Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón–Lozanovskiĭ construction. Factorization theorems for operators in spaces more general than the Lebesgue ${{L}^{p}}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt ${{A}_{p}}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from ${{L}^{p}}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón–Lozanovskiĭ construction are involved in the proofs.

We introduce some non-linear transformations from the set of Hausdorff moment sequences into itself; among them is the one defined by the formula: $T\left( {{\left( {{a}_{n}} \right)}_{n}} \right)\,=\,1/\left( {{a}_{0}}\,+\cdots +\,{{a}_{n}} \right).$ We give some examples of Hausdorff moment sequences arising from the transformations and provide the corresponding measures: one of these sequences is the reciprocal of the harmonic numbers ${{\left( 1+1/2\,+\cdots +\,1/\left( n+1 \right) \right)}^{-1}}.$

We provide a porosity-based approach to the differentiability and continuity of real-valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone $K$ with non-empty interior. We also show that the set of nowhere $K$-monotone functions has a $\sigma $-porous complement in the space of continuous functions endowed with the uniform metric.

We prove a symmetric imprimitivity theorem for commuting proper actions of locally compact groups $H$ and $K$ on a ${{C}^{*}}$-algebra.

We consider some deformations of ${{G}_{2}}$-structures on 7-manifolds. We discover a canonical way to deform a ${{G}_{2}}$-structure by a vector field in which the associated metric gets “twisted” in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy ${{G}_{2}}$. Similarly we consider analogous constructions for Spin(7)-structures on 8-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the Spin(7) case.

Let $\ell $ be a length function on a group $G$, and let ${{M}_{\ell }}$ denote the operator of pointwise multiplication by $\ell $ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\ell }}$ can be used as a “Dirac” operator for $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that if $G$ is a hyperbolic group and if $\ell $ is a word-length function on $G$, then the topology from this metric coincides with the weak-$*$ topology (our definition of a “compact quantum metric space”). We show that a convenient framework is that of filtered ${{C}^{*}}$-algebras which satisfy a suitable “Haagerup-type” condition. We also use this framework to prove an analogous fact for certain reduced free products of ${{C}^{*}}$-algebras.

The original Gelfond–Schnirelman method, proposed in 1936, uses polynomials with integer coefficients and small norms on [0, 1] to give a Chebyshev-type lower bound in prime number theory. We study a generalization of this method for polynomials in many variables. Our main result is a lower bound for the integral of Chebyshev's $\psi $-function, expressed in terms of the weighted capacity. This extends previous work of Nair and Chudnovsky, and connects the subject to the potential theory with external fields generated by polynomial-type weights. We also solve the corresponding potential theoretic problem, by finding the extremal measure and its support.

Let $\rho :\,{{G}_{Q}}\,\to \,\text{G}{{\text{L}}_{n}}\left( {{Q}_{\ell }} \right)$ be a motivic $\ell $-adic Galois representation. For fixed $m\,>\,1$ we initiate an investigation of the density of the set of primes $p$ such that the trace of the image of an arithmetic Frobenius at $p$ under $\rho $ is an $m$-th power residue modulo $p$. Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals $1/m$ whenever the image of $\rho $ is open. We further conjecture that for such $\rho $ the set of these primes $p$ is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain $m$ in the complementary case of modular forms of $\text{CM}$-type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the $\text{CM}$ case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at $p$ in abelian extensions of imaginary quadratic fields unramified away from $p$.