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Local interactions between particles of a collection causes all particles to reorganize in new positions. The purpose of this paper is to construct an energy-based model of self-organizing subgroups, which describes the behavior of singular local moves of a particle. The present paper extends the Hegselmann-Krause model on consensus dynamics, where agents simultaneously move to the barycenter of all agents in an epsilon neighborhood. The Energy-based model presented here is analyzed and simulated on finite metric space. AMS Subject Classifications:81T80; 93A30; 37M05; 68U20
In this work, we extend the Hegselmann and Krause (HK) model, presented in [16] to an arbitrary metric space. We also present some theoretical analysis and some numerical results of the condensing of particles in finite and continuous metric spaces. For simulations in a finite metric space, we introduce the notion "random metric" using the split metrics studies by Dress and al. [2, 11, 12].
Das libor Markt Modell (LMM) ist seit seiner Entwicklung in den Veröffentlichungen von Brace, Gatarek, Musiela (1997), einerseits, und unabhängig von diesen von Miltersen, Sandmann, Sondermann (1997), andererseits, zu dem anerkanntesten Instrument zur Modellierung der Zinsstruktur und der damit verbundenen Preisfindung für relevante Finanzderivate geworden. libor steht dabei für London Inter-Bank Offered Rate, ein täglich in London fixierter Referenz-Zins für kurzfristige Anlagen. Drei- oder sechsmonatige Laufzeiten sind in Verbindung mit dem LMM üblich. Die Forschung zur Verbesserung dieses Modells hat in den letzten Jahren an Zuwachs gewonnen. Beim Versuch den Fehler der Anpassung an die täglich beobachteten Preise von Zinsoptionen wie Caps und Swaptions zu verringern, erhält man in der Folge auch genauere Bewertungen für andere, exotischere, Derivate. Die zugrunde liegende und zentrale Idee des LMM besteht darin, die Forward (Termin) Zinsen direkt als primären (Vektor) Prozess mehrerer libor Sätze zu betrachten und diese simultan zu modellieren, anstatt sie nur herzuleiten aus einem übergeordneten, unendlich dimensionalen Forward Zinsprozess, wie im zeitlich früher entwickelten Heath-Jarrow-Morton Modell. Das überzeugendste Argument für diese Diskretisierung ist, dass die libor Sätze direkt im Markt beobachtbar sind und ihre Volatilitäten auf eine natürliche Weise in Beziehung gebracht werden können zu bereits liquide gehandelten Produkten, eben jenen Caps und Swaptions. Dennoch beinhaltet das Modell eine gravierende Insuffizienz, indem es keine Krümmung der Volatilitätsoberfläche, im Hinblick auf Optionen mit verschiedenen Basiszinsen, abbildet. Wie im einfachen eindimensionalen Black-Scholes Modell prägen sich auch hier die Ungenauigkeiten der Verteilung in fehlenden heavy tails deutlich aus. Smile und Skew Effekte sind erkennbar. Im klassischen liborMarkt Modell wird in Richtung der Basiszinsdimension nur eine affine Struktur erzeugt, welche bestenfalls als Approximation für die erwünschte Oberfläche dienen kann. Die beobachteten Verzerrungen führen naturgemäss zu einer ungenauen Abbildung der Realität und fehlerhaften Reproduktion der Preise in Regionen, die ein wenig entfernt vom Bereich am Geld liegen. Derartig ungewollte Dissonanzen in Gewinn und Verlustzahlen führten z.B. in 1998 zu gravierenden Verlusten im Zinsderivateportfolio der heutigen Royal Bank of Scotland. ...
This work connects Markov chain imbedding technique (MCIT) introduced by M.V. Koutras and J.C. Fu with distributions concerning the cycle structure of permutations. As a final result program code is given that uses MCIT to deliver proper numerical values for these. The discrete distributions of interest are the one of the cycle structure, the one of the number of cycles, the one of the rth longest and shortest cycle and finally the length of a random chosen cycle. These are analyzed for equiprobable permutations as well as for biased ones. Analytical solutions and limit distributions are also considered to put the results on a safe, theoretical base.
Mixed volumes, mixed Ehrhart theory and applications to tropical geometry and linkage configurations
(2009)
The aim of this thesis is the discussion of mixed volumes, their interplay with algebraic geometry, discrete geometry and tropical geometry and their use in applications such as linkage configuration problems. Namely we present new technical tools for mixed volume computation, a novel approach to Ehrhart theory that links mixed volumes with counting integer points in Minkowski sums, new expressions in terms of mixed volumes of combinatorial quantities in tropical geometry and furthermore we employ mixed volume techniques to obtain bounds in certain graph embedding problems.
Tropical geometry is the geometry of the tropical semiring \[\mathbb{T}:=(\mathbb{R}\cup\{\infty\},\min,+).\] Classical algebraic structures correspond to tropical structures. If $I\lhd K[x_1,\ldots,x_n]$ is an ideal in a polynomial ring over a field $K$ with valuation $v$, then the classical algebraic variety correspond to the tropical variety $T(I)$. It is the set of all points $w$, such that the minimum $\min\{v(c_\alpha)+w\cdot\alpha\}$ is achieved twice for all $f=\sum_\alpha c_\alpha x^\alpha\in I$. So tropical geometry relates algebraic geometric problems with discrete geometric problems. In this thesis we obtain a tropical version of the Eisenbud-Evans Theorem which states that every algebraic variety in $\mathbb{R}^n$ is the intersection of $n$ hypersurfaces. We find out that in the tropical setting every tropical variety $T(I)$ can be written as an intersection of only $(n+1)$ tropical hypersurfaces. So we get a finite generating system of $I$ such that the corresponding tropical hypersurfaces intersect to the tropical variety, a so-called tropical basis. Let $I \lhd K[x_1,\ldots,x_n]$ be a prime ideal generated by the polynomials $f_1, \ldots, f_r$. Then there exist $g_0,\ldots,g_{n} \in I$ such that \[ T(I) \ = \ \bigcap_{i=0}^{n}T(g_i)\] and thus $\mathcal{G} := \{f_1, \ldots, f_r, g_0, \ldots, g_{n}\}$ is a tropical basis for $I$ of cardinality $r+n+1$. Tropical bases are discussed by Bogart, Jensen, Speyer, Sturmfels and Thomas where it is shown that tropical bases of linear polynomials of a linear ideal have to be very large. We do not restrict the tropical basis to consist of linear polynomials and therefore we get a shorter tropical basis. But the degrees of our polynomials can be very large. The main ingredient to get a short tropical basis is the use of projections, in particular geometrically regular projections. Together with the fact that preimages of projections of tropical varieties are themselves tropical varieties of a certain elimination ideal we get the desired result. Let $I \lhd K[x_1, \ldots, x_n]$ be an $m$-dimensional prime ideal and $\pi : \mathbb{R}^n \to \mathbb{R}^{m+1}$ be a rational projection. Then $\pi^{-1}(\pi(T(I)))$ is a tropical variety, namely \[ \pi^{-1}(\pi(T(I))) \ = \ T(J \cap K[x_1, \ldots, x_n]) \,\] Here $J$ is an ideal in $K[x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_{n-m-1}]$ derived from the ideal $I$. We show that this elimination ideal is a principal ideal which yields a polynomial in our tropical basis. The advantage of our method is that we find our polynomials by projections and therefore we can use the results of Gelfand, Kapranov and Zelevinsky , of Esterov and Khovanskii , and of Sturmfels, Tevelev and Yu. With mixed fiber polytopes we get the structure and combinatorics of the image of a tropical variety and therefore the structure of the polynomials in our tropical basis. Let $I=\lhd K[x_1,\ldots,x_n]$ an $m$-dimensional ideal, generated by generic polynomials $f_1,\ldots, f_{n-m}$, $\pi:\mathbb{R}^n\to\mathbb{R}^{m+1}$ a projection and $\psi$ a projection presented by a matrix with a rowspace equal to the kernel of $\pi$. Then up to affine isomorphisms, the cells of the dual subdivision of $\pi^{-1} \pi T(I)$ are of the form \[ \sum_{i=1}^p \Sigma_{\psi} (C_{i1}^{\vee}, \ldots, C_{i{k}}^{\vee}) \] for some $p\in\mathbb{N}$ and faces $F_1, \ldots, F_p$ of $T(f_1)\cap\ldots\cap T(f_k)$ and the dual cell of $F_i\subseteq U = T(f_1)\cup\ldots\cup T(f_k)$ is given by $F_i^\vee=C_{i1}^{\vee}+ \ldots+ C_{ik}^{\vee}$ with faces $C_{i1}, \ldots, C_{i k}$ of $T(f_1), \ldots, T(f_{k})$. In case that we project a tropical curve we want to find the number of $(n-1)$-cells of the above form with $p>1$, i.e. the cells which are dual to vertices of $\pi(T(I))$ which are the intersection of the images of two non-adjacent $1$-cells of $T(I)$. Vertices of this type are called selfintersection points. We show that there exist a tropcal line $L_n\subset\mathbb{R}^n$ and a projection $\pi:\mathbb{R}^n\to\mathbb{R}^2$, such that $L_n$ has $\sum_{i=1}^{n-2}i$ selfintersection points. Furthermore we find tropical curves $\mathcal{C}\subset\mathbb{R}^n$, which are transversal intersections of $n-1$ tropical hypersurfaces of degrees $d_1,\ldots,d_{n-1}$ and a projection $\pi:\mathbb{R}^n\to\mathbb{R}^2$, such that $\mathcal{C}$ has at least $(d_1\cdot\ldots\cdot d_{n-1})^2\cdot \sum_{i=1}^{n-2}i) $ selfintersection points. A caterpillar is a certain simple type of a tropical line and for this type we show that it can have at most $\sum_{i=1}^{n-2}i$ selfintersection points.