A Synthesis of Six Recent Studies on Numerosity Abilities in an Ant

Working on the numerosity ability of the ant Myrmica sabuleti, we have already summarized for the readers’ convenience our previous papers in two successive publications. Since that time, we have produced six more papers on the subject, and we thought it was time to present a summary of them. These studies deal with the ants’ ability in expecting the following element in an arithmetic or a geometric sequence, as well as with the required similarity between visual cues and the maximum horizontal and vertical distance between such cues enabling the ants to mentally add them up. The experimental methods that were used in these studies are here only briefly reported and their most important results are concisely related, as the extended information can be found in these six papers here summarized. We present novel tables and figures for illustrating this synthesis.

Ants Can Expect the Size of the Next Element in a Geometric Sequence of Increasing or Decreasing Shapes, Only If This Sequence Is Present

Having shown that the ant Myrmica sabuleti can expect the following number in an arithmetic sequence of increasing or decreasing numbers, we here investigated on their ability in expecting the size of the following element in an increasing or decreasing geometric sequence of shapes, otherwise identical. We found that the ants could anticipatively correctly increment or decrement a geometric sequence when tested in the presence of the learned sequence, but not without seeing the sequence in its learned sequential order. Such a behavior, i.e. perfectly choosing the next element of a sequence when in presence of that sequence but not otherwise, seems appropriate for the use of encountered cues while foraging and returning to the nest.

Rethinking the Proportional Design Principles of Timber-Framed Buddhist Buildings in the Goryeo Era

This study examines how the wooden architecture of the Goryeo Dynasty in Korea evolved in an original way while incorporating Chinese architectural principles. For the Goryeo Era’s timber-framed buildings, eave purlin height was determined according to √2H times the eave column height (H), while the eave column height influenced the proportional location of each purlin, determined by the √2H times decrease rate in the cross-section. Thus, eave column height was proportionately connected to a geometric sequence with a common ratio of √2H. This technical approach, achieved using an L-square ruler and a drawing compass, contributed to determining eave purlin and ridge post placement, bracket system height, and outermost bay width. This study notes that the practical works were consistently preserved in East Asian Buddhist architecture, in that a universal rule of proportion was applied to buildings constructed during the Tang–Song and the Goryeo Dynasties, surmounting differences in local construction methods. These design principles were a vestige of socio-cultural exchange on the East Asian continent and a minimal step toward the establishment of structurally safe framed buildings.

A Multilevel Simulation Method for Time-Variant Reliability Analysis

Crude Monte Carlo simulation (MCS) is the most robust and easily implemented method for performing time-variant reliability analysis (TRA). However, it is inefficient, especially for high reliability problems. This paper aims to present a random simulation method called the multilevel Monte Carlo (MLMC) method for TRA to enhance the computational efficiency of crude MCS while maintaining its accuracy and robustness. The proposed method first discretizes the time interval of interest using a geometric sequence of different timesteps. The cumulative probability of failure associated with the finest level can then be estimated by computing corrections using all levels. To assess the cumulative probability of failure in a way that minimizes the overall computational complexity, the number of random samples at each level is optimized. Moreover, the correction associated with each level is independently computed using crude MCS. Thereby, the proposed method can achieve the accuracy associated with the finest level at a much lower computational cost than that of crude MCS, and retains the robustness of crude MCS with respect to nonlinearity and dimensions. The effectiveness of the proposed method is validated by numerical examples.

Inequalities approach in determination of convergence of recurrence sequences

The paper proves convergence for three uniquely defined recursive sequences, namely, arithmetico-geometric sequence, the Newton-Raphson recursive sequence, and the nested/composite recursive sequence. The three main hurdles for this prove processes are boundedness, monotonicity, and convergence. Oftentimes, these processes lie in the predominant use of prove by mathematical induction and also require some bit of creativity and inspiration drawn from the convergence monotone theorem. However, these techniques are not adopted here, rather, as a novelty, extensive use of basic manipulation of inequalities and useful equations are applied in illustrating convergence for these sequences. Moreover, we established a mathematical expression for the limit of the nested recurrence sequence in terms of its leading term which yields favorable results.

Generating Function for the Figurative Numbers of Regular Polyhedron

In this paper, we are going to demonstrate a method for determining the generating functions of tetrahedral, hexahedral, octahedral, dodecahedral, and icosahedral figurative numbers. The method is based on the differences between the members of the series of the mentioned figurative numbers, as well as on the previously specified generating functions for the sequence ∑n≥0n+1xn and geometric sequence ∑n≥0xn.