Public Articles
Physics, Electricity, 13-14
\begin{equation*} \text{Power}=\frac{\text{Energy Transferred}}{\text{Time}} \end{equation*} Power is measured in watts, and energy in joules. Alternatively, one kilowatt-hour is the energy transferred in one hour from a source of power 1000W; 3.6MJ.
Current is a flow of charge. Charge itself is a fundamental property of matter. Charge can be positive or negative, and opposite charges attract. Charge itself is measured in coulombs, with the charge on one electron being −1.6 × 10−19C.
Conduction is the flow of electric charges through a material. An insulator is a material that will not readily conduct electricity, as it has no free charges - the tightly bound electrons would require a relatively large amount of energy to be freed.
By rubbing together two insulators, electrons are transferred from one material to another, leaving neither material with a neutral charge.
Metals are good conductors due to the fact that some of their atomic electrons are free to move between atoms and carry charge.
Electrostatic phenomena are when there is no flow of continuous charge. A continuous charge is known as a current. \begin{equation*}
Q=It
\end{equation*} The direction of conventional current is from positive to negative. However electrons, being negatively charged, flow from negative to positive.
Current is the rate of flow of charge: With a current of one ampere, one coulomb will pass a given point every second.
An electric current will be set up in a conductor if there is:
An energy source, such as a battery made up of cells
A continuous circuit
Charge is conserved wiothin a circuit. At any given point, the input and output charge must be equal - Kirchoff’s First Law
In a series circuit, current is the same everywhere. In a parallel circuit, the sum of the current in branches is equal to the total current.
Chemistry, O Revision Notes
Ionic solids are held together by opposite charges
Anions and Cations form giant ionic lattice
Eg. NaCl
Each Na+ ion surrounded by six Cl− ions and vice versa
Overall attraction within lattice greater than repulsion
Strong melting and boiling points
Many ionic substances dissolve readily in water
Ions becomes surrounded by water and spread out
Ions lose regular arrangement - becomes random
Ions behave independently of each other
Not all ionic substances dissolve readily in solution, as energy changes are an important factor.
Before an ionic solid can dissolve, the ions must be separated from their lattice. Energy must be supplied to overcome the electrostatic attractions, making it an endothermic process.
ΔLEH is the enthalpy change when one mole of a solid is formed from its separated ions in their gaseous states.
Energy required to form a lattice is negative
Energy required to break a lattice is therefore −ΔLEH
Lattice enthalpy is affected by both the charge and radii of the ions.
A greater charge will increase the magnitude of the lattice enthalpy
A greater atomic radii will decrease the magnitude of the lattice enthalpy
Larger separation between charges leads to smaller attraction
The magnitude of the lattice enthalpy will increase with greater charge density
Thus, substances with large enthalpies are normally insoluble
A2 Physics, Turning Points in Physics
Stream of electrons; current
Produced in discharge tubes
Low pressure tube with a high potential difference
Hot cathode emits electrons by thermionic emission and accelerates them through gas
Gas atoms are ionised, so have a positive charge
Low pressure allows them to accelerate towards cathode with few collisions
Collision with cathode releases more electrons
Gas is therefore conducting
De-excitation of electrons in gas atoms gives off visible and UV photons
Gas at very low pressure will have fewer collisions with electrons
Cathode ray hits glass in tube, exciting glass electrons
Glass then glows due to de-excitation
The emission of electrons from a hot cathode is known as thermionic emission.
Few electrons in the lattice are delocalised throughout crystal lattice
Allow conduction
The work function gives the minimum energy to remove one of these electrons
Heating of cathode transfers energy to free electrons
Overcome attractive electrical force of lattice
Emitted from metal surface
Thermionic emission does not require a gas to be present. An electron beam will be formed in an evacuated tube as the electrons do not collide; this is the bases of a thermionic valve.
Contain an electron gun
Annular anode helps acceleration
The cathode can be heated directly or indirectly.
Direct Heating
Tungsten filament with zirconium dioxide
Zirconium Dioxide reduces work function, resulting in greater rate of electron emission
Indirect Heating
Electric filament heats a separate cathode
Made of nickel, coated in oxides of barium calcium and aluminium
Operates efficiently at lower temperatures
Work done on a charged particle accelerating between a potential difference is given by W = QV. So, for an electron, \begin{gather*} W=eV\\ \therefore E_k=eV\\ \therefore \frac{1}{2}mv^2=eV \end{gather*} This gives the definition of the electronvolt; the energy gained by an electron after it has been accelerated through a potential difference of 1V.
Biomolecular Histology as a Novel Proxy for Ancient DNA and Protein Sequence Preservation
Topical Review: Extracting Molecular Frame Photoionization Dynamics from Experimental Data
and 1 collaborator
Community support intensity in urban settings: an empirical, place based model
The promise and challenges of characterising genome-wide structural variants: A case study in a critically endangered parrot
and 4 collaborators
olddraft
and 1 collaborator
Angular momentum transport within evolved low-mass stars
and 4 collaborators
Asteroseismology of 1.0 − 2.0M⊙ red giants by the Kepler satellite has enabled the first definitive measurements of interior rotation in both first ascent red giant branch (RGB) stars and those on the Helium burning clump. The inferred rotation rates are 10 − 30 days for the ≈0.2M⊙ He degenerate cores on the RGB and 30 − 100 days for the He burning core in a clump star. Using the MESA code we calculate state-of-the-art stellar evolution models of low mass rotating stars from the zero-age main sequence to the cooling white dwarf (WD) stage. We include transport of angular momentum due to rotationally induced instabilities and circulations, as well as magnetic fields in radiative zones (generated by the Tayler-Spruit dynamo). We find that all models fail to predict core rotation as slow as observed on the RGB and during core He burning, implying that an unmodeled angular momentum transport process must be operating on the early RGB of low mass stars. Later evolution of the star from the He burning clump to the cooling WD phase appears to be at nearly constant core angular momentum. We also incorporate the adiabatic pulsation code, ADIPLS, to explicitly highlight this shortfall when applied to a specific Kepler asteroseismic target, KIC8366239.
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