This thesis is concerned with studying the behavior of a gossiping protocol in the specific sense meant by Ericsson; in the following pages I’ll introduce a Markov process which models the spread of information in such systems. The results will be verified by means of a discreet-event simulation. / Gossiping Protocols, are inherently random in behavior.Nonetheless, they are not structure-less. Their asymptotic behavior when implemented in large scales is the matter of focus in this thesis. / Tel: +46709700505 Address: Pinnharvsgatan 3 E lgh 1202 43147 Mölndal Sweden

Accurate and reliable effort estimation is still one of the most challenging processes in software engineering. There have been numbers of attempts to develop cost estimation models. However, the evaluation of model accuracy and reliability of those models have gained interest in the last decade. A model can be finely tuned according to specific data, but the issue remains there is the selection of the most appropriate model. A model predictive accuracy is determined by the difference of the various accuracy measures. The one with minimum relative error is considered to be the best fit. The model predictive accuracy is needed to be statistically significant in order to be the best fit. This practice evolved into model evaluation. Models predictive accuracy indicators need to be statistically tested before taking a decision to use a model for estimation. The aim of this thesis is to statistically evaluate well known effort estimation models according to their predictive accuracy indicators using two new approaches; bootstrap confidence intervals and permutation tests. In this thesis, the significance of the difference between various accuracy indicators were empirically tested on the projects obtained from the International Software Benchmarking Standard Group (ISBSG) data set. We selected projects of Un-Adjusted Function Points (UFP) of quality A. Then, the techniques; Analysis Of Variance ANOVA and regression to form Least Square (LS) set and Estimation by Analogy (EbA) set were used. Step wise ANOVA was used to form parametric model. K-NN algorithm was employed in order to obtain analogue projects for effort estimation use in EbA. It was found that the estimation reliability increased with the pre-processing of the data statistically, moreover the significance of the accuracy indicators were not only tested statistically but also with the help of more complex inferential statistical methods. The decision of selecting non-parametric methodology (EbA) for generating project estimates in not by chance but statistically proved.

This thesis describes methods of secret communication based on latin squares and their close relative, quasigroups. Different types of cryptosystems are described, including ciphers, public-key cryptosystems, and cryptographic hash functions. There is also a chapter devoted to different secret sharing schemes based on latin squares. The primary objective is to present previously described cryptosystems and secret sharing schemes in a more accessible manner, but this text also defines two new ciphers based on isotopic latin squares and reconstructs a lost proof related to row-latin squares. / Denna uppsats beskriver kryptosystem och metoder för hemlighetsdelning baserade på latinska kvadrater och det närliggande konceptet kvasigrupper. Olika sorters chiffer, både symmetriska och asymmetriska, behandlas. Dessutom finns ett kapitel tillägnat kryptografiska hashfunktioner och ett tillägnat metoder för hemlighetsdelning. Huvudsyftet är att beskriva redan existerande metoder för hemlig kommunikation på ett mer lättillgängligt sätt och med nya exempel, men dessutom återskapas ett, till synes, förlorat bevis relaterat till rad-latinska kvadrater samt beskrivs två nya chiffer baserade på isotopa latinska kvadrater.

An area in graph theory is graph colouring, which essentially is a labeling of the vertices or edges according to certain constraints. In this thesis we consider star edge colouring, which is a variant of proper edge colouring where we additionally require the graph to have no two-coloured paths or cycles with length 4. The smallest number of colours needed to colour a graph G with a star edge colouring is called the star chromatic index of G and is denoted <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cchi'_%7Bst%7D(G)" />. This paper proves an upper bound of the star chromatic index of bipartite graphs in terms of the maximum degree; the maximum degree of G is the largest number of edges incident to a single vertex in G. For bipartite graphs Bk with maximum degree <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?k%5Cgeq1" />, the star chromatic index is proven to satisfy<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%20%5Cchi'_%7Bst%7D(B_k)%20%5Cleq%20k%5E2%20-%20k%20+%201" />. For bipartite graphs <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?B_%7Bk,n%7D" />, where all vertices in one part have degree n, and all vertices in the other part have degree k, it is proven that the star chromatic index satisfies <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cchi'_%7Bst%7D(Bk,n)%20%5Cleq%20k%5E2%20-2k%20+%20n%20+%201,%20k%20%5Cgeq%20n%20%3E%201" />. We also prove an upper bound for a special case of multipartite graphs, namely <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?K_%7Bn,1,1,%5Cdots,1%7D" /> with m parts of size one. The star chromatic index of such a graph satisfies<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cchi'_%7Bst%7D(K_%7Bn,1,1,%5Cdots,1%7D)%20%5Cleq%2015%5Clceil%5Cfrac%7Bn%7D%7B8%7D%5Crceil%5Ccdot%5Clceil%5Cfrac%7Bm%7D%7B8%7D%5Crceil%20+%20%5Cfrac%7B1%7D%7B2%7Dm(m-1),%5C,m%20%5Cgeq%205" />. For complete multipartite graphs where m &lt; 5, we prove lower upper bounds than the one above.

Based on the generating matrix to the Tribonacci sequence, the Tribonacci cat map is a discrete chaotic dynamical system, similar to Arnold's discrete cat map, but on three dimensional space. In this thesis, this new mapping is introduced and the properties of its matrix are presented. The main results of the investigation prove how the size of the domain of the map affects its period and explore the orbit lengths of non-trivial points. Different upper bounds to the map are studied and proved, and a conjecture based on numerical calculations is proposed. The Tribonacci cat map is used for applications such as 3D image encryption and colour encryption. In the latter case, the results provided by the mapping are compared to those from a generalised form of the map.

In this research,we study iterations of non-pleasantly ramified polynomials over fields of positive characteristic and subsequently, their lower ramification numbers. Of particular interest for this thesis are polynomials for which both the multiplicity and  the degree of its iterates grow exponentially. Specifically we consider the family  of polynomials such that given a positive integer k the family is given by P(z) = z(1 + z (3^k-1)/2 + z3^k-1). The cubic polynomial z + z2 + z3 is a special case of this family and is particularly interesting.

Let <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D%20=%20(%5Cmathbb%7BP%7D,%20%5Cpreceq)" data-classname="equation_inline" data-title="" /> be a finite poset (partially ordered set) with cardinality <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" data-classname="equation_inline" data-title="" />. A linear extension of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" /> is an order-preserving bijection <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csigma" data-classname="equation_inline" data-title="" />: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D%20%5Crightarrow%20%5Bn%5D" data-classname="equation_inline" data-title="" />, that is, if <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?x%20%5Cpreceq%20y" data-classname="equation_inline" data-title="" /> in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" /> then <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csigma(x)%20%5Cle%20%5Csigma(y)" data-classname="equation_inline" data-title="" />. We define the poset probability <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)" data-classname="equation_inline" data-title="" /> as the proportion of linear extensions where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csigma(%5Calpha)%20%5Cle%20%5Csigma(%5Cbeta)" data-classname="equation_inline" data-title="" />. We are primarily interested in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)" data-classname="equation_inline" data-title="" /> for incomparable elements <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha%20%5Cparallel%20%5Cbeta" data-classname="equation" data-title="" />. The probability has significance in areas such as information theory. Let <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?e(%5Cmathbb%7BP%7D)" data-classname="equation_inline" data-title="" /> denote the total number of linear extensions of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" />. We prove that the poset probability can be evaluated as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)%20=%20%5Cfrac%7B%20%5Csum_%7BT%20%5Cin%20B(%5Calpha,%5Cbeta)%7D%20e(T)%20e(%5Cmathbb%7BP%7D%20%5Csetminus%20(T%20%5Ccup%20%5C%7B%5Calpha%5C%7D))%7D%7Be(%5Cmathbb%7BP%7D)%7D" data-classname="equation" data-title="" /> where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?B(%5Calpha,%5Cbeta)" data-classname="equation_inline" data-title="" /> is the set of order ideals of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" /> without <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" data-classname="equation" data-title="" /> or <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cbeta" data-classname="equation" data-title="" />, where we can add <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" data-classname="equation_inline" data-title="" /> to get a new order ideal of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" />. The practicality of the preceding formula is explored and we show that <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?T%20%5Cin%20B(%5Calpha,%5Cbeta)%20%5CLeftrightarrow%20%5Cleft%5C%7B%20x%20%7C%20x%20%5Cprec%20%5Calpha%20%5Cright%5C%7D%20%5Csubseteq%20T%20%5Ctext%7B%20and%20%7D%20T%20%5Ctext%7B%20order%20ideal%20of%20%7D%0A%5Cleft%5C%7B%20x%20%7C%20%5Calpha%20%5Cnot%20%5Cpreceq%20x,%5C%20%5Cbeta%20%5Cnot%20%5Cpreceq%20x%7D" data-classname="equation" /> The formula is particularly useful for certain classes of posets such as partition posets which are examined in further detail. We apply the formula to prove that, for all partition posets of shape <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Bn,n%5D" data-classname="equation_inline" data-title="" />, the probability obeys <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P((2,a)%20%5Cpreceq%20(1,a+1))%20=%20%5Cfrac%7B%20C_a%20C_%7Bn-a%7D%7D%20%7BC_n%7D" data-classname="equation" data-title="" /> where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?C_n" data-classname="equation_inline" data-title="" /> is the nth Catalan number and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?a%20%3C%20n" data-classname="equation_inline" data-title="" />. We also explore how Monte Carlo methods can be used to estimate <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)" data-classname="equation_inline" data-title="" />.

The matching of a tournament could be a tedious task, especially if there are many teams takingpart. To combat this difficult task this thesis evaluates if, by creating a method, there is a wayto automate this matching. The method proposed is based on only two parameters, shortestdistance and whether the teams have faced each other already. By being based on shortestdistance the method could not only shorten the planning time but also shorten the distance thatthe participants have to travel.To test the proposed method and validate whether it would give a sufficiently good matching itis applied to an already played Bridge tournament. The tournament was played in 2019 andincluded 365 teams and 7 rounds. This tournament was matched manually based partly ondistance and whether the teams have faced each other. Teams were split into clusters andrandomly matched each other within those. Thus, the teams did not often play against the closestteam but never a team very far away. Results are given in the form of maps for each round anda table which compares the distances for both the methods.The method was applied successfully, and each round was, as excepted, a shorter distance thanthe original matching. Thus, proving that an automatic tournament matching can be done. Thereare some improvement and alteration that can be done in the future to find a matching that morelifelike.

In this thesis we will investigate how the SBP-SAT finite difference method behave with and without an interface. As model problem, we consider the heat equation with piecewise constant coefficients. The thesis is split in two main parts. In the first part we look at the heat equation in one-dimension, and in the second part we expand the problem to a two-dimensional domain. We show how the SAT-parameters are chosen such that the scheme is dual consistent and stable. Then, we perform numerical experiments, now looking at the static case. In the one-dimensional case we see that the second order SBP-SAT method with an interface converge with an order of two, while the second order SBP-SAT method without an interface converge with an order of one.

Denna bok riktar sig till gymnasieelever som vill fördjupa sig i ämnet RSA-kryptografi . RSA-kryptografi är en avancerad metod för att kommunicera med hemliga meddelanden och används flitigt inom t.ex. bankvärlden. När du handlar med ditt kort eller använder din e-legitimation används RSA-kryptogra fi för att allt du gör ska vara skyddat och säkert. Vid stora transaktioner mellan olika banker används också RSA-kryptogra fi för att både den som betalar och den som får betalt ska vara säkra att allt går rätt till.Boken är uppdelad i fyra kapitel. Kapitel 3 och 4 är betydligt mer avancerade än kapitel 1 och 2. Kapitel 1 består mestadels av exempel och övningar som behandlar matematiken som krävs för att kunna utföra RSA-kryptogra fi med små tal. Kapitel 2 använder matematiken i kapitel 1 för att genom exempel och övingar metodiskt lära ut hur RSA-kryptogra fi med små tal går till. Kapitel 3 visar matematiken som ligger till grund för att RSA-kryptografi fungerar. Detta visas med hjälp av exempel, satser, förtydligade bevis samt några enstaka övningar. Kapitel 4 förklarar varför RSA-kryptografi är säkert och enkelt att använda. Primtalstester utgör det viktigaste ämnet i detta sista kapitel.