This thesis provides a modern and self–contained study of integral differentiation with respect to axis–aligned rectangles. It focuses on the classical Jessen–Marcinkiewicz–Zygmund (JMZ) Theorem and the later extension by Zygmund. Throughout, our aim is to make the underlying theory and its results accessible to undergraduate–level readers. Every intermediate result is proved in full, ensuring all the essential details are appreciated. After a brief review of the preliminaries, including measure theory and Lp , Lp,∞ and L(log L)k spaces, we then discuss dyadic intervals and the Hardy–Littlewood maximal operator. An important intermediate result is the Lebesgue Differentiation Theorem (LDT). We provide a proof of the LDT that avoids any covering lemmas and instead uses the properties of dyadic intervals. Both the weak– L1 and strong Lp bounds for maximal operators are also presented. Together with the Lebesgue Differentiation Theorem, they form the basis for the main part of the thesis. A detailed and step–by–step reconstruction of the 1935 JMZ Theorem forms the core of this thesis. The theorem extends the Lebesgue Differentiation Theorem from balls and cubes to the more general axis–aligned rectangles. We give a complete proof of the theorem and also show that the condition f ∈L(log+ L)d−1(Rd) is sharp. Further, we revisit Zygmund’s 1967 extension, which considers rectangles with 1 ≤k≤ddistinct side lengths. This naturally leads to a discussion of Zygmund’s Conjecture, which tries to push the limits of Zygmund’s extension of the JMZ Theorem. We analyse the counterexamples by F. Soria and G. Rey, and the examples by A. C´ordoba and F. Soria. Each step in the construction of their examples and counterexamples is explained in detail. Finally, we close the thesis with a few concluding remarks and an outlook for possible future research on the conditions under which Zygmund’s Conjecture holds – or fails.

This thesis explores the Kakeya conjecture for n=2, which states that every subset of Rⁿ containing a unit line segment in every direction has Minkowski dimension n. To tackle this problem we explore what the Minkowski dimension is, and use the Kakeya maximal operator. We start by stating the idea of a Kakeya needle set. We construct a sequence of these sets with arbitrarily small measure, followed by looking at Kakeya sets, which can even have measure equal to zero. We derive a proof of the Kakeya conjecture using the Kakeya maximal operator conjecture and its dual form, where we also prove all necessary implications. Resulting in a complete proof for n=2 with respect to the Minkowski dimension.

Layman's Abstract:
This paper studies apparent patterns in mathematics that break at a certain point and aims to provide a mathematical explanation for these breaks. We focus on two patterns. The first pattern, discovered by D. and J. Borwein, is as follows: $\pi$, $\pi$, $\pi$, ..., $\pi$, $\pi - 0.000000000462...$. These numbers are the outcomes of integrals called the Borwein integrals. At first glance, it is not obvious why this pattern breaks. However, after performing a specific analysis called Fourier analysis, we will find the reason behind the apparent breakdown of the pattern.

Next, we study another very similar pattern: $\pi$, $\pi$, $\pi$, ..., $\pi$, $\pi - 0.000000003589792...$. These numbers are the outcomes of different integrals, which we call the Nahin integrals. Once again, with the help of Fourier analysis we will find a reasonable explanation for why the pattern breaks and what the value of the following numbers will be.

In conclusion, this paper serves as a warning to those who assume a pattern exists based on a first glance. Furthermore, when an apparent pattern does not exist, this paper applies a methodology that can be more broadly used to determine the actual predictable behaviour.

Peer Abstract:
This paper primarily studies the Borwein integrals $B_n$:
\begin{align*}
b_n(x) &= \prod_{k=0}^n \frac{\sin\left(\frac{x}{2k+1}\right)}{\frac{x}{2k+1}}\\
B_n &= \int_{-\infty}^{\infty} b_n \, dx, \quad n = 0, 1, 2, ...
\end{align*}

These integrals are of interest because of their peculiar results, namely $B_0$ up to $B_6$ are all equal to $\pi$. However, $B_7$ is almost, but not quite, equal to $\pi$, equalling approximately $\pi - 0.000000000462$.

First, to try and observe a reason for the breaking of this apparent pattern, we will perform a graphical analysis on the integrands $b_n$. This will prove to be not very insightful, so another approach using Fourier analysis will be applied. To perform such an analysis, the Fourier transform of functions in $L^2$ must first be defined. We do this on the basis of functions in $L^1\cap L^2$. Then, the Fourier transform of $\frac{\sin(\frac{x}{k})}{\frac{x}{k}}, \quad k=1$, is calculated and generalized to an arbitrary $k \in \mathbb{R}$. After this, the Fourier transform of the Borwein integrands is calculated, and their graphs are analyzed.

Interestingly, the Fourier transform of the first Borwein integrand is a Heaviside step function with a 'plateau' of width $\frac{1}{\pi}$, where the function is equal to $\pi$ centered around 0. Each Fourier transform after this is a moving average of the one before, where the moving average window of the $n$th transform is determined by $\frac{1}{\pi(2n+1)}$. This means there is a very simple explanation for where the apparent pattern will break. Namely, if the difference between $\frac{1}{\pi}$ and $\frac{1}{\pi}(\frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n+1})$ becomes negative, then the plateau will vanish as the window becomes larger than the plateau, and even at the center point zero, the function value will become slightly less than $\pi$.

While the value of each Borwein integral decreases, there does exist a limit equal to approximately $\pi - 0.0000704$. Thus, while the pattern does 'break', it never breaks very badly and always remains quite close to $\pi$.

After studying the Borwein integrals and their behaviour at infinity, we will study another sequence of functions. We call these functions the Nahin functions, defined as follows:
\begin{align*}
h_n(x) &= \frac{\sin(4x)}{x} \prod_{k=0}^n \cos\left(\frac{x}{k+1}\right)\\
H_n &= \int_{-\infty}^{\infty} h_n \, dx, \quad n = 0, 1, 2, ...
\end{align*}

Note that $h_n(0) = H_n$. Using the same methodology as for the Borwein integrals, we will calculate the Fourier transform of the Nahin integrands and find a direct link between the Nahin integrands and the Borwein integrals. Namely, the Fourier transform of the Nahin integrands can be written as a factor times the sum of the dilated Fourier transform of $B_0$. The analysis will further result in an explanation for why the Nahin integrals' 'pattern' breaks at $n=30$. When $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n+1}$ is greater than four, the Nahin integrals will no longer be equal to but will be less than $\pi$. Finally, a closed expression will be found for the value of the Nahin integrals, and further research will be suggested to discover the value of this closed expression as $n$ approaches infinity.

District heating leverages centralised, high efficiency combined heat and power (CHP) systems. It uses waste heat to lower energy consumption and reduce greenhouse emissions. The system also supports renewable energy sources like geothermal and biomass, providing a sustainable heating alternative.

This report examines a nonlinear network of pressure and flow challenges. It focuses on enhancing the Newton-Raphson method and refining direct solving techniques for a single time-step. As net- works grow in complexity, efficient and effective solutions become crucial. Various strategies to speed up the Newton-Raphson algorithm in Gradyent’s heat network solver are discussed, where derivative calculations are straightforward. Both direct and iterative methods to improve the algorithm’s steps are explored. The effectiveness of these enhancements is tested and evaluated across networks of different sizes.sc

Let (S, Σ, μ) be a divisible measure space. Let F be a collection of subsets of S that are in Σ. For some applications it can be useful to describe the overlap between the sets in F. The sparse and Carleson constant both describe this overlap in a different way. The closer both of these constants are to 1, the closer the sets in F are to being pairwise disjoint. It has been shown that the sparse and Carleson condition are actually equivalent: we always have that F is Λ-Carleson if and only if F is Λ^-1-sparse. Proving that a Λ^-1-sparse collection is Λ-Carleson is quite simple, but proving that every Λ-Carleson collection also is Λ^-1-sparse turns out to be much harder. Previous proofs of the fact that Λ-Carleson are Λ^-1-sparse, such as the one by Hänninen [14] and Rey [25], have all relied on difficult theory. There is also no known method to exactly find the sets E_Q for each Q ∈ F that we need to satisfy the sparse condition. Rey is able to approximate these sets, but his algorithm has a logarithmic loss that can only be removed when imposing geometric restrictions.

In this paper I will give a proof of the equivalence of the sparse and Carleson condition for any finite collection F that relies only on basic set and optimisation theory. This proof can be extended to infinite collections F with only a minimal restriction. Asides from proving the equivalence, I also describe an algorithm that can find the sets E_Q for each Q ∈ F that we need to satisfy the sparse condition if F is finite and Carleson with respect to a divisible measure μ. Finally, I will describe an algorithm that we can use to find the Carleson constant of a finite collection F if it is unknown.

In this thesis we study for which domain types the Poincare inequality holds for all functions having continuous first derivative. We first consider the classical Poincare inequality, which we prove holds for a very large class of open sets in R^d. We then constructively prove that bounded, open, and connected domains in R^d, which also possess a smooth C¹-boundary, must satisfy the Poincare-Wirtinger inequality. We do this in six successive steps.
First, we show that an arbitrary open rectangle in R^d must satisfy the inequality.Second, we prove that a C¹-diffeomorphism with a sufficient condition, between a set which satisfies the inequality and an open, bounded and connected set implies the open, bounded and connected set also satisfies the Poincare-Wirtinger inequality. Third, we show that there exists such a C¹-diffeomorphism between a domain in the class of open rectangles with one face distorted by a C¹-function and another domain in the class of arbitrary open rectangles in R^d. Fourth, we show the class of all open rectangles with one face distorted by a C¹-function satisfies the Poincare-Wirtinger inequality. Fifth, we show the union of non-disjoint open sets which satisfy the inequality in turn also satisfies the Poincare-Wirtinger inequality. Lastly, we cover the open, bounded and connected domain with a C¹-boundary by a collection of rectangles from the classes of open rectangles with one face distorted by a C¹-function and arbitrary open rectangles to show that the domain satisfies thePoincare-Wirtinger inequality.

Finally, we extend our function space to the first-order Sobolev space and show that we can directly extend our results to this function space.

In this thesis we study the boundedness of a generalization of the Hardy-Littlewood maximal operator, involving rearrangement invariant Banach function space and indices of the spaces.
We first consider a classical proof of boundedness of the Hardy-Littlewood maximal operator on rearrangement invariant Banach function spaces. After establishing necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator, we consider a generalization of the Hardy-Littlewood maximal operator introduced by C. Pérez. We investigate and slightly improve the known sufficient conditions under which this more general maximal operator is bounded on a rearrangement invariant Banach function space. After which we search and find necessary conditions for boundedness in a general setting. In the final section we study Boyd indices and fundamental indices, especially how they are related to boundedness of the more general maximal operator. We also introduce weak fundamental indices and investigate some of their properties and uses. Finally we show how under certain assumptions we can state equivalent necessary and sufficient conditions for boundedness on Lorentz spaces and Orlicz spaces.