Expositiones Mathematicae最新文献

A series of physically motivated operations appearing in the study of composite materials are interpreted in terms of elementary continued fraction transforms of matrix valued, rational Stieltjes functions.

Adapting the ideas of L. Keen and C. Series used in their study of the Riley slice of Schottky groups generated by two parabolics, we explicitly identify ‘half-space’ neighbourhoods of pleating rays which lie completely in the Riley slice. This gives a provable method to determine if a point is in the Riley slice or not. We also discuss the family of Farey polynomials which determine the rational pleating rays and their root set which determines the Riley slice; this leads to a dynamical systems interpretation of the slice. Adapting these methods to the case of Schottky groups generated by two elliptic elements in subsequent work facilitates the programme to identify all the finitely many arithmetic generalised triangle groups and their kin.

Let $F$ be a nonarchimedean local field of characteristic 0 and residue field of order not divisible by 2. We show how to calculate the product of the covolume of a torsion-free lattice in $PGL(2,F)$ and the formal dimension of a discrete series representation of $GL(2,F)$. The covolume comes from a theorem of Ihara, and the formal dimensions are contained in results of Corwin, Moy, and Sally. By a theorem going back to Atiyah, and by triviality of the second cohomology group of a free group, the resulting product is the von Neumann dimension of a discrete series representation considered as a representation of a free group factor.

In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field $F_q$ can be written as a sum of $m$ many $k\text{th}$ powers as long as $q>k^{\frac{2m}{m-1}}$; and we also compute the smallest positive integers $m$ such that every element of $F_q$ can be written as a sum of $m$ many $k\text{th}$ powers for all $q$ too small to be covered by the above mentioned results when $2⩽k⩽37$.

In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset $E$ of a finite field $F_q$ for which $\left|E\right|>\frac{qk}{\sqrt{q-1}}$ must contain at least two distinct elements whose difference is a $k\text{th}$ power.

We study a recently introduced two-person combinatorial game, the $(a,b)$-monochromatic clique transversal game which is played by Alice and Bob on a graph $G$. As we observe, this game is equivalent to the $(b,a)$-biased Maker–Breaker game played on the clique-hypergraph of $G$. Our main results concern the threshold bias $a_1\left(G\right)$ that is the smallest integer $a$ such that Alice can win in the $(a,1)$-monochromatic clique transversal game on $G$ if she is the first to play. Among other results, we determine the possible values of $a_1\left(G\right)$ for the disjoint union of graphs, prove a formula for $a_1\left(G\right)$ if $G$ is triangle-free, and obtain the exact values of $a_1(C_n\square C_m)$, $a_1(C_n\square P_m)$, and $a_1(P_n\square P_m)$ for all possible pairs $(n,m)$.

This paper proves such a new Hilbert’s Nullstellensatz for analytic trigonometric polynomials that if ${\left\{f_j\right\}}_{j=1}^{n\ge 2}$ are analytic trigonometric polynomials without common zero in the finite complex plane $ℂ$ then there are analytic trigonometric polynomials ${\left\{g_j\right\}}_{j=1}^{n\ge 2}$ obeying ${\sum }_{j=1}^{n\ge 2}{f}_j{g}_j=1$ in $ℂ$, thereby not only strengthening Helmer’s Principal Ideal Theorem for entire functions, but also finding an intrinsic path from Hilbert’s Nullstellensatz for analytic polynomials to Pythagoras’ Identity on $ℂ$.

We make several observations relating the Lie algebra $g_2\subset \mathrm{so}\left(7\right)$, associative 3-planes, and $\mathrm{so}\left(4\right)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in g_2$ cannot have rank 2, and if it has rank 4 then its kernel is an associative subspace. We prove a canonical form theorem for elements of $g_2$. Given an associative 3-plane $P$ in $R^7$, we construct a Lie subalgebra $\Theta \left(P\right)$ of $\mathrm{so}\left(7\right)={\Lambda }^2\left(R^7\right)$ that is isomorphic to $\mathrm{so}\left(4\right)$. This $\mathrm{so}\left(4\right)$ subalgebra differs from other known constructions of $\mathrm{so}\left(4\right)$ subalgebras of $\mathrm{so}\left(7\right)$ determined by an associative 3-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.