Coordination compounds of Cu(II), Ni(II) and Co(II) with sulphamethazine-5-bromo-salicylaldimine

Coordination compounds of Cu (II), Ni (II) and Co (II) with sulphamethazine salicylaldimine (an antitubercular) have been prepared with a view to study their antibacterial activity. These complexes are granular, stable and are quantitatively formed and characterised by elemental analysis. Structures have been assigned based on their infrared, electronic absorption spectral and magnetic susceptibility studies. The antibacterial activity was tested against eleven available pathogens and in some cases complexes are found to be more potent.     

Trigonometry (II)

This paper considers geodesic triangies in a Riemannian manifoldM. First we imbed the set of geodesic triangles inM into a big spaceE, then find some equations inE satisfied by tangent vectors of . Finally we give an application of the result.     

具有四项的指数丢番图方程(Ⅱ)

本文中，我们给出了丢番图方程的解ｘ，ｙ，ｚ，ｗ的上界，其中ｐ，ｑ是给定的互素的正整数，ａ，ｂ，ｃ，ｄ是给定的适合ａｂｅｄ≠０的整数，此外，我们将指出在具体情形下如何把上界降低到方程允许的实际的解．最后，我们将用这个方法来解方程１９．５ｘ·１７ｙ＝１２．５ｚ＋４１．１７ｗ＋１４， ５． ３ｘ· １３ｙ ＋ ２０＝ ７． ３ｚ ＋ １４． １３ｗ和 １３· ２ｘ＋ ５· ３ｙ＝ ２５． ２ｚ＋ １１． ３ｗ．     

Cubic Lienard Equations with Quadratic Damping (II)

Abstract   Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienard equations with quadratic damping have at most three limit cycles. This implies that the guess in which the system has at most two limit cycles is false. We give the sufficient conditions for the system has at most three limit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by using numerical simulation. Supported by the National Natural Science Foundation of China and National Key Basic Research Special Found (No. G1998020307).     

具有四项的指数丢番图方程(Ⅱ)

本文中，我们给出了丢番图方程的解ｘ，ｙ，ｚ，ｗ的上界，其中ｐ，ｑ是给定的互素的正整数，ａ，ｂ，ｃ，ｄ是给定的适合ａｂｅｄ≠０的整数，此外，我们将指出在具体情形下如何把上界降低到方程允许的实际的解．最后，我们将用这个方法来解方程１９．５ｘ·１７ｙ＝１２．５ｚ＋４１．１７ｗ＋１４， ５． ３ｘ· １３ｙ ＋ ２０＝ ７． ３ｚ ＋ １４． １３ｗ和 １３· ２ｘ＋ ５· ３ｙ＝ ２５． ２ｚ＋ １１． ３ｗ．     

The Interaction of Shocks with Dispersive Waves II. Incompressible-Integrable Limit

This is the second in a two-part series of articles in which we analyze a system similar in structure to the well-known Zakharov equations from weak plasma turbulence theory, but with a nonlinear conservation equation allowing finite time shock formation. In this article we analyze the incompressible limit in which the shock speed is large compared to the underlying group velocity of the dispersive wave (a situation typically encountered in applications). After presenting some exact solutions of the full system, a multiscale perturbation method is used to resolve several basic wave interactions. The analysis breaks down into two categories: the nonlinear limit and the linear limit, corresponding to the form of the equations when the group velocity to shock speed ratio, denoted by ε, is zero. The former case is an integrable limit in which the model reduces to the cubic nonlinear Schrödinger equation governing the dispersive wave envelope. We focus on the interaction of a “fast” shock wave and a single hump soliton. In the latter case, the ε=0 problem reduces to the linear Schrödinger equation, and the focus is on a fast shock interacting with a dispersive wave whose amplitude is cusped and exponentially decaying. To motivate the time scales and structure of the shock-dispersive wave interactions at lowest orders, we first analyze a simpler system of ordinary differential equations structurally similar to the original system. Then we return to the fully coupled partial differential equations and develop a multiscale asymptotic method to derive the effective leading-order shock equations and the leading-order modulation equations governing the phase and amplitude of the dispersive wave envelope. The leading-order interaction equations admit a fairly complete analysis based on characteristic methods. Conditions are derived in which: (a) the shock passes through the soliton, (b) the shock is completely blocked by the soliton, or (c) the shock reverses direction. In the linear limit, a phenomenon is described in which the dispersive wave induces the formation of a second, transient shock front in the rapidly moving hyperbolic wave. In all cases, we can characterize the long-time dynamics of the shock. The influence of the shock on the dispersive wave is manifested, to leading order, in the generalized frequency of the dispersive wave: the fast-time part of the frequency is the shock wave itself. Hence, the frequency undergoes a sudden jump across the shock layer.In the last section, a sequence of numerical experiments depicting some of the interesting interactions predicted by the analysis is performed on the leading-order shock equations.     

Dinuclear copper (II) and nickel (II) systems with planar M2N2O2 bridge

The tridentate ligand systemb (abbreviated as inkR₂) readily yield copper (II) and nickel (II) species of the formula M₂ (inkR₂)₂(CLO₄)₂. 2xH₂O (x=0–1). Dinuclear formulation is based on variable temperature magnetic susceptibility and conductivity data and on the known structure of some related systems. The Cu₂ (inkR₂) ₂ ²⁺ species are strongly antiferromagnetic (?2J=600–800 cm^?1) while the Ni₂(inkR₂) ₂ ²⁺ species are diamagnetic. The major coordination sphere is planar around each metal (II). The metal ions in a dimer are linked by planar M₂N₂O₂ bridge. The copper (II) and nickel (II) species freely form solid solutions. In these statistical scrambling of copper and nickel occur among the metal ion sites of the dimeric structure. Powder epr spectra of such mixed crystals are indicative of axial geometry around copper (II) ion.     

QUSAI-PRIMITIVE RINFS(II)

Denote R an associative ring,\[\mathcal{M}\] a right modular idea of R,i,e,there exists an \[a \in R\] such that for all \[r \in R\],\[r + ar \in \mathcal{M}\], Let \[\{ {\mathcal{M}_i}\} \] be a given set of modular right ideals of R.Then introduce the following definition: Definition 1.Let \[\mathcal{M}\] be a modular right ideals of R. An element a of \[\mathcal{M}\] is called an \[\mathcal{M}\]-right quasi-regular element,if{i+ai}=\[\mathcal{M}\] for all \[i \in \mathcal{M}\].A right ideal L of R is called \[i \in \mathcal{M}\]-regular right ideal if every element of L is an \[i \in \mathcal{M}\]-right quasiregular element. Definition 2. Let \[i \in \mathcal{M}\] and \[{\mathcal{M}^'}\] be two right ideals of R,\[{\mathcal{M}^'}\] is called \[{\mathcal{M}^'}\]-modular if \[{\mathcal{M}^'} \subset \mathcal{M}\] and if there exist an element \[a \in \mathcal{M}\] such that for all \[i \in \mathcal{M}\],\[i + ai \in {\mathcal{M}^'}\]. Now we introduce the symbol \[{\hat \mathcal{M}}\].Let \[\mathcal{M} \in \sum \].Then if \[\mathcal{M}\] is an \[\mathcal{M}\]-regular right ideal,we put \[\hat \mathcal{M} = \mathcal{M}\];if \[\mathcal{M}\] is not an \[\mathcal{M}\]-regular ideal,we put \[{\hat \mathcal{M}}\] to be an \[\mathcal{M}\]-maximal modular right ideal in \[\mathcal{M}\].Let \[\mathcal{M} \in \sum \].Then if \[\mathcal{M}\] is not an \[\mathcal{M}\]-regular right ideal,we put \[\hat \mathcal{M} = \mathcal{M} \in {{\hat \sum }_\mathcal{M}} = \{ \hat \mathcal{M}|\hat \mathcal{M} is \mathcal{M}\} \]-maximal modular right ideal};if \[\mathcal{M}\] is an \[\mathcal{M}\]-right regular right idal,we put \[{{\hat \sum }_\mathcal{M}} = \mathcal{M}\]. Now we put \[\hat \sum = \{ \hat \mathcal{M}|\hat \mathcal{M} \in {{\hat \sum }_\mathcal{M}},\mathcal{M} \in \sum \} \] and \[\hat J = \cup {L_i}\] (1) for an element \[\mathcal{M} \in \sum \],where \[{L_i}\] are \[\mathcal{M}\]-regular right ideal,and U is set theoretical sum.Furthermore we put \[\hat J = \mathop \cap \limits_{\mathcal{M} \in \sum } {{\hat J}_\mathcal{M}}\] (2) and \[{J_1} = \{ b|b \in \mathop \cap \limits_{\mathcal{M} \in \sum } \mathcal{M},\],b satisfying the following condition}, (3) i,e,if |b)+\[{\mathcal{M}^{{\text{1}}}}{\text{ = }}\mathcal{M} \in \sum \] for an \[\mathcal{M}\]-modular right ideal \[{\mathcal{M}^{{\text{1}}}}\],then it must be \[{\mathcal{M}^{{\text{1}}}}{\text{ = }}\mathcal{M}\],where |b) is the intersection of all right ideals including b. Definition 3.an element \[\mathcal{M}\] of \[\sum \] is called satisfying J1-left idealizer condition,if \[x \in {J_1},y \in \mathcal{M}\],then \[rx + ryx \in \mathcal{M}\] for all \[r \in R\].The \[\sum \] is called satisfying J1-left idealizer condition(briefly,J1-l,i,c) if every \[\mathcal{M}\] \[\mathcal{M}\] of \[\sum \] is satisfying J1-l,i.c. Theorem 1. Suppose that \[\sum = \{ \mathcal{M}\} \] is satisfying J1-l.i.c.and put \[\beta = \hat \mathcal{M}\];\[R = \{ x \in R|Rx \subset \hat \mathcal{M}\} ,\hat \mathcal{M} \in \hat \sum \],then J1 is an ideal and \[{J_1} = \hat J = \sum\limits_{\hat \mathcal{M} \in \hat \sum } {\hat \mathcal{M} = \mathop \cap \limits_{\hat \mathcal{M} \in \hat \sum } } \beta \] Definition 4. Let \[\sum = \{ \mathcal{M}\} \] be satisfying J1-l.i.c.\[\hat \sum = \{ \hat \mathcal{M}|\hat \mathcal{M} \in {{\hat \sum }_\mathcal{M}},\mathcal{M} \in \sum \} \] as stated in (1), then we call ideal \[{J_1} = \mathop \cup \limits_{\hat \mathcal{M} \in \hat \sum } \hat \mathcal{M}\] the \[\sum \]-radioal of R. If J1=0, then R is called \[\sum \]-semisimple ring. Theorem 2. Let \[\sum = \{ \mathcal{M}\} \] be satisfying J1-'.i.c,where J1 is \[\sum \]-radical of R}, and \[\bar \sum = \{ \bar \mathcal{M}\} ,\bar \mathcal{M} = \mathcal{M}/{J_1},\mathcal{M} \in \sum ,\bar \hat \sum = \{ \bar \hat \mathcal{M}\} ,\hat \mathcal{M} \in \hat \sum ,\bar \hat \mathcal{M} = \hat \mathcal{M}/{J_1}\] then the \[{\bar \sum }\]-radical of \[\bar R = R/{J_1}\] is \[{\bar 0}\]. Definition 5. Let \[\sum = \{ \mathcal{M}\} \] be satisfying J1-l.i.c. and \[\hat \sum = \{ \hat \mathcal{M}\} \], then R is called a basic ring if and only if there exists an element \[{\hat \mathcal{M}}\] of such that \[\hat \mathcal{M}:R = 0\]. Let \[\beta \] be an ideal of R, if \[\beta = \hat \mathcal{M}\]\[:R\], \[\hat \mathcal{M} \in \hat \sum \],then \[\beta \] is called a basic ideal of R. Theorem 3. The \[\sum \]-rdical of R is the intersection of all basic ideals of R. Theorem 4. Any \[\sum \]-semisimple ring is isomorphic to a subdirect sum of basic rings. Theorem 5. Let R be an associative ring. Suppose that the set \[\sum \] includes only one element R, then the \[\sum \]-radieal of R, the \[\sum \]-semisimfple and the basic rings become the Jacobson radical, the Jacobson semisimple and the primitive rings respectively. Definition 6. An element \[m \in \mathfrak{M}\] is called strictly cyclic if \[m \in mR\]. \[\mathfrak{M}\] is called special if there exists a subset M of \[\mathfrak{M}\] such that every element \[m \in M\] is strictly cyclic and 0：\[\mathfrak{M} = \mathop \cap \limits_{m \in M} 0:m\] Definition 7. A module \[\mathfrak{M}\] is called a special dense module if and only if (i)\[\mathfrak{M}\] is special, (ii) \[\mathfrak{M}\] is a F-space as stated in [1] ,(\[\mathfrak{M}\]) suppose that\[{u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_n}}}\] be arbitrary finite F-independent elements and \[{u_{{i_1}}}r \ne 0,{u_{{i_j}}} = 0,j \ne 1\] for an element \[r \in R\], then there exists an element \[t \in R\] such that .\[{u_{{i_1}}}tR = \mathfrak{M},{u_{{i_j}}} = 0,j \ne 1\]. Let S be the set of all free elements of \[\mathfrak{M}\] as stated in [1]. It is clear that S is a strictly cyclic set and \[\mathfrak{M}\] is a special module. Now put I to be the class of all speciall dense modules with M = S, Denote \[{\Lambda _s} = \{ {\mathcal{M}_m}\} \] where =\[{\mathcal{M}_m} = 0:m,m \in S\], and \[\sum = \{ \mathcal{M}|\mathcal{M} \in {\Lambda _s},s \subset \mathfrak{M} \in I\} \]； \[{\hat \sum }\] as stated before. Then we can show that \[{J^*} = \mathop \cap \limits_{\mathcal{M} \in \sum } \mathcal{M} = \mathop \cap \limits_{\hat \mathcal{M} \in \hat \sum } \hat \mathcal{M}\] is a \[\sum \] -radical and \[{J^*} \subset J\], where J is Jacobson radical. Definition 8. The above stated \[\sum \]-radical \[{J^*}\] will be called the quasi Jacobson radical. A ring R is Called quasi Jacobson semisimple ring if and only if the quasi Jacobson radical \[{J^*}\] = 0. Theorem 6. Let R be a quasi Jacobson semisimple ring, then R is isomorphic to a subdirect sum of quasi primitive rings.     

Coordination properties of vic-isonitrosoimines in their copper (II) and palladium (II) complexes

Preparation and structural characterization of palladium (II) complexes of ligands III-V and copper (II) complexes of III are reported. The elemental analyses of the complexes show that the metal: ligand ratio is 1:2. The electrical conductance in acetone shows the non-electrolytic nature of the complexes. The diamagnetic character suggests a gross square-planar geometry for the palladium (II) complexes. Copper (II) complexes are paramagnetic with¼eff.～1·90 B.M. Spectral data suggest that in all the complexes the ligand coordinates to the metal (II) symmetrically through isonitroso-nitrogen and imine-nitrogen, forming a five membered chelate ring. Amine-exchange reactions of the complexes are discussed and compared on the basis of their structures.     

具相互作用项的非线性Schr(o)dinger方程的散射

In this paper,we give a simplified proof on the energy scattering for the nonlinear Schr(o)dinger equations with interaction terems by u8e of the interaction Morawetz estimate,which is originally introduced in[4].     

Generalized KKM type theorems in FC-spaces with applications (II)

This paper is a continuum of the preceding paper of author. By applying a coincidence theorem in noncompact FC-space without any convexity structure due to author, a new KKM type theorem is first proved under noncompact setting of FC-spaces. The equivalent relation between the coincidence theorem and the KKM type theorem is also established. As applications of the KKM type theorem, we establish some new existence theorems of solutions for three classes of generalized vector equilibrium problems under noncompact setting of FC-spaces. These theorems improve and generalize many known results in literature.     

与随机控制有关的一类变分方程(Ⅱ)

此篇借助本文（Ｉ）（见本刊Ｖｏｌ．４４,Ｎｏ．４,ｐ．７２７－７３６）的结论证明了一类变分方程解的存在性,分析了本文方法上的独特之处,指出这类变分方程在一类应用条件相当广泛的新随机控制模型的研究中起了重要作用．     

A DIOPHANTINE INEQAULITY (Ⅱ)

This paper proves the existence of infinitely many integer solutions to a Diophantine inequality.     

具有一对零态射的Morita Context环(II)

设$(A,B,V,W,\psi,\phi)$是一个Morita Context,具有一对零态射$\psi=0$, $\phi=0$, $C =\left ( \begin{array} {cc}A & V \\W & B \end{array}\right)$是对应的Morita Context环.本文给出了$C$与$A,B,V,W$之间关于环的$\pi$-正则性、semiclean性、Mophic性和环的Exchgange性、Potent性、GM性的关系.     

On Chen's theorem (II)

Let N be a sufficiently large even integer and S(N) denote the number of solutions of the equation

N=p+P₂,