Estimations of the low dimensional homology of Lie algebras with large abelian ideals

A Lie algebra $L$ of dimension $n \ge1$ may be classified, looking for restrictions of the size on its second integral homology Lie algebra $H_2(L,\mathbb{Z})$, denoted by $M(L)$ and often called Schur multiplier of $L$. In case $L$ is nilpotent, we proved that $\mathrm{dim} \ M(L) \leq \frac{1}{2}(n+m-2)(n-m-1)+1$, where $\mathrm{dim} \ L^2=m \ge 1$, and worked on this bound under various perspectives. In the present paper, we estimate the previous bound for $\mathrm{dim} \ M(L)$ with respect to other inequalities of the same nature. Finally, we provide new upper bounds for the Schur multipliers of pairs and triples of nilpotent Lie algebras, by means of certain exact sequences due to Ganea and Stallings in their original form.Comment: 9 pages, to appear in Bull. Belgian Math. Soc. with structural revision

On the tensor degree of finite groups

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x \otimes y= 1_{_{G \otimes G}}$ in the nonabelian tensor square $G \otimes G$ of $G$. This number, divided by $|G|^2$, is called the tensor degree of $G$ and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335--343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.Comment: 10 pages, accepted in Ars Combinatoria with revision

Commuting powers and exterior degree of finite groups

In [P. Niroomand, R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335-343] it is introduced a group invariant, related to the number of elements $x$ and $y$ of a finite group $G$, such that $x \wedge y = 1_{G \wedge G}$ in the exterior square $G \wedge G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m \wedge k$ of $H \wedge K$ such that $h^m \wedge k = 1_{H \wedge K}$, where $m \ge 1$ and $H$ and $K$ are arbitrary subgroups of $G$.Comment: to appear in the J. Korean Math. Soc. with revision

A note on the Schur multiplier of a nilpotent Lie algebra

For a nilpotent Lie algebra $L$ of dimension $n$ and dim$(L^2)=m$, we find the upper bound dim$(M(L))\leq {1/2}(n+m-2)(n-m-1)+1$, where $M(L)$ denotes the Schur multiplier of $L$. In case $m=1$ the equality holds if and only if $L\cong H(1)\oplus A$, where $A$ is an abelian Lie algebra of dimension $n-3$ and H(1) is the Heisenberg algebra of dimension 3.Comment: Paper in press in Comm. Algebra with small revision

Density-density propagator for one-dimensional interacting spinless fermions with non-linear dispersion and calculation of the Coulomb drag resistivity

Using bosonization-fermionization transformation we map the Tomonaga-Luttinger model of spinless fermions with non-linear dispersion on the model of fermionic quasiparticles whose interaction is irrelevant in the renormalization group sense. Such mapping allows us to set up an expansion for the density-density propagator of the original Tomonaga-Luttinger Hamiltonian in orders of the (irrelevant) quasiparticle interaction. The lowest order term in such an expansion is proportional to the propagator for free fermions. The next term is also evaluated. The propagator found is used for calculation of the Coulomb drug resistivity $r$ in a system of two capacitively coupled one-dimensional conductors. It is shown that $r$ is proportional to $T^2$ for both free and interacting fermions. The marginal repulsive in-chain interaction acts to reduce $r$ as compared to the non-interacting result. The correction to $r$ due to the quasiparticle interaction is found as well. It scales as $T^4$ at low temperature.Comment: 5 pages, 1 eps figure; the new version of the e-print corrects an error, which exists in the original submission; fortunately, all important conclusions of the study remain vali

The exterior degree of a pair of finite groups

The exterior degree of a pair of finite groups $(G,N)$, which is a generalization of the exterior degree of finite groups, is the probability for two elements $(g,n)$ in $(G,N)$ such that $g\wedge n=1$. In the present paper, we state some relations between this concept and the relative commutatively degree, capability and the Schur multiplier of a pair of groups.Comment: To appear in Mediterr. J. Mat

Progress toward the computational discovery of new metal–organic framework adsorbents for energy applications

Metal–organic frameworks (MOFs) are a class of nanoporous material precisely synthesized from molecular building blocks. MOFs could have a critical role in many energy technologies, including carbon capture, separations and storage of energy carriers. Molecular simulations can improve our molecular-level understanding of adsorption in MOFs, and it is now possible to use realistic models for these complicated materials and predict their adsorption properties in quantitative agreement with experiments. Here we review the predictive design and discovery of MOF adsorbents for the separation and storage of energy-relevant molecules, with a view to understanding whether we can reliably discover novel MOFs computationally prior to laboratory synthesis and characterization. We highlight in silico approaches that have discovered new adsorbents that were subsequently confirmed by experiments, and we discuss the roles of high-throughput computational screening and machine learning. We conclude that these tools are already accelerating the discovery of new applications for existing MOFs, and there are now several examples of new MOFs discovered by computational modelling

Quantum degenerate Bose-Fermi mixture of chemically different atomic species with widely tunable interactions

We have created a quantum degenerate Bose-Fermi mixture of 23Na and 40K with widely tunable interactions via broad interspecies Feshbach resonances. Twenty Feshbach resonances between 23Na and 40K were identified. The large and negative triplet background scattering length between 23Na and 40K causes a sharp enhancement of the fermion density in the presence of a Bose condensate. As explained via the asymptotic bound-state model (ABM), this strong background scattering leads to a series of wide Feshbach resonances observed at low magnetic fields. Our work opens up the prospect to create chemically stable, fermionic ground state molecules of 23Na-40K where strong, long-range dipolar interactions will set the dominant energy scale