The notion of augmenting graphs generalizes Berge’s idea of augmenting chains, which was used by Edmonds in his celebrated solution of the maximum matching problem. This problem is a special case of the more general maximum independent set (MIS) problem. Recently, the augmenting graph approach has been successfully applied to solve MIS in various other special cases. However, our knowledge of augmenting graphs is still very limited and we do not even know what the minimal infinite classes of augmenting graphs are. In the present paper, we find an answer to this question and apply it to extend the area of polynomial-time solvability of the maximum independent set problem.

The theorem of Birkhoff-von Neumann (see [2]) on the decomposition of bistochastic matrices (i.e., matrix with nonnegative entries and all row sums and column sums equal to one) has found various applications in scheduling; it is in particular a basic tool in the two-phase method of the preemptive scheduling problem on various machines with different capacities (see [4],[5],[6]). Let us now formulate a variation of the theorem. Given a real matrix A with entries aij unrestricted in sign, we denote by r(A, i)(resp.c(A, j)) the sum σ aij (resp._σi aij) of the entries in row i (resp. in column j). Furthermore let T(A) be defined by T(A) = max( max i{divides}r(A,i){divides} max j{divides}c(A,j){divides}). Matrix A is called regular if {divides}r(A, i) {divides} = {divides} c(A, j){divides} = T(A) for any row i and any column j. Notice that if the entries aij are unrestricted in sign, then A need not be a square matrix.

The scheduling of a sports tournament involves some fundamental issues, such as the type of league, the tournament characteristics, the timing of the scheduling process, and the goals and constraints of each specific problem.

Mathematical models based on theoretical graph concepts help to formulate and solve fundamental scheduling problems. In this chapter, we consider some elementary sports scheduling problems and concentrate on the combinatorial structures involved in their models and solution approaches.

Advancements in additive manufacturing (AM) have accelerated research on multi-materials. Among metal AM techniques, laser powder-bed fusion (L-PBF) stands out for its precision and suitability for lab-scale studies, making it the primary platform for multi-material AM research. However, fusion-based AM processes such as L-PBF face persistent challenges when combining dissimilar alloys, primarily due to the metallurgical incompatibility of alloy pairs. While systems with high solubility are generally easier to process, compound-forming pairs tend to crack due to the formation of brittle intermetallics. Similarly, immiscible systems are prone to poor metallurgical bonding and interfacial cracking. However, the mechanisms underlying the behavior of immiscible systems remain insufficiently addressed, as most insights are derived from ex situ and post-mortem characterization, often resulting in conflicting interpretations. To address this gap, enhancements were made to the miniature L-PBF machine (MiniSLM) originally developed for operando X-ray investigation of the L-PBF process at synchrotron beamlines. The modified setup and newly developed experimental procedures enabled real-time observation of multi-material L-PBF. Using this approach, two multi-material systems were systematically investigated, both previously reported to exhibit cracking during AM. First, cracking in nickel-copper multi-materials were studied using the IN625-CuCrZr pair. Although Ni and Cu are mutually soluble, alloying elements in IN625 induce immiscibility with Cu, forming two liquids with distinct freezing ranges and causing solidification cracking along Ni-rich grain boundaries. Moreover, cracking was found composition-dependent, peaking in the 20-40 wt.% CuCrZr-IN625 range. Additionally, lower heat inputs shifted the peak crack susceptibility toward higher CuCrZr compositions. Second, cracking mechanisms in L-PBF of steel-copper multi-materials were investigated using the 316L-CuCrZr system. Similar to the previous case, immiscibility in this system leads to the formation of two liquids with vastly different freezing ranges, and causes solidification cracking. Additionally, in this system, the Cu-rich liquid segregates between Fe-rich cells and dendrites, later promoting metal-induced embrittlement and liquation cracking. Experiments guided by these observations showed that mitigating phase separation significantly reduces cracking. Building on these findings, several strategies were developed to suppress cracking in the steel-copper system. These included gradual melting of CuCrZr on 316L, in situ alloying of CuCrZr with AlSi12, deposition of CuCrZr on boron steel, and employing laser beam shaping as a potential remedy. While process adjustments effectively reduced cracking, it was concluded that complete prevention requires tailoring feedstock chemistry and interface control through further process modifications. In summary, this thesis provides a phenomenological understanding of crack formation during fusion processing of immiscible multi-materials. It highlights the critical role of liquid immiscibility and phase separation in driving cracking and demonstrates that combining tailored processing strategies with compositional modifications is essential for mitigating these challenges. The findings establish a foundation for producing multi-material systems with improved structural integrity and processing reliability.

This thesis is divided into two main parts. The first two chapters focus on the enantioselective total synthesis of indole alkaloids, while the third chapter describes the development of a Wacker-type oxidation of trisubstituted alkenes involving a fluorinative ring expansion of exocyclic unsaturated carbonyl compounds into 2-fluoro-1,3-dicarbonyl derivatives under Pd(II)/Pd(IV) catalysis. Chapter 1 presents the journey towards the development of an enantioselective total synthesis of (+)-melonine, an unprecedented monoterpene indole alkaloid whose structure was revised in 2021, along with its N4-oxide derivative. After a general introduction on the indole alkaloid total synthesis, with a particular emphasis on strategies to access the 2,2,3-trisubstituted indoline moiety, three distinct synthetic approaches will be discussed. The first two failed strategies were respectively based on accessing the proposed biosynthetic iminium intermediate and on a challenging 7-exo-trig dearomative radical cyclization. The third approach, which successfully provided the natural product, features the following key steps: a) a Johnson-Claisen rearrangement, and a two-step key sequence involving b) a TFA-mediated Boc-deprotection/aziridination/indoline formation domino process, followed by c) a malonate desymmetrization through AlMe3-mediated bridge lactamization. The total synthesis of (+)-melonine was achieved in 15 linear steps, ultimately affording 165 mg of the natural product in a single batch, along with its first X-Ray crystallographic structure. Chapter 2 describes the development of the divergent and enantioselective total synthesis of three heteroyohimbines. Notably, the synthesis successfully exploited Franzéns stereodivergent organocatalyzed domino cascade, involving an enantioselective Michael addition with subsequent diastereoselective Pictet-Spengler reaction. Ley oxidation furnished a concise access to a key lactone intermediate in gram-scale, which is predominantly targeted in the heteroyohimbine total synthesis. This lactone serves as a versatile platform to access (-)-ajmalicine, (+)-mayumbine and ultimately (-)-roxburghine C. Although roxburghine isomers were isolated in the 1970s, only one previous synthetic effort had been reported towards these intriguing octacyclic structures. Remarkably, the total synthesis of (-)-roxburghine C was completed through a diastereoselective intramolecular Pictet-Spengler reaction of a methyl ketone and a chemoselective reduction of amidoester into enaminoester using Vaska's catalyst. This route ultimately provided nearly 200 mg of the bis-indole alkaloid with an overall yield of 12% over the 12-step longest linear sequence. Chapter 3 reports the first examples of a Wacker-type oxidation applied to trisubstituted alkenes. Following a review on Pd-catalyzed domino processes involving a type I dyotropic rearrangement of organo-Pd(IV) species, along with the evolution of the Wacker oxidation, the development of this methodology will be discussed. Exocyclic unsaturated carbonyl compounds were efficiently employed as trisubstituted olefins for this oxidative and fluorinative one-carbon ring expansion. A wide range of functional groups was tolerated and mechanistic studies revealed that a 1,2-alkyl/Pd(IV) dyotropic rearrangement operates, involving an unusual and thermodynamically disfavored Pd(IV)-migration from the alpha to the beta-position of the carbonyl group.

Single-photon avalanche diode (SPAD) detection technologies are increasingly important across various scientific domains, due to their ability to capture high-resolution temporal data. However, they face critical challenges in {\em in-vivo} applications, such as afterpulsing artifacts that degrade signal fidelity, a lack of synchronization between multiple cameras that limits temporal alignment and data coherence, and the generation of extremely high data volumes that strain readout and processing capabilities. This thesis addresses these limitations by developing high-speed, synchronized, and computation-efficient camera architectures. A high-speed interface enables full readout of a $512 \times 512$ binary frame imager at up to 88 kfps. %, with continuous streaming at 50 kfps. A dual-camera system achieves pixel-to-pixel timing alignment error better than 120 ps and streams continuously at 25 kfps. The same synchronization method is applied to a dual TCSPC camera system ($32 \times 32$ resolution), reaching 50 Mcps per camera; it was integrated into a 3D quantum microscope with a 220-ps (FWHM) coincidence peak. The TCSPC gating mechanism is characterized, showing reduced afterpulsing via a hard gating technique, enabling photon collection up to 3 ns earlier. Building on these high-data-rate systems, two advanced camera architectures are introduced: an interleaved-gate design for FLIM optimization that mitigates photobleaching effects and preserves fluorescence response shape, and a GRU-based design for real-time fixed-point neural network execution using SIMD topology, supporting parallel processing of eight pixels. In this implementation, scheduling is performed manually, while instruction generation is automated. Finally, a novel SPAD imager is developed for {\em in-vivo} applications, featuring $64 \times 108$ pixels, 50 ps TCSPC resolution, and the 16 pixels groups' comb organization, where each group shares two TDCs following a Winner-Take-All algorithm. The imager is more efficient than its predecessors, thanks to the capability of better managing photon timestamps over extended periods of time. This is of paramount importance in quantum imaging applications. The image thus achieves a count rate of 901 Mcps, or an average of 141 kcounts per pixel, one of the highest ever achieved in a SPAD camera. These contributions collectively advance the capabilities of single-photon imaging systems in terms of speed, synchronization, and intelligent data processing.

Accelerating the energy transition toward systems based on low-cost renewable feedstocks and diverse energy sources is imperative. Solid oxide fuel cell (SOFC) technology represent a promising solution, offering high conversion efficiency, fuel flexibility, and combined heat and power capabilities. While material development has reached maturity, the field still lacks systematic and comprehensive investigation across multiple scales: from fundamental electrochemical processes in single cells to system-level challenges including control optimization and degradation criteria in compact CHP units. This thesis systematically investigates the electrochemical performance of Ni-GDC electrolyte supported cells (ESCs) from single-cell to system levels, based on which the control optimization algorithm and the degradation criteria are developed. Chapter 1 introduces SOFC technology and its comparative advantages. Chapter 2 employs electrochemical impedance spectroscopy (EIS) with distribution of relaxation times, equivalent circuit model , and complex nonlinear least square fit to deconvolute single-cell processes, identifying dominant resistances in gas conversion (0.1-1 Hz), O$^{2-}$ surface exchange in GDC (0.1-10 Hz), coupled gas diffusion/surface exchange (1-50 Hz), and charge transfer (>50 Hz). Chapter 3 examines degradation mechanisms under biogas reformate conditions with and without sulfur poisoning. EIS analysis indicated that major degradation originated from ohmic resistance and electrode charge transfer resistances. Sulfur poisoning tests demonstrated H$_2$S causes irreversible degradation, while degradation caused by dimethyl sulfide were fully reversible. Active negatrode GDC surface reaction was identified as the main contributor of better sulfur resistance than the Ni-YSZ anode supported cell. At the system level (Chapter 4), a Ό-CHP SOFC system was characterized under operational extremes. EIS and total harmonic distortion analysis identified fuel starvation thresholds (safe utilization factor <81%) and diagnostic markers (0.01-0.1 Hz excitation). Carbon deposition tests and long-term operation confirmed system stability. The hierarchical analysis from single cell to compact system allowed for the control and optimization of the system with clear operating boundary. Chapter 5 presents a constraint-adaptation real-time optimization algorithm that reached set power target within 15 minutes and improved electrical efficiency by 5%. The algorithm showed satisfying robustness against natural gas grid fluctuations. Chapter 6 compares static control strategies (fixed power/voltage/temperature) based on different end-of-life (EoL) criteria, including power loss, voltage reduction, and cumulative energy degradation. Cumulative energy degradation was identified as a reliable EoL criteria under dynamic operation. This work provides fundamental insights into electrochemical process deconvolution for Ni-GDC ESCs and establishes diagnostic methodologies for fault conditions of stacks and systems. The integrated pathway---combining advanced characterization, real-time optimization algorithms, and novel EoL criteria---offers a universal framework to accelerate solid oxide cell technology deployment. The proposed approaches address critical challenges in performance optimization and lifetime prediction, advancing fuel cell and electrolysis technology toward commercial viability.

Thanks to their modular design, metal-organic frameworks (MOFs) hold great promise for versatile and impactful applications. However, challenges remain in achieving rapid, reproducible syntheses, obtaining fully activated materials with permanent porosity, and developing a deeper understanding of their structure-property relationships. This thesis explores the synthesis, activation, structural diversity, and multifunctionality of MOFs, highlighting how their fundamental properties can be leveraged to address environmental challenges. Here, we will focus on CO2 capture and the detection and removal of arsenic from water, reducing the risks to human health and ecosystems. In Chapter 1, we outline the motivation for this work and the associated challenges of MOFs and possible environmental applications. In Chapter 2, we present a joint experimental, computational, and high-throughput approach to improve on the traditional "trial and error" method of MOF synthesis optimization. In this project, we optimize the synthesis conditions of an aluminum (Al)-porphyrin-based MOF (i.e., Al-PMOF), promising for CO2 capture. Despite the challenges in synthesizing porphyrin-based MOFs, this approach proved to be efficient. It allowed us to translate the traditional 16-hour hydrothermal synthesis to a microwave-based method, yielding high crystallinity, effective CO2 capture, and a much shorter reaction time (50 minutes) with fewer than 50 iterations. In Chapter 3, we focus on preparing MOFs for functional use, with emphasis on their activation. We extend our study beyond permanent porosity and explore how activation influences the optical properties of a bismuth (Bi)-ellagate MOF (i.e., SU-101), with promising features for photocatalysis. We investigate several solvent exchange procedures, followed by thermal and vacuum activation, to expose the MOF's reactive under-coordinated metal sites. This improves its absorption in the visible range and alters the photoluminescence (PL) spectra, demonstrating that solvent removal plays a key role in shaping the material's optoelectronic properties. Subsequently, in Chapter 4, we perform a systematic study on pyrene-based-MOFs (i.e., 1,3,6,8-tetrakis(p-benzoic acid)pyrene (TBAPy)), where we synthesize structures with different metals (i.e., Al, Ga, Sc, and In). The orthorhombic system shows a parallel stacking of the ligand, creating favorable CO2 binding sites, but the metal center affects this stacking, and therefore the uptake. Crystallography shows secondary phases that influence adsorption; accounting for them improves prediction accuracy and guides material design. Finally, in Chapter 5, we investigate a porous coordination network (PCN)-222, a zirconium (Zr)-porphyrin MOF. We shaped the MOF's powder into beads using polyethersulfone (PES) and studied the composite for arsenate capture. We show the material's ability to reduce the arsenic concentration below the World Health Organization (WHO) limit of 10 ppb. Moreover, thanks to the sensitive optical properties of the ligand (i.e., meso-tetra(4-carboxyphenyl)porphine (TCPP)), the composite also provides a built-in visual indicator for filter replacement. We further implement the material in a simple device, built in-house, that can probe ligand emission changes depending on the arsenic concentration. These tools can be very valuable for remote populations in need of cheap, portable, and user-friendly testing and water purification methods.