Search

We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations., This paper supersedes arXiv:1907.00919 and contains that paper as a subsection. 35 pages, some color figures

This paper classifies Poisson structures with the reduced simple normal crossing divisor on a product of Fano varieties of Picard number 1. The characterization of even-dimensional projective spaces from the viewpoint of Poisson structures is given by Lima and Pereira. In this paper, we generalize the characterization of projective spaces to any dimension.

A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.

This note corrects an error in our paper “A Galois correspondence for reduced crossed products of unital simple $\text{C}^{\ast }$-algebras by discrete groups”, http://dx.doi.org/10.4153/CJM-2018-014-6. The main results of the original paper are unchanged.

The Orthogonal Least Squares (OLS) algorithm is an efficient sparse recovery algorithm that has received much attention in recent years. On one hand, this paper considers that the OLS algorithm recovers the supports of sparse signals in the noisy case. We show that the OLS algorithm exactly recovers the support of $K$-sparse signal $\boldsymbol{x}$ from $\boldsymbol{y}=\boldsymbol{\unicode[STIX]{x1D6F7}}\boldsymbol{x}+\boldsymbol{e}$ in $K$ iterations, provided that the sensing matrix $\boldsymbol{\unicode[STIX]{x1D6F7}}$ satisfies the restricted isometry property (RIP) with restricted isometry constant (RIC) $\unicode[STIX]{x1D6FF}_{K+1}, and the minimum magnitude of the nonzero elements of $\boldsymbol{x}$ satisfies some constraint. On the other hand, this paper demonstrates that the OLS algorithm exactly recovers the support of the best $K$-term approximation of an almost sparse signal $\boldsymbol{x}$ in the general perturbations case, which means both $\boldsymbol{y}$ and $\boldsymbol{\unicode[STIX]{x1D6F7}}$ are perturbed. We show that the support of the best $K$-term approximation of $\boldsymbol{x}$ can be recovered under reasonable conditions based on the restricted isometry property (RIP).

This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.

Alfred Schild has established conditions that Lorentz transformationsmap world-vectors (ct, x, y, z) with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in SL(z, ℂ). We consider the lattice of subgroups of the group originated in Schild’s paper and obtain generating sets for the full group and its subgroups.

Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications., Comment: minor revision

The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections

Suppose that G is a finite group and k is a field of characteristic p > 0. A ghost map is a map in the stable category of finitely generated kG-modules which induces the zero map in Tate cohomology in all degrees. In an earlier paper we showed that the thick subcategory generated by the trivial module has no nonzero ghost maps if and only if the Sylow p-subgroup of G is cyclic of order 2 or 3. In this paper we introduce and study variations of ghost maps. In particular, we consider the behavior of ghost maps under restriction and induction functors. We find all groups satisfying a strong form of Freyd’s generating hypothesis and show that ghosts can be detected on a finite range of degrees of Tate cohomology. We also consider maps that mimic ghosts in high degrees.

Let X(n), for n ∊ ℕ, be the set of all subsets of a metric space (X, d) of cardinality at most n. The set X(n) equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions r: X(n)→ X(n − 1) for n ≥ 2. It is known that such retractions do not exist if X is the one-dimensional sphere. On the other hand, Kovalev has recently established their existence if X is a Hilbert space, and he also posed a question as to whether or not such Lipschitz retractions exist when X is a Hadamard space. In this paper we answer the question in the positive.

The Qpspaces of holomorphic functions on the disk, hyperbolic Riemann surfaces or complex unit ball have been studied deeply. Meanwhile, there are a lot of papers devoted to theclasses of meromorphic functions on the disk or hyperbolic Riemann surfaces. In this paper, we prove the nesting property (inclusion relations) ofclasses on hyperbolic Riemann surfaces. The same property for Qp spaces was also established systematically and precisely in earlier work by the authors of this paper.

Let G 1, G 2 , …, Gt be arbitrary graphs. The Ramsey number R(G 1 , G 2, …, Gt ) is the smallest positive integer n such that if the edges of the complete graph Kn are partitioned into t disjoint color classes giving t graphs H 1, H 2, …, H t, then at least one Hi has a subgraph isomorphic to Gi. In this paper, we provide the exact value of the R(Tn, Wm) for odd m, n ≥ m−1, where T n is either a caterpillar, a tree with diameter at most four, or a tree with a vertex adjacent to at least leaves, and W n is the wheel on n + 1 vertices. Also, we determine R(C n, W m) for even integers n and m, n ≥ m + 500, which improves a result of Shi and confirms a conjecture of Surahmat et al. In addition, the multicolor Ramsey number of trees versus an odd wheel is discussed in this paper.

Let Q be a finite acyclic quiver, let J be an ideal of kQ generated by all arrows in Q, and let A be a finite-dimensional k-algebra. The category of all finite-dimensional representations of (Q, J2) over A is denoted by rep(Q, J2, A). In this paper, we introduce the category exa(Q, J2, A), which is a subcategory of rep (Q, J2, A) of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in rep(Q, J2, A), via the exact representations plus an extra condition. As a corollary, A is a self-injective algebra if and only if the Gorensteinprojective representations are exactly the exact representations of (Q, J2) over A.

The normalized eigenvalues Ʌi(M, g) of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics g on a fixed surface M. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in 𝕊3 or 𝕊4. In this paper a family of extremal metrics induced by minimal immersions in 𝕊5 is investigated.

This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in {\rm Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic., Comment: 29 pages; to appear in the Canadian Journal of Mathematics

In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.

In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares in the polynomial ring in the topology induced by the ℓ1-norm is equal to Pos([–1; 1]n), the cone consisting of all polynomials that are non-negative on the hypercube [–1,1]n. The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted -seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theoremfrom2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of 2d-powers, proving, in particular, that for any integer d ≥ 1, the closure of the cone of sums of 2d-powers in the topology induced by the -norm is equal to Pos([–1; 1]n).

Let A be a semisimple, unital, and complex Banach algebra. It is well known and easy to prove that A is commutative if and only $e^xe^y=e^{x+y}$ for all $x,y\in A$ . Elaborating on the spectral theory of commutativity developed by Aupetit, Zemánek, and Zemánek and Pták, we derive, in this paper, commutativity results via a spectral comparison of $e^xe^y$ and $e^{x+y}$ .

Let q be a prime number and $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta $ a root of an irreducible trinomial $x^{6}+ax+b$ having integer coefficients. In this paper, we provide some explicit conditions on $a, b$ for which K is not monogenic. As an application, in a special case when $a =0$ , K is not monogenic if $b\equiv 7 \mod 8$ or $b\equiv 8 \mod 9$ . As an example, we also give a nonmonogenic class of number fields defined by irreducible sextic trinomials.

For every pseudovariety $\mathbf {V}$ of finite monoids, let $\mathbf {LV}$ denote the pseudovariety of all finite semigroups all of whose local submonoids belong to $\mathbf {V}$ . In this paper, it is shown that, for every nontrivial semidirectly closed pseudovariety $\mathbf {V}$ of finite monoids, the pseudovariety $\mathbf {LV}$ of finite semigroups is also semidirectly closed if, and only if, the given pseudovariety $\mathbf {V}$ is local in the sense of Tilson. This finding resolves a long-standing open problem posed in the second volume of the classic monograph by Eilenberg.

A sequence $\left \{g_k\right \}_{k=1}^{\infty }$ in a Hilbert space ${\cal H}$ has the expansion property if each $f\in \overline {\text {span}} \left \{g_k\right \}_{k=1}^{\infty }$ has a representation $f=\sum _{k=1}^{\infty } c_k g_k$ for some scalar coefficients $c_k.$ In this paper, we analyze the question whether there exist small norm-perturbations of $\left \{g_k\right \}_{k=1}^{\infty }$ which allow to represent all $f\in {\cal H};$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\left \{g_k\right \}_{k=1}^{\infty }$ such that $g_k\to 0$ as $k\to \infty ,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.

Let $N/K$ be a finite Galois extension of p-adic number fields, and let $\rho ^{\mathrm {nr}} \colon G_K \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ be an r-dimensional unramified representation of the absolute Galois group $G_K$ , which is the restriction of an unramified representation $\rho ^{\mathrm {nr}}_{{{\mathbb Q}}_{p}} \colon G_{{\mathbb Q}_{p}} \longrightarrow \mathrm {Gl}_r({{\mathbb Z}_{p}})$ . In this paper, we consider the $\mathrm {Gal}(N/K)$ -equivariant local $\varepsilon $ -conjecture for the p-adic representation $T = \mathbb Z_p^r(1)(\rho ^{\mathrm {nr}})$ . For example, if A is an abelian variety of dimension r defined over ${{\mathbb Q}_{p}}$ with good ordinary reduction, then the Tate module $T = T_p\hat A$ associated to the formal group $\hat A$ of A is a p-adic representation of this form. We prove the conjecture for all tame extensions $N/K$ and a certain family of weakly and wildly ramified extensions $N/K$ . This generalizes previous work of Izychev and Venjakob in the tame case and of the authors in the weakly and wildly ramified case.

In this paper, we focus on a Kirchhoff-type equation with singular potential and critical exponent. By virtue of the generalized version of Lions-type theorem and the Nehari–Pohožaev manifold, we established the existence of Nehari–Pohožaev-type ground state solutions to the mentioned equation. Some recent results from the literature are generally improved and extended.

For fixed$\alpha \in [0,1]$, consider the set$S_{\alpha ,N}$of dilated squares$\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $modulo$1$. Rudnick and Sarnak conjectured that, for Lebesgue, almost all such$\alpha $the gap-distribution of$S_{\alpha ,N}$is consistent with the Poisson model (in the limit asNtends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of$S_{\alpha ,N}$in a random interval of length$L/N$, provided that$L> N^{1/4+\varepsilon }$. The threshold of$\tfrac {1}{4}$is substantially smaller than the threshold of$\tfrac {1}{2}$(which is the threshold that would be given by a naïve discrepancy estimate).Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations$(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $for a nonlacunary sequence$(a_n)_{n=1}^{\infty } $of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

Let $\pi $ be an automorphic irreducible cuspidal representation of $\mathrm{GL}_{m}$ over $\mathbb {Q}$ . Denoted by $\lambda _{\pi }(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi )$ associated with $\pi $ . Let $\pi _{1}$ be an automorphic irreducible cuspidal representation of $\mathrm{SL}(2,\mathbb {Z})$ . Denoted by $\lambda _{\pi _{1}\times \pi _{1}}(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi _{1}\times \pi _{1})$ associated with $\pi _{1}\times \pi _{1}$ . In this paper, we study the cancellations of $\lambda _{\pi }(n)$ and $\lambda _{\pi _{1}\times \pi _{1}}(n)$ over Beatty sequences.

We investigate integral means over spherical shell of holomorphic functions in the unit ball $\mathbb {B}_n$ of $\mathbb {C}^n$ with respect to the weighted volume measures and their relation with the weighted Hadamard product. The main result of this paper has many consequences which improve some well-known estimates related to the Hadamard product in Hardy spaces and weighted Bergman spaces.

In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

The purpose of this paper is twofold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua–Marcus’ inequalities. Our results are stronger and more general than the existing ones.

In this paper, by applying for a new approach of the so-called Tsinghua principle, we prove the nonexistence of locally conformally flat real hypersurfaces in both the m-dimensional complex quadric $Q^m$ and the complex hyperbolic quadric $Q^{m\ast }$ for $m\ge 3$ .

In the integral Khovanov homology of links, the presence of odd torsion is rare. Homologically thin links, that is links whose Khovanov homology is supported on two adjacent diagonals, are known to only contain $\mathbb{Z}_2$ torsion. In this paper, we prove a local version of this result. If the Khovanov homology of a link is supported in two adjacent diagonals over a range of homological gradings and the Khovanov homology satisfies some other mild restrictions, then the Khovanov homology of that link has only $\mathbb{Z}_2$ torsion over that range of homological gradings. These conditions are then shown to be met by an infinite family of 3-braids, strictly containing all 3-strand torus links, thus giving a partial answer to Sazdanovic and Przytycki's conjecture that 3-braids have only $\mathbb{Z}_2$ torsion in Khovanov homology. We also give explicit computations of integral Khovanov homology for all links in this family., Comment: 20 pages, 11 figures. Section 4 has been simplified

Let $M=$ diag $(\rho _1,\rho _2)\in M_{2}({\mathbb R})$ be an expanding matrix and Let $\{D_n\}_{n=1}^{\infty }$ be a sequence of digit sets with $D_n=\left \{(0, 0)^T,\,\,\,(a_n, 0 )^T, \,\,\, (0, b_n )^T \right \}$ , where $a_n, b_n\in \{-1,1\}$ . The associated Borel probability measure $$ \begin{align*} \mu_{M,\{D_n\}}:=\delta_{M^{-1}D_1}\ast \delta_{M^{-2}D_2}\ast \delta_{M^{-3}D_3}\ast \cdots \end{align*} $$ is called a Moran Sierpinski-type measure. In this paper, we show that $\mu _{M, \{D_n\}}$ is a spectral measure if and only if $3\mid \rho _i$ for each $i=1, 2$ . The special case is the Sierpinski-type measure with $a_n=b_n=1$ for all $n\in {\mathbb N}$ , which is proved by Dai et al. [Appl. Comput. Harmon. Anal. (2020), https://doi.org/10.1016/j.acha.2019.12.001].

Let M be a closed, connected, orientable topological 4-manifold, and G be a finite group acting topologically and locally linearly on M. In this paper we investigate the Borel spectral sequence for the G-equivariant cohomology of M, and establish new bounds on the rank of G for homologically trivial actions with discrete singular set., 22 pages (v2). Accepted for publication in the Canadian Journal of Mathematics

Let $\mathbb {F}_q$ be the finite field of q elements. In this paper, we study the vanishing behavior of multizeta values over $\mathbb {F}_q[t]$ at negative integers. These values are analogs of the classical multizeta values. At negative integers, they are series of products of power sums $S_d(k)$ which are polynomials in t. By studying the t-valuation of $S_d(s)$ for $s < 0$ , we show that multizeta values at negative integers vanish only at trivial zeros. The proof is inspired by the idea of Sheats in the proof of a statement of “greedy element” by Carlitz.

We introduce a weak Lefschetz-type result on Chow groups of complete intersections. As an application, we can reproduce some of the results in [P]. The purpose of this paper is not to reproduce all of [P] but rather illustrate why the aforementioned weak Lefschetz result is an interesting idea worth exploiting in itself. We hope the reader agrees.

Let $\Omega $ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$ , $T_{\Omega }$ be the convolution singular integral operator with kernel $\frac {\Omega (x)}{|x|^d}$ . In this paper, we prove that if $\Omega \in L\log L(S^{d-1})$ , and U is an operator which is bounded on $L^2(\mathbb {R}^d)$ and satisfies the weak type endpoint estimate of $L(\log L)^{\beta }$ type, then the composition operator $UT_{\Omega }$ satisfies a weak type endpoint estimate of $L(\log L)^{\beta +1}$ type.

In this paper, we obtain a variation of the Pólya–Vinogradov inequality with the sum restricted to a certain height. Assume $\chi $ to be a primitive character modulo q, $ \epsilon>0$ and $N\le q^{1-\gamma }$ , with $0\le \gamma \le 1/3$ . We prove that $$ \begin{align*} |\sum_{n=1}^N \chi(n) |\le c (\tfrac{1}{3} -\gamma+\epsilon )\sqrt{q}\log q \end{align*} $$ with $c=2/\pi ^2$ if $\chi $ is even and $c=1/\pi $ if $\chi $ is odd. The result is based on the work of Hildebrand and Kerr.

A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p-inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p-inner parallel bodies. Moreover, the sufficient and necessary conditions of the equality case for the main inequality are obtained, which characterize the so-called tangential bodies.

Many authors define an isometry of a metric space to be a distance-preserving map of the space onto itself. In this note, we discuss spaces for which surjectivity is a consequence of the distance-preserving property rather than an initial assumption. These spaces include, for example, the three classical (Euclidean, spherical, and hyperbolic) geometries of constant curvature that are usually discussed independently of each other. In this partly expository paper, we explore basic ideas about the isometries of a metric space, and apply these to various familiar metric geometries.

We show that the space of linear series of certain multi-degree (including the balanced ones) and rank $r$ on a general binary curve has the expected dimension if nonempty. This generalizes Theorem 24 of Caporaso's paper about binary curves from the case $r\leq 2$ to arbitrary rank, and shows that the space of Osserman-limit linear series on a general binary curve has the expected dimension, which was known for $r\leq 2$. In addition, we show that this space of linear series is still of expected dimension after imposing certain ramification conditions with respect to a sequence of increasing effective divisors supported on two general points lying on different components of the curve., Comment: Minor changes, to appear in Canadian Mathematical Bulletin

In this paper, we study the boundedness and compactness of the inclusion mapping from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to tent spaces. Meanwhile, the boundedness, compactness, and essential norm of Volterra integral operators from Dirichlet type spaces $\mathcal {D}^{p}_{p-1 }$ to general function spaces are also investigated.

Let $\mathcal A$ and $\mathcal B$ be commutative and semisimple Banach algebras and let $\theta \in \Delta (\mathcal B)$ . In this paper, we prove that $\mathcal A\times _{\theta }\mathcal B$ is a type I-BSE algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are so. As a main application of this result, we prove that $\mathcal A\times _{\theta }\mathcal B$ is isomorphic with a $C^*$ -algebra if and only if ${\mathcal A}_e$ and $\mathcal B$ are isomorphic with $C^* $ -algebras. Moreover, we derive related results for the case where $\mathcal A$ is unital.

In this paper, we decompose $\overline {D}(a,M)$ into modular and mock modular parts, so that it gives as a straightforward consequencethe celebrated results of Bringmann and Lovejoy on Maass forms. Let $\overline {p}(n)$ be the number of partitions of n and $\overline {N}(a,M,n)$ be the number of overpartitions of n with rank congruent to a modulo M. Motivated by Hickerson and Mortenson, we find and prove a general formula for Dyson’s ranks by considering the deviation of the ranks from the average: $$ \begin{align*} \overline{D}(a,M) &=\sum\limits_{n=0}^{\infty}\Big(\overline{N}(a,M,n) -\frac{\overline{p}(n)}{M}\Big)q^{n}. \end{align*} $$ Based on Appell–Lerch sum properties and universal mock theta functions, we obtain the stronger version of the results of Bringmann and Lovejoy.

In this paper, we characterize the multiple operator integrals mappings which are bounded on the Haagerup tensor product of spaces of compact operators. We show that such maps are automatically completely bounded and prove that this is equivalent to a certain factorization property of the symbol associated to the operator integral mapping. This generalizes a result by Juschenko-Todorov-Turowska on the boundedness of continuous multilinear Schur multipliers., Comment: 13 pages

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

In this paper, we are interested in obtaining large values of Dirichlet L-functions evaluated at zeros of a class of L-functions, that is, $$ \begin{align*}\max_{\substack{F(\rho)=0\\ T\leq \Im \rho \leq 2T}}L(\rho,\chi), \end{align*} $$ where $\chi $ is a primitive Dirichlet character and F belongs to a class of L-functions. The class we consider includes L-functions associated with automorphic representations of $GL(n)$ over ${\mathbb {Q}}$ .

In this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braids, begun in our earlier papers with Islambouli and Abel. We also present several similar results for other families of semi-infinite and bi-infinite multi-colored braids., 37 pages, 13 figures

This paper provides an addendum to M. Hrusak and J. van Mill “Nearly Countable Dense Homogeneous Spaces.” Canad. J. Math., published online 2013-03-08, http://dx.doi.org/10.4153/ CJM-2013-006-8. It was brought to our attention by Su Gao that the proof of Theorem 5.2 in our paper is incomplete. We are indebted to him for this observation. The aim of this note is to correct this. Theorem 5.2 Let G be a closed subgroup of S∞ and let κ be the number of orbits for the canonical action G × 2N → 2N. Then there is an action of a Polish group H on X = N× [0, 1) such that X has κ H-types of countable dense sets. Proof Let G act on X in the following natural way: (g, (n, t)) 7−→ (g(n), t) for g ∈ G, n ∈ N, t ∈ [0, 1). Put F = { f ∈H (X) : (∀ n ∈ N)( f (n, 0) = (n, 0)) } . Then F is a closed normal subgroup of H (X) and hence is Polish. Moreover, for any two countable dense subsets D and E of N × (0, 1) there exists f ∈ F such that f (D) = E. Treat G also as subgroup of H (X). The Polish semi-direct product group H = G o F acts on X as follows: ((g, f ), x) 7→ ( f ◦ g)(x) for f ∈ F, g ∈ G, x ∈ X. Note that topologically, H = G o F is G× F, but its group operation ∗ is given by (g1, f1) ∗ (g2, f2) = (g1g2, f1g1 f2g 1 ). A typical countable dense subset of X has the form D ∪ A, where D is a countable dense subset of N× (0, 1), and A ⊆ N× {0}. By identifying P(N× {0}) and 2N in the standard way, it is clear that we get what we want. Centro de Ciencias Matematicas, UNAM, A.P. 61-3, Xangari, Morelia, Michoacan, 58089, Mexico e-mail: michael@matmor.unam.mx Faculty of Sciences, Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands e-mail: j.van.mill@vu.nl Received by the editors October 28, 2013. Published electronically November 7, 2013. The first author was supported by a PAPIIT grant IN 102311 and CONACyT grant 177758. The second author is pleased to thank the Centro de Ciencias Matematicas at Morelia for generous hospitality and support. AMS subject classification: 54H05, 03E15, 54E50.

Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y \in \mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$., 32 pages. Final version, to appear in Canadian Journal of Mathematics