In this work, we study error correcting codes and generalize the concepts with a view toward a novel application in the study of DNA sequences. The author investigates the possibility that an error correcting linear code could be included in the human genome through application and research. The author finds that while it is an accepted hypothesis that it is reasonable that some kind of error correcting code is used in DNA, no one has actually been able to identify one. The author uses the application to illustrate how the subject of coding theory can provide a teaching enrichment activity for undergraduate mathematics.

Establishing secret keys from the commonly-observed randomness of reciprocal wireless propagation channels has recently received considerable attention. In this work we propose improved strategies for channel estimation between MIMO or beamforming systems for secret key generation. The amount of mutual information that can be extracted from the channel matrix estimates is determined by the quality of channel matrix estimates. By allocating increased energy to channel estimation for higher gain beamforming combinations at the expense of low-gain combinations, key establishment performance can be increased. Formalizing the notion of preferential energy allocation to the most efficient excitations is the central theme of this dissertation. For probing with beamforming systems, we formulate a theoretically optimal probing strategy that upper bounds the number of key bits that can be generated from reciprocal channel observations. Specifically, we demonstrate that the eigenvectors of the channel spatial covariance matrix should be used as beamformer weights during channel estimation and we optimize the energy allocated to channel estimation for each beamformer weight under a total energy constraint. The optimal probing strategy is not directly implementable in practice, and therefore we propose two different modifications to the optimal algorithm based on a Kronecker approximation to the spatial covariance matrix. Though these approximations are suboptimal, they each perform well relative to the upper bound. To explore how effective an array is at extracting all of the information available in the propagation environment connecting two nodes, we apply the optimal beamformer probing strategy to a vector current basis function expansion on the array volume. We prove that the resulting key rate is a key rate spatial bound that upper bounds the key rate achievable by any set of antenna arrays probing the channel with the same total energy constraint. For MIMO systems we assume the channel is separable with a Kronecker model, and then for that model we propose an improved probing strategy that iteratively optimizes the energy allocation for each node using concave maximization. The performance of this iterative approach is better than that achieved using the traditional probing strategy in many realistic probing scenarios.

Let G be a simple graph on n vertices, and let S(G) be the class of all real-valued symmetric nxn matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The smallest rank achieved by a matrix in S(G) is called the minimum rank of G, denoted mr(G). The maximum nullity achieved by a matrix in S(G) is denoted M(G). For each graph G, there is an associated minimum rank class, MR(G) consisting of all matrices A in S(G) with rank A = mr(G). Although no restrictions are applied to the diagonal entries of matrices in S(G), sometimes diagonal entries corresponding to specific vertices of G must be zero for all matrices in MR(G). These vertices are known as nil vertices (see [6]). In this paper I discuss some basic results about nil vertices in general and nil vertices in cographs and prove that cographs with a nil vertex of a particular form contain two other nil vertices symmetric to the first. I discuss several open questions relating to these results and a counterexample. I prove that for all cographs G without an induced complete tripartite graph with independent sets all of size 3, the zero-forcing number Z(G), a graph theoretic parameter, is equal to M(G). In fact this result holds for a slightly larger class of cographs and in particular holds for all threshold graphs. Lastly, I prove that the maximum of the minimum ranks of all cographs on n vertices is the floor of 2n/3.

[pt] Nessa dissertação vamos abordar a famosa lei do Semicírculo de Wigner, que dá uma descrição do comportamento do espectro de autovalores de matrizes aleatórias simétricas. A demonstração combina ideias e técnicas de Combinatória e Probabilidade, incluindo uma analise cautelosa dos momentos da distribuição de autovalores. / [en] In this dissertation we will approach the famous Wigner s Semicircle Law, which gives a description of the behavior of the eigenvalue spectrum of symmetric random matrices. The proof combines ideas and techniques from Combinatorics and Probability, including a careful analysis of the moments of the eigenvalue distribution.

Courier companies receive delivery orders at different times in advance. Some orders are known long beforehand, some arise with a very short notice. Currently the order delegation, deciding which car is going to drive which order, is performed completely manually by a (TL) where the TL use their experience to guess upcoming orders. If delivery orders could be predicted beforehand, algorithms could create suggestions for vehicle routing and vehicle selection. This thesis used the data set from a Stockholm based courier company. The Stockholm area was divided into zones using agglomerative clustering and K-Means, where the zones were used to group deliveries into time-sliced Origin Destination (OD) matrices. One cell in one OD-matrix contained the number of deliveries from one zone to another during one hour. Long-Short Term Memory (LSTM) Recurrent Neural Networks were used for the prediction. The training features consisted of prior OD-matrices, week day, hour of day, month, precipitation, and the air temperature. The LSTM based approach performed better than the baseline, the Mean Squared Error was reduced from 1.1092 to 0.07705 and the F1 score increased from 41% to 52%. All features except for the precipitation and air temperature contributed noticeably to the prediction power. The result indicates that it is possible to predict some future delivery orders, but that many are random and are independent from prior deliveries. Letting the model train on data as it is observed would likely boost the predictive power. / Budföretag får in leveransorder olika tid i förväg. Vissa order är kända lång tid i förväg, medan andra uppkommer med kort varsel. I dagsläget genomförs orderdelegationen, delegering av vilken bil som kör vilken order, manuellt av en transportledare (TL) där TL använder sin erfarenhet för att gissa framtida order. Om leveransorder skulle kunna förutsägas i förväg kan fordonsrutter och fordonsval föreslås av algoritmer. Denna uppsats använde sig utav ett dataset från ett Stockholmsbaserat budföretag. Stockholmsområdet delades in i zoner med agglomerativ klustring och K-Means, där zoner användes för att gruppera leveranser in i tidsdelade Ursprungsdestinationsmatriser (OD-matriser).  En cell i en OD-matris innehåller antalet leveranser från en zon till en annan under en timme. Neurala nätverk med lång-kortsiktsminne (LSTM) användes för förutsägelsen. Modellen tränades på tidigare OD-matriser, veckodag, timme, månad, nederbörd, och lufttemperatur. Det LSTM-baserade tillvägagångssättet presterade bättre än baslinjen, det genomsnittliga kvadratfelet minskade från 1,1092 till 0,07705 och F1-poängen ökade från 41% till 52%. Nederbörd och lufttemperatur bidrog inte märkbart till förutsägelsens prestation. Resultatet indikerar att det är möjligt att förutse vissa leveransorder, men att en stor andel är slumpmässiga och oberoende från tidigare leveranser. Att låta modellen tränas med nya data när den observeras skulle troligtvis öka prognosförmågan.

We consider random symmetric Toeplitz matrices of size n. Assuming that the entries on the diagonals are independent centered random variables with finite γ-th moment (γ>2), a law of large numbers is established for the largest eigenvalue. Following the approach of Sen and Virág (2013), in the limit of large n, the largest rescaled eigenvalue is shown to converge to the limit 0.8288... . The background theory is explained and some symmetry results on the eigenvectors of the Toeplitz matrix and an auxiliary matrix are presented. A numerical investigation illustrates the rate of convergence and the oscillatory nature of the eigenvectors of the Toeplitz matrix. Finally, the possibility of proving a limiting distribution for the largest eigenvalue is discussed, and suggestions for future research are made. / Vi betraktar stokastiska Toeplitz-matriser av storlek n. Givet att elementen på diagonalerna är oberoende, centrerade stokastiska variabler med ändligt γ-moment (γ>2), fastställer vi ett stora talens lag för det största egenvärdet. Med metoden från Sen och Virág (2013) visar vi att det största omskalade egenvärdet konvergera mot gränsen 0.8288... . Bakgrundsteorin förklaras och några symmetriresultat för Toeplitz-matrisens egenvektorer presenteras. En numerisk undersökning illustrerar konvergenshastigheten och Toeplitz-matrisens egenvektorers periodiska natur. Slutligen diskuteras möjligheten att bevisa en asymptotisk fördelning för de största egenvärderna och förslag för fortsatt forskning läggs fram.

In this dissertation, the prospect of a quantum technology based on a photonic crystal chip hosting a quantum network made of quantum dot spins interacting via single photons is investigated. The mathematical procedure to deal with the Liouville-Von Neumann equation, which describes the time-evolution of the density matrix, was derived for an arbitrary system, giving general equations. Using this theoretical groundwork, a numerical model was then developed to study the spatiotemporal dynamics of entanglement between various qubits produced in a controlled way over the entire quantum network. As a result, an efficient quantum interface was engineered allowing for storage qubits and traveling qubits to exchange information coherently while demonstrating little error and loss in the process; such interface is indispensable for the realization of a functional quantum network. Furthermore, a carefully orchestrated dynamic control over the propagation of the flying qubit showed high-efficiency capability for on-chip single-photon transfer. Using the optimized dispersion properties obtained quantum mechanically as design parameters, a possible physical structure for the photonic crystal chip was constructed using the Plane Wave Expansion and FiniteDifference Time-Domain numerical techniques, exhibiting almost identical transfer efficiencies in terms of normalized energy densities of the classical electromagnetic field. These promising results bring us one step closer to the physical realization of an integrated quantum technology combining both semiconductor quantum dots and subwavelength photonic structures.

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