A note on simultaneous Diophantine approximation on planar curves

Let denote the set of simultaneously - approximable points in and denote the set of multiplicatively ψ-approximable points in . Let be a manifold in . The aim is to develop a metric theory for the sets and analogous to the classical theory in which is simply . In this note, we mainly restrict our attention to the case that is a planar curve . A complete Hausdorff dimension theory is established for the sets and . A divergent Khintchine type result is obtained for ; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on of is full. Furthermore, in the case that is a rational quadric the convergent Khintchine type result is obtained for both types of approximation. Our results for naturally generalize the dimension and Lebesgue measure statements of Beresnevich et al. (Mem AMS, 179 (846), 1–91 (2006)). Moreover, within the multiplicative framework, our results for constitute the first of their type. The research of Victor V. Beresnevich was supported by an EPSRC Grant R90727/01. Sanju L. Velani is a Royal Society University Research Fellow. For Iona and Ayesha on No. 3.     

组织蛋白酶B响应的超极化129Xe MRI探针对肺癌细胞的超灵敏探测

组织蛋白酶B（Cat B）是一种溶酶体半胱氨酸蛋白酶,在细胞代谢中起重要作用.已有研究表明Cat B在肺癌细胞中会过表达.因此,细胞内Cat B水平的检测非常重要.迄今为止,细胞内Cat B的检测方法主要为荧光成像,但该技术受限于渗透性和自发荧光背景干扰.为了解决这些问题,我们设计了一种基于超极化¹²⁹Xe磁共振成像的新型探针.它由一个作为¹²⁹Xe核磁共振（NMR）报告基团的穴番分子笼和一个作为Cat B特异性可裂解基团的酰胺键组成.当探针与Cat B相互作用时,酰胺键的断裂会导致其¹²⁹Xe化学位移发生变化.结合超极化-化学交换饱和转移（Hyper-CEST）技术,可为Cat B提供一种新颖的检测方法.     

Continuous Re‐hyperpolarization of Nuclear Spins Using Parahydrogen: Theory and Experiment

The continuous re‐hyperpolarization of nuclear spins in the liquid state by means of parahydrogen (para‐H₂) and chemical exchange at low magnetic fields was recently discovered and offers intriguing perspectives for many varieties of magnetic resonance. In this contribution, we provide a theoretical assessment of this effect and compare the results to experimental data. A distinct distribution of polarization is found, which shares some features with experimental data and, interestingly, does not directly correspond to the loss of the singlet order of para‐H₂. We derived expressions for the magnetic field and para‐H₂–substrate interaction time, for which the polarization transfer is maximal. This work sheds light onto the effect of continuous hyperpolarization and elucidates the underlying mechanism, which may facilitate the development of an optimized catalyst. As an application, continuous hyperpolarization may enable highly sensitive nuclear magnetic resonance at very low magnetic fields, for example, for the cost‐efficient screening of drugs.     

Analytic solutions in the dyon black hole with a cosmic string: Scalar fields,Hawking radiation and energy flux

Charged massive scalar fields are considered in the gravitational and electromagnetic field produced by a dyonic black hole with a cosmic string along its axis of symmetry. Exact solutions of both angular and radial parts of the covariant Klein–Gordon equation in this background are obtained, and are given in terms of the confluent Heun functions. The role of the presence of the cosmic string in these solutions is showed up. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation spectrum and the energy flux.     

Exact solutions of nonlinear time fractional partial differential equations by sub‐equation method

In this article, the sub‐equation method is presented for finding the exact solutions of a nonlinear fractional partial differential equations. For this, the fractional complex transformation method has been used to convert fractional‐order partial differential equation to ordinary differential equation. The fractional derivatives are described in Jumarie's the modified Riemann–Liouville sense. We apply to this method for the nonlinear time fractional differential equations. With the aid of symbolic computation, a variety of exact solutions for them are obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Casimir effect in the Hořava–Lifshitz gravity with a cosmological constant

We calculate the Casimir energy of a massless scalar field confined between two nearby parallel plates formed by ideal uncharged conductors, placed tangentially to the surface of a sphere with mass M$M$ and radius R$R$. To this end, we take into account a static and spherically symmetric solution of Ho?ava–Lifshitz (HL) gravity, with a cosmological constant term, in lower orders of approximation, considering both weak-field and infrared limits. We show that the Casimir energy, just in the second order weak-field approximation, is modified due to the parameter of the HL gravity as well as to the cosmological constant.     

An Existence Proof for the Gravitating BPS Monopole

We prove the existence of the gravitating BPS monopole in Einstein-Yang-Mills-Higgs (EYMH) theory. Existence is established using a Newtonian perturbation argument which shows that a Yang-Mills-Higgs BPS monopole solution can be be continued analytically in powers of 1/c² to an EYMH solution. Communicated by Sergiu Klainerman submitted 2/04/04, accepted 29/08/05     

Error analysis of finite element approximations of the inverse mean curvature flow arising from the general relativity

This paper proposes and analyzes a finite element method for a nonlinear singular elliptic equation arising from the black hole theory in the general relativity. The nonlinear equation, which was derived and analyzed by Huisken and Ilmanen in (J Diff Geom 59:353–437), represents a level set formulation for the inverse mean curvature flow describing the evolution of a hypersurface whose normal velocity equals the reciprocal of its mean curvature. We first propose a finite element method for a regularized flow which involves a small parameter ɛ; a rigorous analysis is presented to study well-posedness and convergence of the scheme under certain mesh-constraints, and optimal rates of convergence are verified. We then prove uniform convergence of the finite element solution to the unique weak solution of the nonlinear singular elliptic equation as the mesh size h and the regularization parameter ɛ both tend to zero. Computational results are provided to show the efficiency of the proposed finite element method and to numerically validate the “jumping out” phenomenon of the weak solution of the inverse mean curvature flow. Numerical studies are presented to evidence the existence of a polynomial scaling law between the mesh size h and the regularization parameter ɛ for optimal convergence of the proposed scheme. Finally, a numerical convergence study for another approach recently proposed by R. Moser (The inverse mean curvature flow and p-harmonic functions. preprint U Bath, 2005) for approximating the inverse mean curvature flow via p-harmonic functions is also included.     

Some extended analyses concerning the physics and kinematics of wave propagation in moving systems

For studying the details of the physical processes concerning space-time relations of signal exchanges in moving inertial-systems, it is purposeful to make at first a short exposition of the kinematics of electromagnetic wave propagation from a moving source to a stationary or moving field point in the free unbounded space. An introductory analysis on the kinematics of wave propagation from a source to an observer in a moving inertial system follows thereafter, primarily for elucidating the natural causality of the source and the observer remaining centered at the common midpoint of the radiating waves, without undergoing any relative drift. Special emphasis is then put to study the space-time relations of wave propagation from a source to an observer in the moving inertial system. The use of the emitting and receiving signal-cones in the moving system, with their common plane-cut, provides all the details with convincing insights concerning the causality and effect, both in the absolute space and in the Lorentz-space. The essential findings are elucidated with application to numerical examples and corresponding illustrations.