We study complex commutative Banach algebras (and, more generally, continuous inverse algebras) in which the holomorphic functions of a fixed $n$-tuple of elements are dense. In particular, we characterize the compact subsets of ${{\mathbb{C}}^{n}}$ which appear as joint spectra of such $n$ -tuples. The characterization is compared with several established notions of holomorphic convexity by means of approximation conditions.

K. Ding studied a class of Schubert varieties ${{X}_{\lambda }}$ in type A partial flag manifolds, indexed by integer partitions $\text{ }\!\!\lambda\!\!\text{ }$ and in bijection with dominant permutations. He observed that the Schubert cell structure of ${{X}_{\lambda }}$ is indexed by maximal rook placements on the Ferrers board ${{B}_{\lambda \text{ }}}$, and that the integral cohomology groups ${{H}^{*}}\left( {{X}_{\lambda }};\,\mathbb{Z} \right),\,{{H}^{*}}\left( {{X}_{\mu }};\,\mathbb{Z} \right)$ are additively isomorphic exactly when the Ferrers boards ${{B}_{\lambda \text{ }}}$, ${{B}_{\mu }}$ satisfy the combinatorial condition of rook-equivalence.

We classify the varieties ${{X}_{\lambda }}$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder–Bernstein Index of any of these spaces is ${{2}^{{\aleph_{0}}}}$. Our construction is based on a Banach space introduced by W. T. Gowers and B. Maurey in 1997. We also use classical descriptive set theory methods, as in some work of the first author and C. Rosendal, to improve some results of P. G. Casazza and of N. J. Kalton on the Schroeder–Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition.

Let $C$ be the class of convex sequences of real numbers. The quadratic irrational numbers can be partitioned into two types as follows. If $\alpha$ is of the first type and $\left( {{c}_{k}} \right)\,\in C$, then $\sum{{{(-1)}^{\left\lfloor k\alpha \right\rfloor }}}{{c}_{k}}$ converges if and only if ${{c}_{k}}\log k\to 0$. If $\alpha$ is of the second type and $\left( {{c}_{k}} \right)\,\in C$, then $\sum{{{(-1)}^{\left\lfloor k\alpha \right\rfloor }}}{{c}_{k}}$ converges if and only if $\sum{{{c}_{k}}/k}$ converges. An example of a quadratic irrational of the first type is $\sqrt{2}$, and an example of the second type is $\sqrt{3}$. The analysis of this problem relies heavily on the representation of $\alpha$ as a simple continued fraction and on properties of the sequences of partial sums $S\left( n \right)\,=\,{{\sum\nolimits_{k=1}^{n}{\left( -1 \right)}}^{\left\lfloor k\alpha \right\rfloor }}$ and double partial sums $T\left( n \right)\,=\,\sum\nolimits_{k=1}^{n}{\,S\left( k \right)}$.

Two formulas for the multiplicity of the fiber cone $F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$-primary ideal of a $d$-dimensional Cohen–Macaulay local ring $\left( R,m \right)$ are derived in terms of the mixed multiplicity ${{e}_{d-1}}\left( \text{m }|I \right)$, the multiplicity $e\left( I \right)$, and superficial elements. As a consequence, the Cohen–Macaulay property of $F\left( I \right)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen–Macaulay and Gorenstein properties of fiber cones of $\text{m}$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or $l\left( Im \right)Jm) = 1$.

Let $\phi (n)$ be the Euler phi-function, define ${{\phi }_{0}}\left( n \right)\,=\,n$ and ${{\phi }_{k+1}}\left( n \right)\,=\,\phi \left( {{\phi }_{k}}\left( n \right) \right)$ for all $k\ge 0$. We will determine an asymptotic formula for the set of integers $n$ less than $x$ for which ${{\phi }_{k}}\left( n \right)$ is $y$-smooth, conditionally on a weak form of the Elliott–Halberstam conjecture.

In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$-adic groups. We also construct a family of unitary spherical representations for real and complex classical groups.

Purely simple Kronecker modules $M$, built from an algebraically closed field $K$, arise from a triplet $\left( m,\,h,\,\alpha \right)$ where $m$ is a positive integer, $h:\,K\,\cup \,\{\infty \}\,\to \,\{\infty ,\,0,\,1,\,2,\,3,\,.\,.\,.\,\}$ is a height function, and $\alpha$ is a $K$-linear functional on the space $K\left( X \right)$ of rational functions in one variable $X$. Every pair $\left( h,\,\alpha \right)$ comes with a polynomial $f$ in $K\left( X \right)\left[ Y\, \right]$ called the regulator. When the module $M$ admits non-trivial endomorphisms, $f$ must be linear or quadratic in $Y$. In that case $M$ is purely simple if and only if $f$ is an irreducible quadratic. Then the $K$-algebra End $M$ embeds in the quadratic function field ${K\left( X \right)\left[ Y\, \right]}/{(f\,)}\;$. For some height functions $h$ of infinite support $I$, the search for a functional $\alpha$ for which $\left( h,\,\alpha \right)$ has regulator $0$ comes down to having functions $\eta \,:\,I\,\to \,K$ such that no planar curve intersects the graph of $\eta$ on a cofinite subset. If $K$ has characterictic not $2$, and the triplet $\left( m,\,h,\,\alpha \right)$ gives a purely-simple Kronecker module $M$ having non-trivial endomorphisms, then $h$ attains the value $\infty$ at least once on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$ and $h$ is finite-valued at least twice on $\text{K}\,\cup \,\text{ }\!\!\{\!\!\text{ }\infty \text{ }\!\!\}\!\!\text{ }$. Conversely all these $h$ form part of such triplets. The proof of this result hinges on the fact that a rational function $r$ is a perfect square in $K\left( X \right)$ if and only if $r$ is a perfect square in the completions of $K\left( X \right)$ with respect to all of its valuations.

For each real number $\xi$, let $\widehat{{{\lambda }_{2}}}\left( \xi \right)$ denote the supremum of all real numbers $\text{ }\!\!\lambda\!\!\text{ }$ such that, for each sufficiently large $X$ , the inequalities $\left| {{x}_{0}} \right|\,\le \,X,\,\left| {{x}_{0}}\xi \,-\,{{x}_{1}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ and $\left| {{x}_{0}}{{\xi }^{2}}\,-\,{{x}_{2}} \right|\,\le \,{{X}^{-\lambda \text{ }}}$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero, and let $\widehat{{{\omega }_{2}}}\left( \xi \right)$ denote the supremum of all real numbers $\omega $ such that, for each sufficiently large $X$, the dual inequalities $\left| {{x}_{0}}\,+\,{{x}_{1}}\xi \,+\,{{x}_{2}}{{\xi }^{2}} \right|\,\le \,{{X}^{-\omega }}$, $\left| {{x}_{1}} \right|\,\le \,X$ and $\left| {{x}_{2}} \right|\,\le \,X$ admit a solution in integers ${{x}_{0}},\,{{x}_{1}}$ and ${{x}_{2}}$ not all zero. Answering a question of Y. Bugeaud and M. Laurent, we show that the exponents $\widehat{{{\lambda }_{2}}}\left( \xi \right)$ where $\xi$ ranges through all real numbers with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$ form a dense subset of the interval $\left[ 1/2,\,\left( \sqrt{5}\,-\,1 \right)/2 \right]$ while, for the same values of $\xi$, the dual exponents $\widehat{{{\omega }_{2}}}\left( \xi \right)$ form a dense subset of $\left[ 2,\,\left( \sqrt{5}\,+\,3 \right)/2 \right]$. Part of the proof rests on a result of V. Jarník showing that $\widehat{{{\lambda }_{2}}}\left( \xi \right)=1-{{\hat{\omega }}_{2}}{{\left( \xi \right)}^{-1}}$ for any real number $\xi$ with $[\mathbb{Q}(\xi )\,:\mathbb{Q}]\,>\,2$.