In this paper, we study connections on hermitian modules, and show that metric connections exist on regular hermitian modules; i.e. finitely generated projective modules together with a non-singular hermitian form. In addition, we develop an index calculus for such modules, and provide a characterization in terms of the existence of a pseudo-inverse of the matrix representing the hermitian form with respect to a set of generators. As a first illustration of the above concepts, we find metric connections on the fuzzy sphere. Finally, the framework is applied to a class of noncommutative minimal surfaces, for which there is a natural concept of torsion, and we prove that there exist metric and torsion free connections for every minimal surface in this class.

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the *-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given. In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non -trivial Levi-Civita connections are constructed. (c) 2024 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

We introduce q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.

We introduce shift algebras as certain crossed product algebras based on general function spaces and study properties, as well as the classification, of a particular class of modules depending on a set of matrix parameters. It turns out that the structure of these modules depends in a crucial way on the properties of the function spaces. Moreover, for a class of subalgebras related to compact manifolds, we provide a construction procedure for the corresponding fuzzy spaces, i.e. sequences of finite dimensional modules of increasing dimension as the deformation parameter tends to zero, as well as infinite dimensional modules related to fuzzy non -compact spaces.

A graph G is called interval colorable if it has a proper edge coloring with colors 1, 2, 3, ... such that the colors of the edges incident to every vertex of G form an interval of integers. Not all graphs are interval colorable; in fact, quite few families have been proved to admit interval colorings. In this paper we introduce and investigate a new notion, the interval coloring thickness of a graph G, denoted theta int(G), which is the minimum number of interval colorable edge-disjoint subgraphs of G whose union is G. Our investigation is motivated by scheduling problems with compactness require-ments, in particular, problems whose solution may consist of several schedules, but where each schedule must not contain any waiting periods or idle times for all involved parties. We first prove that every connected properly 3-edge colorable graph with maximum degree 3 is interval colorable, and using this result, we deduce an upper bound on theta int(G) for general graphs G. We demonstrate that this upper bound can be improved in the case when G is bipartite, planar or complete multipartite and consider some applications in timetabling.

Let G(k) denote the set of connected k-regular graphs G, k ≥ 2, where the number of vertices at distance 2 from any vertex in G does not exceed k. Asratian (2006) showed (using other terminology) that a graph G ∈ G(k) is Hamiltonian if for each vertex u of G the subgraph induced by the set of vertices at distance at most 2 from u is 2-connected. We prove here that in fact all graphs in the sets G(3), G(4) and G(5) are Hamiltonian. We also prove that the problem of determining whether there exists a Hamilton cycle in a graph from G(6) is NP-complete. Nevertheless we show that every locally connected graph G ∈ G(k), k ≥ 6, is Hamiltonian and that for each non-Hamiltonian cycle C in G there exists a cycle C' of length |V(C)|+l in G, l ∈ {1, 2}, such that V(C) ⊂ V(C). Finally, we note that all our conditions for Hamiltonicity apply to infinitely many graphs with large diameters.

In this paper we present some results for a connected infinite graph G with finite degrees where the properties of balls of small radii guarantee the existence of some Hamiltonian and connectivity properties of G. (For a vertex w of a graph G the ball of radius r centered at w is the subgraph of G induced by the set Mr(w) of vertices whose distance from w does not exceed r). In particular, we prove that if every ball of radius 2 in G is 2-connected and G satisfies the condition dG(u)+dG(v)≥|M2(w)|−1 for each path uwv in G, where u and v are non-adjacent vertices, then G has a Hamiltonian curve, introduced by Kündgen et al. (2017). Furthermore, we prove that if every ball of radius 1 in G satisfies Ore’s condition (1960) then all balls of any radius in G are Hamiltonian.

Q-space trajectory imaging (QTI) allows non-invasive estimation of microstructural features of heterogeneous porous media via diffusion magnetic resonance imaging performed with generalised gradient waveforms. A recently proposed constrained estimation framework, called QTI+, improved QTI’s resilience to noise and data sparsity, thus increasing the reliability of the method by enforcing relevant positivity constraints. In this work we consider expanding the set of constraints to be applied during the fitting of the QTI model. We show that the additional conditions, which introduce an upper bound on the diffusivity values, further improve the retrieved parameters on a publicly available human brain dataset as well as on data acquired from healthy volunteers using a scanner-ready protocol.

Diffusion MRI is a powerful technique sensitive to the microstructure of heterogeneous media. By relating the dMRI signal obtained via general gradient waveforms to the moments of an underlying diffusion tensor distribution, q-space trajectory imaging (QTI) provides several quantities indicative of the structural composition of the medium. Substantial improvements in the reliability of the produced estimates has been achieved via incorporating necessary positivity constraints in the estimation by employing Semidefinite Programming. Here we present the Matlab code implementing said constraints, provide a simple example showing the main functionalities of the package, and point to resources within the package that can be used to reproduce results recently published with this software. The block-based structure of our implementation allows the selection of steps to be performed, and facilitates the incorporation of new constraints in future releases.

The complex orbifold structure of the moduli space of Riemann surfaces of genus g (g≥2) produces a stratification into complex subvarieties named equisymmetric strata. Each equisymmetric stratum is formed by the surfaces where the group of automorphisms acts in a topologically equivalent way. The Riemann surfaces in the equisymmetric strata of dimension one are of two structurally different types. Type 1 equisymmetric strata correspond to Riemann surfaces where the group of automorphisms produces a quotient surface of genus zero, while those of Type 2 appear when such a quotient is a surface of genus one. Type 1 equisymmetric strata have been extensively studied by the authors of the present work in a previous recent paper, we now focus on Type 2 strata. We first establish the existence of such strata and their frequency of occurrence in moduli spaces. As a main result we obtain a complete description of Type 2 strata as coverings of the sphere branched over three points (Belyi curves) and where certain isolated points (punctures) have to be eliminated. Finally, we study in detail the doubly infinite family of Type 2 strata whose automorphism groups have order the product of two primes.

For a bipartite graph G with parts X and Y, an X-interval coloring is a proper edge coloring of G by integers such that the colors on the edges incident to any vertex in X form an interval. Denote by chi(int) (G, X) the minimum k such that G has an X-interval coloring with k colors. Casselgren and Toft (2016) [12] asked whether there is a polynomial P(Delta) such that if G has maximum degree at most A, then chi(int)(G, X) <= P(A). In this short note, we answer this question in the affirmative; in fact, we prove that a cubic polynomial suffices. We also deduce some improved upper bounds on chi(int)(G, X) for bipartite graphs with small maximum degree. (C) 2021 The Author(s). Published by Elsevier B.V.

We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if G is a connected graph with maximum degree Delta(G) >= 4 that is not a complete graph and P subset of V (G) is a set of vertices where either (i) at most Delta(G) - 2 colors are forbidden for every vertex in P, and any two vertices of P are at distance at least 4, or (ii) at most Delta(G) - 3 colors are forbidden for every vertex in P, and any two vertices of P are at distance at least 3, then there is a proper Delta(G)-coloring of G respecting these constraints. In fact, we shall prove that these results hold in the more general setting of list colorings. These results are sharp.

We consider the problem of extending partial edge colorings of iterated cartesian products of even cycles and paths, focusing on the case when the precolored edges satisfy either an Evans-type condition or is a matching. In particular, we prove that if G = Cd2k is the dth power of the cartesian product of the even cycle C2k with itself, and at most 2d-1 edges of G are precolored, then there is a proper 2d-edge coloring of G that agrees with the partial coloring. We show that the same conclusion holds, without restrictions on the number of precolored edges, if any two precolored edges are at distance at least 4 from each other. For odd cycles of length at least 5, we prove that if G = Cd2k +1 is the dth power of the cartesian product of the odd cycle C2k +1 with itself (k >= 2), and at most 2d edges of G are precolored, then there is a proper (2d+1)-edge coloring of G that agrees with the partial coloring. Our results generalize previous ones on precoloring extension of hypercubes [Journal of Graph Theory 95 (2020) 410-444].

Let F be a (possibly improper) edge-coloring of a graph G; a vertex coloring of G is adapted to F if no color appears at the same time on an edge and on its two endpoints. If for some integer k, a graph G is such that given any list assignment L to the vertices of G, with |L(v)| ≥ k for all v, and any edge-coloring F of G, G admits a coloring c adapted to F where c(v) ∈ L(v) for all v, then G is said to be adaptably k-choosable. A (k, d)-list assignment for a graph G is a map that assigns to each vertex v a list L(v) of at least k colors such that |L(x) ∩ L(y)\ ≤ d whenever x and y are adjacent. A graph is (k, d)-choosable if for every (k, d)-list assignment L there is an L-coloring of G. It has been conjectured that planar graphs are (3, l)-choosable. We give some progress on this conjecture by giving sufficient conditions for a planar graph to be adaptably 3-choosable. Since (k, l)-choosability is a special case of adaptable k-choosablity, this implies that a planar graph satisfying these conditions is (3,1)-choosable. 

A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. The star chromatic index chi(st)(G) of G is the minimum number t for which G has a star edge coloring with t colors. We prove upper bounds for the star chromatic index of bipartite graphs G where all vertices in one part have maximum degree 2 and all vertices in the other part has maximum degree b. Let k be an integer (k >= 1); we prove that if b = 2k + 1, then chi(st)(G) <= 3k + 2; and if b = 2k, then chi(st)(G) < 3k; both upper bounds are sharp. We also consider complete bipartite graphs; in particular we determine the star chromatic index of such graphs when one part has size at most 3, and prove upper bounds for the general case. Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most 6; in particular we settle this conjecture for cubic Halin graphs. (C) 2021 The Authors. Published by Elsevier B.V.

Consider a partial Latin square P where the first two rows and first three columns are completely filled, and every other cell of P is empty. It has been conjectured that all such partial Latin squares of order at least 8 are completable. Based on a technique by Kuhl and McGinn we describe a framework for completing partial Latin squares in this class. Moreover, we use our method for proving that all partial Latin squares from this family, where the intersection of the nonempty rows and columns form a Latin rectangle with three distinct symbols, are completable.

We consider the problem of extending and avoiding partial edge colorings of hypercubes; that is, given a partial edge coloring phi of the d-dimensional hypercube Q(d), we are interested in whether there is a proper d-edge coloring of Q(d) that agrees with the coloring phi on every edge that is colored under phi; or, similarly, if there is a proper d-edge coloring that disagrees with phi on every edge that is colored under phi. In particular, we prove that for any d >= 1, if phi is a partial d-edge coloring of Q(d), then phi is avoidable if every color appears on at most d/8 edges and the coloring satisfies a relatively mild structural condition, or phi is proper and every color appears on at most d - 2 edges. We also show that phi is avoidable if d is divisible by 3 and every color class of phi is an induced matching. Moreover, for all 1 <= k <= d, we characterize for which configurations consisting of a partial coloring phi of d - k edges and a partial coloring psi of k edges, there is an extension of phi that avoids psi.

For a connected graph G and a, b is an element of R, the general degree -eccentricity index is defined as DEIa,b(G) = Sigma(v is an element of V(G)) d(G)(a)(v)ecc(G)(b)(v), where V(G) is the vertex set of G, d(G)(v) is the degree of a vertex v and ecc(G)(v) is the eccentricity of v in G, i.e. the maximum distance from v to another vertex of the graph. This index generalizes several well-known 'topological indices' of graphs such as the eccentric connectivity index. We characterize the unique trees with the maximum and the minimum general degree-eccentricity index among all n -vertex trees with fixed maximum degree for the cases a >= 1, b <= 0 and 0 <= a <= 1, b >= 0. This complements previous results on the general degree -eccentricity index for various classes of trees.

We consider the problem of extending and avoiding partial edge colorings of trees; that is, given a partial edge coloring phi of a tree T we are interested in whether there is a proper Delta(T )-edge coloring of T that agrees with the coloring phi on every edge that is colored under phi; or, similarly, if there is a proper Delta(T )-edge coloring that disagrees with phi on every edge that is colored under phi. We characterize which partial edge colorings with at most Delta(T ) + 1 precolored edges in a tree T are extendable, thereby proving an analogue of a result by Andersen for Latin squares. Furthermore we obtain some "mixed" results on extending a partial edge coloring subject to the condition that the extension should avoid a given partial edge coloring; in particular, for all 0 <= k <= Delta(T ), we characterize for which configurations consisting of a partial coloring phi of Delta(T ) - k edges and a partial coloring psi of k + 1 edges of a tree T, there is an extension of phi that avoids psi.

We consider the problem of extending partial edge colorings of Cartesian products of graphs. More specifically, we suggest the following Evans -type conjecture. If G is a graph where every precoloring of at most k precolored edges can be extended to a proper chi 0(G)-edge coloring, then every precoloring of at most k + 1 edges of G ?K2 is extendable to a proper (chi'(G) + 1)edge coloring of G ?K2. In this paper we verify that this conjecture holds for trees, complete and complete bipartite graphs, as well as for graphs with small maximum degree. We also prove versions of the conjecture for general regular graphs where the precolored edges are required to be independent.

We consider the problem of extending partial edge colorings of trees. We obtain analogues of classical results on extending partial Latin squares by characterizing exactly which partial edge colorings with at most Delta(T) precolored edges of a tree T with maximum degree Delta(T) are extendable to proper Delta(T)-edge colorings of T. Furthermore, we prove sharp conditions on when it is possible to extend a partial edge coloring where the precolored edges form a matching or a collection of connected subgraphs. Finally, we consider the problem of avoiding a given (not necessarily proper) partial edge coloring.

A k-improper edge coloring of a graph G is a mapping alpha : E(G) -> N such that at most k edges of G with a common endpoint have the same color. An improper edge coloring of a graph G is called an improper interval edge coloring if the colors of the edges incident to each vertex of G form an integral interval. In this paper we introduce and investigate a new notion, the interval coloring impropriety (or just impropriety) of a graph G defined as the smallest k such that G has a k-improper interval edge coloring; we denote the smallest such k by mu(int)(G). We prove upper bounds on mu(int)(G) for general graphs G and for particular families such as bipartite, complete multipartite and outerplanar graphs; we also determine mu(int)(G) exactly for G belonging to some particular classes of graphs. Furthermore, we provide several families of graphs with large impropriety; in particular, we prove that for each positive integer k, there exists a graph G with mu(int)(G) = k. Finally, for graphs with at least two vertices we prove a new upper bound on the number of colors used in an improper interval edge coloring. (C) 2021 The Author(s). Published by Elsevier B.V.

Given a partial edge coloring of a complete graph K-n and lists of allowed colors for the non-colored edges of K-n, can we extend the partial edge coloring to a proper edge coloring of K-n using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

The knowledge of temperature and emissivity of the receiver are both critical for a solar tower power plant, in order to guarantee an efficient operation of the thermal receiver on the one hand, while monitoring any degradation of the receiver coating on the other hand. To make these measurements, a new thermographic system is currently being developed, using a multispectral camera working in the short wavelength infrared spectrum. This system applies the principle of ratio thermography, using a couple of narrow bandpass filters centered on atmospheric water absorption bands, at 1.4 and 1.9 µm, to reduce the influence of solar reflections on the measurement signal, making it sensitive to atmospheric conditions.

In this thesis, a batch simulation approach is used to identify boundary atmospheric and operating conditions necessary to achieve temperature errors below 2 %, minimizing the influence of solar reflection. Furthermore the influence of atmospheric parameters on the sensitivity of ratio thermography is analyzed, in particular the validity of the gray body assumption.

It is shown that the atmosphere has a critical influence on the measurement accuracy. A humid atmosphere and/or high zenith angle is necessary for making accurate measurements. Furthermore only receiver temperatures above 450^◦C could be measured for the current system configuration, regardless of atmospheric conditions. Assuming negligible solar reflections, the validity of the gray body assumption is shown to be sensitive to the precipitable water vapor. A model based atmospheric compensation is therefore required to further improve the accuracy of ratio thermography. 

This is a literature study, in linear algebra, about positive and nonnegative matrices and their special properties. We say that a matrix or a vector is positive/nonnegative if all of its entries are positive/nonnegative. First, we study some generalities and become acquainted with two types of nonnegative matrices; irreducible and reducible. After exploring their characteristics we investigate and prove the two main theorems of this subject, namely Perron's and Perron-Frobenius' theorem. In short Perron's theorem from 1907 tells us that the spectral radius of a positive matrix is a simple eigenvalue of the matrix and that its eigenvector can be taken to be positive. In 1912, Georg Frobenius generalized Perron's results also to irreducible nonnegative matrices.

The two theorems have a wide range of applications in both pure mathematics and practical matters. In real world scenarios, many measurements are nonnegative (length, time, amount, etc.) and so their mathematical formulations often relate to Perron-Frobenius theory. The theory's importance to linear dynamical systems, such as Markov chains, cannot be overstated; it determines when, and to what, an iterative process will converge. This result is in turn the underlying theory for the page-ranking algorithm developed by Google in 1998. We will see examples of all these applications in chapters four and five where we will be particularly interested in different types of Markov chains. 

The theory in this thesis can be found in many books. Here, most of the material is gathered from Horn-Johnson [5], Meyer [9] and Shapiro [10]. However, all of the theorems and proofs are formulated in my own way and the examples and illustrations are concocted by myself, unless otherwise noted. 

This study investigates the potential of engaging secondary pre-service teachers in a team-teaching approach called Socratic lectures to develop their communicational skills. An intervention was designed centered around a workshop sandwiched between a pre- and a post-presentation on natural logarithmic and derivatives to an audience of undergraduate students. A quantitative and qualitative analysis of 155 pre- and 92 documented post-experiences of the undergraduate students’ shows that the pre-service teachers’ presentations and communicating skills developed positively over the course of the intervention. The accompanying statistical analysis of Likert evaluation items shows asignificant increase of multiple communicational aspects of the pre-service teachers, and the line-by-line analysis of written evaluations support this claim regarding the pre-service teachers’interactivity. Issues about generalizability and future research is also discussed.

A Hamilton cycle in a graph is a cycle that passes through every vertex of the graph. A graph is called Hamiltonian if it contains such a cycle. The problem of determining if a graph is Hamiltonian has been studied extensively, and there are many known sufficient conditions both for Hamiltonicity and for other, related properties.

A large portion of these conditions relate the degrees of vertices of the graph to the number of vertices in the entire graph, and thus they can only apply to a limited set of graphs with high edge density. In a series of papers, Asratian and Khachatryan developed local analogues of some of these criteria. These results do not suffer from the same drawbacks as their global counterparts, and apply to larger classes of graphs.

In this thesis we study this approach of creating local conditions for Hamiltonicity and related properties, and use it to develop local analogues of some classic results. We will also see how these local conditions can allow us to extend theorems on Hamiltonicity to infinite graphs.

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  K_k,l has local list number one, the star graph S_n=K_1,n has local list number 3, and K_2,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.

Q-space trajectory imaging (QTI) enables the estimation of useful scalar measures indicative of the local tissue structure. This is accomplished by employing generalized gradient waveforms for diffusion sensitization alongside a diffusion tensor distribution (DTD) model. The first two moments of the underlying DTD are made available by acquisitions at low diffusion sensitivity (b-values). Here, we show that three independent conditions have to be fulfilled by the mean and covariance tensors associated with distributions of symmetric positive semidefinite tensors. We introduce an estimation framework utilizing semi-definite programming (SDP) to guarantee that these conditions are met. Applying the framework on simulated signal profiles for diffusion tensors distributed according to non-central Wishart distributions demonstrates the improved noise resilience of QTI+ over the commonly employed estimation methods. Our findings on a human brain data set also reveal pronounced improvements, especially so for acquisition protocols featuring few number of volumes. Our methods robustness to noise is expected to not only improve the accuracy of the estimates, but also enable a meaningful interpretation of contrast in the derived scalar maps. The techniques performance on shorter acquisitions could make it feasible in routine clinical practice.

 SchottkyspaceSg,whereg≥2isaninteger,isaconnectedcomplexorbifold of dimension 3(g − 1); it provides a parametrization of the PSL2(C)-conjugacy classes of Schottky groups Γ of rank g. The branch locus Bg ⊂ Sg, consisting of those conjugacy classes of Schottky groups being a finite index proper normal subgroup of some Kleinian group, is known to be connected. If [Γ] ∈ Bg, then there is a Kleinian group K containing Γ as a normal subgroup of index some prime integer p ≥ 2. The structural description, in terms of Klein-Maskit Combination Theorems, of such a group K is completely determined by a triple (t, r, s), where t, r, s ≥ 0 are integers such that g = p(t + r + s − 1) + 1 − r. For each such a tuple (g, p; t, r, s) there is a corresponding cyclic-Schottky stratum F(g, p; t, r, s) ⊂ Bg. It is known that F(g, 2; t, r, s) is connected. In this paper, for p ≥ 3, we study the connectivity of these F(g, p; t, r, s).

This article discusses the Length-Constrained Cycle Partition Problem (LCCP), which constitutes a new generalization of the Travelling Salesperson Problem (TSP). Apart from nonnegative edge weights, the undirected graph in LCCP features a nonnegative critical length parameter for each vertex. A cycle partition, i.e. a vertex-disjoint cycle cover, is a feasible solution for LCCP if the length of each cycle is not greater than the critical length of each vertex contained in it. The goal is to find a feasible partition having a minimum number of cycles. Besides analyzing theoretical properties and developing preprocessing techniques, we propose an elaborate heuristic algorithm that produces solutions of good quality even for large-size instances. Moreover, we present two exact mixed-integer programming formulations (MIPs) for LCCP, which are inspired by well-known modeling approaches for TSP. Further, we introduce the concept of conflict hypergraphs, whose cliques yield valid constraints for the MIP models. We conclude with a discussion on computational experiments that we conducted using (A)TSPLIB-based problem instances. As a motivating example application, we describe a routing problem where a fleet of uncrewed aerial vehicles (UAVs) must patrol a given set of areas.

Let (W,S) be a Coxeter system. We introduce the boolean com-plex of involutions ofWwhich is an analogue of the boolean complex ofWstudied by Ragnarsson and Tenner. By applying discrete Morse theory,we determine the homotopy type of the boolean complex of involutionsfor a large class of (W,S), including all finite Coxeter groups, finding thatthe homotopy type is that of a wedge of spheres of dimension |S|-1. In addition, we find simple recurrence formulas for the number of spheres inthe wedge

We prove that in finite, simply laced types, every Schubert variety indexed by an involution which is not the longest element of some standard parabolic subgroup is singular. 

A twisted derivation is a generalized derivative satisfying a twisted version of the ordinary Leibniz rule for products. In particular, a (σ, τ )-derivation on an algebra A, is a derivation where Leibniz rule is twisted by two endomorphisms σ and τ on A. Such derivations play an important role in the theory of quantum groups, as well as in the context of discretized and deformed derivatives. In this thesis, we develop a (commutative and noncommutative) differential geometry based on (σ, τ )- derivations. To this end, we introduce the notion of (σ, τ )-algebra, consisting of an associative algebra together with a set of (σ, τ )-derivations, to construct connections satisfying a twisted Leibniz rule in analogy with (σ, τ )-derivations. We show that such connections always exist on projective modules and that it is possible to construct connections compatible with a hermitian form. To construct torsion and curvature of (σ, τ )-connections, we introduce the notion of (σ, τ )-Lie algebra and demonstrate that it is possible to construct a Levi-Civita (σ, τ )-connection. Having constructed the framework for studying (σ, τ)-connections, we demonstrate that the framework applied to commutative algebras can help to also give a good understanding of (σ, τ )-derivations on commutative algebras. In particular, we introduce a notion of symmetric (σ, τ )-derivations together with some regularity conditions. For example, we show that strongly regular (σ, τ )-derivations are always inner and there exist a symmetric (σ, τ )-connection on symmetric (σ, τ )- algebras. Finally, we introduce a (σ, τ)-Hochschild cohomology theory which in first degree captures the outer (σ, τ )-derivations of an associative algebra. Along the way, examples including both commutative and noncommutative algebras are presented to illustrate the novel concepts. 

We classify compact Riemann surfaces of genus g, where g−1 is a prime p, which have a group of automorphisms of order ρ(g−1)for some integer ρ≥1, and determine isogeny decompositions of the corresponding Jacobian varieties. This extends results of Belolipetzky and the second author for ρ>6, and of the first and third authors for ρ= 3, 4, 5 and 6. As a corollary we classify the orientably regular hypermaps (including maps) of genus p+1, together with the non-orientable regular hypermaps of characteristic −p, with automorphism group of order divisible by the prime p; this extends results of Conder, Širáň and Tucker for maps.

Let be a finite poset (partially ordered set) with cardinality . A linear extension of is an order-preserving bijection : , that is, if in then . We define the poset probability as the proportion of linear extensions where . We are primarily interested in for incomparable elements . The probability has significance in areas such as information theory. Let denote the total number of linear extensions of . We prove that the poset probability can be evaluated as

where is the set of order ideals of without or , where we can add to get a new order ideal of . The practicality of the preceding formula is explored and we show that

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 , the probability obeys

where is the nth Catalan number and .

We also explore how Monte Carlo methods can be used to estimate .

This essay focuses on configurations consisting of lines and points, as well as circles and points, which have applications in areas such as data correction methods. We present classical examples of these configurations, such as Miguel’s configuration and projective planes. Furthermore, we investigate the three series of circle configurations introduced by Gévay and Pisanski in 2012. These series are based on V-constructions of various unit graphs, polyhedra, and Kneser graphs, which describe relationships between sets. The majority of the theory is derived from [11] and [6]. All images have been created by myself, and those that appear more symmetric were generated using chapGPT.

The sudden occurrence of a global pandemic brought an imbalance to each of our ways of living. Prediction of the COVID-19 progresses is crucial for governments and is deemed essential for the healthcare industry to solve which indicates how mathematical modelling becomes in dire need of use This master thesis aims to use well-established compartmental models like SEIR and modifications of SEIR to simulate scenarios like introducing different restrictions (i.e. isolation of elders, closing schools) by first fitting parameters of the SEIR models to official data. An estimation of the true calculated number of COVID-19 cases in Denmark, Germany, Spain, Sweden and the United Kingdom is made by comparing the number of deceased cases with the number of confirmed cases between the first and second waves of the virus spread. A simulation showing the effect of the British variant B.1.1.7 had on the Swedish third wave is also made.

The estimation made from comparing the first and second waves shows that the number of confirmed cases underestimates the true calculated number of cases with a factor between 7.57 and 36.5 depending on the country in consideration. The restriction simulations show that isolating elders decreases the number of elders that contract the disease but no significant change is seen in the total number of infected individuals. When restrictions like closing schools are simulated the total number of infected decreases significantly but the change in decreased cases among elders is not that high.

The simulations also show that the British mutation B.1.1.7 with its increased infectiousness is responsible for the third wave of the virus spread. Without the existence of the British mutation, the Swedish third wave of virus spread would have been less severe.

Peakons (peaked solitons) are particular solutions admitted by certain nonlinear PDEs, most famously the Camassa-Holm shallow water wave equation. These solutions take the form of a train of peak-shaped waves, interacting in a particle-like fashion. In this article we give an overview of the mathematics of peakons, with particular emphasis on the connections to classical problems in analysis, such as Padé approximation, mixed Hermite-Padé approximation, multi-point Padé approximation, continued fractions of Stieltjes type and (bi)orthogonal polynomials. The exposition follows the chronological development of our understanding, exploring the peakon solutions of the Camassa-Holm, Degasperis-Procesi, Novikov, Geng-Xue and modified Camassa-Holm (FORQ) equations. All of these paradigm examples are integrable systems arising from the compatibility condition of a Lax pair, and a recurring theme in the context of peakons is the need to properly interpret these Lax pairs in the sense of Schwartz's theory of distributions. We trace out the path leading from distributional Lax pairs to explicit formulas for peakon solutions via a variety of approximation-theoretic problems, and we illustrate the peakon dynamics with graphics.

Let p be a parabolic subalgebra of 𝔰⁢𝔩⁢(V) of maximal dimension and let n⊂p be the corresponding nilradical. In this paper, we classify the set of sl(V)-modules whose restriction to U(n) is free of rank 1. It turns out that isomorphism classes of such modules are parametrized by polynomials in dimV−1 variables. We determine the submodule structure for these modules and we show that they generically are simple.

Magnetic Resonance Imaging (MRI) is an essential healthcare technology, with diffusion MRI being a specialized technique. Diffusion MRI exploits the inherent diffusion of water molecules within the human body to produce a high-resolution tissue image. An MRI image contains information about a 3D volume in space, composed of 3D units called voxels.

This thesis assumes the existence of a probability distribution for the diffusivity within a voxel, the diffusion tensor distribution (DTD), with the diffusivity described by a family of diffusion tensors. In 2D, these tensors can be described by 2x2 symmetric positive semidefinite matrices. The objective is to estimate the DTD of a voxel with neural networks for both 1D and 2D diffusion tensors. We assume the DTD to be a discrete distribution, with a finite set of diffusion tensors.

The MRI signal is influenced by several experimental parameters, which for diffusion measurements are summarized in the measurement tensor B. To determine the diffusivity of a voxel, multiple measurement tensors are utilized, producing various MRI signals. From these signals, the network estimates the corresponding DTD of the voxel. The network seeks to employ the earth mover's distance (EMD) as its loss function, given the established use of EMD as a distance between probability distributions. Due to the difficulty of expressing the EMD as a differentiable loss function, the Sinkhorn distance, an entropic regularized approximation of the EMD, is used instead.

Different network configurations are explored through simulations to identify optimal settings, assessed by the EMD loss and the closeness of the Sinkhorn loss to the EMD.

The results indicate that the network achieves satisfactory accuracy when approximating DTDs with a small number of diffusivities, but struggles when the number increases. Future work could explore alternative loss functions and advanced neural network architectures. Despite the challenges encountered, this thesis offers relevant insight into the estimation of diffusion tensor distributions.

The study and modeling of water exchange in complex media using different applications of diffusion and relaxation magnetic resonance (MR) have been of interest in recent years. Most models attempt to describe this process using a first order kinetics expression, which is appropriate to describe chemical exchange; however, it may not be suitable to describe diffusion-driven exchange since it has no direct relationship to diffusion dynamics of water molecules. In this paper, these limitations are addressed through a more general exchange expression that does consider such important properties. This exchange fraction expression features a multi-exponential recovery at short times and a mono-exponential decay at long times, both of which are not captured by the first order kinetics expression. Furthermore, simplified exchange expressions containing partial information of the analyzed system's diffusion and relaxation processes and geometry are proposed, which can potentially be employed in already established estimation protocols. Finally, exchange fractions estimated from simulated MR data and derived here were compared, showing qualitative similarities but quantitative differences, suggesting that the features of the derived exchange fraction in this paper can be partially recovered by employing an existing estimation framework.

Given a set of k colors and a graph G with a subset S of precolored edges (a partial k-edge coloring of G), we consider the problem of determining whether G has a proper edge coloring of G with the same k colors (an extension of the partial coloring) where the colors of edges in S are not changed. If such a coloring exists, then the partial k-coloring is called extendable.

Some scheduling problems as well as some combinatorial problems can be reformulated as partial edge coloring extension problems for corresponding graphs. Partial edge coloring extension problems seem to have been first considered in connection with the problem of completing partial Latin squares. In 1960 Evans stated his conjecture that any partial Latin square of size n with at most n – 1 non-empty cells can be completed to a Latin square of size n. In terms of edge colorings this is equivalent to the statement that any proper partial n-edge coloring of the balanced complete bipartite graph K_n,n with at most n – 1 precolored edges is extendable. This classical conjecture was proved by Smetaniuk (1981), and also independently by Andersen and Hilton (1983). Moreover, Andersen and Hilton completely characterized which partial Latin squares of size n with n non-empty cells that cannot be completed to a Latin square of size n. In addition, Andersen (1985) characterized partial Latin squares of size n with n+1 non-empty cells that are completable to Latin squares of size n.

More recently, the problem of extending a partial edge coloring where the precolored edges form a matching has been considered by Edwards et al. (2018). Casselgren, Markstrom and Pham (2020) studied questions on extending partial edge colorings of the n-dimensional hypercubes Q_n. In particular, they obtained an analogue of the positive solution to Evans' conjecture on completing partial Latin squares by proving that every proper partial edge coloring of at most n – 1 edges of Q_n can be extended to a proper n-edge coloring of Q_n. They also characterized which partial edge colorings of Q_n with precisely n precolored edges are extendable to proper n-edge colorings of Q_n.

In this thesis we study similar partial edge coloring extension problems for trees. Let T be a tree with maximum degree Δ(T). First, we characterize which partial edge colorings with at most Δ(T) precolored edges in T that are extendable to proper Δ(T)-edge colorings, thereby proving an analogue of the aforementioned result by Andersen and Hilton for Latin squares. Then, we prove an analogue for trees of the result of Andersen by characterizing exactly which precolorings of at most Δ(T) + 1 precolored edges in a tree T that are extendable to Δ(T)-edge colorings of T. We also prove sharp conditions on when it is possible to extend a precolored matching or a collection of connected precolored subgraphs of a tree T to a Δ(T)-edge coloring of T. Finally, we consider the problem of avoiding a given (not necessarily proper) partial edge coloring.

In this report we will plunge down in the fascinating world of the generating functions. Generating functions showcase the "power of power series", giving more depth to the word "power" in power series. We start off small to get a good understanding of the generating function and what it does. Also, off course, explaining why it works and why we can do some of the things we do with them. We will see alot of examples throughout the text that helps the reader to grasp the mathematical object that is the generating function.

We will look at several kinds of generating functions, the main focus when we establish our understanding of these will be the "ordinary power series" generating function ("ops") that we discuss before moving on to the "exponential generating function" ("egf"). During our discussion on ops we will see a "first time in literature" derivation of the generating function for a recurrence relation regarding "branched coverings". After finishing the discussion regarding egf we move on the Hermite polynomials and show how we derive their generating function. Which is a generating function that generates functions. Lastly we will have a quick look at the "moment generating function".

This master thesis uses Piecewise Deterministic Markov Processes (PDMPs) to estimate the initial kinematic state of a supersonic missile. PDMPs is a class of samplers designed for Bayesian inferences. These samplers are adept at efficiently sampling complex distributions. Specifically, the Zig-Zag process, a noteworthy PDMP sampler, will be employed to estimate the supersonic missile's state. In this thesis we apply a stochastic differential equation (SDE) that effectively describes the intricate dynamics of a supersonic missile. To ensure a comprehensive representation of the missile's motion, we will incorporate air-resistance models specifically designed for supersonic aerial objects into this SDE. Calculations for the SDE will be performed with Itô calculus. By leveraging the Zig-Zag process in this context, we will demonstrate its effectiveness and suitability for addressing the initial kinematic state estimation problem of supersonic missiles. The experimental results will showcase the accuracy and efficiency of this approach, exploring its potential for real-world applications.

A star edge coloring is any edge coloring which is both proper and contains no cycles or path of length four which are bicolored, and the star chromatic index of a graph is the smallest number of colors for which that graph can be star edge colored. Star edge coloring is a relatively new field in graph theory, and very little is known regarding upper bounds of the star chromatic index of most graph types, one of these families being multipartite graphs. We investigate a method for obtaining upper bounds on the star chromatic index of complete multipartite graphs. The basic idea is to decompose such graphs into smaller complete bipartite graphs and applying known upper bounds for such graphs.This method has also been implemented and we present a hypothesis based on simulations.

Noncommutative geometry has over the past four of decades grown into a rich field of study. Novel ideas and concepts are rapidly being developed, and a notable application of the theory outside of pure mathematics is quantum theory. This thesis will focus on a derivation-based approach to noncommutative geometry using the framework of real calculi, which is a rather direct approach to the subject. Due to their direct nature, real calculi are useful when studying classical concepts in Riemannian geometry and how they may be generalized to a noncommutative setting.

This thesis aims to shed light on algebraic aspects of real calculi by introducing a concept of morphisms of real calculi, which enables the study of real calculi on a structural level. In particular, real calculi over matrix algebras are discussed both from an algebraic and a geometric perspective.Morphisms are also interpreted geometrically, giving a way to develop a noncommutative theory of embeddings. As an example, the noncommutative torus is minimally embedded into the noncommutative 3-sphere.

In analogy with the geometric situation, we study real calculi over projective modules and describe how they are related to free real calculi using real calculus homomorphisms. Moreover, we study real calculi over matrix algebras and discuss several aspects of the classification problem for real calculi in this case. We also use matrix algebras to give concrete examples of real calculi where the module is projective, and how this affects the existence of a Levi-Civita connection. 

Noncommutative geometry extends the traditional connections between algebra and geometry beyond the realm of commutative algebras, allowing for a broader exploration of geometric concepts in noncommutative settings. The geometric perspective facilitates the study and understanding of various mathematical structures, including operator algebras, quantum groups, and noncommutative spaces. Since its inception, noncommutative geometry has experienced remarkable growth, attracting mathematicians from diverse backgrounds who seek to delve into the geometric aspects of noncommutative structures. Through this lens, groundbreaking discoveries have deepened our understanding of fundamental mathematical principles and opened up new avenues of research. This ongoing exploration not only enriches our mathematical knowledge but also finds practical applications in theoretical physics, quantum field theory, and interdisciplinary fields.

The primary focus of this thesis is to offer valuable insights into the derivation-based approach of real calculi, which employs modules over an algebra as an algebraic analogy for vector bundles over differential manifolds. An overarching goal is to give noncommutative counterparts of classical geometric concepts, with a specific emphasis being placed on a noncommutative adaptation of the Levi-Civita connection in (pseudo-)Riemannian geometry. An investigation into the existence of a Levi-Civita connection is conducted in the context of general projective modules, and in cases where it exists a theory of embeddings is developed and used to give a minimal embedding of the noncommutative torus into the noncommutative 3-sphere. The thesis also establishes the concept of morphisms of real calculi, which plays a crucial role in examining the relationship between projective modules and specific free modules in this framework. Moreover, the thesis provides an in-depth examination of matrix algebras, utilizing them as illustrative examples to showcase the process of determining isomorphism classes of real calculi in various scenarios and presenting classes of examples where a Levi-Civita connection does not exist.

We present several characterizations of sigma-compact Hattori spaces, and reject some possible characterization candidates of the spaces.