On quantization of r-matrices for Belavin-Drinfeld Triples

We suggest a formula for quantum universal $R$-matrices corresponding to quasitriangular classical $r$-matrices classified by Belavin and Drinfeld for all simple Lie algebras. The $R$-matrices are obtained by twisting the standard universal $R$-matrix.Comment: 12 pages, LaTe

Lipidic Formulations Inspired by COVID Vaccines as Smart Coatings to Enhance Nanoparticle-Based Cancer Therapy

Recent advances in nanomedicine have led to the introduction and subsequent establishment of nanoparticles in cancer treatment and diagnosis. Nonetheless, their application is still hindered by a series of challenges related to their biocompatibility and biodistribution. In this paper, we take inspiration from the recently produced and widely spread COVID vaccines, based on the combinational use of ionizable solid lipid nanoparticles, cholesterol, PEGylated lipids, and neutral lipids able to incorporate mRNA fragments. Here, we focus on the implementation of a lipidic formulation meant to be used as a smart coating of solid-state nanoparticles. The composition of this formulation is finely tuned to ensure efficient and stable shielding of the cargo. The resulting shell is a highly customized tool that enables the possibility of further functionalizations with targeting agents, peptides, antibodies, and fluorescent moieties for future in vitro and in vivo tests and validations. Finally, as a proof of concept, zinc oxide nanoparticles doped with iron and successively coated with this lipidic formulation are tested in a pancreatic cancer cell line, BxPC-3. The results show an astonishing increase in cell viability with respect to the same uncoated nanoparticles. The preliminary results presented here pave the way towards many different therapeutic approaches based on the massive presence of highly biostable and well-tolerated nanoparticles in tumor tissues, such as sonodynamic therapy, photodynamic therapy, hyperthermia, and diagnosis by means of magnetic resonance imaging

Quantum Lie algebras associated to Uq(gln)U_q(gl_n) and Uq(sln)U_q(sl_n)

Quantum Lie algebras \qlie{g} are non-associative algebras which are embedded into the quantized enveloping algebras $U_q(g)$ of Drinfeld and Jimbo in the same way as ordinary Lie algebras are embedded into their enveloping algebras. The quantum Lie product on \qlie{g} is induced by the quantum adjoint action of $U_q(g)$. We construct the quantum Lie algebras associated to $U_q(gl_n)$ and $U_q(sl_n)$. We determine the structure constants and the quantum root systems, which are now functions of the quantum parameter $q$. They exhibit an interesting duality symmetry under $q\leftrightarrow 1/q$.Comment: Latex 9 page

Representations of the quantum matrix algebra Mq,p(2)M_{q,p}(2)

It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra $M_{ q,p}(2)$ ( the coordinate ring of $GL_{q,p}(2)$) exist only when both q and p are roots of unity. In this case th e space of states has either the topology of a torus or a cylinder which may be thought of as generalizations of cyclic representations.Comment: 20 page

A mean-field kinetic lattice gas model of electrochemical cells

We develop Electrochemical Mean-Field Kinetic Equations (EMFKE) to simulate electrochemical cells. We start from a microscopic lattice-gas model with charged particles, and build mean-field kinetic equations following the lines of earlier work for neutral particles. We include the Poisson equation to account for the influence of the electric field on ion migration, and oxido-reduction processes on the electrode surfaces to allow for growth and dissolution. We confirm the viability of our approach by simulating (i) the electrochemical equilibrium at flat electrodes, which displays the correct charged double-layer, (ii) the growth kinetics of one-dimensional electrochemical cells during growth and dissolution, and (iii) electrochemical dendrites in two dimensions.Comment: 14 pages twocolumn, 17 figure

On Quantum Lie Algebras and Quantum Root Systems

As a natural generalization of ordinary Lie algebras we introduce the concept of quantum Lie algebras ${\cal L}_q(g)$. We define these in terms of certain adjoint submodules of quantized enveloping algebras $U_q(g)$ endowed with a quantum Lie bracket given by the quantum adjoint action. The structure constants of these algebras depend on the quantum deformation parameter $q$ and they go over into the usual Lie algebras when $q=1$. The notions of q-conjugation and q-linearity are introduced. q-linear analogues of the classical antipode and Cartan involution are defined and a generalised Killing form, q-linear in the first entry and linear in the second, is obtained. These structures allow the derivation of symmetries between the structure constants of quantum Lie algebras. The explicitly worked out examples of $g=sl_3$ and $so_5$ illustrate the results.Comment: 22 pages, latex, version to appear in J. Phys. A. see http://www.mth.kcl.ac.uk/~delius/q-lie.html for calculations and further informatio

On the fluctuations of jamming coverage upon random sequential adsorption on homogeneous and heterogeneous media

The fluctuations of the jamming coverage upon Random Sequential Adsorption (RSA) are studied using both analytical and numerical techniques. Our main result shows that these fluctuations (characterized by $\sigma_{\theta_J}$) decay with the lattice size according to the power-law $\sigma_{\theta_J} \propto L^{-1/ \nu}$. The exponent $\nu$ depends on the dimensionality $D$ of the substrate and the fractal dimension of the set where the RSA process actually takes place ($d_f$) according to $\nu = 2 / (2D - d_f)$.This theoretical result is confirmed by means of extensive numerical simulations applied to the RSA of dimers on homogeneous and stochastic fractal substrates. Furthermore, our predictions are in excellent agreement with different previous numerical results. It is also shown that, studying correlated stochastic processes, one can define various fluctuating quantities designed to capture either the underlying physics of individual processes or that of the whole system. So, subtle differences in the definitions may lead to dramatically different physical interpretations of the results. Here, this statement is demonstrated for the case of RSA of dimers on binary alloys.Comment: 20 pages, 8 figure

Freezing Transition in Decaying Burgers Turbulence and Random Matrix Dualities

We reveal a phase transition with decreasing viscosity $\nu$ at \nu=\nu_c>0 in one-dimensional decaying Burgers turbulence with a power-law correlated random profile of Gaussian-distributed initial velocities \sim|x-x'|^{-2}. The low-viscosity phase exhibits non-Gaussian one-point probability density of velocities, continuously dependent on \nu, reflecting a spontaneous one step replica symmetry breaking (RSB) in the associated statistical mechanics problem. We obtain the low orders cumulants analytically. Our results, which are checked numerically, are based on combining insights in the mechanism of the freezing transition in random logarithmic potentials with an extension of duality relations discovered recently in Random Matrix Theory. They are essentially non mean-field in nature as also demonstrated by the shock size distribution computed numerically and different from the short range correlated Kida model, itself well described by a mean field one step RSB ansatz. We also provide some insights for the finite viscosity behaviour of velocities in the latter model.Comment: Published version, essentially restructured & misprints corrected. 6 pages, 5 figure

Universal integrability objects

We discuss the main points of the quantum group approach in the theory of quantum integrable systems and illustrate them for the case of the quantum group $U_q(\mathcal L(\mathfrak{sl}_2))$. We give a complete set of the functional relations correcting inexactitudes of the previous considerations. A special attention is given to the connection of the representations used to construct the universal transfer operators and $Q$-operators.Comment: 21 pages, submitted to the Proceedings of the International Workshop "CQIS-2012" (Dubna, January 23-27, 2012