Measure functions for frames

This paper addresses the natural question: “How should frames be compared?” We answer this question by quantifying the overcompleteness of all frames with the same index set. We introduce the concept of a frame measure function: a function which maps each frame to a continuous function. The comparison of these functions induces an equivalence and partial order that allows for a meaningful comparison of frames indexed by the same set. We define the ultrafilter measure function, an explicit frame measure function that we show is contained both algebraically and topologically inside all frame measure functions. We explore additional properties of frame measure functions, showing that they are additive on a large class of supersets—those that come from so called non-expansive frames. We apply our results to the Gabor setting, computing the frame measure function of Gabor frames and establishing a new result about supersets of Gabor frames.     

Category with a natural cone

Generally, in homotopy theory a cylinder object (or, its dual, a path object) is used to define homotopy between morphisms, and a cone object is used to build exact sequences of homotopy groups. Here, an axiomatic theory based on a cone functor is given. Suspension objects are associated to based objects and cofibrations, obtaining homotopy groups referred to an object and relative to a cofibration, respectively. Exact sequences of these groups are built. Algebraic and particular examples are given. We point out that the main results of this paper were already stated in [3], and the purpose of this article is to give full details of the foregoing.     

等滴光度法测定钼铁中的钼

根据ΔAp=KΔ（bc)p原理，采用“等滴光度法”测定钼铁中的钼，通过试验确定了最佳测定条件。该方法简便、快速，用于钼铁标准样品中钼的测定，测定结果的精密度为0.196%，与标准值之间的相对误差为0.064%（n=4)。     

Preparation and characterization of fused-silica capillary columns coated with <Emphasis Type="Italic">m</Emphasis>-carborane–siloxane copolymers for gas chromatography

The carborane–siloxane copolymers Dexsil 300, a 34.5% bis(dimethylsilyl)-m-carborane–65.5% dimethylsiloxane copolymer, and Dexsil 400, a 24.9% bis(dimethylsilyl)-m-carborane–50.8% dimethyl, 24.3% methylphenylsiloxane copolymer, were coated on fused silica capillary columns and their gas chromatographic properties were evaluated. Their selectivity was evaluated using both Rohrschneider–McReynolds constants and triacylglycerol indices. The bis(dimethylsilyl)-m-carborane unit turned out to be equivalent to two dimethylsiloxy units and one half of a diphenylsiloxy unit. The m-carborane unit was found to cause a 15–25 K shift in the elution temperature between 120 and 360 °C. The working range was from 20 and 0 °C to 380 °C for Dexsil 300 and Dexsil 400, respectively. The column bleeding levels at 380 °C were below 20 and 15 pA for Dexsil 300 and Dexsil 400, respectively.     

Development of a method for estimating an accurate equivalence point in nickel titration of cyanide ions

In a nickel titration of cyanide ions using murexide as indicator, an accurate equivalence point was determined by a non-linear least-squares curve-fitting for a titration curve. This method was developed to establish a standard solution for cyanide ions. In a curve-fitting procedure, a theoretical titration curve was calculated, assuming that nickel ion formed only a 1:4 Ni²⁺:CN⁻ complex with cyanide ions and formed only a 1:1 complex with murexide. Results of the curve-fitting were reasonable at any pH and any indicator concentration studied. The combined standard uncertainty for a concentration of a 1000 mg kg⁻¹ cyanide solution by this method was 0.079%.     

Neutrino Oscillations in Exotic Geometries and the Equivalence Principle Violation

Taking into account the violation of the equivalence principle, we analyze the propagation of neutrinos in the gravitational field generated by wormholes. The analysis is performed for massive and massless neutrinos. By using the constraints on the (negative) mass of wormholes, which is of the order of stellar-substellar masses, we determine constraints on f which characterizes the degree of violation of the equivalence principle.     

Hochschild and ordinary cohomology rings of small categories

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH^∗(kC)→H^∗(|C|,k) and prove it is split surjective, using the factorization category of Quillen [D. Quillen, Higher algebraic K-theory I, in: Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin, 1973, pp. 85-147] and certain techniques from functor cohomology theory. This generalizes the well-known theorems for groups and posets. Based on this result, we construct a seven-dimensional category algebra whose Hochschild cohomology ring modulo nilpotents is not finitely generated, disproving a conjecture of Snashall and Solberg [N. Snashall, Ø. Solberg, Support varieties and Hochschild cohomology rings, Proc. London Math. Soc. 88 (3) (2004) 705-732].     

George Boole and the origins of invariant theory

Historians have repeatedly asserted that invariant theory was born in two papers of George Boole (1841 and 1842). Although several themes and techniques of 19th-century invariant theory are enunciated in this work, in reacting to it (and thereby founding the British school of invariant theory), Arthur Cayley shifted Boole's research program.     

Linearization of vector fields and embedding of diffeomorphisms in flows via Nash–Moser theorem

In the first part of this paper we give suitable spectral properties of the adjoint operators induced by appropriate perturbations of some hyperbolic linear vector fields. These properties are useful to prove general facts based on the Nash–Moser inverse function theorem. In the second part of this work we study circumstances where a global linearization of a vector field X$X$ in a real numerical space is feasible and where some diffeomorphisms which are close to exp(X)$exp\left(X\right)$ can be embedded in a flow.     

Symmetry and transformation properties of iterative ordinary differential equation

Symmetries of linear iterative equations and new conditions on the infinitesimals are obtained. Regarding the expressions of the solutions in terms of the parameters of the source equation, an ansatz is made on the original parameters. We have also obtained an expression for the source parameters of the transformed equation under equivalence transformations. We conducted this work with a special emphasis on second-, third- and fourth-order equations, although some of our results are valid for equations of a general order.