Canonical decomposition of linear differential operators with selected differential Galois groups

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. Its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi-Yau 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.Comment: 40 page

Globally nilpotent differential operators and the square Ising model

We recall various multiple integrals related to the isotropic square Ising model, and corresponding, respectively, to the n-particle contributions of the magnetic susceptibility, to the (lattice) form factors, to the two-point correlation functions and to their lambda-extensions. These integrals are holonomic and even G-functions: they satisfy Fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties. We recall the explicit forms, found in previous work, of these Fuchsian equations. These differential operators are very selected Fuchsian linear differential operators, and their remarkable properties have a deep geometrical origin: they are all globally nilpotent, or, sometimes, even have zero p-curvature. Focusing on the factorised parts of all these operators, we find out that the global nilpotence of the factors corresponds to a set of selected structures of algebraic geometry: elliptic curves, modular curves, and even a remarkable weight-1 modular form emerging in the three-particle contribution $\chi^{(3)}$ of the magnetic susceptibility of the square Ising model. In the case where we do not have G-functions, but Hamburger functions (one irregular singularity at 0 or $\infty$) that correspond to the confluence of singularities in the scaling limit, the p-curvature is also found to verify new structures associated with simple deformations of the nilpotent property.Comment: 55 page

Lateral-directional control of the x-15 airplane

Lateral directional control and stability characteristics of X-15 aircraf

A-infinity algebra of an elliptic curve and Eisenstein series

We compute explicitly the A-infinity structure on the Ext-algebra of the collection $({\mathcal O}_C, L)$, where $L$ is a line bundle of degree 1 on an elliptic curve $C$. The answer involves higher derivatives of Eisenstein series.Comment: 13 pages, 3 figures; v3: added remark on the limit at the cus

Renormalization, isogenies and rational symmetries of differential equations

We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization group.Comment: 36 page

Painleve versus Fuchs

The sigma form of the Painlev{\'e} VI equation contains four arbitrary parameters and generically the solutions can be said to be genuinely ``nonlinear'' because they do not satisfy linear differential equations of finite order. However, when there are certain restrictions on the four parameters there exist one parameter families of solutions which do satisfy (Fuchsian) differential equations of finite order. We here study this phenomena of Fuchsian solutions to the Painlev{\'e} equation with a focus on the particular PVI equation which is satisfied by the diagonal correlation function C(N,N) of the Ising model. We obtain Fuchsian equations of order $N+1$ for C(N,N) and show that the equation for C(N,N) is equivalent to the $N^{th}$ symmetric power of the equation for the elliptic integral $E$. We show that these Fuchsian equations correspond to rational algebraic curves with an additional Riccati structure and we show that the Malmquist Hamiltonian $p,q$ variables are rational functions in complete elliptic integrals. Fuchsian equations for off diagonal correlations $C(N,M)$ are given which extend our considerations to discrete generalizations of Painlev{\'e}.Comment: 18 pages, Dedicated to the centenary of the publication of the Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de Paris by Richard Fuchs in 190

Angular Momentum Profiles of Warm Dark Matter Halos

We compare the specific angular momentum profiles of virialized dark halos in cold dark matter (CDM) and warm dark matter (WDM) models using high-resolution dissipationless simulations. The simulations were initialized using the same set of modes, except on small scales, where the power was suppressed in WDM below the filtering length. Remarkably, WDM as well as CDM halos are well-described by the two-parameter angular momentum profile of Bullock et al. (2001), even though the halo masses are below the filtering scale of the WDM. Although the best-fit shape parameters change quantitatively for individual halos in the two simulations, we find no systematic variation in profile shapes as a function of the dark matter type. The scatter in shape parameters is significantly smaller for the WDM halos, suggesting that substructure and/or merging history plays a role producing scatter about the mean angular momentum distribution, but that the average angular momentum profiles of halos originate from larger-scale phenomena or a mechanism associated with the virialization process. The known mismatch between the angular momentum distributions of dark halos and disk galaxies is therefore present in WDM as well as CDM models. Our WDM halos tend to have a less coherent (more misaligned) angular momentum structure and smaller spin parameters than do their CDM counterparts, although we caution that this result is based on a small number of halos.Comment: 5 pages, 1 figure, Submitted to ApJ

Study of the local field distribution on a single-molecule magnet-by a single paramagnetic crystal; a DPPH crystal on the surface of an Mn12-acetate crystal

The local magnetic field distribution on the subsurface of a single-molecule magnet crystal, SMM, above blocking temperature (T >> Tb) detected for a very short time interval (~ 10-10 s), has been investigated. Electron Paramagnetic Resonance (EPR) spectroscopy using a local paramagnetic probe was employed as a simple alternative detection method. An SMM crystal of [Mn12O12(CH3COO)16(H2O)4].2CH3COOH.4H2O (Mn12-acetate) and a crystal of 2,2- diphenyl-1-picrylhydrazyl (DPPH) as the paramagnetic probe were chosen for this study. The EPR spectra of DPPH deposited on Mn12-acetate show additional broadening and shifting in the magnetic field in comparison to the spectra of the DPPH in the absence of the SMM crystal. The additional broadening of the DPPH linewidth was considered in terms of the two dominant electron spin interactions (dipolar and exchange) and the local magnetic field distribution on the crystal surface. The temperature dependence of the linewidth of the Gaussian distribution of local fields at the SMM surface was extrapolated for the low temperature interval (70-5 K)

Distribution of localized states from fine analysis of electron spin resonance spectra of organic semiconductors: Physical meaning and methodology

We develop an analytical method for the processing of electron spin resonance (ESR) spectra. The goal is to obtain the distributions of trapped carriers over both their degree of localization and their binding energy in semiconductor crystals or films composed of regularly aligned organic molecules [Phys. Rev. Lett. v. 104, 056602 (2010)]. Our method has two steps. We first carry out a fine analysis of the shape of the ESR spectra due to the trapped carriers; this reveals the distribution of the trap density of the states over the degree of localization. This analysis is based on the reasonable assumption that the linewidth of the trapped carriers is predetermined by their degree of localization because of the hyperfine mechanism. We then transform the distribution over the degree of localization into a distribution over the binding energies. The transformation uses the relationships between the binding energies and the localization parameters of the trapped carriers. The particular relation for the system under study is obtained by the Holstein model for trapped polarons using a diagrammatic Monte Carlo analysis. We illustrate the application of the method to pentacene organic thin-film transistors.Comment: 14 pages, 11 figure