Algebraic families of subfields in division rings

We describe relations between maximal subfields in a division ring and in its rational extensions. More precisely, we prove that properties such as being Galois or purely inseparable over the centre generically carry over from one to another. We provide an application to enveloping skewfields in positive characteristics. Namely, there always exist two maximal subfields of the enveloping skewfield of a solvable Lie algebra, such that one is Galois and the second purely inseparable of exponent 1 over the centre. This extends results of Schue in the restricted case. Along the way we provide a description of the enveloping algebra of the p-envelope of a Lie algebra as a polynomial extension of the smaller enveloping algebra.Comment: 9 pages, revised according to referee comments, new titl

Competing superconducting instabilities in the one-dimensional p-band degenerate cold fermionic system

The zero-temperature phase diagram of $p$-orbital two-component fermionic system loaded into a one-dimensional optical lattice is mapped out by means of analytical and numerical techniques. It is shown that the $p$-band model away from half-filling hosts various competing superconducting phases for attractive and repulsive interactions. At quarter filling, we analyze the possible formation of incompressible Mott phases and in particular for repulsive interactions, we find the occurrence of a Mott transition with the formation of fully gapped bond-ordering waves.Comment: published versio

Thermodynamic Analysis of Interacting Nucleic Acid Strands

Motivated by the analysis of natural and engineered DNA and RNA systems, we present the first algorithm for calculating the partition function of an unpseudoknotted complex of multiple interacting nucleic acid strands. This dynamic program is based on a rigorous extension of secondary structure models to the multistranded case, addressing representation and distinguishability issues that do not arise for single-stranded structures. We then derive the form of the partition function for a fixed volume containing a dilute solution of nucleic acid complexes. This expression can be evaluated explicitly for small numbers of strands, allowing the calculation of the equilibrium population distribution for each species of complex. Alternatively, for large systems (e.g., a test tube), we show that the unique complex concentrations corresponding to thermodynamic equilibrium can be obtained by solving a convex programming problem. Partition function and concentration information can then be used to calculate equilibrium base-pairing observables. The underlying physics and mathematical formulation of these problems lead to an interesting blend of approaches, including ideas from graph theory, group theory, dynamic programming, combinatorics, convex optimization, and Lagrange duality

Linear maps on k^I, and homomorphic images of infinite direct product algebras

Let k be an infinite field, I an infinite set, V a k-vector-space, and g:k^I\to V a k-linear map. It is shown that if dim_k(V) is not too large (under various hypotheses on card(k) and card(I), if it is finite, respectively countable, respectively < card(k)), then ker(g) must contain elements (u_i)_{i\in I} with all but finitely many components u_i nonzero. These results are used to prove that any homomorphism from a direct product \prod_I A_i of not-necessarily-associative algebras A_i onto an algebra B, where dim_k(B) is not too large (in the same senses) must factor through the projection of \prod_I A_i onto the product of finitely many of the A_i, modulo a map into the subalgebra \{b\in B | bB=Bb=\{0\}\}\subseteq B. Detailed consequences are noted in the case where the A_i are Lie algebras.Comment: 14 pages. Lemma 6 has been strengthened, with resulting strengthening of other results. Some typos etc. have been correcte

Congenital anomalies from a physics perspective. The key role of "manufacturing" volatility

Genetic and environmental factors are traditionally seen as the sole causes of congenital anomalies. In this paper we introduce a third possible cause, namely random "manufacturing" discrepancies with respect to ``design'' values. A clear way to demonstrate the existence of this component is to ``shut'' the two others and to see whether or not there is remaining variability. Perfect clones raised under well controlled laboratory conditions fulfill the conditions for such a test. Carried out for four different species, the test reveals a variability remainder of the order of 10%-20% in terms of coefficient of variation. As an example, the CV of the volume of E.coli bacteria immediately after binary fission is of the order of 10%. In short, ``manufacturing'' discrepancies occur randomly, even when no harmful mutation or environmental factors are involved. Not surprisingly, there is a strong connection between congenital defects and infant mortality. In the wake of birth there is a gradual elimination of defective units and this screening accounts for the post-natal fall of infant mortality. Apart from this trend, post-natal death rates also have humps and peaks associated with various inabilities and defects.\qL In short, infant mortality rates convert the case-by-case and mostly qualitative problem of congenital malformations into a global quantitative effect which, so to say, summarizes and registers what goes wrong in the embryonic phase. Based on the natural assumption that for simple organisms (e.g. rotifers) the manufacturing processes are shorter than for more complex organisms (e.g. mammals), fewer congenital anomalies are expected. Somehow, this feature should be visible on the infant mortality rate. How this conjecture can be tested is outlined in our conclusion.Comment: 43 pages, 9 figure