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At least 991 records · Page 55

Dynamics of photosynthetic light harvesting systems interacting with N-photon Fock states

Here, we develop a method to simulate the excitonic dynamics of realistic photosynthetic light harvesting systems, including non-Markovian coupling to phonon degrees of freedom, under excitation by N-photon Fock state pulses. This method combines the input–output and the hierarchical equations of motion formalisms into a double hierarchy of density matrix equations. We show analytically that under weak field excitation relevant to natural photosynthesis conditions, an N-photon Fock state input and a corresponding coherent state input give rise to equal density matrices in the excited manifold. However, an N-photon Fock state input induces no off-diagonal coherence between the ground and excited subspaces, in contrast with the coherences created by a coherent state input. We derive expressions for the probability to absorb a single Fock state photon with or without the influence of phonons. For short pulses (or, equivalently, wide bandwidth pulses), we show that the absorption probability has a universal behavior that depends only upon a system-dependent effective energy spread parameter Δ and an exciton–light coupling constant Γ. This holds for a broad range of chromophore systems and for a variety of pulse shapes. We also analyze the absorption probability in the opposite long pulse (narrow bandwidth) regime. We then derive an expression for the long time emission rate in the presence of phonons and use it to study the difference between collective vs independent emission. Finally, we present a numerical simulation for the LHCII monomer (14-mer) system under single photon excitation that illustrates the use of the double hierarchy equations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Simulation and power quality analysis of a Loose-Coupled bipolar DC microgrid in an office building

With distributed generation and battery storage technologies thriving in microgrids, the use of direct current (DC) microgrids in the building sector offers multiple advantages in energy efficiency and power quality compared with alternating current (AC) systems. This study developed a new concept of a loose-coupled bipolar DC building power system. In this work, the concept was used to design a real-world office building in Shenzhen, China. A power system model was developed to study the stability and control of the DC power system and to verify DC power quality. The design and modeling of the DC power control system is discussed in detail. The study developed a few common fault scenarios in DC building microgrids that were simulated in the MATLAB-Simulink environment to validate the design of a loose-coupled bipolar DC system. The results indicate that the proposed loose-coupled bipolar DC system schema, when implemented with proper control algorithms, can achieve good fault-tolerant performance with reliable power quality, even during disruptive system events.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

The fully coupled regionally refined model of E3SM version 2: overview of the atmosphere, land, and river results

Abstract. This paper provides an overview of the United States (US) Department of Energy's (DOE's) Energy Exascale Earth System Model version 2 (E3SMv2) fully coupled regionally refined model (RRM) and documents the overall atmosphere, land, and river results from the Coupled Model Intercomparison Project 6 (CMIP6) DECK (Diagnosis, Evaluation, and Characterization of Klima) and historical simulations – a first-of-its-kind set of climate production simulations using RRM. The North American (NA) RRM (NARRM) is developed as the high-resolution configuration of E3SMv2 with the primary goal of more explicitly addressing DOE's mission needs regarding impacts to the US energy sector facing Earth system changes. The NARRM features finer horizontal resolution grids centered over NA, consisting of 25→100 km atmosphere and land, a 0.125∘ river-routing model, and 14→60 km ocean and sea ice. By design, the computational cost of NARRM is ∼3× of the uniform low-resolution (LR) model at 100 km but only ∼ 10 %–20 % of a globally uniform high-resolution model at 25 km. A novel hybrid time step strategy for the atmosphere is key for NARRM to achieve improved climate simulation fidelity within the high-resolution patch without sacrificing the overall global performance. The global climate, including climatology, time series, sensitivity, and feedback, is confirmed to be largely identical between NARRM and LR as quantified with typical climate metrics. Over the refined NA area, NARRM is generally superior to LR, including for precipitation and clouds over the contiguous US (CONUS), summertime marine stratocumulus clouds off the coast of California, liquid and ice phase clouds near the North Pole region, extratropical cyclones, and spatial variability in land hydrological processes. The improvements over land are related to the better-resolved topography in NARRM, whereas those over ocean are attributable to the improved air–sea interactions with finer grids for both atmosphere and ocean and sea ice. Some features appear insensitive to the resolution change analyzed here, for instance the diurnal propagation of organized mesoscale convective systems over CONUS and the warm-season land–atmosphere coupling at the southern Great Plains. In summary, our study presents a realistically efficient approach to leverage the fully coupled RRM framework for a standard Earth system model release and high-resolution climate production simulations.

54 ENVIRONMENTAL SCIENCES↗

Coupling magnons to an opto-electronic parametric oscillator

Hybrid magnonic systems have emerged as versatile modular components for quantum signal transduction and sensing applications owing to their capability of connecting distinct quantum platforms. To date, the majority of the magnonic systems have been explored in a local, near-field scheme, due to the close proximity required for realizing a strong coupling between magnons and other excitations. This constraint greatly limits the applicability of magnons in developing remotely coupled, distributed quantum network systems. On the contrary, opto-electronic architectures hosting self-sustained oscillations have been a unique platform for long-haul signal transmission and processing. Here, we integrated an opto-electronic oscillator with a magnonic oscillator consisting of a microwave waveguide and a Y 3 Fe 5 O 12 (YIG) sphere, and demonstrated strong and coherent coupling between YIG’s magnon modes and the opto-electronic oscillator’s characteristic photon modes—revealing the hallmark anti-crossing gap in the measured spectrum. In particular, the photon mode is produced on-demand via a nonlinear, parametric process as stipulated by an external seed pump. Both the internal cavity phase and the external pump phase can be precisely tuned to stabilize either degenerate or nondegenerate auto-oscillations. Our result lays out a new, hybrid platform for investigating the long-distance coupling and nonlinearity in coherent magnonic phenomena, which may be found useful in constructing the future “distributed hybrid magnonic systems.”

36 MATERIALS SCIENCE↗

Anthromes and forest carbon responses to global change

Human effects on ecosystems date back thousands of years, and anthropogenic biomes—anthromes—broadly incorporate the effects of human population density and land use on ecosystems. Forests are integral to the global carbon cycle, containing large biomass carbon stocks, yet their responses to land use and climate change are uncertain but critical to informing climate change mitigation strategies, ecosystem management, and Earth system modeling. Using an anthromes perspective and the site locations from the Global Forest Carbon (ForC) Database, we compare intensively used, cultured, and wildland forest lands in tropical and extratropical regions. We summarize recent past (1900-present) patterns of land use intensification, and we use a feedback analysis of Earth system models from the Coupled Model Intercomparison Project Phase 6 to estimate the sensitivity of forest carbon stocks to CO 2 and temperature change for different anthromes among regions. Modeled global forest carbon stock responses are positive for CO 2 increase but neutral to negative for temperature increase. Across anthromes (intensively used, cultured, and wildland forest areas), modeled forest carbon stock responses of temperate and boreal forests are less variable than those of tropical forests. Tropical wildland forest areas appear especially sensitive to CO 2 and temperature change, with the negative temperature response highlighting the potential vulnerability of the globally significant carbon stock in tropical forests. The net effect of anthropogenic activities—including land-use intensification and environmental change and their interactions with natural forest dynamics—will shape future forest carbon stock changes. These interactive effects will likely be strongest in tropical wildlands.

54 ENVIRONMENTAL SCIENCES↗

Synthesis and Photophysical Properties of Light-Harvesting Gold Nanoclusters Fully Functionalized with Antenna Chromophores

The development of efficient light-harvesting systems is important to understand the key aspects of solar-energy conversion processes and to utilize them in various photonic applications. In this study, atomically well-defined gold nanoclusters are reported as a new platform to fabricate artificial light-harvesting systems. An efficient amide coupling method is developed to synthesize water-soluble Au 22 clusters fully protected with pyrene chromophores by taking advantage of their facile phase-transfer reaction. The synthesized Au 22 clusters with densely packed 18 pyrene chromophores (Au 22 –PyB 18 ) exhibit triple-emission in blue, green, and red wavelength regions arising respectively from pyrene monomer, pyrene excimer, and Au 22 emission, producing bright white light emission together. The photoluminescence of Au 22 is enhanced by more than tenfold, demonstrating that pyrenes at the periphery efficiently channel the absorbed energy to the luminescent Au 22 at the center. A combination of femtosecond transient absorption and anisotropy measurements of Au 22 –PyB 18 explicitly reveals three main decay components of 220 fs, 3.5 ps, and 160 ps that can be assigned to energy migration between pyrenes and energy transfer processes from pyrene monomer and excimer to the central Au 22 , respectively.

14 SOLAR ENERGY↗

Integrating Solar Energy, Desalination, and Electrolysis

Solar electricity enables the advancement and deployment of technologies that are strongly influenced by clean energy availability and cost. The economics of both desalination and hydrogen production from water electrolysis are dominated by the cost of energy, and the availability of inexpensive solar energy creates markets and offers incentives to the desalination and electrolyzer industries. In this report the production of high-purity water and hydrogen from seawater is focused on. Current electrolyzers require deionized water so they need to be coupled with desalination units. It is shown that such coupling is cost effective in hydrogen generation, and it also offers benefits to thermal desalination, which can utilize waste heat from electrolysis. Furthermore, such coupling can be optimized when electrolyzers operate at high current density, using low-cost solar and/or wind electricity, as such operation increases both hydrogen production and heat generation. Results of technoeconomic modeling of polymer electrolyte membrane electrolyzers define thresholds of electricity pricing, current density, and operating temperature that make clean electrolytic hydrogen cost competitive with hydrogen from steam methane reforming (SMR). By using 2020 hourly electricity pricing in California and Texas, it is estimated that hydrogen can be produced from seawater in coupled desalination-electrolyzer systems at prices near $2 kg -1 H 2 , reaching cost parity with hydrogen produced from SMR.

14 SOLAR ENERGY↗

Effective description of sub-maximal chaos: stringy effects for SYK scrambling

It has been proposed that the exponential decay and subsequent power law saturation of out-of-time-order correlation functions can be universally described by collective ‘scramblon’ modes. We develop this idea from a path integral perspective in several examples, thereby establishing a general formalism. After reformulating previous work on the Schwarzian theory and identity conformal blocks in two-dimensional CFTs relevant for systems in the infinite coupling limit with maximal quantum Lyapunov exponent, we focus on theories with sub-maximal chaos: we study the large-q limit of the SYK quantum dot and chain, both of which are amenable to analytical treatment at finite coupling. In both cases we identify the relevant scramblon modes, derive their effective action, and find bilocal vertex functions, thus constructing an effective description of chaos. The final results can be matched in detail to stringy corrections to the gravitational eikonal S-matrix in holographic CFTs, including a stringy Regge trajectory, bulk to boundary propagators, and multi-string effects that are unexplored holographically.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Probing supersymmetric black holes with surface defects

It has long been conjectured that the large N deconfinement phase transition of $\mathcal{N}$ = 4 SU(N) super-Yang-Mills corresponds via AdS/CFT to the Hawking-Page transition in which black holes dominate the thermal ensemble, and quantitative evidence of this has come through the recent matching of the superconformal index of $\frac{1}{16}$-BPS states to the supersymmetric black hole entropy. Here we introduce the half-BPS Gukov-Witten surface defect as a probe of the superconformal index, which also serves as an order parameter for the deconfinement transition. This can be studied directly in field theory as a modification of the usual unitary matrix model or in the dual description as a D3-brane probe in the background of a (complex) supersymmetric black hole. Using a saddle point approximation, we determine our defect index in the large N limit as a simple function of the chemical potentials and show independently that it is reproduced by the renormalized action of the brane in the black hole background. Along the way, we also comment on the Cardy limit and the thermodynamics of the D3-brane in the generalized ensemble. The defect index sharply distinguishes between the confining and the deconfining phases of the gauge theory and thus is a supersymmetric non-perturbative order parameter for these large N phase transitions which deserves further investigation. Finally, our work provides an example where the properties of a black hole coupled to an external system can be analyzed precisely.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Linking classical PRA models to a dynamic PRA

Here, this paper presents a series of methods designed to incorporate classical Probabilistic Risk Assessment (PRA) models such as Event Trees (ETs) and Fault Trees (FTs) into dynamic PRA. In contrast to classical PRA, dynamic PRA couples stochastic methods with system simulators to determine the risks associated with complex systems such as nuclear power plants. Compared with classical PRA methods, they can evaluate with higher resolution the safety impact of timing and sequencing of events on the progression of the accident. As part of a dynamic PRA analysis, it is not uncommon that parts of the system to be analyzed might not require a computationally expensive simulation model. These parts could be in fact modeled by employing classical PRA models (e.g., a FT). Here, we present a set of methods and tools that can be used to link the most common classical PRA models (ETs, FTs, reliability block diagrams and Markov models) to simulation codes such as RELAP5-3D: creating a “hybrid PRA.” In order to show the potential of such an hybrid PRA we employ this method to verify ET modeling assumptions (e.g., success criteria) using a large break loss of coolant accident initiating event as a test case. In this respect, we link a set of FTs from the original PRA to the RELAP5-3D code and perform a hybrid PRA. The FTs are employed to model the control logic of several safety systems and to propagate component failures throughout the system. Provided the generated dynamic PRA data, we show how conservative assumptions in the original PRA can be identified and how such original PRA can be modified by updating success criteria captured by the set of RELAP5-3D simulation runs.

97 - MATHEMATICS AND COMPUTING↗

Exponential time differencing for the tracer equations appearing in primitive equation ocean models

The tracer equations are part of the primitive equations used in ocean modeling and describe the transport of tracers, such as temperature, salinity or chemicals, in the ocean. Depending on the number of tracers considered, several equations may be added to and coupled to the dynamics system. In many relevant situations, the time-step requirements of explicit methods imposed by the transport and mixing in the vertical direction are more restrictive than those for the horizontal, and this may cause the need to use very small time steps if a fully explicit method is employed. To overcome this issue, we propose an exponential time differencing (ETD) solver where the vertical terms (transport and diffusion) are treated with a matrix exponential, whereas the horizontal terms are dealt with in an explicit way. In this work, we investigate numerically the computational speed-ups that can be obtained over other semi-implicit methods, and we analyze the advantages of the method in the case of multiple tracers.

42 ENGINEERING↗

Regional patterns in hydrologic response, a new three-component metric for hydrograph analysis and implications for ecohydrology, Northwest Volcanic Aquifer Study Area, USA

Spatial patterns of hydrologic response were examined for the Northwest Volcanic Aquifer Study Area (NVASA). The utility of established hydrograph-separation methods for assessing hydrologic response in permeable volcanic terranes was assessed and a new three-component metric for hydrograph analysis was developed. The new metric, which partitions streamflow into subcomponents defined by the timescales of hydrologic response (e.g., fast-runoff, intermediate-interflow and slow-baseflow), was used to gain a fundamental understanding of the regional hydrology, investigate sub-regional differences, influencing factors, and ecohydrological implications. The combined effects of NVASA’s physiography, climate and geology create a strongly coupled surface-groundwater system that produces copious baseflow and limited quantities of runoff and interflow. Patterns of hydrologic response are influenced by the type and rate of precipitation and permeability of the underlying geology. Under variable precipitation conditions the hydrologic response of volcanic terranes with similar permeability and subsurface-storage capacity can be significantly different. From a water management and ecohydrology perspective, understanding regional patterns of hydrologic response and sub-regional differences is fundamental. Results indicate that minimum-flow methods provide the most conservative estimate of baseflow and may be the most robust for filtering out snowmelt bias in baseflow estimates. Baseflow contributes ~75% of the perennial streamflow across the NVASA and represents a critical component of the regional water supply that provides critical cold-water habitat.

54 ENVIRONMENTAL SCIENCES↗

Drought-induced changes in groundwater-surface water exchange at Lake Mead area

This study focuses on the Lake Mead region in the southwestern United States, a key water reservoir serving over 25 million people and agricultural lands across several states. The area has experienced recurring anthropogenic droughts since the early 2000s. We investigate the hydrological response of the coupled surface-groundwater system to the 2020–2022 drought, one of the most severe on record. To this end, we use Sentinel-1 Interferometric Synthetic Aperture Radar (InSAR) data to quantify vertical land motion caused by the elastic response of the crust to water-mass loss in the lake vicinity. Next, we apply an inverse elastic load modeling framework to quantify water loss. We further assess possible hydraulic connectivity between Lake Mead and adjacent groundwater reservoirs. We detect ground uplift of up to 8 mm/yr near the lake center, likely due to crustal rebound from reduced water-mass loading. We estimated the total water storage loss at 3.03 ± 0.25 km 3 /yr across a 3150 km 2 area surrounding Lake Mead. Groundwater accounts for approximately a third of that, being 0.94 ± 0.32 km 3 /yr. In addition, observed time lags of 6–98 days between lake and groundwater level responses, corresponding to a lateral diffusivity of 3.2–86 m 2 /s, suggest spatially variable connectivity between the lake and aquifers. These findings highlight that drought impacts propagate through the subsurface within interconnected systems, resulting in reduced buffering capacity of groundwater resources following droughts, and emphasizing the need for more integrated surface and groundwater management strategies to enhance resilience under climate and anthropogenic stressors.

58 GEOSCIENCES↗

Conservative DG method for the micro-macro decomposition of the Vlasov–Poisson–Lenard–Bernstein model

The micro-macro (mM) decomposition approach is considered for the numerical solution of the Vlasov– Poisson–Lenard–Bernstein (VPLB) system, which is relevant for plasma physics applications. In the mM approach, the kinetic distribution function is decomposed as $f$ = $\mathscr{E}$[$ρ_f$] + g , where $\mathscr{E}$ is a local equilibrium distribution, depending on the macroscopic moments $ρ_f$ = $∫_{\mathbb{R}}$ $efdv$ = $\langle$$ef$$\rangle$$_{\mathbb{R}}$, where e = (1, $v$, $\frac{1}{2}$ $v^2$) T , and $g$, the microscopic distribution, is defined such that $\langle$$eg$$\rangle$$_{\mathbb{R}}$ = 0. We aim to design numerical methods for the mM decomposition of the VPLB system, which consists of coupled equations for $ρ_f$ and $g$. To this end, we use the discontinuous Galerkin (DG) method for phase-space discretization, and implicit-explicit (IMEX) time integration, where the phase-space advection terms are integrated explicitly and the collision operator is integrated implicitly. We give special consideration to ensure that the resulting mM method maintains the $\langle$$eg$$\rangle$$_{\mathbb{R}}$ = 0 constraint, which may be necessary for obtaining (i) satisfactory results in the collision dominated regime with coarse velocity resolution, and (ii) unambiguous conservation properties. The constraint-preserving property is achieved through a consistent discretization of the equations governing the micro and macro components. Here, we present numerical results that demonstrate the performance of the mM method. The mM method is also compared against a corresponding DG-IMEX method solving directly for $f$.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Modular Electrochemical Synthesis Using a Redox Reservoir Paired with Independent Half-Reactions

Electrochemical synthesis of high-value chemicals using renewable electricity could lead to sustainability advantages over conventional chemical manufacturing. The tight coupling between paired anodic and cathodic reactions in conventional electrochemical processes, however, constrains the products and their efficiencies. Here, we demonstrate a strategy to achieve modular electrochemical synthesis (ModES) of different chemicals by decoupling and pairing the oxidative and reductive half-reactions with a heterogeneous redox reservoir (RR). Specifically, sodium nickel hexacyanoferrate (NaNiHCF) can serve as a robust and reversible RR that enables the modular production of several strong oxidants, hydrogen peroxide (H 2 O2) and sodium persulfate (Na 2 S 2 O 8 ) or active chlorine, in separate RR-coupled electrochemical processes. The system’s excellent performance and operational stability is demonstrated by continuous operation over 100 cycles (~36 h) to produce H 2 O 2 and Na 2 S 2 O 8 with a high voltage efficiency. The concept presented here highlights the prospects for using RRs to enable efficient, modular, and on-demand electrochemical synthesis of valuable chemicals.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Large electron-phonon drag asymmetry and reverse heat flow in the topological semimetal θ-TaN

A broad range of unusual transport behaviors have been discovered in topological semimetals. However, to date, the effect on the thermopower from intrinsic momentum exchange between electrons and phonons has received little attention. Here we report that huge electron-phonon drag enhancements of the thermopower of the to- pological semimetal, θ-phase tantalum nitride (θ-TaN), can occur that persist even up to room temperature. Our first principles calculations also identify a surprising asymmetry in which the large drag-enhanced thermopowers found slightly above the material’s chemical potential disappear just below it. The large thermopower en- hancements result from anomalous drag contributions from high frequency acoustic phonons with unusually small decay rates. The apparent vanishing drag results from (i) the emergence of an exceptionally high electrical conductivity promoted by the steep linear electronic dispersions extending below one of the topological nodal points; (ii) a remarkable cancellation in which momentum transferred from a charge current creates oppositely directed phonon heat currents of nearly equal magnitude, thereby masking the drag contributions. This extraordinary transport behavior is a consequence of an unusual interplay between intrinsic electron and phonon material properties in θ-TaN. Overall, our work gives new insights into the fundamental physical properties of coupled electron-phonon systems and motivates further exploration of drag effects in semimetals.

36 MATERIALS SCIENCE↗

Electromagnetic production of kaons on the nucleon

Studies of the electromagnetic production of strange quarks started in the 1950s as something of a curiosity that puzzled experimentalists and theorists alike. Eventually, a nascent understanding of these processes began to take shape through the first pioneering experiments dedicated to explore photo- and electroproduction that were carried out in the period from the 1950s to the 1980s. As the datasets increased, concomitant advances in theoretical models were realized. However, these initial studies also made clear that more precise data was essential to continue to move forward. A paradigm shift occurred in the 1990s with the development of second-generation facilities at ELSA, MAMI, SPring-8, and JLab. High-intensity, high duty-factor accelerators, coupled with novel detector systems and advances in computing and readout electronics, brought nuclear physics experiments forward by orders of magnitude in counting statistics compared to the first-generation efforts. This was an utter boon to strangeness physics investigations, and to date, more than 50 dedicated experiments in kaon photo- and electroproduction have been completed at facilities around the world, leading to a host of experimental observables that have enabled significant advances in the exploration of strongly interacting systems that decay via $s\bar{s}$ quark pair creation. These data have proven to be an essential complementary pathway to study the spectrum and structure of the excited states of the nucleon, and the search for missing and exotic baryon configurations. As well, investigations in these channels are requisite for exploring hypernuclear production as a probe of the $YN$ interaction and for studies of the electromagnetic form factors of strange mesons. This review was designed to provide the first-ever in-depth overview of both the experimental and theoretical progress in the field of the electromagnetic production of strangeness. This work looks back over 70 years of past developments, discusses ongoing work and near-term plans, and details future possibilities being considered for third-generation facilities. Extensive lists of the available datasets and theoretical models are provided, together with a comprehensive supporting bibliography of the field. Throughout this work, the primary impacts of these explorations are highlighted, along with connections to a wide range of related phenomenological applications. An important goal of this review is to provide a complete, (reasonably) self-contained guide into this field prepared at a level that is relevant for both new and seasoned scientists, whether experimentalists, phenomenologists, or theorists, to better understand what has been accomplished by so many dedicated folks-each building on what has come before-and to appreciate the exciting future potential for continued studies in this area.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

Development of a Universal Second-Tier Newborn Screening LC–MS/MS Method for Amino Acids, Lysophosphatidylcholines, and Organic Acids

First-tier MS-based newborn screening by flow injection analysis can have high presumptive positive rates, often due to isomeric/isobaric compounds or poor biomarker specificity. These presumptive positive samples can be analyzed by second-tier screening assays employing separations such as liquid chromatography–mass spectrometry (LC–MS/MS), which increases test specificity and drastically reduces false positive referrals. The ability to screen for multiple disorders in a single multiplexed test simplifies workflows and maximizes public health laboratories’ resources. In this study, we developed and validated a highly multiplexed second-tier method for dried blood spots using a hydrophilic interaction liquid chromatography (HILIC) column coupled to an MS/MS system. The LC–MS/MS method was capable of simultaneously detecting second-tier biomarkers for maple syrup urine disease, homocystinuria, methylmalonic acidemia, propionic acidemia, glutaric acidemia type 1, glutaric acidemia type 2, guanidinoacetate methyltransferase deficiency, short-chain acyl-CoA dehydrogenase deficiency, adrenoleukodystrophy, and Pompe disease.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗