Encoding many-valued logic in {\lambda}-calculus

We extend the well-known Church encoding of two-valued Boolean Logic in $\lambda$-calculus to encodings of $n$-valued propositional logic (for $3\leq n\leq 5$) in well-chosen infinitary extensions in $\lambda$-calculus. In case of three-valued logic we use the infinitary extension of the finite $\lambda$-calculus in which all terms have their B\"ohm tree as their unique normal form. We refine this construction for $n\in\{4,5\}$. These $n$-valued logics are all variants of McCarthy's left-sequential, three-valued propositional calculus. The four- and five-valued logic have been given complete axiomatisations by Bergstra and Van de Pol. The encodings of these $n$-valued logics are of interest because they can be used to calculate the truth values of infinitary propositions. With a novel application of McCarthy's three-valued logic we can now resolve Russell's paradox. Since B\"ohm trees are always finite in Church's original $\lambda{\mathbf I}$-calculus, we believe their construction to be within the technical means of Church. Arguably he could have found this encoding of three-valued logic and used it to resolve Russell's paradox.Comment: 15 page

Impact of Controlling the Site Distribution of Al Atoms on Catalytic Properties in Ferrierite-Type Zeolites

Zeolites with the ferrierite (FER) topology are synthesized using a combination of tetramethylammonium (TMA) cations with differently sized cyclic amines (pyrrolidine (Pyr), hexamethyleneimine (HMI), and 1,4- diazabicyclo[2.2.2]octane (DAB)). Using these organic structure-directing agents (SDAs), low Si/Al ratios and concentrated synthesis mixtures favor the crystallization of FER materials. Increasing the size of the cyclic amine or decreasing the aluminum content leads to the crystallization of other phases or the creation of excessive amounts of connectivity defects. TMA cations play a decisive role in the synthesis of the FER materials, and their presence allows the use of HMI to synthesize FER. Proton MAS NMR is used to quantify the accessibility of pyridine to acid sites in these FER samples, where it is found that the FER + HMI + TMA sample contains only 27% acid sites in the 8-MR channels, whereas FER + Pyr and FER + Pyr + TMA contain 89% and 84%, respectively. The constraint index (CI) test and the carbonylation of dimethyl ether (DME) with carbon monoxide are used as probe reactions to evaluate how changes in the aluminum distribution in these FER samples affect their catalytic behavior. Results show that the use of Pyr as an SDA results in the selective population of acid sites in the 8-MR channels, whereas the use of HMI generates FER zeolites with an increased concentration of acid sites in the 10-MR channels

The Magnus expansion and some of its applications

Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods, a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature.Comment: Report on the Magnus expansion for differential equations and its applications to several physical problem

Invertible Ideals and Gaussian Semirings

In the first section of this paper, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Pr\"{u}fer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Pr\"{u}fer semirings in terms of some identities over its ideals such as $(I + J)(I \cap J) = IJ$ for all ideals $I$, $J$ of $S$. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Pr\"{u}fer and Gaussian semirings are equivalent. At last we end this paper by giving a plenty of examples of proper Gaussian and Pr\"{u}fer semirings.Comment: Final versio

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An In Vitro Comparison of the Rake Angles Between K3 and ProFile Endodontic File Systems

The purpose of this study was to compare rake angles of the ProFile and K3 file systems. Twenty-five 40/0.06 taper files were obtained for each system. Five files from the same manufacturer were placed perpendicularly into a vial of Epoxicure Resin and left to set for 24 h. The set-ups were removed from the vials and each were sectioned 5 mm from the tip of the files and polished. A photomicrograph was taken of each file with 100× magnification. Five sets of ProFile and five sets of K3 files were processed in this manner. Images were captured digitally, and rake angles of each file were measured. Multivariate ANOVA found a significant difference (p \u3c 0.001) among the three negative rake angles of the ProFile system compared with the K3 system