The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour. Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices). In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice. To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy. The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function. The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations. We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

Since the introduction of Pollard's rho method for integer factorisation in 1975 there has been great interest in understanding the dynamics of quadratic maps over finite fields. One avenue for this, and indeed the heuristic on which Pollard bases the proof of the method's efficacy, is the idea that quadratic maps behave roughly like random maps. We explore this heuristic from the perspective of comparing the number of periodic points. We find that empirically random maps appear to model the number of periodic points of quadratic maps well, and moreover prove that the number of periodic points of random maps satisfy an interesting asymptotic behaviour that we have observed experimentally for quadratic maps.

A graph is an ordered pair composed by a set of vertices and a set of edges, the latter consisting of unordered pairs of vertices. Two vertices in such a pair are each others neighbors. Two edges are adjacent if they share a common vertex. Denote the amount of edges that share a specific vertex as the degree of the vertex. A proper edge-coloring of a graph is an assignment of colors from some finite set, to the edges of a graph where no two adjacent edges have the same color. A bipartition (X,Y) of a set of vertices V is an ordered pair of two disjoint sets of vertices such that V is the union of X and Y, where all the vertices in X only have neighbors in Y and vice versa. A bipartite graph is a graph whose vertices admit a bipartition (X,Y). Let G be one such graph. An X-interval coloring of G is a proper edge coloring where the colors of the edges incident to each vertex in X form an interval of integers. Denote by χ'int(G,X) the least number of colors needed for an X-interval coloring of G. In this paper we prove that if G is a bipartite graph with maximum degree 3n (n is a natural number), where all the vertices in X have degree 3, then <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathit%7B%5Cchi'_%7Bint%7D%5Cleft(G,X%5Cright)%5Cleq%7D%0A%5C%5C%0A%5Cmathit%7B%5Cleft(n-1%5Cright)%5Cleft(3n+5%5Cright)/2+3%7D%0A%5C%5C%0A%5Cmathit%7Bif%20n%20is%20odd,%7D%0A%5C%5C%0A%5Cmathit%7Bor%7D%0A%5C%5C%0A%5Cmathbf%7B3n%5E%7B2%7D/2+1%7D%0A%5C%5C%0A%5Cmathit%7Bif%20n%20is%20even%7D.%0A" />

This thesis examines the problem of counting and describing walks in graphs, and the problem when such walks have modular restrictions on how many timesit visits each vertex. For the special cases of the path graph, the cycle graph, the grid graph and the cylinder graph, generating functions and regular languages for their walks and walks with modular restrictions are constructed. At the end of the thesis, a theorem is proved that connects the generating function for walks in a graph to the generating function for walks in a covering graph.

Vid beräkning i addition krävs kunskap att tolka siffrors positioner och värden i tal. Kunskaper om det decimala talsystemets struktur och platsvärde är grundläggande för att kunna göra beräkningar i addition. Syftet med denna litteraturstudie var att genomföra en litteraturöversikt om hur platsvärdet förhåller sig till addition och hur undervisning kan bedrivas om det decimala talsystemet. Litteratursökningen som gjordes genererade i artiklar som mestadels undersökte elever i de lägre åldrarna. Resultatet i litteraturöversikten baserades på tio artiklar som har analyserats av skribenterna. Resultatet visade att det fanns ett förhållande mellan platsvärde och förmågan att beräkna addition. Elever i denna litteraturstudie har visat svårigheter med att förstå platsvärde i det decimala talsystemet. Svårigheterna gick att förebygga genom varierade undervisningssekvenser som belyste olika aspekter av det decimala talsystemet. Undervisningsformer med tavla och konkret material visade sig vara de mest förekommande metoderna. Slutsatsen från skribenterna blev att undervisningsformen i sig inte är avgörande för att förstå det decimala talsystemet. Det matematiska innehållet och hur det förmedlas av läraren är betydande för hur eleverna lär sig platsvärde i det decimala talsystemet.

We have found an operation between permutations in the symmetric group Sn upon which we have experimentally derived results that can be linked to desirable properties in cryptography, mainly in the domain of one-way functions. From it, we have implemented a beta version of an algorithm for a hashing function by exploiting the operation’s low computational cost for speed and its properties for security. Its design makes it resistant to length extension attacks and the encoding of blocks into permutations suggests that any differential cryptanalysis technique that is based on bit conditions should be useless against it. More precisely, when measuring the evolution of differences in the compression function, bit-based distances such as the exclusive-or distance should be replaced by another type of distance, still to be determined in future research. In this work we will present the algorithm and introduce a new framework of cryptanalysis for collision and preimage attacks in order to somehow measure its security. Once this is done, we will run comparison tests against MD5 and SHA256 in order to externally evaluate our algorithm in terms of speed, weaknesses and strength.

We study the local list number of graphs introduced by Lennerstad and Eriksson. A labelling of a graph on n vertices is a bijection from vertex set to the set {1,…, n}. Given such a labelling c a vertex u is distance consistent if for all vertices v and w |c(u)-c(v)|=|c(u)-c(w)|+1 implies d(u,w)≤ d(u,v). A graph G is k-distance consistent if there is a labelling with k distance-consistent vertices. The local list number of a graph G is the largest k such that G is  k-distance consistent. We determine the local list number of cycles, complete bipartite graphs and some trees as well as prove bounds for some families of trees. We show that the local list number of even cycles is two, and of odd cycles is three. We also show that, if  k, l≥ 3 , the complete bipartite graph  Kk,l has local list number one, the star graph Sn=K1,n has local list number 3, and K2,k  has local list number 2. Finally, we show that for each n≥3 and each k such that 3≤k≤n there is a tree with local list number k. / Vi studerar det lokala listtalet introducerat av Lennerstad och Eriksson. En märkning av en graf på n hörn är en bijektion från hörnmängden till mängden {1, . . . , n}. Givet en sådan märkning c är ett hörn u avståndskonsistent om för alla hörn v och w |c(u) − c(v)| = |c(u) − c(w)| = 1 implicerar d(u, w) ≤d(u, v). En graf G är k-avståndskonsistent om det nns en märkning med k avståndskonsistenta hörn. Det lokala listtalet av en graf G är det största k sådan att G är k -avståndskonsistent. Vi bestämmer den lokala listtalet av cykler, kompletta bipartita grafer och vissa träd och visar som gränser för några familjer av träd. Vi visar att det lokla listtalet av jämna cykler är två, och av udda cykler är tre. Vi visar också att, om k, l ≥ 3, den kompletta bipartita grafen Kk,l har lokalt listtal ett, stjärngrafen Sn = K1,n har lokalt listtal 3, och K2,k har lokalt listtal 2. Slutligen, visar vi att för varje n≥3 och varje k sådant att 3 ≤ k≤n finns ett träd med lokalt listtal k.

We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables. The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions. The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2. The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.

In this thesis, we discuss important results for the numerical range of general square matrices. Especially, we examine analytically the numerical range of complex-valued $2 \times 2$ matrices. Also, we investigate and discuss the Gershgorin region of general square matrices. Lastly, we examine numerically the numerical range and Gershgorin regions for different types of square matrices, both contain the spectrum of the matrix, and compare these regions, using the calculation software Maple.

Interactions between combinatorics and statistical mechanics have provided many fruitful insights in both fields. A compelling example is Kuperberg’s solution to the alternating sign matrix conjecture, and its following generalisations. In this thesis we investigate two models from statistical mechanics which have received attention in recent years. The first is the fully packed loop model. A conjecture from 2001 by Razumov and Stroganov opened the field for a large ongoing investigation of the O(1) loop model and its connections to a refinement of the fully packed loop model. We apply a combinatorial bijection originally found by de Gier to an older conjecture made by Propp. The second model is the hard particle model. Recent discoveries by Fendley et al. and results by Jonsson suggests that the hard square model with cylindrical boundary conditions possess some beautiful combinatorial properties. We apply both topological and purely combinatorial methods to related independence complexes to try and gain a better understanding of this model.