The Genuine Savings Criterion and the Value of Population

Arrow, Dasgupta and Maler demonstrate thatin any dynamic model of the economy with changing population, population should properly be one of the state variables of the system. It enters both in the maxim and, at least under total utilitarianism, and into the production function in one way or another. If population growth is exponential and there are constant returns to scale, then a simple transformation to per capita variables can be used to eliminate one state variable. However, this simple transformation cannot be made if growth is not exponential, as it obviously is not and cannot be. If the growth of population is exogenous, then introducing it into the system does not affect the optimal policy. However, if one asks whether the system is sustainable, in the sense of at least maintaining total welfare (integral of discounted utilities), then the criterion is that the value of the rates of change of the state variables is non-negative, so that the shadow price of population becomes relevant. In this paper, we derive explicit formulas in a simple model, showing that the rate of growth of per capita capital is not the correct formula but must have other terms added to it. We also study the question under an alternative criterion of long-run average utilitarianism.

Cosmic rays from trans-relativistic supernovae

We derive constraints that must be satisfied by the sources of ~10^{15} to ~10^{18} eV cosmic rays, under the assumption that the sources are Galactic. We show that while these constraints are not satisfied by ordinary supernovae (SNe), which are believed to be the sources of <10^{15} eV cosmic rays, they may be satisfied by the recently discovered class of trans-relativistic supernovae (TRSNe), which were observed in association with gamma-ray bursts. We define TRSNe as SNe that deposit a large fraction, f_R>10^{-2}, of their kinetic energy in mildly relativistic, \gamma\beta>1, ejecta. The high velocity ejecta enable particle acceleration to ~10^{18} eV, and the large value of f_R (compared to f_R~10^{-7} for ordinary SNe) ensures that if TRSNe produce the observed ~10^{18} eV cosmic ray flux, they do not overproduce the flux at lower energies. This, combined with the estimated rate and energy production of TRSNe, imply that Galactic TRSNe may be the sources of cosmic rays with energies up to ~10^{18}eV .Comment: Accepted to ApJ. Expanded abstract, introduction, discussio

Potential of Interplanetary Torques and Solar Modulation for Triggering Terrestrial Atmospheric and Lithospheric Events

The Sun is forced into an orbit around the barycenter of the solar system because of the changing mass distribution of the planets. Solar-planetary-lunar dynamic relationships may form a new basis for understanding and predicting cyclic solar forcing functions on the Earth's climate.Comment: Invited Paper at the Fourth UN/ESA Workshop on Basic Space Science, Cairo, Egypt, July 1994. 7 pages LaTeX. Accepted for publication in the journal Earth, Moon, and Planet

Entropy and weak solutions in the thermal model for the compressible Euler equations

Among the existing models for compressible fluids, the one by Kataoka and Tsutahara (KT model, Phys. Rev. E 69, 056702, 2004) has a simple and rigorous theoretical background. The drawback of this KT model is that it can cause numerical instability if the local Mach number exceeds 1. The precise mechanism of this instability has not yet been clarified. In this paper, we derive entropy functions whose local equilibria are suitable to recover the Euler-like equations in the framework of the lattice Boltzmann method for the KT model. Numerical examples are also given, which are consistent with the above theoretical arguments, and show that the entropy condition is not fully guaranteed in KT model. The negative entropy may be the inherent cause for the non-physical oscillations in the vicinity of the shock. In contrast to these Karlin's microscopic entropy approach, the corresponding subsidiary entropy condition in the LBM calculation could also be deduced explicitly from the macroscopic version, which provides some insights on the numerical instability of the lattice Boltzmann model for shock calculation.Comment: 27 pages,6 figure

Spartan Daily, May 4, 1998

Volume 110, Issue 65https://scholarworks.sjsu.edu/spartandaily/9281/thumbnail.jp

A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations

This paper is a continuation of the work on unbounded Toeplitz-like operators T_\Om with rational matrix symbol \Om initiated in Groenewald et. al (Complex Anal. Oper. Theory 15, 1(2021)), where a Wiener-Hopf type factorization of \Om is obtained and used to determine when T_\Om is Fredholm and compute the Fredholm index in case T_\Om is Fredholm. Due to the high level of non-uniqueness and complicated form of the Wiener-Hopf type factorization, it does not appear useful in determining when T_\Om is invertible. In the present paper we use state space methods to characterize invertibility of T_\Om in terms of the existence of a stabilizing solution of an associated nonsymmetric discrete algebraic Riccati equation, which in turn leads to a pseudo-canonical factorization of \Om and concrete formulas of T_\Om^{-1}.Comment: 19 page

A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics

In a recent paper (Groenewald et al.\ {\em Complex Anal.\ Oper.\ Theory} \textbf{15:1} (2021)) we considered an unbounded Toeplitz-like operator $T_\Omega$ generated by a rational matrix function $\Omega$ that has poles on the unit circle $\mathbb{T}$ of the complex plane. A Wiener-Hopf type factorization was proved and this factorization was used to determine some Fredholm properties of the operator $T_\Omega$, including the Fredholm index. Due to the lower triangular structure (rather than diagonal) of the middle term in the Wiener-Hopf type factorization and the lack of uniqueness, it is not straightforward to determine the dimension of the kernel of $T_\Omega$ from this factorization, and hence of the co-kernel, even when $T_\Omega$ is Fredholm. In the current paper we provide a formula for the dimension of the kernel of $T_\Omega$ under an additional assumption on the Wiener-Hopf type factorization. In the case that $\Omega$ is a $2 \times 2$ matrix function, a characterization of the kernel of the middle factor of the Wiener-Hopf type factorization is given and in many cases a formula for the dimension of the kernel is obtained. The characterization of the kernel of the middle factor for the $2 \times 2$ case is partially extended to the case of matrix functions of arbitrary size.Comment: 27 page

Influence of Fermion Velocity Renormalization on Dynamical Mass Generation in QED3_3

We study dynamical fermion mass generation in (2+1)-dimensional quantum electrodynamics with a gauge field coupling to massless Dirac fermions and non-relativistic scalar bosons. We calculate the fermion velocity renormalization and then examine its influence on dynamical mass generation by using the Dyson-Schwinger equation. It is found that dynamical mass generation takes place even after including the scalar bosons as long as the bosonic compressibility parameter $\xi$ is sufficiently small. In addition, the fermion velocity renormalization enhances the dynamically generated mass.Comment: 6 pages, 3 figures, Chinese Physics Letter, Vol 29, page 057401(2012

Ti(3)C(2) MXene co-catalyst on metal sulfide photo-absorbers for enhanced visible-light photocatalytic hydrogen production

Scalable and sustainable solar hydrogen production through photocatalytic water splitting requires highly active and stable earth-abundant co-catalysts to replace expensive and rare platinum. Here we employ density functional theory calculations to direct atomic-level exploration, design and fabrication of a MXene material, Ti3C2 nanoparticles, as a highly efficient co-catalyst. Ti3C2 nanoparticles are rationally integrated with cadmium sulfide via a hydrothermal strategy to induce a super high visible-light photocatalytic hydrogen production activity of 14,342 μmol h-1 g-1 and an apparent quantum efficiency of 40.1% at 420 nm. This high performance arises from the favourable Fermi level position, electrical conductivity and hydrogen evolution capacity of Ti3C2 nanoparticles. Furthermore, Ti3C2 nanoparticles also serve as an efficient co-catalyst on ZnS or ZnxCd1-xS. This work demonstrates the potential of earth-abundant MXene family materials to construct numerous high performance and low-cost photocatalysts/photoelectrodes.Jingrun Ran, Guoping Gao, Fa-Tang Li, Tian-Yi Ma, Aijun Du and Shi-Zhang Qia