We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the “obvious” limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson–Schensted–Knuth correspondence. Then we use this result to generalize the mirabolic Robinson–Schensted–Knuth correspondence defined by Travkin, to the case of two partial flags and a line.

We define the filtrated $\text{K}$-theory of a ${{\text{C}}^{*}}$-algebra over a finite topological space $X$ and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over $X$ in terms of filtrated $\text{K}$-theory.

For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification.

We also exhibit an example where filtrated $\text{K}$-theory is not yet a complete invariant. We describe two ${{\text{C}}^{*}}$-algebras over a space $X$ with four points that have isomorphic filtrated $\text{K}$-theory without being $\text{KK}\left( X \right)$-equivalent. For this space $X$, we enrich filtrated $\text{K}$-theory by another $\text{K}$-theory functor to a complete invariant up to $\text{KK}\left( X \right)$-equivalence that satisfies a Universal Coefficient Theorem.

In 1960, Sobolev proved that for a finite reflection group $G$, a $G$-invariant cubature formula is of degree $t$ if and only if it is exact for all $G$-invariant polynomials of degree at most $t$. In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type $B$, to other finite reflection groups.

We study negative powers of Laguerre differential operators in ${{\mathbb{R}}^{d}},\,d\,\ge \,1$. For these operators we prove two-weight $[{{L}^{p}}\,-\,{{L}^{q}}$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices. we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the convexity principle to cover the continuous range of $\alpha \,\in \,\,{{[-1/2,\infty )}^{d}}$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on ${{\mathbb{R}}^{d}}$ and isomorphic to $\mathbb{Z}_{2}^{d}$. The two weight ${{L}^{p}}\,-\,{{L}^{q}}$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

Darmon cycles are a higher weight analogue of Stark–Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on ${{\Gamma }_{0}}\left( N \right)$ of even weight ${{k}_{0}}\,\ge \,2$. They are conjectured to be the restriction of global cohomology classes in the Bloch–Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross–Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur– Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$.

Let $\left( \text{ }\!\!\chi\!\!\text{ ,}\,d,\,\mu \right)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu \left( \left\{ x \right\} \right)\,=\,0$ for all $x\,\in \,\text{ }\!\!\chi\!\!\text{ }$. In this paper, we show that the boundedness of a Calderón–Zygmund operator $T$ on ${{L}^{2}}\left( \mu \right)$ is equivalent to that of $T$ on ${{L}^{p}}\left( \mu \right)$ for some $p\,\in \,\left( 1,\,\infty \right)$, and that of $T$ from ${{L}^{1}}\left( \mu \right)$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. As an application, we prove that if $T$ is a Calderón–Zygmund operator bounded on ${{L}^{2}}\left( \mu \right)$, then its maximal operator is bounded on ${{L}^{p}}\left( \mu \right)$ for all $p\,\in \,\left( 1,\,\infty \right)$ and from the space of all complex-valued Borel measures on $\text{ }\!\!\chi\!\!\text{ }$ to ${{L}^{1,\,\infty }}\left( \mu \right)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $c<{{\aleph }_{\omega }}$, we construct a weakly tight family under the hypothesis $\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$. The case when $\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require $\mathfrak{b}<{{\aleph }_{\omega }}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b}<{{\aleph }_{\omega }}$ is used to treat the case $\mathfrak{s}=\mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

Extending the work of Larson and Todorcevic, we show that there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form $\text{PFA}\left( S \right)\left[ S \right]$, where $S$ is a coherent Souslin tree.

In this paper, we establish the regularity of strong solutions to nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.

The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is ${{C}^{2}}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

Let $\rho :\,G\,\to \,\text{GL}\left( V \right)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let ${{\sigma }_{1}},\ldots ,{{\sigma }_{n}}$ be a system of generators of the algebra of invariant polynomials $\mathbb{C}{{\left[ V \right]}^{G}}$. We study the problem of lifting mappings $f:\,{{\mathbb{R}}^{q}}\,\supseteq \,U\,\to \,\sigma \left( V \right)\,\subseteq \,{{\mathbb{C}}^{n}}$ over the mapping of invariants $\sigma \,=\,\left( {{\sigma }_{1}},\ldots ,{{\sigma }_{n}} \right):\,V\,\to \,\sigma \left( V \right)$. Note that $\sigma \left( V \right)$ can be identified with the categorical quotient $V//G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $C\subseteq {{C}^{\infty }}$ satisfying some mild closedness properties that guarantee resolution of singularities in $C$, e.g., the real analytic class, then $f$ admits a lift of the same class $C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\text{SB}{{\text{V}}_{\text{loc}}}$, i.e., special functions of bounded variation. If $\rho $ is a real representation of a compact Lie group, we obtain stronger versions.

An extension theorem for holomorphic mappings between two domains in ${{\mathbb{C}}^{2}}$ is proved under purely local hypotheses.

In his last letter to Hardy, Ramanujan defined 17 functions $F\left( q \right),\,\left| q \right|\,<\,1$, which he called mock $\theta $-functions. He observed that as $q$ radially approaches any root of unity $\zeta $ at which $F\left( q \right)$ has an exponential singularity, there is a $\theta $-function ${{T}_{\zeta }}\left( q \right)$ with $F\left( q \right)\,-\,{{T}_{\zeta }}\left( q \right)\,=\,O\left( 1 \right)$. Since then, other functions have been found that possess this property. These functions are related to a function $H\left( x,\,q \right)$, where $x$ is usually ${{q}^{r}}$ or ${{e}^{2\pi ir}}$ for some rational number $r$. For this reason we refer to $H$ as a “universal” mock $\theta $-function. Modular transformations of $H$ give rise to the functions $K,\,{{K}_{1}},\,{{K}_{2}}$. The functions $K$ and ${{K}_{1}}$ appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell–Lerch sums (also called generalized Lambert series). Some relations (mock theta “conjectures”) involving mock $\theta $-functions of even order and $H$ are listed.

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form

$${\nabla }'P\left( x \right)\nabla u+\text{HR}u+\text{{S}'G}u\text{+ F}u\text{=}\text{f}\,\text{+}\,\text{{T}'g}\,\text{in}\,\Theta$$

$$u=\varphi \,\,on\,\,\partial \Theta$$

The principal part ${\xi }'P\left( x \right)\xi$ of the above equation is assumed to be comparable to a quadratic form $\mathcal{Q}\left( x,\xi \right)={\xi }'Q\left( x \right)\xi$ that may vanish for non-zero $\xi \in {{\mathbb{R}}^{n}}$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $Q{{H}^{1}}\left( \Theta \right)={{W}^{1,2}}\left( \Theta ,Q \right)$ and $QH_{0}^{1}\left( \Theta \right)=W_{0}^{1,2}\left( \Theta ,Q \right)$ as defined in previous works. E.T. Sawyer and R.L. Wheeden (2010) have given a regularity theory for a subset of the class of equations dealt with here.

We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal{M}$ can be computed from the decomposability number, $\text{dens}\left( \mathcal{M} \right)$, and the minimal cardinality of a generating set, $\text{gen}\left( \mathcal{M} \right)$. Applications include the equivalence of the well-known generator problem, “Is every separably-acting von Neumann algebra singly-generated?”, with the formally stronger questions, “Is every countably-generated von Neumann algebra singly-generated?” and “Is the gen invariant monotone?” Modulo the generator problem, we determine the range of the invariant $\left( \text{gen}\left( \mathcal{M} \right),\,\text{dens}\left( \mathcal{M} \right) \right)$ , which is mostly governed by the inequality $\text{dens}\left( \mathcal{M} \right)\,\le {{\mathfrak{C}}^{\text{gen}\left( \mathcal{M} \right)}}$.

Analytical study of the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solutions with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray–Hopf solution of the Boussinesq system are established as the regularizing parameter $\alpha$ vanishes. The proofs are done in the frequency space and use energy methods, the Arselà-Ascoli compactness theorem and a Friedrichs-like approximation scheme.