• Volume 116, Issue 1

February 2006,   pages  1-119

• Arithmetic Properties of the Ramanujan Function

We study some arithmetic properties of the Ramanujan function 𝜏(𝑛), such as the largest prime divisor 𝑃(𝜏 (𝑛)) and the number of distinct prime divisors 𝜔(𝜏(𝑛)) of 𝜏(𝑛) for various sequences of 𝑛. In particular, we show that 𝑃(𝜏 (𝑛)) ≥ $(\log n)^{33/31+𝜎(1)}$ for infinitely many 𝑛, and

$$P(𝜏(p)𝜏(p^2)𝜏(p^3))>(1+𝜎(1))\frac{\log\log p\log\log\log p}{\log\log\log\log p}$$

for every prime 𝑝 with $𝜏(p)≠ 0$.

• Standard Monomial Bases and Geometric Consequences for Certain Rings of Invariants

Consider the diagonal action of $SL_n(K)$ on the affine space $X = V^{\oplus m} \oplus (V^∗)^{\oplus q}$ where $V = K^n, K$ an algebraically closed field of arbitrary characteristic and $m, q > n$. We construct a `standard monomial’ basis for the ring of invariants $K[X]^{SL_n(K)}$. As a consequence, we deduce that $K[X]^{SL_n(K)}$ is Cohen–Macaulay. We also present the first and second fundamental theorems for $SL_n(K)$-actions.

• Zero Cycles on Certain Surfaces in Arbitrary Characteristic

Let 𝑘 be a field of arbitrary characteristic. Let 𝑆 be a singular surface defined over 𝑘 with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation $\overline{S}$ is finite dimensional. We give numerical conditions under which the Chow group of zero cycles of 𝑆 is finite dimensional.

• The Jacobian of a Nonorientable Klein Surface, II

The aim here is to continue the investigation in [1] of Jacobians of a Klein surface and also to correct an error in [1].

• Boundary Regularity of Correspondences in $\mathbb{C}^n$

Let 𝑀,𝑀′ be smooth, real analytic hypersurfaces of finite type in $\mathbb{C}^n$ and $\hat{f}$ a holomorphic correspondence (not necessarily proper) that is defined on one side of 𝑀, extends continuously up to 𝑀 and maps 𝑀 to 𝑀′. It is shown that $\hat{f}$ must extend across 𝑀 as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich–Pinchuk extension result for CR maps.

• Matrix Multiplication Operators on Banach Function Spaces

In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups for solving the abstract Cauchy problem.

• On Characterisation of Markov Processes Via Martingale Problems

It is well-known that well-posedness of a martingale problem in the class of continuous (or r.c.l.l.) solutions enables one to construct the associated transition probability functions. We extend this result to the case when the martingale problem is well-posed in the class of solutions which are continuous in probability. This extension is used to improve on a criterion for a probability measure to be invariant for the semigroup associated with the Markov process. We also give examples of martingale problems that are well-posed in the class of solutions which are continuous in probability but for which no r.c.l.l. solution exists.

• Formulation of the Problem of Sonic Boom by a Maneuvering Aerofoil as a One-Parameter Family of Cauchy Problems

For the structure of a sonic boom produced by a simple aerofoil at a large distance from its source we take a physical model which consists of a leading shock (LS), a trailing shock (TS) and a one-parameter family of nonlinear wavefronts in between the two shocks. Then we develop a mathematical model and show that according to this model the LS is governed by a hyperbolic system of equations in conservation form and the system of equations governing the TS has a pair of complex eigenvalues. Similarly, we show that a nonlinear wavefront originating from a point on the front part of the aerofoil is governed by a hyperbolic system of conservation laws and that originating from a point on the rear part is governed by a system of conservation laws, which is elliptic. Consequently, we expect the geometry of the TS to be kink-free and topologically different from the geometry of the LS. In the last section we point out an evidence of kinks on the LS and kink-free TS from the numerical solution of the Euler’s equations by Inoue, Sakai and Nishida [5].

• # Proceedings – Mathematical Sciences

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