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Abstract: Given a closed subset $K$ in ${\mathbb R}$, the rational $K$-truncated moment problem ($K$-RTMP) asks to characterize the existence of a positive Borel measure $\mu$, supported on $K$, such that a linear functional ${\mathcal L}$, defined on all rational functions of the form ${f \over q}$, where $q$ is a fixed polynomial with all real zeros of even order and $f$ is any real polynomial of degree at most $2k$, is an integration with respect to ▫$\mu$▫. The case of a compact set ▫$K$▫ was solved by Chandler in 1994, but there is no argument that ensures that $\mu$ vanishes on all real zeros of $q$. An obvious necessary condition for the solvability of the $K$-RTMP is that ${\mathcal L}$ is nonnegative on every $f$ satisfying $f|_K \ge 0$. If ${\mathcal L}$ is strictly positive on every $0 \ne f|_K \ge 0$, we add the missing argument in Chadler's solution and also bound the number of atoms in a minimal representing measure. We show by an example that nonnegativity of ${\mathcal L}$ is not sufficient and add the missing conditions to the solution. We also solve the $K$-RTMP for unbounded $K$ and derive the solutions to the strong truncated Hamburger moment problem and the truncated moment problem on the unit circle as special cases.
Keywords: truncated moment problem, rational truncated moment problem, representing measure, moment matrix, localizing moment matrix, preordering
Published in DiRROS: 19.12.2024; Views: 36; Downloads: 13
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Published in DiRROS: 19.12.2024; Views: 30; Downloads: 8
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Abstract: Lack of knowledge on autumn phenology of deciduous trees still exists for high-latitude regions. We studied the leaf and wood growth autumn phenology of mountain birch in a sub-arctic climate (northern Sweden) and compared them with the same dynamics for silver birch in a temperate climate (southern Norway and Belgium). The first autumn phenophase for mountain birch was the decline of the remotely sensed Terrestrial Chlorophyll Index (TCI) at the end of July. This was followed by wood growth cessation, onset of chlorophyll degradation and of loss of canopy greenness, and the latter accompanied by onset of anthocyanin production and flavonoids degradation. The earlier timing of TCI decline than chlorophyll degradation was probably due to the different scales of measurements (ecosystem level vs. tree leaves, respectively). In 2020, the decline in canopy greenness started in the same period at the three studied sites, showing an unexpected early timing for Belgium, likely due to the very warm late summer conditions and drought stress or intraseasonal legacy effects. Accordingly, wood growth cessation also occurred unexpectedly earlier in Belgium than in Norway. The end of senescence was inversely related to latitude. Our study presents, for the first time, the autumn timeline of a deciduous species at the northern treeline, and indicates that the timing of autumn phenology of birch populations does not consistently follow the latitudinal gradient but varies according to the phenophase, the scale of measurements and the current year meteorological conditions.
Keywords: Betula pendula, Betula pubescens, deciduous trees, leaf senescence, terrestrial chlorophyll index, wood phenology
Published in DiRROS: 19.12.2024; Views: 35; Downloads: 15
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Published in DiRROS: 19.12.2024; Views: 40; Downloads: 9
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Keywords: gender equality, marine biotechnology, blue economy
Published in DiRROS: 19.12.2024; Views: 20; Downloads: 7
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