Radiation effects of poly(vinylidene fluoride) (PVDF) (I)

The relationship between the crystallinity of poly (vinylidene fluoride) (PVDF) and electron radiation effects is studied with differential scanning calorimeter and X-ray diffraction. The form of crystal of PVDF is not changed. In the low dose zone (lower than 400 kGy), the crystallinity of PVDF increases slightly with the increase of absorbed dose, and then decreases slowly, whereas the decrease of the crystallinity is accelerated with the increase of the absorbed dose in high dose zone. Gel and Sol properties of irradiated samples of PVDF are also studied with extraction of N, N-dimethyl formamide. In a low dose range, it fits Charlesby-Pinner formula very well; however, with the increase of the dose, it begins to deviate from this formula because of the obstruction of the crystalline region of PVDF.     

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.     

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.     

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.     

Time and temperature dependent deformation of poly(ether ether ketone) (PEEK)

The stress-strain behavior in tension and the effect of temperature on the creep of poly(ether ether ketone) (PEEK) have been studied. At room temperature, 130° below the glass-transition temperature, the material does not become brittle, and the specimens show necking in tension over a wide range of elongation rates. The stress and strain at yield and the strain at break are almost linear functions of the logarithmic elongation rate. The values of stress and strain at yield increase slightly with increasing elongation rate, while the strain at break decreases markedly. The short-term creep tests were conducted at temperatures extending from 20 to 200°C. The glass-transition temperature was found to be about 155°C. The creep of PEEK is greatest at temperatures above 130°C. In the glass region the time dependence of the deformation is much weaker. It has been found that the time-temperature relation for PEEK corresponds well with its thermorheological simplicity in the temperature range investigated. The data on the temperature shift factor below and above the glass-transition temperature may be fitted separately to the Arrhenius and Williams-Landel-Ferry (WLF) equations, respectively. The long-term creep tests show that PEEK has excellent creep resistance at room temperature. After 14-month tests at a stress level of 30 MPa the total strain exceeds the instantaneous elastic strain only by a factor of 1.15.Institute of Polymer Mechanics. Latvian Academy of Sciences, 23 Aizkraukles St., LV-1006 Riga. Latvia. Department of Polymeric Materials, Chalmers University of Technology. S-412 96 Gothenburg, Sweden. Published in Mekhanika Kompozitnykh Materialov, No. 6, pp. 734–746, November–December, 1997.     

A DIOPHANTINE INEQAULITY (Ⅱ)

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

Adjoining cofinitary permutations (II)

 We construct several forcing models in each of which there exists a maximal cofinitary group, i.e., a maximal almost disjoint group, G≤Sym(ℕ), such that G is also a maximal almost disjoint family in Sym(ℕ). We also ask several open questions in this area in the fourth section of this paper. Received: 25 December 2000 / Revised version: 10 December 2001 Published online: 5 November 2002 Current address:Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109. USA.e-mail: yizhang@umich.edu The author's research on this subject was partially supported by a visiting grant from the Institute Mittag–Leffler, Royal Academy of Science, Sweden and the grant no. 40734 of Academy of Finland. Mathematics Subject Classification (2000): 03E35, 20A15, 20B07, 20B35     

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,     

不可约M－矩阵的刻画（II）

该文给出了（非）奇异Ｚ－矩阵是（非）奇异不可约Ｍ－矩阵的一些充分必要条件．     

Maximal exponents of polyhedral cones (II)

Let K be a proper (i.e., closed, pointed, full convex) cone in Rⁿ. An n×n matrix A is said to be K-primitive if there exists a positive integer k such that ; the least such k is referred to as the exponent of A and is denoted by γ(A). For a polyhedral cone K, the maximum value of γ(A), taken over all K-primitive matrices A, is called the exponent of K and is denoted by γ(K). It is proved that the maximum value of γ(K) as K runs through all n-dimensional minimal cones (i.e., cones having n+1 extreme rays) is n²-n+1 if n is odd, and is n²-n if n is even, the maximum value of the exponent being attained by a minimal cone with a balanced relation for its extreme vectors. The K-primitive matrices A such that γ(A) attain the maximum value are identified up to cone-equivalence modulo positive scalar multiplication.     

Algebre senza base finita (II)

Sunto. Si finiscono di passare in rassegna i teoremi più importanti della teoria delle algebre a base finita, quale è esposta dalloScorza, lasciando cadere l'ipotesi della limitazione della base. Dei teoremi che restano validi si dà la dimostrazione solo quand'essa è necessariamente diversa da quella delloScorza: nel caso di teoremi la cui dimostrazione verte essenzialmente nella finitezza della base sono dati esempi di algebre per cui essi cadono in difetto. Estratto dalla tesi di laurea (Pisa, Scuola Normale Superiore, 1942).     

The Compact Quantum Group Uq（2） （Ⅱ）

In this paper, we first prove that the θ-deformation Uθ（2） of U（2） constructed by Connes and Violette is our special case of the quantum group Uq（2） constructed in our previous paper. Then we will show that the set of truces on the C^＊ -algebra Uθ, θ irrational, is determined by the set of the truces on a subalgebra of Uθ.     

On Generalizations of Prime Ideals (II)

Let R be a commutative ring with identity. We say that a proper ideal P of R is (n ? 1, n)-weakly prime (n ≥ 2) if 0 ≠ a ₁…a _n  ∈ P implies a ₁…a _i?1 a _i+1…a _n  ∈ P for some i ∈ {1,…, n}, where a ₁,…, a _n  ∈ R. In this article, we study (n ? 1, n)-weakly prime ideals. A number of results concerning (n ? 1, n)-weakly prime ideals and examples of (n ? 1, n)-weakly prime ideals are given. Rings with the property that for a positive integer n such that 2 ≤ n ≤ 5, every proper ideal is (n ? 1, n)-weakly prime are characterized. Moreover, it is shown that in some rings, nonzero (n ? 1, n)-weakly prime ideals and (n ? 1, n)-prime ideals coincide.