A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation

A bijective map $r: X^2 \longrightarrow X^2$, where $X = \{x_1, ..., x_n \}$ is a finite set, is called a \emph{set-theoretic solution of the Yang-Baxter equation} (YBE) if the braid relation $r_{12}r_{23}r_{12} = r_{23}r_{12}r_{23}$ holds in $X^3.$ A non-degenerate involutive solution $(X,r)$ satisfying $r(xx)=xx$, for all $x \in X$, is called \emph{square-free solution}. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions $(X,r)$ and the associated Yang-Baxter algebraic structures -- the semigroup $S(X,r)$, the group $G(X,r)$ and the $k$- algebra $A(k, X,r)$ over a field $k$, generated by $X$ and with quadratic defining relations naturally arising and uniquely determined by $r$. We study the properties of the associated Yang-Baxter structures and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial type, are equivalent. This implies that the Yang-Baxter algebra $A(k, X,r)$ is Poincar\'{e}-Birkhoff-Witt type algebra, with respect to some appropriate ordering of $X$. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof-Schedler.Comment: 34 page

Dressing Symmetries of Holomorphic BF Theories

We consider holomorphic BF theories, their solutions and symmetries. The equivalence of Cech and Dolbeault descriptions of holomorphic bundles is used to develop a method for calculating hidden (nonlocal) symmetries of holomorphic BF theories. A special cohomological symmetry group and its action on the solution space are described.Comment: 14 pages, LaTeX2

Assessing the Economic Gains from Telecom Competition

This paper develops and simulates a dynamic model of strategic telecom competition. The goal is to understand how regulatory policy, particularly relative to lease charges for local network elements, affects telecom competition, investment, retail prices, and consumer welfare. The model assumes two products, local voice service and data (broadband), and three types of players the regional Bell operating companies, referred to as incumbent local exchange carriers (ILECs), cable companies (Cables), and competitive local exchange carriers (CLECs). The game begins with a) ILECs established in each county with respect to the provision of local voice and data services and b) Cables established in roughly half of the counties with respect to the provision of data.There are one-time fixed costs of entering a county, product- and period-specific costs of operating in a county, and marginal costs of supplying each product. Economies of scope reduce the fixed entry and operating costs of supplying both products in a given county at a given point in time. Finally, in supplying telecom services in a given county, CLECs may enter by leasing ILEC infrastructure at specified access rates. The requirement that ILECs allow CLECs to lease their local network facilities was established in the Telecommunications Act of 1996 as part of a quid pro quo that promised ILECs entry into the long distance market. But the ILECs continue to contest the quid. The ILECs support their position by suggesting that leased access reduces telecom investment and output and raises telecom prices. Our model considers the entire range of options available to each of the players, but it reaches the opposite conclusion. Indeed, we find thatif UNE-P rates were set at the Supreme Court-approved total element long-run incremental cost (TELRIC) levels, telecom investment and employment outlays would increase by over one fifth in counties containing the majority of the U.S. population and by over 30 percent in counties containing almost a third of the population. The present value of telecom outlays over the next 5 and 20 years would rise by $71 billion and$155 billion, respectively. On average, the switch from actual to TELRIC UNE rates would lower local phone rates across the country's 3108 counties by $57 per year, generating annual total savings to consumers of$15 billion. Almost two fifths of the population would experience reductions in local phone rates of 20 percent or more. Over one fifth would experience rate reductions of 30 percent or more. These findings of price reductions are based on a fairly conservative parameterization of our model with respect to the specification of true ILEC and CLEC incremental long-run production costs.

Cell-penetrating peptide conjugates of peptide nucleic acids (PNA) as inhibitors of HIV-1 Tat-dependent trans-activation in cells

The trans-activation response (TAR) RNA stem–loop that occurs at the 5′ end of HIV RNA transcripts is an important antiviral target and is the site of interaction of the HIV-1 Tat protein together with host cellular factors. Oligonucleotides and their analogues targeted to TAR are potential antiviral candidates. We have investigated a range of cell penetrating peptide (CPP) conjugates of a 16mer peptide nucleic acid (PNA) analogue targeted to the apical stem–loop of TAR and show that disulfide-linked PNA conjugates of two types of CPP (Transportan or a novel chimeric peptide R(6)-Penetratin) exhibit dose-dependent inhibition of Tat-dependent trans-activation in a HeLa cell assay when incubated for 24 h. Activity is reached within 6 h if the lysosomotropic reagent chloroquine is co-administered. Fluorescein-labelled stably-linked conjugates of Tat, Transportan or Transportan TP10 with PNA were inactive when delivered alone, but attained trans-activation inhibition in the presence of chloroquine. Confocal microscopy showed that such fluorescently labelled CPP–PNA conjugates were sequestered in endosomal or membrane-bound compartments of HeLa cells, which varied in appearance depending on the CPP type. Co-administration of chloroquine was seen in some cases to release fluorescence from such compartments into the nucleus, but with different patterns depending on the CPP. The results show that CPP–PNA conjugates of different types can inhibit Tat-dependent trans-activation in HeLa cells and have potential for development as antiviral agents. Endosomal or membrane release is a major factor limiting nuclear delivery and trans-activation inhibition

From Solar Proton Burning to Pionic Deuterium through the Nambu-Jona-Lasinio model of light nuclei

Within the Nambu-Jona-Lasinio model of light nuclei (the NNJL model), describing strong low-energy nuclear interactions, we compute the width of the energy level of the ground state of pionic deuterium. The theoretical value fits well the experimental data. Using the cross sections for the reactions nu_e + d -> p + p + e^- and nu_e + d -> p + n + nu_e, computed in the NNJL model, and the experimental values of the events of these reactions, detected by the SNO Collaboration, we compute the boron neutrino fluxes. The theoretical values agree well with the experimental data and the theoretical predictions within the Standard Solar Model by Bahcall. We argue the applicability of the constraints on the astrophysical factor for the solar proton burning, imposed by helioseismology, to the width of the energy level of the ground state of pionic deuterium. We show that the experimental data on the width satisfy these constraints. This testifies an indirect measurement of the recommended value of the astrophysical factor for the solar proton burning in terrestrial laboratories in terms of the width of the energy level of the ground state of pionic deuterium.Comment: 10 pages, no figures, Late