Complemented subspaces of spaces obtained by interpolation

The final version of this paper appears in: "Journal of the London Mathematical Society" 44(2) (1991): 503-513. Print.If Z is a quotient of a subspace of a separable Banach space X, and V is any separable Banach space, then there is a Banach couple (A0,A1) such that A0 and A1 are isometric to X⊕V, and any intermediate space obtained using the real or complex interpolation method contains a complemented subspace isomorphic to Z. Thus many properties of Banach spaces, including having non-trivial cotype, having the Radon-Nikodym property, and having the analytic unconditional martingale difference sequence property, do not pass to intermediate spaces

Homodyned and Heterodyned Vibrational Sum Frequency Generation Spectroscopy of Electrochemical Interfaces

Since its development in the 1980s the second-order nonlinear optical technique of vibrational sum frequency generation spectroscopy (SFG) evolved into a versatile tool to detect and characterize molecules at interfaces. To obtain complete information on the molecules’ absolute orientation and to disentangle the interference of a possible non-resonant substrate’s response which distorts the spectral line shape, however, phase-resolved measurements need to be applied. In such heterodyned SFG experiments the signal pulse which is generated by the nonlinear interaction between two incoming short pulses and a sample subsequently interferes with a so called local oscillator (LO) pulse. To yield accurate results, relative phase stability between the signal and LO as well as a precise control of their individual timings must be warranted. The simultaneous fulfillment of this two requirements so far have restricted heterodyned SFG experiments to the study of solid/air and liquid/air interfaces. This thesis presents a way to overcome these limitations by integrating a timing control scheme into a collinear high accuracy phase-resolved SFG spectrometer. The versatility of this approach is tested at the solid/liquid interface and extended to potential dependent measurements since among others understanding of electrochemical processes at the electrode/electrolyte interface will be critical in the development of more efficient batteries and fuel cells to tackle the challenges presented by the climate crisis. The obtained heterodyned SFG spectra allow an insight into the parameters that influence the non-resonant substrate’s response, such as an applied potential bias and the presence of specifically and non-specifically adsorbed ions and molecules. In addition a method is presented how to use phase-resolved SFG spectra to determine the phase of the local field Fresnel factors, which so far had to be modeled. The heterodyned spectra are compared to their homodyned analogs to discuss which new information in fact can be obtained and what limits still need to be overcome.Eine verbreitete Methode zur Charakterisierung von molekularen Prozessen an Grenzflächen ist die Summenfrequenz(SFG)-Schwingungsspektroskopie, die auf einem nichtlinearen optischen Effekt zweiter Ordnung beruht. Die ursprüngliche Methode detektiert die Intensität des SFG Signals. Dabei gehen jedoch Informationen über die absolute Orientierung der Moleküle an der Grenzfläche verloren. Außerdem kann ein nichtresonanter Beitrag des unterliegenden Substrats durch Interferenz die Linienform des Spektrums beeinflussen, was die Interpretation erschwert. Daher wurde unter Zuhilfenahme eines lokalen Oszillator (LO) Pulses die phasenaufgelöste SFG-Spektroskopie entwickelt. Zur Gewährleistung einer akkuraten Phasenauflösung muss eine konstante Phasenbeziehung zwischen Signal und LO vorherrschen. Außerdem verlangt die Methode eine präzise Kontrolle der individuellen Pulszeiten. Die gleichzeitige Einhaltung beider Voraussetzungen ist experimentell aufwendig, was bisher einen Einsatz über die Grenzflächen fest/gasförimg und flüssig/gasförmig hinaus verhinderte. Die vorliegende Arbeit erweitert den Einsatzbereich auch auf die fest/flüssig Grenzfläche und erlaubt damit auch phasenaufgelöste in situ Messungen elektrochemischer Prozesse an der Elektrodenoberfläche. Das Verständnis dieser Prozesse ist ein wichtiger Schritt zur Entwicklung effektiverer Batterien und Elektroden für Brennstoffzellen, wichtige Bestandteile zur Eindämmung des Klimawandels. Zur Realisierung solcher Messungen wird ein Schema zur Pulszeitkontrolle vorgestellt und in ein kollineares hoch präzises phasenaufgelöstes SFG-Spektrometer eingeführt. Die entstandenen phasenaufgelösten SFG-Spektren erlauben einen Einblick in die Abhängigkeit des nichtresonanten SFG Beitrags des Substrat von Faktoren wie einer angelegten elektrischen Spannung und der Anwesenheit gebundener/ungebundener Moleküle /Ionen an der Substratgrenzfläche. Desweiteren wird eine Methode vorgestellt, die es erlaubt die Phase der lokalen Feldfaktoren zu bestimmen, die bisher nur theoretisch berechnet werden konnte. Die phasenaufgelösten SFG-Spektren werden mit herkömmlichen SFG-Spektren verglichen um den zusätzlichen Informationsgewinn herauszuarbeiten, aber auch um noch offene Fragen zu erkennen

Approximating Nash Equilibria and Dense Bipartite Subgraphs via an Approximate Version of Carathéodory's Theorem

We present algorithmic applications of an approximate version of Caratheodory's theorem. The theorem states that given a set of vectors X in R^d, for every vector in the convex hull of X there exists an ε-close (under the p-norm distance, for 2 ≤ p < ∞) vector that can be expressed as a convex combination of at most b vectors of X, where the bound b depends on ε and the norm p and is independent of the dimension d. This theorem can be derived by instantiating Maurey's lemma, early references to which can be found in the work of Pisier (1981) and Carl (1985). However, in this paper we present a self-contained proof of this result. Using this theorem we establish that in a bimatrix game with n x n payoff matrices A, B, if the number of non-zero entries in any column of A+B is at most s then an ε-Nash equilibrium of the game can be computed in time n^O(log s/ε^2}). This, in particular, gives us a polynomial-time approximation scheme for Nash equilibrium in games with fixed column sparsity s. Moreover, for arbitrary bimatrix games---since s can be at most n---the running time of our algorithm matches the best-known upper bound, which was obtained by Lipton, Markakis, and Mehta (2003). The approximate Carathéodory's theorem also leads to an additive approximation algorithm for the densest k-bipartite subgraph problem. Given a graph with n vertices and maximum degree d, the developed algorithm determines a k x k bipartite subgraph with density within ε (in the additive sense) of the optimal density in time n^O(log d/ε^2)

Zermelo Navigation and a Speed Limit to Quantum Information Processing

We use a specific geometric method to determine speed limits to the implementation of quantum gates in controlled quantum systems that have a specific class of constrained control functions. We achieve this by applying a recent theorem of Shen, which provides a connection between time optimal navigation on Riemannian manifolds and the geodesics of a certain Finsler metric of Randers type. We use the lengths of these geodesics to derive the optimal implementation times (under the assumption of constant control fields) for an arbitrary quantum operation (on a finite dimensional Hilbert space), and explicitly calculate the result for the case of a controlled single spin system in a magnetic field, and a swap gate in a Heisenberg spin chain

Approaching religion through linguistics: methodological thoughts on a linguistic analysis of 'religion' in political communication

The constructions of ‘religion’ in general language are seldom themselves in the focus of empirical research. Aiming to retrieve the inherent knowledge that lies within these constructions, this article suggests a term-based textual analysis to focus on the linguistic use of ‘religion’. This method invites us to question the unity of texts through an analysis of textual semantics. It offers the chance to ask about the formation of the concept. The article initially shows how this approach differs from comparative and policy-oriented studies by differentiating between criticism and critique. It then develops the idea of a term-based textual analysis. Using examples from the policy field of foreign aid, the text illustrates how much inherent knowledge there is in the usage of ‘religion’ in political communication and calls for a general reconsideration of the way ‘religion’ is approached in empirical research

Correntropy: Implications of nonGaussianity for the moment expansion and deconvolution

Accepted versio