Graphical and computational methods for determining the stability constants of mono- and polynuclear complexes with a common intersection point of the family of formation curves

Aqueous polynuclear systems have been analyzed, for which the family of formation curves intersects at a common point. The analyzed graphical and computational method for determining the stability constants can be used as initial values within the iterative calculation process. In some cases, the stability constants are calculated using only the coordinates of the common intersection point. The obtained equations could be of special interest when the experimental data can be interpreted in several models. In these cases, given the large volume of experimental data, the calculation is simple and the model can certainly be chosen with high safety. The obtained equations may also be applied for critical evaluation of tabular data if the coordinates of the intersection point are known. A series of real polynuclear systems have been analyzed and useful conclusions have been made.

Lie Group Based Type Synthesis Using Transformation Configuration Space for Reconfigurable Parallel Mechanisms With Bifurcation Between Spherical Motion and Planar Motion

This paper investigates novel reconfigurable parallel mechanisms with bifurcation between planar subgroup SE(2) and rotation subgroup SO(3) based on a transformation configuration space. Having recollected necessary theoretical fundamentals with regard to compositional submanifolds and kinematic bonds, the motion representation of the platform of reconfigurable parallel mechanisms are investigated. The transformation configuration space of a reconfigurable parallel mechanism with motion branches is proposed with respect to the fact that the intersection of Lie subgroups or submanifolds is the identity element or a non-identity element. The switch conditions of the transformation configuration space are discussed, leading to establishment of a theoretical foundation for realizing a switch of motion branches. The switch principle of reconfigurable parallel mechanisms is further investigated with respect to the common motion between SE(2) parallel-mechanism motion generators and SO(3) parallel-mechanism motion generators. Under this principle, the subchains with common motion generators are synthesized and divided into two types of generators. The first type of generators generates kinematic chains with a common intersection of three joint axes, and the second type of generators generates a common intersection of two joint axes. Following this, two types of reconfigurable parallel mechanisms with three identical subchains are synthesized, resulting in 11 varieties in which platforms can be switched between SE(2) and SO(3) after passing through the singularity configuration space.

Dynamis eis Soterian

This study shows that the syntagma δύναµις εἰς σωτηρίαν was widely used in ancient Greek literature of the Classical, Hellenistic, and Greco-Roman periods. A semantic context analysis reveals that “danger” is the common intersection of all contexts in which the syntagma δύναµις εἰς σωτηρίαν occurs. In a modified way it also appears in texts of the New Testament (Rom 1:16; 1 Pet 1:5): By using δύναµις (θεοῦ) εἰς σωτηρίαν in a context focused on danger, Paul (as well as the author of 1 Peter) indicates that his use of the syntagma is consistent with the pagan, non-biblical use of δύναµις εἰς σωτηρίαν.

Explainable Certain Answers

When a dataset is not fully specified and can represent many possible worlds, one commonly answers queries by computing certain answers to them. A natural way of defining certainty is to say that an answer is certain if it is consistent with query answers in all possible worlds, and is furthermore the most informative answer with this property. However, the existence and complexity of such answers is not yet well understood even for relational databases. Thus in applications one tends to use different notions, essentially the intersection of query answers in possible worlds. However, justification of such notions has long been questioned. This leads to two problems: are certain answers based on informativeness feasible in applications? and can a clean justification be provided for intersection-based notions? Our goal is to answer both. For the former, we show that such answers may not exist, or be very large, even in simple cases of querying incomplete data. For the latter, we add the concept of explanations to the notion of informativeness: it shows not only that one object is more informative than the other, but also says why this is so. This leads to a modified notion of certainty: explainable certain answers. We present a general framework for reasoning about them, and show that for open and closed world relational databases, they are precisely the common intersection-based notions of certainty.

Multicolour Sunflowers

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersectionCof all of them, and |C| is smaller than each of the sets. A longstanding conjecture due to Erdős and Szemerédi (solved recently in [7, 9]; see also [22]) was that the maximum size of a family of subsets of [n] that contains no sunflower of fixed sizek> 2 is exponentially smaller than 2nasn→ ∞. We consider the problems of determining the maximum sum and product ofkfamilies of subsets of [n] that contain no sunflower of sizekwith one set from each family. For the sum, we prove that the maximum is$$(k-1)2^n+1+\sum_{s=0}^{k-2}\binom{n}{s}$$for alln⩾k⩾ 3, and for thek= 3 case of the product, we prove that the maximum is$$\biggl(\ffrac{1}{8}+o(1)\biggr)2^{3n}.$$We conjecture that for all fixedk⩾ 3, the maximum product is (1/8+o(1))2kn.

An Accurate Method for Determining Cutter-Workpiece Engagements in Five-Axis Milling With a General Tool Considering Cutter Runout

Cutter runout is universal and inevitable in milling process and has a direct impact on the shape of the in-process geometry. However, most of the works on the cutter-workpiece engagement (CWE) extraction neglect the cutter runout impact, which will result in a loss of precision. In this paper, an accurate method is presented to obtain CWE boundaries in five-axis milling with a general tool integrating the cutter runout impact. First, each flute's rotary surface is analytically derived. Then, by intersecting the section circle corresponding to the current flute with each of the rotary surface formed by previous flutes, a set of candidate feasible contact arcs (CFCAs) are obtained, and the valid feasible contact arc (VFCA) is defined as the common intersection of these CFCAs. Next, by intersecting the VFCA with the workpiece surfaces, the partial arc which locates inside the workpiece volume is extracted as the engagement arc. Finally, the CWE map is plotted by mapping a set of engagement arcs to a 2D space. To validate the proposed method, the CWE maps with/without integrating the cutter runout impact in five-axis milling of an axial compressor blisk are extracted and compared. The results reveal that the shape of CWE boundaries is changed a lot owing to the cutter runout impact. A cutting force comparison experiment has been carried out to show that the proposed method will lead to higher prediction accuracy especially in the finish milling process with low immersion angle.

A NOVEL METHOD FOR THE DETERMINATION OF THE POINT OF ZERO NET PROTON CHARGE OF COLLOIDAL KAOLINITE IN AQUEOUS SOLUTIONS

In this work, a novel method to determine the point of zero net proton charge (PZNPC) of colloidal kaolinite in aqueous solutions was presented through the measurement of zeta potential of the particles in various concentrations of NaCl solution and the mathematic regression of the zeta potential versus pH curves. This method is based on the observation that there are two common intersection points (CIP) in the two curves of zeta potential versus pH of colloidal kaolinite with a low and a high NaCl concentration. The observation might be attributed to that NaCl would compress the electric double layer of kaolinite particles and affect the efficiency of the protonation/deprotonation of the HD-faces. The experimental result showed that the PZNPC of kaolinite is 5.37.

Workspace Analysis and Optimization of 3-RRR Planar Parallel Manipulator

Predicting shape and finding out exact area of the workspace of 3-DOF planar parallel manipulator having three rotary joints has been a research topic for long time. Optimal design based on such calculations provides a conclusive design methodology. In this work, geometrical method is used o get the exact shape of the workspace and expression of area for the shape obtained. The 3-DOF parallel manipulator is treated as a combination of three independent serial manipulators and its workspace as the common intersection area of these three serial manipulators. The three workspaces are then translated by length equal to radius of the end effecter and the common intersection area is obtained as the workspace of the parallel manipulator under study. The area is determined using the mathematical expression for the shape obtained. Based on direct search algorithm, a methodology is developed to optimize the area. Thus, following, this methodology, an user with specification of area of the workspace can be provided with an optimized parallel manipulator taking into account the inventory of link lengths and the cost of the links.