This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Let $K$ be a totally real number field of degree $n$. We show that the number of totally positive integers (or more generally the number of totally positive elements of a given fractional ideal) of given trace is evenly distributed around its expected value, which is obtained from geometric considerations. This result depends on unfolding an integral over a compact torus.

Let $\pi $ be a square integrable representation of ${G}'=\text{S}{{\text{L}}_{n}}(D)$, with $D$ a central division algebra of finite dimension over a local field $F$of non-zero characteristic. We prove that, on the elliptic set, the character of $\pi $ equals the complex conjugate of the orbital integral of one of the pseudocoefficients of $\pi $. We prove also the orthogonality relations for characters of square integrable representations of ${G}'$. We prove the stable transfer of orbital integrals between $\text{S}{{\text{L}}_{n}}(F)$ and its inner forms.

Meromorphic continuation of the Eisenstein series induced from spherical, cuspidal data on parabolic subgroups is achieved via reworking Bernstein's adaptation of Selberg's third proof of meromorphic continuation.

In this paper, we consider the quantum version of a Hamiltonian model describing friction. This model consists of a particle which interacts with a bosonic reservoir representing a homogeneous medium through which the particle moves. We show that if the particle is confined, then the Hamiltonian admits a ground state if and only if a suitable infrared condition is satisfied. The latter is violated in the case of linear friction, but satisfied when the friction force is proportional to a higher power of the particle speed.

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 study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

We characterize the pairs of weights $\left( v,\,w \right)$ for which the operator $Tf\left( x \right)=g\left( x \right)\int{_{s\left( x \right)}^{h\left( x \right)}}\,f\,$ with $s$ and $h$ increasing and continuous functions is of strong type $\left( p,\,q \right)$ or weak type $\left( p,\,q \right)$ with respect to the pair $\left( v,\,w \right)$ in the case $0\,<\,q\,<\,p$ and $1\,<\,p\,<\,\infty$. The result for the weak type is new while the characterizations for the strong type improve the ones given by H. P. Heinig and G. Sinnamon. In particular, we do not assume differentiability properties on $s$ and $h$ and we obtain that the strong type inequality $\left( p,q \right),q\,<p$, is characterized by the fact that the function

$$\Phi \left( x \right)\,=\,\sup \,{{\left( \int_{c}^{d}{{{g}^{q}}w} \right)}^{1/p}}\left( \int_{s\left( d \right)}^{h\left( c \right)}{{{v}^{1-{p}'}}} \right){{\,}^{1/{p}'}}$$

belongs to ${{L}^{r}}\left( {{g}^{q}}w \right)$, where ${1}/{r\,=\,{\,1}/{q\,-\,{1}/{p}\;}\;}\;$ and the supremum is taken over all $c$ and $d$ such that $c\le x\le d$ and $s\left( d \right)\,\le \,h\left( c \right)$.

In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\,\to \,C(X)$ is an epimorphism in the category $\mathcal{C}\mathcal{R}$ of commutative rings (with units). We call such an embedding a $\mathcal{C}\mathcal{R}$-epic embedding and we say that $X$ is absolute $\mathcal{C}\mathcal{R}$-epic if every embedding of $X$ is $\mathcal{C}\mathcal{R}$-epic. We continue this investigation. Our most notable result shows that a Lindelöf space $X$ is absolute $\mathcal{C}\mathcal{R}$-epic if a countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta X$-neighbourhood of $X$. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all $\sigma $-compact spaces and all Lindelöf $P$-spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute $\mathcal{C}\mathcal{R}$-epic.

Given a semidirect product $G\,=\,N\,\rtimes \,H$ where $N$ is nilpotent, connected, simply connected and normal in $G$ and where $H$ is a vector group for which $ad(\mathfrak{h})$ is completely reducible and $\mathbf{R}$-split, let $\tau $ denote the quasiregular representation of $G$ in ${{L}^{2}}(N)$. An element $\psi \,\in \,{{L}^{2}}(N)$ is said to be admissible if the wavelet transform $f\,\mapsto \,\left\langle f,\,\tau (\cdot )\psi \right\rangle $ defines an isometry from ${{L}^{2}}(N)$ into ${{L}^{2}}(G)$. In this paper we give an explicit construction of admissible vectors in the case where $G$ is not unimodular and the stabilizers in $H$ of its action on $\hat{N}$ are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of ${{L}^{2}}(G)$ into $G$-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

On associe à toute structure de proto-bigèbre de Lie sur un espace vectoriel $F$ de dimension finie des structures d’algèbre de Lie d’homotopie définies respectivement sur la suspension de l’algèbre extérieure de $F$ et celle de son dual ${{F}^{*}}$. Dans ces algèbres, tous les crochets $n$-aires sont nuls pour $n\,\ge \,4$ du fait qu’ils proviennent d’une structure de proto-bigèbre de Lie. Plus généralement, on associe à un élément de degré impair de l’algèbre extérieure de la somme directe de $F$ et ${{F}^{*}}$, une collection d’applications multilinéaires antisymétriques sur l’algèbre extérieure de $F$ (resp. ${{F}^{*}}$), qui vérifient les identités de Jacobi généralisées, définissant les algèbres de Lie d’homotopie, si l’élément donné est de carré nul pour le grand crochet de l’algèbre extérieure de la somme directe de $F$ et de ${{F}^{*}}$.

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.

We consider the $k$-osculating varieties ${{O}_{k,\,n.d}}$ to the (Veronese) $d$-uple embeddings of ${{\mathbb{P}}^{n}}$. We study the dimension of their higher secant varieties via inverse systems (apolarity). By associating certain 0-dimensional schemes $Y\,\subset \,{{\mathbb{P}}^{n}}$ to $O_{k,n,d}^{s}$ and by studying their Hilbert functions, we are able, in several cases, to determine whether those secant varieties are defective or not.

We derive a weighted ${{L}^{2}}$-estimate of the Witten spinor in a complete Riemannian spin manifold $({{M}^{n}},\,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.

The $H$-space that represents Brown-Peterson cohomology $\text{B}{{\text{P}}^{k}}\left( - \right)$ was split by the second author into indecomposable factors, which all have torsion-free homotopy and homology. Here, we do the same for the related spectrum $P\left( n \right)$, by constructing idempotent operations in $P\left( n \right)$-cohomology $P{{(n)}^{k}}\left( - \right)$ in the style of Boardman-Johnson-Wilson; this relies heavily on the Ravenel-Wilson determination of the relevant Hopf ring. The resulting $\left( i\,-\,1 \right)$-connected $H$-spaces ${{Y}_{i}}$ have free connective Morava $K$-homology $k{{(n)}_{*}}({{Y}_{i}})$, and may be built from the spaces in the $\Omega$-spectrum for $k\left( n \right)$ using only ${{v}_{n}}$-torsion invariants.

We also extend Quillen's theorem on complex cobordism to show that for any space $X$, the $P{{\left( n \right)}_{*}}$-module $P{{(n)}^{*}}\,(X)$ is generated by elements of $P{{(n)}^{i}}(X)$ for $i\,\ge \,0$. This result is essential for the work of Ravenel-Wilson-Yagita, which in many cases allows one to compute BP-cohomology from Morava $K$-theory.

Call a non-Moufang Bol loop minimally non-Moufang if every proper subloop is Moufang and minimally nonassociative if every proper subloop is associative. We prove that these concepts are the same for Bol loops which are nilpotent of class two and in which certain associators square to 1. In the process, we derive many commutator and associator identities which hold in such loops.

In this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of such modules is given by spaces of jets of tensor fields.

We investigate large sieve inequalities such as

$$\frac{1}{m}\underset{j=1}{\overset{m}{\mathop{\sum }}}\,\,\psi \left( \log \,|P({{e}^{i\tau j}})| \right)\,\le \,\frac{C}{2\pi }\,\int_{0}^{2\pi }{\psi }\,\left( \log [e\,|P({{e}^{i\tau }})|] \right)\,d\tau ,$$

where $\psi$ is convex and increasing, $P$ is a polynomial or an exponential of a potential, and the constant $C$ depends on the degree of $P$, and the distribution of the points $0\,\le \,{{\tau }_{1}}\,<\,{{\tau }_{2}}\,<\,\cdots \,<\,{{\tau }_{m}}\,\le \,2\pi$. The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials.

This paper describes new results on the growth and zeros of the Ruelle zeta function for the Julia set of a hyperbolic rational map. It is shown that the zeta function is bounded by exp$({{C}_{k}}{{\left| s \right|}^{\delta }})$ in strips $|\,\text{Re}\,s\,|\,\le \,K$, where $\delta$ is the dimension of the Julia set. This leads to bounds on the number of zeros in strips (interpreted as the Pollicott–Ruelle resonances of this dynamical system). An upper bound on the number of zeros in polynomial regions $\left\{ \left| \,\operatorname{Re}s \right|\,\le \,{{\left| \,\operatorname{Im}s \right|}^{\alpha }} \right\}$ is given, followed by weaker lower bound estimates in strips $\{\text{Re}\,s\,>\,-C,\,|\,\text{IM}\,s|\,\le \,r\}$, and logarithmic neighbourhoods $\{|\text{Re}s|\le {{}_{\rho }}\log |\operatorname{Im}s|\}$. Recent numerical work of Strain-Zworski suggests the upper bounds in strips are optimal.