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205,962 | [
"https://physics.stackexchange.com/questions/205962",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/74666/"
] | From my textbook:` "When you touch a positively charged object charge spreads over your body."
My understanding is that the protons in a solid conductor are immobile since they are contained within the nucleus, and the electrons are the mobile charge carriers. How, then, is positive charge transferred? Wouldn't it ma... | If there is a relative absence of electrons on the nuclei then this absence acts as if it were a positive charge. In fact, the absences can even sometimes behave like bona-fide particles; they are called "holes" in semiconductor physics.
It would make a lot more sense if electric charge were the negative of what it is... | <ul>
<li>A <em>neutral</em> material has equal amounts of positive protons and negative electrons. </li>
<li>A <em>negatively charged</em> material has more electrons than protons. </li>
<li>A <em>positively charged</em> material has less electrons than protons. </li>
</ul>
No matter the charge carrier itself, the mat... | https://physics.stackexchange.com |
125,172 | [
"https://cs.stackexchange.com/questions/125172",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/120587/"
] | Wikipedia states on the page of the halting problem, "For any program f that might determine if programs halt, a "pathological" program g called with an input can pass its own source and its input to f and then specifically do the opposite of what f predicts g will do."
Suppose we have 2 neural networks, approximating... | Turing proved that the halting problem is undecidable.
No algorithm that you can come up with will solve the halting problem.
| It is not quite clear to me what you have in mind with infinite size + depth limits for your neural net, but once you have sorted that out you should end up with the following game:
Player I plays a program p, and Player II plays a program q.
Player 1 wins iff p halts on input q, and answer Yes if q halts on input q,... | https://cs.stackexchange.com |
127,791 | [
"https://physics.stackexchange.com/questions/127791",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/55945/"
] | This is a very well known problem, but I can't find an answer in the specific case I'm looking for.
Let's consider two balls :
<ul>
<li>Ball 1 weighs 10 kg</li>
<li>Ball 2 weighs 1 kg</li>
<li>Balls have identical volumes (so Ball 1 is much more dense)</li>
<li>Balls have identical shapes (perfect spheres)</li>
</ul>... | I am sorry to say, but your colleague is right.
Of course, air friction acts in the same way. However, the friction is, in good approximation, proportional to the square of the velocity, $F=kv^2$. At terminal velocity, this force balances gravity,
$$ m g = k v^2 $$
And thus
$$ v=\sqrt{\frac{mg}{k}}$$
So, the termi... | Ball 1 will drop faster in air, but both balls will drop at the same speed in vacuum.
In vacuum, there is only the gravitational force on each ball. That force is proportional to mass. The acceleration of a object due to a force is inversely proportional to its mass, so the mass cancels out. Each ball will accelera... | https://physics.stackexchange.com |
10,816 | [
"https://softwareengineering.stackexchange.com/questions/10816",
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"https://softwareengineering.stackexchange.com/users/4954/"
] | I have just completed a course on computability and logic which was an interesting course. The lecturer recommend a few books on his slides, which include "Gödel, Escher, Bach". I can see the book is quite famous, and looks very interesting. But I have a few questions to ask regarding its content.
<ol>
<li>Is the cont... | <blockquote>
Is the content still valid today? I guess most theoretical stuff don't change over night, but is there some major points which does not hold today which I should be aware of?
</blockquote>
The content is logic and math. It doesn't change in any substantial way, not only over night. It will be valid fo... | Gödel, Escher, Bach is a very unusual book. A lot of concepts are presented and need a LOT of digestion to appreciate. All you need to know is presented in the book, and you might find the explanation of Gödels Theorem (which is done quite well) and the various Turing-concepts fascinating.
Also the way he narrates a... | https://softwareengineering.stackexchange.com |
540,305 | [
"https://physics.stackexchange.com/questions/540305",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/222480/"
] | Let's assume I have a cavity with one main frequency and two different sideband frequencies characterized by the annihilation operators <span class="math-container">$a$</span>, <span class="math-container">$a_1$</span> and <span class="math-container">$a_2$</span>. If I want to calculate the number of photons in the ca... | There is no such law, because it isn't true. It isn't even remotely true.
Probably the most extreme counter example would be the earth itself, which hurtles around the sun at about 67,000 mph. I'd like to see your bee do this! (without riding on the back of the Earth, of course!)
And, of course, bees do not travel ... | The other answer is good and correct. As this is not a pure physics question I want to add something that relates to this, though: the scaling law. This is a very basic explanation why smaller animals are stronger relative to their size. This could explain why smaller animals tend to be faster or jump further (relative... | https://physics.stackexchange.com |
273,592 | [
"https://softwareengineering.stackexchange.com/questions/273592",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/154645/"
] | Our logging module has a buffer that often gets full and we are searching for a solution to shorten our logs, without making any compormise in performance.
The code is written in C++ and currently we are using __FUNCTION__ to extract the function name for a trace, but on windows environments it also appends the whole ... | You could use <code>__LINE__</code> instead, and although the function names will no longer be present in your logs, you can easily resolve them when analysing the logs.
An alternative is a static object in each function, declared at the start, that wraps trhe logger functions so you use it to write the log lines. The... | Maintain a map in which you accumulate function names and associate them with unique integer ids.
You should be able to optimize the performance of the map by taking into consideration the fact that the function name produced by the compiler for a certain function will always have the same address. (There will be only... | https://softwareengineering.stackexchange.com |
420,520 | [
"https://softwareengineering.stackexchange.com/questions/420520",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/27426/"
] | I've quite frequently seen benchmarks where the tester discarded the highest <em>and</em> the lowest time out of <em>N</em> runs.
Discarding the highest time I understand; it's probably high because of some other processes running suddenly demanding more CPU.
But doesn't the lowest time indicate the best possible perfo... | The lowest timing might indeed represent the "true" timing without outside interference, or it might be a measurement error. E.g. the boosting behaviour of a CPU might speed up the first run in a larger benchmarking suite, or a less congested network might speed up a net-dependent benchmark during certain tim... | Outliers indicate unusual situations. Outliers are interesting in science, because they give you something to investigate, but they are useless in benchmarks.
If you have 10000 benchmark runs, and 9999 of them took one hour, but 1 of them took 1 second, then it is useless for your customers to tell them it will take 1 ... | https://softwareengineering.stackexchange.com |
327,504 | [
"https://electronics.stackexchange.com/questions/327504",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/143453/"
] | When Intel or Apple outsources their electronics to Asia, what prevents the outsource vendor from copying the design and making a knock-off product?
| It doesn't matter where the chips are fabbed. Many Intel chips are designed in China and India anyway. Or in Israel. What prevents the complex chips from pirating is their built-in complexity and built-in security.
To begin, the chips are fabricated from tape-out files that are used to create photomasks. One can't get... | Approximately 75% of Intel's semiconductor fabrication is performed in the USA and only old generations (65nm) in China.
Apple also do not trust China for foundry work but did trust Samsung and now turning to a Taiwanese foundry, TSMC to make future chips with 10nm fabs for A10X, A11.
| https://electronics.stackexchange.com |
115,829 | [
"https://stats.stackexchange.com/questions/115829",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/16496/"
] | I have a question about the following statement about entropy:
<blockquote>
If a source provides us with a sequence chosen from 4 symbols (say A, C, G, T), then the maximum average information per symbol is 2 bits. If the source provides blocks of 3 of these symbols, then the maximum average information is 6 bits pe... | By block, I assume they mean something like [AAA] or [ATC]: a set of several concatenated symbols.
We know, from the definition of joint entropy, that:
$$H(X,Y,Z) = - \sum_x \sum_y \sum_z P(x,y,z) \log_2(P(x,y,z))$$
so you could calculate it directly, assuming you knew the joint distributions for $X$, $Y$, and $Z$.
... | The symbols could be both different and the same (any of the 64 triplets: AAA, AAC, ..., TTT). If the MAI (maximal average information) of one symbol is 2 bits, the MAI of 3 symbols is 3*2=6 bits. Another way to calculate the information is $ -(64*1/64*log_2(1/64))=6 $
| https://stats.stackexchange.com |
11,452 | [
"https://security.stackexchange.com/questions/11452",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/6892/"
] | Since one of the 10 domains on the CISSP is legal/governance, I figured this would be the best place to ask.
I recently did a security assessment for a non-profit. I found that their systems needed some attention. I want to make it clear to them that just because your network has no sensitive data on it, you are still... | It is still mostly the case that liability for computer breaches (not including disclosure of private information) is legally blameless. There are plenty of laws cropping up all over about data privacy, but as far as protection of computing resources themselves there isn't really a standard. "Gross negligence" on the I... | Legal issues are very tricky, they change (some times dramatically) between different countries and jurisdictions. Even when there are clear laws and regulation (for example, privacy protection in the UK/EU), the penalties/fines are usually quite small, and the chances of 'getting caught' even lower. I am aware of some... | https://security.stackexchange.com |
225,026 | [
"https://softwareengineering.stackexchange.com/questions/225026",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/97450/"
] | I have read that we should only catch exceptions that can be handled, which makes catching the base exception class (C# in this case) a bad idea (on top of other reasons). I am currently part of a project in which I so far have yet to see anything but the base exception being caught. I mentioned that it is considered b... | Certain language features like
<ul>
<li>Garbage Collection</li>
<li>Exception Systems</li>
<li>Lazy Evaluation</li>
</ul>
are not generally useful in a real-time system. One should probably choose a language without these features, and try to prove certain properties like maximum memory usage, or maximum response tim... | To answer your question, one has to understand what exceptions are, and how they work.
Exceptions are usually thrown when such errors occurs, where user's assistance is required. In such cases, it doesn't matter how long it takes to unwind the stack and handle the exception.
Without catch handlers, the program stops... | https://softwareengineering.stackexchange.com |
244,571 | [
"https://security.stackexchange.com/questions/244571",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/43921/"
] | I was reading an article which said that if you install custom root cert from a third party then they can decipher all communication between you and others.
But that doesn't make sense. What I understand is that root cert allows SSL mechanism to verify if a certificate provided by connecting party is legit or not.
So u... | <blockquote>
I was reading an article which said that if you install custom root cert from a third party then they can decipher all communication between you and others.
</blockquote>
I have no idea what you were reading (citations would be helpful). But you are right in that it is not sufficient to just have a custom ... | If you use https for example to buy stuff from Amazon, Amazon will send you a certificate to prove it’s them, and that works because nobody other than Amazon can get an “Amazon” certificate from one of the companies whose root certificates your device trust.
But if you let me install a root certificate on your device t... | https://security.stackexchange.com |
24,446 | [
"https://robotics.stackexchange.com/questions/24446",
"https://robotics.stackexchange.com",
"https://robotics.stackexchange.com/users/33409/"
] | I want to make an URDF from a Staubli TX2-140. I have the .STL of the all arm. Does someone know how to separate it to get individual links .STL?
Thanks
| You answered yourself.
Your underlying model of point moving through space usually assumes the center of gravity to match the camera frame.
If your inertial sensor have a translational offset from the camera attached to a rotating rigid body, you would measure extra tangential and radiant acceleration components that ... | It's important to know the offset between the camera and the IMU because the IMU's information is only useful when it can be correleated with the camera data.
In a lot of the trivial cases this difference could be negligible. However depending on your requirements for accuracy you will need to compensate.
Lets use a th... | https://robotics.stackexchange.com |
14,187 | [
"https://datascience.stackexchange.com/questions/14187",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/22012/"
] | I have noticed that such terms as model <strong>hyperparameter</strong> and model <strong>parameter</strong> have been used interchangeably on the web without prior clarification. I think this is incorrect and needs explanation. Consider a machine learning model, an SVM/NN/NB based classificator or image recognizer, ju... | Hyperparameters and parameters are often used interchangeably but there is a difference between them. You can call something a 'hyperparameter' if it cannot be learned within the estimator directly. However, 'parameters' is a more general term. When you say 'passing the parameters to the model', it generally means a co... | In addition to the answer above.
<em>Model parameters</em> are the properties of the training data that are learnt during training by the classifier or other ml model. For example in case of some NLP task: word frequency, sentence length, noun or verb distribution per sentence, the number of specific character n-grams... | https://datascience.stackexchange.com |
56,669 | [
"https://softwareengineering.stackexchange.com/questions/56669",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/15177/"
] | I'm wondering whether or not I've done the right thing as a contractor. Basically I'm in to my 3rd month, and my current client messed up the payment in the first month, and I just found out that they are again late in paying me for my 2nd month.
It wouldn't be so bad if I wasn't in a bit of a financial situation due ... | I don't believe you did the right thing, honestly. If you truly enjoy the job and want to work for these people, you would not have resorted to burning a bridge with them. Pay issues are always tricky, and it's not terribly professional for them to have botched your pay in the first place, but the question also relates... | I have been running a web development company for about 2 years. I have made it a rule for myself to always take 50% in advance and 50% <strong>before</strong> the project is handed over. This way I never run into problems. Companies tried to make me bend my rule for them ( some by trying to act cheeky ) and I'll share... | https://softwareengineering.stackexchange.com |
486,347 | [
"https://electronics.stackexchange.com/questions/486347",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/225886/"
] | If I've got two batteries of the same capacity, one is 12V and the other is, say, 18V with a step down buck converter, will I essentially be able to power my load at 12V for longer? I know that a battery's voltage will decrease as it discharges, so it seems to me that a better way to power a load with a more stable vol... | The other guys are right in theory or abstract. And certainly, whichever battery has higher energy storage will likely last longer.
But from a product design perspective it is a good idea to use a battery pack that is a bit higher in voltage than what you need and buck down to your desired voltage. So I think you are ... | What matters in not <em>voltage</em>, it's <strong><em>energy</em></strong>.
The energy in a battery is quoted in Wh (watt-hours). Many still don't specify the watt-hour capacity but do quote the Ah (amp-hour) capacity. With this and the battery voltage you can calculate the Wh capacity from:
<span class="math-contai... | https://electronics.stackexchange.com |
87,644 | [
"https://physics.stackexchange.com/questions/87644",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/34549/"
] | In vertical circular motion we conserve energy for calculating velocities at a point (if initial velocity given). But, energy can only be conserved when forces are conservative. Tension is not a conservative force. Does it not affect the particles velocity? Is the tension's role only to provide centripetal accelerati... | By definition, the tension in the string can only supply a force <strong><em>towards the center</em></strong> of the circular motion, so in one sense the tension is a centripetal force.
However, at any moment, the force of gravity can be decomposed into radial and tangential components. It is the <strong><em>sum</em>... | Yes, tension only affects the <em>direction</em> of the particle's velocity. This is because it is always perpendicular to the velocity, and because work is actually the dot product of force and displacement:
$$
W = F \cdot s = |F| \times |s| \times \cos(\theta)
$$
, a force perpendicular to the displacement does no ... | https://physics.stackexchange.com |
369,191 | [
"https://softwareengineering.stackexchange.com/questions/369191",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/302679/"
] | I'm working on a larger solo project and right now, and I have several classes in which I do not see any reason to create an instance of.
My dice class right now, for example, stores all of its data statically and all of its methods are static too. I don't need to initialize it because when I want to roll the dice and... | There is nothing wrong with static classes that are truly <em>static</em>. That is to say, there is no internal state to speak of that would cause the output of the methods to change.
If <code>Dice.roll()</code> is simply returning a new random number from 1 to 6, it's not changing state. Granted, you may be shari... | To give an example of the limitations of a <code>static</code> class, what if some of your gamers want to get a slight bonus on their die rolls? And they are willing to pay big bucks! :-)
Yes, you could add another parameter, so <code>Dice.roll(bonus)</code>,
Later you need D20s.
<code>Dice.roll(bonus, sides)</cod... | https://softwareengineering.stackexchange.com |
1,996,061 | [
"https://math.stackexchange.com/questions/1996061",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/369451/"
] | I have a function $f(x)=cos(x/4)e^{-\frac{x^2}{4}}$ that has a fixed point in between [0,1] and I know that a point is fixed if $f(x)=x$. Solving it for $x$ is a manually is messy, through a computer program i get $x=0.825..$, though I think I am only suppose to show that there is one (and only one) fixed point in that... | <h3>1. To show that there is a fixed point of <span class="math-container">$f(x)=\cos(\frac x4)e^{-x^2/4}$</span> in <span class="math-container">$[0,1]$</span></h3>
We may show that there is a fixed point of <span class="math-container">$f(x)$</span> if there is an <span class="math-container">$x\in[0,1]$</span> such... | If $g:[0,1] \to [0,1]$ is continuous, Then $g$ has a fixed point. Try to prove this !
Your function $f$ therefore has a fixed point, this means that $h(x):=f(x)-x$ has a zero in [0,1].
Observe that $h'(x)<0$ for all $ x \in [0,1]$. Thus $h$ has only one zero in [0,1]. Hence $f$ has only one fixed point in [0,1]
| https://math.stackexchange.com |
135,731 | [
"https://cs.stackexchange.com/questions/135731",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/132196/"
] | I am trying to figure out the solution of the recurrence relation
<span class="math-container">$$T(n) = 5T(n/3) + n \log n$$</span> using the Master Method.
I am guessing that <span class="math-container">$f(n) = O(n^{1.46 - \varepsilon})$</span>, but I am confused in the part that <span class="math-container">$\frac{n... | When <span class="math-container">$f(n)$</span> is said to be polynomially smaller than <span class="math-container">$g(n)$</span> it just means that there is some constant <span class="math-container">$\varepsilon > 0$</span> such that <span class="math-container">$f(n) \in O(g(n) n^{-\varepsilon})$</span>.
This is... | It is well known, that for any fixed number <span class="math-container">$a>0$</span> we have <span class="math-container">$\log(n)=O(n^a)$</span>.
Then, for a small enough epsilon, that is for <span class="math-container">$\epsilon<0.46$</span> we can define <span class="math-container">$a=0.46-\epsilon>0$</s... | https://cs.stackexchange.com |
436,364 | [
"https://electronics.stackexchange.com/questions/436364",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/220888/"
] | I know how we can amplify a given signal using BJT transistors by biasing them. But I would like to know what is that crux property which enables BJT transistor to act like an amplifier. Is it the constant nature of reverse saturation current or is it the definite relationship between the base and the collector current... | A transistor on its own does not make it an amplifier.
The transistor needs a circuit around it to do the actual (signal) amplification.
Depending on the circuit a transistor can amplify current changes and/or voltage changes and that means <strong>power amplification</strong>. Power amplification means that you need... | The crux property that allows <strong>anything</strong> to act as an amplifier is that it can control a high power signal, using a low power input.
In the case of a transistor, it's the fact that a low power base current or gate voltage can change a large collector or drain current.
There's a whole host of other devi... | https://electronics.stackexchange.com |
974,960 | [
"https://math.stackexchange.com/questions/974960",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/58414/"
] | If $x=t^2\sin3t$ and $y=t^2\cos3t$, find $\frac{dy}{dx}$ in terms of $t$. This is how I tried solving it:
$$
\frac{dx}{dt} = 2t\sin3t + 3t^2\cos3t \\
\frac{dy}{dt} = 2t\cos3t - 3t^2\sin3t \\
\frac{dy}{dx} = \frac{2t\cos3t - 3t^2\sin3t}{2t\sin3t + 3t^2\cos3t}
$$
But the answer listed is:
$$
\frac{2-3t\tan3t}{2\tan3... | You are on the right trace. Just divide $t\cos 3t$ in the numerator and denominator of $\frac{dy}{dx}$.
| Your answer and the given answer are the same: just divide the numerator and denominator of your answer by $t\cos 3t$ to find the other answer.
| https://math.stackexchange.com |
9,782 | [
"https://dba.stackexchange.com/questions/9782",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/1348/"
] | How to design this simple DB?
I need to add some things to a DB I use to keep track of servers.
Table looks something like this.
<pre><code>Matrix
| Name | Description | Etc | Etc |
------------------------------------------------
| Server01 | First Server | Data | Data |
</code></pre>
I wan... | I'd probably start with something like this:
<pre><code>create table Matrix (
id integer primary key auto_increment,
Name varchar(30) not null,
Description varchar(30) not null
);
create table Updates ( -- Note, singular not possible, conflicts with keyword 'update'.
id integer... | Based on what you've got here, it looks like you should be able to JOIN Matrix and Updates together based-on the "Name" field. If you wanted to see all servers from Matrix that had updates, you could to something like this:
<pre><code>SELECT m.Name, m.Description, u.*
FROM Matrix m
INNER JOIN Updates u ON u.Name = m.... | https://dba.stackexchange.com |
386,273 | [
"https://softwareengineering.stackexchange.com/questions/386273",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/320543/"
] | I am getting into SonarQube and everything looks quite and simple so far, but I am not sure what is the final purpose of CCQ overall.
Yes it gives you a lot of tips about whats going on inside your code, and no - it seems to not be perfect everytime.
Are CCQ tools expected to deny a branch merge if there is on exampl... | Quality gates – checks that must pass before some changes can be merged – are a useful way to detect quality problems early. These gates can include any kinds of quality checks, including running test suites or using static analysis tools. The idea is that finding and fixing problems early is much easier (and therefore... | This answer is only an opinion.
To enforce a certain code quality it can be useful to use systems like SonarQube as a guard to prohibit violations merged in your working branch.
But you need to find a compromise among all your developers which rules to enforce this way. If you are too strict you will annoy your develo... | https://softwareengineering.stackexchange.com |
29,013 | [
"https://mechanics.stackexchange.com/questions/29013",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/16919/"
] | What's the preferred, non-chemical/non-additive method for removing oil sludge from an engine that's still installed in the vehicle? I'll have the head and oil pan removed. How do I clean out the oil galleries and other passages in the block without damaging the surface finishes?
| Lots of elbow grease.
Really, if you don't want to use chemicals, you are stuck with using cleaning utensils like putty knife, toothbrush, rags, and whatnot.
Personally, there really isn't a problem with running the engine with a can of SeaFoam (or the like) in it for a period of time (under 100 miles), then dumping... | We use a pressure washer for this job. We remove the oil pan and valve covers. We then pressure wash from every angle we can reach. It makes a big mess, cover up everything you do not want to clean up later. We then reinstall the oil pan an valve covers, fill with oil, add a new oil filter. Then the engine is started a... | https://mechanics.stackexchange.com |
40,928 | [
"https://softwareengineering.stackexchange.com/questions/40928",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/1973/"
] | What is the standard way of describing how software products are released and the proportionate order of magnitude to which the changes relative to the software product are conveyed?
Is
<ol>
<li>Release </li>
<li>Update</li>
<li>Patch </li>
<li>Bug Fix</li>
</ol>
redundant?
or Is
<ol>
<li>Update</li>
<li>Patch <... | Where I work we use the following terminology:
<strong>Snapshot</strong>
Developers share these around with each other (example-0.0.1-SNAPSHOT) and are built over and over while developers get to grips with the various user stories and tasks.
These never go beyond the development servers. To get beyond 0.0.1-SNAPSH... | I personally enjoy a 4-numbered version, however there's no standard for versioning.
The last digit I auto-increment everytime I deploy, so I'm sure I always have a new version without having to worry too much about the significance. Usually I'll increment the third number only when I add a noticeable feature or be... | https://softwareengineering.stackexchange.com |
340,626 | [
"https://physics.stackexchange.com/questions/340626",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/157594/"
] | I want to ask a question about the meaning of the names of some dielectric properties. I'm not a native English speaker, so I don't know why the name of the properties are susceptibility and permittivity. It could be trivial, but I want to understand why they are named this way.
As far as I understand, susceptibility ... | It's mostly about frequency. The feel sensors in your skin have low frequency capability. At high frequencies, they feel only the average.
The average of a sustained sound is 0. You said yourself "<b>±</b>200 Pa". The ± part means that there are both positive and negative pressure peaks. Averaged to... | Whenever you're dealing with human experience, you must avoid mistaking a stimulus with the sensation it causes. In a room at 60 degrees Fahrenheit, a metallic table feels colder than a wooden table, even though their temperatures are the same; that's because you don't experience the temperature of an object, but rath... | https://physics.stackexchange.com |
152,878 | [
"https://softwareengineering.stackexchange.com/questions/152878",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/54893/"
] | When I have a function that might, or might not receive a certain parameter, is it better to overload the function, or to add optional args?
If each one has it's ups and downs - when would I use each?
| If the language supports them properly (e.g. type-safety, if applicable), I would prefer optional arguments for the following reasons:
<ul>
<li>They convey your intent better so noone suspects that your function overload will do something different (which it probably shouldn't anyway).</li>
<li>Less code to maintain, ... | Assuming a constructor kind of situation: I often choose a fluent builder pattern to prevent situations with many options.
Eg.<code>Ordering.natural().onResultOf(function).reverse().compound(Ordering.natural().onResultOf(function2))</code> is an example of calling a fluent builder interface implemented in Guava.
Of... | https://softwareengineering.stackexchange.com |
2,965,085 | [
"https://math.stackexchange.com/questions/2965085",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/475010/"
] | This was a question on an exam I recently took that I didn't do so great on, so I'm trying to understand this problem thoroughly. What flaws are there?
<blockquote>
<strong>Suppose that <span class="math-container">$\{a_n\}$</span> and <span class="math-container">$\{b_n\}$</span> are two convergent sequences that b... | I am not sure whether your current proof is wrong but I am confident that it has some unneeded extra parts. This proof should be <span class="math-container">$3$</span> or <span class="math-container">$4$</span> sentences.
For a simpler approach try this without invoking <span class="math-container">$\sup$</span> or <... | <span class="math-container">$$c_{2n}=b_{2n}$$</span>
<span class="math-container">$$c_{2n+1}=a_{2n+1}$$</span>
<span class="math-container">$$\lim_{n\to+\infty}b_{2n}=\lim_{n\to+\infty}a_{2n+1}=L$$</span>
thus
<span class="math-container">$$\lim_{n\to+\infty} c_{2n}=\lim_{n\to+\infty} c_{2n+1}=L$$</span>
and
<spa... | https://math.stackexchange.com |
205,671 | [
"https://math.stackexchange.com/questions/205671",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/41851/"
] | How would one go about showing the polar version of the Cauchy Riemann Equations are sufficient to get differentiability of a complex valued function which has continuous partial derivatives?
I haven't found any proof of this online.
One of my ideas was writing out $r$ and $\theta$ in terms of $x$ and $y$, then taki... | <em>I happen to have some notes on this question. What follows here is the usual approach, it's just multivariate calculus paired with the Cauchy Riemann equations. I have an idea for an easier way, I'll post it as a second answer in a bit if it works.</em>
If we use polar coordinates to rewrite $f$ as follows:
$$ f(x... | A less standard approach: Take $\{e_{r},e_{\theta}\}$ as a basis in polar coordinates then the Jacobian matrix for any function on $\mathbb{R}^2$ has the form:
$$ [df] = \left[ \begin{array}{cc} U_r & U_{\theta} \\ V_r & V_{\theta} \end{array} \right] $$
<strong>Define</strong> complex-differentiability via com... | https://math.stackexchange.com |
471,217 | [
"https://stats.stackexchange.com/questions/471217",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/274071/"
] | I'm totally aware of that if we look at some loss process <span class="math-container">$L_t$</span>, then <span class="math-container">$\text{VaR}(\alpha)$</span> is a quantile of the loss distribution. If we assume that <span class="math-container">$L_t=-X_t$</span> is the negative returns and they follow a GARCH-mode... | I would say the quoted statement is ambiguous and possibly misleading.
Heteroskedasticity does not affect forecasting but serial correlation would
make point forecast invalid.
In general, forecast implications of residual diagnostics are:
<ol>
<li><strong><em>No heteroskedasticity and no serial correlation</em></stro... | The short answer is that, usually, autocorrelation does not impact the estimates of coefficients, but impacts the variances. That's why he's saying that <em>point forecasts</em> will not change, but confidences will. Also, in time series regression the residuals are almost always correlated.
In other words in a model <... | https://stats.stackexchange.com |
10,650 | [
"https://chemistry.stackexchange.com/questions/10650",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/4805/"
] | This question is based on the Brønsted-Lowry model. I saw many answers to similar questions online. However none of them have given a detailed answer that is based on molecular structure (the answers all told the OP to rely on a reference table for a way to identify strong and weak acids).
| All chemical processes are reversible at least to some extent.
Keeping this in mind, a strong acid must have a weak conjugate <em>base</em> (I think you mean strong acids have weak conjugate bases).
Why? Let's try to prove by contradiction. If a strong acid <em>had</em> a <em>strong conjugate base</em>, then the bas... | Let us solve the mystery of relative 'strength' and 'weakness' of the acids and bases logically by understanding what makes it strong or weak.
It can be said that a weak acid is the molecule which partially dissociate, thus it gives less $\ce{H+}$ ions on dissociation. This is because more energy is needed to break... | https://chemistry.stackexchange.com |
14,722 | [
"https://quant.stackexchange.com/questions/14722",
"https://quant.stackexchange.com",
"https://quant.stackexchange.com/users/4729/"
] | So I was trading the option contracts on NLY (Annaly Capital Managment) today. The stock took a big dip today which piqued my interests in selling some OTM puts. Since the options market on this particular stock is not a very liquid you have very wide bid-ask spread making it tough if you want to get your order filled.... | There's nothing nefarious going on here, it's simply a market maker ensuring they get their fair share (Hah!). Pro-rata allocation at many options exchanges mean market makers simply have to match your price to get a piece of the action. This is a good demonstration as to why that is bad for market structure since the ... | When market is illiquid and market maker want bid price X in book where first order, for example your best bid is at X-Y, then market maker quoting not X, because it not makes any sense, it quotes X-Y+tick and have same fill probability but makes p = X - (X-Y+tick) = Y-tick more profit. You should trade by some order ... | https://quant.stackexchange.com |
474,087 | [
"https://electronics.stackexchange.com/questions/474087",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/239286/"
] | When we start charging the capacitors the charge on the other plate is always opposite & equal then how come we say current lead voltage by 90°, should not we say that it lag/lead by 180°.
| Actually your text has a proper physical interpretation.
When a capacitor is charged with sinusoidal current and the potentials of both plates are observed independently against a distant reference point in the space, we really can say that the AC components of the potentials have opposite phase angles or as well one... | In a capacitor, current is the derivative of voltage, which means that current is change rate of voltage. (exactly <span class="math-container">\$I=C*dU/dt\$</span>)
If you differentiate a sine wave, you get another sine with 90deg phase lag (aka a cosine).
So the 90deg thing has nothing to do with opposite charges.... | https://electronics.stackexchange.com |
209,806 | [
"https://physics.stackexchange.com/questions/209806",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/94031/"
] | Does the amplitude of the light ray decrease when it moves from a rarer to a denser medium?
I think that since amplitude depends upon the energy of the light ray, it should decrease. This is because of the kinetic energy of the light wave decreases (velocity decreases as light travels from rarer to denser medium), hen... | Fresnel's equations sum up the behaviour of light across medium boundaries in terms of linear dielectric response. This problem was solved in the 19th century.
| The amplitude of the electric field in the medium depends on the medium's permittivity, which is not directly related to its density.
<blockquote>
Kinetic energy of the light wave decreases(velocity decreases as light travels from rarer to denser medium), hence the energy of the wave falls.
</blockquote>
This is n... | https://physics.stackexchange.com |
331,585 | [
"https://physics.stackexchange.com/questions/331585",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/63691/"
] | Let $P$ be a particle of mass $m$ moving frictionlessly on the inside of an inverted cone with apex $O$ and angle of ascent $\alpha$. $P$ is subjected only to the influence of gravity and the normal force provided by the inside surface of the cone. Suppose further that that the initial velocity vector is horizontal and... | The answer is the same reason the planets don't fall into the sun: as the particle gets closer to the center, its angular velocity goes up by angular momentum conservation. At a certain point, this costs so much energy that it's impossible to get any closer; we say the particle is 'repelled by the angular momentum barr... | In real world of course it would spiral in and down because of friction and losing energy to heat and slowing down. Also if you imagine the ball rolling around at a given height in a perfect circle the speed would need to be chosen perfectly to balance forces ... Gravity plus the reaction force ( perpendicular to the ... | https://physics.stackexchange.com |
2,083,990 | [
"https://math.stackexchange.com/questions/2083990",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/326673/"
] | Cauchy-Shwarz Inequality is:
$$(a_1b_1 + a_2b_2 + \cdots + a_nb_n)^2 \leq (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + \cdots + b_n^2)$$
However, it can be manipulated as:
$$\sqrt{(a_1-b_1)^2 + (a_2-b_2)^2 + \cdots + (a_n-b_n)^2} \leq \sqrt{a_1^2 + a_2^2 + \cdots + a_n^2} + \sqrt{b_1^2 + b_2^2 + \cdots + b_n^2}$$
I'm t... | Hint:
Replacing $b_i$ by $-b_i$ one can transform
$$
\sqrt{(a_1-b_1)^2+\cdots+(a_n-b_n)^2}\leq \sqrt{a_1^2+\cdots+a_n^2}+\sqrt{b_1^2+\cdots+b_n^2}
$$
into
$$
\sqrt{(a_1+b_1)^2+\cdots+(a_n+b_n)^2}\leq \sqrt{a_1^2+\cdots+a_n^2}+\sqrt{b_1^2+\cdots+b_n^2}
$$
Then,
\begin{align*}
\sqrt{(a_1+b_1+c_1)^2+\cdots+(a_n+b_n+c_... | Let $\vec{a}(a_1,a_2,...,a_n)$, $\vec{b}(b_1,b_2,...,b_n)$,...,$\vec{z}(z_1,z_2,...,z_n)$.
Hence, $|\vec{a}|+|\vec{b}|+...+|\vec{z}|\geq|\vec{a}+\vec{b}+...+\vec{z}|$ and we are done
because in our case we got $|\vec{a}|+|\vec{b}|\geq|\vec{a}+\vec{b}|$.
| https://math.stackexchange.com |
498,694 | [
"https://physics.stackexchange.com/questions/498694",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/227641/"
] | Personally, If I saw this law I would say that momentum is a scalar quantity, because I know that (scalar quantity X scalar quantity = scalar quantity).
Since momentum is a vector quantity, So why there is no any vector quantity in this law to make momentum as an vector quantity (vector quantity X scalar quantity = ve... | That is not the momentum vector, it is the modulus.
If you recall the definition of kinetic energy,
<span class="math-container">$$K=\frac{1}{2} m v^2 = \frac{1}{2} \frac{m^2}{m}v^2 = \frac{1}{2 m} (mv)^2=\frac{p^2}{2m}$$</span>
So <span class="math-container">$K=p^2/2m$</span>, that is
<span class="math-container"... | <span class="math-container">$\vec p = m\,\vec v \Rightarrow \vec p \cdot \vec p = m^2\,\vec v\cdot \vec v \Rightarrow p^2 = m^2 v^2$</span> where <span class="math-container">$p$</span> and <span class="math-container">$v$</span> are the magnitudes of the vector quantities.
Which produce the relationship <span cla... | https://physics.stackexchange.com |
48,470 | [
"https://dba.stackexchange.com/questions/48470",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/10850/"
] | I have Instant File Initialization enabled on our SQL Server, so the 'empty' part of the database file is not zeroed before being allocated. My understanding is that this means that the file could contain 'deleted' data.
So now I want to send a copy of a database (probably a backup file) outside of the company. But ... | A SQL Server BACKUP only backs up <em>extents</em> that are being used to hold data. The unused <em>extents</em> are left behind by the backup. When a page is used for data it will be formatted for use as needed, so that page would be free of old data.
Therefore, all you should need to do is backup the database and... | Have you considered restoring a copy of it, enabling TDE which will force it to 0 out, then disabling TDE again? I haven't tested this but I think it'll work, we'll have to test it out though.
Something else too, have you considered restoring a copy of it with IFI disabled, creating a new filegroup with new files, pr... | https://dba.stackexchange.com |
11,493 | [
"https://engineering.stackexchange.com/questions/11493",
"https://engineering.stackexchange.com",
"https://engineering.stackexchange.com/users/2231/"
] | So let's say I have some heat generating machine be it a combustion engine or a fridge cooled by a fluid. I don't have a single radiator that is big enough for the amount of heat being generated but I do have a couple of smaller ones so I guess I could connect them together.
Should I put the radiators in series (one co... | The efficiency of any radiator (heat exchanger) is a function of the temperature difference between the two fluids in question. All else being equal, a heat exchanger with a greater temperature differential will transfer more heat.
Each radiator will have a temperature gradient across it. (Here I'm talking about how m... | <strong>Assumptions:</strong>
<ul>
<li>"Radiator" means a forced-air air-to-fluid heat exchanger.</li>
<li>Radiators in either setup will pull from their own fresh air source (not from the exhaust of another radiator).</li>
<li>Ignoring radiator design, natural convection, and internal fluid turbulence effects.</li>
<... | https://engineering.stackexchange.com |
35,025 | [
"https://engineering.stackexchange.com/questions/35025",
"https://engineering.stackexchange.com",
"https://engineering.stackexchange.com/users/26446/"
] | Problem statement: 2 kg of air at 500 kPa expands adiabatically in a closed system until its volume (V) is doubled and it's temperature become equal to that of surrounding which is at 100 kPa, 5°C. For the process determine
<ol>
<li>maximum work </li>
<li>change in availability</li>
<li>irreversibilty</li>
</ol>
For ... | <ol>
<li>If you simply integrate the internal energy change from pressure-volume changes you are assuming that going from state 1 -> 2 is fully reversible (isentropic). The way the question has been created makes it irreversible, so there is an increase in entropy in your system. If the system changed reversibly your t... | 1) The maximum mechanical work of a process is not the same as the actual work done on/by the system. The former is calculated using the assumptions:
<ul>
<li>reversible</li>
<li>ideal gas</li>
</ul>
In the reversible case,
<span class="math-container">$$ \Delta U = 0 \rightarrow w = -q\ \ \mathrm{IUPAC\ form,\ for\... | https://engineering.stackexchange.com |
335,912 | [
"https://electronics.stackexchange.com/questions/335912",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/8784/"
] | People tell me I shall not use my CB-radio station without an antenna connected or it will result in damage of the device. Is this true and why is this?
| You could potentially break some radio transmitters by operating them without the antenna connected.
Several things combine to make this possible.
First, it's difficult to make power at RF frequencies, so the power devices are often fairly fragile, and run near their limiting conditions.
Secondly, radio signals passing... | Just to complement the excellent answer of Neil_UK and stress the fact that at RF frequencies voltages and currents don't really behave as those nice entities you know from KCL and KVL.
You must drop Kirchhoff's laws and get your hands dirty with transmission lines theory, where the same concepts of voltage and curre... | https://electronics.stackexchange.com |
7,899 | [
"https://stats.stackexchange.com/questions/7899",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/3376/"
] | I need to draw a complex graphics for visual data analysis.
I have 2 variables and a big number of cases (>1000). For example (number is 100 if to make dispersion less "normal"):
<pre><code>x <- rnorm(100,mean=95,sd=50)
y <- rnorm(100,mean=35,sd=20)
d <- data.frame(x=x,y=y)
</code></pre>
1) I need to plot ra... | Does the picture below look like what you want to achieve?
<img src="https://i.stack.imgur.com/NDRc2.png" alt="enter image description here">
Here's the <strong>updated</strong> R code, following your comments:
<pre><code>do.it <- function(df, type="confidence", ...) {
require(ellipse)
lm0 <- lm(y ~ x, dat... | For point 1 just use the <code>cex</code> parameter on plot to set the point size.
For instance
<pre><code>x = rnorm(100)
plot(x, pch=20, cex=abs(x))
</code></pre>
To have multiple graphs in one plot use <code>par(mfrow=c(numrows, numcols))</code> to have an evenly spaced layout or <code>layout</code> to make more c... | https://stats.stackexchange.com |
37,054 | [
"https://security.stackexchange.com/questions/37054",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/25204/"
] | In addition to the standard best practices on password logins. Has anyone considered a captcha like system but instead of just typing out what is in the captcha, the captcha will be directions on what you must do to your password during the login.
For example if my password was "password123" on the login page if I use... | Interesting idea, but there are a number of reasons why this does not add as much security as you hope.
First, software based keyloggers typically have luxury of being able to not only receive what you type, but also see what you see. In particular, if a bank introduced such a system then a short while later malware w... | Furthermore not adding too much security as stated, the changes to the current password systems to allow your mechanism would make the password management weaker.
Current systems doesn't compare your answer with saved password but they compare encrypted answer with encrypted saved password (the system is unable to kno... | https://security.stackexchange.com |
470,242 | [
"https://stats.stackexchange.com/questions/470242",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/225557/"
] | Suppose we have <span class="math-container">$X_1,...,X_n$</span> iid the shifted exponential distribution:
<span class="math-container">$$f(x)=\lambda e^{-\lambda(x-\theta)}, x\ge \theta$$</span>
I have figured out both the MLE for <span class="math-container">$\lambda$</span> and <span class="math-container">$\thet... | For consistency, by the weak law of large numbers <span class="math-container">$\bar X_n \stackrel{\text p}\to \frac 1\lambda + \theta$</span> and <span class="math-container">$X_\min \stackrel{\text p}\to \theta$</span> so by Slutsky
<span class="math-container">$$
\bar X_n - X_\min \stackrel{\text p}\to \frac 1\lamb... | Although you are also asking about the estimator <span class="math-container">$\hat{\lambda}$</span>, I am going to note some things about <span class="math-container">$\hat{\theta}$</span>. In this particular case it is quite easy to obtain the <em>exact</em> distribution of this estimator. Since you have a series o... | https://stats.stackexchange.com |
9,596 | [
"https://stats.stackexchange.com/questions/9596",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/71/"
] | The health screening for immigration to the US includes a mandatory chest x-ray screening for all adults, with the aim of detecting signs of tuberculosis. If the x-ray indicates the possibility of TB, the applicant then submits sputum samples for culturing. That's a lot of x-rays - and many of them don't result in po... | You haven't specified what "modeling" you plan on, but it sounds like you're asking about how to select <em>independent</em> variables among $A$, $B$, and $C$ for the purpose of (say) regressing a fourth <em>dependent</em> variable $W$ on them.
To see that this approach <em>can</em> go wrong, consider three independen... | If you have only 3 IVs, why do you want to reduce them?
That is, is your sample very small (so that 3 IVs risks overfitting)? In this case, consider partial least squares
Or are the measurements very expensive (so, in future, you'd like to measure only one IV)? In this case, I'd consider looking at the different re... | https://stats.stackexchange.com |
273,673 | [
"https://softwareengineering.stackexchange.com/questions/273673",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/168501/"
] | Say you have some basic code where similar operations will take place in nearby lexical scopes. Take for example some simple pseudo code:
<pre><code>variable = "foo"
# Do something with variable
if (True) {
variable = "bar"
# Do something else with variable
}
for i in range 1..100 {
variable = i
# Do a... | If the variable in question represents the same thing for both functions, I can't see why it would be a problem. If you're arbitrarily using <code>variable</code> to mean "any variable within a function that can do anything" then yes, it is a problem. Name your variables in the context to which they are used.
| There are two entirely different things at play here: <strong><em>variable reuse</em></strong> and <strong><em>variable name reuse</em></strong> (redeclaration.) Your sample pseudocode does not make it clear which one of the two cases you are referring to, so I will mention both.
This is variable reuse:
<pre><code>i... | https://softwareengineering.stackexchange.com |
67,164 | [
"https://stats.stackexchange.com/questions/67164",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/13864/"
] | I have a set of data, called $S$, on an interval of values $[a,b]$ (which is not $[0,1]$). I am searching for a distribution that fits the data. I have tried the normal, lognormal, gamma, etc. None of them fit the data well.
I get the best fit with the beta distribution, after normalizing the data to the interval [0... | The transformed beta distribution you use is explicitly bounded on the interval $[a,b]$. Any values outside of that interval must have probability 0 by the law of total probability. You are explicitly making the statement that an observation outside of that interval is impossible. Therefore, it's not surprising that w... | If your distribution is defined to exist on the interval $[a, b]$ where $a$ and $b$ are both known then it is both legal and sensible to consider a beta distribution after scaling to $[0,1]$.
But the bounds really must apply; you can't at the same time have even one value outside that interval.
Also, the bounds mus... | https://stats.stackexchange.com |
3,501,715 | [
"https://math.stackexchange.com/questions/3501715",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/739267/"
] | Somebody can help me with this polynomial problem? I tried something but i am not really sure if i can finish with that. Thank you!
<blockquote>
Let <span class="math-container">$P$</span> a polynomial with integer coefficients for which exists <span class="math-container">$2$</span> integer numbers, one odd, one ev... | Suppose that <span class="math-container">$P$</span> has an integer zero; let us call it <span class="math-container">$n$</span>. Let <span class="math-container">$Q(x)=P(x+n)$</span>. Then <span class="math-container">$Q(x)$</span> is also a polynomial with integer coefficients and furthermore <span class="math-contai... | Since <span class="math-container">$P(\text{some even})=\text{odd}$</span>, the constant term must be odd, since the non-constant terms will always evaluate to an even number with an even argument. By the rational root theorem, then, any integer zero of <span class="math-container">$P$</span> must be odd.
Now consider... | https://math.stackexchange.com |
369,772 | [
"https://softwareengineering.stackexchange.com/questions/369772",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/205106/"
] | So, from my understanding, Html Helpers are supposed to be for single elements or small snippets of html. However, I'm not really sure why I think this, or how I came to abide by the practice. I couldn't find anything from Microsoft that covers this topic either.
The reason I ask is that recently I've been asked to bu... | Yes. I would say that the static HTML helper style of control is a bit unusual and seen as bad practice unless you have a good reason to use it.
The reason being that you have to build the html in code rather than templating. which makes it troublesome to change and maintain.
The alternative would be a set of viewmod... | If you're questioning your code, then it probably means someone else will too. Always keep in mind the SOLID principles, specifically Single Responsibility. I've build large markup pages in the past, what I believe works the best is to have smaller components that make up the page. With Razor you can leverage partials.... | https://softwareengineering.stackexchange.com |
75,064 | [
"https://mathoverflow.net/questions/75064",
"https://mathoverflow.net",
"https://mathoverflow.net/users/799/"
] | I'm interested in finding an algebra with elements x,y which are identified by every finite-dimensional module. I'm primarily interested in the case that everything is over the complex field, but answers over other fields would also be interesting.
| The Weyl algebra, generated by $x$ and $y$ subject to the single relation $xy-yx=1$, is an example. A silly one, one may add: it does not have any finite dimensional representation!
There are more interesting examples: let $\mathfrak g$ be the Lie algebra with basis $x_i$, $y_i$, with $i\in\mathbb Z$, and $z$, such th... | A similar example to Mariano's is the algebra generated by x,y subject to the relation $xy=1$. In a finite dimensional representation, yx must of course map to the same element as 1. However, there is a faithful infinite dimensional rep where x,y go to a unilateral shift and its adjoint so yx is not 1.
| https://mathoverflow.net |
107,327 | [
"https://datascience.stackexchange.com/questions/107327",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/131342/"
] | I asked this question on stack overflow and was told that this is a better place for it.
I am confused with the terms validation and testing, is validating the model same as testing it? is it possible to use testing data for validation?
what even confuses me more is that when to use validation? is it a necessary step f... | Usually you first split your dataset into train/test set, and then if your model training process requires a validation set, you can further split your train-set into the final train-set and the validation-set. A simple rule is that the test set never shows up in your model development process, including when you devel... | You can use the testing data to perform hyperparameters optimization to see which hyperparameters of your model pipeline work the best. The validation data is then only used once to see how the whole model pipeline works on out of sample data. For this process the test dataset cannot be used again as this data was alre... | https://datascience.stackexchange.com |
3,363,223 | [
"https://math.stackexchange.com/questions/3363223",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | Trying to show for all <span class="math-container">$n$</span> there is a positive <span class="math-container">$x_0>0$</span> such that for all <span class="math-container">$x\geq x_0$</span>:
<span class="math-container">$$e^{-x}\leq \frac{n!}{x^n}$$</span>
The function <span class="math-container">$e^{-x}-\frac{... | From the series expansion <span class="math-container">$e^{x} \geq \frac {x^{n}} {n!}$</span> for all <span class="math-container">$x >0$</span>. This inequality is the same as the one you are trying to prove. Any <span class="math-container">$x_0>0$</span> is fine.
| Over <span class="math-container">$\mathbb{R}^+$</span>, the only stationary point of <span class="math-container">$f(x)=x^n e^{-x}$</span> occurs at <span class="math-container">$x=n$</span>, since this is the only solution of <span class="math-container">$f'(x)=0$</span>.
This stationary point is an absolute maximum... | https://math.stackexchange.com |
3,240,277 | [
"https://math.stackexchange.com/questions/3240277",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/670036/"
] | Let F be the distribution function of the probability <span class="math-container">$\mathbb{P}$</span> on <span class="math-container">$\mathbb{R}$</span> (induced by some random variable <span class="math-container">$X$</span>).
Prove: <span class="math-container">$\mathbb{P}((a,b]) =\mathbb{P}(a < X \leq b) = F(b)... | You certainly have an error as <span class="math-container">$\mathbb{P}(a < X \leq b) = \mathbb{P}(\{X > a\} \cap \{X \leq b\})$</span> with an intersection rather than a union, so perhaps (assuming <span class="math-container">$a \le b$</span> and so <span class="math-container">$\{X \leq a\} \subset \{X \le b\}... | You have to use continuity at right of <span class="math-container">$F$</span>. Let <span class="math-container">$a_n\to a^+$</span>. I denote <span class="math-container">$\mathbb P_X(A)=\mathbb P\{X\in A\}$</span>. Then
<span class="math-container">$$\mathbb P_X(a,b]=\mathbb P_X\left(\bigcup_{n\in\mathbb N}[a_n,b]\... | https://math.stackexchange.com |
418,267 | [
"https://physics.stackexchange.com/questions/418267",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/127589/"
] | I've heard, that Schrodinger initially tried to interpret the wavefunction that he obtained for an electron as a charge density, but it wasn't correct. I know, that nowadays the modulus squared of his wavefunction is interpreted as a probability density of obtaining the particle in a given space interval, but what were... | As long as you don’t collapse the wave function, its behavior would be similar to a charge density indeed. The density would have a linear relation to $|Ψ|^2$, so that the total charge gives you $-e$.
However, when you measure its position, the wave function collapses, and that would not happen with a charge density. T... | I assume your question is, since the modulus square of a wave function is the probability density, then if a particle is charged, its probability density is also the charge density, with up to a constant in front.
Now if you would like to interpret wave function $\psi$ as probability density, not only its imaginary pa... | https://physics.stackexchange.com |
7,440 | [
"https://mechanics.stackexchange.com/questions/7440",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/2618/"
] | The brake fluid reservoir level drops very fast in my 2007 4WD V6 Toyota Tacoma, probably within 3-5 days of filling it up. The funny thing is, it drops to just below where it says minimum but not further down, which makes me exclude the possibility of brake lines leaking somewhere downstream in the system and suspect... | As another pointed out, your master cylinder could be faulty. I had mine resleeved with
stainless steel. Any brake & clutch specialists like repco can do this. To remove and replace is not complex.
| I just had same issue on Corolla 2006. Dropped brake fluid fast, but master cylinder area dry. The spot on driveway where it would pool was under master cylinder. It ended up being the right caliper, then leaking along frame upper side and escaping coincidentally under the cylinder. I've done lots of repairs in the dri... | https://mechanics.stackexchange.com |
59,095 | [
"https://mathoverflow.net/questions/59095",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3233/"
] | Inspired by other questions i have two questions about modules over division rings: given a division ring $D$ with center $Z(D)=K$. One has the notion of dimension for left modules (vector spaces) $V$ over $D$, which is well behaved like in linear algebra.
But does the following also hold: Given a left vector space $W... | The answer to both questions is positive if $D$ is finite-dimensional over its center $K$, and negative, in general, otherwise.
Q1. Suppose $V$ is a $D$-bimodule over $K$ (i.e., a $D\otimes_K D^{op}$-module), while $W$ is a left $D$-module of dimension $n$, as in your question. Then $W$ is isomorphic to a direct sum... | Concerning Q2, $D$ and $D^* $ are in general not isomorphic as $K$-vector spaces: suppose that $D$ has countable infinite dimension as a vector space over $K$. The dual vector space $D^*$ has then <em>uncountable</em> dimension over $K$. If $K$ is a countable field (say, the rational numbers) it follows that $D$ has a ... | https://mathoverflow.net |
245,276 | [
"https://physics.stackexchange.com/questions/245276",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/55560/"
] | <blockquote>
Let $\hat Q$ be an operator with a complete set of orthonomal eigenvectors:
$$
\hat Q |e_n\rangle=q_n|e_n\rangle\ \ (n=1,2,3,...)
$$
Show that $\hat Q$ can be written in terms of its <strong>spectral decomposition</strong>:
$$
\hat Q=\sum_n q_n|e_n\rangle\langle e_n|.
$$
</blockquote>
Firstly in c... | Hint:
if $|e_i\rangle$ is the basis any vector $$|\alpha \rangle= \sum a_i |e_i \rangle$$
So, $\langle e_i|\alpha \rangle = a_i$ (Fourier inverse)
Apply $Q$ on both sides
$$Q|\alpha \rangle=\sum_i a_i Q|e_i \rangle$$
So,
$$Q|\alpha \rangle=\sum_i a_i q_i|e_i \rangle$$
$$Q|\alpha \rangle=\sum_i \langle e_i|\alp... | If I understood your question correctly there seems to be a confusion with the two summations.
Since the basis is orthonormal we can say that, for any two basis vectors $|e_m\rangle$ and $|e_n\rangle$.
$$\langle e_m|e_n\rangle = \delta_{mn}$$
i.e $$\delta = 1, m=n$$ and $$\delta = 0, m\neq n$$
Lets expand the last eq... | https://physics.stackexchange.com |
50,100 | [
"https://stats.stackexchange.com/questions/50100",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/20856/"
] | Given U is an uniform random variable on [0,1] and F is a cdf with inverse $F^{-1}$. I want to prove that $F^{-1}(U)$ has distribution F. Is the following proof correct:
$F_{X}^{-1}(u) = x$ <=> $F_{X}(x) = u$
Proof:
$F_{X}(x) = F_{U}(u)$ =>
$\int_{0}^{u} P_{u}(u)du = \int_{0}^{1} 1 du = u.$
Is this correct?
| Suppose that the given CDF $F(\cdot)$ is a one-to-one mapping from $(-\infty,\infty)$ to $(0,1)$. Then, the inverse function $F^{-1}$ is a one-to-one map from $(0,1)$ to
$(-\infty,\infty)$ with the property that
if $F(x) = u$ for a given $x \in (-\infty,\infty)$, then $0 < u < 1$,
and $F^{-1}(u) = x$.
Now $X=F... | Start like this: Let $X = F^{-1}(U)$. Then $F_X(x) = P(X\leq x) = P(F^{-1}(U)\leq x) =...$
| https://stats.stackexchange.com |
4,253,623 | [
"https://math.stackexchange.com/questions/4253623",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/952495/"
] | I am currently studying series and I have a question which is more of a detail...Let’s say that I have the following series: <span class="math-container">$$\sum_{n=0}^{\infty}\frac{(-1)^n 100}{2n+3}$$</span>
and I want to check for absolute convergence, studying this series, <span class="math-container">$$\sum_{n=0}^{\... | Yes, you can. <span class="math-container">$\left|\frac{(-1)^n 100}{2n+3}\right| = \frac{100}{2n+3}$</span>
, so <span class="math-container">$\sum\limits_{n=0}^{\infty} \left|\frac{(-1)^n 100}{2n+3}\right|$</span> and <span class="math-container">$\sum\limits_{n=0}^{\infty} \frac{100}{2n+3}$</span> are just the same s... | As already noticed, no further step is strictly needed but we could add one more step as follows
<span class="math-container">$$\sum_{n=0}^{\infty} \left|\frac{(-1)^n 100}{2n+3}\right|= \sum_{n=0}^{\infty} \frac{\left|(-1)^n\right| 100}{2n+3}= \sum_{n=0}^{\infty} \frac{100}{2n+3}$$</span>
to emphasize we are acting on ... | https://math.stackexchange.com |
355,876 | [
"https://softwareengineering.stackexchange.com/questions/355876",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/73508/"
] | Something that bugs me with sprint planning, is where training fits in. Let's say you have a requirement to learn JQuery for a small web application. There would seem to be a number of possible approaches - each with their own potential pitfalls.
<strong>Specific task</strong>
A task is added to the sprint to learn t... | I think you're over-thinking it. Treat it like anything else that isn't directly tied to developing the product (eating lunch, taking a break, attending a presentation, a short work-week due to a holiday, ...)
That is to say, be aware of it during sprint planning and adjust the number of stories you pull in to account... | Within the context of Scrum, I see a few different possibilities:
<ol>
<li><strong>The training is in the form of an official course with required/expected attendance at specific dates.</strong> In this case, I would treat those course dates as planned vacation days as far as the sprint planning goes. This means that ... | https://softwareengineering.stackexchange.com |
36,174 | [
"https://stats.stackexchange.com/questions/36174",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/11634/"
] | My understanding is the the random forest picks randomly <em>mtry</em> variables to build each decision tree. So if mtry=ncol/3 then each variables will be used on average in 1/3 of the trees. And 2/3 of the trees will not use them.
But what if I know that a single variable is probably very important, would it be good... | Note that <em>mtry</em> is the number of variables randomly sampled as candidates at each split. And from this candidates the best is choosen to perform splitting. Thus the proportion you have mentioned is not satisfied completely. More important variables appear more frequently, and less important – less frequently. S... | Since June 2015, a new promising RF algorithm on R-CRAN called 'ranger' has this feature. It is tweaked with, <strong>split.select.weights</strong>: <em>"Numeric vector with weights between 0 and 1, representing the probability to select variables for splitting."</em>
| https://stats.stackexchange.com |
1,135,890 | [
"https://math.stackexchange.com/questions/1135890",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/213382/"
] | I know that the the kernel of a vector L : V → W between two vector spaces V and W, is the set of all elements v of V for which L(v) = 0. From here on I have no idea what to do.
| The statement
<blockquote>
$\ker(A)$ is a subset of $\ker(A^2)$
</blockquote>
means
<blockquote>
if $\def\v{{\bf v}}\v$ is in $\ker(A)$, then $\v$ is in $\ker(A^2)$,
</blockquote>
that is,
<blockquote>
if $A\v={\bf0}$ then $A^2\v={\bf0}$.
</blockquote>
Assuming $A\v={\bf0}$, can you explain why $A^2\v={\bf0... | Here's a hint: If $v\in\ker A$, then that means $Av = 0$ as you note. You want to show that if $v\in\ker A$, i.e. $Av = 0$, then $A^2 v = 0$ or equivalently $A(Av) = 0$. Do you see how to proceed?
By the way, you mean to say "is a subset of", not "is an element of" in your title.
| https://math.stackexchange.com |
116,800 | [
"https://mathoverflow.net/questions/116800",
"https://mathoverflow.net",
"https://mathoverflow.net/users/30098/"
] | Hello there, i need to solve this problem:
I have 2 different bi-partite weighted graph, g1 and g2 and i would like to measure their similarity, g1 and g2 may have different number of vertex and edges and they are a result of a clustering algorithm over different data-sets.
Ideas,hints,thoughts are HIGHLY appreciated... | Is it enough to have something which is defined or do you also want it to be relatively easy to compute?
We can say (as you do) that distance is $0$ when and only when the two graphs are identical in the sense that they have equal numbers of vertices and edges and corresponding edges have the same weight. But it can b... | You might try tackling the problem from the other end in more detail:
given a (large) set S of relational structures, what functions from SxS to
the real interval [0,1] will serve as similarity measures?
If you can place a partial order on S and ask that the measure preserve
some aspects of that order, that will limit... | https://mathoverflow.net |
335,031 | [
"https://mathoverflow.net/questions/335031",
"https://mathoverflow.net",
"https://mathoverflow.net/users/104633/"
] | If <span class="math-container">$A \in M_n(\mathbb{C})$</span>, then <span class="math-container">$A$</span> is called <em>reducible</em> if there is a permuation matrix <span class="math-container">$P$</span> such that
<span class="math-container">$$
P^\top A P =
\begin{bmatrix}
A_{11} & A_{12} \\
0 & A_{22}... | The matrix <span class="math-container">$P^\top AP$</span> is doubly stochastic. Suppose that <span class="math-container">$A_{11}$</span> is <span class="math-container">$m_{1} \times m_{1}.$</span> Note that the sum of the entries of each column of <span class="math-container">$A_{11}$</span> is <span class="math-con... | Here is a proof of the fact claimed by Perfect and Mirsky.
<strong>Proof.</strong> Let <span class="math-container">$A$</span> be doubly stochastic and let <span class="math-container">$B = P^TAP$</span> be a Frobenius normal form of <span class="math-container">$A$</span>, which is given as in the question. Then <spa... | https://mathoverflow.net |
581,337 | [
"https://math.stackexchange.com/questions/581337",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/92974/"
] | This is one of the problem in my text book where the section which the problem is stated talks about the mean value theorem and Rolle's theorem. By looking at this, I have no idea where to start. Can I have some hints??
| $\cos(x)$ is differentiable in $\mathbb{R}$.
By Mean value theorem $|\cos(x) - \cos(y)| = |sin(\xi)||x - y|$ where $\xi \in (x,y)$ or $(y,x)$.
$|\sin(x)| < 1$ $\forall$ $x \in \mathbb{R}$.
Now get your answer.
| This follows also from the standard $|\sin(x)| \leq |x|$:
$$\left| cos(x) - \cos(y) \right|=2 \left| \sin(\frac{x+y}{2}) \sin(\frac{y-x}{2}) \right| \leq 2 \cdot 1 \cdot \left|\frac{y-x}{2} \right|$$
| https://math.stackexchange.com |
166,768 | [
"https://softwareengineering.stackexchange.com/questions/166768",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/65973/"
] | Hello fellow programmers,
I am reading a book on C# and the author is comparing Abstract classes and Interfaces. He claims that if you have the following "abstract class:"
<pre><code>abstract class CloneableType
{
public abstract object Clone();
}
</code></pre>
Then you cannot do this:
<pre><code>public class MiniV... | The difference between the abstract classes and interfaces in your case could allow the developer to well-define the pattern of your design with abstract class create "is a" relationship and interface create "can do" relationship.
Put it this way, if you have a Toy car and it would be not make sense to make them "is a... | <strong>Short Answer:</strong> Author is just emphasizing that a class may inherit only from one, and ONLY <strong>one Base class</strong>. However, it may have as <strong>many Interfaces</strong> as needed. This is true for C#, VB.NET languages.
However, you may have set of classes that inherit one from another. You ... | https://softwareengineering.stackexchange.com |
304,019 | [
"https://softwareengineering.stackexchange.com/questions/304019",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/139874/"
] | This is supposed to return an infinite <code>list</code> of x's. However a list is created using an element, then the operator <strong>':'</strong> and then a list.
The recursive definition of <code>repeat' x = x:repeat' x</code> seems to never get to the point where an actual list is created as it seams to continuous... | Probably your confusion comes from the fact that you are used to eager evaluation, whereas Haskell uses lazy evaluation.
For example, if you were to use the definition
<pre><code>repeat' x = x : repeat' x
</code></pre>
to evaluate the expression <code>repeat' 10</code> eagerly, then you would get
<pre><code>repeat'... | You seem to have two confusions here: first, how does Haskell ever complete a recursive definition such as <code>repeat x = x:repeat x</code>, and second, how does it know that this does, in fact, define a list.
The first is answered, as the other responses have done, by saying "laziness". Writing <code>repeat x = x:... | https://softwareengineering.stackexchange.com |
7,089 | [
"https://electronics.stackexchange.com/questions/7089",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/206/"
] | I am using DxDesigner for the first time (9.2). When running a Design Rules Check (DRC) there are many rules that can be applied. Many of them make sense to check, e.g. "Property name exceeds maximum length". Others are giving me trouble, e.g. Un-driven Net is an error if I enable it when a net is driven by a transf... | Here's my working list.
Migration are all on:
<pre><code>drc-001 Property can't be mapped to Common Properties ^(~?[a-zA-Z_0-9+-@.# ]+)$ Error
drc-002 Invalid net name format ^(~?[a-zA-Z_0-9+-.]+)$ Error
drc-003 Invalid property value format Error
drc-004 Invalid symbol name format ^(~?... | I can answer for Altium, and maybe you can translate to DxDesigner.
I pay heavy attention to the group of rules labeled "Electrical": clearance, short-circuit, un-routed net, and un-connected pin. I also pay attention to minimum trace width, minimum annular ring, minimum hole size, silkscreen over component pads, mini... | https://electronics.stackexchange.com |
425,232 | [
"https://softwareengineering.stackexchange.com/questions/425232",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/360359/"
] | I'm building at a Build and Deployment pipeline and looking for clarification on a couple points. In addition, I'm trying to implement Trunk Based Development with short-lived branches.
The process I have thus far:
<ol>
<li>Local development is done on the <code>main</code> branch.
</li>
<li>Developer, before pushing t... | Both
<h3>Unit Tests</h3>
Unit test are test for a small piece of code within a larger binary. These tests shouldn't care about the docker image or any other issue.
Though as its intended to run within docker, within a given base image, it does stand to reason that you should run these tests within that platform to elim... | The purpose of the container is to remove variance in the environment. Unit tests shouldn't have any dependence on the environment. You would expect them to run the same either way.
You have integration tests, which should catch any errors anyway. So really its going to come down to practical considerations.
<ul>
<li>D... | https://softwareengineering.stackexchange.com |
1,372,749 | [
"https://math.stackexchange.com/questions/1372749",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/180934/"
] | According to Wikipedia (page about lognormal distribution), if $X \thicksim N(\mu, \sigma^2)$ then $Y=e^X \thicksim \mathrm{logN}(\mu, \sigma^2)$.<br>
But the support of $\mathrm{logN}$ is just $(0,+\infty)$<br>
So assuming $N(0,\cdot)$ I would have a fair chance that an instance of $X$ be negative. What would happen ... | If $x$ is a negative number then $e^x$ is a positive number less than $1$.
The logarithm of a positive number less than $1$ is a negative number.
$e^\text{something negative}$ is a positive number less than $1$. It has a logarithm. Its logarithm is negative.
| The terminology may be a bit confusing, but when we say $Y$ is lognormally distributed, that means the logarithm of $Y$ is normally distributed. In other words if $X \sim \operatorname{Normal}(\mu,\sigma^2)$, then $\log Y = X$, and $Y = e^X \sim \operatorname{LogNormal}(\mu,\sigma^2)$. Then it becomes clear that if $... | https://math.stackexchange.com |
84,991 | [
"https://electronics.stackexchange.com/questions/84991",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/29617/"
] | I have a ST32F103RET6 MCU. I got the datasheet (and refrence manual). When I read the datasheet (and refrence manual), I found this section:
<img src="https://i.stack.imgur.com/3hfJV.png" alt="image">
Now I have several questions about it:
<ol>
<li>Why does the datasheet (and refrence manual) mention this interface?... | That item is referring to the chip's interface to external memories and external memory-mapped peripherals (such as LCD controllers).
The 6800/8080 reference is describing two different "styles" of control signals for memory/IO busses: The 6800 used a R/W line and a common enable line, while the 8080 used separate "re... | From the datasheet:
<blockquote>
The FSMC can be configured to interface seamlessly with most graphic LCD controllers. It
supports the Intel 8080 and Motorola 6800 modes, and is flexible enough to adapt to
specific LCD interfaces. This
LCD parallel interface capability
makes it easy to build cost-
effect... | https://electronics.stackexchange.com |
85,511 | [
"https://mathoverflow.net/questions/85511",
"https://mathoverflow.net",
"https://mathoverflow.net/users/20521/"
] | Can you use the Crank-Nicolson method to get a numerical approximation to the fisher-kolmogorov equation?
If not what would be the easiest way to model the equation using matlab?
Thanks and sorry its so basic but i'm new to these sorts of numerical approximations
| Crank-Nicolson will work fine, though a fully implicit scheme may actually perform a bit better especially if you have discontinuous initial conditions. Since Crank-Nicolson is effectively half explicit, discontinuities in the initial conditions $u(x,0)$ propagate into a lot of noise echoing around your solution grid.... | The easiest solution really depends on the initial conditions and the coefficients. If your coefficients are not too large, quite frankly the easiest way to do so would be a Method of Lines. Approximate the LaPlacian to order 2 in space by the tridiagonal matrix (1,-2,1) and now you have a system of ODEs. In MATLAB if ... | https://mathoverflow.net |
1,135,064 | [
"https://math.stackexchange.com/questions/1135064",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/64953/"
] | That is a simple question and I can't start a simple desenveloment.
Just $k=5$ we have $4^5 = 1024 > 5^4 = 625$
for $k+1$: $4^{k+1} > (k+1)^4\Rightarrow 4^k > (k+1)^4/4$
And How can i proceed after that?
Best,
| Start with the induction assumption,
$$n^4 <4^n$$ multiply both sides by $4$ to get
$$4n^4 < 4^{n+1}$$
now show that
$$(n+1)^4 \leq 4 n^4$$ in which $n\geq 5$ is used.
An easy way to show this last is to prove
$$\left(1+\frac{1}{n}\right)^4 < 4$$ since
$$1+\frac{1}{n} \leq 1+\frac{1}{5}$$ we just have... | A <strong>different</strong> approach using calculus (not induction)
Let $f: \mathbb R_+^* \supset \mathbb N^* \to \mathbb R$ such that $f(x) = \ln x / x$ which has negative derivative for every $x>e$. So
$$n \geq 5 > 4 > e \iff f(n) < f(4) \iff \ln n / n < \ln 4 / 4 \iff \ln (4^n) > \ln (n^4) \iff ... | https://math.stackexchange.com |
9,106 | [
"https://dba.stackexchange.com/questions/9106",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/5070/"
] | I'm attempting to run a (Windows) batch file from within a SQL Server agent job under SQL Server 2008 R2. In the Job Steps, I've created a new step, set its type to "Operating system (CmdExec)". I initially tried to put just the batch file name in the "Command" field. It gives an error that it can't find the file.
Th... | If memory serves me correctly:
<pre><code>cmd.exe /c "c:\test.bat"
</code></pre>
Quotes are needed if there are any spaces in the path or file name
| only <code>"c:\test.bat"</code> will do the job
| https://dba.stackexchange.com |
350,883 | [
"https://softwareengineering.stackexchange.com/questions/350883",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/196048/"
] | <strong>I wanted to know if the way I deal with source files that need to be deleted from version control could be regarded as bad practice.</strong>
I want to explain it to you based on that example:
I recently got very angry because I had to tediously sort out Java classes in a programme that were basically dead c... | The problem with adding a comment to a file that it should be deleted, instead of deleting it in source control and putting the explanation there, is the assumption that if developers do not read commit messages that they will surely read comments in source code.
From an outsider's perspective, this methodology seems ... | <strong>Yes it is bad practice.</strong>
You should put the explanation for the deletion in the commit message when you commit the deletion of the files.
Comments in source files should explain the code <em>as it currently looks</em>.
Commit messages should explain why the <em>changes</em> in the commit were made, so... | https://softwareengineering.stackexchange.com |
399,454 | [
"https://softwareengineering.stackexchange.com/questions/399454",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/346711/"
] | I have a microservice, lets call it microservice A.
It talks to another microservice's REST endpoint, lets call that one microservice B.
I'm monitoring the health of microservice A using the Springboot actuator health endpoint.
I've had a code review comment that says that microservice A 'should' also be checking the... | I think this is unusual. If service B has failed the health check then presumably calls to service B will result in the appropriate HTTP status code (404, 500 etc), and if that's the case then service A can raise it's own exception or handle the outage of service B gracefully (if that's possible).
So there is no need... | To answer this question, we could compare the approaches.
<strong>Reasons to Monitor Services</strong>
<ol>
<li><strong>Monitoring system signalling as a failure of micro service B.</strong> This helps the ops (or dev) team to down to business to quickly fix the issue.</li>
<li><strong>Monitoring system signalling as... | https://softwareengineering.stackexchange.com |
65,941 | [
"https://softwareengineering.stackexchange.com/questions/65941",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/133/"
] | I've a question about the Delegate Pattern :
If I want to delegate the parsing of a xml file, for instance, that I want to be converted into a Key-value Dictionary. Let's take this classical example...
What's the best practice about the creation of that Dictionary Object?
<ul>
<li>Is it the "main" object that crea... | I dont think the delegation pattern concerns its self with this. Delegation is about behaviour, what you appear to be talking about is ownership.
The ownership question really depends on your domain. Your object may have different life times, and as such may be able to legitimately maintain ownership for you, or alter... | I think it is better to have the delegate create the obect to fill and return.
This allows for polymorphism in the solution where the caller does not have knowledge about the actual implementation details other than that the object created implements the interface that is part of the contract.
It is up to the delega... | https://softwareengineering.stackexchange.com |
4,080,946 | [
"https://math.stackexchange.com/questions/4080946",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/865005/"
] | I have a short question. Would <span class="math-container">$\mathbb{C}^{2 \times 2}$</span> represent a Cartesian product of four complex planes? Or, could it represent the set of all <span class="math-container">$2 \times 2$</span> matrices holding complex numbers? Are both definitions synonymous?
My apologies if the... | <span class="math-container">$$\underline{x}\in\Bbb C^4=[x_1, x_2, x_3, x_4]^T :x_j\in\Bbb C$$</span>
<span class="math-container">$$\underline{x}\in\Bbb C^{2\times 2}=\bigg[\begin{matrix} x_1 \ x_2 \\x_3 \ x_4\end{matrix}\bigg]:x_j\in\Bbb C$$</span>
Essentially, <span class="math-container">$\Bbb F^n=\overbrace{\Bbb F... | Given a field <span class="math-container">$F$</span> and two natural numbers <span class="math-container">$m$</span> and <span class="math-container">$n$</span>, <span class="math-container">$F^{m\times n}$</span> is the set of all <span class="math-container">$m\times n$</span> matrices with entries in <span class="m... | https://math.stackexchange.com |
37,710 | [
"https://datascience.stackexchange.com/questions/37710",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/57724/"
] | <code>Naive Bayes</code> is called naive because it makes the naive assumption that features have zero correlation with each other. They are independent of each other. Why does naive Bayes want to make such an assumption?
| By doing so, the joint distribution can be found easily by just multiplying the probability of each feature whilst in the real world they may not be independent and you have to find the correct joint distribution. It is naive due to this simplification.
| Naive bayes make such assumptions to simplify the calculations. You can take a look at bayesian belief network which do not make such assumptions
| https://datascience.stackexchange.com |
150,168 | [
"https://mathoverflow.net/questions/150168",
"https://mathoverflow.net",
"https://mathoverflow.net/users/36922/"
] | This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent sheaves be constructed categorically from the category of vector bundles? I am thinking of Coh being some kind of 'abelian... | Here are a few comments that might be useful. I don't think there is a chance that this can work unless the scheme in question has the resolution property (meaning every coherent sheaf is a quotient of a locally free sheaf of finite rank). Otherwise the category of locally free sheaves does not even form a generator of... | Here's a negative answer: there can be no self-dual way to pass from the category of finitely generated projective modules to the category of all finitely generated modules (over, say, a Noetherian ring). To see this, note that the category of finitely generated projective modules is self-dual via the functor $Hom(-,R... | https://mathoverflow.net |
65,962 | [
"https://mathoverflow.net/questions/65962",
"https://mathoverflow.net",
"https://mathoverflow.net/users/5952/"
] | Let us call a group object $G$ in a category $\mathcal C$ <em>rigid</em>, if it has the following property: For every group object $X$ in $\mathcal C$, every morphism $G\to X$ in $\mathcal C$ respecting the unit sections is already a morphism of group objects.
As a formal consequence, rigid group objects are commutat... | Even the circle is not rigid in this sense! With such a weak notion of "group up to homotopy" you can make a multiplication on $G=K(\mathbb Z,1)\times K(\mathbb Z,2)$ such that there is no nontrivial homomorphism from $K(\mathbb Z,1)$.
Edit: Let me explain and generalize, showing that a point is the only example.
For... | Let $A$ be any discrete abelian group. The classifying space $BA$ is then a topological abelian group. If $A=\mathbb{Z}^n$ then $BA$ is an $n$-torus, as in the question. For any connected based space $X$, we have $[X,BA]=H^1(X;A)=\text{Hom}(\pi_1(X),A)$. Now suppose that $X$ is an $H$-space. For any based map $f:X... | https://mathoverflow.net |
195,186 | [
"https://mathoverflow.net/questions/195186",
"https://mathoverflow.net",
"https://mathoverflow.net/users/66471/"
] | I am trying to find the answer of
$$\int dU \ |Tr(U^m)|^2$$
where $m\in\mathbb{N}$ and $U$'s are unitary matrices in $\textit{U}(n)$ and $dU$ is a normalized Haar measure. In the case $m=1$, the answer seems to be $1$.
I don't know where to start. Does anyone have an idea? Is there a clean answer for $m>1$ ?
| $$\int_{{\rm U}(n)} dU\,|{\rm Tr}\,(U^m)|^2={\rm min}\,(n,m).$$
see Theorem 2.1.b of <A HREF="http://statweb.stanford.edu/~cgates/PERSI/papers/functionals.pdf" rel="nofollow">Diaconis and Evans</A> (2001). [*]
[*] This 2001 reference corrects an earlier paper by <A HREF="http://statweb.stanford.edu/~cgates/PERSI/pape... | This is an elaboration on my comment.
Let's start in a more general setting since the comment was not about the groups $U(n)$:
Let $G$ be a compact group and $H$ a normalized Haar measure on it.
Let $p_m:G\to G$ be defined by $p_m(x)=x^m$, $m\in\mathbb Z$.
Now for any continuous function $f:G\to\mathbb R$ we have
$$
\... | https://mathoverflow.net |
892,749 | [
"https://math.stackexchange.com/questions/892749",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/162476/"
] | Let $x,y,z$ be the positive real numbers, if $x^2+y^2+z^2=1$, then how can we find the minimal value of this function $f(x,y,z)=\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}$.
| I will show that,
$$ \frac{xy}{z}+ \frac{yz}{x}+\frac{xz}{y} \ge \sqrt{3}$$
This is equivalent to,
$$ (\frac{xy}{z}+ \frac{yz}{x}+\frac{xz}{y})^2 \ge 3$$
Which is same as,
$$ \frac{x^2y^2}{z^2}+\frac{y^2z^2}{x^2}+\frac{x^2z^2}{y^2} + 2(x^2+y^2+z^2) \ge 3 = 3(x^2+y^2+z^2) $$
So it suffice to prove that,
$$ \frac{x... | <strong>Hint</strong> First, note that $$\dfrac{xz}{y}+\dfrac{yz}{x}+\dfrac{xy}{z}=xyz\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right).$$ Using $a+b+c\geq 3\sqrt[3]{abc}$, try to estimate lower bounds for $xyz$ and $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}$. If you are stuck in the second, consider $$(x^2+y^2+z^... | https://math.stackexchange.com |
45,246 | [
"https://datascience.stackexchange.com/questions/45246",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/57731/"
] | I want to create a deep learning model (CNN) for binary classification, can I used the softmax function instead of the sigmoid function in binary classification?
Adding the classification layer to the model, will be like this
<pre><code>model.add(Dense(2, activation='softmax'))
</code></pre>
| For binary classification, it should give the same results, because softmax is a generalization of sigmoid for a larger number of classes.
| The answer is not <strong>always</strong> a yes. You can always formulate the binary classification problem in such a way that both sigmoid and softmax will work. <strong>However you should be careful to use the right formulation.</strong>
<strong>Sigmoid</strong> can be used when your last dense layer has <strong>a si... | https://datascience.stackexchange.com |
14,052 | [
"https://physics.stackexchange.com/questions/14052",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/4984/"
] | I am on a boat docked at Cape Charles, VA, about 30 or 40 miles from the center of Hurricane Irene. This understandably got me thinking about the force of wind on the boat. Since air friction is proportional to the speed squared (except for some friction types I'm sure someone will be kind enough to remind me of), is t... | On the most basic level I would say that both your suspicion and your reasoning are correct, although some caveats should follow.
First I should address this on the basis of simple kinematics. Yes, the energy of an air molecule is $m v^2$ but we need to formalize when and if this will translate into pressure. Imagin... | Well, I think of it this way. Imagine that your boat and the wind are the only things present in a giant empty space, and let the wind blow head on onto the bow of your boat at a constant speed. Since in this space there are only the wind and your boat (and yourself on it), you can't conclude if it is the wind blowing ... | https://physics.stackexchange.com |
483,851 | [
"https://electronics.stackexchange.com/questions/483851",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/243984/"
] | Disclaimer: this is beginner question.
I can't figure out why o_address gives an inferred latch warning in post-synthesis and how to fix it.
When I try to assign it it gives another warning: design project has port o_address driven by constant 0.
I have also tried to assign it in every single case but it still doesn... | The latch is inferred because your "when" has no "else".
If there is a reset (RST) o_address is set to zero. Fine.<br>
But when the reset goes away what then?<br>
It has to stay that way according to your code so a latch is inferred.
So make sure you assign to o_address in the 'others' section and the latch goes a... | <blockquote>
<blockquote>
o_address <= "00000000000000";
</blockquote>
</blockquote>
is the only assignment it ever gets. It is never assigned another value anywhere. Once it gets set that way, it stays that way forever, hence constant.
The latch is because when others does nothing so all values needs to b... | https://electronics.stackexchange.com |
31,376 | [
"https://mathoverflow.net/questions/31376",
"https://mathoverflow.net",
"https://mathoverflow.net/users/7499/"
] | Given a real oriented vector bundle E over the base space B of rank n, such that the Euler characteristic class in the n-th cohomology group of B vanishes, is it true that there exists a global nowhere-vanishing section of the bundle?
Any idea where to find a proof or a counterexample?
Thanks!
| Hi Dima,
I think that the answer to your question is no: as it is pointed out in the book "Differential Forms in Algebraic Topology" of R. Bott and W. Tu, cohomological invariants are too coarse to ensure the existence of geometrical objects.
More precisely, Example 23.16 of the book of Bott and Tu shows that $S^4$ h... | If $B$ is triangulated, $e(E)\in H^n(B,Z)$ is only the obstruction to have a non-vanishing section <em>on the $n$-skeleton</em> of $B$, but if $\dim B>n$, it is possible that none of these sections extends to the $n+1$ skeleton : the obstruction lies in $H^{n+1}(B,\pi_{n}(S^{n-1}))$, and may be non-zero if $n>2$.... | https://mathoverflow.net |
270,597 | [
"https://mathoverflow.net/questions/270597",
"https://mathoverflow.net",
"https://mathoverflow.net/users/3199/"
] | Suppose we have a finitely presented group $G$ with solvable word problem. (For instance, the command RWSGroup in Magma terminates giving us a finite [but possibly gigantic] rewrite system.) Is there then an algorithm to determine whether $G$ has an element of order 2?
If so, where is the best place to find such an i... | Using a mixture of computation and thought I believe that I have established that this group is indeed torsion-free. I don't know of any general approach to solving that particular problem. Even if the group is hyperbolic (which this example is not, because it has free abelian subgroups of rank $2$), I am not aware of ... | For your given group presentation, I think one may reduce
the problem of finding torsion elements a bit.
We may rewrite the relators as:
$xyx^{-1}= y^{-1}z^{-2}, xyx^{-1} = z^2 y^{-1}, xz^2x^{-1}=z^{-2}, xzx^{-1}=y^{-2}z$.
We get the resulting relations $xyx^{-1}=y^{-1}z^{-2}=z^2y^{-1}, xz^2x^{-1}=z^{-2}=(y^{-2}z)... | https://mathoverflow.net |
4,290,112 | [
"https://math.stackexchange.com/questions/4290112",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/666297/"
] | If <span class="math-container">$$5 \cos^{-1}(\frac{1-x^2}{1+x^2})+7 \sin^{-1}(\frac{2x}{1+x^2})\\ -4 \tan^{-1}(\frac{2x}{1-x^2})-\tan^{-1}x=5 \pi,$$</span> then what is <span class="math-container">$x?$</span>
Attempt: let <span class="math-container">$x = \cot t = \frac{\cos t }{\sin t};$</span> then <span class="mat... | so <span class="math-container">$$5\cos^{-1}(\frac{1-x^2}{1+x^2})+7\sin^{-1}(\frac{2x}{1+x^2})-4\tan^{-1}(\frac{2x}{1-x^2})-\tan^{-1}x=5\pi.$$</span>
let us assume <span class="math-container">$x=\tan t$</span> for <span class="math-container">$t\in(-\frac{\pi}{2},\frac{\pi}{2})$</span>
Proceeding further we will have ... | If <span class="math-container">$x=\tan(t)$</span> for <span class="math-container">$t\in \left(-\frac\pi2,\frac\pi2\right),$</span> this becomes
<span class="math-container">$$5\cos^{-1}(\cos2t)+7\sin^{-1}(\sin2t)\\-4\tan^{-1}(\tan2t)-\tan^{-1}(\tan t)=5\pi.$$</span>
<span class="math-container">$\tan^{-1}(\tan(t))=t$... | https://math.stackexchange.com |
59,432 | [
"https://physics.stackexchange.com/questions/59432",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/2146/"
] | In the Lagrangian for a charged particle in an electromagnetic field
$$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$
the energy of the particle is contained in the kinetic term, the rest being interaction terms of the particle with the electromagnetic field. If it's travelling at some velocity, th... | I think you are asking for a finite well of width L that is from $ -L/2< x< L/2$. Why do we only use
$\psi(x) = A e^{+\kappa x} $ for $ x<-L/2 $
and
$\psi(x) = B e^{-\kappa x} $ for $ x>+L/2 $.
The reason is we want to be able to interpret the wavefunction as the probability density for finding the... | You need to ensure that the wavefunction is normalisable: $$\int\limits_{-\infty}^{\infty} \mathrm{d}x\, |\psi(x)|^2 = 1. $$
This ensures that the wave function yields a valid probability distribution upon application of the Born rule. If you use the normal convention of taking positive $x$ values to the right and nega... | https://physics.stackexchange.com |
4,445 | [
"https://physics.stackexchange.com/questions/4445",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/1708/"
] | A jar is filled with two types of balls, red and green. Red balls have radius $r_1$ and mass $m_1$, green balls have radius $r_2$ and mass $m_2$.
If initially the balls are randomly placed throughout the jar, and we are to shake this jar, generally one type of balls will tend to gather at the top. Why does this happen... | "Composite" observables such as the angular momentum have no "unique" complementary observables. In fact, the three components of the angular momentum are not really "independent" in the sense of spanning a proper configuration space. Only two functions of the three angular momentum components - conventionally $j^2$ an... | Intuitively, spin in the Z direction should have an angle around the Z axis as the complementary variable. Attempts to construct this "angle" operator (or "phase" operator conjugate to the number operator for harmonic oscillators) end up failing some of the desiderata, because the boundedness, continuity, and self-adj... | https://physics.stackexchange.com |
266,353 | [
"https://softwareengineering.stackexchange.com/questions/266353",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/82855/"
] | Every VCS that I've seen includes a means (and sometimes requires you) to make a log of what a particular commit does. If this history is being properly kept, what is the purpose of manually keeping the same history in a regular file?
Note that I'm not positing the changelog is useless, I just don't see a use for it ... | Basically, you're completely right. All the information should be in the VCS as well, and as commenters said, changelogs are often generated out of VCS information.
However, prepared changelogs come with some remarkable advantages:
<ul>
<li>A changelog does not require to connect to the VCS, maybe install the proper ... | When used properly, VCS has too much detail. Every little change is recorded, when a changelog is generally for the big user-visible changes. In short, the target audience for a changelog is not the same as for the commit messages of the VCS.
If you are disciplined, a changelog can be an auto-generated subset of what... | https://softwareengineering.stackexchange.com |
2,053,795 | [
"https://math.stackexchange.com/questions/2053795",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/391383/"
] | I want to know the value of $a$ that makes $f(x)$ continuous
$$f(x)=\frac{x+3}{x^2+ax+9}$$
| If $f(x)$ must be continuous for $x\in \mathbb{R}$
$$x^2+ax+9>0$$
this restriction is fulfilled for
$$a^2-36<0$$
because the quadratic polynomial does not have any real roots, $\textit{i.e.}$ its discriminant is strictly negative.
| We know $f(x)$ is continuous on the whole real line $\mathbb{R}$ iff the denominator is never zero. So we are looking for values of $a$ such that $x^2 + ax + 9$ has no real solutions.
We can tell how many real roots a quadratic equation $p(x) = x^2 + \alpha x + \beta$ has by looking at it's discriminant $\alpha^2 - 4\... | https://math.stackexchange.com |
226,451 | [
"https://mathoverflow.net/questions/226451",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | How do I see that the set $\mathfrak{N}_4$ consisting of all unoriented cobordism classes of smooth closed $4$-manifolds contains at least four distinct elements?
| Thom proved that the unoriented bordism is a polynomial ring over $\mathbb{F}_2$ generated by elements $x_i$ with $i$ running over all numbers not of the form $2^k-1$. Thus, $\mathfrak{N}_4 = \mathbb{F}_2 \cdot \{x_2^2, x_4\}$. Thom moreover proves that the even $x_i$ can be represented by $\mathbb{RP}^i$.
Thus, our ... | You have to find 4 manifolds with different stiefle-whitney numbers.
| https://mathoverflow.net |
132,135 | [
"https://cs.stackexchange.com/questions/132135",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/128344/"
] | The language <span class="math-container">$L=\{W\in\{0,1\}^{*} \mid W=0^{x}1^{y} \text{ where } x\geq0, y>0 \text{ are integers and } y\nmid x\}$</span> is not regular.
How would one prove this using Pumping Lemma?
I thought about it for quite a while but couldn't quite figure it out. Lots of number theory is necess... | You just need to show that
<span class="math-container">$$
\lim_{n \to \infty} \frac{\log^r n}{n^p} = 0.
$$</span>
This is trivial if <span class="math-container">$r \le 0$</span> since <span class="math-container">$\frac{\log^r n}{n^p} = \frac{1}{n^p \log^{-r} n}$</span> and <span class="math-container">$\lim_{n \to \... | Let's consider only <span class="math-container">$r>0$</span>. For <span class="math-container">$\ln(n)^r \in o(n^p),p>0$</span> is enough to show <span class="math-container">$\ln(n) \in o(n^\alpha)$</span>, for <span class="math-container">$\frac{p}{r}=\alpha>0$</span>, because <span class="math-container">... | https://cs.stackexchange.com |
1,134,379 | [
"https://math.stackexchange.com/questions/1134379",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/180936/"
] | $(\sin x)^7 = a\sin 7x+b\sin 5x+c\sin 3x+d\sin x$. Find $d$. $x$ is an angle, and $a, b, c, d$ are all constants.
I am not sure where to start! The only idea I have is to possibly break up $(\sin x)^7$ into smaller parts such as $(\sin x)^2 * (\sin x)^2 * (\sin x)^3$.
Hints only, please.
| <strong>Hint</strong>. Using Euler's formula and related things,
$$\eqalign{
(\sin x)^7
&=\Bigl(\frac{e^{ix}-e^{-ix}}{2i}\Bigr)^7\cr
&=\cdots\quad\langle\hbox{expand by binomial theorem}\rangle\cr
&=\frac{1}{(2i)^7}(e^{7ix}-e^{-7ix}+\langle\hbox{more terms}\rangle)\cr
&=-\frac{1}{64}\sin7x+\l... | We do a much simpler problem, which can be a step in the solution of your problem. The key is the product to sum formula
$$\sin a\cos b=\frac{\sin(a+b)+\sin(a-b)}{2}.\tag{1}$$
We also use the double-angle formula
$$\sin^2 x=\frac{1-\cos(2x)}{2}.\tag{2}$$
Now we do $(\sin x)^3$. This is $\frac{\sin x}{2}-\frac{\sin x\... | https://math.stackexchange.com |
1,836 | [
"https://physics.stackexchange.com/questions/1836",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/293/"
] | I think that something is invisible if it's isolated particles are smaller than the wavelength of visible light. Is this correct?
Why is air invisible? What about other gases and fumes which are visible?
| I think the pithy answer is that our eyes adapted to see the subset of the electromagnetic spectrum where air has no absorption peaks. If we saw in different frequency ranges, then air would scatter the light we saw, and our eyes would be less useful.
| I don't think one can just state that particle size smaller than the wavelength of light implies no interaction with light. You have to look at the quantum mechanical modes of the atoms/molecules. If they have modes with frequencies in the wvaelength range you are interested in, then you will get interaction/absorption... | https://physics.stackexchange.com |
2,724,345 | [
"https://math.stackexchange.com/questions/2724345",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/355377/"
] | How can I find the closed form for the following please?
$$f(r) = \frac{1}{R_x+R_C}\int^{R_x+R_c}_{0}\frac{1}{1+sr^{-4}} dr$$
where $R_x,R_c$ are constants
| A couple things: the integral doesn't depend on $r$; i.e. you won't get out a function $f(r)$ because the $r$ is integrated out. Also, there is (possibly) a singularity in the range of integration if $s < 0$ so I'll assume that $s > 0$ (if $s = 0$ then the integral is trivial).
Put $T = R_x + R_c$. Then the inte... | After User8128's answer, assuming that you enjoy complex numbers, let $a,b,c,d$ to be the roots of $x^4+1=0$ and write $x^4+1=(x-a)(x-b)(x-c)(x-d)$.
Using partial fraction decomposition, we then have
$$\frac 1{x^4+1}=\frac{1}{(a-b) (a-c) (a-d) (x-a)}+\frac{1}{(b-a) (b-c) (b-d) (x-b)}+\frac{1}{(c-a) (c-b) (c-d) (x-c)}+... | https://math.stackexchange.com |
68,531 | [
"https://dsp.stackexchange.com/questions/68531",
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"https://dsp.stackexchange.com/users/44793/"
] | I am designing a 2nd order IIR digital filter :
My tf equation with coefficients is
<pre><code>b = [0 1.209e09]
a= [9.2175 -2.6952 1.0000]
sys=tf(b,a,0.1,'Variable','z^-1')
bode(sys)
</code></pre>
I have couple of questions :
<ol>
<li>How to get unity scaling .. as a formula... so that i can always multiply it wit... | It's important to specify at which frequency you want unity gain. But assuming you mean DC (<span class="math-container">$\omega=0$</span>), because that filter has a low pass characteristic, the DC gain of an IIR filter is given by
<span class="math-container">$$G_{DC}=\frac{\sum_kb[k]}{\sum_ka[k]}\tag{1}$$</span>
It'... | <pre><code>b = [0 1.209e09]
a= [9.2175 -2.6952 1.0000]
% original -----------------------
sys = tf(b,a,0.1,'Variable','z^-1');
% fixes --------------------------
sys=sys/dcgain(sys);
% scale coefficients by b(2)
sys1=tf(b/b(2),a/b(2),0.1,'Variable','z^-1')
sys1=sys1/dcgain(sys1);
bode(sys,'-', sys1,'--')
</co... | https://dsp.stackexchange.com |
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