• Volume 105, Issue 1

February 1995,   pages  1-122

• The structure of generic subintegrality

In order to give an elementwise characterization of a subintegral extension of ℚ-algebras, a family of generic ℚ-algebras was introduced in . This family is parametrized by two integral parameters p ⩾ 0,N ⩾ 1, the member corresponding top, N being the subalgebraR = ℚ [{γn¦n ⩾ N}] of the polynomial algebra ℚ[x1,…,xp, z] inp + 1 variables, where$$\gamma _n = z^n + \sum\nolimits_{i = 1}^p {(_i^n )} x_i z^{n - i}$$. This is graded by weight (z) = 1, weight (xi) =i, and it is shown in  to be finitely generated. So these algebras provide examples of geometric objects. In this paper we study the structure of these algebras. It is shown first that the ideal of relations among all the γn’s is generated by quadratic relations. This is used to determine an explicit monomial basis for each homogeneous component ofR, thereby obtaining an expression for the Poincaré series ofR. It is then proved thatR has Krull dimension p+1 and embedding dimensionN + 2p, and that in a presentation ofR as a graded quotient of the polynomial algebra inN + 2p variables the ideal of relations is generated minimally by$$\left( \begin{gathered} N + p \\ 2 \\ \end{gathered} \right)$$ elements. Such a minimal presentation is found explicitly. As corollaries, it is shown thatR is always Cohen-Macaulay and that it is Gorenstein if and only if it is a complete intersection if and only ifN + p ⩽ 2. It is also shown thatR is Hilbertian in the sense that for everyn ⩾ 0 the value of its Hilbert function atn coincides with the value of the Hilbert polynomial corresponding to the congruence class ofn.

• Flat connections, geometric invariants and energy of harmonic functions on compact Riemann surfaces

A geometric invariant is associated to the space of flat connections on a G-bundle over a compact Riemann surface and is related to the energy of harmonic functions.

• Fibred Frobenius theorem

We give a version of Frobenius Theorem for fibred manifolds whose proof is shorter than the “short proofs” of the classical Frobenius Theorem. In fact, what shortens the proof is the fibred form of the statement, since it permits an inductive process which is not possible from the standard statement.

• On infinitesimalh-conformal motions of Finsler metric

The conformai theory of Finsler spaces was initiated by Knebelman in 1929 and lately Kikuchi  gave the conditions for a Finsler space to be conformai to a Minkowski space. However under theh-condition, the third author  obtained the conditions for a Finsler space to beh-conformal to a Minkowski space.

The purpose of the paper is to investigate the infinitesimalh-conformal motions of Finsler metric and its application to anH-recurrent Finsler space. We obtain the following results.

1. Theorem 2.1. If an HR-Fn space is a Landsberg space, then the tensorFhjki is recurrent.

2. Proposition 3.3. An infinitesimalh-conformal motion satisfies$$L_x G_{jk}^i = \rho _j \delta _k^i + \rho _k \delta _j^i - \rho ^i g_{jk} - \phi _1 l^i l_j l_k ,$$$$L_x G_j^i = \rho \delta _j^i + \phi _j y^i - y_j \rho ^i .$$

3. Proposition 3.6. An infinitesimalh-conformal motion satisfiesLx Pjki = ρCjki.

4. Theorem 3.7. In order that an infinitesimalh-conformal motion preserves Landsberg spaces, it is necessary and sufficient that the transformation be an infinitesimal homothetic motion.

5. Theorem 3.8. An infinitesimalh-conformal motion preserves *P-Finsler spaces.

6. Theorem 3.10. An infinitesimalh-conformal motion preservesh-conformally flat Finsler spaces.

7. Theorem 4.1. An infinitesimal homothetic motion preservesH-recurrent Finsler spaces.

8. Theorem 4.2. If anH-recurrent Finsler space admits an infinitesimal homothetic motion, then Lie derivatives of the tensorFhjki, and all its successive covariant derivatives byxi oryi vanish.

• A bibasic hypergeometric transformation associated with combinatorial identities of the Rogers-Ramanujan type

During the last five decades, a number of combinatorial generalizations and interpretations have occurred for the identities of the Rogers-Ramanujan type. The object of this paper is to give a most general known analytic auxiliary functional generalization which can be used to give combinatorial interpretations of generalizedq-identities of the Rogers-Ramanujan type. The derivation realise the theory of basic hypergeometric series with two unconnected bases.

• Some theorems on the general summability methods

In this paper a new theorem which covers many methods of summability is proved. Several results are also deduced.

• Symmetrizing a Hessenberg matrix: Designs for VLSI parallel processor arrays

A symmetrizer of a nonsymmetric matrix A is the symmetric matrixX that satisfies the equationXA =AtX, wheret indicates the transpose. A symmetrizer is useful in converting a nonsymmetric eigenvalue problem into a symmetric one which is relatively easy to solve and finds applications in stability problems in control theory and in the study of general matrices. Three designs based on VLSI parallel processor arrays are presented to compute a symmetrizer of a lower Hessenberg matrix. Their scope is discussed. The first one is the Leiserson systolic design while the remaining two, viz., the double pipe design and the fitted diagonal design are the derived versions of the first design with improved performance.

• Control of interconnected nonlinear delay differential equations inW2(1)

Our main interest in this paper is the resolution of the problem of controllability of interconnected nonlinear delay systems in function space, from which hopefully the existence of an optimal control law can be deduced later. We insist that each subsystem be controlled by its own variables while taking into account the interacting effects. This is the recent basic insight of  on ordinary differential systems. Controllability is deduced for the composite system from the assumption of controllability of each free subsystem and a growth condition of the interconnecting structure. Conditions for a free system’s controllability are given. One application is presented. The insight it provides for the growth of global economy has important policy implications.

• A note on integrable solutions of Hammerstein integral equations

We derive a set of sufficient conditions for the existence of solutions of a Hammerstein integral equation.

• On over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in a compressible fluid

A study is made of over-reflection of acoustic-gravity waves incident upon a magnetic shear layer in an isothermal compressible electrically conducting fluid in the presence of an external magnetic field. The reflection and transmission coefficients of hydromagnetic acoustic-gravity waves incident upon magnetic shear layer are calculated. The invariance of wave-action flux is used to investigate the properties of reflection, transmission and absorption of the waves incident upon the shear layer, and then to discuss how these properties depend on the wavelength, length scale of the shear layers, and the ratio of the flow speed and phase speed of the waves. Special attention is given to the relationship between the wave-amplification and critical-level behaviour. It is shown that there exists a critical level within the shear layer and the wave incident upon the shear layer is over-reflected, that is, more energy is reflected back towards the source than was originally emitted. The mechanism of the over-reflection (or wave amplification) is due to the fact that the excess reflected energy is extracted by the wave from the external magnetic field. It is also found that the absence of critical level within the shear layer leads to non-amplification of waves. For the case of very large vertical wavelength of waves, the coefficients of incident, reflected and transmitted energy are calculated. In this limiting situation, the wave is neither amplified nor absorbed by the shear layer. Finally, it is shown that resonance occurs at a particular value of the phase velocity of the wave.