The Axiom of Choice and the Partition Principle from Dialectica Categories

The method of morphisms is a well-known application of Dialectica categories to set theory (more precisely, to the theory of cardinal invariants of the continuum). In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents (within $\textbf{ZF}$) of either the Axiom of Choice ($\textbf{AC}$) or Partition Principle ($\textbf{PP}$)—which is a consequence of $\textbf{AC}$ whose precise status of its relationship with$\textbf{AC}$ itself is an open problem for more than a hundred years.

Design Curve Construction Based on Tolerance Limit Concept

Design curves, such as fatigue design S-N curves, are usually constructed by analyzing test data, which often exhibit large scatter. There are several methods available to construct a design curve and some of these methods, with varying degrees of conservativeness, accuracy, and simplicity, have been adopted by engineering standards, codes and guidelines, such as the American Society of Mechanical Engineers (ASME) Code. However, to meet increasing engineering demands, a simplified and user-friendly engineering method with rigorous mathematical and physical basis is still urgently needed to accurately manage the margin of safety and decrease the cost. In this paper, the current engineering practices for constructing a design curve are briefly reviewed, followed by the introduction of the tolerance limit concept because of its ability to relate the design curve well to sample size, failure probability, and confidence level. Recognizing the physical unsoundness of the hyperbolic shape of the design curves constructed with the Owen's tolerance limit approach, a new simple design curve construction method is developed based on the “equal partition principle.” Finally, the predicted results from various methods are compared and the advantage of the new method is demonstrated with several worked examples.

Study on a Nearest Neighbor Algorithm for Point Clouds Based on the Space Partition Principle

In the product modeling based on the reverse engineering, the point cloud data smoothing and multi-view point cloud data registration will be related to search some nearest neighbor points. The search speed will determine the efficiency of product modeling in some cases. The paper analysis the nearest neighbor point query algorithms, KD tree and Range tree, based on the space partition principle. The tree structure creation and query method are described by pseudo-code in the paper. Finally, the experimental results involving different sizes point clouds demonstrate that KD tree and Range tree have their own advantages in space storage and time complexity. Two data strictures all meet the efficiency of the search algorithm.

Design Curve Construction Based on Tolerance Limit Concept

The design curves, such as fatigue design S-N curves, required in engineering designs, are usually constructed by analyzing test data, which often exhibit large scatter. There are many methods available to construct a design curve and many of these methods, with varying degrees of conservativeness, accuracy, and simplicity, have been adopted by engineering standards, codes and guidelines, such as the ASME and ASTM codes and standards. However, to meet the increasing engineering demands, a simplified, user friendly engineering method with rigorous mathematical and physical basis is still urgently needed to accurately manage the margin of safety on one hand and decrease the cost on the other hand. In this paper the current engineering practices for constructing a design curve is reviewed. It is followed by the tolerance limit concept for general regression cases, because of its capability to relate the design curve to sample size, probability, and confidence level. A simple approximate solution is derived for Owen’s tolerance limit approach, which previously could be solved only with a very complex procedure. Finally, recognizing the physical unsoundness of the hyperbolic shape of the design curves constructed with the Owen’s tolerance limit approach, a new simple design curve construction method is proposed based on the “equal partition principle”. The predicted results from the new method are compared with that of other methods and the advantage of the proposed procedure over other methods is demonstrated with several worked examples. Linear design curve construction with heteroscedastic characteristics (variable variance) and nonlinear design curve are also discussed.

STUDIES ON REACTIONS RELATING TO CARBOHYDRATES AND POLYSACCHARIDES: XLIII. CYCLIC ACETAL AND KETAL FORMATION FROM α-PHENYL GLYCEROL AS FURTHER EXAMPLES OF THE "RING PARTITION PRINCIPLE"

The work represents an extension of the "Hibbert-Michael ring partition theory" to the interaction of α-phenyl glycerol with acetone and p-nitrobenzaldehyde respectively. The α-phenyl glycerol was prepared in the form of a pure crystalline product from cinnamyl alcohol and on condensation with acetone yielded the two expected, isomeric, five-membered (dioxolane) ring compounds. These two isopropylidene derivatives were isolated and their structures definitely determined by the usual hydrolysis and methylation technique. In this manner the corresponding α- and γ-methyl ethers of α-phenyl glycerol were isolated, the identity of which had been previously settled by direct synthesis employing well-established reactions.The β-methyl ether of α-phenyl glycerol was obtained by methylating crystalline 1:3 p-nitrobenzylidene α-phenyl glycerol and then hydrolyzing the ether.Condensation of p-nitrobenzaldehyde with α-phenyl glycerol should yield, according to the "ring partition principle", one six- and two five-membered cyclic acetals.The crystalline isomer separating from the crude condensation product was shown to be the six-membered cyclic acetal. Removal of this left a viscous oil containing the five-membered acetals which, on methylation, and subsequent hydrolysis, yielded a small amount of the γ-methyl ether of α-phenyl glycerol, thus indicating the presence of some 1:2 p-nitrobenzylidene glycerol in the original reaction product. Due to the large number of theoretically possible five-membered rings it was not found possible to isolate, or prove the presence of, both five-membered acetals in the oil left after removal of the crystalline six-membered acetal. Presumably both of the structural five-membered acetals were formed, but due to their mutual solubility relations it was not possible to bring about a separation of crystalline forms.