• Volume 99, Issue 1

April 1989,   pages  1-101

• Zassenhaus conjecture forA5

We develop a criterion for rational conjugacy of torsion units of the integral group ringℤG of a finite groupG, as also a necessary condition for an element ofℤG to be a torsion unit, and apply them to verify the Zassenhaus conjecture in case whenG=A5.

• Aperiodic rings, necklace rings and Witt vectors—II

In an earlier paper the author and Dr K Wehrhahn have introduced the concept of the aperiodic ring Ap(A) of a commutative ringA. It is a commutative ring equipped with two families of operatorsVr: Ap(A)→Ap(A),Fr: Ap(A)→Ap(A) for every integerr≥1, called the Verschiebung and Frobenius operators. Let$$D(A) = \{ \sum\nolimits_{k \geqslant 1} {V_k \underline {S(a_k )} \left| {a_k \in A} \right.} \}$$, where for anya∈A,$$\underline {S(a)}$$ is the elementS(a, 1),S(a, 2),S(a, 3),...)of Ap (A). Let W(A) denote the ring of Witt vectors ofA. Let χ: W(A)→Ap(A) denote the map$$\sum\nolimits_{k \geqslant 1} {V_k \underline {S(a_k )} }$$. We prove that χ is a ring homomorphism preserving the Verschiebung and Frobenius operators with image χ=D(A). Moreover χ:W(A)→D(A) is an isomorphism if and only if the additive group ofA is torsion-free.

• Classification of isolated complete intersection singularities

In this article we prove that an isolated complete intersection singularity (V,0) is characterized by a module of finite lengthA(V) (cf. §1 for definition) associated to it. The proof uses the theory of finitely determined map germs and generalises the corresponding result by Yau and Mather [4], for hypersurfaces.

• Quasiminimal invariants for foliations of orientable closed surfaces

The Katok bound for the dimension of the cone of invariant measures for “quasiminimal” orientable foliations of closed oriented surfaces is extended to the nonquasiminimal case, in particular allowing for more general singularities. Equivalence of the Katok bound a bound for the dimension of the cone of invariant measures for a minimal interval exchange is established.

• Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in ℝ2

We prove the existence of a positive solution of the following problem −Δu=f(r,u) inDu&gt;0u=0, on ∂D whereD is the unit disc in ℝ2 andf is a superlinear function with critical growth.

• A softer, stronger Lidskii theorem

We provide a new approach to Lidskii’s theorem relating the eigenvalues of the differenceA—B of two self-adjoint matrices to the eigenvalues ofA andB respectively. This approach combines our earlier work on the spectral matching of matrices joined by a normal path with some familiar techniques of functional analysis. It is based, therefore, on general principles and has the additional advantage of extending Lidskii’s result to certain pairs of normal matrices. We are also able to treat some related results on spectral variation stemming from the work of Sunder, Halmos and Bouldin.

• On unbounded subnormal operators

A minimal normal extension of unbounded subnormal operators is established and characterized and spectral inclusion theorem is proved. An inverse Cayley transform is constructed to obtain a closed unbounded subnormal operator from a bounded one. Two classes of unbounded subnormals viz analytic Toeplitz operators and Bergman operators are exhibited.

• Thermohaline convection with cross-diffusion in an anisotropic porous medium

Using normal mode technique it has been shown that (i) values of the anisotropy parameter are important in deciding the mode of convection in a doubly diffusive fluid saturating a porous medium. (ii) Depending on the values of the Soret and Dufour parameters, an increase in anisotropy parameter either promotes or inhibits instability, (iii) cross-diffusion induces instability even in a potentially stable set-up and (iv) for certain values of the Dufour and Soret parameters there is a discontinuity in the critical thermal Rayleigh number, which disappears if the porous medium has horizontal isotropy.

• # Proceedings – Mathematical Sciences

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