A representation formula for weakly compact starshaped sets

Gegenstand dieser Arbeit ist die numerische Analysis von Optimierungsproblemen mit partiellen Differentialgleichungen (PDEs), deren Zustand oder Steuerung punktweisen Ungleichungsbeschränkungen unterliegt. Wir interessieren uns insbesondere für nichtkonvexe Probleme mit semilinearer Zustandsgleichung. Es liegt in der Natur derartiger Probleme, dass eine Lösung oftmals nur numerisch gefunden werden kann. Man interessiert sich deshalb für den Fehler zwischen einer (lokalen) Lösung des kontinuierlichen Problems und einer zugehörigen (lokalen) diskreten Lösung. Eine umfassende Diskussion des kontinuierliche Problems ist dabei eine Grundvoraussetzung. Insbesondere bei vorhanden punktweisen Zustandsbeschränkungen treten spezifische Schwierigkeiten sowohl analytischer als auch numerischer Art auf, denen man entweder direkt oder mit Hilfe von Regularisierungsansätzen begegnen kann. Mit dieser Dissertation leisten wir auf verschiedene Weise neue Beiträge zur Diskussion von Optimalsteuerungsproblemen mit punktweisen Zustandsschranken aber auch reinen Kontrollschranken. Nach kurzer Einführung in die Thematik und Bereitstellung gewisser Grundlagen beschäftigen wir uns in Kapitel 3 mit einem elliptischen semiinfiniten Optimalsteuerungsproblem. Bekannte Resultate zu notwendigen und hinreichenden Optimalitätsbedingungen, die vergleichsweise hohe Regularität von Lösungen elliptischer PDEs und die Endlichdimensionalität des Steuerungsraumes lassen eine direkte Diskussion von a priori Diskretisierungsfehlerabschätzungen für dieses Problem zu. Unser Hauptergebnis in diesem Kapitel ist eine a priori Fehlerschranke der Ordnung O(h^2|ln h|) für lokale Lösungen einer Finite-Elemente-Diskretisierung des Optimalsteuerungsproblems mit Gitterweite h in einem zweidimensionalen Ortsgebiet. Dazu stellen wir gewisse Annahmen an die Struktur der aktiven Menge. In Kapitel 4 betrachten wir ein parabolisches Optimalsteuerungsproblem mit punktweise beschräkten Steuerungsfunktionen, semilinearer Zustandsgleichung und punktweisen Zustandsbeschränkungen im gesamten Orts-Zeit-Gebiet. Im Gegensatz zu dem in Kapitel 3 diskutierten elliptischen Problem sind hier u.a. hinreichende Bedingungen zweiter Ordnung nur für eindimensionale Ortsgebiete verfügbar. Auch läßt die Verwendung von Steuerungsfunktionen an Stelle endlich vieler Parameter keine sinnvollen a priori Annahmen an die Struktur der aktiven Menge zu. Wir regularisieren daher das Problem mit der auf Meyer, Rösch und Tröltzsch zurückgehenden Lavrentievregularisierung, und können so unter anderem auf bekannte Resultate zurückgreifen, die eine höhere Regularität der Lagrangeschen Multiplikatoren sichern und eine tiefergehende Analysis ermöglichen. Wir beweisen ein Konvergenzresultat für lokale Lösungen des regularisierten Problems und weisen die lokale Eindeutigkeit regularisierter Lösungen nach. In Kapitel 5 untersuchen wir die Finite-Element-Diskretisierung eines kontrollbeschränkten parabolischen Optimalsteuerungsproblems mit semilinearer Zustandsgleichung. Wir beweisen Fehlerordnungen für diskrete lokale Lösungen in der L^2-Norm. Dabei erweitern wir Resultate, die für linear-quadratische Probleme bekannt sind, auf den nichtkonvexen Fall. Es müssen insbesondere Beschränktheitsresultate in der L^infty-Norm der semidiskreten und diskreten Zustände gezeigt werden, die unabhängig von den Diskretisierungsparametern gelten. Außerdem erfordert die Diskussion lokal optimaler Lösungen, dass Konvergenzresultate und quadratische Wachstumsbedingungen in denselben Normen betrachtet werden.

Let $E$ and $F$ be locally convex spaces and $G$ their completed $\varepsilon$-tensor product. It is shown that if $S$ and $T$ are weakly almost periodic equicontinuous semigroups of operators on $E$ and $F$ respectively, then, under mild restrictions on $E$ or $F, S \otimes T$ is a weakly almost periodic equicontinuous semigroup of operators on $G$, and the almost periodic and flight vector subspaces of $G$ are related in a natural way to the corresponding subspaces of $E$ and $F$ via the $\varepsilon$-tensor product. Furthermore, if $E$ and $F$ both decompose into a direct sum of these subspaces then so does $G$.

Optimization problems that involve discrete variables are exposed to the conflict between being a powerful modeling tool and often being hard to solve. Infinite-dimensional processes, as e.g. described by differential equations, underlying the optimization may lead to the need to solve for distributed discrete control variables. This work analyzes approximation arguments that replace the need for solving the optimization problem by the need for first solving a relaxation and second computing appropriate roundings to regain discrete controls. We provide sufficient conditions on rounding algorithms and their grid refinement strategies that allow to prove approximation of the relaxed controls by the discrete controls in weaker topologies, a feature due to the infinite-dimensional vantage point. If the control-to-state mapping of the underlying process exhibits suitable compactness properties, state vector approximation follows in the norm topology as well as, under additional assumptions, optimality principles of the computed discrete controls. The conditions are verified for representatives of the family of Sum-Up Rounding algorithms. We apply the arguments on different classes of mixed-integer optimization problems that are constrained by partial differential equations. Specifically, we consider discrete control inputs, which are distributed in the time domain, for evolution equations that are governed by a differential operator that generates a strongly continuous semigroup, discrete control inputs, which are distributed in multi-dimensional spatial domains, for elliptic boundary value problems and discrete control inputs, which are distributed in space-time cylinders, for evolution equations that are governed by differential operators such that the corresponding Cauchy problem satisfies maximal parabolic regularity. Furthermore, we apply the arguments outside the scope of partial differential equations to a signal reconstruction problem. Computational results illustrate the findings.

One of the standard goals of in nite-dimensional topology is the topological classi cation of linear metric spaces and their convex subsets. The origins of this problem go back to the questions of Fr echet [Fr] and Banach [Ba] whether all in nite-dimensional separable Fr echet (Banach) spaces are homeomorphic (we explain our terminology at the end of the introduction). These questions have been a rmatively answered by the combined results of Anderson and Kadec (see [BP]) which may be stated as follows: every separable Fr echet space is homeomorphic to a Hilbert space. This result was generalized by Toru nczyk [To] for the nonseparable case. Since then the main interest in the eld has turned to topological classi cation of incomplete or nonlocally convex linear metric spaces. Quite recently, Cauty [Ca2] has constructed an example of a separable linear complete metric space X which is not an AR. In particular, X is not homeomorphic to any convex subset of a locally convex linear metric space. The topological classi cation of incomplete locally convex metric linear spaces, or even incomplete pre-Hilbert spaces, is much more complex than for the complete ones. First, there exist pre-Hilbert spaces of an arbitrary Borel class. Moreover, the Borel type invariant does not classify such spaces, and in the class of -compact spaces there exist uncountably many pairwise nonhomeomorphic such spaces [BP], [Ca1]. Until recently, there was hope that, in spite of these di culties, the classi cation problem could be restricted to subspaces of Hilbert spaces. The result of Bessaga and Dobrowolski [BD] stating that every locally convex linear -compact metric space is homeomorphic to a pre-Hilbert space looked like a promising rst step. The main objective of this paper is to show that, in general, it is not possible to restrict ones attention to subspaces of Hilbert spaces only. We show that the classes of pre-Hilbert,

In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N > 1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N = 2. We further show that this theorem does not hold for N > 2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical position expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity.

The various properties of topological spaces in the classical Baire category theorems which imply the Baire property also imply the stronger property of pseudo-completeness. In contrast to some of these properties and to the Baire property, J. C. Oxtoby has shown that pseudo-completeness is productive. The following main result places pseudo-completeness in the context of linear topological spaces: Let E and F be linear topological spaces and g be a continuous linear mapping of E into F. If E is pseudo-complete, g is almost open, and the completion of g[E] has a continuous metric, then g[E] is complete. The proof of this result uses the difference theorem, but not an open mapping theorem. The hypotheses lead to a discussion of conditions for a linear mapping to be almost open and for a linear topological space to have a continuous metric. An example shows that, although a translation invariant continuous metric on a linear topological space E extends to a translation invariant continuous pseudo-metric on the completion of E, this extension need not be a metric, even if it induces a normed topology on E. Othes examples show that a pseudo-complet e linear topological space need not be complete in its natural uniformity, and that the almost open condition of the main result may not be omitted and is not implied by the combination of the other conditions and the conclusion. Pseudo-complete topological spaces, introduced by Oxtoby [8], successfully bind together the classical Baire category theorems. They have been the subject of recent work; Aarts and Lutzer [1], for example, have shown that a metrizable space is pseudo-comple te if and only if it has a dense subspace which is topologically complete. In the context of linear topological spaces, Todd [13] shows that, under certain conditions, a linear topological space which is pseudocomplete is complete in the natural uniformity. Theorem 3.1 extends this result for the image of a pseudo-complete linear topological space under a continuous linear almost open mapping; in contrast. Example 5.5 gives pseudo-complete locally convex linear topological spaces which are not complete, and hence are not Ptak spaces. From open mapping theorems arise restrictions on continuous images of certain types of spaces. As an application of his open mapping theorem, Banach ([2], p. 38), showed that the continuous linear image of a complete metric linear space in a space of the same

Locally decodable codes are error correcting codes with the extra property that, in order to retrieve the correct value of just one position of the input with high probability, it is sufficient to read a small number of positions of the corresponding, possibly corrupted codeword. A breakthrough result by Yekhanin showed that 3-query linear locally decodable codes may have subexponential length. The construction of Yekhanin, and the three query constructions that followed, achieve correctness only up to a certain limit which is $1 - 3 delta$ for nonbinary codes, where an adversary is allowed to corrupt up to delta fraction of the codeword. The largest correctness for a subexponential length 3-query binary code is achieved in a construction by Woodruff, and it is below 1 - 3 delta. We show that achieving slightly larger correctness (as a function of $delta$) requires exponential codeword length for 3-query codes. Previously, there were no larger than quadratic lower bounds known for locally decodable codes with more than 2 queries, even in the case of 3-query linear codes. Our results hold for linear codes over arbitrary finite fields and for binary nonlinear codes. Considering larger number of queries, we obtain lower bounds for q-query codes for q>3, under certain assumptions on the decoding algorithm that have been commonly used in previous constructions. We also prove bounds on the largest correctness achievable by these decoding algorithms, regardless of the length of the code. Our results explain the limitations on correctness in previous constructions using such decoding algorithms. In addition, our results imply tradeoffs on the parameters of error correcting data structures.

Why individuals identify with Europe? In the absence of direct personal experiences with the EU and with people of other member states, how can the identification with the European project be developed? Despite the great number of EU competences, legislation and policy programmes, they are mostly implemented by member states. Are these EU policies amplifying citizens’ exposure to the idea of the EU? And finally, are citizens aware of the aims and perceive the benefits of the Cohesion Policy? If so, does it contribute to strengthen the shared political values under the European Social Model and, consequently, is it a driver of the identification with Europe? Giving answer to these questions is the main objective of this deliverable. This work tries to give some light to answer, at least partially, these questions by means of a key assumption. For most people in Europe, their experience with the EU takes place in the national political arena. In fact, one of the major EU policies is the Cohesion Policy that accounts for some 350 billion euros in the 2007-2013 programming period, about a third of total EU budget. Still, this policy is managed mostly locally. When individuals think on Europe they usually do it from a local point of view. The consequences of European integration depend not only on EU policies but also on how national, regional and local governments manage them. The Cohesion Policy is basically a regional development policy. Consequently, the territorial dimension is a key aspect that needs to be considered. In fact, most imbalances in economic and social (e.g. education) terms take place within every country and even within every region. In this respect, the perception of the Cohesion Policy and their impact on the development of a European identity could well vary between individuals in rural and urban areas. Therefore, we provide a specific look at the urban – rural divide of the way mechanisms and determinants impact on citizens’ European identification.

We consider a linear algebraic system A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters p, q varying within given intervals [p], [q]. The so-called parametric tolerable solution set is studied for unboundedness. Basing on a characterization of the parametric tolerable solution set as a convex polyhedron, we present necessary and sufficient conditions (in both general and computable forms) for a nonempty parametric tolerable solution set to be unbounded. Every parametric tolerable solution set is represented as a sum of a linear subspace and a bounded convex polyhedron. The latter implies better estimations (outer and inner) for the unbounded parametric tolerable solution set. Numerical examples illustrate the discussed methodology and the solution sets.

In the last decade, planning of closed-loop supply chains in different strategic, tactical, and operational levels has attracted many interests due to economic reasons, environmental challenges, and government legislations. This paper presents a novel linear programming model for the integrated production and distribution planning in closed-loop supply chains under uncertainty. The proposed model involves multi-product and multi-period which considers multiple transportation modes, direct or indirect shipments, advertising costs, and several customer zones for different types of products and also attempts to integrate production and distribution plans in the forward and reverse sides of the closed-loop supply chain, simultaneously. To deal with uncertain input data, a robust optimization counterpart based on polyhedral uncertainty set is developed to obtain optimal solutions immunizing the problem for any realization of uncertain parameters in the given polyhedral uncertainty set. Computation results for a numerical example under different scenarios are discussed to give insights about the features of the proposed robust optimization model in handling the uncertainty of parameters. Finally, some sensitivity analyses are performed to show the behaviour of the robust and deterministic models respect to changes of uncertainty levels of parameters as well as the amounts of important parameters such as demands and returns.

Objective To evaluate the treatment of acne vulgaris with acupuncture by different methods of point selection.Methods 51 cases of acne vulgaris were randomly recruited into local point selection group,distal point selection group and combination of local and distal point selection group.10 cases of healthy people were recruited for normal control group.Serum arachidonic acid (AA),free testosterone (FT),and sex hormone binding globulin (SHBG) levels were observed in all treatment groups.Results After the treatment,serum AA was (20.56±5.28)μg/L,(18.86±5.70) μg/L and (14.62±3.06) μg/L,while SHBG was (333.31±87.96) nmol/L,(332.02±87.82) nmol/L and (273.02±61.74) nmol/L respectively in local point selection group,distal point selection group and combination of local and distal point selection group,both showing no significant difference than the normal control group (P＞0.05).There was no significant difference in FT value between the combination of local and distal point selection group (5.62± 1.98) nmol/L and the normal control group (P＞0.05).The decrease of AA in the combination of local and distal point selection group was between than the local point selection group (P＜0.01).Conclusion Acupuncture can effectively improve the level of inflammatory factors and androgen of the acne vulgaris.Different methods of point selection of acupuncture on the improvement of some indicators were quite different,which laid the foundation for the best compatibility of acupuncture treatment in acne vulgaris. Key words: Acne vulgaris; Different selectionofpoints; Acupuncture; AA; FT; SHBG

If X is a discrete topological space, the points of its Stone-ˇCech compactification βX can be regarded as ultrafilters on X, and this fact is a useful tool in analysing the properties of βX. The purpose of this paper is to describe the compactification X of a metric space in terms of the concept of near ultrafilters. We describe the topological space ˜X and we investigate conditions under which ˜ S will be a semigroup compactification if S is a semigroup which has a metric. These conditions will always hold if the topology of S is defined by an invariant metric, and in this case our compactification ˜ S coincides with SLUC.

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-hard to find a cheapest biconnected spanning graph, where edge costs are the Euclidean distances of the respective points. We propose a different type of subgraph, based on forbidding (due to failure) the use of a vertex. A minimum spanning multi-tree is a spanning graph that contains for each possible forbidden vertex a spanning tree that is minimum among the spanning trees that do not use the forbidden vertex. We propose a worst-case time optimal algorithm for computing a minimum spanning multi-tree for a planar Euclidean point set. A minimum spanning multi-tree is cheap, even though it embeds a linear number of MSTs: Its cost is more than the MST cost only by a constant factor. Furthermore, we propose a linear time algorithm for computing a cheap vertex failure tolerant graph, given the Delaunay triangulation. This graph bounds the cost of the minimum spanning multi-tree from above.

Local oxygen transfer velocities measured in a linear wind-wave tunnel with respect to wind speed and fetch are presented in this thesis. For this, a non-intrusive laser-induced fluorescence (LIF) method was developed to measure vertical oxygen concentration profiles in the water-sided mass boundary layer. The fluorophore used is a water soluble ruthenium complex, which is quenched according to the Stern-Volmer equation. This equation, which originally describes the quenching only for a weak excitation, was generalized for arbitrary laser irradiance. Measurements confirm this generalization and yield a new value for the Stern-Volmer constant.
\n
\nThe LIF method was applied with high spatial and temporal resolution of 6.2 µm and 1.2 kHz, respectively, in order to resolve the mass boundary layer and fast processes. To obtain mean oxygen concentration profiles with high precision, an algorithm was developed to detect the water surface in the recorded images. The measured mean concentration profiles show a transition in the self-similar shape with the onset of waves. The results for a flat water surface are in agreement with the surface renewal model. For a wavy water surface, the small eddy model and the surface renewal model both describe the data equally well. Vanishing oxygen concentration fluctuations at the flat water surface were measured, which is in agreement with existing models for a rigid interface. The local transfer velocities obtained from mean concentration profiles are best parametrized with the friction velocity. In this work, the great potential of LIF measurements to probe transfer velocities locally is demonstrated.

In a dialogue procedure the decision maker has to determine in each step the aspiration and reservation level expressing his wishes (goals). This leads to an optimization problem which is not easy to solve in the nonconvex case (the known starting point is not feasible). We propose a modified standard embedding (one parametric optimization). This problem will be discussed from the point of view of parametric optimization (non-degenerate critical points, singularities, pathfollowing methods to describe numerically a connected component in the set of stationary points and in the set of generalized critical points, respectively, and jumps (descent methods) to other connected components in these sets). This embedding is much better for computing a goal realizer or replaying that the goal was not realistic than the embeddings considered in the literature before, but in the worst case we have to find all connected components and this is an open problem.