We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.

We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry.

We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.

Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.

Abstract prepared by Francesco Paolo Gallinaro

E-mail: francesco.gallinaro@mathematik.uni-freiburg.de

URL: https://etheses.whiterose.ac.uk/31077/

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 $e^{x+y}$ .

We establish bounds for triple exponential sums with mixed exponential and linear terms. The method we use is by Shparlinski [‘Bilinear forms with Kloosterman and Gauss sums’, Preprint, 2016, arXiv:1608.06160] together with a bound for the additive energy from Roche-Newton et al. [‘New sum-product type estimates over finite fields’, Adv. Math.293 (2016), 589–605].

The relationship between species richness and sample area has been characterized in many natural communities but has rarely been examined in crop–weed communities. We determined the species–area relationship in short-term (≤4 yr) and long-term (>15 yr) moldboard-plowed (MP), chisel-plowed (CP), and no-tillage (NT) fields cropped to corn and in short-term MP, CP, and NT fields cropped to soybean. A total of 10 corn fields and 10 soybean fields were sampled for species richness in 14 nested sample areas that ranged from 0.0625 to 512 m2. The influence of sample area on frequency of species occurrence was also determined. Species richness was greater in long-term NT fields than in tilled or short-term NT fields. The species–area relationship in tilled and short-term NT fields was best described by an exponential function. In contrast, a power function was the best fit for the species–area relationship in long-term NT fields. The functional minimum area required to represent 75% of the total weed species in tilled and short-term NT fields was 32 m2. A functional minimum area could not be determined in long-term NT fields because species richness continued to increase over the range of sample areas. Regression functions predicted that sample areas of 1 m2 would contain less than 50% of the observed maximum species richness in these fields. Sample areas of 36 m2 in tilled and short-term NT fields and 185 m2 in long-term NT fields were predicted to measure 75% of observed maximum species richness in these fields. Pigweed species and common lambsquarters occurred at high frequencies and were detected in most sample areas. This information could be used to better define sample area requirements and improve sampling procedures for species richness of weed communities.

We find all real-valued general solutions $f:S\rightarrow \mathbb{R}$ of the d’Alembert functional equation with involution

$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$

$x,y\in S$

$S$

$\unicode[STIX]{x1D70E}~:~S\rightarrow S$

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$

$\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$

$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$

$x,y\in \mathbb{R}^{n}$

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$

Wu [‘An order characterization of commutativity for $C^{\ast }$ -algebras’, Proc. Amer. Math. Soc.129 (2001), 983–987] proved that if the exponential function on the set of all positive elements of a $C^{\ast }$ -algebra is monotone in the usual partial order, then the algebra in question is necessarily commutative. In this note, we present a local version of that result and obtain a characterisation of central elements in $C^{\ast }$ -algebras in terms of the order.

We show that if the pair $(f,g)$ of functions mapping a linear space $X$ over the field $\mathbb{K}=\mathbb{R}\text{ or }\mathbb{C}$ into $\mathbb{K}$ satisfies the composite equation

$$\begin{eqnarray}f(x+g(x)y)=f(x)f(y)\quad \text{for }x,y\in X\end{eqnarray}$$

$f$

$f$

$g$

It is well known that an exponential real function, which is Lebesgue measurable (Baire measurable, respectively) or bounded on a set of positive Lebesgue measure (of the second category with the Baire property, respectively), is continuous. Here we consider bounded on ‘big’ set solutions of an equation generalizing the exponential equation as well as the Goła̧b–Schinzel equation. Moreover, we unify results into a more general and abstract case.