Recently, there has been a notable rise in the use of colored lighting for both indoor andoutdoor spaces. This trend necessitates a clear understanding of the principles behindselecting lighting levels that are both ergonomic and energy efficient. The objective ofthis study was to establish guidelines for planning colored light. An experiment wasconducted where the perceived brightness of three different near-monochromatic lightswere compared to white light. The stimuli covered a narrow visual field. 33 personsaged 18-40Y participated. Through the measurement of the participants' perception ofthe amount of colored light required -to achieve the same level of brightness as withwhite light- the study was able to determine a percentage-based relationship betweencolored and white light. The result showed that there were clear differences in theperceived brightness of the different colored lights, in line with earlier research withsimilar conditions. This implicates that the results may be used as a foundation whenplanning colored light.

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

This thesis is devoted to the study of operators on discrete structures. The operators are supposed to be self-adjoint and obey a certain translation invariance property. The discrete structures are given as Cayley graphs via finitely generated groups. Here, sofic groups and amenable groups are in the center of our considerations. Note that every finitely generated amenable group is sofic. We investigate the spectrum of a discrete self-adjoint operator by studying a sequence of finite dimensional analogues of these operators. In the setting of amenable groups we obtain these approximating operators by restricting the operator in question to finite subsets Qn , n ∈ N. These finite dimensional operators are self-adjoint and therefore admit a well-defined normalized eigenvalue counting function. The limit of the normalized eigenvalue counting functions when |Qn | → ∞ (if it exists) is called the integrated density of states (IDS). It is a distribution function of a probability measure encoding the distribution of the spectrum of the operator in question on the real axis. In this thesis, we prove the existence of the IDS in various geometric settings and for different types of operators. The models we consider include deterministic as well as random situations. Depending on the specific setting, we prove existence of the IDS as a weak limit of distribution functions or even as a uniform limit. Moreover, in certain situations we are able to express the IDS via a semi-explicit formula using the trace of the spectral projection of the original operator. This is sometimes referred to as the validity of the Pastur-Shubin trace formula. In the most general geometric setting we study, the operators are defined on Cayley graphs of sofic groups. Here we prove weak convergence of the eigenvalue counting functions and verify the validity of the Pastur-Shubin trace formula for random and non-random operators . These results apply to operators which not necessarily bounded or of finite hopping range. The methods are based on resolvent techniques. This theory is established without having an ergodic theorem for sofic groups at hand. Note that ergodic theory is the usual tool used in the proof of convergence results of this type. Specifying to operators on amenable groups we are able to prove stronger results. In the discrete case, we show that the IDS exists uniformly for a certain class of finite hopping range operators. This is obtained by using a Banach space-valued ergodic theorem. We show that this applies to eigenvalue counting functions, which implies their convergence with respect to the Banach space norm, in this case the supremum norm. Thus, the heart of this theory is the verification of the Banach space-valued ergodic theorem. Proceeding in two steps we first prove this result for so-called ST-amenable groups. Then, using results from the theory of ε-quasi tilings, we prove a version of the Banach space-valued ergodic theorem which is valid for all amenable groups. Focusing on random operators on amenable groups, we prove uniform existence of the IDS without the assumption that the operator needs to be of finite hopping range or bounded. Moreover, we verify the Pastur-Shubin trace formula. Here we present different techniques. First we show uniform convergence of the normalized eigenvalue counting functions adapting the technique of the Banach space-valued ergodic theorem from the deterministic setting. In a second approach we use weak convergence of the eigenvalue counting functions and additionally obtain control over the convergence at the jumps of the IDS. These ingredients are applied to verify uniform existence of the IDS. In both situations we employ results from the theory of large deviations, in order to deal with long-range interactions.

Aufkonversions-Nanokristalle (UCNC) zeichnen sich als einzigartige Lumineszenzreporter aus, die Nah-infrarotes Anregungslicht in Photonen höherer Energie umwandeln. Für die gezielte Anpassung von Eigenschaften, bedarf es ein tiefes Verständnis der Prozesse der Aufwärtskonversionslumineszenz (UCL) und deren Abhängigkeit von Material und Partikeldesign. Diese Doktorarbeit untersucht die UCL-Prozesse von Yb3+,Er3+ dotierten SrF2-UCNC und zielt darauf ab, die UCL-Eigenschaften der bisher unterschätzten kubischen Wirtsgitter zu verstehen und zu steigern. Hierbei wird die fluorolytische Sol-Gel-Synthese als neuartige Syntheseroute für UCNC vorgestellt. Vorteile wie ausgezeichnete Reproduzierbarkeit, viele Freiheitsgrade bei der Temperaturbehandlung und Partikelgestaltung werden anhand von SrF2 UCNC demonstriert. Die UCNC wurden mittels UCL-Spektren, UCL-Quantenausbeuten, leistungsdichte-abhängiger relativer spektraler Verteilung sowie der Lumineszenzabklingkinetiken unter Einbeziehung kristalliner Eigenschaften wie der Kristallphase, der Kristallitgröße, der Gitterparameter und der Teilchengröße untersucht. Die Abhängigkeit der UCL-Eigenschaften von der Dotierungsmenge wurde mit einer umfassenden Dotierungsreihe beschrieben und der optimale Dotierungsbereich (Yb3+,Er3+) von kleinen, ungeschalten SrF2-UCNC eingegrenzt. Bei der Studie dotierter Kerne mit passivierenden Schalen wurde der Einfluss von Temperaturbehandlung auf die UCL-Mechanismen und die Kern-Schale-Vermischung untersucht. Anhand von unterschiedlich kalzinierten UCNC Pulvern wurde die Empfindlichkeit der UCL gegenüber der Änderung kristalliner Eigenschaften, wie Kristallphase, Kristallinität, und Kristallitgröße betrachtet. Zusammen liefern die Dotierungs-, die Kern-Schale- und die Kalzinierungsstudie wertvolle Einblicke in das gitterspezifische Verhalten der UCL-Eigenschaften als Funktion der Energiemigration und der Kristalleigenschaften. / Upconversion nanocrystals (UCNC) represent a unique type of luminescence reporters that convert near-infrared excitation light into higher energy photons. Tailoring UCNC with specific luminescence properties requires an in-depth understanding of upconversion luminescence (UCL) processes and their dependence on material and particle design. This Ph.D. thesis focuses on the UCL processes of Yb3+,Er3+ doped SrF2-UCNC and aims to understand and enhance the UCL properties of the previously underestimated cubic host lattices. Herein, fluorolytic sol-gel synthesis is introduced as a novel synthetic route for UCNC. Advantages such as excellent reproducibility, high flexibility in temperature treatment and particle design are demonstrated using SrF2 UCNC. The UCNC were characterized by UCL spectra, UCL quantum yields, excitation power density-dependent relative spectral distribution, and luminescence decay kinetics involving crystalline properties such as crystal phase, crystallite size, lattice parameters, and particle size. The dependence of UCL properties on doping amount was described in a comprehensive doping study, and the optimal doping range (Yb3+,Er3+) of small, unshelled SrF2-UCNC was identified. In a core-shell study of doped core UCNC with passivating shells, the influence of temperature treatment on UCL mechanisms and core-shell mixing was investigated. Further, using different calcined UCNC powders, the sensitivity of UCL to the change of crystalline properties, such as crystal phase, crystallinity, and crystallite size, was assessed. Together, the doping, core-shell, and calcination studies provide valuable insight into the lattice-specific behavior of UCL properties as a function of energy migration and crystal properties.