The role of impacting processes in the chemical evolution of the atmosphere of primordial Earth

The role of impacting processes in the chemical evolution of the atmosphere of primordial Earth is discussed. The following subject areas are covered: (1) Earth's initial atmosphere; (2) continuous degassing; (3) impact processes and the Earth's protoatmosphere; and (4) the evolution of an impact-generated atmosphere

Early stages in the evolution of the atmosphere and climate on the Earth-group planets

The early evolution of the atmospheres and climate of the Earth, Mars and Venus is discussed, based on a concept of common initial conditions and main processes (besides known differences in chemical composition and outgassing rate). It is concluded that: (1) liquid water appeared on the surface of the earth in the first few hundred million years; the average surface temperature was near the melting point for about the first two eons; CO2 was the main component of the atmosphere in the first 100-500 million years; (2) much more temperate outgassing and low solar heating led to the much later appearance of liquid water on the Martian surface, only one to two billion years ago; the Martian era of rivers, relatively dense atmosphere and warm climate ended as a result of irreversible chemical bonding of CO2 by Urey equilibrium processes; (3) a great lack of water in the primordial material of Venus is proposed; liquid water never was present on the surface of the planet, and there was practically no chemical bonding of CO2; the surface temperature was over 600 K four billion years ago

Finite type modules and Bethe Ansatz for quantum toroidal gl(1)

We study highest weight representations of the Borel subalgebra of the quantum toroidal gl(1) algebra with finite-dimensional weight spaces. In particular, we develop the q-character theory for such modules. We introduce and study the subcategory of `finite type' modules. By definition, a module over the Borel subalgebra is finite type if the Cartan like current \psi^+(z) has a finite number of eigenvalues, even though the module itself can be infinite dimensional. We use our results to diagonalize the transfer matrix T_{V,W}(u;p) analogous to those of the six vertex model. In our setting T_{V,W}(u;p) acts in a tensor product W of Fock spaces and V is a highest weight module over the Borel subalgebra of quantum toroidal gl(1) with finite-dimensional weight spaces. Namely we show that for a special choice of finite type modules $V$ the corresponding transfer matrices, Q(u;p) and T(u;p), are polynomials in u and satisfy a two-term TQ relation. We use this relation to prove the Bethe Ansatz equation for the zeroes of the eigenvalues of Q(u;p). Then we show that the eigenvalues of T_{V,W}(u;p) are given by an appropriate substitution of eigenvalues of Q(u;p) into the q-character of V.Comment: Latex 42 page

Form factors and action of U_{\sqrt{-1}}(sl_2~) on infinite-cycles

Let ${\bf p}=\{P_{n,l}\}_{n,l\in\Z_{\ge 0}\atop n-2l=m}$ be a sequence of skew-symmetric polynomials in $X_1,...,X_l$ satisfying $\deg_{X_j}P_{n,l}\le n-1$, whose coefficients are symmetric Laurent polynomials in $z_1,...,z_n$. We call ${\bf p}$ an $\infty$-cycle if $P_{n+2,l+1}\bigl|_{X_{l+1}=z^{-1},z_{n-1}=z,z_n=-z} =z^{-n-1}\prod_{a=1}^l(1-X_a^2z^2)\cdot P_{n,l}$ holds for all $n,l$. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables $\alpha_a=-\log X_a$ are the integration variables and $\beta_j=\log z_j$ are the rapidity variables. To each $\infty$-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the $\infty$-cycles. In this paper, we define an action of $U_{\sqrt{-1}}(\widetilde{\mathfrak{sl}}_2)$ on the space of $\infty$-cycles. There are two sectors of $\infty$-cycles depending on whether $n$ is even or odd. Using this action, we show that the character of the space of even (resp. odd) $\infty$-cycles which are polynomials in $z_1,...,z_n$ is equal to the level $(-1)$ irreducible character of $\hat{\mathfrak{sl}}_2$ with lowest weight $-\Lambda_0$ (resp. $-\Lambda_1$). We also suggest a possible tensor product structure of the full space of $\infty$-cycles.Comment: 27 pages, abstract and section 3.1 revise