Probability Thermodynamics and Probability Quantum Field

In this paper, we introduce probability thermodynamics and probability quantum fields. By probability we mean that there is an unknown operator, physical or nonphysical, whose eigenvalues obey a certain statistical distribution. Eigenvalue spectra define spectral functions. Various thermodynamic quantities in thermodynamics and effective actions in quantum field theory are all spectral functions. In the scheme, eigenvalues obey a probability distribution, so a probability distribution determines a family of spectral functions in thermodynamics and in quantum field theory. This leads to probability thermodynamics and probability quantum fields determined by a probability distribution. There are two types of spectra: lower bounded spectra, corresponding to the probability distribution with nonnegative random variables, and the lower unbounded spectra, corresponding to probability distributions with negative random variables. For lower unbounded spectra, we use the generalized definition of spectral functions. In some cases, we encounter divergences. We remove the divergence by a renormalization procedure. Moreover, in virtue of spectral theory in physics, we generalize some concepts in probability theory. For example, the moment generating function in probability theory does not always exist. We redefine the moment generating function as the generalized heat kernel, which makes the concept definable when the definition in probability theory fails. As examples, we construct examples corresponding to some probability distributions. Thermodynamic quantities, vacuum amplitudes, one-loop effective actions, and vacuum energies for various probability distributions are presented

The covering number for some Mercer kernel Hilbert spaces

AbstractIn the present paper, we investigate the estimates for the covering number of a ball in a Mercer kernel Hilbert space on [0,1]. Let Pl(x) be the Legendre orthogonal polynomial of order l, al>0 be real numbers satisfying ∑l=0+∞lal<+∞. Then, for the Mercer kernel functionK(x,t)=∑l=0+∞alPl(x)Pl(t),x,t∈[0,1],we provide the upper estimates of the covering number for the Mercer kernel Hilbert space reproducing from K(x,t). For some particular al we give the lower estimates. Meanwhile, a kind of l2-norm estimate for the inverse Mercer matrix associated with the Mercer kernel K(x,t) is given

A Temperature-Precipitation Based Leafing Model and Its Application in Northeast China

Plant phenology models, especially leafing models, play critical roles in evaluating the impact of climate change on the primary production of temperate plants. Existing models based on temperature alone could not accurately simulate plant leafing in arid and semi-arid regions. The objective of the present study was to test the suitability of the existing temperature-based leafing models in arid and semi-arid regions, and to develop a temperature-precipitation based leafing model (TP), based on the long-term (i.e., 12–27 years) ground leafing observation data and meteorological data in Northeast China. The better simulation of leafing for all the plant species in Northeast China was given by TP with the fixed starting date (TPn) than with the parameterized starting date (TPm), which gave the smallest average root mean square error (RMSE) of 4.21 days. Tree leafing models were validated with independent data, and the coefficient of determination (R2) was greater than 0.60 in 75% of the estimates by TP and the spring warming model (SW) with the fixed starting date. The average RMSE of herb leafing simulated by TPn was 5.03 days, much lower than other models (>9.51 days), while the average R2 of TPn and TPm were 0.68 and 0.57, respectively, much higher than the other models (<0.22). It indicates that TPn is a universal model and more suitable for simulating leafing of trees and herbs than the prior models. Furthermore, water is an important factor determining herb leafing in arid and semi-arid temperate regions

N-[(E)-4-Pyridylmethyl­ene]-4-[(E)-4-(4-pyridylmethyl­eneamino)benz­yl]aniline tetra­hydrate

The title compound, C25H20N4·4H2O, crystallizes with the organic mol­ecule lying on a twofold rotation axis through the methyl­ene bridge C atom; there are also two water molecules in the asymmetric unit. The crystal structure is stabilized by C—H⋯O, O—H⋯O and O—H⋯N hydrogen bonds, linking the water mol­ecules to each other and to the pyridine N atom