Bulgarian solitaire is played on \(n\) cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, \(\sigma\)-Bulgarian ...solitaire, the number of cards you pick from a pile is some function \(\sigma\) of the pile size, such that you pick \(\sigma(h)\le h\) cards from a pile of size \(h\). Here we consider a special class of such functions. Let us call \(\sigma\) well-behaved if \(\sigma(1)=1\) and if both \(\sigma(h)\) and \(h-\sigma(h)\) are non-decreasing functions of \(h\). Well-behaved \(\sigma\)-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of \(n\) cards exists it is unique. Moreover, if piles are sorted in order of decreasing size (\(\lambda_1 \ge \lambda_2\ge \dots\)) then a configuration is convex if and only if it is a stable configuration of some well-behaved \(\sigma\)-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions (\(\sigma_1, \sigma_2, \dots\)) may tend to a limit shape \(\phi\). We show that every convex \(\phi\) with certain properties can arise as the limit shape of some sequence of well-behaved \(\sigma_n\). For the special case when \(\sigma_n(h)=\lceil q_n h \rceil\) for \(0 < q_n \le 1\), these limit shapes are triangular (in case \(q_n^2 n\rightarrow 0\)), or exponential (in case \(q_n^2 n\rightarrow \infty\)), or interpolating between these shapes (in case \(q_n^2 n\rightarrow C>0\)).
This thesis explores adapting self-supervised representation learning to visual domains beyond natural scenes, focusing on medical imaging. The research addresses the central question: "How can ...self-supervised representation learning be specifically adapted for detecting liver cancer in histopathology images?" The study utilizes the PAIP 2019 dataset for liver cancer segmentation and employs a self-supervised approach based on the VICReg method. The evaluation results demonstrated that the ImageNet-pretrained model achieved superior performance on the test set, with a clipped Jaccard index of 0.7747 at a threshold of 0.65. The VICReg-pretrained model followed closely with a score of 0.7461, while the model initialized with random weights trailed behind at 0.5420. These findings indicate that while ImageNet-pretrained models outperformed VICReg-pretrained models, the latter still captured essential data characteristics, suggesting the potential of self-supervised learning in diverse visual domains. The research attempts to contribute to advancing self-supervised learning in non-natural scenes, provides insights into model pretraining strategies, and introduces novel non-linear data augmentation techniques.
This is a report where an approach to use a patch antenna as the transmit device when sending radio waves in a RFID system has been investigated. The project is successful in the sense that the ...antenna is working as imagined, the antenna parameters may however not be satisfactory. The antenna read range may be a little to insufficient when the RFID tags are worn by humans which is one of the underlying requirement for the system this antenna was design for. Tags which are rotated from a vertical alignment also reduce the effectiveness of the antenna even more to a point which is not acceptable. Suggestions for how to further improve the antenna are given and addresses the issues mentioned above. The report first contains a brief introduction to antennas in general and also information about patch antennas specifically as that was the antenna chosen to be constructed and tested for this system as it theoretically seemed very fitting. A working antenna was constructed and tested in a real environment together with simulations of the antenna to further examine it. The finished antenna is evaluated with possible advantages and drawbacks being discussed together with mentioning how then antenna could be improved for better performance.
This thesis deals with processes on integer partitions and their limit shapes, with focus on deterministic and stochastic variants on one such process called Bulgarian solitaire. The main scientific ...contributions are the following. Paper I: Bulgarian solitaire is a dynamical system on integer partitions of n which converges to a unique fixed point if n=1+2+...+k is a triangular number. There are few results about the structure of the game tree, but when k tends to infinity the game tree itself converges to a structure that we are able to analyze. Its level sizes turns out to be a bisection of the Fibonacci numbers. The leaves in this tree structure are enumerated using Fibonacci numbers as well. We also demonstrate to which extent these results apply to the case when k is finite. Paper II: Bulgarian solitaire is played on n cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, σ-Bulgarian solitaire, the number of cards you pick from a pile is some function σ of the pile size, such that you pick σ(h) < h cards from a pile of size h. Here we consider a special class of such functions. Let us call σ well-behaved if σ(1) = 1 and if both σ(h) and h − σ(h) are non-decreasing functions of h. Well-behaved σ-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of n cards exists it is unique. Moreover, if piles are sorted in order of decreasing size then a configuration is convex if and only if it is a stable configuration of some well-behaved σ-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions (σ1, σ2, ...) may tend to a limit shape Φ. We show that every convex Φ with certain properties can arise as the limit shape of some sequence of well-behaved σn. For the special case when σn(h) = ceil(qnh) for 0 < qn ≤ 1 (where ceil is the ceiling function rounding upward to the nearest integer), these limit shapes are triangular (in case qn2n → 0), or exponential (in case qn2n → ∞), or interpolating between these shapes (in case qn2n → C > 0). Paper III: We introduce pn-random qn-proportion Bulgarian solitaire (0 < pn,qn ≤ 1), played on n cards distributed in piles. In each pile, a number of cards equal to the proportion qn of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability pn, independently of other candidate cards. This generalizes Popov's random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed p as n tends to infinity. Here we let both pn and qn vary with n. We show that under the conditions qn2pnn/log n → ∞ and pnqn → 0 as n → ∞, the pn-random qn-proportion Bulgarian solitaire has an exponential limit shape. Paper IV: We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.
We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain ...goes only in one direction, either up or down in the poset (an \emph{up chain} or \emph{down chain}). The second type toggles between two adjacent rank levels (an \emph{up-and-down chain}). We introduce two compatibility concepts between the up-directed transition probabilities (an \emph{up rule}) and the down-directed (a \emph{down rule}), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.
We consider a stochastic version of Bulgarian solitaire: A number of cards are distributed in piles; in every round a new pile is formed by cards from the old piles, and each card is picked ...independently with a fixed probability. This game corresponds to a multi-square birth-and-death process on Young diagrams of integer partitions. We prove that this process converges in a strong sense to an exponential limit shape as the number of cards tends to infinity. Furthermore, we bound the probability of deviation from the limit shape and relate this to the number of rounds played in the solitaire.
We introduce \emph{\(p_n\)-random \(q_n\)-proportion Bulgarian solitaire} (\(0<p_n,q_n\le 1\)), played on \(n\) cards distributed in piles. In each pile, a number of cards equal to the proportion ...\(q_n\) of the pile size rounded upward to the nearest integer are candidates to be picked. Each candidate card is picked with probability \(p_n\), independently of other candidate cards. This generalizes Popov's random Bulgarian solitaire, in which there is a single candidate card in each pile. Popov showed that a triangular limit shape is obtained for a fixed \(p\) as \(n\) tends to infinity. Here we let both \(p_n\) and \(q_n\) vary with \(n\). We show that under the conditions \(q_n^2 p_n n/{\log n}\rightarrow \infty\) and \(p_n q_n \rightarrow 0\) as \(n\to\infty\), the \(p_n\)-random \(q_n\)-proportion Bulgarian solitaire has an exponential limit shape.
