The three-state toric homogeneous Markov chain model has Markov degree two

We prove that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means, that a certain class of toric ideals are generated by quadratic binomials. This was conjectured by Haws, Martin del Campo, Takemura and Yoshida, who proved that they are generated by binomials of degree six or less.Comment: Updated language and notation. 13page

Europe’s Lack of Structural Transformation and Necessary Policy Changes of EMU

Primary goal of stabilization policy in the Treaty of European Union is price stability. That goal may be in conflict with the goal of full employment in the member states, particularly, then the union are hit by an asymmetric shock. Assuming perfect capital mobility a initial adverse shock (Krugman 1993) may have permanent effects by releasing a self-reinforcing process, which will result in lower relative growth. Given the specification of a model that captures the crucial element of efficient structural transformation it is easy to conclude the lack of necessary structural transformation within EMU. In addition, the basic foundation of economic policy by EMU, as manifested by the Treaty of European Union, is by latter research put into question. Therefore this paper suggest, it is necessary that the Treaty of European Union must be supplemented, changed, or both.European Monetary Union; Structural flexibility; Optimal transformation; Phillips curve; Maastricht Treaty; Stability and Growth Pact.

Tverberg's theorem and graph coloring

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are encoded with graphs. When two points are adjacent in the graph, they are not in the same part. If the restrictions are too harsh, then the topological Tverberg theorem fails. The colored Tverberg theorem corresponds to graphs constructed as disjoint unions of small complete graphs. Hell studied the case of paths and cycles. In graph theory these partitions are usually viewed as graph colorings. As explored by Aharoni, Haxell, Meshulam and others there are fundamental connections between several notions of graph colorings and topological combinatorics. For ordinary graph colorings it is enough to require that the number of colors q satisfy q>Delta, where Delta is the maximal degree of the graph. It was proven by the first author using equivariant topology that if q>\Delta^2 then the topological Tverberg theorem still works. It is conjectured that q>K\Delta is also enough for some constant K, and in this paper we prove a fixed-parameter version of that conjecture. The required topological connectivity results are proven with shellability, which also strengthens some previous partial results where the topological connectivity was proven with the nerve lemma.Comment: To appear in Discrete and Computational Geometry, 13 pages, 1 figure. Updated languag

Polytopes from Subgraph Statistics

Polytopes from subgraph statistics are important in applications and conjectures and theorems in extremal graph theory can be stated as properties of them. We have studied them with a view towards applications by inscribing large explicit polytopes and semi-algebraic sets when the facet descriptions are intractable. The semi-algebraic sets called curvy zonotopes are introduced and studied using graph limits. From both volume calculations and algebraic descriptions we find several interesting conjectures.Comment: Full article, 21 pages, 8 figures. Minor expository update

Algebraic Statistics

This thesis on algebraic statistics contains five papers. In paper I we define ideals of graph homomorphisms. These ideals generalize many of the toric ideals defined in terms of graphs that are important in algebraic statistics and commutative algebra.   In paper II we study polytopes from subgraph statistics. Polytopes from subgraph statistics are important for statistical models for large graphs and many problems in extremal graph theory can be stated in terms of them. We find easily described semi-algebraic sets that are contained in these polytopes, and using them we compute dimensions and get volume bounds for the polytopes.  In paper III we study the topological Tverberg theorem and its generalizations. We develop a toolbox for complexes from graphs using vertex decomposability to bound the connectivity.  In paper IV we prove a conjecture by Haws, Martin del Campo, Takemura and Yoshida. It states that the three-state toric homogenous Markov chain model has Markov degree two. In algebraic terminology this means that a certain class of toric ideals are generated by quadratic binomials.  In paper V we produce cellular resolutions for a large class of edge ideals and their powers. Using algebraic discrete Morse theory it is then possible to make many of these resolutions minimal, for example explicit minimal resolutions for powers of edge ideals of paths are constructed this way.Denna avhandling om algebraisk statistik innehåller fem artiklar. I artikel I definieras ideal av grafhomomorfier. Dessa ideal generaliserar ett flertal konstruktioner av ideal från grafer som är viktiga i algebraisk statistik samt kommutativ algebra. I artikel II behandlas polytoper från delgrafsstatistik. Dessa är viktiga för att förstå statistiska modeller som beskriver stora grafer och många problem om ytterlighetsgrafer kan formuleras med dem. Bland verktygen som används är att beskriva semi-algebraiska mängder i polytoperna och genom detta bestämma deras dimension samt begränsa volymen. I artikel III behandlas den topologiska tverbergssatsen med generaliseringar. Grafkomplexen förstås genom att begränsa sammanhängandegraden medelst hörnnedbrytbarhet. I artikel IV bevisas att ideal tillhörande markovkedjor med tre tillstånd är genererade i grad två, vilket förmodats av Haws, Martin del Campo, Takemura och Yoshida. I artikel V skapas cellulära upplösningar för en stor klass av kantideal samt deras potenser. Med algebraisk diskret morseteori görs dessa upplösningar minimala för kantideal från stigar.

Modelling Power Spikes with Inhomogeneous Markov-Switching Models

Interest in modelling electricity prices has, despite its relatively short history, resulted in widespread types of models that tend to be too intricate to incorporate most price characteristics. This thesis pursues a flexible approach that comprehends stylized facts of electricity prices while it still handles complexity in order to facilitate calibration and forecasting. Although time-varying transitions of non-linear Markov-switching models add a new dimension to the problem, the extension is pivotal to encompass the timing of power spikes. Simulation studies provide a comparison between the maximum likelihood estimator and the EM algorithm and validate the precision of the estimators. A comprehensive study of the model framework in the independent regime setting that is applied to real data from the German and Nordic markets confirms the hypothesis that extensive models with exogenous variables outperform time-invariant counterparts. Improvements of electricity price dynamics and other issues involved in the process of modelling electricity prices as well as potential future research topics are also suggested and discussed