In recent years, the large increase in connected devices and the data that is collected by these devices has caused a heightened interest in distributed processing. Many practical distributed networks are of heterogeneous nature. Because of this, algorithms operating within these networks need to be simple, robust against network dynamics and resource efficient. Additionally, if privacy preservation methods are properly implemented, they add to the power of distributed processing by making it possible to leverage the data of many different users, without infringing the privacy of the individuals involved. In this study we focus on the primal-dual method of multipliers (PDMM), which is a promising dis- tributed optimisation algorithm that seems to be suitable for distributed optimisation in heterogeneous networks. Most theoretical work that can be found in existing literature focuses on synchronous ver- sions of PDMM. However, in heterogeneous networks, asynchronous algorithms are favourable over synchronous algorithms. So far, simulation results have indicated that asynchronous PDMM converges and can even converge in the presence of transmission losses. In this work we analyse the properties of stochastic PDMM, which is a general framework that can model variations of PDMM such as asynchronous PDMM and PDMM with transmission losses. We build upon previous empirical results of PDMM and formulate theoretical proofs to substantiate these results. After defining stochastic PDMM and proving its convergence, we compare a number of PDMM variations that have been mentioned throughout the literature. Lastly, we derive a lower bound for the variance of the auxiliary variable in the context of stochastic PDMM, assuming uniform updating probabilities. This lower bound indicates that subspace based privacy preservation is applicable to certain instances of stochastic PDMM, like asynchronous PDMM. The main result of this work is a theoretical proof that shows that stochastic PDMM converges almost surely if the updating probabilities of each auxiliary variable are nonzero. Two important conclusions that follow from this proof are the almost sure convergence of asynchronous PDMM and unicast PDMM with transmission losses. Another useful result is the fact that subspace based privacy preservation is effective when using asynchronous PDMM.

This thesis aimed to aid ErasmusMC in the workforce planning process of deciding how to implement training among the nursing personnel in order to build a more flexible workforce, so the hospital can be prepared for fluctuating and unexpected demand, while minimizing the costs of doing so. To accomplish this, a mathematical model was developed taking into account the structure of the nursing personnel within the hospital, and the structure of how training among nurses is implemented. Since the planning was aimed to be done on a daily basis, the contractual obligations between nurses and the hospital had to be respected as well. The resulted model was tested using real data provided by Erasmus MC, and with the mathematical optimization solver Gurobi. Consequently, a Genetic Algorithm (GA) approach was proposed; this heuristic accomplished to outperform Gurobi when applied to the biggest two instances considered in this project.

Hearing aids as a form of audio preprocessing is increasingly common in everyday life. The goal of this thesis is to implement a blind approach to the cocktail party problem and challenge some of the regular assumptions made in literature. We approach the problemas wideband FD-BSS. From this field of research, the common assumption of contineous activity is dropped. Instead a number of users detection is implemented as a preprocessing step and ensure the appropriate number of demixing vectors for each time frequency bin. The validity of the standard mixing model used for STFT’s is challenged by looking at the response of a linear array. Source separation is achieved by demixing vectors based on the GSVD, derived in a model-based approach. While most ermutation solvers offer an a posteriori solution for all users, we looked at finding local solutions for a single user. Combining this with the user identification called the alignment step, we conclude that the permutation problem can be reduced to selecting a demixing vector for each discrete time-frequency instance. The correlation coefficient proves to be a sufficient metric to couple reconstructions to the original data as it selects most of the active time-frequency bins. In the far-field case, our approach performs in a comparable but not superior manner. We did find that our method is much more robust against inaccuracies introduced when narrowband channels are assumed but not actually available. This is strongly exemplified by our experiment of a changing DFT-size. The Frobinius norm was suggested as a measure of distance between the estimate STFT and the orignial signals time frequency domain description but it resulted in counter intuitive results which didn’t correspond with other metrics used in this thesis. It is expected that there are effects induced by changing the size of the STFT which are not accounted for. Our demixing vectors achieve comparable inteligibility, measured by STOI, as the compared techniques and it is more robust against smaller sample sizes than the theoretically SINR optimal MVDR.

Nowadays, enormous amounts of data are produced on a daily basis. Whether it is a cute family picture, a funny cat video, or a scientic paper, all of the data is stored. The challenge in data storage nowadays is about nding a way to store a lot of data, in such a way that it will stay preserved for many years without too much maintenance and such that if errors occur, the data can still be retrieved. DNA is a great choice for this, since the DNA from extinct species that lived 10,000 years ago can still be retrieved, it does not need maintenance and it is estimated that it can store 5 PB per gram [2]. However, in reading and writing DNA, substitution, insertion and deletion errors occur, so the data needs to be protected against these errors. Therefore, several coding methods have already been invented and researched. Takahasi et al. [1] designed a full automated system for writing, storing and reading data, which consisted only of the word hello, using DNA. This thesis focuses on the coding method used by [1], namely a Hamming code, and compares it to the implementation of a DNA based Reed-Solomon code, applied to the same data. An analysis is made based on the net information density, GC-weight, homopolymer runs and the error detection and correction properties. As expected, there is a trade-o between the net information density and the error detection and correction properties. Although the net information density of the Reed-Solomon code is lower, it can correct more errors and it has the potential of also being applied to a bigger data set.

Stochastic resonance (SR) is a phenomenon where the performance of a nonlinear system subjected to noise is better than it is without noise. This phenomenon can be found both in natural and artificial systems, especially in threshold-based systems such as comparators or a population of neurons. As noise always exists and interacts with the system, it is advantageous to design a system that purposefully uses SR to boost its performance. One possible use of SR is in the signal reconstruction. In this application, noise is added to the input signal and is processed by a comparator to produce a 1-bit output signal. This output can be averaged to recover the amplified input signal. In this thesis, three challenges regarding the use of SR in signal reconstruction are addressed. Firstly, to use SR not only to reconstruct the original signal, but also to implement mathematical operators, specifically a multiplier and an adder, that take two or more input signals. Secondly, to define the relevant metric(s) that can be used to measure the performance of the operators. Lastly, to build a system out of the proposed SR-based operators. The SR-based mathematical operators are implemented on the system level with a comparator as its fundamental building block. To analyze the behavior of the operators, formulas for noise and distortion power are derived. MATLAB simulation is then used to verify the theoretical analysis. Finally, these SR-based mathematical operators are used to build a Teager Energy Operator (TEO) for action potentials (APs) detection. An SR-based multiplier and adder can be implemented with the use of an XNOR logic gate and a binary half adder, respectively. The theoretical formulas are successful in predicting the noise and distortion behavior of the operators. In conclusion, it is possible to implement mathematical operators, specifically multipliers and adders, which use noise to boost their performance. The noise and distortion behavior of these operators can be predicted mathematically. Signal-to-noise ratio (SNR) and signal-to-noise-and-distortion ratio (SNDR) are used to measure the performance of the operators. Systems, specifically a TEO, can be built using the SR-based operators.

LoRaWAN is amongst the most prominent LPWAN (Low Power Wide Area Networking) technologies and provides an operation mode, called Class B, where devices can be reached with limited latency. One of the most important features of LoRaWAN Class B is multicast, which allows a single transmission to be shared by multiple multicast group member end-devices. This enables key IoT functionalities such as FOTA (Firmware Over The Air) and other applications that need to transfer multiple packets to a set of end-devices. This thesis will delve into LoRaWAN Class B multicast and study its scalability -- the most important behavior to study in all IoT technologies. An ns-3 module for LoRaWAN Class B multicast has been extended and used for in-depth analysis and drawing scalability insights. The study also explores methods to improve performance as well as enhance scalability.

Climate change challenges the resiliency of our cities, and lack of spaces in densely paved areas makes it impossible to implement adaptation and mitigation strategies on the street level. However, buildings have often unused roofs, which can be converted into sustainable roofing options. Green roofs are systems which allow for the growth of different types of vegetation, mitigating heat and capturing water. Blue roofs are layers for water collection, which provide temporary storage and slow release of rainwater. Yellow roof is another name to indicate roofs with photovoltaic (PV) panels. Some areas of the city may be more subject to flooding and heat. Some buildings may instead receive more solar radiation. Therefore, prioritizing which option to install where may be the key to the success of sustainable roofing placement. In this study, we construct an optimization model that evaluates the benefits of the roofing option in each potential location and chooses the combination of roofs and roofing options which maximizes the total societal benefits. The reduction of flooding and heat risks and the production of clean energy are modeled considering both climate variables and the city’s characteristics. However, the derived data and parameters are uncertain by nature. Climate data is used in ensembles and is integrated into the problem by formulating the model into a stochastic and robust version. We take Schiedam as our case study area and solve the model. We also perform uncertainty analysis, making use of clustering and classification tree regression. This analysis allows to understand which parameters are driving which type of solutions, and to understand how robust the constructed model is. Moreover, we formulate the same problem in the shape of a multi-time step model, where the decision maker can make the optimal placement decision every year. In this case, it is possible to simplify the decision process into a one-branch tree for which we present a methodology. The results show that our model is robust in terms of placement choice when parameters change, and we discover which parameters drive the main variations. The total benefits are instead much more reliant on the uncertainty of the problem.

Code-Based Cryptography is a branch of the Post-Quantum Cryptography research area. As such, its focus is on developing algorithms that can be used in the current communication systems to secure them against an adversary powered in the (near) future by a quantum computer. A code-based type cryptosystem is a public key cryptosystem that is resistant or slightly reduces its security level against attacks by the known quantum algorithms. The biggest drawback of this otherwise secure cryptosystem is its large public key. This thesis considers a specific type of linear codes, large minimum distance self-dual codes, and punctured codes derived from them that can provide the same security level as the original McEliece system with approximately a 30% smaller public key. Estimation of the bit security level of a cryptosystem using a small example of such a code for its private key confirms that increasing the minimum weight of the code significantly reduces the public key size of the system. Further, we determine the parameters of putative self-dual codes with a large minimum weight providing classical bit security of 80, 128 and 256 bits (quantum 67, 101, and 183 bits), respectively. For the parameters corresponding to the classical 80 bits security of the McEliece Cryptosystem, a particular example of a binary high minimum distance self-dual code is constructed. It is the first code of its type and length. A punctured code of this example is used for the private key of the McEliece cryptosystem. A new decoding algorithm is introduced, which is suitable for the specific construction of the new self-dual code. Moreover, we present a decryption strategy that decodes the complete code instead of the punctured private key. All this results in a McEliece type cryptosystem with 80 bits security classical (68 bits quantum) and reduced public key size of around 38.5% compared to the original system. Reducing the key size makes the quantum safe McEliece Cryptosystem more attractive for practical use.

It is predicted that in about 100 years most of the earth's fossil fuels will have been depleted. Currently, fossil fuels still make up 80% of the Dutch energy production. Thus, to handle this depletion, the research on renewable energy production and its usage is being stimulated by governments. With the use of fossil fuel energy production, it is easy to increase production to meet the unexpected peaks in energy demand by burning more fuels. However, with renewable sources this is not possible. Thus, the way that the produced energy is being used needs to be altered. Furthermore, the amount of renewable energy that is privately generated has increased over the last couple of years. The energy that has been generated for private use and is not needed at that time, can either be sent back into the grid for other users or charged to a battery for later personal use. The electricity network that will regulate the buying and selling of energy is called a smart grid. When using a battery to store privately generated energy, the decisions that are made for the (dis)charging of the battery are of great influence on the total energy cost at the end of the month. When implementing a battery in a household or company that privately generates energy, these decisions need to be made within a fixed time limit of 15 minutes. In this thesis, four mathematical optimisation methods are compared to each other on result and run time. These methods are dynamic programming, local search, tabu search, and simulated annealing. Dynamic programming gives the solution with the lowest possible cost, but does not always have the lowest average run time. The total cost of the solution resulting from local search does not come close enough to the lowest possible cost generated by dynamic programming to be a viable alternative to dynamic programming. Tabu search is an extension of local search, it could result in a solution with a total cost close enough to the lowest possible cost if it runs more iterations than local search. However, due to this the average run time will exceed the run time of dynamic programming. Therefore, it is also not a viable alternative for dynamic programming. Simulated annealing has a shorter run time than dynamic programming when using a forecast time of 1 day or less. The total cost of the solutions come very close to those of dynamic programming. Therefore, while the run time of dynamic programming still fits within the available time limit, it is advisable to use this method to determine the charging decisions for a private battery in a smart grid. However, the simulations that were run in for this thesis do not encompass the entire real-life case. If after the expansion of the problem to be implemented in real-life the run time of dynamic programming were to exceed the available time period, then simulated annealing would be a good alternative for implementation.

The four-colour conjecture (4CC) is a question that asks whether any map can be coloured using only four colours, with the constraint that neighboring countries must have distinct colours. This conjecture has re- mained unanswered for over 170 years, with a rich history of various attempts to prove it. In 1879, Kempe put forward a proof, but it was invalidated by Heawood after 11 years. Heawood did succeed, however, in prov- ing the weaker five-colour theorem. It wasn’t until 1976 that the first genuine proof was discovered by Appel and Haken. This proof sparked controversy due to its reliance on approximately 1200 hours of computer computation, making it unverifiable by hand. The purpose of this report is to provide a comprehensive overview of the historical background and re- cent advancements in the four-colour theorem. It will delve into the numerous failed attempts to prove the conjecture and discuss the groundbreaking proof by Appel and Haken. Additionally, it will explore a recent endeavor by Dr. Xiang, who approached the problem from a different perspective but also encountered a fallacy in his work. The main contribution of this thesis involves a new proof that builds upon the research of Dr. Yeh, who attempted to prove the four-colour theorem by transforming it into a system of linear equa- tions. The missing crucial steps will be addressed for whose correction new ideas will be introduced, using an integral version of Farkas’ lemma and superadditivity. The ideas being researched do not finalize the proof but contribute towards a possible elegant proof in the future. Finally, related topics such as maps on different surfaces and maps with disconnected regions will be considered, broadening the scope of the discussion.

This bachelor thesis is about binary error-correcting codes. A binary code is a collection words with the same length n that consists only of zeroes and ones. The error-correcting quality of a code is determined by the Hamming distance d of a code. A classical question in coding theory is: What is the maximum number of codewords in a code with length n and Hamming distance d? This maximum is denoted with A(n,d). Determining A(n,d) is quite difficult and often we have to make due with lower and upper bounds on A(n,d). In this thesis we compare six different bounds for finding an upper bound on A(n,d), give a detailed description of each bound, a proof and a clarifying example. The main focus of this thesis lies on the linear programming bound (with extra constraints). For d=4 we dive deeper into the linear programming bound and the Krawtchouck polynomials it uses. Finally we draw some conclusions from the comparisons of the different types of bounds.

Dit onderzoek gaat over de mogelijkheden en onmogelijkheden met betrekking tot constructies met passer en ongemarkeerde liniaal. Hier werken we met de euclidische meetkunde en zullen we dus ook beginnen met de vijf postulaten van Euclides en een aantal toepassingen ervan. Er zal besproken worden wat het betekent voor een punt of constructie om construeerbaar te zijn, hiervoor zal gedefinieerd moeten worden wat het betekent voor een getal om construeerbaar te zijn. Vervolgens komen de volgende meetkundige problemen aan bod: de driedeling van een hoek, de verdubbeling van een kubus, de kwadratuur van een cirkel en de construeerbaarheid van regelmatige veelhoeken. Om te laten zien wat wel en niet mogelijk is wordt er gebruik gemaakt van moderne algebra, zo kan een constructie probleem omgeschreven worden naar een concreet algebraïsch probleem. Hiervoor maken we gebruik van de theorie rondom lichaamsuitbreidingen en de Galois theorie.

While the Minimum Euclidean Distance detection is known to be optimal for channels affected by Gaussian noise, it has been shown that Minimum Pearson Distance detection (MPD) may perform better when the channel is also affected by an unknown offset, though for a good performance some adaptations for classical binary block codes are necessary. It is shown for cosets of first order Reed-Muller codes R(1,m) containing words of weight d/2, where d is the code's distance, that the minimum Pearson distance is always low for m≤4. However, it is possible to find cosets where the minimum Pearson distance is higher for m≥5.

In a military environment, tactical networks enable information sharing between all the different entities in the field. In this environment, multiple groups of people from different organizations, and with different goals and policies have to share information. The information has to be shared without the risk of leaking information to unauthorized entities. Cryptography algorithms are used to encrypt information with a key to remain in control of when, where and to whom it is shared. All information is encrypted based on the concept of content-based encryption. In this unreliable environment, the cryptographic keys used to secure the data have to be available to continue collecting and processing information. A key management architecture should be in place, to facilitate the generation and distribution of these keys. The purpose of this key management architecture is to provide the entities in the field with specific keys such that information access policies can be enforced. The challenge here is that in tactical networks, network partitionings are expected to happen. Therefore, the same keys have to be redundantly available at multiple locations to prevent a single point of failure. In a connected network, the keys can constantly be synchronized between these locations. However, the problem of key de-synchronization occurs if the network is split for some time, keys are changed on both sides, and then the network is recombined. This leads to possible conflicting keys because synchronization was temporarily not possible. The key management architecture must be able to handle such conflicts and reintegrate them as necessary. In this thesis, we present a decentralized key management architecture with a solution for the key de-synchronization problem. We propose to use Conflict-free replicated data types, to store the keys at multiple locations and prevent conflicts. Conflict-free replicated data types is a concept to store and replicate data across multiple instances. This data type is characterized by the possibility to update the data in all instances independently, and concurrently, without coordination between the instances. Additionally, three approaches for the coordination of key creation are proposed with different levels of consistency and availability. The architecture and the three approaches are compared in experiments to evaluate the differences and prove the feasibility of the designs.

Subgraph matching is a fundamental problem in various fields such as machine learning, computer vision, image processing, and bioinformatics, where detecting specific substructures within an object is often crucial. In these domains, not only structure plays an essential role, but also the feature information on nodes should be incorporated, thus highlighting the necessity for comprehensive analytical approaches. In this thesis, we propose two novel subgraph matching frameworks using the Fused Gromov-Wasserstein (FGW) distance, namely the Subgraph Optimal Transport (SOT) and the Sliding Subgraph Optimal Transport (SSOT). Both frameworks integrate a dummy node strategy to handle the discrepancy between two graphs of different sizes. The SSOT extends upon the SOT by incorporating a sliding window framework and Wasserstein pruning to enhance the performance, especially for sparse large graphs. Our frameworks can be easily implemented and are adaptable for problems of exact matching, top-k approximate matching, and inexact matching. We further propose a normalized FGW distance to cater to the practical interests and enhance the performance evaluation. We adopt the Frank-Wolfe algorithm for optimization and develop computation-reducing techniques by isolating the dummy node. By conducting experiments on both synthetic and real-world datasets, we demonstrate that the SOT method achieves excellent performance on small graphs, and the SSOT method improves the accuracy over the SOT on large graphs. Both these two methods show the ability to outperform the state-of-the-art methods in noisy environments in terms of accuracy and efficiency.

With cockpit crew costs being the second largest costs of an airline, making optimal use of the available crew is very important. The productivity of the crew is limited by the labor agreement and law regulations, which prevent the crew members from working irregularly or excessively. In this thesis we present several methods to solve the crew scheduling problem as to calculate the crew productivity based on the labor agreements and law regulations. The crew scheduling problem is decomposed into the crew pairing problem and the crew rostering problem. A set covering approach is used to solve the traditional crew pairing problem and a matching algorithm is used to solve the crew pairing problem that arises when we allow flights being retimed. The crew rostering problem is tackled by a minimum cost flow network method and a column generation approach. All the methods are tested on a variety of flight schedules deviating in the number of night flights included.

In this thesis, we design and assess a multi-slice resource allocation framework that is based on machine learning techniques (subset of artificial intelligence techniques). The proposed framework employs two machine learning techniques namely, artificial neural networks and reinforcement learning for resource management in sliced RAN. Alternative multi-slice resource allocation methods that involve only artificial neural networks but not reinforcement learning are also defined.

Every day an extremely large amount of messages get sent over the internet, radio, WiFi, etc. These messages need to be encoded in order to protect the important data. However just like many of these systems messages can get corrupted over a certain channel due to noise, fluctuations in power or temperature etc. [2] Therefore cyclic redundancy checks (CRC), which are bits that are appended to message words to protect the data, are used to detect these errors. This is done by using CRC polynomials. An occurring undetected error is possible in any case, therefore we need to study closely the probability of an undetected error occurring, that is the probability that the received, erroneous data is accepted as the transmitted message. The code words that are the results of adding cyclic redunancy checks (CRC) can also be combined with an error-correcting codes like BCH codes. As shown in [3] combining CRC polynomials with error-correcting can reduce the previously mentioned probability, but one CRC polynomial can perform better. In this thesis these CRC polynomials are studied in combination with certain different codes and are evaluated based on their performance with respects to their undetected error probability. Two examples of codes are given, in the first example a CRC polynomial is found that does not perform well in the system without using an error-correcting code, but performs the best in the system with an error-correcting code. An analysis is done which shows that by looking at consecutive roots of the generator polynomial, optimal CRC polynomials can be found. The second example uses a different code and finds another optimal CRC polynomial

The world can’t keep up with its own data and therefore needs new ways to store it. Research is being done into new types of experimental data storage like DNA sampling or racetrack memory, which could completely revolutionize the way storage works. However, such storage media are prone to certain type of errors like deletion errors where information is lost. Consequently, there has been a surge in interest in developing deletion correcting codes to address this challenge. We are interested in the maximal size of a code correcting up to a certain number of deletions. Nevertheless, constructing these codes can be quite intricate and therefore we are often limited to working with bounds. In this thesis, we will study and evaluate upper bounds for these codes. We will compare bounds that were derived analytically to bounds that have been computed with the use of linear programs. We will compute upper bounds for 1-, 2- and 3-deletion correcting codes. Moreover, we will show how to compute these bounds with a linear program, by posing the problem of finding a largest deletion correcting code as finding the transversal number on a hypergraph. It will become clear that the bounds computed through linear programming are much stronger than any bound derived analytically.

Binary erasure-repairing codes protect information stored on multiple servers by adding parity servers. A characteristic of a code is its cooperative locality; a measure of the amount of servers that need to be accessed to repair erased servers. This report discusses the cooperative locality of Hamming codes and shortened Hamming codes, using the row space of parity-check matrices of these codes. In some cases, an equality for this locality is found, in other cases a bound is given.