On the Arithmetic of Endomorphism Ring End$$({Z_p} \times {Z_{{p^m}}})$$

For a prime p, let \({E_{P,{P^m}}} = \left\{ {\left( {\begin{array}{*{20}{c}}aamp;b \\ {{p^{m – 1}}c}amp;d \end{array}} \right)\left| {a,b,c \in {Z_p},d} \right. \in {Z_{{p^m}}}} \right\}\). We first establish a ring isomorphism from \({Z_{p,{p^m}}}\)onto \({E_{p,{p^m}}}\). Then we provide a way to compute -d and d^–1 by using arithmetic in Z_p and \({Z_{{p^m}}}\), and characterize the invertible elements of \({E_{p,{p^m}}}\). Moreover, we introduce the minimal polynomial for each element in \({E_{p,{p^m}}}\)and give its applications.     

Further results on generalized LCM matrices

Let S={x_1,x_2,...,x_n } be a set of n distinct positive integers and f be an arithmetic function.By(f[S])(resp.( f[S])),we denote the n*n matrix whose i,j entry is Σ[x_i,x_j]|l l∈S f(l) (resp.Σx∈Sf(x)-Σ x_i,|l l∈S f(l)-Σ x_j,|l l∈S f(l)+Σ[x_i,x_j]|l l∈S f(l)).In this paper,we first investigate the structures of the matrices ( f[S]) and( f[S]),then we give the formulae for the determinants of these matrices.These extend the results obtained by Bege in 2011.Finally,we give two examples to demonstrate the validity of our main results.     

Global existence and bounded estimate of solutions of the BBM-Burgers equation

By the construction of a Cauchy sequence in a Banach space and the global bounded estimate of solution,we obtain the global existence and the bounded estimate of solution of BBM-Burgers equation without the viscous term u_t +∑ n j=1 f_j(u)_xj-γ_1△u_t+γ_3△~2u=0with large initial date in 4-dimension space.     

The boundedness of singular integral operator along an open curve

It is well known that, the singular integral operatorS defined as: ifL is a closed smooth contour in the complex plane C, thenS is a bounded linear operator fromH ^μ(L) intoH ^μ(L): ifL is an open smooth curve, thenS is just a linear operator fromH ^* intoH ^*. In this paper, we define a Banach space , and prove that is a bounded linear operator, then verify the boundedness of other kinds of singular integral operators. Supported by the National Science Foundation of China Wang Xiaolin: born in Aug. 1950, Professor     

Landau定理在高维复射影空间的推广

利用到复射影空间Pn(C)的全纯映射的正规性和值分布理论,结合Zalcman引理,对单位圆盘到高维复射影空间中全纯曲线的Landau定理进行了研究,得到了如下结果:设f:?→Pn(C)为全纯曲线D1,D2,…,D2t+1为Pn(C)上的2t+1个超曲面且位于t?次一般位置.若对于每一个j=1,2,…,2t+1,f(c)...     

Hadamard缺项幂级数及双曲完备极小曲面

基于具有Hadamard缺项的特殊幂级数，研究了一类位于$\mathbb{R}^3 $中2个平行平面之间的双曲型完备极小曲面族。首先得到如下结果：若$h(z) = \displaystyle\sum\limits_{j = 1}^\infty {{a_j}{z^{{n_j}}}}$是一个具有Hadamard缺项的幂级数，其中，$z \in {\mathbb{C}}$，$j = 1,2, \cdots $，且满足给定的3个特殊条件，则对于单位圆盘 $ \Delta $内的任意发散曲线$ \gamma $，有$\displaystyle\int_\gamma {{{\left| {h''(z)} \right|}^2}\left| {{\rm{d}}z} \right|} = \infty$。同时列举出了满足上述条件的具体的解析函数，其次通过选择适当的Weierstrass表示对，并利用上述结论，构造出了位于$\mathbb{R}^3 $中2个平行平面之间的双曲型完备极小曲面族及其具体形式。     

The Hàjek-Rènyi inequlity for Banach space valued random variable sequences and its application

In this paper we prove the following Hàjek-Rènyi inequality: Let 0≤p≤1, then for any Banach spaceB, anyL ^p integrableB valued random variable sequence {D _n , n≥1}, any real number sequence {b _n , n≥1} with 0<b _n, ↑ ∞, any integern≥1, there exist a constantC=C _p>0 (only depending onp) such that

Binding number,minimum degree and bipancyclism in bipartite graphs

Let G =(V_1,V_2,E) be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B(G),is defined to be n if G =K_n and min i∈{1,2}|N(S)|n min |N(S)|/|S|otherwise.We call G bipancyclic if it contains a cycle of every even length m for 4 ≤ m ≤ 2n.A theorem showed that if G is a balanced bipartite graph with 2n vertices,B(G) 3 / 2 and n 139,then G is bipancyclic.This paper generalizes the conclusion as follows:Let 0 c 3 / 2 and G be a 2-colmected balanced bipartite graph with 2n(n is large enough) vertices such that B(G) c and δ(G)(2-c)n/(3-c)+2/3.Then G is bipancyclic.     

Synthesis of biodiesel from rapeseed oil using K2O/ γ -Al2O3 as nano-solid-base catalyst

Nano-solid-base catalyst K₂O/γ-Al₂O₃ was prepared and adopted for the synthesis of biodiesel by transesterification of rapeseed oil with methanol. The particle diameter of the catalyst was about 50 nm, which was measured by transmission electron microscopy (TEM). The variables affecting the yield of biodiesel during transesterification, such as mass ratio of KNO₃ to γ-Al₂O₃, calcination temperature, calcination time, and catalyst content, molar ratio of methanol to oil, reaction temperature and reaction time were investigated. The catalyst obtained by calcining a mixture of KNO₃ and γ-Al₂O₃ at 600 °C for 3 h, was found to be the optimum one, which gave the highest catalytic activity in reaction. With 3% (m _catalyst/m _oil) catalyst, when the transesterification carried out at a molar ratio of methanol to oil 12:1, a reaction temperature of 70 °C, and a reaction time of 3 h, yield of 94% was achieved. Foundation item: Supported by the National High-Technology Research and Development Program of China (863 Program) (2007AA100703)     

Generalized Galerkin approximations for pseudoinverses and operator equations of the first kind

0 IntroductionLeqtuiliHnebaer afo rHimlbaenrtd sap baoceun.dLeedtlianaenardfufnbceti oan baolu onndeHd ,serse--spectively. Our goal is to seekv∈Hsuchthata(u,v) =f(u) , u∈ H (1)As a rule, we call the expression (1) a variational equationandva solutiontothis equation.By virtue of the Riesz repre-sentationtheorem,there exists a uniquely determined boundedlinear operatorA:H→Hsuchthata(u,v) =〈u,Av〉, u,v∈ H,where〈·,·〉is the inner product onH.If the sesquilinearformais also with posit…     

Texture evolution of commercial pure Ti during cold rolling and recrystallization annealing

X-ray diffraction (XRD) was employed to analyze the texture evolution of commercial pure (CP) Ti during cold rolling and recrystallization annealing. The texture components were measured by electron backscattered diffraction (EBSD) after recrystallization annealing. The CP Ti tends to form a texture with the basal pole tilted 30°-40° away from the normal direction toward the transverse direction. The texture of the initial hot-rolled plate can be classified into three kinds, i.e., the pyramid texture (1013)[5230] and (2021)[1015], the basal plane texture (0001)[2110], and the stronger prism texture (1120)[0001]. After cold rolling and annealing (700℃, 60 min), the main texture components are the cold-rolled texture (1125)[1123] and the recrystallized texture (1013)[5230]. The texture (2021)[1015] is inherited from the texture of the initial hot-rolled plate with the decrease of orientation density gradually. The volume contents of the cold-rolled texture {2115} <0110> and the recrystallized texture {1013} <1210> are calculated by EBSD. After recrystallization annealing, the specimen is rich in the recrystallized texture and inherits some of texture components from the cold-rolled texture. When the annealing time is prolonged, the anisotropic value decreases.     

A topological approach to the correlation of standard formation enthalpy with path index of benzenes derivatives

A mathematical model formulated as ΔfHθm(g)=a+sum from i=1 to 5bipi(i≠2) was constructed for the relationshipbetween standard formation enthalpy ΔfHθm(g) and path index Pi of substituted benzenes derivatives. An empiric equation for the calculation of ΔfHθm(g) was worked out. The calculated values of standard formation enthalpy based on this model are excellently consistent with those from experimental for 55 organic compounds. The model is shown to be simple and of practical usefulness, particularly when required experimental data are unavailable.     

Super-Edge-Connectivity of G(k,d,s)(s≥k/2)

A graph G is super-edge-connected,for short super-λ,if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree.Alphabet overlap graph G(k,d,s)is undirected,simple graph with vertex set V={v|v=1()kv…v;vi∈{1,2,…,d},i=1,…,k}.Two vertices u=(u1…uk)and v=(v1…vk)are adjacent if and only if us+i=vi or vs+i=ui(i=1,…,k-s).In particular G(k,d,1)is just an undirected de Bruijn graph.In this paper,we show that the diameter of G(k,d,s)is k s,the girth is 3.Finally,we prove that G(k,d,s)(s≥k/2)is super-λ.     

Exponential inequality for a class of NOD random variables and its application

In this paper,an exponential inequality for weighted sums of identically distributed NOD (negatively orthant dependent) random variables is established,by which we obtain the almost sure convergence rate of which reaches the available one for independent random variables in terms of Berstein type inequality. As application,we obtain the relevant exponential inequality for Priestley-Chao estimator of nonparametric regression estimate under NOD samples,from which the strong consistency rate is also obtained.     

Water solubility in orthopyroxene: Dependence on temperature and Al content

The water solubility in Al-Fe-Mg orthopyroxene [(Mg,Fe,Al)(Si,Al)O₃: X _Fe = 0.1] was investigated as a function of temperature and Al contents. Experiments were performed at 10 kbar with temperatures ranging from 800 to 1200°C under water-saturated conditions. Water contents in the (Mg,Fe)SiO₃-H₂O-Al₂O₃ system were determined using unpolarized Fourier transform infrared spectroscopy. The present results show that water solubility in Al-bearing orthopyroxene decreases systematically with temperature from approximately 1 weight % at 800°C to 568 ± 58 ppm at 1200°C and increase significantly with increasing Al₂O₃ contents under the same annealing temperature and pressure. Combined with published results on the dependence of hydroxyl solubility on water fugacity and pressure, the present results can be described by the relation $C_{OH} = A(T)f_{H_2 O}^{n = 0.5} \exp \left( { - \frac{{\Delta H^{1bar} + \Delta V^{solid} P}} {{RT}}} \right)$ where A = 0.0024 ± 0.0015 ppm/bar^0.5, ΔH ^1bar = ?103.348 ± 9.768 kJ/mol, and solid ΔV ^solid = 9.2 ± 1.1 cm³/mol. This equation implies that the incorporation mechanism of water in aluminous orthopyroxene involves the isolated OH groups. Based on the experimentally established solubility model used in this study, it is suggested that water solubility decreases with increasing temperature under typical upper mantle pressure. The predicted temperature dependence of water solubility is in good agreement with the previous experimental observations in Al-bearing orthopyroxene, but the opposite dependence is observed in Al-free systems. Moreover, our estimation of the water solubility in upper-mantle minerals as a function of depth for a typical oceanic geotherm might be of potential importance in interpreting the geophysical observations.     

带有强阻尼时滞项的m-Laplacian型波方程解的爆破

研究带有强阻尼时滞项的m-Laplacian型波方程: ${{u}_{tt}}-{{\Delta }_{m}}u-\Delta u+g*\Delta u-{{\mu }_{1}}\Delta {{u}_{t}}\left( x, t \right)-{{\mu }_{2}}\Delta {{u}_{t}}\left( x, t-\tau \right)={{\left| u \right|}^{p-2}}u$ 解的爆破:当初始能量0 < E(0) < E₁时, 利用能量函数构造凹函数L₁(t), 得到微分不等式$\frac{\text{d}{{L}_{1}}\left( t \right)}{\text{d}t}\ge {{\xi }_{0}}L_{1}^{1+\nu }\left( t \right)\ \left( {{\xi }_{0}}>0, \nu >0, t\ge 0 \right)$, 在(0, t)上对此微分不等式积分, 从而可知存在有限时间T*>0, 使得当时间t趋于T*时, 该m-Laplacian型波方程的解爆破; 当初始能量E(0) < 0时, 构造凹函数L₂(t), 通过同样的方法得到该方程的解存在有限时间爆破.     

A sufficient condition fork′(z)(k(z)∈H q,q≥1) to be ofH 1

We proved ifk(z)∈H ^q (q≥1),g(z) is analytic on |z|=1, , thenk′(z)∈H ¹, especially, ifq=1, thenk(z) is an analytic function on the closed unit disk |z|≤1. Peng Zhigang: born in June 1970, Ph. D     

On the boundary value problem for harmonic maps of the Poincaré disc

The boundary value problem for harmonic maps of the Poincare disc is discussed. The emphasis is on the non-smoothness of the given boundary values in the problem. Let T . be a subspace of the universal Teichmüller space, defined as a set of normalized quasisymmetric homeomorphisms h of the unit circle S onto itself where h admits a quasiconformal extension to the unit disc D with a complex dilatation μ satisfying where ρ(z)|dz|2 is the Poincare metric of D. Let B . be a Banach space consisting of holomorphic quadratic differentials φ in D with norms It is shown that for any given quasisymmetric homeomorphism h : S1→S1∈ T . , there is a unique quasiconformal harmonic map of D with respect to the Poincare metric whose boundary corresponding is h and the Hopf differential of such a harmonic map belongs to B .     

Ranks of repeated-root constacyclic code

By the discussion of division in \(F_{^{^{^{_2 m} } } } \left[ u \right]/\left\langle {u^4 } \right\rangle \), the minimal spanning set and the rank of a (1 + u + u ²) - constacyclic code with an arbitrary length N = 2 ^e n over \(\Phi \) are determined based on the factorization of (x ⁿ - 1) over \({F_{{2^m}}}\).     

The bubble leakage mechanism for vertical upflows by the phase separation concept

Recently,our research group proposed the phase separation condenser tube,in which a mesh cylinder was inserted to form the flow structure of‘‘gas near the tube wall and liquid in the tube core’’,significantly enhance the condensation heat transfer.But the bubble leakage towards the core region may worsen the heat transfer enhancement.In order to prevent the bubble leakage,the critical criterion was proposed based on the Young–Laplace equation,considering the inertia force,viscous force and pulsating flow.It was found that the critical criterion depends on the dimensionless parameter G*,the We number and a coefficient C.The numerical model was developed in terms of the volume of fluid method to predict the two-phase laminar flow in the phase separation condenser tube.The results show that the bubble leakage takes place at the bubble tip,which is agreed with the experimental observations.The critical curve distinguishing the non-bubble-breaking and bubble-breaking was obtained by comparing the bubble dynamics at different G*and We.The coefficient C was determined.The critical criterion for the bubble leakage is given as G We0:22We0:99349 G4:7 103 We0:00651t196:39We0:006514cosa DHW,providing the design and operation guidance for the phase separation condenser tube.