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The digits of Pierce expansion adhere to the law of large numbers. It is known that the Hausdorff dimension of the set of exceptions to the law of large numbers is $1$. We offer an elementary proof of this fact by adapting Jun Wu's method used in the context of Engel expansions., Comment: 15 pages

In his 1994 work, Shallit introduced a rule for determining leap years that generalizes both the historically used Julian calendar and the contemporary Gregorian calendar. This rule depends on a so-called intercalation sequence. According to what we term Shallit's law of leap years, almost every point of the interval $[0,1]$ with respect to Lebesgue measure has the same limsup and liminf, respectively, of a quotient defined in terms of the number of leap years determined by the rule using the Pierce expansion digit sequence as an intercalation sequence. In this paper, we show that the set of exceptions to this law is dense and has full Hausdorff dimension in $[0,1]$, and that the exceptional set intersected with any non-empty open subset of $[0,1]$ has full Hausdorff dimension in $[0,1]$. As a more general result, we establish that for certain subsets of $[0,1]$ concerning the limiting behavior of Pierce expansion digits, intersecting with a non-empty open subset of $[0,1]$ preserves the Hausdorff dimension., Comment: 20 pages; Theorem 1.1(i) revised; typos corrected

The continued fraction mapping maps a number in the interval $[0,1)$ to the sequence of its partial quotients. When restricted to the set of irrationals, which is a subspace of the Euclidean space $\mathbb{R}$, the continued fraction mapping is a homeomorphism onto the product space $\mathbb{N}^{\mathbb{N}}$, where $\mathbb{N}$ is a discrete space. In this short note, we examine the continuity of the continued fraction mapping, addressing both irrational and rational points of the unit interval., Comment: 11 pages

In this paper, we investigate the convergence exponent of Pierce expansion digit sequences. We explore some basic properties of the convergence exponent as a real-valued function defined on the closed unit interval, as well as those of the level sets of the function. Additionally, we further study subsets of the closed unit interval on which the series of positive $s$th powers of the reciprocals of the Pierce expansion digits diverges., Comment: 27 pages, results extended

The digits of Pierce expansion obey the law of large numbers, the central limit theorem, and the law of the iterated logarithm in the Lebesgue measure sense. We calculate the Hausdorff dimensions of the exceptional sets of each of the three laws. We further determine the Hausdorff dimensions of certain sets arising in Pierce expansions., Comment: 46 pages; proof of Lemma 4.6 revised; typos corrected

We introduce the error-sum function of Pierce expansions. Some basic properties of the error-sum function are analyzed. We also examine the fractal property of the graph of it by calculating the Hausdorff dimension, the box-counting dimension, and the covering dimension of the graph., Comment: 31 pages; typos corrected

For multi-limbed robots, motion planning with posture and force constraints tends to be a difficult optimization problem due to nonlinearities, which also present extended solve times. We propose a multi-stage optimization framework with data-driven inter-stage coupling constraints to address the nonlinearity. Both clustering and evolutionary approaches to find the McCormick envelope relaxations are used to find the problem-specific parameters. The learned constraints are then used in the prior stages, which provides advanced knowledge of the following stages. This leads to improved solve times and interpretability of the results. The planner is validated through multiple walking and climbing tasks on a 10 kg hexapod robot.

Similarity index measures the topological proximity of node pairs in a complex network. Numerous similarity indices have been defined and investigated, but the dependency of structure on the performance of similarity indices has not been sufficiently investigated. In this study, we investigated the relationship between the performance of similarity indices and structural properties of a network by employing a two-state random network. A node in a two-state network has binary types that are initially given, and a connection probability is determined from the state of the node pair. The performance of similarity indices affects the number of links and the ratio of intra-connections to inter-connections. Similarity indices have different characteristics depending on their type. Local indices perform well in small-size networks and do not depend on whether the structure is intra-dominant or inter-dominant. In contrast, global indices perform better in large-size networks, and some such indices do not perform well in an inter-dominant structure. We also found that link prediction performance and the performance of similarity are correlated in both model networks and empirical networks. This relationship implies that link prediction performance can be used as an approximation for the performance of the similarity index when metadata for node types are unavailable. This relationship may help to find the appropriate index for given networks., Comment: 17 pages, 8 figures

A linear accelerator, called RAON, has been being developed as a part of Rare Isotope Science Project (RISP) by Institute for Basic Science (IBS) [1]. The linear accelerator utilizes an electron cyclotron resonance (ECR) ion source for providing intense highly charged ion beams to the linear accelerator. 28GHz ECR ion source can extract heavy ion beams from proton to uranium. A superconducting magnet system for 28GHz ECR ion source is composed of hexapole coils and four solenoid coils made with low Tc superconducting wires of NbTi [2]. The electromagnetic force acts on the superconducting magnets due to the magnetic field and flowing current in case of not only normal state but also quench state [3]. In case of quench on hexapole coils, unbalanced flowing current among the hexapole coils is generated and it causes unbalanced electromagnetic force. Coil motions and coil strains in quench state are larger than those in normal state due to unbalanced electromagnetic force among hexapole coils. Therefore, analysis on electromagnetic characteristics of superconducting magnet for 28GHz ECR ion source according to the series resistance of protection circuit in case of quench should be conducted. In this paper, analysis on electromagnetic characteristics of superconducting hexapole coils for 28GHz ECR ion source according to the series resistance of protection circuit in case of quench is performed by using finite elements method (FEM) simulation.

Link prediction is a technique that uses the topological information in a given network to infer the missing links in it. Since past research on link prediction has primarily focused on enhancing performance for given empirical systems, negligible attention has been devoted to link prediction with regard to network models. In this paper, we thus apply link prediction to two network models: The Watts-Strogatz (WS) model and Barab\'asi-Albert (BA) model. We attempt to gain a better understanding of the relation between accuracy and each network parameter (mean degree, the number of nodes and the rewiring probability in the WS model) through network models. Six similarity indices are used, with precision and area under the ROC curve (AUC) value as the accuracy metrics. We observe a positive correlation between mean degree and accuracy, and size independence of the AUC value.

The role of the cloud-radiation interaction in the simulation of the Madden-Julian oscillation (MJO) is investigated. A special focus is on the enhancement of column-integrated diabatic heating due to the greenhouse effects of clouds and moisture in the region of anomalous convection. The degree of this enhancement, the greenhouse enhancement factor (GEF), is measured at different precipitation anomaly regimes as the negative ratio of anomalous outgoing longwave radiation to anomalous precipitation. Observations show that the GEF varies significantly with precipitation anomaly and with the MJO cycle. The greenhouse enhancement is greater in weak precipitation anomaly regimes and its effectiveness decreases monotonically with increasing precipitation anomaly. The GEF also amplifies locally when convection is strengthened in association with the MJO, especially in the weak precipitation anomaly regime (less than 5 mm day(exp -1)). A robust statistical relationship is found among CMIP5 climate model simulations between the GEF and the MJO simulation fidelity. Models that simulate a stronger MJO also simulate a greater GEF, especially in the weak precipitation anomaly regime (less than 5 mm day(exp -1)). Models with a greater GEF in the strong precipitation anomaly regime (greater than 30 mm day(-1)) represent a slightly slower MJO propagation speed. Many models that lack the MJO underestimate the GEF in general and in particular in the weak precipitation anomaly regime. The results herein highlight that the cloud-radiation interaction is a crucial process for climate models to correctly represent the MJO.