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We study continuous paths in the Chabauty topology on the set $\mathcal{D}_n$ of torsion free discrete subgroups of the isometry group of $n$-dimensional hyperbolic space. We prove a combination theorem for paths in $\mathcal{D}_n$, which allows us to construct an exotic path of discrete subgroups along which no two subgroups are isomorphic. We also introduce a technique we refer to as "chromatography" to prove a decomposition theorem that characterizes paths of convex cocompact groups in $\mathcal{D}_3$.

Column chromatography is an important process in downstream biopharmaceutical manufacturing that enables high-selectivity separation of proteins through various modalities, such as affinity, ion exchange, hydrophobic interactions, or a combination of the aforementioned modes. Current mechanistic models of column chromatography typically abstract particle-level phenomena, in particular adsorption kinetics. A mechanistic model capable of incorporating particle-level phenomena would increase the value derived from mechanistic models. To this end, we model column chromatography via sphere packing, formulating three versions, each with increasing complexity. The first, homogeneous circle packing, is recast as maximum independent set and solved by the Quantum Approximate Optimization Algorithm on a quantum computer. The second, heterogeneous circle packing, is formulated as a graphical optimization problem and solved via classical simulations, accompanied by a road map to a quantum solution. An extension to the third, heterogeneous sphere packing, is formulated mathematically in a manner suitable to a quantum solution. Finally, detailed resource scaling is conducted to estimate the quant

We propose a theoretically designed laser resonance chromatography (LRC) experiment on Rf$^+$ (Z = 104) drifting in He buffer gas. To this end, we first developed a four-level rate equation model that simulates the optical pumping of Rf$^+$ from its ground state, $^2$D$_{3/2}$ (7s$^2$6d$^1$), to the metastable $^4$F$_{3/2}$ (7s$^1$6d$^2$) state via laser resonant excitation of the intermediate $^4$F$_{3/2}$ (7s$^1$6d$^1$7p$^1$) state prior to electronic state chromatography. This model predicts a 93% pumping efficiency that suffices to enable efficient laser resonance chromatography of this ion. We then performed accurate relativistic Multi-Reference Configuration-Interaction (MRCI) calculations to model the interaction of Rf$^+$ with He in the ground $^2$D$_{3/2}$ (7s$^2$6d$^1$), low-lying $^2$D$_{5/2}$ (7s$^2$6d$^1$), and metastable $^4$F$_{3/2}$ (7s$^1$6d$^2$) states. These ion-atom interaction potentials were used to calculate the state-specific ion mobilities. For gas temperatures above 100 K and small applied electric fields, the reduced ion mobilities of the ground and metastable states differ significantly. In particular, at room temperature the difference between the reduc

We study the rate of convergence to the steady state in the True Moving Bed model of linear chromatography, as a function of the six parameters that appear in the model. The model is a system of eight linear partial differential equations of hyperbolic type, coupled through the equations themselves and also through boundary conditions. We prove that the rate of convergence is given by a dominant eigenvalue, whose existence we prove by means of the Krein-Rutman Theorem, and by comparison arguments. We show how to construct a (not at all simple) characteristic function, whose roots are the eigenvalues. We also study the asymptotic profile of the solutions for large times, although this part is not purely analytical, but a combination of analytical and numerical techniques. Beyond the theoretical results, these models also offer explicit quantitative information: we apply all our results to a Case Study, namely the separation of omeprazole enantiomers. Finally, we consider a simpler limit case, where all the calculations become explicit.

We report first results obtained during the commissioning of the Laser Resonance Chromatography (LRC) apparatus, which is conceived to enable atomic structure investigations in the region of the heaviest elements beyond nobelium. In our studies we first established optimum conditions for the operation of the different components of the setup, including the radio-frequency quadrupole ion buncher and the cryogenic drift tube, which was operated with helium buffer gas at relatively low electric fields. We used laser ablated hafnium, lutetium, and ytterbium cations to assess the chromatography performance of the drift tube at a gas temperature of $295$K. Arrival time distributions of singly charged lutetium revealed two distinct ion mobilities of this ion in the ground and metastable state in helium with a relative difference of about $19$%. By using $^{219}$Rn ions from a $^{223}$Ra recoil source the overall efficiency of the apparatus is found to be $(0.6\pm0.1)$%. The findings help to establish LRC on lutetium, which is the lighter chemical homolog of lawrencium.

The Mars Spectrometry 2: Gas Chromatography challenge was sponsored by NASA and run on the DrivenData competition platform in 2022. This report describes the solution which achieved the second-best score on the competition's test dataset. The solution utilized two-dimensional, image-like representations of the competition's chromatography data samples. A number of different Convolutional Neural Network models were trained and ensembled for the final submission.

A radiochemical method for the isolation of plutonium isotopes from environmental samples, based on the use of specific chromatography resins for actinides (TEVA, Eichrom Industries), has been set up in our laboratory and optimised for their posterior determination by alpha spectrometry (AS) or accelerator mass spectrometry (AMS). The proposed radiochemical method has replaced in our lab a well established one based on the use of a relatively un-specific anion-exchange resin (AG1X8, Biorad), because it is clearly less time consuming, reduces the amounts and molarities of acid wastes produced, and reproducibly gives high radiochemical yields. In order to check the reliability of the proposed radiochemical method for the determination of plutonium isotopes in different environmental matrixes, twin aliquots of a set of samples were prepared with TEVA and with AG 1X8 resins and measured by AS. Some samples prepared with TEVA resins were measured as well by AMS. As it is shown in the text, there is a comfortable agreement between AS and AMS, which adequately validates the method.

Relativistic effects strongly influence the electronic structures of the heaviest elements, thereby shaping their chemical and physical properties. Studying ion mobility within a noble gas environment reveals how the ion-neutral interactions depend on the ion's electronic configurations, thus providing an avenue for exploring these effects. An ion mobility spectrometer with a cryogenic drift tube was developed to precisely measure the low-field reduced mobility of heavy lanthanide and actinide cations. The apparatus was characterized by optimizing the bunching operation of ions with a miniature RF coulomb buncher and evaluating the chromatography performance of the drift tube operated with helium buffer gas at a temperature of 298K. Systematic ion mobility measurements of lutetium cations (Lu$^{+}$) drifting in helium gas were carried out as a case study. The electronic state chromatography of Lu$^{+}$ has been demonstrated. The low-field reduced ion mobility for the ground and lowest meta-stable state of Lu$^{+}$ have been examined. In addition, the variation of both states' reduced mobility and the quenching of meta-stable population has been investigated under different reduced

Column chromatography is a laboratory and industrial technique used to separate different substances mixed in a solution. Mathematically, it can be modelled using non-linear partial differential equations whose main ingredients are the adsorption isotherms, which are non-linear functions modelling the affinity between the different substances in the solution and the solid stationary phase filling the column. The goal of this work is twofold. Firstly, we aim to extend the techniques of Donat, Guerrero and Mulet (Appl. Numer. Math. 123 (2018) 22-42) to other adsorption isotherms. In particular, we propose a family of generalized Langmuir-type isotherms and prove that the correspondence between the concentrations of solutes in the liquid phase (the primitive variables) and the conserved variables is well defined and admits a global smooth inverse that can be computed numerically. Secondly, to establish the well-posedness of the mathematical model, we study the eigenstructure of the Jacobian of the mentioned correspondence and use this characteristic information to get oscillation-free sharp interfaces on the numerical approximate solutions. To do so, we determine the structure of the

We applied Bayesian Optimal Experimental Design (OED) in the estimation of parameters involved in the Equilibrium Dispersive Model for chromatography with two components with the Langmuir adsorption isotherm. The coefficients estimated were Henry's coefficients, the total absorption capacity and the number of theoretical plates, while the design variables were the injection time and the initial concentration. The Bayesian OED algorithm is based on nested Monte Carlo estimation, which becomes computationally challenging due to the simulation time of the PDE involved in the dispersive model. This complication was relaxed by introducing a surrogate model based on Piecewise Sparse Linear Interpolation. Using the surrogate model instead the original reduces significantly the simulation time and it approximates the solution of the PDE with high degree of accuracy. The estimation of the parameters over strategical design points provided by OED reduces the uncertainty in the estimation of parameters. Additionally, the Bayesian OED methodology indicates no improvements when increasing the number of measurements in temporal nodes above a threshold value.

We propose a laser resonance chromatography (LRC) experiment on $^{229}$Th$^{3+}$, with the goal of detecting the ion's electronic ground $5f$ $^2$F$_{5/2}$ state and metastable $7s$ $^2$S$_{1/2}$ state by means of their ion mobilities. To this end, we first model the ion-neutral interaction potentials for the two electronic states with a relativistic Fock space coupled cluster method and complete basis-set extrapolation scheme. The interaction potentials are used to simulate the state-specific reduced ion mobilities in terms of the operating temperature and the external electric field. The ion mobilities differ by more than 7% at 300 K and moderate field strengths; thus, separation of the $^{229}$Th$^{3+}$ metastable state lies within the reach of LRC experiments targeting optical probing and monitoring of the nuclear clock transition in this isotope.

Diffusion models generate high-resolution images through iterative stochastic processes. In particular, the denoising method is one of the most popular approaches that predicts the noise in samples and denoises it at each time step. It has been commonly observed that the resolution of generated samples changes over time, starting off blurry and coarse, and becoming sharper and finer. In this paper, we introduce "resolution chromatography" that indicates the signal generation rate of each resolution, which is very helpful concept to mathematically explain this coarse-to-fine behavior in generation process, to understand the role of noise schedule, and to design time-dependent modulation. Using resolution chromatography, we determine which resolution level becomes dominant at a specific time step, and experimentally verify our theory with text-to-image diffusion models. We also propose some direct applications utilizing the concept: upscaling pre-trained models to higher resolutions and time-dependent prompt composing. Our theory not only enables a better understanding of numerous pre-existing techniques for manipulating image generation, but also suggests the potential for designing

Modulation is the key element of the comprehensive two-dimensional gas chromatography separation. Forward fill/flush flow modulation is cost effective, robust and suitable for analysis of a wide range of samples. Even though this modulation system is well known, studies regarding its optimization are sparse. In this work, based on hundreds of experiments involving multiple column sets and modulation conditions, an approach was proposed that permits to facilitate the choice of the forward fill/flush flow modulation parameters. A score function was developed that allows to predict the forward fill/flush flow modulation process efficiency as judged by the modulated peak shape. The score function was based on the physical rules for optimized and quantitative forward fill/flush flow modulation proposed in our previous work which state that the sum of the fill and flush modulation distance should be close to the modulation channel length and that the ratio of the flush and fill distance should be sufficiently high for efficient channel flushing. The score function was embedded in a freely available tool in the form of a forward fill/flush flow modulation calculator which allows the user

Stationary phase plays a crucial role in the operation of a protein chromatography column. Conventional resins composed of acrylic polymers and their derivatives contribute to heterogeneity of the packing of stationary phase inside these columns. Alternative polymer combinations through customized surface functionalization schemes which consist of multiple steps using static coating techniques are well known. In comparison, it is hypothesized that a single-step scheme is sufficient to obtain porous adsorbents as stationary phase for tuning surface morphology and protein immobilization. To overcome the challenge of heterogeneous packing and ease of fabrication at a laboratory scale, a change in the form factor of separation materials has been proposed in the form of functional copolymer surfaces. In the present work, an amphiphilic, block copolymer, poly(methyl methacrylate-co-methacrylic acid) has been chosen and fully characterized for its potential usage in protein chromatography. Hydrophilicity of the acrylic copolymer and abundance of carboxyl groups inherently on the copolymer surface have been successfully demonstrated through contact angle measurements, Fourier transform inf

Determining the adsorption isotherms is an issue of significant importance in preparative chromatography. A modern technique for estimating adsorption isotherms is to solve an inverse problem so that the simulated batch separation coincides with actual experimental results. However, due to the ill-posedness, the high non-linearity, and the uncertainty quantification of the corresponding physical model, the existing deterministic inversion methods are usually inefficient in real-world applications. To overcome these difficulties and study the uncertainties of the adsorption-isotherm parameters, in this work, based on the Bayesian sampling framework, we propose a statistical approach for estimating the adsorption isotherms in various chromatography systems. Two modified Markov chain Monte Carlo algorithms are developed for a numerical realization of our statistical approach. Numerical experiments with both synthetic and real data are conducted and described to show the efficiency of the proposed new method.

Polyynes are linear carbon chains characterized by alternated single and triple bonds and terminated by hydrogen or other terminal substituents. They can be synthesized by pulsed laser ablation in liquid (PLAL) as a scalable, cost-effective, and fast physical technique. Water can be employed as a solvent for PLAL to avoid toxicity problems and to reduce costs compared to organic solvents. However, in this case, the production yield of polyynes reached is extremely low and prevents further characterization and implementation in new functional materials. In this work, we synthesized polyynes by pulsed laser ablation in water and we optimized the process parameters to improve the yield of polyynes by PLAL. Then, we developed a procedure entirely based on reversed-phase high-performance liquid chromatography (RP-HPLC) which effectively enables the concentration, separation and collection of polyynes according to their length. Since the polyynes sample is an aqueous solution, we could inject it directly into the RP-HPLC column without the dilution step required in the case of a sample in an organic solvent. Thanks to our single RP-HPLC procedure, it is possible to highly increase the co

The possibility of carbohydrate separation in BEH HILIC (Ethylene Bridged Hybride, Hydrophilic Interaction Liquid Chromatography) column was studied by ultra-performance liquid chromatography (UPLC) with evaporative light scattering detector (ELSD) and mobile phase containing amine compounds as modifiers. The chromatography conditions and ELSD parameters were optimized to separate five typical carbohydrates and applied to analysis of four infant milk powders. The linear ranges of carbohydrate determination were 20-300mg/L for fructose and glucose, 20-250mg/L for sucrose and lactose, and 35-180mg/L for fructo-oligosaccharide. The LODs were 16.4mg/L for fructose and glucose, 17.3mg/L for sucrose, 20.0mg/L for lactose, and 46.7mg/L for fructo-oligosaccharide. Relative standard deviations (RSDs) ranged between 3.45-4.23%, 1.46-4.17%, 4.14-5.60%, 1.39-4.09%, and 2.49-3.61% for fructose, glucose, sucrose, lactose, and fructo-oilgosaccharide, respectively and recoveries ranged between 95.0 and 105.4%