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1. Endpoint contributions to excited-state modular Hamiltonians

   https://doi.org/10.1007/JHEP12(2020)128  

   Kabat, Daniel ; Lifschytz, Gilad ; Nguyen, Phuc ; Sarkar, Debajyoti ( December 2020 , Journal of High Energy Physics)

   A bstract We compute modular Hamiltonians for excited states obtained by perturbing the vacuum with a unitary operator. We use operator methods and work to first order in the strength of the perturbation. For the most part we divide space in half and focus on perturbations generated by integrating a local operator J over a null plane. Local operators with weight n ≥ 2 under vacuum modular flow produce an additional endpoint contribution to the modular Hamiltonian. Intuitively this is because operators with weight n ≥ 2 can move degrees of freedom from a region to its complement. The endpoint contribution is an integral of J over a null plane. We show this in detail for stress tensor perturbations in two dimensions, where the result can be verified by a conformal transformation, and for scalar perturbations in a CFT. This lets us conjecture a general form for the endpoint contribution that applies to any field theory divided into half-spaces.
2. Criteria for Embedded Eigenvalues for Discrete Schrödinger Operators

   https://doi.org/10.1093/imrn/rnz262  

   Liu, Wencai ( November 2019 , International Mathematics Research Notices)

   In this paper, we consider discrete Schrödinger operators of the form, $$\begin{equation*} (Hu)(n) = u({n+1})+u({n-1})+V(n)u(n). \end{equation*}$$We view $H$ as a perturbation of the free operator $H_0$, where $(H_0u)(n)= u({n+1})+u({n-1})$. For $H_0$ (no perturbation), $\sigma _{\textrm{ess}}(H_0)=\sigma _{\textrm{ac}}(H)=[-2,2]$ and $H_0$ does not have eigenvalues embedded into $(-2,2)$. It is an interesting and important problem to identify the perturbation such that the operator $H_0+V$ has one eigenvalue (finitely many eigenvalues or countable eigenvalues) embedded into $(-2,2)$. We introduce the almost sign type potentials and develop the Prüfer transformation to address this problem, which leads to the following five results. 1: We obtain the sharp spectral transition for the existence of irrational type eigenvalues or rational type eigenvalues with even denominators.2: Suppose $\limsup _{n\to \infty } n|V(n)|=a<\infty .$ We obtain a lower/upper bound of $a$ such that $H_0+V$ has one rational type eigenvalue with odd denominator.3: We obtain the asymptotical behavior of embedded eigenvalues around the boundaries of $(-2,2)$.4: Given any finite set of points $\{ E_j\}_{j=1}^N$ in $(-2,2)$ with $0\notin \{ E_j\}_{j=1}^N+\{ E_j\}_{j=1}^N$, we construct the explicit potential $V(n)=\frac{O(1)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}_{j=1}^N$.5: Given any countable set of points $\{ E_j\}$ in $(-2,2)$ with $0\notin \{ E_j\}+\{ E_j\}$, andmore »any function $h(n)>0$ going to infinity arbitrarily slowly, we construct the explicit potential $|V(n)|\leq \frac{h(n)}{1+|n|}$ such that $H=H_0+V$ has eigenvalues $\{ E_j\}$.

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3. Florida mangrove saltmarsh reference surface soils

   https://doi.org/10.6073/pasta/0e08cbe07c84488cb7b9dd16669946d0  

   Lewis, David Bruce ( January 2021 )

   Abstract

   Site description. This data package consists of data obtained from sampling surface soil (the 0-7.6 cm depth profile) in black mangrove (Avicennia germinans) dominated forest and black needlerush (Juncus roemerianus) saltmarsh along the Gulf of Mexico coastline in peninsular west-central Florida, USA. This location has a subtropical climate with mean daily temperatures ranging from 15.4 °C in January to 27.8 °C in August, and annual precipitation of 1336 mm. Precipitation falls as rain primarily between June and September. Tides are semi-diurnal, with 0.57 m median amplitudes during the year preceding sampling (U.S. NOAA National Ocean Service, Clearwater Beach, Florida, station 8726724). Sea-level rise is 4.0 ± 0.6 mm per year (1973-2020 trend, mean ± 95 % confidence interval, NOAA NOS Clearwater Beach station). The A. germinans mangrove zone is either adjacent to water or fringed on the seaward side by a narrow band of red mangrove (Rhizophora mangle). A near-monoculture of J. roemerianus is often adjacent to and immediately landward of the A. germinans zone. The transition from the mangrove to the J. roemerianus zone is variable in our study area. An abrupt edge between closed-canopy mangrove and J. roemerianus monoculture may extend for up to several hundred meters in some locations, while other stretches of ecotone present a gradual transition where smaller, widely spaced trees are interspersed into the herbaceous marsh. Juncus roemerianus then extends landward to a high marsh patchwork of succulent halophytes (including Salicornia bigellovi, Sesuvium sp., and Batis maritima), scattered dwarf mangrove, and salt pans, followed in turn by upland vegetation that includes Pinus sp. and Serenoa repens. Field design and sample collection. We established three study sites spaced at approximately 5 km intervals along the western coastline of the central Florida peninsula. The sites consisted of the Salt Springs (28.3298°, -82.7274°), Energy Marine Center (28.2903°, -82.7278°), and Green Key (28.2530°, -82.7496°) sites on the Gulf of Mexico coastline in Pasco County, Florida, USA. At each site, we established three plot pairs, each consisting of one saltmarsh plot and one mangrove plot. Plots were 50 m^2 in size. Plots pairs within a site were separated by 230-1070 m, and the mangrove and saltmarsh plots composing a pair were 70-170 m apart. All plot pairs consisted of directly adjacent patches of mangrove forest and J. roemerianus saltmarsh, with the mangrove forests exhibiting a closed canopy and a tree architecture (height 4-6 m, crown width 1.5-3 m). Mangrove plots were located at approximately the midpoint between the seaward edge (water-mangrove interface) and landward edge (mangrove-marsh interface) of the mangrove zone. Saltmarsh plots were located 20-25 m away from any mangrove trees and into the J. roemerianus zone (i.e., landward from the mangrove-marsh interface). Plot pairs were coarsely similar in geomorphic setting, as all were located on the Gulf of Mexico coastline, rather than within major sheltering formations like Tampa Bay, and all plot pairs fit the tide-dominated domain of the Woodroffe classification (Woodroffe, 2002, &#34;Coasts: Form, Process and Evolution&#34;, Cambridge University Press), given their conspicuous semi-diurnal tides. There was nevertheless some geomorphic variation, as some plot pairs were directly open to the Gulf of Mexico while others sat behind keys and spits or along small tidal creeks. Our use of a plot-pair approach is intended to control for this geomorphic variation. Plot center elevations (cm above mean sea level, NAVD 88) were estimated by overlaying the plot locations determined with a global positioning system (Garmin GPS 60, Olathe, KS, USA) on a LiDAR-derived bare-earth digital elevation model (Dewberry, Inc., 2019). The digital elevation model had a vertical accuracy of ± 10 cm (95 % CI) and a horizontal accuracy of ± 116 cm (95 % CI). Soil samples were collected via coring at low tide in June 2011. From each plot, we collected a composite soil sample consisting of three discrete 5.1 cm diameter soil cores taken at equidistant points to 7.6 cm depth. Cores were taken by tapping a sleeve into the soil until its top was flush with the soil surface, sliding a hand under the core, and lifting it up. Cores were then capped and transferred on ice to our laboratory at the University of South Florida (Tampa, Florida, USA), where they were combined in plastic zipper bags, and homogenized by hand into plot-level composite samples on the day they were collected. A damp soil subsample was immediately taken from each composite sample to initiate 1 y incubations for determination of active C and N (see below). The remainder of each composite sample was then placed in a drying oven (60 °C) for 1 week with frequent mixing of the soil to prevent aggregation and liberate water. Organic wetland soils are sometimes dried at 70 °C, however high drying temperatures can volatilize non-water liquids and oxidize and decompose organic matter, so 50 °C is also a common drying temperature for organic soils (Gardner 1986, &#34;Methods of Soil Analysis: Part 1&#34;, Soil Science Society of America); we accordingly chose 60 °C as a compromise between sufficient water removal and avoidance of non-water mass loss. Bulk density was determined as soil dry mass per core volume (adding back the dry mass equivalent of the damp subsample removed prior to drying). Dried subsamples were obtained for determination of soil organic matter (SOM), mineral texture composition, and extractable and total carbon (C) and nitrogen (N) within the following week. Sample analyses. A dried subsample was apportioned from each composite sample to determine SOM as mass loss on ignition at 550 °C for 4 h. After organic matter was removed from soil via ignition, mineral particle size composition was determined using a combination of wet sieving and density separation in 49 mM (3 %) sodium hexametaphosphate ((NaPO_3)_6) following procedures in Kettler et al. (2001, Soil Science Society of America Journal 65, 849-852). The percentage of dry soil mass composed of silt and clay particles (hereafter, fines) was calculated as the mass lost from dispersed mineral soil after sieving (0.053 mm mesh sieve). Fines could have been slightly underestimated if any clay particles were burned off during the preceding ignition of soil. An additional subsample was taken from each composite sample to determine extractable N and organic C concentrations via 0.5 M potassium sulfate (K_2SO_4) extractions. We combined soil and extractant (ratio of 1 g dry soil:5 mL extractant) in plastic bottles, reciprocally shook the slurry for 1 h at 120 rpm, and then gravity filtered it through Fisher G6 (1.6 μm pore size) glass fiber filters, followed by colorimetric detection of nitrite (NO_2^-) &#43; nitrate (NO_3^-) and ammonium (NH_4^&#43;) in the filtrate (Hood Nowotny et al., 2010,Soil Science Society of America Journal 74, 1018-1027) using a microplate spectrophotometer (Biotek Epoch, Winooski, VT, USA). Filtrate was also analyzed for dissolved organic C (referred to hereafter as extractable organic C) and total dissolved N via combustion and oxidation followed by detection of the evolved CO_2 and N oxide gases on a Formacs HT TOC/TN analyzer (Skalar, Breda, The Netherlands). Extractable organic N was then computed as total dissolved N in filtrate minus extractable mineral N (itself the sum of extractable NH_4-N and NO_2-N &#43; NO_3-N). We determined soil total C and N from dried, milled subsamples subjected to elemental analysis (ECS 4010, Costech, Inc., Valencia, CA, USA) at the University of South Florida Stable Isotope Laboratory. Median concentration of inorganic C in unvegetated surface soil at our sites is 0.5 % of soil mass (Anderson, 2019, Univ. of South Florida M.S. thesis via methods in Wang et al., 2011, Environmental Monitoring and Assessment 174, 241-257). Inorganic C concentrations are likely even lower in our samples from under vegetation, where organic matter would dilute the contribution of inorganic C to soil mass. Nevertheless, the presence of a small inorganic C pool in our soils may be counted in the total C values we report. Extractable organic C is necessarily of organic C origin given the method (sparging with HCl) used in detection. Active C and N represent the fractions of organic C and N that are mineralizable by soil microorganisms under aerobic conditions in long-term soil incubations. To quantify active C and N, 60 g of field-moist soil were apportioned from each composite sample, placed in a filtration apparatus, and incubated in the dark at 25 °C and field capacity moisture for 365 d (as in Lewis et al., 2014, Ecosphere 5, art59). Moisture levels were maintained by frequently weighing incubated soil and wetting them up to target mass. Daily CO_2 flux was quantified on 29 occasions at 0.5-3 week intervals during the incubation period (with shorter intervals earlier in the incubation), and these per day flux rates were integrated over the 365 d period to compute an estimate of active C. Observations of per day flux were made by sealing samples overnight in airtight chambers fitted with septa and quantifying headspace CO_2 accumulation by injecting headspace samples (obtained through the septa via needle and syringe) into an infrared gas analyzer (PP Systems EGM 4, Amesbury, MA, USA). To estimate active N, each incubated sample was leached with a C and N free, 35 psu solution containing micronutrients (Nadelhoffer, 1990, Soil Science Society of America Journal 54, 411-415) on 19 occasions at increasing 1-6 week intervals during the 365 d incubation, and then extracted in 0.5 M K_2SO_4 at the end of the incubation in order to remove any residual mineral N. Active N was then quantified as the total mass of mineral N leached and extracted. Mineral N in leached and extracted solutions was detected as NH_4-N and NO_2-N &#43; NO_3-N via colorimetry as above. This incubation technique precludes new C and N inputs and persistently leaches mineral N, forcing microorganisms to meet demand by mineralizing existing pools, and thereby directly assays the potential activity of soil organic C and N pools present at the time of soil sampling. Because this analysis commences with disrupting soil physical structure, it is biased toward higher estimates of active fractions. Calculations. Non-mobile C and N fractions were computed as total C and N concentrations minus the extractable and active fractions of each element. This data package reports surface-soil constituents (moisture, fines, SOM, and C and N pools and fractions) in both gravimetric units (mass constituent / mass soil) and areal units (mass constituent / soil surface area integrated through 7.6 cm soil depth, the depth of sampling). Areal concentrations were computed as X × D × 7.6, where X is the gravimetric concentration of a soil constituent, D is soil bulk density (g dry soil / cm^3), and 7.6 is the sampling depth in cm.
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4. Weyl sums and the Lyapunov exponent for the skew-shift Schrödinger cocycle

   Han, Rui ; Lemm, Marius ; Schlag, Wilhelm ( October 2019 , Journal of spectral theory)

   We study the one-dimensional discrete Schr\"odinger operator with the skew-shift potential $2\lambda \cos\2π(j^2\omega+jy+x)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $\lambda>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(\lambda)$ at small $\lambda$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(\lambda)$ is fully consistent with $L(\lambda)$ being positive and satisfying the usual Figotin-Pastur type asymptotics $L(\lambda)\sim C\lambda^2$ as $\lambda\to 0$. The analogous quantity behaves completely differently in the Almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $\lambda<1$. The main technical work consists in establishing good lower bounds on the exponential sums (quadratic Weyl sums) that appear in our perturbation series.
5. Microfluidic Synthesis of Elastomeric Microparticles: A Case Study in Catalysis of Palladium-Mediated Cross-Coupling

   Bennett, Jeffrey A ; Genzer, Jan ; Abolhasani, Milad ( October 2018 , 2018 AIChE Annual Meeting)

   Metal-mediated cross-coupling reactions offer organic chemists a wide array of stereo- and chemically-selective reactions with broad applications in fine chemical and pharmaceutical synthesis.1 Current batch-based synthesis methods are beginning to be replaced with flow chemistry strategies to take advantage of the improved consistency and process control methods offered by continuous flow systems.2,3 Most cross-coupling chemistries still encounter several issues in flow using homogeneous catalysis, including expensive catalyst recovery and air sensitivity due to the chemical nature of the catalyst ligands.1 To mitigate some of these issues, a ligand-free heterogeneous catalysis reaction was developed using palladium (Pd) loaded into a polymeric network of a silicone elastomer, poly(hydromethylsiloxane) (PHMS), that is not air sensitive and can be used with mild reaction solvents (ethanol and water).4 In this work we present a novel method of producing soft catalytic microparticles using a multiphase flow-focusing microreactor and demonstrate their application for continuous Suzuki-Miyaura cross-coupling reactions. The catalytic microparticles are produced in a coaxial glass capillary-based 3D flow-focusing microreactor. The microreactor consists of two precursors, a cross-linking catalyst in toluene and a mixture of the PHMS polymer and a divinyl cross-linker. The dispersed phase containing the polymer, cross-linker, and cross-linking catalyst is continuously mixed and thenmore »formed into microdroplets by the continuous phase of water and surfactant (sodium dodecyl sulfate) introduced in a counter-flow configuration. Elastomeric microdroplets with a diameter ranging between 50 to 300 micron are produced at 25 to 250 Hz with a size polydispersity less than 3% in single stream production. The physicochemical properties of the elastomeric microparticles such as particle swelling/softness can be tuned using the ratio of cross-linker to polymer as well as the ratio of polymer mixture to solvent during the particle formation. Swelling in toluene can be tuned up to 400% of the initial particle volume by reducing the concentration of cross-linker in the mixture and increasing the ratio of polymer to solvent during production.5 After the particles are produced and collected, they are transferred into toluene containing palladium acetate, allowing the particles to incorporate the palladium into the polymer network and then reduce the palladium to Pd0 with the Si-H functionality present on the PHMS backbones. After the reduction, the Pd-loaded particles can be washed and dried for storage or switched into an ethanol/water solution for loading into a micro-packed bed reactor (µ-PBR) for continuous organic synthesis. The in-situ reduction of Pd within the PHMS microparticles was confirmed using energy dispersive X-ray spectroscopy (EDS), X-ray photoelectron spectroscopy (XPS) and focused ion beam-SEM, and TEM techniques. In the next step, we used the developed µ-PBR to conduct continuous organic synthesis of 4-phenyltoluene by Suzuki-Miyaura cross-coupling of 4-iodotoluene and phenylboronic acid using potassium carbonate as the base. Catalyst leaching was determined to only occur at sub ppm concentrations even at high solvent flow rates after 24 h of continuous run using inductively coupled plasma mass spectrometry (ICP-MS). The developed µ-PBR using the elastomeric microparticles is an important initial step towards the development of highly-efficient and green continuous manufacturing technologies in the pharma industry. In addition, the developed elastomeric microparticle synthesis technique can be utilized for the development of a library of other chemically cross-linkable polymer/cross-linker pairs for applications in organic synthesis, targeted drug delivery, cell encapsulation, or biomedical imaging. References 1. Ruiz-Castillo P, Buchwald SL. Applications of Palladium-Catalyzed C-N Cross-Coupling Reactions. Chem Rev. 2016;116(19):12564-12649. 2. Adamo A, Beingessner RL, Behnam M, et al. On-demand continuous-flow production of pharmaceuticals in a compact, reconfigurable system. Science. 2016;352(6281):61 LP-67. 3. Jensen KF. Flow Chemistry — Microreaction Technology Comes of Age. 2017;63(3). 4. Stibingerova I, Voltrova S, Kocova S, Lindale M, Srogl J. Modular Approach to Heterogenous Catalysis. Manipulation of Cross-Coupling Catalyst Activity. Org Lett. 2016;18(2):312-315. 5. Bennett JA, Kristof AJ, Vasudevan V, Genzer J, Srogl J, Abolhasani M. Microfluidic synthesis of elastomeric microparticles: A case study in catalysis of palladium-mediated cross-coupling. AIChE J. 2018;0(0):1-10.« less