Overview of (pro-)Lie group structures on Hopf algebra character groups

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: The Butcher group from numerical analysis and character groups which arise from the Connes--Kreimer theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai

Nonlinear quantum gravity on the constant mean curvature foliation

A new approach to quantum gravity is presented based on a nonlinear quantization scheme for canonical field theories with an implicitly defined Hamiltonian. The constant mean curvature foliation is employed to eliminate the momentum constraints in canonical general relativity. It is, however, argued that the Hamiltonian constraint may be advantageously retained in the reduced classical system to be quantized. This permits the Hamiltonian constraint equation to be consistently turned into an expectation value equation on quantization that describes the scale factor on each spatial hypersurface characterized by a constant mean exterior curvature. This expectation value equation augments the dynamical quantum evolution of the unconstrained conformal three-geometry with a transverse traceless momentum tensor density. The resulting quantum theory is inherently nonlinear. Nonetheless, it is unitary and free from a nonlocal and implicit description of the Hamiltonian operator. Finally, by imposing additional homogeneity symmetries, a broad class of Bianchi cosmological models are analyzed as nonlinear quantum minisuperspaces in the context of the proposed theory.Comment: 14 pages. Classical and Quantum Gravity (To appear

Structural Instability in Polyacene : A Projector Quantum Monte Carlo Study

We have studied polyacene within the Hubbard model to explore the effect of electron correlations on the Peierls' instability in a system marginally away from one-dimension. We employ the projector quantum Monte Carlo method to obtain ground state estimates of the energy and various correlation functions. We find strong similarities between polyacene and polyacetylene which can be rationalized from the real-space valence-bond arguments of Mazumdar and Dixit. Electron correlations tend to enhance the Peierls' instability in polyacene. This enhancement appears to attain a maximum at $U/t \sim 3.0$ and the maximum shifts to larger values when the alternation parameter is increased. The system shows no tendency to destroy the imposed bond-alternation pattern, as evidenced by the bond-bond correlations. The cis- distortion is seen to be favoured over the trans- distortion. The spin-spin correlations show that undistorted polyacene is susceptible to a SDW distortion for large interaction strength. The charge-charge correlations indicate the absence of a CDW distortion for the parameters studied.Comment: 13 pages, 10 figures available on reques

About curvature, conformal metrics and warped products

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2 \mu}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth functions and $\mu$ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If $(B,g_B)$ is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave-convex nonlinearities and singular partial differential equations of the Lichnerowicz-York type among others.Comment: 32 pages, 3 figure

Structural and Electronic Instabilities in Polyacenes: Density Matrix Renormalization Group Study of a Long--Range Interacting Model

We have carried out Density Matrix Renormalization Group (DMRG) calculations on the ground state of long polyacene oligomers within a Pariser-Parr-Pople (PPP) Hamiltonian. The PPP model includes long-range electron correlations which are required for physically realistic modeling of conjugated polymers. We have obtained the ground state energy as a function of the dimerization $\delta$ and various correlation functions and structure factors for $\delta=0$. From energetics, we find that while the nature of the Peierls' instabilityin polyacene is conditional and strong electron correlations enhance the dimerization. The {\it cis} form of the distortion is favoured over the {\it trans} form. However, from the analysis of correlation functions and associated structure factors, we find that polyacene is not susceptible to the formation of a bond order wave (BOW), spin density wave (SDW) or a charge density wave (CDW) in the ground state.Comment: 31 pages, latex, 13 figure

Initial Steps Towards a Clinical FLASH Radiotherapy System: Pediatric Whole Brain Irradiation with 40 MeV Electrons at FLASH Dose Rates

In this work, we investigated the delivery of a clinically acceptable pediatric whole brain radiotherapy plan at FLASH dose rates using two lateral opposing 40-MeV electron beams produced by a practically realizable linear accelerator system. The EGSnrc Monte Carlo software modules, BEAMnrc and DOSXYZnrc, were used to generate whole brain radiotherapy plans for a pediatric patient using two lateral opposing 40-MeV electron beams. Electron beam phase space files were simulated using a model of a diverging beam with a diameter of 10 cm at 50 cm SAD (defined at brain midline). The electron beams were collimated using a 10-cm-thick block composed of 5 cm of aluminum oxide and 5 cm of tungsten. For comparison, a 6-MV photon plan was calculated with the Varian AAA algorithm. Electron beam parameters were based on a novel linear accelerator designed for the PHASER system and powered by a commercial 6-MW klystron. Calculations of the linear accelerator's performance indicated an average beam current of at least 6.25 µA, providing a dose rate of 115 Gy/s at isocenter, high enough for cognition-sparing FLASH effects. The electron plan was less homogenous with a homogeneity index of 0.133 compared to the photon plan's index of 0.087. Overall, the dosimetric characteristics of the 40-MeV electron plan were suitable for treatment. In conclusion, Monte Carlo simulations performed in this work indicate that two lateral opposing 40-MeV electron beams can be used for pediatric whole brain irradiation at FLASH dose rates of >115 Gy/s and serve as motivation for a practical clinical FLASH radiotherapy system, which can be implemented in the near future

Therapeutic potential of HIV protease-activable CASP3

Development of a therapeutic application of CASP3/caspase 3/CPP32, an executor of apoptosis, has been challenging because regulation of its activation is complicated. This study aimed to inhibit cancer cell growth and human immunodeficiency virus type 1 (HIV-1) propagation through a CASP3 mutant, CASP3*, activable by HIV-1-encoded aspartate protease. Active CASP3* was delivered to leukemic cells using a protein transduction vehicle, the lentivirus-like nanoparticle (LENA), which should contain thousands of CASP3*-Gag protein molecules and release the activated CASP3* into the target cell cytoplasm. CASP3*-LENA induced apoptosis in various types of leukemic cells. In addition to being effective against leukemic cells, constitutive expression of CASP3* restricted HIV-1 propagation in SUP-T1 cells. The attenuation of HIV-1 replication in SUP-T1/CASP3* cells was attributed to the elimination of HIV-1-infected cells by apoptosis. These data suggest that CASP3* has therapeutic potential against both lymphoid malignancies and HIV-1 infection