I've learnt in classroom that the requirements for electrical current to flow in a circuit are as follows:   there must be a closed conducting path between negative and positive terminals there must be a potential difference across the terminals   A copper wire is solid meaning all the atoms are very close to each other and don't move around very much and copper atom have valency electrons to donate around easily.  In a liquid solution the ions are constantly moving around finding new neighbors each time, my question is suppose we introduce 2 opposite charge electrodes into the solution how can the first requirement be met in these random chaos?  If current can flow under the above mentioned condition can it do so in plasmas which is much more spacious, random and chaotic than liquid?  Note: I'm more interested in the quantum behavior like how the ion finds suitable neighbors is it magnetism at work as it is drawn to a positive ion since itself is negatively charge.  I've provide a drawing to help in my understanding, it is not to scale and I know about uncertainty principle.
The current is maximum through those segments of a circuit that offer the least resistance. But how do electrons know beforehand that which path will resist their drift the least?
Switch statement fallthrough in C#? On Chrome on Android, code blocks on the mobile site have no syntax highlighting, are not a monospaced font, and words are wrapped.  Mobile site on mobile browser:    Desktop site on mobile browser:    Inline code has the same issue, it isn't a monospaced font.  It has been this way for as long as I can remember. Wouldn't be surprised if this is a dupe. I want to know all sets of single-variate functions that fulfill the following qualities: $\forall_{f:R^2 \to R^2} \forall_{g \in CAF} \forall_{h \in CAF} \exists_{k\in CAF} k(x) = f(g(x),h(x))$ where CAF is the set I am wishing to find. I know that the set of constants and the set of piece wise constants each satisfy that equation. Do any others? Value of $x=\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}} $ and location of second digit $1$ of $x$ What is the value of the series   $\sum\limits_{n=1}^\infty {\frac{1}{10^n-1}}$   and can one find closed forms for the sums   $\sum\limits_{n=1}^N {\frac{1}{10^n-1}}$?   I found that $\sum_{n=1}^N {\frac{1}{10^n-1}} $  =  $\sum_{n=1}^N (\lim_{m\to\infty  }{\frac{10^{-n}-10^{-nm}}{1-10^{-n}}})$  =  $0.111\ldots+0.0101\ldots+\ldots$ but I can't go farther.     The original question is: "Where does the second '1' appear?" Third grade spiral in TikZ - dimension too large I'm trying to draw a smooth third grade spiral (r^3 = a^3\cdot \varphi) over $60\times 60$mm grid and I'm getting the "Dimension too large"-error if I set $a &gt; 320$ but I need to fill the grid area completely:  \begin{tikzpicture}[x=1mm, y=1mm]      \draw[step=1, line width=0.1mm, opacity=0.3] (-30, -30) grid (30, 30);      \draw [domain=0:50, variable=\t, samples=1000]          plot ({\t r}:{pow(320*\t, 1/3)}); \end{tikzpicture}     Is there any way to avoid this error? How does one experimentally determine chirality, helicity, spin and angular momentum of a fundamental particle? Conversion Error setting value for 'null Converter' - Why do I need a Converter in JSF? "This webpage has a redirect loop" in Chrome but working fine on FF and IE? What should I look for when buying chain lube / lubricant? In the past, I personally have only used WD-40.   However, I have occasionally had my bicycle looked over at the locally run, free service sessions and most of the mechanics there use a hand applied type of greasy gel (I forget the name).  Questions:   Is WD-40 suitable for bicycle chains? Would alternatives provide better chain longevity? When shopping around for lubricant, what are the most important things to look for? What do you use and why?  (optional added extra) Any tips for reducing chain wear? How do I save files edited with vim? How does the loopback interface work I'm trying to create a pmatrix whose components are rather complicated fractions (Christoffel symbols), and the line spacing is too small making the whole thing a bit cramped and hard to read.  Is there any easy way to increase the spacing (about 1.5 the default would be perfect)? polyglossia or bidi bug: [\d\.] gets inverted to [\.\d] After bidi was updated to v. 19.6 (2016-05-23), using at least one RTL language with polyglossia leads to number and following dot being inverted. Cf.  \documentclass{book} \usepackage{polyglossia} \setotherlanguage{hebrew} \begin{document} 6. \end{document}   with and without \setotherlanguage{hebrew}. Note that it is not necessary to actually use that language.  I do not know whether this is a bug in bidi or an incompatible change which polyglossia needs to adapt to. Because of this I have not opened an issue on polyglossia's github page. Sadly, bidi's contact information are outdated – the sourceforge referred to on CTAN is unused and the github page that is mentioned in the package's readme and PDF does not seem to exist any more either.  The problem might be something along the lines of     Does anybody have the previous version of bidi at hand?  TIA, Dario I just received a PhD invitation letter from a German University which mentions that my remuneration will be 2/3 of full TV-L 13 position. I do not know what a TV-L 13 position is (Google is of no help here), and didn't want to be rude, so I am asking here first. Is this a common position or specific to Germany? If anyone knows, currently how much remuneration corresponds to a TV-L position? Rising edge pulse detector from logic gates The circuits I describe are entirely made of 7400 series logic gates (7402, 7404 and 7408 ic). I'm trying to build a rising (positive) edge pulse detector using logic gates.  The following circuit should work in theory:   (see )  I do not expect the short output to be seen through a LED so, to test it, I make it trigger an SR latch to its up state:    However, it doesn't have effect on the latch. So I added an inductor to delay the input to the NOT gates in the pulse detector circuit and it worked:   But now I can see the short flashing output through a LED which I should not because it would mean it is too long to work with a circuit like that:   Which intends to toggle the D Latch output on each clock rising edge (Note that this is a D Latch not a D Flip-flop)  And anyway there is no place for inductors in integrated circuits so there must be a way to do this only with logic gates. Can someone solve this mystery?  BTW It does not show in my schematics but I did put 10K pull-down resistors where there might be floating pins. I'm training a neural network and the training loss decreases, but the validation loss doesn't, or it decreases much less than what I would expect, based on references or experiments with very similar architectures and data. How can I fix this?    As for question        to which this question is inspired, the question is intentionally left general so that other questions about how to reduce the generalization error of a neural network down to a level which has been proved to be attainable, can be closed as duplicates of this one.  See also dedicated thread on Meta: Finding maximal sublist of linearly independent vectors When is the present perfect tense used instead of the past tense? When is the present perfect tense used instead of the past tense?    I know that the present perfect tense is used when some adverbs (e.g., never, ever) are present in the sentence; the same is true for sentences like the following one.     When you returned, I have been at home since 3:00 PM.   In which other cases should I use the present perfect? Do the following sentences require it?     I have walked downtown everyday for a year.        I have been at home since 3:00 PM. The polar of a convex body $C$ is defined as:  For a convex body $C$, its polar $C^*$ is given by $$C^* = \{x\in\mathbb{R}^n: \langle x,c\rangle \le 1, \forall c\in C\}$$  If I start with an orthonormal basis $(e_i)_1^n$, positive constants $(\alpha_j)_1^n &gt; 0$ and the ellipsoid $$\varepsilon = \left\{x\in\mathbb{R}^n: \sum\limits_{j=1}^n \frac{\langle x,e_j\rangle^2}{\alpha_j^2} \leq 1  \right\}$$ how do I show that its polar ellipsoid is $$\varepsilon^* = S_1 = \left\{y\in\mathbb{R}^n: \sum_{j=1}^n \alpha_j^2 \langle y,e_j\rangle^2 \le 1\right\}$$ Using the definition, I saw that $$\varepsilon^* = S_2 = \{z\in\mathbb{R}^n: \langle z,x\rangle \le 1, \text{for all }x\in \varepsilon\}$$ Also, $\langle z,x\rangle = \sum_i\langle z,e_i\rangle\langle x,e_i\rangle$ I used Cauchy Schwartz to do one side of the proof, i.e. $S_1\subseteq S_2$: $$\left(\sum_i\langle y,e_i\rangle\langle x,e_i\rangle\right)^2 \leq \left(\sum_i\alpha_i^2 \langle y,e_i\rangle^2\right)\left(\sum_i \frac{\langle x,e_i\rangle^2}{\alpha_i^2}\right) \le 1 \implies \langle y,x\rangle \le 1$$ I need help with the other direction, i.e. $S_2\subseteq S_1$. Which other inequality might help? I have seen this construction quite often:      Online ads have been around since the dawn of the Web, but only in   recent years have they become the rapturous life dream of Silicon   Valley.   What is the rule there?. When your sentence doesn't start with pronoun + verb, invert them as verb + pronoun?. I know it sounds awkward but is it possible (grammatically correct) to use something similar to:      Online ads have been around since the dawn of the Web, but only in   recent years they have become...   And in any case, does this only work with have (or has)? Maybe it works fine with 'had' but I can't think of an example right now.