Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ p -improving estimate. For all 1 < p < 2, there is a constant C p so that for all integer N and functions f supported on [0, N ], there holds {N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known.
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This content will become publicly available on May 2, 2025
Moving beyond silver in point-of-use drinking water pathogen control
Managing drinking water-associated pathogens that can cause infections in immunocompromised individuals is a persistent challenge, particularly for healthcare facilities where occupant exposures carry a substantial health risk.
more » « less- Award ID(s):
- 1935378
- PAR ID:
- 10511710
- Publisher / Repository:
- Royal Society of Chemistry
- Date Published:
- Journal Name:
- Environmental Science: Water Research & Technology
- Volume:
- 10
- Issue:
- 5
- ISSN:
- 2053-1400
- Page Range / eLocation ID:
- 1009 to 1018
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Dual light-excited ketone/transition-metal catalysis is a rapidly developing field of photochemistry. It allows for versatile functionalizations of C–H or C–X bonds enabled by triplet ketone acting as a hydrogen-atom-abstracting agent, a single-electron acceptor, or a photosensitizer. This review summarizes recent developments of synthetically useful transformations promoted by the synergy between triplet ketone and transition-metal catalysis.
1 Introduction
2 Triplet Ketone Catalysis via Hydrogen Atom Transfer
2.1 Triplet Ketones with Nickel Catalysis
2.2 Triplet Ketones with Copper Catalysis
2.3 Triplet Ketones with Other Transition-Metal Catalysis
3 Triplet Ketone Catalysis via Single-Electron Transfer
4 Triplet Ketone Catalysis via Energy Transfer
5 Conclusions
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null (Ed.)Abstract We show that, under certain geometric conditions, there are no nonconstant quasiminimizers with finite p th power energy in a (not necessarily complete) metric measure space equipped with a globally doubling measure supporting a global $$p$$ p -Poincaré inequality. The geometric conditions are that either (a) the measure has a sufficiently strong volume growth at infinity, or (b) the metric space is annularly quasiconvex (or its discrete version, annularly chainable) around some point in the space. Moreover, on the weighted real line $$\mathbf {R}$$ R , we characterize all locally doubling measures, supporting a local $$p$$ p -Poincaré inequality, for which there exist nonconstant quasiminimizers of finite $$p$$ p -energy, and show that a quasiminimizer is of finite $$p$$ p -energy if and only if it is bounded. As $$p$$ p -harmonic functions are quasiminimizers they are covered by these results.more » « less
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Abstract Metabolomic studies allow a deeper understanding of the processes of a given ecological community than nucleic acid–based surveys alone. In the case of the gut microbiota, a metabolic profile of, for example, a fecal sample provides details about the function and interactions within the distal region of the gastrointestinal tract, and such a profile can be generated in a number of different ways. This unit elaborates on the use of 1D1H NMR spectroscopy as a commonly used method to characterize small‐molecule metabolites of the fecal metabonome (meta‐metabolome). We describe a set of protocols for the preparation of fecal water extraction, storage, scanning, measurement of pH, and spectral processing and analysis. We also compare the effects of various sample storage conditions for processed and unprocessed samples to provide a framework for comprehensive analysis of small molecules from stool‐derived samples. © 2020 Wiley Periodicals LLC
Basic Protocol 1 : Extracting fecal water from crude fecal samplesAlternate Protocol 1 : Extracting fecal water from small crude fecal samplesBasic Protocol 2 : Acquiring NMR spectra of metabolite samplesAlternate Protocol 2 : Acquiring NMR spectra of metabolite samples using Bruker spectrometer running TopSpin 3.xAlternate Protocol 3 : Acquiring NMR spectra of metabolite samples by semiautomated processBasic Protocol 3 : Measuring sample pHSupport Protocol 1 : Cleaning NMR tubesBasic Protocol 4 : Processing raw spectra dataBasic Protocol 5 : Profiling spectraSupport Protocol 2 : Spectral profiling of sugars and other complex metabolites -
Caratheodory’s theorem says that any point in the convex hull of a set $P$ in $R^d$ is in the convex hull of a subset $P'$ of $P$ such that $|P'| \le d + 1$. For some sets P, the upper bound d + 1 can be improved. The best upper bound for P is known as the Caratheodory number [2, 15, 17]. In this paper, we study a computational problem of finding the smallest set $P'$ for a given set $P$ and a point $p$. We call the size of this set $P'$, the Caratheodory number of a point p or CNP. We show that the problem of deciding the Caratheodory number of a point is NP-hard. Furthermore, we show that the problem is k-LDT-hard. We present two algorithms for computing a smallest set $P'$, if CNP= 2,3. Bárány [1] generalized Caratheodory’s theorem by using d+1 sets (colored sets) such that their convex hulls intersect. We introduce a Colorful Caratheodory number of a point or CCNP which can be smaller than d+1. Then we extend our results for CNP to CCNP.more » « less
