固定边矩形板的弯曲问题

The problem of bending of orthotropic rectangular plates with clamped edges on elastic foundation may be reduced to the following differential equation and boundary conditions (?⁴w)/(?x⁴)+2λ(?⁴w)/(?x²?y²)+(?⁴w)/(?y⁴)+kw=q/D. w=0, (?w)/(?x)=0 at x=±a, w=0, (?w)/(?y)=0, at y=±b. In the case of isotropic plates, λ = 1. In this paper a perturbation method is proposed for the solution of this problem fay expanding w in power series of λ: w=w₀+w₁λ+w₂λ²+……. It is proved that this series is convergent when -1 ≤λ≤1.     

A new approach to Ermakov systems and applications in quantum physics

Milne–Pinney equation [(x)\ddot]=-w²(t)x+ k/x³\ddot x=-\omega^2(t)x+ k/{x^3} is usually studied together with the time-dependent harmonic oscillator [(y)\ddot]+w²(t) y=0\ddot y+\omega^2(t) y=0 and the system is called Ermakov system, and actually Pinney showed in a short paper that the general solution of the first equation can be written as a superposition of two solutions of the associated harmonic oscillator. A recent generalization of the concept of Lie systems for second order differential equations and the usual techniques of Lie systems will be used to study the Ermakov system. Several applications of Ermakov systems in Quantum Mechanics as the relation between Schroedinger and Milne equations or the use of Lewis–Riesenfeld invariant will be analysed from this geometric viewpoint.     

Low Temperature Properties for Correlation Functions in Classical N-Vector Spin Models

We obtain convergent multi-scale expansions for the one-and two-point correlation functions of the low temperature lattice classical N - vector spin model in d S 3 dimensions, N S 2. The Gibbs factor is taken as exp[-b(1/2 ||?f||² +l/8 || |f|² - 1 ||² + v/2||f- h||²)], \exp [-\beta (1/2 ||\partial \phi||^2 +\lambda/8 ||\, |\phi|^2 - 1 ||^2 + v/2||\phi - h||^2)], where f(x), h ? R^N\phi(x), h \in R^N, x ? Z^dx \in Z^d, |h|=1, b < ￥|h|=1, \beta < \infty, l 3 ￥\lambda \geq \infty are large and 0 < v h 1. In the thermodynamic and v ˉ 0v \downarrow 0 limits, with h = e₁, and j L ‘^½ ‘, the expansion gives áf₁(x)? = 1+0(1/b^1/2)\langle \phi_1(x)\rangle = 1+0(1/\beta^{1/2}) (spontaneous magnetization), áf₁(x)f_i(y)? = 0\langle \phi_1(x)\phi_i(y)\rangle=0, áf_i (x)f_i (y)? = c₀ D^-1(x,y)+R(x,y)\langle \phi_i (x)\phi_i (y)\rangle = c_0 \Delta^{-1}(x,y)+R(x,y) (Goldstone Bosons), i = 2, 3, ?, Ni= 2, 3,\,\ldots, N, and áf₁(x)f₁(y)?^T=R￠(x,y)\langle \phi_1(x)\phi_1(y)\rangle^T=R'(x,y), where |R(x,y)||R(x,y)|, |R￠(x,y)| < 0(1)(1+|x-y|)^d-2+r|R'(x,y)|< 0(1)(1+|x-y|)^{d-2+\rho} for some „ > 0, and c₀ is aprecisely determined constant.     

Co2+-Ti4+和Cu2+-Nb5+离子取代BaFe12O19单晶体中Fe3+离子的研究

本文对用Co²⁺-Ti⁴⁺和Cu²⁺-Nb⁵⁺离子取代BaFe₁₂O₁₉单晶体中Fe³⁺离子进行了研究,以Bi₂O₃作为助熔剂生长出了BaFe_12-2xCo_x²⁺Ti_x⁴⁺O₁₉(x=0;0.04;0.09;0.13;0.27和0.68)以及BaFe_12-x[Nb_1/3⁵⁺Cu_2/3²⁺]_xO₁₉(x=0;0.28;0.44和0.60)这两系列的单晶体,测定了100—300K温度范围内样品的磁化强度σ与单轴各向异性常数K₁,我们发现,对Co²⁺-Ti⁴⁺取代的样品,当x≤0.09时,其σ与K₁随x的增加而缓慢增加;当x>0.09时,其σ与K₁随x的增加而迅速降低,至x=1.1时,K₁变为零,对Nb⁵⁺-Cu²⁺取代的样品,其σ值在整个成份范围内基本保持不变,且有缓慢增加趋势;而K₁值则随x增加而单调下降,提出了取代离子在M型六角铁氧体中可能的分布模型来解释我们的结果。 关键词：     

Lagrangian formulation of a generalized Lane-Emden equation and double reduction

Abstract

We classify the Noether point symmetries of the generalized Lane-Emden equation y″+ ny′/x+ f(y)?=?0 with respect to the standard Lagrangian L = xⁿy′²/2 — xⁿ ∫f(y)dy for various functions f(y). We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f(y). One of them is f(y) = αy^r , where r ≠ 0,1. The case r?=?5 was considered previously and only a one-parameter family of solutions was presented. Here we provide a complete integration not only for r?= 5 but for other r values. We also give the Lie point symmetries for each case. In two of the new cases, the single Noether symmetry is also the only Lie point symmetry.     

Asymptotics for Large Time of Global Solutions to the Generalized Kadomtsev–Petviashvili Equation

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP where † = 1 or † = m 1. When „ = 2 and † = m 1, (KP) is known as the KPI equation, while „ = 2, † = + 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case „ = 3, † = m 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if „ S 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: ||u(t)||_￥ ￡ C (1 + |t|)^-1 (log(2+|t|))^k, ||u_x(t)||_￥ ￡ C (1 + |t|)^-1 \|u(t)\|_\infty \le C (1 + |t|)^{-1} (\log (2+|t|))^{\kappa}, \|u_x(t)\|_\infty \le C (1 + |t|)^{-1} for all t ] R, where s = 1 if „ = 3 and s = 0 if „ S 4. We also find the large time asymptotics for the solution.     

X-ray and pulsed NMR studies of lithium nitride chlorides in anti-fluorite type structures

Lithium nitride chloride, Li_2-2xN_0.5-xCl_0.5+x′ have been found to possess anti-fluorite type structures in the range 0.05 <x< 0.1, with Fm3m, and with the lattice parameter in the range 5.354 ≈ 5.375 Å. The ⁷Li relaxation was studied in the temperature range 178–625 K, and the activation energy for Li⁺ hopping motion, E_a = 0.19 eV was obtained for x = 1. Small variation of E_a with composition or with occupancy of the Li⁺ site was confirmed.     

Lithium ion conductivity of the B-site substitution in Li3xLa0.67−xRE y III Ti1−2yPyO3 system (REIII=Sc3+, Y3+, Nd3+, Sm3+, Eu3+, Yb3+)

The B-site substituted perovskite solid solution systems Li_3xLa_0.67−xRE_yTi_1−2yP_yO₃ (RE=Sc, Y, Nd, Sm, Eu, Yb) have been investigated. Perovskite solid solutions formed in the range of x=0.10, y<0.10 for RE=Sc³⁺, Y³⁺, Yb³⁺, x=0.10, y≤0.05 for RE=Nd³⁺, Sm³⁺, Eu³⁺. Li_0.3La_0.57Nd_0.05Ti_0.9P_0.05O₃ has the highest bulk conductivity of 4.31×10⁻⁴ S·cm⁻¹ and the highest total conductivity of 2.52×10⁻⁴ S·cm⁻¹ at room temperature in all prepared compositions. The compositions have low activation energies of about 24–30 kJ/mol in the temperature ranges of 298–523 K. SEM studies showed that the sample made by solid-state reaction has a sphere-like morphology and a rough particle with particle size of about 50 μm. The research results also indicated that the reaction temperature decreases and the electrochemical stabilities of the titanate-based perovskite-type solid solutions are improved by using RE³⁺ and P⁵⁺ replaced Ti⁴⁺ on B-site in the Li_3xLa_0.67−xTiO₃ parent.     

Airy函数乘积的积分

本文提出对∫xⁿA^(p)B^(q)dx型不定积分普遍方法,其中A,B都是含有参变量λ的Airy方程y″=(λ+x)y的解,p,q是微分阶数,n是非负整数。当p,q≤1,A与B所含的参数不相同时,本文导出了上述积分普遍公式。当A=B时,又提出了与Albright完全不相同的求积方法。作为它的应用,本文讨论了处于三角势阱中电子的波函数正交归一问题,并计算了势阱中能级间的跃迁几率。 关键词：     

Superconductivity and ferromagnetism in the orthorhombic system (TbxLu1−x)RuB2

The pseudoternary orthorhombic system Tb_xLu_1?xRuB₂) (0 ? x ? 0.1 andx = 1.0) has been investigated by static magnetization, ac susceptibility and resistivity measurements down to 1.5 K. Superconductivity occurs for 0 ? x ? 0.1, while ferromagnetic order occurs for x = 1.0. For each sample studied, the Tb ion maintains a configurationally stable trivalent state. The initial linear depression of the superconducting transition temperature yields a coupling constant value N(0)Γ² between conduction electrons and magnetic Tb³⁺ moments of 3.5 x 10^-4 ev-atom-states/spin-direction.     

Effect of lithium ion content on the lithium ion conductivity of the garnet-like structure Li<Subscript>5+x</Subscript>BaLa<Subscript>2</Subscript>Ta<Subscript>2</Subscript>O<Subscript>11.5+0.5x</Subscript> (<Emphasis Type="Italic">x</Emphasis> = 0–2)

We report systematic studies on the transport properties by varying the lithium oxide content of the garnet-based solid electrolyte Li_5+xBaLa₂Ta₂O_11.5+0.5x (x=0, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00) for understanding the ionic conductivity dependence on the crystal lattice parameter and carrier concentration. Powder X-ray diffraction data of Li_5+xBaLa₂Ta₂O_11.5+0.5x (x=0, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00) indicate the existence of the garnet-like structure for any of the compositions. The cubic lattice parameter was found to increase with increasing x and reaches a maximum at x=1.00, then decreases slightly with a further increase in x. Impedance measurements obtained at 50 °C indicate a maximum of the grain-boundary resistance (R_gb) contribution to the total resistance (R_b+R_gb) at x=0.0 and a considerable decrease with increase in lithium concentration. The total (bulk + grain-boundary) and bulk ionic conductivity increase with increasing lithium content and reach a maximum at x=1.00 and then decrease slightly with further increase in x. Among the investigated compounds, Li₆BaLa₂Ta₂O₁₂ exhibits the highest total (bulk + grain-boundary) and bulk ionic conductivity of 1.5×10^-4 and 1.8×10^-4 S/cm at 50 °C, respectively. The results obtained in the present investigation of the Li_5+xBaLa₂Ta₂O_11.5+0.5x (x=0–2) series clearly revealed that the lithium content plays a major role in decreasing the grain boundary resistance contribution to the total resistance and also in increasing the ionic conductivity of the garnet-like compound. PACS 66.10.Ed; 82.45.Gj; 82.47.Aa     

Magnetic interaction in Mg, Ti, Nb doped manganites

An effect of Mn substitution with Me=Mg²⁺, Ti⁴⁺, Nb⁵⁺ in manganites has been investigated by preparing La_0.7Sr_0.3(Mn_1-xMe_x)O₃ and La_1-xSr_x(Mn_{1 - x/2}Nb_x/2)O₃ series. It was established that substitution of manganese with magnesium up to x = 0.16 leads to a collapse of a long-range ferromagnetic order whereas La_0.7Sr_0.3(Mn ^{3 +} _0.85Nb ^{5 +} _0.15)O₃ is ferromagnet with T _C = 123 K and exhibits a large magnetoresistance below Curie point despite an absence of four-valent manganese. Hypothetical magnetic phase diagrams are constructed for La_0.7Sr_0.3(Mn_1-xMe_x)O₃ and La_1-xSr_x(Mn_{1 - x/2}Nb_x/2)O₃. Our results show that Mn³⁺-O-Mn³⁺ exchange interaction is ferromagnetic in the orbitally disordered manganites as well as an increase of Mn⁴⁺ content above 50% from a total amount of manganese ions leads to formation of a spin glass state due to a competition between antiferromagnetic Mn⁴⁺-O-Mn⁴⁺ and ferromagnetic Mn³⁺-O-Mn⁴⁺(Mn³⁺) superexchange interactions. Received 24 January 2002 Published online 9 July 2002     

Bose-Einstein condensation in an exactly soluble system of interacting particles

The model investigated recently by Tóth, a lattice gas of bosons with hard-core repulsion on a complete graph, is studied here by diagonalizing the Hamiltonian. The thermodynamic free energy per site is shown to bef, where $$\beta f = \beta \rho ^2 + \mathop {Min}\limits_{0 \leqslant x \leqslant Min(\rho , 1 - \rho )} \left\{ {\beta x(1 - x) + x\log x + (1 - x)\log (1 - x)} \right\}$$ whereβ is the inverse temperature andρ∈[0, 1] is the number of particles per site. This formula is equivalent to the one obtained by Tóth. There is a phase transition atβ +β ^* (ρ) = (1?2ρ)^?1 log[(1?ρ)/ρ]. Ifβ?β ^*(ρ), Bose-Einstein condensation is shown to be present, the condensate density (number of condensed particles per site) in the thermodynamic limit being [ρ?x ^*][l?ρ?x ^*], wherex ^* is the minimizing value ofx, satisfyingβ ^*(x ^*)=β.     

Large Time Asymptotics for the Kadomtsev–Petviashvili Equation

We study the large time asymptotic behavior of solutions to the Kadomtsev–Petviashvili equations $$\left\{\begin{array}{ll} u_{t} + u_{xxx} + \sigma \partial_{x}^{-1}u_{yy} = -\partial_{x}u^{2}, \quad \quad (x, y) \in {\bf R}^{2}, t \in {\bf R},\\ u(0, x, y) = u_{0}( x, y), \, \quad \quad \qquad \qquad (x, y) \in {\bf R}^{2},\end{array}\right.$$ where σ = ±1 and \({\partial_{x}^{-1} = \int_{-\infty}^{x}dx^{\prime} }\) . We prove that the large time asymptotics of the derivative u _x of the solution has a quasilinear character.     

Orthogonal polynomial solutions of the Fokker-Planck equation

We have tabulated the form of the coefficientsg ₁(x) andg ₂(x) as well as the boundary values [a, b] of the Fokker-Planck equation $$\frac{{\partial P(x, t)}}{{\partial t}} = - \frac{\partial }{{\partial x}}[g_1 (x)P(x, t)] + \frac{{\partial ^2 }}{{\partial x^2 }}[g_2 (x)P(x, t)],a \leqslant x \leqslant b$$ for which the solution can be written as an eigenfunction expansion in the classical orthogonal polynomials. We also discuss the problem of finding solutions in terms of the discrete classical polynomials for the differential difference equations of stochastic processes.     

Boundary regularity for some nonlinear elliptic degenerate equations

We consider the nonlinear elliptic degenerate equation (1) $$ - x^2 \left( {\frac{{\partial ^2 u}}{{\partial x^2 }} + \frac{{\partial ^2 u}}{{\partial y^2 }}} \right) + 2u = f(u)in\Omega _a ,$$ where $$\Omega _a = \left\{ {(x,y) \in \mathbb{R}^2 ,0< x< a,\left| y \right|< a} \right\}$$ for some constanta>0 andf is aC ^∞ functions on ? such thatf(0)=f′(0)=0. Our main result asserts that: ifu∈C \((\bar \Omega _a )\) satisfies (2) $$u(0,y) = 0for\left| y \right|< a,$$ thenx ^?2 u(x,y)∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) and in particularu∈C ^∞ \(\left( {\bar \Omega _{a/2} } \right)\) .     

Structural and electrical properties of iron molybdenum phosphate glasses

Summary  Iron molybdenum phosphate glasses [xMoO₃ · (0.6 -x)P₂O₅ · 0.4Li₂O] :yFe₂O₃ with 0 ≤x ≤ 0.6 andy = 0.03 (mol%) prepared in ambient atmosphere using the melt quenching technique were studied by using DC electrical conductivity,⁵⁷Fe M?ssbauer and infrared spectroscopies. The DC conductivity depends on the MoO₃ concentrationx. It was observed that, with increasingx, the ratio Fe²⁺ /(Fe³⁺ + Fe²⁺) and the DC conductivity increase. Infrared spectroscopy and X-ray powder diffraction indicate that a Li₂ MoO₄ crystalline phase is present for high MoO₃ content samples (x = 0.5, 0.6). This work was partly sponsored by FINEP, CNPq (Brazilian agencies) and UECE (Universidade Estadual do Cearà).     

Mössbauer effect study of Ni1−xCuxMnyFe2−yO4 system

The Mössbauer effect technique has been employed for the study of magnetic properties of spinel series Ni_1?xCu_xMn_yFe_2?yO₄ with 0.0≤x≤1.0, and y=0.6. The substitution of Mn³⁺ and Cu²⁺ ions results in a slight decrease of the hyperfine field at B‐ as well as A‐sites. The area ratio of Fe³⁺ ions at the A‐ and B‐site at 77 K indicates that Cu²⁺, Ni²⁺ and Mn³⁺ ions occupy the octahedral sites in an evidence for complete inverse spinel in this system. The temperature dependence of the hyperfine parameters has been studied for composition with x=0.5 where Nèel point T_N and Debye temperature θ_D are found to be 650 and 679 K, respectively. The temperature dependence of the sublattice magnetization σ(T) obeys a one‐third‐power law in the range 0.5N<0.99.     

MyM′1-yFClxBr1-x:Sm2+的电子跃迁过程和光谱烧孔效率

通过测量无机光谱烧孔系列材料MyM′_1-yFCl_xBr_1-x:Sm²⁺（Ｍ＝Ｍｇ，Ｃａ，Ｓｒ，Ｂａ）中４ｆ５ｄ带的激发光谱随组分ｘ和ｙ的变化，⁵Ｄ_J—⁷Ｆ₀（Ｊ＝２，１，０）跃迁的荧光衰减随组分与温度的变化，对其烧孔的电子跃迁过程及其对烧孔效率的影响进行了研究．得出结论：在此系列材料中，随着Ｂｒ含量和小半径的碱土离子的增加，Ｓｍ相似文献