u>0,x∈Ω,u|∂Ω=∞, where Ω is a bounded domain with a smooth boundary in RN(N≥2), b∈Cα(Ω̄) which is positive in Ω and may be vanishing on the boundary and f is normalized regularly varying at infinity with index p>1. Our analysis is based on the sub-supersolution method and Karamata regular variation theory.

Two dinuclear lanthanide (Ln) complexes, formulated as [phen2Ln2(HCOO)4(HCOO)2−2x (NO3)2x ] (1, Ln = Gd and x = 0.52; 2, Ln = Er and x = 0.90; phen = 1,10-phenanthroline), were synthesized and characterized. They are isostructural. The dinuclear molecule consists of two Ln3+ bridged by four formate groups and chelated by phen and formate/nitrate ligands, and the Ln3+ possesses a coordination environment of distorted tri-capped trigonal prism of LnO7N2. Both compounds behave as paramagnets between 2 and 300 K, but display two static field induced magnetic relaxation processes. One is slow and of spin-lattice type, and it results from the lifting of Kramer’s degeneracy of the ground-states of both Gd3+ and Er3+, and the other is fast, and it might be spin-spin type.