Ideals and solutions of nonlinear field equations

Families of horizontal ideals of contact manifolds of finite order are studied. Each horizontal ideal is shown to admit ann-dimensional module of Cauchy characteristic vectors that is also a module of annihilators (in the sense of Cartan) of the contact ideal. Since horizontal ideals are generated by 1-forms, any completely integrable horizontal ideal in the family leads to a foliation of the contact manifold by submanifolds of dimensionn on which the horizontal ideal vanishes. Explicit conditions are obtained under which an open subset of a leaf of this foliation is the graph of a solution map of the fundamental ideal that characterizes a given system of partial differential equations of finite order withn independent variables. The solution maps are obtained by sequential integration of systems of autonomous ordinary differential equations that are determined by the Cauchy characteristic vector fields for the problem. We show that every smooth solution map can be obtained in this manner. Let {_Vi¦1in} be a basis for the module of Cauchy characteristic vector fields that are in Jacobi normal form. If a subsidiary balance ideal admits each of then vector fieldsV _i as a smooth isovector field, then certain leaves of the foliation generated by the corresponding closed horizontal ideal are shown to be graphs of solution maps of the fundamental ideal. A subclass of these constructions agree with those of the Cartan-Kähler theorem. Conditions are also obtained under which every leaf of the foliation is the graph of a solution map. Solving a given system ofr partial differential equations withn independent variables on a first-order contact manifold is shown to be equivalent to the problem of constructing a complete system of independent first integrals. Properties of systems of first integrals are analyzed by studying the collection ISO[A _ij ] of all isovectors of the horizontal ideal. We show that ISO[A _ij ] admits the direct sum decomposition ^*[A _ij ]W[A _ij ] as a vector space, where ^*[A _ij ] is the module of Cauchy characteristics of the horizontal ideal. ISO[A _ij ] also forms a Lie algebra under the standard Lie product,^*[A _ij ] andW[A _ij ] are Lie subalgebras of ISO[A _ij ], and [A _ij ] is an ideal. A change of coordinates that resolves (straightens out) the canonical basis for ^*[A _ij ] is constructed. This change of coordinates is used to reduce the problem of solving the given system of PDE to the problem of root extraction of a system ofr functions ofn variables, and to establish the existence of solutions to a second-order system of overdetermined PDE that generate the subspaceW[A _ij ]. Similar results are obtained for second-order contact manifolds. Extended canonical transformations are studied. They are shown to provide algorithms for calculating large classes of closed horizontal ideals and a partial analog of classical Hamilton-Jacobi theory.     

Generalized logic and the representation of rings

Following the progression towards weaker logics, a number of authors have considered the notion of a sheaf over a quantale or, equivalently, a quantale valued set. In this paper, we use ideas from enriched category theory to motivate the definition of a quantic sheaf. Given a localic subquantale of Q, a quantic sheaf over Q gives a sheaf in the usual sense. As an application, we derive a series of sheaf representations for commutative rings including the familiar Pierce representation.     

Novel alkyl-modified anionic siloxanes as pseudostationary phases for electrokinetic chromatography. III. Performance in organic-modified buffers

Anionic water-soluble siloxanes modified with different amounts of alkyl chains have been used as pseudostationary phases in electrokinetic chromatography. Ionic siloxane polymers with attached alkyl chains of C8 and C12 and having different alkyl chain densities have been employed previously to achieve selective and efficient separations with a range of electrophoretic mobilities and methylene selectivities. In this study, the performance of three alkyl-modified siloxanes is examined in different organic-modified buffers and at differing amounts of organic modifier. The organic modifiers used are acetonitrile and methanol. The siloxanes are stable in these organic solvents and show good mobility and good methylene selectivities even at high concentration of organic solvent. Siloxanes have also been used to separate a mixture of 14 polynuclear aromatic hydrocarbons in an acetonitrile-modified buffer.     

Antibacterial Activity of Pharmaceutical-Grade Rose Bengal: An Application of a Synthetic Dye in Antibacterial Therapies

Rose bengal has been used in the diagnosis of ophthalmic disorders and liver function, and has been studied for the treatment of solid tumor cancers. To date, the antibacterial activity of rose bengal has been sporadically reported; however, these data have been generated with a commercial grade of rose bengal, which contains major uncontrolled impurities generated by the manufacturing process (80–95% dye content). A high-purity form of rose bengal formulation (HP-RBf, >99.5% dye content) kills a battery of Gram-positive bacteria, including drug-resistant strains at low concentrations (0.01–3.13 μg/mL) under fluorescent, LED, and natural light in a few minutes. Significantly, HP-RBf effectively eradicates Gram-positive bacterial biofilms. The frequency that Gram-positive bacteria spontaneously developed resistance to HP-RB is extremely low (less than 1 × 10⁻¹³). Toxicity data obtained through our research programs indicate that HP-RB is feasible as an anti-infective drug for the treatment of skin and soft tissue infections (SSTIs) involving multidrug-resistant (MDR) microbial invasion of the skin, and for eradicating biofilms. This article summarizes the antibacterial activity of pharmaceutical-grade rose bengal, HP-RB, against Gram-positive bacteria, its cytotoxicity against skin cells under illumination conditions, and mechanistic insights into rose bengal’s bactericidal activity under dark conditions.     

A high-throughput search for direct methanol fuel cell anode electrocatalysts of type PtxBiyPbz

We used a high-throughput method to screen for direct methanol fuel cell anode electrocatalysts in the Pt-Bi-Pb system. Previous studies showed that PtBi and PtPb (both NiAs structure type) were active electrocatalysts for the oxidation of formic acid, but only PtPb was active in oxidizing methanol. We synthesized thin films with continuous composition spreads of the three elements by magnetron sputtering at deposition temperatures from ambient to 510 °C. A fluorescence method was then used to identify compositions that were active toward methanol oxidation. Only films deposited between temperatures of 160 and 400 °C showed electrocatalytic activity. The areas that were active for methanol oxidation showed predominantly the NiAs structure type according to XRD, with optimal activity for compositions near PtBi_0.01Pb_0.53.     

Field transfer decay and interferometry in coupled waveguides with stochastic sources

Coupled waveguides, such as those used as optical filters, switches, or couplers, are investigated in the context of coherence theory. A stochastic source is introduced to the standard coupled-mode model. Decay in the power transferred between the waveguides as a function of distance is predicted. Application of this result in the form of an interferometric spectrometer is explored.     

Quaternionic theta constants

We investigate the six quaternionic theta constants introduced by Freitag and Hermann. More precisely we investigate their restrictions to the Hermitian resp. Siegel half-space of degree 2. It turns out that these theta constants generate the graded ring of symmetric Hermitian modular forms for the principal congruence subgroup of level 1 + i over the Gaussian number field resp. of Siegel modular forms for the principal congruence subgroup of level 2 and even weight. As an application we obtain a simple construction of Igusa’s Siegel modular form of degree 2 and weight 30 with respect to the non-trivial character.     

App-based scaffolds for writing two-column proofs

This paper presents apps designed to assist students in understanding and developing proofs in geometric theorems. These technologies focus on triangle congruence, triangle similarity and properties of parallelograms. Focus group discussions and initial testing of the apps revealed that the apps offered a more engaging medium for learning proving and were capable of facilitating proof-writing skills in geometry.     

Analysis of and numerical schemes for a mouse population model in Hantavirus epidemics

This paper considers a non-linear system of ordinary differential equations, which arises in the study of hantavirus epidemics. The system has the property that the total population obeys the logistic equation. For this system, we use linearization and the dynamical properties of the logistic equation to analyze the dynamics of the subpopulation system. In view of the usual numerical instabilities associated with standard finite difference methods used for simulating such systems, we construct non-standard finite difference (NSFD) schemes, which preserve the dynamic properties of the system, and may therefore be used for its simulation.     

Self‐Sufficient Formaldehyde‐to‐Methanol Conversion by Organometallic Formaldehyde Dismutase Mimic

The catalytic networks of methylotrophic organisms, featuring redox enzymes for the activation of one‐carbon moieties, can serve as great inspiration in the development of novel homogeneously catalyzed pathways for the interconversion of C₁ molecules at ambient conditions. An imidazolium‐tagged arene–ruthenium complex was identified as an effective functional mimic of the bacterial formaldehyde dismutase, which provides a new and highly selective route for the conversion of formaldehyde to methanol in absence of any external reducing agents. Moreover, secondary amines are reductively methylated by the organometallic dismutase mimic in a redox self‐sufficient manner with formaldehyde acting both as carbon source and reducing agent.