On the reduction of the degree of linear differential operators

Let L be a linear differential operator with coefficients in some differential field k of characteristic zero with algebraically closed field of constants. Let k^a be the algebraic closure of k. For a solution y, Ly=0, we determine the linear differential operator of minimal degree M and coefficients in k^a, such that My=0. This result is then applied to some Picard-Fuchs equations which appear in the study of perturbations of plane polynomial vector fields of Lotka-Volterra type

Temperature induced solubility transitions of various poly(2-oxazoline)s in ethanol-water solvent mixtures

The solution behavior of a series of poly(2-oxazoline)s with different side chains, namely methyl, ethyl, n-propyl, isopropyl, n-butyl, isobutyl, pentyl, hexyl, heptyl, octyl, nonyl, phenyl and benzyl, are reported in ethanol-water solvent mixtures based on turbidimetry investigations. The LCST transitions of poly(2-oxazoline) s with propyl side chains and the UCST transitions of the poly(2-oxazoline) s with more hydrophobic side chains are discussed in relation to the ethanol-water solvent composition and structure. The poly(2-alkyl-2-oxazoline) s with side chains longer than propyl only dissolved during the first heating run, which is discussed and correlated to the melting transition of the polymers

Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms

We develop a theory of Tannakian Galois groups for t-motives and relate this to the theory of Frobenius semilinear difference equations. We show that the transcendence degree of the period matrix associated to a given t-motive is equal to the dimension of its Galois group. Using this result we prove that Carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent.Comment: 39 page

Automated Feature Mining for Two-Dimensional Liquid Chromatography Applied to Polymers Enabled by Mass Remainder Analysis

A fast algorithm for automated feature mining of synthetic (industrial) homopolymers or perfectly alternating copolymers was developed. Comprehensive two-dimensional liquid chromatography-mass spectrometry data (LC × LC-MS) was utilized, undergoing four distinct parts within the algorithm. Initially, the data is reduced by selecting regions of interest within the data. Then, all regions of interest are clustered on the time and mass-to-charge domain to obtain isotopic distributions. Afterward, single-value clusters and background signals are removed from the data structure. In the second part of the algorithm, the isotopic distributions are employed to define the charge state of the polymeric units and the charge-state reduced masses of the units are calculated. In the third part, the mass of the repeating unit (i.e., the monomer) is automatically selected by comparing all mass differences within the data structure. Using the mass of the repeating unit, mass remainder analysis can be performed on the data. This results in groups sharing the same end-group compositions. Lastly, combining information from the clustering step in the first part and the mass remainder analysis results in the creation of compositional series, which are mapped on the chromatogram. Series with similar chromatographic behavior are separated in the mass-remainder domain, whereas series with an overlapping mass remainder are separated in the chromatographic domain. These series were extracted within a calculation time of 3 min. The false positives were then assessed within a reasonable time. The algorithm is verified with LC × LC-MS data of an industrial hexahydrophthalic anhydride-derivatized propylene glycol-terephthalic acid copolyester. Afterward, a chemical structure proposal has been made for each compositional series found within the data

A Difference Version of Nori's Theorem

We consider (Frobenius) difference equations over (F_q(s,t), phi) where phi fixes t and acts on F_q(s) as the Frobenius endomorphism. We prove that every semisimple, simply-connected linear algebraic group G defined over F_q can be realized as a difference Galois group over F_{q^i}(s,t) for some i in N. The proof uses upper and lower bounds on the Galois group scheme of a Frobenius difference equation that are developed in this paper. The result can be seen as a difference analogue of Nori's Theorem which states that G(F_q) occurs as (finite) Galois group over F_q(s).Comment: 29 page

Properties of generalized univariate hypergeometric functions

Based on Spiridonov's analysis of elliptic generalizations of the Gauss hypergeometric function, we develop a common framework for 7-parameter families of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric functions. In each case we derive the symmetries of the generalized hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic) and of type E_6 (trigonometric) using the appropriate versions of the Nassrallah-Rahman beta integral, and we derive contiguous relations using fundamental addition formulas for theta and sine functions. The top level degenerations of the hyperbolic and trigonometric hypergeometric functions are identified with Ruijsenaars' relativistic hypergeometric function and the Askey-Wilson function, respectively. We show that the degeneration process yields various new and known identities for hyperbolic and trigonometric special functions. We also describe an intimate connection between the hyperbolic and trigonometric theory, which yields an expression of the hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric hypergeometric functions.Comment: 46 page

Non integrability of a self-gravitating Riemann liquid ellipsoid

We prove that the motion of a triaxial Riemann ellipsoid of homogeneous liquid without angular momentum does not possess an additional first integral which is meromorphic in position, impulsions, and the elliptic functions which appear in the potential, and thus is not integrable. We prove moreover that this system is not integrable even on a fixed energy level hypersurface.Comment: 14 pages, 8 reference