Let $P\left( z \right)$ be a polynomial of degree not exceeding $n$ and let $W\left( z \right)\,=\,\prod\nolimits_{j=1}^{n}{\left( z\,-\,{{a}_{j}} \right)}$ where $\left| {{a}_{j}} \right|\,>\,1$, $j\,=\,1,\,2,\,.\,.\,.\,,\,n$. If the rational function $r\left( z \right)\,=\,{P\left( z \right)}/{W\left( z \right)}\;$ does not vanish in $\left| z \right|\,<\,k$, then for $k\,=\,1$ it is known that

$$\left| {{r}^{'}}\left( z \right) \right|\le \frac{1}{2}\left| {{B}^{'}}(z) \right|_{\left| z \right|=1}^{\text{Sup}}\left| r(z) \right|$$

where $B\left( Z \right)\,=\,{{{W}^{*}}\left( z \right)}/{W\left( z \right)}\;$ and ${{W}^{*}}\left( z \right)\,=\,{{z}^{n}}\overline{W\left( {1}/{\overline{z}}\; \right)}$. In the paper we consider the case when $k\,>\,1$ and obtain a sharp result. We also show that

$$\underset{\left| z \right|=1}{\mathop{\text{Sup}}}\,\left\{ \left| \frac{{{r}^{\prime }}\left( z \right)}{{{B}^{\prime }}\left( z \right)} \right|+\left| \frac{{{\left( {{r}^{*}}\left( z \right) \right)}^{\prime }}}{{{B}^{\prime }}\left( z \right)} \right| \right\}=\underset{\left| z \right|=1}{\mathop{\text{Sup}}}\,\left| r\left( z \right) \right|$$

where ${{r}^{*}}\left( z \right)\,=\,B\left( z \right)\overline{r\left( {1}/{\overline{z}}\; \right)}$, and as a consquence of this result, we present a generalization of a theorem of O’Hara and Rodriguez for self-inversive polynomials. Finally, we establish a similar result when supremum is replaced by infimum for a rational function which has all its zeros in the unit circle.

A new infinite product ${{t}_{n}}$ was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about ${{t}_{n}}$ by establishing new connections between the modular $j$-invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, ${{t}_{n}}$ generates the Hilbert class field of $\mathbb{Q}\left( \sqrt{-n} \right)$. This shows that ${{t}_{n}}$ is a new class invariant according to H. Weber’s definition of class invariants.

A sequence of positive rationals generates a subgroup of finite index in the multiplicative positive rationals, and group product representations by the sequence need only a bounded number of terms, if and only if certain related sequences have densities uniformly bounded from below.

Let $X$ be a compact nonsingular real algebraic variety and let $Y$ be either the blowup of ${{\mathbb{P}}^{n}}\left( \mathbb{R} \right)$ along a linear subspace or a nonsingular hypersurface of ${{\mathbb{P}}^{m}}\left( \mathbb{R} \right)\,\times \,{{\mathbb{P}}^{n}}\left( \mathbb{R} \right)$ of bidegree (1, 1). It is proved that a ${{\mathcal{C}}^{\infty }}$ map $f:\,X\,\to \,Y$ can be approximated by regular maps if and only if ${{f}^{*}}\left( {{H}^{1}}\left( Y,\,{\mathbb{Z}}/{2}\; \right) \right)\,\subseteq \,H_{a\lg }^{1}\left( X,\,{\mathbb{Z}}/{2}\; \right)$, where $H_{a\lg }^{1}\left( X,\,{\mathbb{Z}}/{2}\; \right)$ is the subgroup of ${{H}^{1}}\left( X,\,{\mathbb{Z}}/{2}\; \right)$ generated by the cohomology classes of algebraic hypersurfaces in $X$. This follows from another result on maps into generalized flag varieties.

Let $\text{Alg}\left( \mathcal{L} \right)$ be the algebra of all bounded linear operators on a normed linear space $\mathcal{X}$ leaving invariant each member of the complete lattice of closed subspaces $\mathcal{L}$. We discuss when the subalgebra of finite rank operators in $\text{Alg}\left( \mathcal{L} \right)$ is non-zero, and give an example which shows this subalgebra may be zero even for finite lattices. We then give a necessary and sufficient lattice condition for decomposing a finite rank operator $F$ into a sum of a rank one operator and an operator whose range is smaller than that of $F$, each of which lies in $\text{Alg}\left( \mathcal{L} \right)$. This unifies results of Erdos, Longstaff, Lambrou, and Spanoudakis. Finally, we use the existence of finite rank operators in certain algebras to characterize the spectra of Riesz operators (generalizing results of Ringrose and Clauss) and compute the Jacobson radical for closed algebras of Riesz operators and $\text{Alg}\left( \mathcal{L} \right)$ for various types of lattices.

Let $x\,=\,\left( {{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}} \right)\,\in \,{{\mathbb{R}}^{n}}$ and ${{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x\,=\,\left( {{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{1}}}}{{x}_{1}},\,.\,.\,.\,,\,{{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{n}}}}{{x}_{n}} \right)$, where $\text{ }\lambda \,>\text{0}$ and $1\,\le \,{{\alpha }_{1}}\,\le \,\cdot \,\cdot \,\cdot \,\le \,{{\alpha }_{n}}$. Denote $\left| \alpha \right|\,=\,{{\alpha }_{1}}+\,\cdot \,\cdot \,\cdot \,+{{\alpha }_{n}}$. We characterize those functions $A\left( x \right)$ for which the parabolic Calderón commutator

1

$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{T}_{A}}f\left( x \right)\equiv \text{p}\text{.v}\text{.}\int_{{{\mathbb{R}}^{n}}}{K\left( x-y \right)\left[ A\left( x \right)-A\left( y \right) \right]}f\left( y \right)dy$$

is bounded on ${{L}^{2}}\left( {{\mathbb{R}}^{n}} \right)$, where $K\left( {{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x \right)\,=\,{{\text{ }\!\!\lambda\!\!\text{ }}^{-\,\left| \alpha \right|\,-\,1}}K\left( x \right)$, $K$ is smooth away fromthe origin and satisfies a certain cancellation property.

Let ${{\mathcal{S}}_{r}}$ be the collection of all axially symmetric functions $f$ in the Sobolev space ${{H}^{1}}\left( {{\mathbb{S}}^{2}} \right)$ such that $\int_{{{\mathbb{S}}^{2}}}{{{x}_{i}}{{e}^{2f\left( x \right)}}\,dw\left( \text{x} \right)}$ vanishes for $i\,=\,1,\,2,\,3$. We prove that

$$\underset{f\in {{\mathcal{S}}_{r}}}{\mathop \inf }\,\frac{1}{2}\int_{{{\mathbb{S}}^{2}}}{{{\left| \nabla f \right|}^{2}}\,dw\,+\,2\,\int_{{{\mathbb{S}}^{2}}}{f\,dw\,-\,\log \,\int_{{{\mathbb{S}}^{2}}}{{{e}^{2f}}\,dw\,>\,-\infty ,}}}$$

and that this infimum is attained. This complements recent work of Feldman, Froese, Ghoussoub and Gui on a conjecture of Chang and Yang concerning the Moser-Aubin inequality.

An integral formula is derived for the spherical functions on the symmetric space ${G}/{K\,=\,{\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;}\;$. This formula allows us to state some results about the analytic continuation of the spherical functions to a tubular neighbourhood of the subalgebra a of the abelian part in the decomposition $G\,=\,KAK$. The corresponding result is then obtained for the heat kernel of the symmetric space ${\text{S}{{\text{O}}_{0}}\left( p,\,q \right)}/{\text{SO}\left( p \right)\,\times \,\text{SO}\left( q \right)}\;$ using the Plancherel formula.

In the Conclusion, we discuss how this analytic continuation can be a helpful tool to study the growth of the heat kernel.

A line singularity is a function germ $f:\,\left( {{\text{C}}^{n+1}},\,0 \right)\,\to \,\text{C}$ with a smooth 1-dimensional critical set $\sum \,=\,\left\{ \left( x,\,y \right)\,\in \,\text{C}\,\times \,{{\text{C}}^{n}}\,|\,y\,=\,0 \right\}$. An isolated line singularity is defined by the condition that for every $x\,\ne \,0$, the germ of $f$ at $\left( x,\,0 \right)$ is equivalent to $y_{1}^{2}\,+\,\cdot \,\cdot \,\cdot \,+\,y_{n}^{2}$. Simple isolated line singularities were classified by Dirk Siersma and are analogous of the famous $A\,-\,D\,-E$ singularities. We give two new characterizations of simple isolated line singularities.