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1.
2.
The growth-fragmentation equation describes a system of growing and dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its asymptotic profile at an exponential speed? These questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this work, we present a probabilistic approach: we use a Feynman–Kac formula to relate the solution of the growth-fragmentation equation to the semigroup of a Markov process, and characterise the rate of decay or growth in terms of this process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual. 相似文献
3.
4.
Pu Duan Bo Zhi Luke Coburn Christy L. Haynes Klaus Schmidt-Rohr 《Magnetic resonance in chemistry : MRC》2020,58(11):1130-1138
The composition of fluorescent polymer nanoparticles, commonly referred to as carbon dots, synthesized by microwave-assisted reaction of citric acid and ethylenediamine was investigated by 13C, 13C{1H}, 1H─13C, 13C{14N}, and 15N solid-state nuclear magnetic resonance (NMR) experiments. 13C NMR with spectral editing provided no evidence for significant condensed aromatic or diamondoid carbon phases. 15N NMR showed that the nanoparticle matrix has been polymerized by amide and some imide formation. Five small, resolved 13C NMR peaks, including an unusual ═CH signal at 84 ppm (1H chemical shift of 5.8 ppm) and ═CN2 at 155 ppm, and two distinctive 15N NMR resonances near 80 and 160 ppm proved the presence of 5-oxo-1,2,3,5-tetrahydroimidazo[1,2-a]pyridine-7-carboxylic acid (IPCA) or its derivatives. This molecular fluorophore with conjugated double bonds, formed by a double cyclization reaction of citric acid and ethylenediamine as first shown by Y. Song, B. Yang, and coworkers in 2015, accounts for the fluorescence of the carbon dots. Cross-peaks in a 1H─13C HETCOR spectrum with brief 1H spin diffusion proved that IPCA is finely dispersed in the polyamide matrix. From quantitative 13C and 15N NMR spectra, a high concentration (18 ± 2 wt%) of IPCA in the carbon dots was determined. A pronounced gradient in 13C chemical-shift perturbations and peak widths, with the broadest lines near the COO group of IPCA, indicated at least partial transformation of the carboxylic acid of IPCA by amide or ester formation. 相似文献
5.
Sameerah Jamal 《Journal of Differential Equations》2019,266(7):4018-4026
A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions. 相似文献
6.
In this paper, we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation with the critical situation that on g and p, where , is periodic and is bounded. 相似文献
7.
Cemil Ibis Sibel Sahinler Ayla Elvira Babayeva 《Phosphorus, sulfur, and silicon and the related elements》2020,195(6):474-480
AbstractIn the present study, the reactions of 2,3-dichloro-1,4-naphthoquinone (DCNQ) with amino-1,2-propanediol and some thiols were investigated. Novel N-, N,S-, and S,O- substituted derivatives were obtained and the structures of all compounds were characterized by spectroscopic methods (FT-IR, 1H NMR, 13C NMR, Mass spectroscopy) and microanalysis. The absorption behaviors of novel compounds were also investigated with UV-Vis spectroscopy in different solvents, such as ethanol, tetrahydrofuran and chloroform. 相似文献
8.
This paper deals with the Cauchy–Dirichlet problem for the fractional Cahn–Hilliard equation. The main results consist of global (in time) existence of weak solutions, characterization of parabolic smoothing effects (implying under proper condition eventual boundedness of trajectories), and convergence of each solution to a (single) equilibrium. In particular, to prove the convergence result, a variant of the so-called ?ojasiewicz–Simon inequality is provided for the fractional Dirichlet Laplacian and (possibly) non-analytic (but ) nonlinearities. 相似文献
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We investigate congruence classes and direct congruence classes of m-tuples in the complex projective space ℂP
n
. For direct congruence one allows only isometries which are induced by linear (instead of semilinear) mappings. We establish
a canonical bijection between the set of direct congruence classes of m-tuples of points in ℂP
n
and the set of equivalence classes of positive semidefinite Hermitean m×m-matrices of rank at most n+1 with 1's on the diagonal. As a corollary we get that the direct congruence class of an m-tuple is uniquely determined by the direct congruence classes of all of its triangles, provided that no pair of points of
the m-tuple has distance π/2. Examples show that the situation changes drastically if one replaces direct congruence classes by
congruence classes or if distances π/2 are allowed. Finally we do the same
kind of investigation also for the complex hyperbolic space ℂH
n
. Most of the results are completely analogous, however, there are also some interesting differences.
Received: 15 January 1996 相似文献