Paramagnetic resonance of Gd3+ in SmCl3·6H2O single crystals

The Paramagnetic Resonance of Gd³⁺ in SmCl₃·6H₂O single crystals, grown from solution, is studied at room temperature. A six line spectrum for H//Z and a seven line spectrum for H//X corresponding to ΔM=± 1 transitions are observed. Their angular variation in ZX plane from?=0° to?=90°, is studied and the spin-Hamiltonian analysis is presented. The probable amount of admixture of the next higher electronic state⁶P_7/2 with the ground state⁸S_7/2 is also estimated.     

EPR spectrum of gadolinium in hydrated praseodymium nitrate

The paramagnetic resonance spectrum of Gd³⁺ in Pr(NO₃)₃·6H₂O single crystals, is studied at room temperature. A seven line spectrum for H//Z as well as for H//X corresponding to ΔM=±1 transitions is observed along with a number of low field transitions (ΔM?2). The spin-Hamiltonian analyses is presented.     

The unit ball of the complex P(3H){{\mathcal P}(^3H)}

Let H be a two-dimensional complex Hilbert space and P(³H){{\mathcal P}(^3H)} the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, B_P(³H){{\mathsf B}_{{\mathcal P}(^3H)}} , from which we deduce that the unit sphere of P(³H){{\mathcal P}(^3H)} is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B_P(³H){{\mathsf B}_{{\mathcal P}(^3H)}} remains extreme as considered as an element of B_L(³H){{\mathsf B}_{{\mathcal L}(^3H)}} . Finally we make a few remarks about the geometry of the unit ball of the predual of P(³H){{\mathcal P}(^3H)} and give a characterization of its smooth points.     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).     

A regularity criterion for the solutions of 3D Navier-Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R³. More precisely, let u=(u¹,u²,u³) be a weak solution in (0,T)×R³. Then u becomes a classical solution if any two functions of _∂1u¹, _∂2u² and _∂3u³ belong to Lθ(0,T;Lr(R³)) provided with , .     

U(1)-invariant special Lagrangian 3-folds. I. Nonsingular solutions

This is the first of three papers studying special Lagrangian 3-submanifolds (SL 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^−iθz₂,z₃), using analytic methods.Let N be such a U(1)-invariant SL 3-fold. Then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy. When a is nonzero, u,v are always smooth and N is always nonsingular. But if a=0, there may be points (x,0) where u,v are not differentiable, which correspond to singular points of N.This paper focusses on the nonsingular case, when a is nonzero. We prove analogues for our nonlinear Cauchy-Riemann equation of well-known results in complex analysis. In particular, we prove existence and uniqueness for solutions of two Dirichlet problems derived from it. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in , with two kinds of boundary conditions. The sequels extend these to the case a=0, study the singularities of the SL 3-folds that arise, and construct special Lagrangian fibrations of open sets in .     

Rotational analysis of the 0-0 band of the A3 ∏ inv-X3 ∑ − system of ND

The 0-0 band of the A³ ∏ _inv?X³ ∑ ^? system of ND was excited in an electrodeless microwave oscillator (2450 Mc./s.) discharge and photographed on a 6·6 meter concave grating spectrograph in the second order at a dispersion of 0·56 A/mm. Twenty-five out of the predicted twenty-seven branches have been identified in the rotational structure of the 0-0 band. From a rotational analysis, the following rotational constants have been determined: (i)X ³ ∑ ^? state—

     

U(1)-invariant special Lagrangian 3-folds. III. Properties of singular solutions

This is the last of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case a nonzero, and proved existence and uniqueness for solutions of two Dirichlet problems derived from the nonlinear Cauchy-Riemann equation. This yields existence and uniqueness of a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions. The second paper extended these results to weak solutions of the Dirichlet problems when a=0, giving existence and uniqueness of many singular U(1)-invariant SL 3-folds in , with boundary conditions.This third paper studies the singularities of these SL 3-folds. We show that under mild conditions the singularities are isolated, and have a multiplicityn>0, and one of two types. Examples are constructed with every multiplicity and type. We also prove the existence of large families of U(1)-invariant special Lagrangian fibrations of open sets in , including singular fibres.     

LOWER BOUNDS FOR SUP＋INF AND SUP＊INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

We give two results about Harnack type inequalities. First, on Riemannian surfaces, we have an estimate of type sup ＋ inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian constant A of V tends to 0 （in certain meaning）, we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.     

U(1)-invariant special Lagrangian 3-folds. II. Existence of singular solutions

This is the second of three papers studying special Lagrangian 3-submanifolds (SLV 3-folds) N in invariant under the U(1)-action e^iθ:(z₁,z₂,z₃)?(e^iθz₁,e^−iθz₂,z₃), using analytic methods. If N is such a 3-fold then |z₁|²−|z₂|²=2a on N for some . Locally, N can be written as a kind of graph of functions satisfying a nonlinear Cauchy-Riemann equation depending on a, so that u+iv is like a holomorphic function of x+iy.The first paper studied the case when a is nonzero. Then u,v are smooth and N is nonsingular. It proved existence and uniqueness for solutions of two Dirichlet problems derived from the equations on u,v. This implied existence and uniqueness for a large class of nonsingular U(1)-invariant SL 3-folds in , with boundary conditions.In this paper and its sequel we focus on the case a=0. Then the nonlinear Cauchy-Riemann equation is not always elliptic. Because of this there may be points (x,0) where u,v are not differentiable, corresponding to singular points of N. This paper is concerned largely with technical analytic issues, and the sequel with the geometry of the singularities of N. We prove a priori estimates for derivatives of solutions of the nonlinear Cauchy-Riemann equation, and use them to show existence and uniqueness of weak solutions u,v to the two Dirichlet problems when a=0, which are continuous and weakly differentiable. This gives existence and uniqueness for a large class of singular U(1)-invariant SL 3-folds in , with boundary conditions.     

Exhaustive weakly wandering sequences

A sequence of integers {n_i : i = 0, 1…} is an exhaustive weakly wandering sequence for a transformation T if for some measurable set W, X=^∞_i=0TⁿⁱW(disj. We introduce a hereditary Property (H) for a sequence of integers associated with an infinite ergodic transformation T, and show that it is a sufficient condition for the sequence to be an exhaustive weakly wandering sequence for T. We then show that every infinite ergodic transformation admits sequences that possess Property (H), and observe that Property (H) is inherited by all subsequences of a sequence that possess it. As a corollary, we obtain an application to tiling the set of integers with infinite subsets.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Convex-transitivity and function spaces

It is shown that if X is a convex-transitive Banach space and 1p<∞, then L^p([0,1],X) and are convex-transitive. Here is the closed linear span of the simple functions in the Bochner space L^∞([0,1],X). If H is an infinite-dimensional Hilbert space and C₀(L) is convex-transitive, then C₀(L,H) is convex-transitive. Some new fairly concrete examples of convex-transitive spaces are provided.     

Uniqueness of weak solutions in critical space of the 3‐D time‐dependent Ginzburg‐Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3‐D time‐dependent Ginzburg‐Landau equations for super‐conductivity with initial data (ψ₀, A₀)∈ L² under the hypothesis that (ψ, A) ∈ L^s(0, T; L^r,∞) × (0, T; with Coulomb gauge for any (r, s) and satisfying + = 1, + = 1, ≥ , ≥ and 3 < r ≤ 6, 3 < ≤ ∞. Here L^r,∞ ≡ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ₀ ∈ , A₀ ∈ L³ (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Classification Theorems for General Orthogonal Polynomials on the Unit Circle

The set of all probability measures σ on the unit circle splits into three disjoint subsets depending on properties of the derived set of {|_n|²dσ}_n0, denoted by Lim(σ). Here {_n}_n0 are orthogonal polynomials in L²(dσ). The first subset is the set of Rakhmanov measures, i.e., of σ with {m}=Lim(σ), m being the normalized (m( )=1) Lebesgue measure on . The second subset Mar( ) consists of Markoff measures, i.e., of σ with mLim(σ), and is in fact the subject of study for the present paper. A measure σ, belongs to Mar( ) iff there are >0 and l>0 such that sup{|a_n+j|:0jl}>, n=0,1,2,…,{a_n} is the Geronimus parameters (=reflectioncoefficients) of σ. We use this equivalence to describe the asymptotic behavior of the zeros of the corresponding orthogonal polynomials (see Theorem G). The third subset consists of σ with {m}Lim(σ). We show that σ is ratio asymptotic iff either σ is a Rakhmanov measure or σ satisfies the López condition (which implies σMar( )). Measures σ satisfying Lim(σ)={ν} (i.e., weakly asymptotic measures) are also classified. Either ν is the sum of equal point masses placed at the roots of zⁿ=λ, λ , n=1,2,…, or ν is the equilibrium measure (with respect to the logarithmic kernel) for the inverse image under an m-preserving endomorphism z→zⁿ, n=1,2,…, of a closed arc J (including J= ) with removed open concentric arc J₀ (including J₀=). Next, weakly asymptotic measures are completely described in terms of their Geronimus parameters. Finally, we obtain explicit formulae for the parameters of the equilibrium measures ν and show that these measures satisfy {ν}=Lim(ν).