For a family of graphs $\mathcal{H}$, the maximum size of a collection of graphs $\mathcal{F}$ for which the symmetric difference $\oplus$ of two distinct graphs $G_1, G_2$ has the property $G_1 \oplus G_2 \not\in \mathcal{H}$ (or $\in \mathcal{H}$) is denoted by $D_{\mathcal{H}}(n)$ (or $M_{\mathcal{H}}(n)$ respectively).
We also denote the independence ratio by $d_{\mathcal{H}}(n) = D_{\mathcal{H}}(n) / 2^{\binom{n}{2}}$.

This thesis gives a new approach to get an upper bound of $d_{\mathcal{H}}(n)$ where $\mathcal{H}$ is the family of graphs that contain cycles of length $2k$, i.e. $\mathcal{H} = \{H : C_{2k} \subseteq H\}.$
The proof of which is based on a different proof by Noga Alon concerning the upper bound of $d_{\mathcal{H}}(n)$ where $\mathcal{H}$ is the family of graphs that contain stars with $2k$ edges, i.e. $\mathcal{H} = \{H : K_{1,2k} \subseteq H\}.$
A bound on the number of $C_{2k}$-free graphs proven by Morris and Saxton indirectly already solved this problem, but Alon's approach could be expanded upon for different types of graph families for which the upper bound is not yet known.
In this thesis we also perform a spectral analysis of the Cayley graph corresponding to the families $\mathcal{H} = \{H : L \subseteq H\}$ and $\mathcal{H} = \{H : L \cong H\}$, in order to give upper bounds on $D_{\mathcal{H}}(n)$.
These upper bounds are compared to known lower bounds.

This thesis is about the following hat guessing game first described by Winkler. Consider a group of n players situated at the vertices of a graph G. An adversary gives each player a hat coloured one of q possible colours. The players are unable to see the colour of their own hat, but each player can see the hat colour of their neighbours in G. The players are asked to simultaneously make a guess on the colour of their own hat, according to a predetermined guessing strategy based solely on the hat colours each player can see. The players win if at least one of them correctly guesses their hat colour, otherwise the adversary wins. Is it possible for the players to devise a strategy which guarantees they win regardless of how the adversary places the hats? Clearly, if the players have a winning strategy on the graph G for q colours, then the same strategy must also be winning for any number of colours less than q. This motivates us to define the hat guessing number HG(G) of G, a parameter first introduced by Farnik.

Definition: Let HG(G) be the maximum q such that the players have a winning strategy when playing the hat guessing game on the graph G with q colours.

Bosek et al. showed an upper bound on the hat guessing number using a partition into independent sets. We show that the same bound holds when V₁, ..., V_l partition the vertices into sets that induce directed acyclic graphs. This implies that for any arc a of the complete graph K_n we have that HG(K_n-a) = n-1, whereas HG(K_n) = n. Furthermore, we give a family of graphs for which choosing a partition into the minimum number of independent sets may be arbitrarily worse than choosing a different partition into independent sets, when applying the bound.

Szczechla have shown the hat guessing number for cycles. They formulate strategies to show that HG(C₄) ≥ 3 and HG(C_3n) ≥ 3 in a complex coordinate system. We reformulate those same strategies and show that they are winning without using this coordinate system.

The winning strategy for C₄ can be generalised to obtain the following result. For an odd prime power q, let M a maximum matching in the complete graph K_q+1. Then HG(K_q+1-M) = q.

On a more general and technical note, we define a notion of equivalence of strategies to try revealing some of the inherent symmetries of the problem. For example, reassigning individual strategies according to an automorphism of the graph does not change whether the collective strategy is winning. We also obtain a notion of uniqueness of a winning strategy, which we conjecture to be a strong, but rare, property.

The Antarctic Circumpolar Current (ACC) is the only ocean current that flows continuously around the globe, and it plays a crucial role in global climate. It encircles Antarctica and keeps warm waters away, helping to maintain the continent’s cold environment. This report studies a simplified mathematical model to capture the ACC’s steady-state behaviour, incorporating the effects of density stratification. The model studied is originally developed by Ding, who reformulated the ocean dynamics of Arctic gyres as a second-order non-linear differential equation for a stream function u(t) that describes the flow field. This includes the effect of Earth’s rotation (Coriolis effect), represented by ω, a prescribed vorticity distribution F (u), and a stratified density function ρ(u).  

In this report, the model is adapted to describe the ACC. This is achieved by imposing Dirichlet boundary conditions representing the two latitudinal limits. This leads to a BVP defined on a finite domain, providing an appropriate mathematical framework for capturing the flow of the ACC. After simplifying the differential equation to an integral equation, our analytical analysis of the model showed that under reasonable assumptions, an unique solution exists. Before computing a numerical solution, the assumptions introduced in the analytic analysis were verified for the specific choices of F and ρ, confirming their regularity and positivity. The equation was then solved numerically using a collocation method, yielding a stream function profile, consistent with observed ACC behaviour. The numerical solution was validated against an exact solution in a special case (constant F and ρ), showing excellent agreement (error < 10−6). Stability tests confirmed that the solution changes only slightly under small perturbations in boundary conditions.  

Finally, the influence of different density stratification profiles was explored. Gentle stratification (constant or linear ρ(u)) led to a stable, symmetric jet, whereas strongly non-linear stratification (quadratic or exponential ρ(u)) resulted in oscillatory or unstable solutions. This highlights the model’s sensitivity to stratification.  

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.

Quaternions prove to be a useful tool when determining rotation groups of regualr polytopes and other object in three and four dimensions. In this thesis it will be studied how the unit quaternions relate to the special orthogonal groups in three and four dimension. Thereafter, this theory is applied to the normal subgroups of the binary octahedral groups as well as the cube and its dual the octahedron. The normal subgroups of the binary octahedral group form 4-dimensional objects, for example the 4-orthoplex or 16-cell and the hyperdiamond or 24-cell.

Symmetries that keep the origin in place, consist of orientation preserving and orientation
flipping isometries, called rotations and reflections respectively. The special orthogonal group in n dimensions, containing all real n × n orthogonal matrices with positive determinant, is precisely the group containing all rotations in n dimensions. The map Φ : H_1→SO(3) defined by q → Φ_q, where Φ_q (p) = qpq^−1 for p∈H is a rotation of ImH ≅ R^3, is a two-to-one surjective homomorphism. The map Ψ : H_1 × H_1 →SO(4), defined by q →Ψ_(q_1,q_2), where
Ψ_(q_1,q_2) (p) = q_1pq_2^−1 is a rotation of H ≅ R^4, is a two-to-one surjective homomorphism.

The n-cube is the generalization of square and cube in general dimensions. Using Φ the binary octahedral group 2O can be established, which describes all quaternions that describe rotational symmetries of the cube and its dual the octahedron. 2O contains 48 unit quaternions, which is twice as many elements as the number of rotational symmetries of the cube and octahedron, due to the two-to-one relation of Φ.

The binary octahedral group 2O describes an object in four dimensions. Its normal subgroups do as well, the most interesting are Q_8, which describes a 4-orthoplex or 16-cell and has the hypercube or 8-cell as dual, and 2T, which describes a hyperdiamond or 24-cell. The binary rotation groups of these polytopes and the object that 2O itself describes, are studied in particular. These binary rotation groups can all be constructed in the same manner.
Write N ∈ {Q_8,2T,2O}, then 2O_N := {(q_1,q_2) ∈ 2O × 2O : q_1N = q_2N} ⊂ H_1 × H_1 is equal to the entire binary rotation group of N. This means that for the rotation group of N, denote this as O_N ⊂ SO(4), it holds that O_N ≅ 2O_N/{±(1, 1)}, where the isomorphism is given by the restriction of Ψ : H_1 × H_1→SO(4) to 2O_N. For the remaining normal subgroups {1} and {±1}, this construction only gives a subgroup of the entire binary rotation group.

One of the main focuses of this thesis is a special kind of symmetrical arrangement called a "Steiner system." Imagine trying to organize a group of objects in such a way that every possible subset of them fits together in a very specific and symmetrical pattern. These patterns are not just mathematical coincidences; they have real-world applications, such as in coding theory.

More precisely, among these Steiner systems, there are some particularly interesting ones known as "Witt designs." These designs are important because their symmetrical properties are used in constructing certain types of error-correcting codes. These codes can correct a certain amount of errors within information that is sent or stored, making it more reliable.

To better understand these Witt designs, the thesis explores two main ways to create them. One method uses something called the Golay code, which is a specific type of error-correcting code. The other method involves a mathematical group called the projective special linear group. Both of these constructions are explained in detail, with proofs provided to show that they indeed form the desired symmetrical patterns.

In short, the Gibbs phenomenon describes the oscillating behaviour followed by an overshoot or undershoot of a Fourier partial sum expansion compared to the original function near jump discontinuities. In particular, the fact that this overshoot or undershoot will never decrease below 9% of the height of the “jump”.
First, some historical remarks about the discovery of the Gibbs phenomenon are given and several fundamental elements of Fourier analysis are introduced. By thoroughly analyzing the Saw-tooth wave function, an example of a function that exhibits the Gibbs phenomenon, we provide some figures in order to visualize this phenomenon. Furthermore, we address and fill in the gaps of the existing mathematical proof to advance the theoretical understanding of the Gibbs phenomenon. This analysis is then generalized to apply to a broader class of functions with jump discontinuities, by first considering a function with a jump discontinuity at 0 and then proving the occurrence of the phenomenon at a general point of jump discontinuity.
Furthermore, a literature research is conducted to evaluate different methods for resolving the Gibbs phenomenon. This thesis includes an in-depth analysis of filtering methods and spectral reprojection methods. Specifically, the Gegenbauer reconstruction method is described and its theoretical foundations in removing the Gibbs phenomenon is examined.
Finally, a real-life application of the Gibbs phenomenon is given by looking at its occurrence in Magnetic Resonance Imaging.
Overall, this thesis not only advances the theoretical understanding of the Gibbs phenomenon but also provides an insight into its resolution and application in Magnetic Resonance Imaging.

This thesis explores and proves the theorem of De Rham using sheaf cohomology. The theorem of De Rham states that for a smooth manifold, the singular cohomology is isomorphic to the De Rham cohomology. Intuitively, this theorem states that the number of holes in a smooth manifold coincides with the failure of the fundamental theorem of calculus for closed differential forms defined on the manifold. This theorem allows us to make analytical conclusions based on the geometry of a surface.

Initially, the homological algebra of abelian groups is studied to quantify the holes in a smooth manifold and the failure of the fundamental theorem of calculus. An isomorphism of these two cohomologies is locally established. Subsequently, sheaves and their corresponding cohomology theory are introduced to lift this local isomorphism to a global one. Ultimately, using a double diagram of sheaves, the singular and De Rham cohomologies are shown to be isomorphic, thereby proving the theorem of De Rham.

Staphylococcus aureus and Pseudomonas aeruginosa are two species of bacteria that are involved in numerous conditions, including lung infections and chronic wound infections. The aim of this project was to study the short-term interactions that occur when P. aeruginosa first encounters an established S. aureus colony, which it then seeks to break apart whilst mixing with S. aureus. Limoli et al. have studied these interactions using experiments, and have thus identified several key aspects involved in these interactions, such as the mechanisms that P. aeruginosa employs to approach the S. aureus colony. The means by which we intended to study interactions between S. aureus and P. aeruginosa is a model that was made by previous members of the Idema group and that was based on the experiments by Limoli et al. In this report, we discuss this model and the biological background relevant to it. We also document the problems that we encountered while trying to run simulations using an existing implementation of this model.

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.

Within this report, an analytical research will be performed on the effectivity of Tuned Mass Dampers (TMDs) and Viscoelastic Dampers (VEDs), and a conclusion will be drawn as to which is most effective in damping the oscillations of an earthquake. First of all, a simple model of second-order differential equations describing the oscillations of each floor within a building of n floors is derived, in which no frictional forces are taken into account. The solution to this model is derived for a building of two floors, which demonstrates the characteristics of oscillations in this model and which shows how resonance is represented mathematically, and what patterns of oscillations look like. Then, a TMD and VED are added to the model of one floor to expose what their working principles are. Subsequently, the methods of adding these dampers to the models for two, five and ten floors are given. For varying earthquake-frequencies, the maximum displacement within 5 seconds is determined, which shows the effect of TMDs and VEDs for different positions within the building. The conclusion from these results is that the effect of the TMDs is always that at the eigenfrequencies, the oscillations of the building are no longer resonating. However, the building will start resonating for other eigenfrequencies due to the TMD. The VED is more consistent, though its damping-effect on resonating frequencies is smaller than for the TMDs. The conclusion is that VEDs are most effective, though more dampers have to be added to improve its effectiveness. After this, a new model is introduced, which does contain a friction force: air resistance. Based on the new model, it is concluded that even oscillations for resonating frequencies are eventually damped out, though they do still grow large. Again, TMDs and VEDs are added to the model, and similar conclusions about the effectivity of the dampers as for the first model were drawn.

In the past few years, the search for good quantum low density parity check (qLDPC) codes suddenly took flight, and many different constructions of these codes have since been presented, including many product constructions. As these code constructions have a natural interpretation in the language of homology, this thesis studies the interplay between homological algebra and various recent product constructions of qLDPC codes.

First, we provide an overview of the theory of singular homology, cellular homology, and homological algebra over vector spaces, and use this theory to analyse two product constructions: the hypergraph product construction and the distance balancing procedure. These constructions can be interpreted as tensor products of chain complexes over vector spaces. We present new proofs for results from the literature regarding the distance and the number of encoded qubits of these product codes.

Secondly, we survey the theory of homology over ring modules, and use this theory to interpret and analyse another product code construction (the lifted product code construction), which is a generalisation of the hypergraph product construction. We prove a Künneth theorem for these codes, and use this theorem to prove a formula for the number of qubits such codes encode.

Thirdly, we investigate the homology of fibre bundles. To this end, we provide an overview of the theory of covering spaces and of fibre bundles, as well as an overview of the theory of homology with local coefficients and of spectral sequences. We then calculate the homology of a specific class of fibre bundles using three different methods. After this discussion, we consider two product code constructions: the fibre bundle product construction and the balanced product construction. We explicate their mathematical foundations, and use these insights to prove two new results for the number of qubits that these product codes encode. Finally, we explain under which conditions these two code constructions and the lifted product construction coincide.

Lastly, we consider a specific example of fibre bundle product codes: the twisted toric code. We determine an analytic expression for the distance of such a code and verify this expression using numerical simulations. Furthermore, we perform an extensive numerical study on these codes to determine how performing a twist alters the scaling of the logical error rate of such codes. We present a new analytic method to explain the scaling of these codes on a small domain, and verify the validity of these calculations by comparing them with the results of our numerical simulations.

Context: In astronomy, collision detection is the computational problem of finding when planets, asteroids or satellites collide. Computer simulations make it possible to show very precisely how bodies move through space, which makes it possible to detect collisions. However, these simulations require a lot of computing power if there are many bodies, which is of course undesirable.
Aims: The aim of this report is to provide more insight into a collision detection method that does not use the computationally expensive simulations, but rather fast mathematical techniques.
Methods: It can be shown that the collision detection problem between two planets is equivalent to finding the integer point (𝑘, 𝑙) between two parallel lines, closest to the origin. The solution uses the continued fraction of the ratio of the orbital periods of the planets, as the slopes of the lines are this ratio. The convergents of the continued fraction form basis vectors with which the point (𝑘, 𝑙) is searched for. This is called lattice basis reduction.
Results: Where brute force always takes 𝑘 steps to find the integer point (𝑘, 𝑙), the method with continued fractions and lattice basis reduction often finds the point in about √𝑘 steps. The worst case however is still 𝑘 steps, the same as brute force. This only happens if there are no basis transformations, so luckily 𝑘 is then often small.
Conclusions: When there are many bodies between which collisions need to be detected, lattice basis reduction provides a fast method. For the basis vectors, the convergents of the continued fraction of the ratio of the orbital periods of the planets must be used. If the solution cannot be reached exactly, (𝑘, 𝑙) can also be estimated.

In this thesis, we use a variation of a commutator technique to prove that l^p-stability is independent of p, for p greater than or equal to one, and for convolution-dominated matrices indexed by relatively separated sets in groups of polynomial growth. Moreover, from the inverse-closedness of the Schur matrices we deduce a Wiener type Lemma for the matrices in the intersection of the weighted convolution-dominated matrices, over all polynomial weights.
Finally, applications of the convolution-dominated matrices are presented. We prove the inverse-closedness of a non-commutative space generated by a discrete series representation restricted to a lattice in a nilpotent Lie group. In addition, we apply the aforementioned result on l^p-stability to show that if \pi(\Lambda)g is a p-frame for the coorbit space Co(L^p) for some p in [1,\infty], then \pi(\Lambda)g is a q-frame for the coorbit space Co(L^q) for each q in [1,\infty], where (\pi, H) is a discrete series representation of a group G of polynomial growth, \Lambda is a relatively separated set in G, and g is a vector in H such that the matrix coefficient V_g g is in the Amalgam space W_{w_a}(G). Moreover, we prove that the frame operator of the frame \pi(\Lambda)g is invertible on the coorbit spaces Co(L^p) for each p in [1,\infty], under the assumption that g is a vector in H, such that V_g g belongs in W_{w_a}(G) for each polynomial weight w_a.

In this thesis the Conjugate Gradient method used in the forecasting of the number of students in the Netherlands is investigated. The addition of international students to the forecasting led to mathematical problem of which a solution could not be computed. While the numerical method is working as intended, the work presented in this thesis shows that the unsolvability lies with a linear system that has no solution. The cause of this is attributed to the use of a selection process in the linear system. Leading to a hugely singular matrix and a forecasting problem without a solution. Several methods were tried to change the use of the selection process but without any results .

Implementing a correction to vesicle fluctuation analysis that accomodates non-zero exposure times, analysing the effects of the changes.

In this thesis, we give a primal-dual interior point method specialized to clustered low-rank semidefinite programs. We introduce multivariate polynomial matrix programs, and we reduce these to clustered low-rank semidefinite programs. This extends the work of Simmons-Duffin [J. High Energ. Phys. 1506, no. 174 (2015)] from univariate to multivariate polynomial matrix programs, and to more general clustered low-rank semidefinite programs.  Clustered low-rank semidefinite programs are programs with low-rank constraint matrices where the positive semidefinite variables are only used within clusters of constraints. The free variables can be used in any constraint, and can be used to connect clusters. The solver uses this structure to speed up the computations in two ways. First, the low rank structure is used to reduce matrix products to products of the form uT M v, where M is a matrix and u and v are vectors, as already suggested by Löfberg and Parrilo in [43rd IEEE CDC (2004)]. Second, an additional block-diagonal structure is introduced due to the clusters. This gives the possibility to do operations such as the Cholesky decomposition block-wise.   We apply this to sphere packing and spherical cap packing. For sphere packing, the speed of the solver is compared to the often used arbitrary precision solver SDPA-GMP, showing the theoretical speedup in time complexity. We give a new three-point bound for the maximum density when packing spherical caps of $N$ sizes on the unit sphere.    

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.

The concept of entanglement is one of the distinguishing features in quantum mechanics. Information about one particle can determine the state of another particle. This information and entanglement in subsystems is quantified by the entanglement entropy. The entanglement entropy has become an important research topic in recent years. Entanglement entropy is expected to grow with the boundary size for systems described by Hamiltonians with local interactions, described by the area law. For one dimension there exists a bound quantifying this area law S = O (2^1/ΔE) in terms of the energy gap ΔE between the ground-state energy and the first excited-state energy. It is an open problem whether the area law holds in higher spatial dimensions. Therefore it is of great interest to study the entanglement entropy in systems that violate the area law. In this thesis spin chain models with local interactions are studied for arbitrary spin. Hamiltonians with a unique ground state are constructed by mapping spin chains onto (coloured) Dyck and (coloured) Motzkin paths. We show that these models express logarithmic or power law violations of the area law for the bipartite entanglement entropy. These results are presented in the table below. The known bound of the area law in one dimensional systems is dependent on the energy gap of these models. The energy gap of the Motzkin-path model is investigated by constructing the an orthogonal excited state with small energy. In doing so we associate the Hamiltonian with the Laplacian of a graph. We present an alternative proof for the bound on the energy gap of the colourless Motzkin-path model than S. Bravyi et al and R. Movassagh et al. We show that the energy gap ΔE scales in terms of the chain length n as ΔE = O(n^-2) in the limit n →∞. Therefore ΔE → 0 in this limit, resulting in area-law violations of the entanglement entropy while maintaining the known bound on the entanglement entropy.

In line with the growing trend of using machine learning to improve solving of combinatorial optimisation problems, one promising idea is to improve node selection within a mixed integer programming branch-and-bound tree by using a learned policy. In contrast to previous work using imitation learning, our policy is focused on learning which of a node's children to select. We present an offline method to learn such a policy in two settings: one that is approximate by committing to pruning of nodes; one that is exact and backtracks from a leaf to use a different strategy. We apply the policy within the popular open-source solver SCIP. Empirical results on four MIP datasets indicate that our node selection policy leads to solutions more quickly than the state-of-the-art in the literature, but not as quickly as the state-of-practice SCIP node selector. While we do not beat the highly-optimised SCIP baseline in terms of solving time on exact solutions, our approximation-based policies have a consistently better optimality gap than all baselines if the accuracy of the predictive model adds value to prediction. Further, the results also indicate that, when a time limit is applied, our approximation method finds better solutions than all baselines in the majority of problems tested.