Neumann-like integrable models

A countable class of integrable dynamical systems, with four-dimensional phase space and conserved quantities in involution (Hn,In)$(H_n,I_n)$ are exhibited. For n=1$n=1$ we recover Neumann system on T^∗S²$T^{\ast }{S}^2$. All these systems are also integrable at the quantum level.     

An Electron Paramagnetic Resonance Study of the Orbital Ordering Compound MgTi<Subscript>2</Subscript>O<Subscript>4</Subscript>

We report the electron paramagnetic resonance (EPR) studies of MgTi₂O₄ in the 300–140 K range. Above the transition temperature T _t (~258 K), the EPR results indicate that MgTi₂O₄ is paramagnetic. The parameters of the EPR spectra show an anomalous change at T _t. The clear EPR lines can be observed in temperature between T _t and 220 K. Besides that the EPR intensity, g value, and EPR linewidth increase with decreasing temperature; in temperature range below 220 K, no clear EPR line can be detected. The EPR spectra results demonstrate that magnetic spin-singlet state and the orbital density wave of MgTi₂O₄ system are formed gradually with decreasing temperature at low temperature range.     

Firing-Induced Microstructural Properties of Quasi-Diamagnetic Carbonate-Based Porous Ceramics: a <Superscript>1</Superscript>H NMR Relaxation Correlation Study

This study deals with the application of two-dimensional proton nuclear magnetic resonance relaxometry (2D ¹H NMR-R) to the characterization of porous ceramics nearly free of magnetic compounds. Different microstructural properties were obtained by firing a diamagnetic mixture of kaolin, calcium, and magnesium carbonate over a wide range of maximum temperatures (600–1100 °C) and firing times at the maximum temperature (soaking times) (0–10 h). The 2D ¹H NMR-R method relies on the correlated measurement of ¹H longitudinal (T ₁) and transverse (T ₂) relaxation times of pore-filling water by which the properties of the interconnected pore space may be investigated. In the absence of significant magnetic susceptibility effect due to para- and ferro-magnetic compounds, the 2D ¹H NMR-R maps allow studying the conjoint effects on pore size distribution and inter-pore coupling due to the variations in both time and temperature of firing. The NMR experiments were performed with a low-field ¹H NMR sensor, which allows non-destructive and in situ analysis. For ceramic specimens fired at 600 and 700 °C, the fraction of smallest pores increases with firing time at the expenses of those with intermediate size. The pore shrinkage occurring at this stage, and likely associated with the transformation of kaolinite in metakaolinite, is affected in a similar way by soaking time and firing temperature, in line with the concept of equivalent firing temperature. At temperatures from 800 to 1100 °C, the structural modifications involving interconnectivity and average pore size are driven primarily by firing temperature and, secondarily, by soaking time. The 2D ¹H NMR-R results are confirmed by more traditional, but destructive, mineralogical, and structural analyses like X-ray powder diffraction, helium pycnometry, mercury intrusion porosimetry, and nitrogen adsorption/desorption method.     

Site-Specific Detection of Free Radicals in Membranes Using an Amphiphilic Spin Trap

The detection of free radicals and related species has attracted considerable attention in recent years due to their critical roles in physiological and pathological processes. Among the various methods for the detection of free radicals, electron spin resonance coupled with spin-trapping technique has been an effective approach for the characterization and quantification of free radicals due to its high specificity. In this study, we designed and synthesized a novel amphiphilic spin trap, 2-(diethoxyphosphoryl)-2-heptadecanyl-3,4-dihydro-2H-pyrrole-1-oxide (DEPHdPO), from 5-(diethoxyphosphoryl)-5-methyl-1-pyrroline-N-oxide with a long hydrocarbon chain at the C-5 position of the pyrroline ring, providing the amphiphilic character. The free-radical-trapping ability of DEPHdPO was evaluated by capturing hydroxyl radicals (^·OH), superoxide anions (\( {\text{O}}_{2}^{ \cdot - } \)), and carbon-centered free radicals in a model membrane prepared from sodium dodecyl sulfate (SDS). The results indicate that the hydrophobic hydrocarbon chain of DEPHdPO can be inserted into the inner core of SDS micelles, and the hydrophilic nitronyl functional moiety is located on the surface layer. Thus, various free radicals, including ^·OH radicals, \( {\text{O}}_{2}^{ \cdot - } \) anions, and carbon-centered radicals could be site-specifically detected near the membrane surface. Moreover, DEPHdPO could be successfully located on the surface of thylakoid membranes, and the nearby photo-initiated \( {\text{O}}_{2}^{ \cdot - } \) anions could be trapped site-specifically.     

Global continuum of positive solutions for discrete <Emphasis Type="Italic">p</Emphasis>-Laplacian eigenvalue problems

We discuss the discrete p-Laplacian eigenvalue problem,

$$\left\{ \begin{gathered} \Delta (\phi _p (\Delta u(k - 1))) + \lambda a(k)g(u(k)) = 0,k \in \{ 1,2,...,T\} , \hfill \\ u(0) = u(T + 1) = 0, \hfill \\ \end{gathered} \right.$$

where T > 1 is a given positive integer and φ _p (x):= |x| ^p?2 x, p > 1. First, the existence of an unbounded continuum C of positive solutions emanating from (λ, u) = (0, 0) is shown under suitable conditions on the nonlinearity. Then, under an additional condition, it is shown that the positive solution is unique for any λ > 0 and all solutions are ordered. Thus the continuum C is a monotone continuous curve globally defined for all λ > 0.

     

A well-balanced and asymptotic-preserving scheme for the one-dimensional linear Dirac equation

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the \(L^2\) norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of \(c\), the speed of light. Moreover, when \(c\) diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a “diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that \(c \Delta x\rightarrow 0\)). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed.     

Superstability of the Cauchy equation with squares in finite-dimensional normed algebras

Our purpose is to provide an affirmative answer to Moszner’s problem [cf. Moszner (Ann Univ Paed Crac Stud Math XI:69–94, 2012), p. 93] concerning the superstability of the Cauchy equation with squares

$$f(x + y)^2 = (f(x) + f(y))^2$$

in the class of functions mapping an Abelian semigroup into a finite-dimensional normed algebra without divisors of zero.

     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.     

A Rolle Type Theorem for Cyclicity of Zeros of Families of Analytic Functions

Let \({\{ {f_{\lambda ;j}}\} _{\lambda \in V;1 \leqslant j \leqslant k}}\) be families of holomorphic functions in the open unit disk \({\text{D}} \subset {\Bbb C}\) ? ? depending holomorphically on a parameter λ ∈ V ? ? ⁿ . We establish a Rolle type theorem for the generalized multiplicity (called cyclicity) of zeros of the family of univariate holomorphic functions \({\left\{ {\sum\nolimits_{j = 1}^k {{f_{\lambda ;j}}} } \right\}_{\lambda \in V}}\) at 0 ∈ D. As a corollary, we estimate the cyclicity of the family of generalized exponential polynomials, that is, the family of entire functions of the form \(\sum\nolimits_{k = 1}^m {{P_k}(z){e^{{Q_k}(z)}}} \), z ∈ ?, where P _k and Q _k are holomorphic polynomials of degrees p and q, respectively, parameterized by vectors of coefficients of P _k and Q _k .