Cyclic subsemigroups of symmetric inverse semigroups

A generalization of the cycle notation for permutations is introduced for partial one-one transformations (charts). Notational representation theorems for charts that generalize those of permutations are given. Notational multiplication of charts is developed and then applied to yield a transparent proof of Frobenius' result which bounds the idempotent in the cyclic subsemigroup. Lastly, the well known result that the structure of the cyclic subgroups of the finite symmetric groups is determined from combinations of disjoint cycles is generalized to the cyclic subsemigroups of the finite symmetric inverse semigroups.     

HPLC Method for the Simultaneous Determination of the UV-Filters Butyl Methoxy Dibenzoylmethane and Octocrylene in the Presence of Their Photodegradants

Butyl methoxy dibenzoylmethane (BMDM) and octocrylene (OC), common UV-filters in sunscreen products are often used in combination. Together they provide broad spectrum photoprotection from exposure to both UVA- and UVB-light. These UV-filters may, however, undergo photodegradation and generate photodegradants, resulting in a potential loss of photoprotection. It is thus a concern that the photostability testing as described by the ICH Guideline Q1B is not a requirement for sunscreen products in Australia, Europe or the USA. UV-filter photodegradants have in addition been shown to be toxic, highlighting the importance of their separation from the parent UV-filters. An HPLC method was developed and validated to quantitatively determine a combination of these UV-filters in the presence of their photodegradants. Reverse-phase chromatography was employed, using a C18 column and an isocratic mobile phase consisting of methanol/water/acetic acid (89/10/1 v/v). Validation according to the ICH guidelines for linearity, accuracy, precision, sensitivity, specificity and robustness was confirmed. The developed and validated method was then successfully applied to the determination of BMDM and OC in an aqueous cream base, typically used in sunscreens, after photostability testing, according to the ICH Guideline Q1B. In addition, the diketo-enol ratio of BMDM in methanol-d ₄ was determined by NMR and the two major photodegradants were identified by FTMS and LC–MS.     

Paroxysmal starting and stopping of circulating waves in excitable media

Levels of intracellular Ca2+ were monitored using fluorescence from Ca2+-sensitive dyes in chick embryonic heart cells cultured in an annular geometry. There was spontaneous starting and stopping of reentrant waves of activity. The results are modeled using modified FitzHugh-Nagumo equations representing pacemakers embedded in a conducting medium. These results provide a potential mechanism for spontaneous abnormal cardiac rhythms in which there are rapid heart beats (tachycardias) that repetitively start and stop.     

The process of front-to-side collision of motor vehicles in terms of energy balance

Nonlinear Dynamics - The reconstruction of a road accident can be treated as the resolution of an “inverse problem” in mechanics using analytical or numerical models. In the road...     

Thermal fluctuations in pinned elastic systems: field theory of rare events and droplets

Using the functional renormalization group (FRG) we study the thermal fluctuations of elastic objects (displacement field u, internal dimension d) pinned by a random potential at low temperature T, as prototypes for glasses. A challenge is how the field theory can describe both typical (minimum energy T = 0) configurations, as well as thermal averages which, at any non-zero T as in the phenomenological droplet picture, are dominated by rare degeneracies between low lying minima. We show that this occurs through an essentially non-perturbative thermal boundary layer (TBL) in the (running) effective action Γ [u] at T > 0 for which we find a consistent scaling ansatz to all orders. The TBL describes how temperature smoothes the singularities of the T = 0 theory and contains the physics of rare thermal excitations (droplets). The formal structure of this TBL, which involves all cumulants of the coarse grained disorder, is first explored around d = 4 using a one-loop Wilson RG. Next, a more systematic exact RG (ERG) method is employed, and first tested on d = 0 models where it can be pushed quite far. There we obtain precise relations between TBL quantities and droplet probabilities (those are constrained by exact identities which are then checked against recent exact results). Our analysis is then extended to higher d, where we illustrate how the TBL scaling remains consistent to all orders in the ERG and how droplet picture results can be retrieved. Since correlations are determined deep in the TBL (by derivatives of Γ [u] at u = 0), it remains to be understood (in any d) how they can be retrieved (as u = 0⁺ limits in the non-analytic T = 0 effective action), i.e., how to recover a T = 0 critical theory. This formidable “matching problem” is solved in detail for d = 0, N = 1 by studying the (partial) TBL structure of higher cumulants when points are brought together. We thereby obtain the β-function at T = 0, all ambiguities removed, displayed here up to four loops. A discussion of the d > 4 case and an exact solution at large d are also provided.     

Intramolecular hydrogen bonding in 3‐imino‐propenylamine: Theoretical investigations

Intramolecular H‐bonds existing for derivatives of 3‐imino‐propenylamine have been studied using the B3LYP/6‐311++G** level of theory. The nature of these interactions, known as resonance‐assisted hydrogen bonds, has been discussed. Vibrational frequencies for α‐derivatives were calculated at the same level of theory. The topological properties of the electron density distributions for N? H···N intramolecular bridges have been analyzed in terms of the Bader theory of atoms in molecules (AIM). Calculation for 3‐imino‐propenylamine derivatives in water solution were also carried out at B3LYP/6‐311++G** level of theory. Finally, the analysis of hydrogen bond in this molecule and their derivatives by quantum theory of natural bond orbital methods fairly support the ab initio results. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

Preparation and structure of three solvatomorphs of the polymer [Co(dbm)2(4ptz)]n: spin canting depending on the supramolecular organization

The syntheses and X-ray structures of three isomeric 1D coordination polymers are reported. The complex [Co(dbm)2(MeOH)2] (1) was used as a precursor in these reactions. The preparation and structure of 1 is also presented; this mononuclear complex is in the cis configuration because this allows the formation of a network of intermolecular hydrogen bonds in the solid state. Reaction of 1 with 2,4,6-tris-(4-pyridyl)-1,3,5-triazine (4ptz) yields the polymers [Co(dbm)2(4ptz)]n.nTHF (2a), [Co(dbm)2(4ptz)]n.0.75nTHF.0.5nEt2O (2b), and [Co(dbm)2(4ptz)]n.3nDMF (2c) in the form of zigzag chains, instead of the expected honeycomb architectures. This is because of the establishment of extended pi-pi stacking throughout these solids, which could not have occurred otherwise. Compounds 2a, 2b, and 2c are solvatomorphs, and formation of either one of them depends on the exact conditions of crystallization, which lead to significant differences in the supramolecular organization of the chains. Bulk magnetic measurements on 2a reveal weak antiferromagnetic exchange within the chains and small ordering throughout the solid that results in the manifestation of the phenomenon of spin canting, whereas for 2c the different supramolecular organization causes the antiferromagnetic exchange not to result in spin canting.     

Surfactant control of gas uptake: effect of butanol films on HCl and HBr entry into supercooled sulfuric acid

The entry of HCl into 60-68 wt % D(2)SO(4) and HBr into 68 wt % acid containing 0-0.18 M 1-butanol was monitored by measuring the fractions of impinging HCl and HBr molecules that desorb as DCl and DBr after undergoing H --> D exchange within the deuterated acid. The addition of 0.18 M butanol to the acid creates butyl films that reach approximately 80% surface coverage at 213 K. Surprisingly, this butyl film does not impede exchange but instead enhances it: the HCl --> DCl exchange fractions increase from 0.52 to 0.74 for 60 wt % D(2)SO(4) and from 0.14 to 0.27 for 68 wt % D(2)SO(4). HBr --> DBr exchange increases even more sharply, rising from 0.22 to 0.65 for 68 wt % D(2)SO(4). We demonstrate that this enhanced exchange corresponds to enhanced uptake into the butyl-coated acid for HBr and infer this equivalence for HCl. In contrast, the entry probability of the basic molecule CF(3)CH(2)OH exceeds 0.85 at all acid concentrations and is only slightly diminished by the butyl film. The OD groups of surface butanol molecules may assist entry by providing extra interfacial protonation sites for HCl and HBr dissociation. The experiments suggest that short-chain surfactants in sulfuric acid aerosols do not hinder heterogeneous reactions of HCl or HBr with other solute species.