• Volume 122, Issue 4

November 2012,   pages  485-684

• Ramanujan's Modular Equations of Degree 5

We provide alternative derivations of theta function identities associated with modular equations of degree 5. We then use the identities to derive the corresponding modular equations.

• Enumerating Set Partitions According to the Number of Descents of Size 𝑑 or more

Let $P(n,k)$ denote the set of partitions of $\{1,2,\ldots,n\}$ having exactly 𝑘 blocks. In this paper, we find the generating function which counts the members of $P(n,k)$ according to the number of descents of size 𝑑 or more, where $d\geq 1$ is fixed. An explicit expression in terms of Stirling numbers of the second kind may be given for the total number of such descents in all the members of $P(n,k)$. We also compute the generating function for the statistics recording the number of ascents of size 𝑑 or more and show that it has the same distribution on $P(n,k)$ as the prior statistics for descents when $d\geq 2$, by both algebraic and combinatorial arguments.

• Discreteness Criteria based on a Test Map in $PU(n, 1)$

The discreteness of isometry groups in complex hyperbolic space is a fundamental problem. In this paper, the discreteness criteria of a 𝑛-dimensional subgroup 𝐺 of $SU(n,1)$ are investigated by using a test map which may not be in 𝐺.

• Rees Valuations

We study dicritical divisors and Rees valuations.

• From Graphs to Free Products

We investigate a construction (from Kodiyalam Vijay and Sunder V S, J. Funct. Anal.260 (2011) 2635–2673) which associates a finite von Neumann algebra $M(\Gamma, \mu)$ to a finite weighted graph $(\Gamma, \mu)$. Pleasantly, but not surprisingly, the von Neumann algebra associated to a flower with 𝑛 petals’ is the group on Neumann algebra of the free group on 𝑛 generators. In general, the algebra $M(\Gamma, \mu)$ is a free product, with amalgamation over a finite-dimensional abelian subalgebra corresponding to the vertex set, of algebras associated to subgraphs with one edge’ (or actually a pair of dual edges). This also yields `natural’ examples of (i) a Fock-type model of an operator with a free Poisson distribution; and (ii) $\mathbb{C}\oplus\mathbb{C}$-valued circular and semi-circular operators.

• On Dominator Colorings in Graphs

A dominator coloring of a graph 𝐺 is a proper coloring of 𝐺 in which every vertex dominates every vertex of at least one color class. The minimum number of colors required for a dominator coloring of 𝐺 is called the dominator chromatic number of 𝐺 and is denoted by $\chi d(G)$. In this paper we present several results on graphs with $\chi d(G)=\chi(G)$ and $\chi d(G)=\gamma(G)$ where $\chi(G)$ and $\gamma(G)$ denote respectively the chromatic number and the domination number of a graph 𝐺. We also prove that if $\mu(G)$ is the Mycielskian of 𝐺, then $\chi d(G)+1\leq\chi d(\mu(G))\leq\chi d(G)+2$.

• Uncertainty Inequalities for the Heisenberg Group

We establish the Heisenberg–Pauli–Weyl uncertainty inequalities for Fourier transform and the continuous wavelet transform on the Heisenberg group.

• Equivalent Moduli of Continuity, Bloch's Theorem for Pluriharmonic Mappings in $\mathbb{B}^n$

In this paper, we first establish a Schwarz–Pick type theorem for pluriharmonic mappings and then we apply it to discuss the equivalent norms on Lipschitz-type spaces. Finally, we obtain several Landau’s and Bloch’s type theorems for pluriharmonic mappings.

• Ricci Flow of Warped Product Metrics with Positive Isotropic Curvature on $S^{p+1}\times S^1$

We study the asymptotic behaviour of the ODE associated to the evolution of curvature operator in the Ricci flow of a doubly warped product metric on $S^{p+1}\times S^1$ with positive isotropic curvature.

• Frobenius Pull Backs of Vector Bundles in Higher Dimensions

We prove that for a smooth projective variety 𝑋 of arbitrary dimension and for a vector bundle 𝐸 over 𝑋, the Harder–Narasimhan filtration of a Frobenius pull back of 𝐸 is a refinement of the Frobenius pull back of the Harder–Narasimhan filtration of 𝐸, provided there is a lower bound on the characteristic 𝑝 (in terms of rank of 𝐸 and the slope of the destabilizing sheaf of the cotangent bundle of 𝑋). We also recall some examples, due to Raynaud and Monsky, to show that some lower bound on 𝑝 is necessary. We also give a bound on the instability degree of the Frobenius pull back of 𝐸 in terms of the instability degree of 𝐸 and well defined invariants of 𝑋.

• Real Hypersurfaces of a Complex Space Form

In this paper we are interested in obtaining a condition under which a compact real hypersurface of a complex projective space $CP^n$ is a geodesic sphere. We also study the question as to whether the characteristic vector field of a real hypersurface of the complex projective space $CP^n$ is harmonic, and show that the answer is in negative.

• A Note on Gaussian Distributions in $\mathbb{R}^n$

Given any finite set $\mathcal{F}$ of $(n-1)$-dimensional subspaces of $\mathbb{R}^n$ we give examples of nonGaussian probability measures in $\mathbb{R}^n$ whose marginal distribution in each subspace from $\mathcal{F}$ is Gaussian. However, if $\mathcal{F}$ is an infinite family of such $(n-1)$-dimensional subspaces then such a nonGaussian probability measure in $\mathbb{R}^n$ does not exist.

• Solution of the Problem of the Identified Minimum for the Tri-Variate Normal

Let $X=(X_1,X_2,X_3)$ be a non-singular tri-variate normal vector with zero means. Let $T=\min\{X_1,X_2,X_3\}$, and $I=i \mathrm{ iff } T=X_i,i=1,2,3$. The problem of the identified minimum $(I,T)$ is then to find if its joint distribution determines uniquely 𝑋. This problem is solved here in the affirmative. To the best of our knowledge, it was first solved in the bivariate normal case (and partially in the tri-variate normal case) in 1978 in [1].

• A Coupled Far-Field Formulation for Time-Periodic Numerical Problems in Fluid Dynamics

Consider uniform flow past an oscillating body generating a time-periodic motion in an exterior domain, modelled by a numerical fluid dynamics solver in the near field around the body. A far-field formulation, based on the Oseen equations, is presented for coupling onto this domain thereby enabling the whole space to be modelled. In particular, examples for formulations by boundary elements and infinite elements are described.

• Subject Index

• Author Index

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019