Public Articles
The question of proximity. Demographic ageing places the 15-minute-city theory under stress
Chromosome-level genome assembly of a triploid poplar Populus alba ’Berolinensis’
and 17 collaborators
A 3D View of Orion: I. Barnard’s Loop
and 12 collaborators
Barnard’s Loop is a famous arc of H\(\alpha\) emission located in the Orion star-forming region. Here, we provide evidence of a possible formation mechanism for Barnard’s Loop and compare our results with recent work suggesting a major feedback event occurred in the region around 6 Myr ago. We present a 3D model of the large-scale Orion region, indicating coherent, radial, 3D expansion of the OBP-Near/Briceño-1 (OBP-B1) cluster in the middle of a large dust cavity. The large-scale gas in the region also appears to be expanding from a central point, originally proposed to be Orion X. OBP-B1 appears to serve as another possible center, and we evaluate whether Orion X or OBP-B1 is more likely to be the cause of the expansion. Recent 3D dust maps are used to characterize the 3D topology of the entire region, which shows Barnard’s Loop’s correspondence with a large dust cavity around the OPB-B1 cluster. The molecular clouds Orion A, Orion B, and Orion \(\lambda\) reside on the shell of this cavity. Simple estimates of gravitational effects from both stars and gas indicate that the expansion of this asymmetric cavity likely induced anisotropy in the kinematics of OBP-B1. We conclude that feedback from OBP-B1 has affected the structure of the Orion A, Orion B, and Orion \(\lambda\) molecular clouds and may have played a major role in the formation of Barnard’s Loop.
Preserving tracer correlations in atmospheric transport models
and 2 collaborators
American Sociological Review
Kazuo Ishiguro and “Godi Media”: A Reading of his Select Novels and the Post-2014 Indian Media
Integral transforms of the Hilfer-type fractional derivatives
and 2 collaborators
In this paper, some important properties concerning the Hilfer-type fractional derivatives are discussed. Integral transforms for these operators are derived as particular cases of the Jafari transform. These integral transforms are used to derive a fractional version of the fundamental theorem of calculus.
Keywords: Integral transforms, Jafari transform, κ-gamma function, κ-beta function, κ-Hilfer fractional derivative, κ-Riesz fractional derivative, κ-fractional operators, (κ, ρ)-Hilfer fractional derivative.
Predictive Learning of Error Recovery with a Sensorised Passivity-based Soft Anthropomorphic Hand
and 2 collaborators
Manipulation strategies based on the passive dynamics of soft-bodied interactions provide robust performances with limited sensory information. They utilise the kinematic structure and passive dynamics of the body to adapt to objects of varying shapes and properties. However, these soft passive interactions make the state of the robotic device influenced by the environment, making control generation and state estimation difficult. This work presents a closed-loop framework for dynamic interaction-based grasping that relies on two novelties: (i) a wrist-driven passive soft anthropomorphic hand that can generate robust grasp strategies using one-step kinaesthetic teaching and (ii) a learning-based perception system that uses temporal data from sparse tactile sensors to predict and adapt to failures before it happens. With the anthropomorphic soft design and wrist-driven control, we show that controllers can be generated robust to novel objects and location uncertainty. With the learning-based high-level perception system and 32 sensing receptors, we show that failures can be predicted in advance, further improving the robustness of the entire system by more than doubling the grasping success rate. From over 1000 real-world grasping trials, both the control and perception framework are also seen to be transferable to novel objects and conditions.
Corresponding author(s) Email: [email protected]
LEADS+ Developmental Model: Proposing a new model based on an integrative conceptual review
and 4 collaborators
淺談羅必達法則 (L’Hôpital’s Rule)
相信大家都還記得剛開始學極限時,其中一個經典的題型就是$\displaystyle \lim_{x \to 1} \frac{x^2 - 1}{x - 1}$這種被稱為不定型(Indeterminate Form)的極限問題。當時我們的做法很簡單,就是把公因式消掉, $$ \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = \lim_{x \to 1} \frac{(x + 1)(x - 1)}{x - 1} = \lim_{x \to 1} (x + 1) = 2 $$ 但是,如果今天我們遇到像是$\displaystyle \lim_{x \to 1} \frac{\ln x}{x-1}$這種問題時,以前的做法就不管用了。此時,如果我們求救於Google大神,大神就會告訴我們:「使用L’ôpital Rule吧!」
什麼叫做不定型極限問題?簡單來說,就是這種直接把x的極限值代入原始方程式之後,會產生$\dfrac{0}{0}$或是$\dfrac{\infty}{\infty}$的問題,一律稱之為不定型的極限問題。
從小學三、四年級開始,我們就知道$\dfrac{0}{0}$是沒有定義的。不定型問題之所以麻煩,倒不是因為$\dfrac{0}{0}$沒有定義,而在於這種問題各種結論—極限不存在、極限存在且為定值,以及極限是無窮大—都有可能發生。(這裡我們將無窮大這種不收斂的極限獨立出來討論。)底下我們來看一個簡單的例子。
Find (a) $\displaystyle \lim_{x \to 0} \frac{x^2}{x}$ (b) $\displaystyle \lim_{x \to 0} \frac{x}{x^2}$ (c) $\displaystyle \lim_{x \to 0^+} \frac{x}{x^2}$
顯然(a)的極限是0,(b)的極限不存在,而(c)的極限為無窮大。
總的來說,舉凡$\displaystyle \lim_{x \to a} \frac{f(x)}{g(x)}$滿足(i)f(x)→0以及g(x)→0或是(ii)f(x)→ ± ∞以及g(x)→ ± ∞,我們就將其稱為不定型的極限問題。
L’Hôpital’s Rule的定理敘述如下。
Suppose f and g are differentiable and g′(x)≠0 on an open interval I that contains a (except possibly at a). Suppose that $$ \lim_{x \to a} f(x) = 0 \quad \text{and} \quad \lim_{x \to a} g(x) = 0 $$ or that $$ \lim_{x \to a} f(x) = \pm \infty \quad \text{and} \quad \lim_{x \to a} g(x) = \pm \infty $$ Then $$ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} $$ if the limit on the right side exists (or is ±∞).
以微積分教科書上常見的例子而言,我們可以簡單理解如下:
若將x代入原方程式會產生$\dfrac{0}{0}$或是$\dfrac{\infty}{\infty}$的話,原始問題就可以轉成分子分母各自微分後的極限問題。但這裡有一點務必特別留心:轉換後的問題,其極限必須存在。(這裡我們視無窮大為極限存在。)
回到前面$\displaystyle \lim_{x \to 1} \frac{\ln x}{x-1}$的例子,雖然分子分母沒有公因式可以對消,但透過羅必達法則,我們還是可以得到 $$ \lim_{x \to 1} \frac{\ln x}{x-1} = \lim_{x \to 1} \frac{1/x}{1} = 1 $$ 的結論。
當然了,羅必達法則也不是萬靈丹,還是有些問題,利用以往的方法可以解決,使用羅必達法則反而做不出來。例如像是 $\displaystyle \lim_{x \to 1^+} \frac{x - 1}{\sqrt{x^2 - 1}}$ 這樣的題目。如果我們使用羅必達法則的話,會陷入無窮迴圈的窘境。 \begin{align*} \lim_{x \to \infty} \frac{x + 2}{\sqrt{x^2 + 4x}} &\overset{L'H}{=} \lim_{x \to \infty} \frac{1}{(x + 2)/\sqrt{x^2 + 4x}} = \lim_{x \to \infty} \frac{\sqrt{x^2 + 4x}}{x + 2} \\ &\overset{L'H}{=} \lim_{x \to \infty} \frac{x + 2}{\sqrt{x^2 + 4x}} = \cdots \end{align*} 那麼,正確的做法應該是什麼呢?這個就當作大家的複習功課囉!XDD
這裡我們附上一個簡易版的證明。完整版的證明需要用到柯西均值定理(Cauchy’s Mean Value Theorem),由於比較抽象,有興趣的同學請參考各大微積分課本。
假設f(a)=g(a)=0、f′和g′都是連續函數,並且g′(a)≠0;則 \begin{align*} \lim_{x \to a} \frac{f'(x)}{g'(x)} &= \frac{f'(a)}{g'(a)} = \frac{\displaystyle\lim_{x \to a} \dfrac{f(x) - f(a)}{x - a}}{\displaystyle\lim_{x \to a} \dfrac{g(x) - g(a)}{x - a}} \\ &= \lim_{x \to a} \frac{f(x) - f(a)}{g(x) - g(a)} \\ &= \lim_{x \to a} \frac{f(x)}{g(x)} \end{align*} 是不是輕輕鬆鬆就得到羅必達法則了呢!
關於羅必達法則,我們就簡單介紹到這兒;其他的變形問題(如:∞ − ∞、∞ ⋅ 0、00...)及應用,請大家自行參考書本裡的相關章節。
Applications of AWK
Deluxe Tables with Authorea
Example of an “astronomer friendly” deluxe table formatted with Authorea. This is the example posted by Jess K on astrobetter in How to Make Awesome Latex Tables. For this table we are using the LaTeXML engine, have a look at this quick tutorial if you are a prospective Authorea power LaTeX user.
cccccccc 1/3 Bright & & 4.24 ⋅ 10−4 & 6.19 && 96.97 & &
& [0.3,3] & & & 1.77 & & 96.7 & 1.9σ
2/3 Dim & & 4.26 ⋅ 10−4 & 4.48 & & 52.77
Louisville Firearms-Related Crime Trends and Their Implications for Businesses and Their Patrons
Obat Kuat Pria Murah Lengkap Wa082221214145
Turn-key research writing and publishing with Authorea for groups and teams
and 2 collaborators
Turing-Hopf bifurcation of a diffusive Holling-Tanner model with nonlocal effect and digestion time delay
Optical-Nanofiber-Enabled Gesture-Recognition Wristband for Human-Machine Interaction with the Assistance of Machine Learning
and 12 collaborators
A Review on Forensic DNA Analysis
Artificial Intelligence Enabled Reagent-free Imaging Hematology Analyzer
and 6 collaborators
Step by step how to conduct evidence appraisal using Gradepro
Production of biogas using waste products (Water Hyacinth and Cow Dung)
and 1 collaborator
High-performance Biopolymer Cryogels for Transient Sensing Ecology-drones
and 6 collaborators
Einstein-Poincare synchronization and general transformations of coordinates
Einstein-Poincare synchronization and deformation of space-time by gravity are sufficient for obtaining Lorentz transformation starting from Galileo transformation.