Barycentric formulae for cardinal (SINC-)interpolants

Summary We present a barycentric representation of cardinal interpolants, as well as a weighted barycentric formula for their efficient evaluation. We also propose a rational cardinal function which in some cases agrees with the corresponding cardinal interpolant and, in other cases, is even more accurate.In numerical examples, we compare the relative accuracy of those various interpolants with one another and with a rational interpolant proposed in former work.Dedicated to the memory of Peter HenriciThis work was done at the University of California at San Diego, La Jolla     

Blow-up formulae for SO(3)-Donaldson polynomials

We calculate first through seventh order terms of a blow-up formula for -Donaldson polynomials, using equivariant cohomology and techniques similar to those of [Oz]. These formulas are used to extend the results of [B2] to any rank one negative definite four manifold and to describe the extension of the equivalence proved in [M] of the Donaldson polynomial and O'Grady's algebro-geometric analogues to t he four dimensional classes. Received: 12 April 1994; in final form 15 November 1995     

The commutation formulae in a finsler space. II

Summary In a previous paper [2] some commutation formulae have been established on the principle of mathematical induction. The present paper includes the commutation formulae involving the covariant derivatives of Cartan and of Berwald. In this paper, too, the method of mathematical induction has been used.     

Barycentric formulae for cardinal (SINC-) interpolants by Jean-Paul Berrut

Summary. A formula for the efficient evaluation of the (truncated) cardinal series is known to be numerically unstable near the interpolation abscissae. Here it is shown how the series can be evaluated in an entirely stable manner. Received February 14, 2000 / Published online October 16, 2000     

More on the generalized macaulay theorem — II

Let k₁ ? k₂? ? ? k_n be given positive integers and let S denote the set of vectors x = (x₁, x₂, … ,x_n) with integer components satisfying 0 ? x₁ ? k_ni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l.     

On Tanaka formulae for (α, d,β)-superprocesses

In this paper, Tanaka formulae for (α, d,β)-superprocesses in the dimensions where the local time exists are established under the optimal initial condition.     

Summation formulae on trigonometric functions

Trigonometric sums over the angles equally distributed on the upper half plane are investigated systematically. Their generating functions and explicit formulae are established through the combination of the formal power series method and partial fraction decompositions.     

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.     

Quadrature formulae on half-infinite intervals

We develop two classes of quadrature rules for integrals extended over the positive real axis, assuming given algebraic behavior of the integrand at the origin and at infinity. Both rules are expressible in terms of Gauss-Jacobi quadratures. Numerical examples are given comparing these rules among themselves and with recently developed quadrature formulae based on Bernstein-type operators.Work supported, in part, by the National Science Foundation under grant CCR-8704404.     

The realization of functions of the Pk by formulae (k ≥ 3)

In this note we study the realization of the functions of many-valued logic by formulae in a special basis subject to the condition that the functions of the basis are of arbitrary weight. An asymptotic bound is obtained for Shannon's function in this case.Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 99–108, January, 1972.In conclusion, the author wishes to express his deep gratitude to S. V. Yablonskii for his attention to this paper.     

Determinantal formulae for matrices with sparse inverses,II: asymmetric zero patterns

In an earlier paper, formulae for det A as a ratio of products of principal minors of A were exhibited, for any given symmetric zero-pattern of A⁻¹. These formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of A⁻¹. However, just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries, the symmetry of the zero pattern is not essential for these formulae. We describe here how analogous formulae for det A may be obtained in the asymmetric-zero-pattern case by introducing a directed spanning tree. We also examine the converse question of determining all possible zero patterns of A⁻¹ which guarantee that a certain determinantal formula holds.     

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.     

Extension formulae on almost complex manifolds

We give the extension formulae on almost complex manifolds and give decompositions of the extension formulae.As applications,we study(n,0)-forms,the(n,0)-Dolbeault cohomology group and(n,q)-forms on almost complex manifolds.     

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.