We use the theory of Jacobi-like forms to construct modular forms for a congruence subgroup of $\text{SL}\left( 2,\,\mathbb{R} \right)$ which can be expressed as linear combinations of products of certain theta functions.

This paper characterizes when a Delone set $X$ in ${{\mathbb{R}}^{n}}$ is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set $X$, let ${{N}_{X}}\left( T \right)$ count the number of translation-inequivalent patches of radius $T$ in $X$ and let ${{M}_{X}}\left( T \right)$ be the minimum radius such that every closed ball of radius ${{M}_{X}}\left( T \right)$ contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to $X$ being an ideal crystal.

Explicitly, for ${{N}_{X}}\left( T \right)$, if $R$ is the covering radius of $X$ then either ${{N}_{X}}\left( T \right)$ is bounded or ${{N}_{X}}\left( T \right)\,\ge \,T/2R$ for all $T\,>\,0$. The constant $1/2R$ in this bound is best possible in all dimensions.

For ${{M}_{X}}\left( T \right)$, either ${{M}_{X}}\left( T \right)$ is bounded or ${{M}_{X}}\left( T \right)\ge T/3$ for all $T\,>\,0$. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set $X$ has ${{M}_{X}}\left( T \right)\,\ge \,c\left( n \right)T$ for all $T\,>\,0$, for a certain constant $c\left( n \right)$ which depends on the dimension $n$ of $X$ and is $>\,1/3$ when $n\,>\,1$.

We study ergodic properties of a family of interval maps that are given as the fractional parts of certain real Möbius transformations. Included are the maps that are exactly $n$-to-1, the classical Gauss map and the Renyi or backward continued fraction map. A new approach is presented for deriving explicit realizations of natural automorphic extensions and their invariant measures.

The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations ${{X}^{2}}-D{{Y}^{2}}\,=\,c$ and ${{x}^{2}}-D{{y}^{2}}=-c$ when $c\,\in \,\mathbb{Z}$, and $D\,\in \,\mathbb{N}$ is not a perfect square. This continues work in [6]–[8].

In this paper we study specializations and one-sided bimodules of simple Jordan superalgebras.

D. H. Lehmer initiated the study of the distribution of totatives, which are numbers coprime with a given integer. This led to various problems considered by P. Erdős, who made a conjecture on such distributions. We prove his conjecture by establishing a theorem on the ordering of residues.

In this paper, we give an integral formula for the eigenfunctions of the ring of differential operators related to the root system ${{B}_{2}}$.

H. O. Kim has shown that contrary to the case of ${{H}^{p}}$-space, the Smirnov class $M$ defined by the radial maximal function is essentially smaller than the classical Smirnov class of the disk. In the paper we show that these two classes have the same corresponding locally convex structure, i.e. they have the same dual spaces and the same Fréchet envelopes. We describe a general form of a continuous linear functional on $M$ and multiplier from $M$ into ${{H}^{p}},\,0\,<\,p\,\le \,\infty $.

We study the representations of extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ where $q$ is $N$-th primitive root of unity ($({{\mathbb{C}}_{q}}$ is the quantum torus in two variables). We first prove that $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$ for a suitable number of copies is a quotient of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$. Thus any finite dimensional irreducible module for $\oplus \,s{{\ell }_{\ell +1}}\left( \mathbb{C} \right)$lifts to a representation of $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$. Conversely, we prove that any finite dimensional irreducible module for $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)$ comes from above. We then construct modules for the extended affine Lie algebras $s{{\ell }_{\ell +1}}\left( {{\mathbb{C}}_{q}} \right)\oplus \mathbb{C}{{d}_{1}}\oplus \mathbb{C}{{d}_{2}}$ which is integrable and has finite dimensional weight spaces.

Let $b$ be an integer with $b\,>\,1$. In this note, we prove that the number of non-zero digits in the base $b$ representation of $n!$ grows at least as fast as a constant, depending on $b$, times $\log \,n$.

In this note we show that the image of the transfer for permutation representations of finite groups is generated by the transfers of special monomials. This leads to a description of the image of the transfer of the alternating groups. We also determine the height of these ideals.

A new construction of the ordinary residue of differential forms is given. This construction is intrinsic, i.e., it is defined without local coordinates, and it is geometric: it is constructed out of the geometric structure of the local and global cohomology groups of the differentials. The Residue Theorem and the local calculation then follow from geometric reasons.

We embed the moduli space $Q$ of 5 points on the projective line ${{S}_{5}}$-equivariantly into $\mathbb{P}\left( V \right)$, where $V$ is the 6-dimensional irreducible module of the symmetric group ${{S}_{5}}$. This module splits with respect to the icosahedral group ${{A}_{5}}$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of ${{A}_{5}}$ on $Q$ to those on the regular icosahedron.

Sufficient conditions are given in order to prove the lowest unknown case of the grade conjecture. The proof combines vanishing results of local cohomology and the ${{S}_{2}}$ condition.

We study a class of subgroups of $\text{PS}{{\text{L}}_{2}}\left( \mathbb{Z} \right)$ which can be characterized in different ways, such as congruence groups, modular forms, modular curves, elliptic surfaces, lattices and graphs.

We consider two dynamical systems associated with a substitution of Pisot type: the usual $\mathbb{Z}$-action on a sequence space, and the $\mathbb{R}$-action, which can be defined as a tiling dynamical system or as a suspension flow. We describe procedures for checking when these systems have pure discrete spectrum (the “balanced pairs algorithm” and the “overlap algorithm”) and study the relation between them. In particular, we show that pure discrete spectrum for the $\mathbb{R}$-action implies pure discrete spectrum for the $\mathbb{Z}$-action, and obtain a partial result in the other direction. As a corollary, we prove pure discrete spectrum for every $\mathbb{R}$-action associated with a two-symbol substitution of Pisot type (this is conjectured for an arbitrary number of symbols).

We give a new measure-theoretical proof of the uniform distribution property of points in model sets (cut and project sets). Each model set comes as a member of a family of related model sets, obtained by joint translation in its ambient (the ‘physical’) space and its internal space. We prove, assuming only that the window defining the model set is measurable with compact closure, that almost surely the distribution of points in any model set from such a family is uniform in the sense of Weyl, and almost surely the model set is pure point diffractive.

Quantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type ${{\text{A}}_{1}}$, $\text{C}$ and $\text{BC}$. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type $\text{C}$.

A well-known theorem of Sarason [11] asserts that if $\left[ {{T}_{f}},\,{{T}_{h}} \right]$ is compact for every $h\,\in \,{{H}^{\infty }}$, then $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$. Using local analysis in the full Toeplitz algebra $T\,=\,T\left( {{L}^{\infty }} \right)$, we show that the membership $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ can be inferred from the compactness of a much smaller collection of commutators $\left[ {{T}_{f}},\,{{T}_{h}} \right]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper ${{C}^{*}}$-subalgebra $T\left( \mathcal{L} \right)$ of $T$ which has the same essential commutant as that of $T$. Thus the image of $T\left( \mathcal{L} \right)$ in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no separable subalgebra $\mathcal{S}$ of $T$ is capable of conferring the membership $f\,\in \,{{H}^{\infty }}\,+\,C\left( T \right)$ through the compactness of the commutators $\left\{ \left[ {{T}_{f,}}\,S \right]\,:\,S\,\in \,\mathcal{S} \right\}$.