The Riemann zeta function, ζ(s) = Σ n^-s, was initially only studied for real s>1. It is not difficult to see this can be extended to complex valued s provided the real part of s is greater than 1. We will study the zeta function on this domain first, and find the Euler product, Π 1/(1-p^-s), where p is prime, which links the zeta function to the primes. To do so, we will study the theory of infinite products.
Next we wish to find an analytic continuation of ζ(s) to the entire complex plane, and we will do so using the Euler-Maclaurin summation formula. To prove the formula holds we will study Bernoulli numbers and polynomials, which also allows us to find specific values of the zeta function. Using the analytic continuation we will then obtain the functional equation, which allows us to study the function in more detail. The functional equation gives us the trivial zeroes, at which point we are able to study the critical strip 0<Re(s)<1, the critical line s=½+it, and attempt to find the non-trivial zeroes.
We investigate the Riemann hypothesis and prove certain classical results, such as Hardy's theorem, which states there are infinitely many zeroes on the critical line. Finally, we discuss certain related theorems from the literature and end by a brief discussion of the consequences of the Riemann hypothesis, as well as what the Generalised Riemann Hypothesis states. ... Read more

This report introduces spherical harmonics, functions defined on the surface of a sphere that play a central role in mathematical analysis, especially in problems with spherical symmetry. They appear in many fields, such as 3D representation within computer graphics, simulation light behaviour and angular momentum within quantum mechanics.

We begin by developing the theory from first principles. We look at what homogeneous harmonics polynomials are and explain how spherical harmonics arise by restricting these polynomials to the unit sphere. Using this we discuss properties such as orthogonality and dimension. We also discuss zonal harmonics, which are symmetric around a chosen axis.

In three dimensions, we solve Laplace’s equation in spherical coordinates to derive explicit formulas for spherical harmonics. Associated Legendre polynomials will play a key role here. This directly connects with angular momentum, which will also be looked at in this report. This report aims to give students an introduction on spherical harmonics and how they can be used. ... Read more

The Rogers-Ramanujan identities are among the most remarkable results in the theory of integer partitions and 𝑞-series. Discovered independently by Rogers, Ramanujan, and Schur, these identities have since appeared in a wide range of mathematical disciplines, including number theory, combinatorics, representation theory, and statistical mechanics. Despite their concise form, the identities have inspired dozens of proofs and interpretations, each revealing new layers of structure and insight.

This thesis investigates four core aspects of the Rogers-Ramanujan identities. First, it presents and explains Ramanujan’s original analytic proof. Second, it explores the combinatorial interpretation of the identities in terms of integer partitions. Third, the thesis provides a chronological overview of key combinatorial bijective proofs, highlighting how these have evolved and deepened our understanding of the identities. Lastly, it presents Igor Pak’s combinatorial proof of the first Rogers-Ramanujan identity.

By combining historical context with mathematical exposition, this work aims to clarify the richness of the Rogers-Ramanujan identities and to contribute to the ongoing effort to understand them more fully. ... Read more

We will study the Clebsch-Gordan coefficients of the modular double of the quantum group Uq(sl(2, R)). This will be done by studying and taking a good look at how B. Ponsot and J. Teschner showed how to compute the Clebsch-Gordan coefficients [1]. Moreover, we will also take an introductory look at the concept of quantum groups by looking at some general theory on Hopf ∗-algebras and their representations. The Clebsch-Gordan coefficients can roughly be described as a relation between a basis of a tensor product U ⊗V of two simple Uq(sl(2, R))-modules and a basis of the decomposition of U ⊗V into simple modules. We will show that this relation can be explicitly described by an integral transformation. Since this describes a relation between modules of a quantum group, the first part of this thesis will give the necessary information to introduce the reader to the concept of quantum groups and their modules. This will be done by introducing Hopf algebras and their modules and then look at their quantum deformations. This first part will also introduce several examples of algebras, Hopf algebras and quantum groups to make the reader get used to the concept of Hopf algebras and quantum groups. ... Read more

In this thesis, we derive a multivariate analogue of Ruijsenaars’s 2F1-generalisation R. We use Hopf algebra representation theory of the modular double of sl.2/, a Hopf algebra structure strongly related to quantum groups, to relate the function R to overlap coefficients of eigenfunctions. Using properties of the algebra and the representation, we derive an Askey-Wilson type difference equation. We moreover recover Ruijsenaars’s unitary transformation kernel E.

Expanding on the Hopf algebra structure, we extend our derivations to the multivariate version of R. Employing representation theory, we obtain multivariate difference equations. Furthermore, we demonstrate that the multivariate function enables the definition of a unitary transformation on multivariate functions in L2..0; ∞/N/. ... Read more

Sophus Lie (1842-1899) known as the founder of the theory of transformation groups, originally aimed to study solutions of differential equations via their symmetries. Over the decades this theory has evolved into the theory of Lie groups. These Lie groups are of an analytic and geometric nature, but Sophus Lie's principal discovery was that these groups can be studied by their "infinitessimal generators" leading to a linearization of the group. The group structure endows this linearized space with a special bracket operation, [x,y]=xy-yx, which gives rise to Lie algebras.

The main applications for Lie algebras stem from physics, notably in quantum mechanics and particle physics. It turns out that representations of Lie algebras are the way to describe symmetries of physical systems. So, it becomes an important task to figure out what all the possible representations are. Thus, our main goal for this thesis is to classify all finite-dimensional semisimple Lie algebra representations. ... Read more

A combinatorial proof of Wigner’s Semicircle Law for the Gaussian Unitary Ensemble (GUE) is presented in this report. The distribution of eigenvalues of different samples of general Wigner matrices is shown to converge to the semicircle distribution, with the aid of histograms created in Python. The type of convergence that is shown is that of the averaged moments of the eigenvalue
distribution of sample GUE matrices to the moments of the semicircle distribution, as the size of the matrices grow large. This is done by using a method known as the ‘method of moments’. The concepts of random matrices, Catalan numbers, mixed moments of standard Gaussian random variables, (non-
crossing) pairings, Wick’s formula and permutation cycles are introduced in this method. The aim of this report is to provide a detailed proof of Wigner’s Semicircle Law in expectation, understandable for bachelor level mathematics students.
... Read more

In this thesis, we will be studying Lie groups and their connection to certain orthogonal polynomials. We will look into the classical Krawtchouk, Meixner and Laguerre polynomials, and the multivariate Krawtchouk and Meixner polynomials as defined by Iliev. Using representations of the Lie groups SU(2) and SU(1,1), it will be shown that the three classical polynomials can be described as matrix coefficients of the representations. Using this connection of the polynomials to Lie groups, we derive various properties of the polynomials from the unitarity of the representation and the associated Lie algebra representation. Next, the representations are generalised to higher dimensional spaces. Doing so, a new connection is shown between the Lie group SU(d+1) and the multivariate Krawtchouk polynomials, extending the known theory for the univariate polynomials. Another new result that will be established is the connection between the multivariate Meixner polynomials and Lie theory. This will be done by defining a representation of SU(1,d) in the Bergman space of the d-dimensional unit ball. Similar as for the univariate polynomials, we will derive the orthogonality, recurrence relations and difference equations from the associated Lie theory. ... Read more

De Wilson- en Racahpolynomen zijn hypergeometrische orthogonale polynomen die helemaal bovenaan staan in het Askey-schema. Deze polynomen zijn de meest algemene hypergeometrische orthogonale polynomen in één variabele en generaliseren de andere hypergeometrische orthogonale polynomen in het Askey-schema.

In deze scriptie wordt ingegaan op twee specifieke eigenschappen van de Wilson- en Racahpolynomen: de drieterms recurrente betrekking en de orthogonaliteitsrelatie. Deze eigenschappen worden met analytische en algebraïsche methoden bestudeerd.

Bij de analytische methode wordt eerst algemene theorie van hypergeometrische functies en orthogonale polynomen bestudeerd. Er worden identiteiten, transformaties en aaneengesloten relaties voor hypergeometrische functies afgeleid. Hiermee kan de drieterms recurrente betrekking van de Wilson- en Racahpolynomen worden afgeleid. Met behulp van de residuenstelling van Cauchy en de theorie van hypergeometrische functies kan de orthogonaliteitsrelatie van beide polynomen worden verkregen.

Bij de algebraïsche methode wordt de Racah-Wilsonalgebra bestudeerd. De Racah-Wilsonalgebra voldoet aan een zogenaamde laddereigenschap waarmee een keten van eigenvectoren kan worden geconstrueerd. Hiermee is het mogelijk om een eindig dimensionale irreducibele representatie te krijgen. Door een inproduct te definiëren op de bases van eigenvectoren van de generatoren van Racah-Wilsonalgebra, kan met behulp van overlapcoëfficiënten een drieterms recurrente betrekking worden afgeleid. Door vervolgens enkele transformaties toe te passen, kan worden aangetoond dat deze drieterms recurrente betrekking overeenkomt met de drieterms recurrente betrekking van de Racahpolynomen. Ten slotte laten we met dit gekozen inproduct zien dat de Racahpolynomen orthogonale polynomen zijn.
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In this thesis we provide an elementary introduction in finite dimensional representation theory of the Lie groups SU(2) and SU(3) for undergraduate students in physics and mathematics. We will also give two application of representation theory of these two groups in physics: the spin and quark models. We begin with first discussing representation theory for finite groups to create intuition for representations. We will explain notions such as intertwining maps and complete reducibility and we will mention some application of representation theory of finite groups inquantum mechanics. Hereafter, we begin with representation theory for Lie groups and Lie algebras, especially the groups SO(3) and SU(2), as these groups will play an important role in the description of spin. One of the main results is that SU(2) is the universal cover of SO(3). Furthermore, we give a description of spin by means of representation theory of SO(3) and its Lie algebra so(3). We will show that half integer representations of the Lie algebra so(3) cannot be exponentiated to representations of the Lie group SO(3), but it can be exponentiated to its universal cover SU(2). Moreover, we study the irreducible representations of SO(3) inside the Hilbert space L₂(R³). We will argue that one of the simplest quantum Hilbert spaces of a particle L₂(R³), can be modified to the completion of the tensor product L₂(R³) ⊗V, where, V is a finite dimensional Hilbert space that incorporates the internal degrees of freedom: spin. V carries an irreducible projective representation of SO(3). We will also discuss the addition of angular momentum of two particles in quantum mechanics. For this, we show how the tensor product of irreducible representations V and W of so(3) decomposes into SO(3) invariant subspaces of L2(R3). Hereafter, we will turn to representation theory of the Lie group SU(3) for setting up the mathematical framework for analysing the quark model. We will proof that there is a one-to-one correspondence between the irreducible representations of sl(3;C)and SU(3). We will also proof the theorem of the highest weight by which we can classify all the irreducible representations of SU(3) and sl(3;C) by their highest weight. We will also introduce the notion of the Weyl group and show that the Weyl group is a symmetry of weights of the finite dimensional representation of sl(3;C). Other properties of these representation, such as the dimension of the irreducible representations of sl(3;C) will be provided. Lastly, the quark model is discussed by means of representation theory of SU(3). We will show how this model can be used to classify two type of particles which also interact by means of the strong force: baryons and mesons. We show that we can classify the lightest mesons and baryons in so-called multiplets by the irreducible representations of SU(3). However, we will also introduce a modification of the strong force which further refines this model. A topic for further study would be how the symmetry group SU(3) describing Quantum Chromo Dynamics (QCD) can be used for the description of mesons and baryons. ... Read more

In this paper we will naturally extend the concept of Fourier analysis to functions on arbitrary groups. We will generalise the idea of a convolution and try to find a formula for Fourier coefficients in such way that the coefficients of the convolution can easily be calculated. In the first section we will start off in familiar territory as we work our way through the Abelian groups. On the cyclic groups the comparison with the torus and the Fourier series is easily made and this enables us to easily copy the functions from the Fourier series and use them on our group. We then expand this idea by comparing the other groups to Fourier series on multiple variables. Here we can again copy the functions over and after some calculations we end up with our desired theorems. Then we will continue working on groups in general but sadly for the non-Abelian groups the idea of comparing it to the Fourier series does not work. To remedy this problem we will introduce representations, homomorphisms between the group and invertible matrices. After introducing the concept of a representation we will show some remarkable theorems from Representation theory, such as Maschke’s theorem and Schur’s lemma. With the help of these theorems we can find the irreducible representations, whose matrix entries from an orthogonal basis. These representations are what we will use to transform the convolution into matrix multiplication. In the last chapter we will go into more specifics on the representations of the symmetric group. The representations on this group can be found with the help of the Young tableaux. Among these tableaux we will find the Specht Modules, on which the group action of Sn action will give rise to the irreducible representations. To conclude we will show how to turn these irreducible representations of the symmetric group into matrices. ... Read more

In this work, Clebsch-Gordan coefficients are studied from both a quantum mechanical and a mathematical perspective. In quantum mechanics, Clebsch-Gordan coefficients arise when two quantum systems with a certain angular momentum are combined and the total angular momentum is to be found. We start by discussing the relevant postulates of quantum mechanics. From there we move on to a discussion of quantum angular momentum, the combining of quantum systems, and Clebsch-Gordan coefficients. Finally, an algorithm for calculating the Clebsch-Gordan coefficients is developed. A firm grasp of linear algebra is required to understand this quantum mechanical perspective. The mathematical perspective is rooted in representation theory. The Clebsch-Gordan coefficients show the decomposition of the tensor product of irreducible representations of the three dimensional rotation group into the direct sum of irreducible representations. We explain the necessary theory to understand how this works. This includes a discussion of irreducible unitary representations, spherical harmonics, direct sum representations, and tensor product representations. Some experience with group theory is required to understand the mathematical perspective. ... Read more

In this thesis, we introduce the quantum groups Uq(SL(2,C)) and Aq(SL(2,C)) as Hopf algebras. We study their representations, including their similarities and differences with the classical theory. We show that the eigenvectors of Koorwinder's twisted primitive elements of Uq(SU(2)) are dual q-Krawtchouk polynomials. We use this explicit expression to define generalised matrix elements and spherical functions in Aq(SL(2,C)). Then we use the Haar functional to show that these generalised matrix elements are Askey-Wilson polynomials with two continuous and two discrete parameters. Next, we show a new result. Namely, two twisted primitive elements of Uq(SL(2,C)) generate Zhedanov's Askey-Wilson algebra AW(3). Consequently, AW(3) is embedded as a subalgebra into Uq(SL(2,C)). We use this to show that overlap functions of twisted primitive elements in Uq(SU(2)) are q-Racah polynomials. With that, we derive a summation formula connecting q-Racah and dual q-Krawtchouk polynomials. ... Read more

context: The long-term evolution of a self-gravitating astrophysical disk can be modeled using secular perturbation theory. Recently, Batygin published a paper where he claims that such a disk with a special density can be described by a Schrödinger equation by using this method. aims: In this thesis, we will study the secular perturbation theory applied to an astrophysical disk with the same density as Batygin, using the Laplace-Lagrange equations. We will take the continuum limit of those equations, and try to find a wave equation like Batygin. methods: We first apply the Laplace-Lagrange equations to a disk with a large number of planets. Then we take the continuum limit of an infinite number of planets. We then compare the numeric results of the discrete disk to the analytic results of the continuum limit. results: The eigenmodes of the system are well approximated by damped sinusoids. Mode number n changes sign n times. The eigenvalues are linear in the mode number. conclusions: The eigenmodes do not satisfy a wave equation. ... Read more

A model is designed for solving gravitational profiles of self-gravitating and rotating planets via the use of Poisson's equation for total gravity, i.e., the sum of the gravitational and rotational potential. Poisson's equation is a partial differential equation that is solved with the usage of Green's functions. This is a major advantage because Earth's surface is an equipotential, which leads to a Dirichlett boundary condition. We apply this methodology to a spherical Earth, where the Green's function is closed. It is demonstrated that, when this Green’s functions based methodology is applied to a homogeneous density, a linear spherical density and the PREM density profiles (where all density profiles are spherically symmetric), then discontinuities occur in the accelerationof gravitation across the surface of the Earth, in the range of 0.01 %- 0.13 %. Such an effect is a symptom of incompatibility with the laws of physics for geostaticalequilibrium, and it is certainly significant as compared to the numerical accuracy of the solution. Moreover, the discrepancy is dependent on geographical latitude, as was to be expected, because it somehow reflects the centrifugal effect. The conclusion is made that this method is indeed capable of detecting, and even quantifying, the incompatibility of spherically symmetric mass distributions with the fundamental laws of physics for self-gravitating and rotating planets. This opens a way -and this is one of the main results- to computing corrections to spherically symmetric mass distributions, such as the PREM model. Based on this method, a further way forward to this end is suggested. ... Read more

This thesis presents an insight in the Riemann zeta function and the prime number theorem at an undergraduate mathematical level. The main goal is to construct an explicit formula for the prime counting function and to prove the prime number theorem using the zeta function and a Tauberian theorem. The Riemann zeta function can be continued analytically to the whole complex plane except at s = 1. Two proofs of this continuation were given by Bernhard Riemann in his famous article ``Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse'' from 1859. Those proofs are studied in detail in this thesis after introducing all the required foreknowledge on the gamma function. The prime counting function π(x) counts the number of primes less than or equal to x. An explicit formula for π(x) in terms of the nontrivial zeros of the zeta function will be constructed in a similar way as Riemann did in his article. Finally, the prime number theorem will be proved. This theorem describes the asymptotic distribution of the primes among the natural numbers. Using the analytic continuation of the zeta function and a Tauberian theorem, the prime number theorem can be proved quite easily with only basic theory from complex analysis. ... Read more

A Taxi Dispatch Problem involves assigning taxis to requests of passengers who are waiting at different locations for a trip. In today's economy and society, the Taxi Dispatch Problem and other transport problems can be found everywhere. Not only in transporting people, but also in food delivery from restaurants and package delivery for all kind of companies. Even though the applications are different, they still have something in common: serving as much as requests as possible, because that means the highest income. In this thesis, we consider the problem in the actual taxi field. A taxi driver often chooses to serve the passenger that is closest, because he makes no money while the taxi is vacant. However, for companies such as Uber, this is probably not the best solution. They have an overview of the locations of the taxis and passengers, and therefore, are able to make an optimal assignment between the taxis and requests. Sometimes, waiting a little longer for new requests leads to an even better solution. Trying to optimize the problem for the long-run and predict where passengers appear and where taxis end up is perfectly suited for Reinforcement Learning (RL), a subfield of Machine Learning. To be able to solve an optimization problem such as a Taxi Dispatch Problem, there needs to be a goal. For a company, this is maximizing the income or profit, and a popular way to do that is by minimizing the travel time. This thesis takes a different approach by looking at the problem from the passenger's perspective, as satisfied passengers lead to more passengers. In this thesis, the goal is to find an optimal policy for assigning taxis to passengers such that the total waiting time over all passengers is minimized, by using Reinforcement Learning. In order to do that, we formulate the problem in terms of the elements of an RL problem, with the RL method Q-Learning as the learning algorithm and ϵ-Greedy as policy. Together with some restrictions and assumptions, we implement this in Java and use this program to make the agent learn and generate results, where the agent is the one that is responsible for assigning passengers to taxis and needs to learn how to make this assignment such that the total waiting time is minimized. We use the program to find the optimal policy. ... Read more

Orthogonality relations of q-Meixner polynomials, polynomials in terms of basic hypergeometric series, will be proved by using spectral measures and a difference operator. ... Read more

The Racah polynomial Rn(λ(x)) is a polynomial of degree n and is variable in λ(x). In this thesis two properties of this polynomial will be studied. One is the orthogonal property of the Racah polynomial. And the other is that the Racah polynomial can also be described as a polynomial of degree x and variable over λ(n). The Racah polynomials will be studied with the use of a representation of the Lie algebra of sl(2;C) and hypergeometric series. To do this, this Lie algebra will first be defined and then we will work towards defining the tensor product of three representations of the Lie algebra sl(2;C). From the tensor product, a series representation for the Racah polynomials will be found, which can be rewritten to a hypergeometric series. Then, the orthogonal property of sl(2;C) will be used to study the orthogonal property of the Racah polynomials. And the polynomial will be rewritten as a polynomial of degree x with the use of some identities of the hypergeometric series. ... Read more

In this thesis we consider the reconstruction of albedo maps of exoplanets. This is done with a new variant of spin-orbit tomography that has been described in [Cowan and Agol, 2008] and more in depth in [Fujii and Kawahara, 2012]. This method reconstructs the albedo map from the reflected-light curve, the total intensity of the light that originates from the host star and is reflected by the planet. In the mentioned papers, the surface map of the planet is modeled as a sum of finite sized surface elements with constant albedo, and the relation between this approximation of the map and the light-curve in the time domain is determined. In this report, we use that the signal is quasi periodic due to diurnal and annual motion, and work with the Fourier peaks of the light-curve. We also approximate the map in a different way, writing it as the sum of spherical harmonics, and neglecting spherical harmonics with high spatial frequencies. This has the advantage that the relation can be worked out analytically (for edge-on and face-on observations) without the use of complex mathematics, and that both the surface map and the light-curve contain a daily frequency. We derive an equation for the reflective light-curve under the assumption that the surface map is not a function of time (no clouds), and that the reflection is Lambertian (equal in magnitude in all directions). This transformation is found to be a linear function of the surface map. This equation is worked out for edge-on and face-on observations with arbitrary axial tilt, which describes the orientation of the spin axis with respect to the observer and the orbital plane. Furthermore, we describe how to invert this relation if the axial tilt is known to the observer. We also aimed at recovering the map if the axial tilt is unknown to the observer, since this would make sure that the reconstruction does not rely on other observations. In contrast to what was found in papers like [Fujii and Kawahara, 2010] and [Fujii and Kawahara, 2012], we did not succeed in this. A number of methods were used for this. The first two looked at the problem from a mathematical perspective: the minimization of the distance between the measured light-curve and the light-curve from the reconstructed map, and Tikhonov regularization. The two failed because both the column space and the singular values respectively are not a function of the axial tilt. The third method that has been treated and tested involved the maximization of the ‘amount’ of positive albedo on the reconstructed map, but a test showed that the distinction that this method makes is in the same order of magnitude as the numerical error, thus proving that this method was not useful as well. Further study might show what causes the results of the two methods to differ in this respect ... Read more