70 search hits
-
Über eine Erweiterung der Methode von Soshnikov zur Untersuchung des größten Eigenwerts auf unsymmetrische Verteilungen
(2013)
-
Felix Grimme
- Seit der Entdeckung des Halbkreisgesetzes durch Wigner werden reell-symmetrische Zufallsmatrix-Ensembles untersucht. Soshnikov hat in einer bahnbrechenden Arbeit gezeigt, dass für Wigner-Ensembles $A_n=(\xi_\ij)_{1\le i\le j\le n}$ mit symmetrisch verteilten Einträgen die Verteilung des größten Eigenwerts in einer geeigneten Skalierung für $n\to\infty$ universelles Verhalten zeigt und schwach gegen die Tracy-Widom-Verteilung, die Verteilung des Gauß'schen orthogonalen Ensembles, konvergiert. Für den Beweis nutzt Soshnikov die Momentenmethode. Hierbei wird die Analyse der Verteilungsfunktion des größten Eigenwerts auf die Analyse von Erwartungswerten von Spuren hoher Matrixpotenzen zurückgeführt (die Exponenten wachsen mit $n^{2/3}$). Die Spuren werden via $\tr A_n^{p_n}=\sum_{(i_0,\ldots,i_{p-1})\in[n]^p}\xi_{i_0,i_1}\xi_{i_1,i_2}\ldots\xi_{i_{p-1},i_0}$ als Summe über geschlossene Pfade kombinatorisch interpretiert. In der Analyse gilt es herauszufinden, welche Klassen von Pfaden (die mit den Momenten der Matrixeinträge in Verbindung stehen) die Spuren in der Asymptotik $n\to\infty$ dominieren. Es stellt sich heraus das dies Pfade sind, die jede ihrer Kanten genau zweimal durchlaufen. Das bedeutet, dass die Spuren asymptotisch nur von den für alle Matrixeinträge gleichen zweiten Momenten abhängen, sie sind also asymptotisch für alle betrachteten Ensembles universell.
Diese Methode wird in der vorliegenden Arbeit auf Wigner-Ensembles mit nicht notwendig symmetrischen Verteilungen der Einträge erweitert. Die Kombinatorik ist in diesem Fall komplexer. Resultat der Arbeit ist, dass die Methode von Soshnikov funktioniert, wenn folgende Bedingungen erfüllt sind:
die ersten und dritten Momente der Einträge sind~0
für die 97.\ Momente existiert eine in~$n$ gleichmäßige Schranke.
-
Integrated size and price optimization for a fashion retailer
(2013)
-
Miriam Kießling
- This thesis is the result of a collaboration with a German fashion retailer which lasted
for several years. The aim was the development of a decision-support system for the
supply of the about 1300 branches in Germany.
There are some specialties about the situation at our industrial partner: The branches
are supplied by prepackaged size-assortments of a product which we call lot-types.
With the objective to economize handling cost, these lot-types are already composed
at the respective low-wage country where the article is also produced. The expense at
the German central warehouse is further reduced by allowing only four or five different lot-types for the delivery of one product. Moreover, each branch is supplied by a
certain quantity of a single lot-type.
For the most fashion articles replenishment is not possible. The sales success of
a product is a priori unknown. Historical sales data can only be used on a higher
aggregation level, e.g., the average historical demand on the commodity group level.
Demand estimation is therefore very vague. Under- and oversupplies are unavoidable.
Influence over the sales process is possible by marking down prices. To compensate
for an oversupply of a product, weekly the price can be reduced to predefined price
steps which depend on the starting price of the product. Mark-downs for an article are
performed simultaneously for all branches and sizes.
Within the cooperation mathematical problem formulations with the aim to minimize measures for the deviation of supply from estimated demand had been developed.
In these measures the selling process is not or only very vaguely regarded.
Now we include the possibility of marking down prices during the selling time
already when deciding on the supply. The result is the two-stage stochastic program
ISPO: The so-called first stage decision is the determination of a supply policy. The
second stage decision, or recourse, is the decision on mark-downs during the selling
time. ISPO yields an expected revenue maximizing supply strategy and corresponding
optimal mark-down strategies for the considered scenarios.
ISPO it too complex to solve it via standard approaches. Customized methods had
to be devised to solve ISPO. On the one side we present an exact solver for benchmarking. On the other side a fast heuristic was developed for practical use at our partner.
The basic idea of our exact solver is to enumerate all possible mark-down strategies.
With this it is possible to reduce ISPO to a former formulation for the optimization of
supply, which can be solved via standard approaches.
In practice enumeration of all valid mark-down strategies for the purpose of solving
ISPO is for reasons of time impossible. Therefore the idea is extended to a customized
Branch&Bound approach. In this context we derived dual bounds for general two-stage stochastic programs which are based on the so-called wait-and-see solution from
stochastic programming. We show that in general our bounds are tighter.
The heuristic, beginning with a valid second stage decision, determines an optimal
first stage decision and alternates between solving the first stage and the second stage
until convergence is reached. The optimality gap is small enough to justify a practical
use at the industrial partner.
In practice the by ISPO proposed mark-down strategies are not applied; instead
latest sales figures are exploited. According to these and an updated demand estimation
weekly a new optimal mark-down strategy for the remaining selling time of the product
is determined. For this purpose we propose an algorithm which relies on dynamic
programming and tries to exclude non-optimal solutions a priori by dominance checks.
ISPO, more precisely our heuristic approach, together with the weekly adaption of
the mark-down strategy forms our decision support system for integrated size and price
optimization DISPO.
We tested DISPO in a five-month field study, performed as a statistical experiment,
at our partner where pairs of similar branches were compared. At one branch of each
pair, the test branch, supply and mark-down decisions came from ISPO. With respect
to latest sales figures the mark-down decisions were weekly updated via our dynamic
programming approach. At the other branch, the control branch, these decisions were
not integrated: Supply was determined according to a strategy resulting from a former
model that disregarded the selling process and mark-downs were handled manually by
our partner. For the branches at which the decisions of ISPO were implemented an
average raise of 1.5 percentage points of relative revenue was observed.
-
THE INDEX THEOREM FOR QUASI-TORI
(2013)
-
Tsz On Mario Chan
- The Index theorem for holomorphic line bundles on complex tori
asserts that some cohomology groups of a line bundle vanish according
to the numbers of negative and positive eigenvalues of the associated
hermitian form. In this thesis, this theorem is generalized to quasi-tori,
i.e. connected complex abelian Lie groups which are not necessarily
compact. In view of the Remmert–Morimoto decomposition of
quasi-tori as well as the Künneth formula, it suffices to consider only
Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic
functions. The Index theorem is generalized to holomorphic line
bundles, both linearizable and non-linearizable, on Cousin-quasi-tori
using L2-methods coupled with the Kazama–Dolbeault isomorphism
and Bochner–Kodaira formulas.
-
Geometrische Konstruktionen linearer Codes über Galois-Ringen der Charakteristik 4 von hoher homogener Minimaldistanz
(2012)
-
Michael Kiermaier
- In dieser Arbeit werden vier neue unendliche Serien von linearen Codes über Galois-Ringen der Charakteristik 4 konstruiert.
Hinsichtlich der Minimaldistanz übertreffen die Gray-Bilder der konstruierten Codes alle bekannten vergleichbaren linearen Codes.
In den Konstruktionen wird die Theorie der projektiven Hjelmslev-Geometrien, der Assoziationsschemata sowie der symmetrischen Bilinearformen in endlichdimensionalen GF(2)-Vektorräumen benutzt.
-
The Integrated Size and Price Optimization problem
(2012)
-
Miriam Kießling
Sascha Kurz
Jörg Rambau
- We present the Integrated Size and Price Optimization Problem (ISPO)
for a fashion discounter with many branches. Based on a two-stage
stochastic programming model with recourse, we develop an exact
algorithm and a production-compliant heuristic that produces small
optimality gaps. In a field study we show that a distribution of
supply over branches and sizes based on ISPO solutions is
significantly better than a one-stage optimization of the
distribution ignoring the possibility of optimal pricing.
-
Shape Calculus Applied to Elliptic Optimal Control Problems
(2012)
-
Michael Frey
- This thesis is devoted to the analysis of a very simple, pointwisely state-constrained optimal control problem of an elliptic partial differential equation. The transfer of an idea from the field of optimal control of ordinary differential equations, which proved fruitful with respect to both theoretical treatment and design of algorithms, is the starting point. On this, the state inequality constraint, which is regarded as an equation inside the active set, is differentiated in order to obtain a control law.
A geometrical splitting of the constraints is necessary to carry over this approach to the chosen model problem. The associated assertions are rigorously ensured. The subsequent derivation of a control law in the sense of the abovementioned idea yields an equivalent reformulation of the model problem. The active set appears as an independent and equal optimization variable in this new formulation. Thereby a new class of optimization problem is established, which forms a hybrid of optimal control and shape-/topology optimization: set optimal control. This class is integrated into the very abstract framework of optimization on vector bundles; for that purpose some important notions from the field of calculus on manifolds are introduced and related with shape calculus.
First order necessary conditions of the set optimal control problem are derived by means of two different approaches: on the one hand a reduced approach via the elimination of the state variable, which uses a formulation as bilevel optimization problem, is pursued, and on the other hand a formal Lagrange principle is presented.
A comparison of the newly obtained optimality conditions with those known form literature yields relations between the Lagrange multipliers; in particular, it becomes apparent that the new approach involves higher regularity. The comparison is embedded to the theory of partial differential-algebraic equations, and it is shown that the new approach yields a reduction of the differential index.
Upon investigation of the gradient and the second covariant derivative of the objective functional different Newton- and trial algorithms are presented and discussed in detail. By means of a comparison with the well-established primal-dual active set method different benefits of the new approach become apparent. In particular, the new algorithms can be formulated in function space without any regularization. Some numerical tests illustrate that an efficient and competitive solution of state-constrained optimal control problems is achieved.
The whole work gives numerous references to different mathematical disciplines and encourages further investigations. All in all, it should be regarded as a first step towards a more comprehensive perspective on state-constrained optimal control of partial differential equations.
-
Irreducible symplectic complex spaces
(2012)
-
Tim Kirschner
- In Chapter 1 we define period mappings of Hodge-de Rahm type for certain submersive, yet not necessarily locally topologically trivial, morphisms of complex manifolds. Generalizing Griffiths's theory, we interpret the differential of such period mappings as the composition of the Kodaira-Spencer map and a map derived from the sheaf cohomological cup product and the contraction of vector fields with differential forms.
In Chapter 2 of the text, we consider a submersive morphism $f\colon X\to S$ of complex spaces which is compactified by a proper, flat, and Kähler morphism $\bar f\colon \bar X\to S$. Taking into account the codimension of $\bar X\setminus X$ in $\bar X$, we draw conclusions about the degeneration behavior of the relative Frölicher spectral sequence of the morphism $f$ and about the local freeness of the modules $\mathrm{R}^qf_*(\Omega^p_f)$; our results can be viewed as relative generalizations of a theorem of Takeo Ohsawa.
In our final Chapter 3, we employ the upshots of the preceding two chapters in order to deduce a local Torelli theorem for irreducible symplectic complex spaces. As an application of the local Torelli theorem, we prove that irreducible symplectic complex spaces whose codimension of the singular locus does not deceed $4$ satisfy the so-called Fujiki relation.
-
Das Optimierungslabor – ein Erfahrungsbericht
(2012)
-
Miriam Kießling
Tobias Kreisel
Sascha Kurz
Jörg Rambau
Konrad Schade
Cornelius Schwarz
- Seit mehreren Jahren besuchen uns Schülerinnen und Schüler an der
Universität zu Anlässen wie dem Tag der Mathematik, dem Girls’ Day,
der MINT-Universität oder einfach auf Initiative ihrer
Klassenleitungen. Sie möchten einen Einblick in die Welt der
Mathematik über die Schulmathematik hinaus bekommen. Doch wie lässt
sich die Brücke vom Schulstoff zu den Inhalten der
Universitätsmathematik schlagen? Und: findet man einen
Themenschwerpunkt, bei dem ein aktives Mitmachen trotz fehlender
Vorkenntnisse in Anbetracht begrenzter Zeit möglich wird?
In der diskreten Optimierung lassen sich Problem-Modellierung und
Problem-Lösung sehr gut trennen. Selbst forschungsnahe Modelle der
ganzzahligen linearen Optimierung (MILP-Modelle) basieren auf sehr
elementaren Überlegungen, wie die Entscheidungsmöglichkeiten, Ziele
und Restriktionen eines Alltagsproblems in Variablen,
Bewertungsfunktionen, Gleichungen und Ungleichungen ausgedrückt werden
können. Wie dann optimale Lösungen gefunden werden, erfordert zwar
tiefergehende Mathematik, es gibt aber Software dafür, in der das
Wissen aus Teilen des Mathematik-Studiums und der mathematischen
Forschung kondensiert vorliegt.
Unser Vermittlungsziel: Schülerinnen und Schüler wissen nach dem
Besuch, dass man verschiedenste Probleme angreifen kann, indem man sie
in die Sprache der Mathematik übersetzt, denn in Software gegossenes
mathematisches Know-How kann dann diese Probleme lösen, ohne etwas
über die Probleme selbst zu wissen. Unsere Idee für eine Maßnahme:
Ein Optimierungslabor. Die Schülerinnen und Schüler isolieren in
Teamarbeit die wesentlichen logischen Merkmale von Sudokulösen,
Rucksackpacken, Routenplanung u.v.a.m. Dann übersetzen sie die
Problemstellungen in die Sprache der Mathematik (hier: MILP-Modelle)
und lassen sie (unterstützt durch unser Team) von Computerprogrammen
lösen (MILP-Löser), die nichts anderes als diese Sprache verstehen.
Schließlich übersetzen sie die mathematischen Lösungen wieder in die
Sprache der Problemstellung. Erfahrungen mit der Modellierung auf
Basis linearer Gleichungssysteme können dabei aus dem Schulunterricht
eingebracht werden.
In diesem Bericht wollen wir unsere Erfahrungen mit konkreten Details
der Umsetzung schildern.
-
An exact column-generation approach for the lot-type design problem
(2012)
-
Sascha Kurz
Miriam Kießling
Jörg Rambau
- We consider a fashion discounter distributing its many branches with
integral multiples from a set of available lot-types. For the problem of
approximating the branch and size dependent demand using those lots
we propose a tailored exact column generation approach assisted by fast
algorithms for intrinsic subproblems, which turns out to be very efficient
on our real-world instances.
-
Galois representations of orthogonal rigid local systems
(2012)
-
Michael Schulte
- We use the middle convolution introduced by Katz to construct a families of lisse sheaves on the affine line without two points. These correspond to continuous representations of the etale fundamental group, which can be specialized to compatible systems of Galois representations. This leads to the second maximally unipotent family.
Because of the geometric origin, we can show using a theorem of Barnet-Lamb, Gee, Geraghty and Taylor that they are potentially automorphic.
