Crystal structure of zirconia by Rietveld refinement

The crystal structures and phase transformation of zirconia ceramics have been investigated by means of X-ray powder diffraction and Rietveld powder diffraction profile fitting technique. A structural transition from monoclinic to tetragonal occurs when Y₂O₃ and₂ are doped into zirconia. The space group of the tetragonal structure is P4₂/nmc,Z=2. The lattice parameters are α=0.362 6(5)nm,c=0.522 6(3)nm for CeO₂ doped zirconia and α=0.360 2(8)nm,=0.517 9(1)nm for Y₂O₃ doped zirconia, respectively. In each unit cell, there are two kinds of equivalent positions, i.e. 2b and 4d, which are occupied by Zr⁴⁺, M(M=Y³⁺, Ce⁴⁺) cations and O^2- anions, respectively. The crystallographic correlation among the cubic, tetragonal and monoclinic structures of ZrO₂ is discussed.     

On «bilocal» and periodicity problems for hyperbolic partial differential equations

The problem stated in the paper is to find solutions of equation (1) in the strip 0≤y≤a, and satisfying conditions (2), (3), α₁, α₁, α₂,z, δ andf being functions which satisfy suitable conditions. This problem is equivalent with that to find solutions of (4), (5). If δ=0, the problem is one of generalized periodicity. Theorem 1–3 give sufficient conditions for the existence of required solutions.     

TheT-ideal generated by the standard identitys 3[x 1,x 2,x 3]

LetK = T_o(s₃), {c_n} its codimensions, {l_n} its colengths and {Χ_n} its sequence of co-characters. For 9≦n, c_n =2n - 1 or c_n =n(n + l)/2- 1, 3≦l_n ≦4 and χ_n =[n] + 2[n-1,1] + α[n-2,2] + β[2²,1^n?4] where α + β≦l.     

Synthesis and characterization of C3N4 hard films

C₃N₄ films have been synthesized on both Si and Ft substrates by microwave plasma chemical vapor deposition (MPCVD) method. X-ray spectra were calculated for single phase α-C₃N₄ and β-C₃N₄ respectively. The experimental X-ray spectra of films deposited on both Si and Pt substrates showed all the strong peaks of α-C₃N₄ and β-C₃N₄ so the films are mixtures of α-C₃N₄ and β-C₃N₄. The N/C atomic ratio is in the range of 1.0–2.0. X-ray photoelectron spectroscopy (XPS) analysis indicated that the binding energy of Is and N ls are 286.2 eV and 399.5 eV respectively, corresponding to polarized C-N bond. Fourier transform infrared absorption (FT-IR) and Raman spectra support the existence of C-N covalent bond in the films. Nano-indentation hardness tests showed that the bulk modulus of a film deposited on Pt is up to 349 GPa     

The real interpolation method on couples of intersections

Suppose that (X ₀, X ₁) is a Banach couple, X ₀ ∩ X ₁ is dense in X ₀ and X ₁, (X₀,X₁)_θq (0 < θ < 1, 1 ≤ q < ∞) are the spaces of the real interpolation method, ψ ∈ (X ₀ ∩ X ₁), ψ ≠ 0, is a linear functional, N = Ker ψ, and N _i stands for N with the norm inherited from X _i (i = 0, 1). The following theorem is proved: the norms of the spaces (N₀,N₁)_θ,q and (X₀,X₁)_θ,q are equivalent on N if and only if θ ? (0, α) ∪ (β_∞, α₀ ∪ (β₀, α_∞) ∪ (β, 1), where α, β, α₀, β₀, α_∞, and β _∞ are the dilation indices of the function k(t)=K(t,ψ;X ₀ ^* ,X ₁ ^* ).     

Norm attaining operators and renormings of Banach spaces

We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc _o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+ε)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+ε)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.     

Eine Bemerkung zu den Nielsen-Transformationen

LetG=H ₁* _A H ₂ a free product with amalgamation with $$H_1 = \left\langle {s_1 ,...,s_m |s_1^{\alpha _1 } = ... = s_m^{\alpha _m } = 1} \right\rangle ,\alpha _i \geqslant 2,m \geqslant 2,$$ a free product ofm cyclic groups. If we ask for the generation ofG, the following questions are significant: 1) For which α _i andm there is a set {x ₁, ...,x _m } of. generators ofH ₁ withx ₁=(s ₁...s _m )^α, α≥2, and what can we say about α andx ₂, ...,x _m ? 2) for which α _i andm there is a set {x ₁, ...,x _m } of generators ofH ₁ withx ₁=(s ₁...s _m )^α x ₂=h(s ₁...s _m )^β h ^?1, α>0, β>0,h∈H ₁, and what can we say about α, β,h andx ₃, ...,x _m ? In this note we give a complete solution of these questions.     

Thinning of point processes,revisited

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A remark on the transformation formula of theta functions associated to positive definite quadratic forms

Let L be a positive Z-lattice with level N = cd, (c, d) = 1. Then the Fourier expansion at cusp $\text{1}\text{d}$ of the theta function associated to L is a theta function associated to L₁, where a lattice L₁ is defined by Z_pL₁ = Z_pL for $\text{p?c}$, Z_pL₁ = the dual of Z_pL for p | c.     

On the complexity of testing membership in the core of min-cost spanning tree games

LetN = {1,...,n} be a finite set of players andK _N the complete graph on the node setN∪{0}. Assume that the edges ofK _N have nonnegative weights and associate with each coalitionS∪N of players as costc(S) the weight of a minimal spanning tree on the node setS∪{0}. Using transformation from EXACT COVER BY 3-SETS, we exhibit the following problem to beNP-complete. Given the vectorxε?^itN withx(N) =c(N). decide whether there exists a coalitionS such thatx(S) >c(S).     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

Synthesis and characterization of C3N4 crystal (I)

A successful experimental synthesis of pure crystalline β- and α-C₃N₄ films on Si(100) substrate was carried out by bias-assisted hot filament chemical vapor deposition (bias-HFCVD). It is found that a mixed-phase C₃-_I-Si_xN_y buffer layer was formed between the Si substrate and the C-N film. A “lattice match selection” was proposed to study the growth mechanism of C₃N₄ clusters composed of many crystal columns with hexagonal facets. Project supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences.     

On finite x-decomposable groups for X = {1, 2, 4}

A normal subgroup N of a finite group G is called an n-decomposable subgroup if N is a union of n distinct conjugacy classes of G. Each finite nonabelian nonperfect group is proved to be isomorphic to Q ₁₂, or Z ₂ × A ₄, or G = ??a, b, c | a ¹¹ = b ⁵ = c ² = 1, b ^?1 ab = a ⁴, c ^?1 ac = a ^?1, c ^?1 bc = b ^?1?? if every nontrivial normal subgroup is 2- or 4-decomposable.     

Alternating sequences

Let a=(α₁, α₂, α₃, …) be a sequence of positive integers. The sequence (c₁, c₂, …, c₃) is a-alternating if 1 ? c₁ < c₂ < … < c_k ? n and in additionthe first α₁ elements have the same parity, the next α₂ elements have opposite parity, the next α₃ elements have the parity of the first group, and so on. The final group of elements of like parity is permitted to have fewer elements than the required number. Let $\text{?(a; n, k)}$ denote the number of a-alternating sequences of length k. An explicit formula for $\text{? (a;n, k)}$ is obtained.     

Spatio-Temporal Dynamical Systems in Inner-Shell Photoionization in Free Molecules,Clusters, and Solids

Dynamic spatial hole localization and symmetry breaking phenomena are examined. Absorption of X-ray synchrotron and free-electron–laser radiation in matter is accompanied by strong dynamic corehole localization and temporary trap of the electron ejected from a deep level within the finite size potential barrier. As a result the symmetry of core excited states is reduced in comparison with ground state as the inversion symmetry is being broken. This is a very general property of coreexcited polyatomic compounds with equivalent atoms as their equivalence implies their equal probability of excitation averaged over large timescale but not simultaneous core excitation. Different approaches to rationalizing the symmetry breaking phenomena are presented and discussed with the emphasis on the quasiatomic dynamic corehole localization model. By examining the experimental ultrafast probe of photoabsorption processes we demonstrate an important role of spatio-temporal (nanometric-femtosecond) dynamically localized coreexcited moieties in molecule, clusters and solids. The photoelectron angular distributions from N and O 1s levels in fixed-in-space N₂ and CO₂ molecules, the photoelectron induced rotational heating of N₂, the Auger decay spectra of N₂ and the near S 1s edge X-ray absorption fine structure of free SF₆ molecules are discussed in more detail.     

Synthesis of C3N4 crystals under high pressure and high temperature

C₃N₄ crystals with the size of several micrometers have been synthesized from C₃N₄H₄ in the presence of nickel-based alloy or cobalt as catalyst under high pressure of 7 GPa and temperature of about 1400°C for 10 min. Scanning electron microscopy, energy-disperse X-ray analysis, and X-ray diffraction were used to examine the grown crystals. The general rule on selecting the starting materials for synthesis of carbon nitride crystals at high pressures and high temperatures is suggested.     

Estimates and regularization for solutions of some ill-posed problems of elliptic and parabolic type

In this paper, we examine, in a systematic fashion, some ill-posed problems arising in the theory of heat conduction. In abstract terms, letH be a Hilbert space andA: D (A)?H→H be an unbounded normal operator, we consider the boundary value problemü(t)=Au(t), 0<t<∞,u(0)=u ₀∈D(A), \(\mathop {\lim }\limits_{t \to 0} \left\| {u\left( t \right)} \right\| = 0\) . The problem of recoveringu ₀ whenu(T) is known for someT>0 is not well-posed. Suppose we are given approximationsx ₁,x ₂,…,x _N tou(T ₁),…,u(T _N) with 0<T, <…<T _N and positive weightsP _i,i=1,…,n, \(\sum\limits_{i = 1}^N {P_i = 1} \) such that \(Q_2 \left( {u_0 } \right) = \sum\limits_{i = 1}^N {P_i } \left\| {u\left( {T_i } \right) - x_i } \right\|^2 \leqslant \varepsilon ^2 \) . If ‖u _t(0)‖≤E for some a priori constantE, we construct a regularized solution ν(t) such that \(Q\left( {\nu \left( 0 \right)} \right) \leqslant \varepsilon ^2 \) while \(\left\| {u\left( 0 \right) - \nu \left( 0 \right)} \right\| = 0\left( {ln \left( {E/\varepsilon } \right)} \right)^{ - 1} \) and \(\left\| {u\left( t \right) - \nu \left( t \right)} \right\| = 0\left( {\varepsilon ^{\beta \left( t \right)} } \right)\) where 0<β(t)<1 and the constant in the order symbol depends uponE. The function β(t) is larger thant/m whent _k andk is the largest integer such that \((\sum\limits_{k = 1}^N {P_i (T_i )} )< (\sum\limits_{k = 1}^N {P_i (T_i )} = m\) , which β(t)=t/m on [T _k, m] and β(t)=1 on [m, ∞). Similar results are obtained if the measurement is made in the maximum norm, i.e.,Q _∞(u ₀)=max{‖u(T _i)?x _i‖, 1≤i≤N}.     

Investigation of Raman spectrum for nano-SnO2

The dependence of the microstructural change and lattice space symmetry of nano-SnO₂ on the annealing temperature has been studied systematically using Raman spectroscopy and X-ray diffraction. Comparing the results of nano-SnO₂ with the results of amorphous film and single crystal of SnO₂, it is found that the new Raman peaks N₁ and N₂ are in accordance with Matossi’s force constant model completely. When the annealing temperature is near 673K, the local lattice disorders and the density of vacant lattice decrease rapidly in the nano-SnO₂ grains. The lattice distortion and the new Raman peaks disappear almost at the same time. The possible mechanisms of the microstructural change and the new Raman peaks N₁ and N₂ are discussed. Project supported by the Foundation of State Science and Technology Commission and the Natural Science Foundation of Anhui Province.     

Some extensions of the Laplace problem

Let ?₁(a,b) be a regular lattice with the fundamental cell a rectangle with sides a,b, let ?₂(a,b,?) be a regular lattice with the fundamental cell a parallelogram with sides a,b and angle ? and let ?₃(a,b,c) be a regular lattice with the fundamental cell \(C_{0}^{( 3) }\) as in Fig. 7. In this paper we compute the probability that a random rectangle r of constant side l,m intersects a side of the lattice. In particular when the rectangle r becomes a segment of length l, (m=0) we obtain the Laplace probability.     

A representation of the Kantorovich-Functional

This paper establishes the representation of the generalizedN-dimensional Wasserstein distance (Kantorovich-Functional) $$W_c (P_1 ,...,P_N ): = \inf \left\{ {\int_{S^N } {c(x_1 ,...,x_N )d\mu } (x_1 ,...,x_N ):\pi _i \mu = P_i ,i = 1,...,N} \right\}$$ in the form ofW _c(P ₁,...,P _N )=sup{∑ _i=1 ^N }∫sf _i dP _i . The conditions we impose onP _i ,c andf _i enable us to follow those classical lines of arguments which lead to the Kantorovich-Rubinstein Theorem: By elementary methods we show how the result for an arbitrary metric space (S, d) can be derived from the case of finiteS. We also apply this result and the techniques of its proof in order to obtain a fairly simple proof of Strassen's Theorem.