This paper aims to improve the Particle Mesh (PM) method for Newtonian Dynamics (ND) and MOdified Newtonian Dynamics (MOND). The PM method involves discretizing the mass density on a grid using Gaussian smoothing, which is used to calculate the gravitational potential using Fast Fourier Transforms (FFTs). A major downside of Gaussian smoothing is that the mass density of two particles overlaps whenever they are in close proximity, which leads to incorrect orbital dynamics. Hence, the linearity of ND is used to correct the accelerations of particles whenever this overlap in mass density occurs. These short-range corrections involve the use of analytical expressions to eliminate the inaccurate influence of the PM method, correcting the interactions between particle pairs to direct particle-particle interactions. These corrections can introduce instabilities whenever two particles are too close in proximity and the time steps are too large. This issue is mitigated by using a softening parameter in the corrections. In MOND, the linearity in the accelerations is removed, which means that these short-range corrections cannot always be used. They may only be used in the limit of high accelerations, for which MOND reduces to ND. The short-range corrections are tested for circular and elliptical two-body systems, showing a significant reduction in error of their expected orbital trajectories. To test the limitations of the PM method under non-linear accelerations in MOND where no corrections may be applied, a wide binary system is used to determine the minimum distance between particles that still yields somewhat accurate. This also demonstrates that without the use of short-range corrections large grids are required, highlighting a major inefficiency of PM methods. Lastly, the corrections are tested in both ND and MOND using the Plummer and Miyamoto-Nagai models, which represent a sphere and a disk, respectively. The Plummer model showed significant improvements in stability in both ND and MOND. However, the Miyamoto-Nagai model shows less optimistic results in MOND due to the difficulty in placing the particles at a minimum separating particles whenever the accelerations are non-lineair. Overall, careful use of these short-range corrections allows for significant improvements in the PM method for both ND and MOND. Further research into optimizing the computational performance of FFTs and short-range corrections using tree codes is still available. Additionally, a time-adaptive PM can be implemented to prevent possible instabilities due to large time steps when using the corrections.

Sixty years after John Bell published his paper in which he showed that local hidden variables with statistical independence are incompatible with quantum mechanics, its implications for the nature of the universe are still hotly debated among physicists and philosophers. In his paper he showed that a universe with local hidden variables and statistical independence should satisfy Bell's inequality, but this is violated by quantum mechanics. Testing Bell's inequality turned out to be difficult, owing to a number of loopholes. These were closed in experiments in 2015 which violated Bell's inequality, putting local hidden variables with statistical independence to rest. In this paper we discuss two conceptually simple versions of Bell's inequality as a game (Maudlin and GHZ) and simulate them using Python. Using local means these games are unlikely to be won, but by utilizing the non-locality of quantum mechanics the game is expected to be won. In our simulations we find that the probability of winning decays exponentially as a function of the length of the game for local strategies in Maudlin's version and we prove that the memory loophole (also known as daily updating) will not help the participants to win. In simulating GHZ as a game we find that the game length follows a geometric distribution.

In this thesis the theory of Markov processes and creation and annihilation operators will be used to derive the time evolution of a discrete reaction-diffusion system. More specifically, we make use of transition rates to construct the generator of a process. We then transform this generator through suitable quantum mechan- ical operators to arrive at our result, namely that the Poisson dis- tribution of a reaction-diffusion system stays Poisson distributed in time with varying rate. This applies for the univariate case, with only a simple birth and death process, as well as the multi- variate case. Furthermore, simulations support this result while also showing that an interacting particle system with pairwise an- nihilation does not evolve according to the Poisson distribution.

All organisms are built out of cellular tissue. Being able to recognise abnormalities in these tissues could be useful in recognizing cancerous cells. In this thesis we construct a mathematical model for cellular tissue based on its spatial structure. We consider cells as elements of the network. Touching cells are considered connected. Cells grow at different growth speeds. We determine the point in time when this network is fully connected, meaning there is a path between every pair of cells through touching cells. This point indicates the start of the last phase of the cellular growth, where friction restricts cell movement. We first use a Poisson point process to generate the locations of the cells. To make the model more similar to cellular tissue, we introduce determinantal point processes which have short-ranged repulsion, meaning points repel each other and thus spread. We compare the repulsion of Poisson and Determinantal point processes with real cellular tissue. We conclude that determinantal point processes have significantly higher repulsion than Poisson point processes. We also conclude that the repulsion in cellular tissue is higher than both point processes. Using simulations, we show that the model with determinantal point processes reaches connectivity significantly earlier than the Poisson model. We conclude that the analytically derived connectivity time point from the Poisson model can be used as an upper bound for the determinantal model.

In this thesis, the hydrodynamic limit (HDL) for two trapping models is studied, the Random Waiting Time Model (RWTM) and the fractional kinetics process (FKP), on a discrete lattice Zd . The RWTM is studied for dimension d ≥ 1 and E[wi ],∞where wi denotes the waiting time at position i . On the other hand, the FKP is studied for d ≥ 3 and a nonexisting first finite moment. Instead, it is assumed that the waiting times wi follow a power law distribution. Two main results are presented in this thesis. Firstly, it is proven that the HDL for the RWTMconverges to the solution of the heat equation. The solution is deterministic. The proof consists of showing that the expectation value of the empirical density fields converge to the aforementioned solution by using the duality property and Doob’s theorem, and that the variance is finite and decays to 0. Additionally, it is proven that a rescaled random walk converges to a Brownian motion by using Lévy’s characterisation of Brownianmotion. Secondly, it is proven that the HDL for the FKP converges to a random measure of the solution of the fractional heat equation, defined in the Caputo sense. Hence, the solution is random. The proof consists of using similar techniques of the proof of the RWTM and of using that the limit of a sequence of random speed measures is again random. Before all of this, an introduction toMarkov processes, their semigroups, martingales, and generators is presented. Additionally, an introduction to random walks, Brownianmotion, and duality is included.

The (nonlinear) behaviour of laser beams can be described with Nonlinear Schrödinger equations (NLS). The purpose of this thesis is to shed light on two mathematical papers that give existence and uniqueness results for an NLS equation called the soliton equation. The contribution is to expand the level of detail with which the analysis and some proofs are presented, and to unify the notation where possible.

This report has been made to gain more insight and knowledge of the Keplerian method for collision detection simulations by simplifying N-body systems to a list or particles with orbital parameters. Collision detection is an essential part in modeling the evolution of protoplanetary disk becoming planetary systems by the merging of planetesimal objects. Presumably Kep- lerian systems allow for a computationally efficient algorithm, having its computational time influenced by orbital parameters, unlike the efficient octree method which only scales with the number of particles.
The aim of this research was to create a collision detection model for a simplified proto- planetary disk system using Kepler systems and analyse the results of the model to check the reliability of this simulation method, i.e. if the quality of the results are trustworthy, perform- ing consistently well, and are accurate. In addition, the method was tested on its computational efficiency by investigating the run time of separate parts of the algorithm.
In the initialisation of the algorithm a particle list is created and sorted by increasing apoap- sis. Then by a sweep and prune filter the list is checked for possible collision pairs. For pairs that have a small enough minimal orbital intersection distance that they can collide, the colli- sion time is calculated and added to a time-ranked collision list. The algorithm keeps merging the pairs on top op the list, creating a new particle with new possible collision pairs and new collision times, and updates the time-ranked collision list after each merge. An empty collision list implies the end of the simulation. Various parameters as well as length of the lists are saved within the algorithm for subsequent examination.
The results of the Keplerian model found showed similarities to other simulation studies, and include the perceived critical eccentricity of 0.02, preservation of shape of body radius distribution, and the formation of large bodies. Simulations showed three stages of collisions, characterised by collisions occurring homogeneously throughout the disk, large body colli- sions, a final stage of small body collisions. The performance of the algorithm could be sum- marised by the number of checks by the sweep and prune filter #checks ∝ N 2 and #checks ∝ ε for eccentricity ε < 0.1, the number of pairs in the collision pair list was found to be propor- tional to the number of starting particles N, the body radius s, the inverse of the inclination 1/I, and the eccentricity ε, so #pairs ∝ N2, s, 1/I, ε. For N < 1000 the computation time of the algorithm appeared to be proportional to N2, but for N < 10.000 proportional to to a higher power. For ε < 10−9 and N < 10.000 the computation time was found to be runtime ∝ s.
The conclusion of this research is that the Keplerian model could be used to model N-body systems for collision detection because of its behavioural resemblance to observable astrophys- ical systems. The computation time of the Keplerian system scales with a higher order per N than for instance the three code algorithm, but has showed to be influenced by orbital parame- ters in which this method differentiates itself from other simulation methods, with a remarkable influence on the total computation time by the body radius s of the system. More research could be done for higher orders of N to better model reality as well as providing more insight on the proportionality of the computation time. Recommendations to the algorithm consist of adding planetary spin as a particle parameter, and methods to incorporate apsidal precession in a com- petent manner.

In this thesis the behaviour of a Bose Einstein condensate is explored that consists of bosons that annihilate. In order to do this a system where bose einstein condensation occurs is modeled as a Zero Range process which is a special case of a Markov process.
First we made a single slow site Zero range process with uniform rates and see how this would be influenced by annihilating particles. For this system it is shown that, in the large system limit, the number of particles in the slow site is given by a differential equation. A series of realization of different sizes of this system are done to support that the fraction of the particles in the ground state converges
to the differential equation.
In order to relate this mathematical approach to physical Bose Einstein condensates it is established that the equilibrium distributions of multiple bosons in one system is the same as the detailed balance of a zero range process where the transition rates depend on the occupation of a site, with an indication function, that is equal to 0 when there are no particles at that site and some constant 𝑐 if there is one
or more particles at that site.
In statistical physics we need to take the energy of the state of the system into account. If the particles do not have interaction with each other this is the sum over all particles of the energy of the particle site. As a zero range model only has rates with one particle hop we can see that the difference in energy of the system is equal to the difference in energy of the sites of hop. In order to stay in detailed
balance the rate for the specific hop must be exp(BΔ𝐸) times the opposite rate, where B = 1 / 𝑘_b 𝑇.
Now we can go from any system with a set of spin orbitals and energies of those spin orbitals and design a zero range model where the spin orbitals correspond to sites of the zero range model with rates between those sites, such that the systems detailed balance distribution is the same distribution as the equilibrium of the physical system. In this zero range model an annihilation term is introduced.
This can be any function of occupation for the site, but in this report a simple relation that decreases when Δ𝐸 increases is used.
In order to see the effect on such systems the system is numerically approached for a 100 particles system in a 3 dimensional harmonic potential. This system is too complicated to get an differential equation that can be easily solved. Therefore it is approximated. This numerical approximation appears to have similar behaviour to the numerical approximation done in [3]. However the results are not identical.
The advantage of the model in this thesis is that an annihilation therm can be implemented. This approximation of a 3 dimensional harmonic oscillator with annihilating particles is done. If annihilation is slow enough, the exited sites form a different equilibrium compared to a Bose Einstein condensate of non annihilating particles. This happens as long as there is a condensate in the ground state.

The aim of this thesis was to test the efficiency in practice of an analytical propagator with collision detection for N-body Keplerian systems. This can be used to simulate the evolution of a protoplanetary disk, which gives insight into how planetary systems form. The analytic propagator calculates collisions one by one, while a numerical propagator would compute each time step. The idea of using the analytic propagator is that collisions are rare in astronomical scales, such that jumping from collision to collision and calculating it, is more efficient than calculating all the time steps that are between collisions. Simplifying the orbits of the planetesimals into perfect Keplerian orbits, analytical solutions exist which are used by the analytic propagator.

In this thesis, the runtimes of simulations were measured as well as other properties directly related to the runtime. The overall efficiency of the algorithm with respect to N seemed to be O(N³), which is one power less than previously predicted. The prediction was that the runtime of the full simulation is O(N²ε + N⁴s³/Ia³). Here ε is the maximum eccentricity, s/a is the ratio of a planetesimal's radius to the semi-major axis of its orbit, and I is the maximum inclination. This was calculated by estimating the total number of collisions to be O(N²s²/Ia²) and the runtime for each collision to be O(N²s/a). But the number of collisions turns from quadratic to linear in N, implying that above a certain N almost all planetesimals collide, which reduces the power of N by one. For comparison, the octree code has an algorithmic efficiency of O(N logN) per time step, and the number of steps for a fixed integration time grows as O(N^4/3 log N).

Economic engineering models individual agents as inertia elements and can be viewed as a microeconomic theory based on analogs with classical mechanics. In this thesis the economic engineering concept of modeling individual agents is used to model economic systems consisting of many agents, e.g. an entire country. This is done using classical statistical physics. In statistical physics the microscopic movement of individual gas particles as described by classical mechanics and the macroscopic properties of gases as described by thermodynamics are linked. Using this insight, microeconomics and macroeconomics are linked within the economic engineering framework.

The critical contribution of this thesis is finding the analog of Gibbs' interpretation of entropy, calling it the amount of diversification. This is done as follows. A thermodynamic system in equilibrium is seen as the analog of a Pareto optimal economy or macroeconomic equilibrium. The 2nd law of thermodynamics guarantees the existence of thermodynamic equilibrium and it follows that the entropy is maximized in equilibrium. Clausius interpreted the entropy as an "arrow of time" that pushes the system towards equilibrium. In this thesis Adam Smith's invisible hand is then viewed as an economic analog of Clausius' entropy. Gibbs gives a statistical interpretation to entropy. By calling the amount of diversification the analog of the Gibbs entropy, it follows that a Pareto optimal economy is fully diversified. The amount of diversification contains both the distribution of economic rent over agents and the portfolio diversification of agents for different goods.

Based on the diversification analog, the thesis develops several further analogs. The analog of the partition function is called the opportunity function and gives the opportunities for extracting profits from an economic system by trading. From this the economic engineering analog of the free energy follows. The temperature and chemical potential are given economic engineering analogs as well, namely the level of welfare and the disposable income per capita respectively. The thesis is finalized with applications of the theory developed.

Synchronisation refers to the tendency of coupled oscillators to move in phase with one another over time, departing from their own natural frequency. This phenomenon occurs in almost all branches of science, engineering, and social life. Synchronisation is a well understood concept in classical mechanics. Quantum synchronisation involves attempting to extend this concept to quantum mechanics, where there are number of phenomena such as damping that are more difficult to realise. This thesis focuses on two things. Firstly, it explores the bridge between classical and quantum synchronisation. Both the equivalencies as well as the difficulties that must be overcome. This is initially done by looking at a simple classical model for two coupled oscillators and extending this to a quantum model, and subsequently by exploring the derivation of a master equation for a damped quantum harmonic oscillator with non-linear damping. Secondly, it explore the synchronisation of four coupled quantum van der Pol oscillators. To this purpose, the six different systems (tree, chain, loop, tower, spade, all-to-all) with four coupled classical van der Pol oscillators first analysed and the Arnold tongues are determined, showing for what parameters synchronisation is expected to occur. The equivalent systems with four quantum van der Pol oscillators are explored to determine whether the classical Arnold tongues give a good indication of when quantum synchronisation is expected to occur. An alternative measure for synchronisation is also investigated for these systems based on the relative entropy of coherence. The Arnold tongues for each of the four classical oscillator systems were determined. For some systems (chain and loop) the Arnold tongues were much smaller than for the three oscillator system while for others (tree, all-to-all) they were only slightly smaller. The quantum Arnold tongues showed very similar behaviour to the classical Arnold tongues. Furthermore, when investigating the correlations between different pairs of oscillators, it was found that the correlation decreased as the number of connections separating two oscillators increased. In the chain system, an expected symmetry was found. The relative entropy of coherence is found to be a good measure of synchronisation, better than the more frequently used complex correlator. This is because the relative entropy can take into account the entire system, while the correlator describes the synchronisation between two individual oscillators in the system.

In this report, we have taken a look at three different types of trebuchets: the seesaw, the traction trebuchet or mangonel, and the counterweight trebuchet. We have created models to describe their motion by making use of Lagrangian mechanics and have compared them based on their maximum ranges. Moreover, we have also calculated a lower bound for the dimensions of themain beam needed to prevent it from breaking based on a static state. We have seen that the mangonel and trebuchet improved on the seesaw by increasing the range roughly sixfold which was expected. The maximum range of the trebuchet, however, slightly decreased compared to the mangonel. This was something we did not expect but think is due to the fact that the dimensions of the catapults were not optimised. Lastly, we found that the beam needed to be at least 18.7 cm to prevent it from breaking. However, for the real catapults, the beam would most likely need to be thicker to prevent it from breaking.

Over the past century various different discrepancies in the expected and observed behaviour of galaxies and galaxy clusters were found. Together this is called the missing mass problem and the most well known theory trying to explain these differences states that there is additional undetectable mass in the form of dark matter. Modified Newtonian Dynamics (MOND) is another theory that tries to explain these discrepancies in a different way then by introducing dark matter. Instead the theory changes Newtons law of gravity for low accelerations, smaller than Milgroms constant a0. In this bachelor thesis we look at a discrete model to simulate MOND in galaxy clusters. To simplify calculations we will not be looking at full MOND, which holds for all accelerations but instead we look at the so called deep MOND, which only holds for accelerations much smaller than a0. We use two different versions of the Poisson equation, the standard version for Newtonian dynamics and a modified version for MOND, to compute the gravitational potential fields caused by a mass density distribution. This is done by using the discrete Fourier transform on an discretized region of space. The initial mass density distribution consists of a number  of galaxies ranging between 50 and 1000 per cluster. Each galaxy is modelled as a sphere of constant density. To calculate the potential with MOND we use an iterative process starting from the potential we get using Newtonian dynamics. In this iterative process we made use of the Helmholtz decomposition. From the MOND potential we can compute an apparent mass distribution, which is the mass distribution that would result in the same MOND potential using Newtonian dynamics. This apparent matter distribution we use to predict at what distance of a galaxy most apparent dark matter is located. Lastly we also look at the apparent mass distributions when the galaxy cluster is projected on a 2d plane. All of these calculations were made in Python on a discrete grid of 256 × 256 × 256 points.
When looking at the total amount of apparent mass in concentric spheres around galaxies in our cluster we saw that this increases in three distinct phases. The middle phases, where a linear increase was seen, had a slope in the same order of magnitude as the theoretical value. We found that the average apparent mass density is the highest in the center of galaxies and decreases very quickly at higher distance to the galaxy. When the distance becomes high enough to reach neighbouring galaxies in the same cluster the average apparent mass density stabilizes and becomes almost constant, but still slightly decreases. For the projected mass density a similar pattern was found.

A one-dimensional model of respiratory deposition is developed based on an Eulerian approach. The model simulates aerosol deposition in all generations of the respiratory tract by numerically solving the aerosol general dynamics equation (GDE) for a range of aerosol diameters. The lung geometry is described by Weibel’s morphometric model, with a time varying alveolar geometry to accommodate inhalation dynamics. The model is first valuated by comparing it with numerical and experimental results. Afterwards a series of parameter studies is performed by changing breathing conditions, particle parameters and lung geometries. An increase in Tidal volume and decrease of breathing period resulted in an increase of the total deposition fraction for coarse particles and a decrease of the total deposition fraction for ultrafine particles. An increase in the particle density resulted in an increase in the total deposition fraction. A decrease in the airway diameter generally resulted in an increase of the total deposition fraction. This difference was most noticeable in the tracheobronchial region. Decreasing the airway diameter in the tracheobronchial region mostly effects coarse particles while decreasing the airway diameter in the alveolar region mostly effects ultrafine particles.

As quantum computers are developing, they are beginning to become useful for practical applications, for example in the field of quantum metrology. In this work, a variational quantum algorithm is used to find an optimal probe state for measuring parameters in a noisy environment. This is achieved by optimizing a cost on a quantum computer, based on the Fisher information of the parameters to be estimated. These parameters are then estimated using maximum likelihood estimators. In a simulation, a probe state was found that performed better than the best possible state for noiseless measurements, although this could not be reproduced on an actual quantum computer.

This thesis is divided into five parts (covering chapter 1–5), that together try to give the reader a basic understanding of the symmetries of and the mathematical structure behind the standard model. The first two chapters cover some Lie and representation theory of the Lie groups U(1), SU(2) and SU(3), which play a central role in the standard model. In the third chapter, the structure of the standard model is given, based purely on observed symmetries and (quantum numbers of) particles. In this chapter, the mathematics of the first two chapters is not used yet. In the fourth chapter, the Lie theory of chapter 1 is used to treat the gauge theory of U(1), SU(2) and SU(3), where for SU(2) also spontaneous symmetry breaking is incorporated. Finally, the fifth and last theoretical chapter gives examples of how representation theory and gauge theory can together structure the particles of the standard model and can dictate which particles and particle interactions are physically allowed. The thesis ends with some recommendations for further study of the topic at hand and for further research to symmetries beyond the standard model.

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.

Knowledge of particle deposition is important in clinical settings or when discussing environmental effects of aerosols on humans.
Particle deposition in the human respiratory tract is determined by breathing patterns and lung morphology, as well as particle properties and deposition mechanisms.
In this study we develop a 1-dimensional model that numerically solves the general dynamic aerosol equation in the human respiratory tract.
The model can be used to calculate deposition fractions for a range of initial parameters.
We use Weibel's morphometric model to describe the lung geometry.

The model is validated by comparing it with previous numerical results, and running sensitivity tests to examine its consistency with parametric variations.
The model proved to be computationally efficient, requiring just seconds to run a simulation.
We use this to perform a number of parametric studies, most notably changing the tidal volume and the breathing rate.
For both of these, an increase in either the volume or the rate decreased the deposition fraction across the spectrum of particle sizes, apart from at the tails of the distribution.
We also examine the effect of particle density on the deposition fraction, which increases with an increasing density.
The source code is published along with this thesis, allowing anyone to perform arbitrary parametric studies of their own.

This thesis gives a thorough description of the mathematical tool called reflection positivity, which can be used to prove the occurrence of phase transitions in physical models. A major result, although already known, is a theorem that gives tractable conditions on the Hamiltonian such that the Boltzmann functional is reflection positive. In this thesis, the theorem is used to give conditions on free parameters in four different models, such that the model is reflection positive with respect to certain chosen reflections. The evaluated models are (a) the antiferromagnetic quantum Heisenberg model; (b) the spin ice model; (c) the 6-vertex model and (d) the 16-vertex model. For the Heisenberg model we found that for reflections in a reflection plane there are certain parameter values such that the Boltzmann functional is reflection positive, this is an already known and published result. For the spin ice model we found that for the spin invariant reflection there is no symmetry that yields a reflection positive Boltzmann functional, this is a new result. For both the 6-vertex and 16-vertex model we showed that, for certain energy values, the Boltzmann functional is reflection positive with respect to reflections in the diagonal, which are also new results. In the case of the 16-vertex model, this boils down to checking whether or not a matrix is positive semidefinite. Using this result we showed that energy values that allow for the existence of magnetic monopoles do not yield a reflection positive Boltzmann functional. A topic for further research is investigating the occurrence of phase transitions in the models that are shown to be reflection positive, for which chessboard estimates seem to be a promising approach. Furthermore, in this thesis it was not rigorously proved that the spin ice model with a spin inverting reflection gives a reflection positive Boltzmann functional for rotational symmetry or symmetry in a reflection plane. This is believed to be true, but does require further investigation.

In this thesis we study genetic regulatory networks using a minimal nonlinear
model from literature and extend this to multi-compartment gene-to-gene interaction networks using both numerical simulations and random matrix theory.
When we add perturbations to a system of two interaction networks, the largest
eigenvalue becomes larger, which means the system is more likely to be unstable.
It turns out that for a certain region of values of random matrix variables,
we can predict the change of their eigenvalues caused by a perturbation with
perturbation theory. We quantify the stability of the genetic regulatory network
models, where the dynamics is largely governed by random matrices, with maximal Lyapunov exponents (MLE's). We have been able to compute these MLE's
when the model is considered with certain constraints. At last, we research the
correlation between the connectivity of two networks describing such models
and their KL divergence. Here we conclude that the KL divergence between
two networks goes up when the connectivity of one of the networks gets larger.
We conclude that for possible follow-up research more advanced programs are
needed with longer running simulations.