Complete spectrum of the infinite-UU Hubbard ring using group theory

We present a full analytical solution of the multiconfigurational strongly-correlated mixed-valence problem corresponding to the $N$-Hubbard ring filled with $N-1$ electrons, and infinite on-site repulsion. While the eigenvalues and the eigenstates of the model are known already, analytical determination of their degeneracy is presented here for the first time. The full solution, including degeneracy count, is achieved for each spin configuration by mapping the Hubbard model into a set of Huckel-annulene problems for rings of variable size. The number and size of these effective Huckel annulenes, both crucial to obtain Hubbard states and their degeneracy, are determined by solving a well-known combinatorial enumeration problem, the necklace problem for $N-1$ beads and two colors, within each subgroup of the $C_{N-1}$ permutation group. Symmetry-adapted solution of the necklace enumeration problem is finally achieved by means of the subduction of coset representation technique [S. Fujita, Theor. Chim. Acta 76, 247 (1989)], which provides a general and elegant strategy to solve the one-hole infinite-$U$ Hubbard problem, including degeneracy count, for any ring size. The proposed group theoretical strategy to solve the infinite-$U$ Hubbard problem for $N-1$ electrons, is easily generalized to the case of arbitrary electron count $L$, by analyzing the permutation group $C_L$ and all its subgroups.Comment: 31 pages, 4 figures. Submitte

Degree Sequences and the Existence of kk-Factors

We consider sufficient conditions for a degree sequence $\pi$ to be forcibly $k$-factor graphical. We note that previous work on degrees and factors has focused primarily on finding conditions for a degree sequence to be potentially $k$-factor graphical. We first give a theorem for $\pi$ to be forcibly 1-factor graphical and, more generally, forcibly graphical with deficiency at most $\beta\ge0$. These theorems are equal in strength to Chv\'atal's well-known hamiltonian theorem, i.e., the best monotone degree condition for hamiltonicity. We then give an equally strong theorem for $\pi$ to be forcibly 2-factor graphical. Unfortunately, the number of nonredundant conditions that must be checked increases significantly in moving from $k=1$ to $k=2$, and we conjecture that the number of nonredundant conditions in a best monotone theorem for a $k$-factor will increase superpolynomially in $k$. This suggests the desirability of finding a theorem for $\pi$ to be forcibly $k$-factor graphical whose algorithmic complexity grows more slowly. In the final section, we present such a theorem for any $k\ge2$, based on Tutte's well-known factor theorem. While this theorem is not best monotone, we show that it is nevertheless tight in a precise way, and give examples illustrating this tightness.Comment: 19 page

Neutron Star Masses and Radii as Inferred from kilo-Hertz QPOs

Kilo-Hertz (kHz) Quasi-periodic oscillations (QPOs) have been discovered in the X-ray fluxes of 8 low-mass X-ray binaries (LMXBs) with the Rossi X-ray Timing Explorer (RXTE). The characteristics of these QPOs are remarkably similar from one source to another. In particular, the highest observed QPO frequencies for 6 of the 8 sources fall in a very narrow range: 1,066 to 1,171 Hz. This is the more remarkable when one considers that these sources are thought to have very different luminosities and magnetic fields, and produce very different count rates in the RXTE detectors. Therefore it is highly unlikely that this near constancy of the highest observed frequencies is due to some unknown selection effect or instrumental bias. In this letter we propose that the highest observed QPO frequency can be taken as the orbital frequency of the marginally stable orbit. This leads to the conclusions that the neutron stars in these LMXBs are inside their marginally stable orbits and have masses in the vicinity of 2.0 solar masses. This mass is consistent with the hypothesis that these neutron stars were born with about 1.4 solar masses and have been accreting matter at a fraction of the Eddington limit for 100 million years.Comment: 7 pages, uses aas2pp4.sty, Accepted by ApJ