qid int64 1 4.65M | metadata listlengths 3 3 | prompt stringlengths 31 25.8k | chosen stringlengths 17 28.2k | rejected stringlengths 19 40.5k | domain stringclasses 28
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|---|---|---|---|---|---|
103,046 | [
"https://stats.stackexchange.com/questions/103046",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/48211/"
] | I am attempting to analyze biological data, to see whether the number of events in a given time interval is more/less than expected based on the overall frequency. How would one approach this?
An example of how I would frame this:
Out of 100 ms, 16/44 events occur in 15 ms, and 28/44 events occur in the remaining 85 ... | <blockquote>
to see whether the number of events in a given time interval is more/less than expected based on the overall frequency.
An example: Out of 100 ms, 16/44 events occur in 15 ms, and 28/44 events occur in the remaining 85 ms. Do more events occur in the 15 ms interval than expected based on the overall freque... | If you know the times of the events to a precision about 100 times that of the average interval or greater, it is possible to use event interval analysis to examine changes in rate. In such an analysis, the intervals between events are used as the primary data, and a uniform distribution of events is the typical null h... | https://stats.stackexchange.com |
100,681 | [
"https://physics.stackexchange.com/questions/100681",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/36631/"
] | Say you have two objects colliding, and there is some elasticity between them. Some of the kinetic energy of the objects will change into elastic potential energy when they collide, but when they bounce off each other again, the energy does not return to being kinetic. What happens with it? Does it stay stored as elast... | Sounds like you're getting at the "coefficient of elasticity," which is a value in [0,1] which represents what percent of the pre-collision kinetic energy is found after the collision. In homogeneous materials, the remainder of the energy is typically lost to deformation or heat (phonons) as you suggest.
you could ima... | <blockquote>
but when they bounce off each other again, the energy does not return to being kinetic
</blockquote>
The potential energy actually does go back to kinetic energy. But often it only partially goes to the translational motion of whole body, while other part of the energy comes into internal motion of atom... | https://physics.stackexchange.com |
1,868 | [
"https://dsp.stackexchange.com/questions/1868",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/39/"
] | I would like to be able to quickly determine whether a given 2D kernel of integer coefficients is separable into two 1D kernels with integer coefficients. E.g.
<pre><code> 2 3 2
4 6 4
2 3 2
</code></pre>
is separable into
<pre><code> 2 3 2
</code></pre>
and
<pre><code> 1
2
1
</code></pre>
The ... | I have taken <code>@Phonon</code>'s answer and modified it somewhat so that it uses the GCD approach on just the top row and left column, rather than on row/column sums. This seems to handle pathological cases a little better. It can still fail if the top row or left column are all zeroes, but these cases can be checke... | Maybe I'm trivializing the problem, but it seems like you could:
<ul>
<li>Break the $N$-by-$M$ matrix $\mathbf{A}$ into rows $\mathbf{a_i}$, $i = 0, 1, \ldots , N-1$.</li>
<li>For each row index $j > 0$:
<ul>
<li>Elementwise divide $\mathbf{a_j}$ by $\mathbf{a_0}$ to yield the ratio of each element in row $j$ to i... | https://dsp.stackexchange.com |
32,088 | [
"https://ai.stackexchange.com/questions/32088",
"https://ai.stackexchange.com",
"https://ai.stackexchange.com/users/20721/"
] | Hyperparameter tuning is the process of selecting the optimal hyperparameters for an ANN.
Now, my guess is that, if we have sufficient data (say, 1.4 million for, say, 6 features), the model can be optimally trained and we don't need a hyperparameter tuner (like Keras-Tuner), because, while training, the data itself wi... | Unfortunately, even with large amounts of training data, hyperparameter choices can strongly influence the performance of a trained model.
What you can usually drop when you have large amounts of training data is regularisation. If your training examples cover the function space you are learning really well, then it is... | You don't NEED a hyperparameter tuner, but it can help in various situations. For example, if your model is not training well, perhaps using a tuner can help.
It's hard to say in which hyperparameters you would be turning over in your specific model, but for some specific hyperparameters if you choose a bad value your... | https://ai.stackexchange.com |
116,782 | [
"https://datascience.stackexchange.com/questions/116782",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/108053/"
] | I have math formula and some data and I need to fit the data to this model.
The math is like. <span class="math-container">$y(x) = ax^k + b$</span> and I need to estimate the <span class="math-container">$a$</span> and the <span class="math-container">$b$</span>.
I have triet gradient descend to estimate these params b... | If you know <span class="math-container">$k$</span>, which it seems you do, then this is just a linear regression. In fact, with just one feature (the <span class="math-container">$x^k$</span>), this is a simple linear regression, and easy equations apply without you having to resort to matrices.
<span class="math-cont... | Just to add to Dave's fine answer, I would like to describe how I would implement the fitting in practice:
In Excel or Google Sheets, I would import my <span class="math-container">$y$</span> and <span class="math-container">$x$</span> data into two columns. I would then create a third column that is <span class="math-... | https://datascience.stackexchange.com |
326,059 | [
"https://softwareengineering.stackexchange.com/questions/326059",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/73535/"
] | I already have experience working with MVC based web apps and started reading recently about REST.
But went into confusion when re-thinking about how existing web app not using any kind of framework/pattern could be refactored to use these two.
So the application (entirely written in PHP) already has a web UI but it'... | Let's assume that if you're talking about a MVC design then you're talking about a server side implementation. The (UI) controller will accept a request, invoke business logic (model), render a view and return it to the client.
Let's assume that if you're talking about a REST version of a website, you are moving to a... | <h3>First what is a RESTful API</h3>
To clear up a few things RESTful APIs (aka a service) are a means to have your applications communicate. You create a service and then many clients that can make request to it.
What is a client? These are many things like the front-end web UI, a reporting engine, the invoicing syste... | https://softwareengineering.stackexchange.com |
2,940,385 | [
"https://math.stackexchange.com/questions/2940385",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/584258/"
] | The conjugate of a function <span class="math-container">$f$</span> is given (for some <span class="math-container">$y \in \operatorname{dom}(f)$</span>) as:
<span class="math-container">$$f^*(y) = \sup_{x \in \operatorname{dom}(f)}\left(y^Tx - f(x)\right)$$</span>
It is known that <span class="math-container">$f^*$<... | For any given <span class="math-container">$x \in \operatorname{dom} f$</span>, the function:
<span class="math-container">$$y \mapsto y^\top x - f(x)$$</span>
is an affine function, which is convex and lower semicontinuous (and, indeed, continuous). The epigraph of these functions is therefore closed and convex.
Then... | As I wrote this question I recalled the following fact, and have attempted to prove this property of conjugate functions using it.
<blockquote>
If <span class="math-container">$h(y,\,x)$</span> is convex in <span class="math-container">$y$</span> for each <span class="math-container">$x \in \mathcal{A}$</span>, then... | https://math.stackexchange.com |
176,477 | [
"https://security.stackexchange.com/questions/176477",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/120534/"
] | This question concerns dictionary attacks conducted:
<ul>
<li>Over the Internet, using programs like THC Hydra</li>
<li>Via protocols such as HTTP, FTP and SMTP</li>
</ul>
I believe I'm right in thinking that: a) due to the sophisticated layers of security they tend to employ, such an attack cannot be run successfull... | <strong>No</strong>. There are two ways of zip encryption, a classic one, which is weaker, and a newer one based on AES.
In both cases the password is needed in order to decrypt the contents (i.e. it's not just UI, where you could be asked for a password without the program actually requiring it to read the file). So ... | No, nowadays zip files are protected by AES. This will hide all of the plaintext in a way that you can only recover it when you have the key. The only other thing you can see is the size of the plaintext as that is as good as identical to the size of the ciphertext.
Hexadecimals are just a readable representation of t... | https://security.stackexchange.com |
48,136 | [
"https://chemistry.stackexchange.com/questions/48136",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/25206/"
] | Some
sources that I studied say that it is possible to produce ethane
through the electrolysis of ethanoic acid. Would this work with vinegar
(5-10% acid)? Also, other sources say that ethanoate salts should be used.
I can easily obtain such salts (sodium hydrogen carbonate (baking soda)
+ ethanoic acid (vinegar))... | Ethane is formed at the anode through $\ce{CH3COO- -> CH3. + CO2}$ and then $\ce{2 CH3. -> C2H6}$.
Since acetate solutions have a higher concentration of acetate anions than acetic acid I would use an acetate solution to get a higher concentration of acetate at the anode in favor of the formation of ethane.
| Indeed that is quite possible. The reaction was first described by Hermann Kolbe and is hence called "Kolbe's Electrolysis"
It is a decarboxylative dimerisation of two carboxylate ions (carboxylate salts--either sodium, or potassium are used instead of carboxylic acids for precisely this reason as they disassociate mo... | https://chemistry.stackexchange.com |
94,849 | [
"https://electronics.stackexchange.com/questions/94849",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/34820/"
] | I'm working in my Electrical Circuits homeworks. One of the exercises says: "Use the Fourier Transform method to calculate \$v_0(t)\$"
The circuit is this:
<img src="https://i.stack.imgur.com/cs6ht.png" alt="Circuit">
where \$v_g=36sgn(t)\$, and \$sgn(t)\$ is the sign of \$t\$ (it returns 1, -1 or 0). Well, my proble... | <blockquote>
However, I don't know how to calculate ω
</blockquote>
There's no reason to calculate \$\omega\$. Your approach should be to find the frequency domain representation of the output voltage, \$V_o(j \omega)\$, and transform <em>that</em> back to the time domain.
In the frequency domain and by voltage di... | The voltages and currents in the circuit do not oscillate at a single frequency $\omega$. Your Fourier transform table is telling you that $v_g(t)$ in this case can be written as the superposition
$$ v_g(t) = \frac{1}{2\pi}\int\limits_{-\infty}^\infty V_g(\omega) e^{j\omega t}\, d\omega = \frac{1}{2\pi}\int\limits_{-\... | https://electronics.stackexchange.com |
446,253 | [
"https://stats.stackexchange.com/questions/446253",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/145263/"
] | I'm doing a logistic regression with 100 observations (of which 20 events), and a few predictors. One of the (binary) predictors is expected (theoretically) to be a quite strong predictor, but its value is missing for 30 records. By theory, we expect this variable to not be correlated with any of the other predictors, ... | Dropping observations with missing data is not a good idea, even under MCAR, because of the loss of statistical power that will result.
Setting a category for missingness is also a bad idea, as it will introduce bias, as you indeed say.
A much better idea is to use multiple imputation. Use all the the variables, eve... | If you're interested in inference and the missing variable is only a control, just drop it. You assume that it's uncorrelated with the others, so it won't change their estimates. It might widen the standard errors a bit, but you can also got another model dropping the missing observations. Fitting that one is a good t... | https://stats.stackexchange.com |
7,728 | [
"https://mechanics.stackexchange.com/questions/7728",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/3987/"
] | I'm considering either buying a Wavetrac LSD (very expensive, so that's discouraging) or rebuilding my current Detroit Locker.
The Locker seems to run fine except there's a lot of play in it. Would this be as simple as shimming the locker or would something more involved be required? I'm assuming this is because of th... | Thanks for your advise, Allan. Got the car up on the lift, and was greeted with a nice large hole in the exhaust - large enough to fit two fingers into, and one of many. The botched weld job didn't help, either.
<img src="https://i.stack.imgur.com/hUCZf.jpg" alt="Hole in the exhaust">
<img src="https://i.stack.imgur.... | Do not put your bare hand over the tailpipe, it will get burnt. If you cannot hear an exhaust leak, puff puffing, you can pressurise the exhaust system with a rag over the tailpipe to make an exhaust leak more pronounced and tracible. The downside to this is that you can destroy the catalytic converter, so it is really... | https://mechanics.stackexchange.com |
513,955 | [
"https://stats.stackexchange.com/questions/513955",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/243780/"
] | While Reading chapter 15 of ESL about Random Forests I had some confusion with this phrase:
<blockquote>
The idea in random forests (Algorithm 15.1) is to improve
the variance reduction of bagging by reducing the correlation between the
trees, without increasing the variance too much.
</blockquote>
If I interpret this ... | Yes, the variance in the individual trees can increase since they use a bootstrap sample of the training dataset, and a subset of the columns. However, using a multiple number of trees will decrease the overall variance of the algorithm.
| Say you have true labels
<pre><code>TL : 1 0 0 1 0
</code></pre>
Say you have 3 trees predicting
<pre><code>T1 : 1 1 0 1 0
T2 : 1 0 1 1 1
T3 : 1 0 0 1 0
</code></pre>
Bagging is computing the average from the upper tree threes.
<pre><code>AV : 1 0 0 1 0
</code></pre>
This reduces the overall variance, thus single trees... | https://stats.stackexchange.com |
283,014 | [
"https://electronics.stackexchange.com/questions/283014",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/137330/"
] | I've managed to blow a capacitor while doing maintenance on my 1967 electric organ. It's a 80uf 25V. I've done some searching and found that I can replace it with one that has a higher voltage rating but the same farad value. Is there anything else that I need to know before I buy a replacement? I've read some things a... | Very likely any 80 µF capacitor you can find today rated at 25 V or more will work fine in this situation. Such a capacitor from 1967 is electrolytic, so already polarized. Replacing it with a 80 µF 35 V, for example, modern capacitor should no result in any problems.
| A 80µF electrolytic capacitor from 1967 could be easily 120uF back then, and 40µF today. It probably had a much higher internal resistance than modern capacitors. I think any good quality 100µF 25V is okay. You may of course use 100µF 35V or higher voltage. If you still prefer 80µF, that is okay, but I really think tha... | https://electronics.stackexchange.com |
88,645 | [
"https://softwareengineering.stackexchange.com/questions/88645",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/9298/"
] | Looking at most (if not all) dynamic languages (e.g. Python, PHP, Perl and Ruby), they are all interpreted. Correct me if I'm wrong. Is there any example of dynamic language that goes through compilation phase? Is dynamic language identical with interpreted language?
| <blockquote>
Looking at most (if not all) dynamic languages [i.e Python, PHP, Perl and Ruby], they are all interpreted.
</blockquote>
Not true. You can compile Python source. That's one existential proof.
There are interpreters for statically-typed languages, and compilers for dynamically-typed languages. The two... | C# 4.0 supports dynamic types (late-binding) and it is compiled.
| https://softwareengineering.stackexchange.com |
89,550 | [
"https://physics.stackexchange.com/questions/89550",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/35507/"
] | Given a photon dropping from $\lambda_1$ to $\lambda_2$, its energy will drop from $\frac{hc}{(\lambda_1)}$ to $\frac{hc}{(\lambda_2)}$. However, I was wondering if there is any significance in the energy of the change in wavelength itself. For $\Delta \lambda = \lambda_2 - \lambda_1$, this change in wavelength has an ... | One common way that this happens is through spontaneous parametric down-conversion.
From Wiki: <em>an important process in quantum optics, used especially as a source of entangled photon pairs, and of single photons. A nonlinear crystal is used to split photons into pairs of photons that, in accordance with the law of... | If $\Delta \lambda$ is much smaller compared to either $\lambda_1$ or $\lambda_2$(it doesn't really matter, it should be much smaller than both), then we can make the following approximation:
$$|\Delta E|= \left|\Delta \left( \frac {hc}{\lambda}\right)\right|\approx\left|\frac {hc\Delta\lambda}{\lambda^2}\right|$$
| https://physics.stackexchange.com |
53,156 | [
"https://mechanics.stackexchange.com/questions/53156",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/14098/"
] | On a recent business trip where I was, unfortunately, driving my wife's 2006 Toyota Camry, the Check Engine Light (CEL) came on after about 90 minutes of driving. I had no tools with me, but stopped within a few minutes at a garage and had them pull the codes -- and there were none! I then restarted the car and the l... | The CEL can be triggered for a number of reasons, depending on what the manufacturer wanted. Only some of those reasons are emissions-related. Everything in the OBDII code list is, in some way, related to emissions.
I'd guess that <em>something</em> happened which triggered the light, but the condition didn't persist.... | The ECM runs checks on diagnostic data which is what the light is based off of. If there’s an prevalent issue within a given drive cycle, a light will sound off. For instance, if the engine’s dirty, and a cylinder misfires, the computer will take note, and a code will be lit. If it was a single occurrence, and later st... | https://mechanics.stackexchange.com |
4,605,450 | [
"https://math.stackexchange.com/questions/4605450",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/1134069/"
] | This is a problem from this year´s local Olympiad exam from Catalunya (2022/2023).
I have tried solving this problem without success. The only useful thing I came up with is that it might be solved by inequalities as its intervals are positive integers and not real numbers. I think I must have forgotten some important ... | There are no integer solutions because
<span class="math-container">$$(n+2)^2 = n^2+4n+4 \lt n^2+5n+6 < n^2 + 6n + 9 = (n+3)^2$$</span>
and there are no other integers between <span class="math-container">$n+2$</span> and <span class="math-container">$n+3$</span>.
| Alternate solution: we solve over the integers.
<span class="math-container">$n^2 + 5n + 6 = (n + 2)(n + 3)$</span>. Now <span class="math-container">$n + 2$</span> and <span class="math-container">$n + 3$</span> are comprime, so if their product is a square, both of these factors must also be squares (or both must be ... | https://math.stackexchange.com |
151,489 | [
"https://physics.stackexchange.com/questions/151489",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/60433/"
] | How can kinetic energy be conserved in an elastic collision as collision is said to occur between two bodies if they physically collide against each other or if the path of one of then is affected by the force exerted by the other? If they collide, path of the object will change so velocity should change as velocity is... | When one says that "kinetic energy is conserved in an elastic collision" that means that the total kinetic energy of the system of particles involved in the collision doesn't change. It does not mean that the kinetic energy of each particle is unchanged. For a two particle system, the kinetic energy of each will change... | If only internal forces are doing work (no work done by external forces), then there is no change in the total amount of mechanical energy. The total mechanical energy is said to be conserved. In these situations, the sum of the kinetic and potential energy is everywhere the same.
| https://physics.stackexchange.com |
599,593 | [
"https://physics.stackexchange.com/questions/599593",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/276163/"
] | When studying quantum systems, we often use "hopping" Hamiltonians to represent the system energy. For example, we might represent a 1D ring of N sites with a single particle as:
<span class="math-container">$H = -t\left[\sum_{i}^{N-1}(\sigma^+_i\sigma^-_{i+1}+\sigma^-_i\sigma^+_{i+1}+)+\sigma^+_L\sigma^-_1+\... | I think the hopping term can be understood as the kinetic energy of the system. For example in the Hubbard model, there is a term <span class="math-container">$-t\sum_{<ij>\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma})$</span> in which each term means destructing a fermion with spin <s... | The other answers relating hopping to kinetic energy are good but I want to emphasize <em>why</em> hopping Hamiltonians are physically meaningful to begin with and can actually be exact in principle (though not in practice).
Tight binding models with electron hopping have their physical origin in <em><strong>Wannier fu... | https://physics.stackexchange.com |
87,774 | [
"https://softwareengineering.stackexchange.com/questions/87774",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/15080/"
] | I'm working on a project that uses a dataset produced from software informally licensed as "non-commercial use only". I'm developing an application that uses that dataset as an input to another algorithm. Our own software has a permissive free software license in which we don't restrict commercial usage.
Are there l... | In the US, datasets are not patentable. They are, though, covered by copyright. I believe the extent of such depends on the nature of the collection (is it lists of facts, or lists of data someone created) and what you are doing with it. So you'd want to present your attorney with the exact nature of your data set a... | From what I gather, the software you used was was "non-commercial use only" and not the dataset. Since you are giving the dataset away for free, I wouldn't really call that commercial use. That being said, this is one of those things that it is better to ask an IP lawyer about.
| https://softwareengineering.stackexchange.com |
310,272 | [
"https://dba.stackexchange.com/questions/310272",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/249585/"
] | I want to set a boolean on every entry in a table matching an array of row ids passed into a stored procedure. Help me crack the proper syntax to accomplish this? The following is as close as I've gotten.
<pre><code>CREATE PROCEDURE tester (id_list bigint[])
AS
$$
UPDATE some_table
SET touched = true
WHERE id IN (unnes... | Use the <code>ANY</code> operator:
<pre><code>UPDATE some_table
SET touched = true
WHERE id = ANY (id_list);
</code></pre>
| a_horse_with_no_name's ANY answer is tidier, but I also found out adding a SELECT in front of the unnest function also works:
<pre><code>CREATE PROCEDURE tester (id_list bigint[])
AS
$$
UPDATE some_table
SET touched = true
WHERE id IN (SELECT unnest(id_list));
$$ LANGUAGE sql;
</code></pre>
| https://dba.stackexchange.com |
2,422 | [
"https://robotics.stackexchange.com/questions/2422",
"https://robotics.stackexchange.com",
"https://robotics.stackexchange.com/users/2483/"
] | For robotic manipulator like the one on the picture:
<img src="https://www.learnartificialneuralnetworks.com/images/rcfig1.jpg" alt="robot arm">
are the configuration space and joint space equivalent? I am trying to understand difference between the two...
| In order to answer my own question <em>Configuration space</em> and <em>Joint space</em> must be defined.
<em>Configuration space</em> of a rigid body is a minimum set of parameters that can determine position of each point in that body or <em>Configuration space</em> is set of all possible configurations of that body... | In robotics the <em>configuration space</em> is <strong>exactly</strong> the <em>joint space</em> of the manipulator.
Differently, to indicate the space where the forward kinematic law maps the joints configuration into, we use the terms <em>task space</em> and <em>operational space</em>, equivalently.
| https://robotics.stackexchange.com |
62,712 | [
"https://dba.stackexchange.com/questions/62712",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/36791/"
] | I have a Microsoft SQL Server 2008 R2 database where the <code>DateTime</code> column is not a <code>datetime</code> data type, but <code>varchar(50)</code>. Our website logs that data and I don't really have control over it (I'm sure a proper <code>datetime</code> column would be preferable). I'm trying to do a query ... | The issue is that you cannot <code>CONVERT</code> or <code>CAST</code> a <code>VARCHAR</code> ISO8601 datetime with an offset to a <code>DATETIME</code>.
From SQL Server 2008 onwards, the <code>DATETIMEOFFSET</code> datatype was introduced to handle datetimes with offsets.
As answered elsewhere, you would need to <co... | Why don't you just use?:
<pre><code>WHERE DateTime >= '2014-01-01'
AND DateTime < CONVERT(CHAR(19), DATEADD(second, 1, GETDATE()), 126)
</code></pre>
<strong>Advantages:</strong><br>
- No values in the <code>varchar</code> column will be converted so you'll get no errors<br>
- Efficiency as indexes can be u... | https://dba.stackexchange.com |
264,258 | [
"https://math.stackexchange.com/questions/264258",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/24695/"
] | Why are Fourier series considered to be the subset of Fourier transform? It should have been other way around. Because a non periodic pulse is a subset of periodic pulse with period infinite. So Fourier transform (non periodic signals) is a subset of Fourier series (periodic signals).
| The set of periodic functions is surely a subset of the set of all functions (most of which are aperiodic). So the Fourier series – which allow us to describe periodic functions differently – are a "subset" (in this sentence, the word "subset" is stranger than before) of the Fourier transform – which is a tool to encod... | I would say that neither is more general, they're different things.
Fourier transform can be defined in a very abstract way, for arbitrary locally compact abelian groups, and sometimes even more general objects. In case of locally compact abelian groups, it turns a function defined on a group into a function defined o... | https://math.stackexchange.com |
86,287 | [
"https://softwareengineering.stackexchange.com/questions/86287",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/17694/"
] | With the addtion of Extension methods to C# we've seen a lot of them crop up in our group. One debate revolves around extension methods like this one:
<pre><code>public static class StringExt
{
/// <summary>
/// Shortcut for string.Format.
/// </summary>
/// <param name="str"></par... | It seems pretty pointless to me.
It's more code that you have to maintain - and in a year's time no one will remember the debates you had and half your code will use <code>string.Format</code> the other half will use this method.
It's also distracting you from the real task of developing software that solves <em>real... | FxCop complains when using <code>String.Format(...)</code> without an <code>IFormatProvider</code>. So it makes sense to have extension methods like or <code>FormatInvariant(...)</code> or <code>FormatLocal(...)</code> because it saves callers from being forced to fiddle around with the <code>System.Globalization</code... | https://softwareengineering.stackexchange.com |
129,174 | [
"https://mathoverflow.net/questions/129174",
"https://mathoverflow.net",
"https://mathoverflow.net/users/12310/"
] | If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, <em>what is the maximum number $k(n)$ of linearly independent vector fields?</em> Rewriting $n=2^{4a+b}(2s+1)$, Adams computes $k(n)=2^b+8a-1$.
... | <em>Thanks Misha for the reference! (Just rewriting it here to complete this thread).</em><br>
It seems that an off-the-cuff calculation (what I refer to as "down-to-earth") probably isn't going to suffice; there is some intricate stuff going on in the proofs involving the homotopy groups of the rotation groups of the... | The Radon-Hurwitz number $k(n)$ is the largest $k$ such that there exists an orthogonal multiplication $\mathbb R^k\times \mathbb R^n\to \mathbb R^n$; so for an ONB $x_1,\dots, x_k$
of $\mathbb R^k$ and a unit vector $y\in \mathbb R^n$ the vectors $y, x_1.y, x_2.y,\dots x_k.y$ are orthogonal in $\mathbb R^n$. This desc... | https://mathoverflow.net |
74,001 | [
"https://physics.stackexchange.com/questions/74001",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/28063/"
] | Are earthquakes getting less frequent across the centuries? I know that more seismic stations have register more earthquakes in the last century, but that doesn't imply there were more. I am interested on the geophysical side of it.
The logic is that things settle down, so there is less stuff to shuttle. Add to it tha... | There is no evidence that the global earthquake frequency has changed significantly over that last few centuries. Because the physical mechanism responsible for earthquakes is motion along faults, we believe there have been earthquakes for as long as the earth has had a crust and lithosphere comparable to today.
Th... | Yes, indeed. Earthquakes have been on a steady rise since the earliest seismographs. Hard data from the United States Geological Service (USGS), Richter 6 earthquakes and above, 1895 thru 2015:
<ul>
<li>1895 - 1905 = 11.</li>
<li>1905 - 1915 = 48.</li>
<li>1915 - 1925 = 281.</li>
<li>1925 - 1935 = 524.</li>
<li>1935 -... | https://physics.stackexchange.com |
1,837,532 | [
"https://math.stackexchange.com/questions/1837532",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/72935/"
] | Looking for assistance in translating this definition into more laymen terms? In other words, can someone explain it to me like I'm a 5 year old?
<blockquote>
<strong>Definition</strong>. A sequence ($s_n$) is said to diverge to $+\infty$ and we write $\lim (s_n) = +\infty$ provided that
for every $M$ in $\mathb... | Capital N and M in these sort of definitions tend to represent large numbers (including extremely negative).
$\epsilon$ and $\delta$ represent numbers near zero.
"For every $M$ in $\mathbb R$ there exists a number $N$ such that $n>N$ implies that $s_n>M$."
For every $M$... that means every. Since it is captial... | Pick any $M$ within the real numbers. Make it as big as you want.
Then a sequence $s_0, s_1, s_2, ...$ diverges to positive infinity if there exists some number $N$ such that $s_{N+1}, s_{N+2}, s_{N+3}, ...$ (in other words, all $s_n$ beyond $s_N$) are all greater than $M$.
This value of $N$ will depend on the parti... | https://math.stackexchange.com |
7,726 | [
"https://electronics.stackexchange.com/questions/7726",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/-1/"
] | Is there any way to get -12 V (DC) using only:
<ul>
<li>+12 V DC</li>
<li>Ground</li>
<li>OPAMPs</li>
<li>Resistors</li>
<li>Capacitors</li>
<li>Inductors</li>
<li>Diodes</li>
</ul>
| <ol>
<li>Label the wire which is currently ground "-12V".</li>
<li>Label the wire which was previously 12V "GND".
(There is no step 3)</li>
</ol>
To test: Connect a DMM's ground lead to the wire you've labeled "GND". Connect the positive lead to the wire you've labeled "-12V". The display will read -12 volts.
<br>
... | LM7660 or equivalent *7660 part.
Have a look how it works, it is fairly easy to implement with an op-amp and a few external components.
I'm not going to do <em>all</em> your homework, though.
| https://electronics.stackexchange.com |
34,724 | [
"https://mathoverflow.net/questions/34724",
"https://mathoverflow.net",
"https://mathoverflow.net/users/8201/"
] | <h3>Overview</h3>
For integers n ≥ 1, let T(n) = {0,1,...,n}<sup>n</sup> and B(n)= {0,1}<sup>n</sup>. Note that |T(n)|=(n+1)<sup>n</sup> and |B(n)| = 2<sup>n</sup>.
A certain set S(n) ⊂ T(n), defined below, contains B(n). The question is about the growth rate of |S(n)|. Does it grow exponentially, like |B(n)|, ... | Your sequence is bounded by $(125+\epsilon)^n$. Obviously, this isn't close to a good bound, but it answers the question.
We start by bounding a different question: Let $\Gamma_n$ be the convex hull of $(0,0)$, $(0,n)$ and $(n,n)$. (So $\Gamma$ is rotated $180^{\circ}$ with respect to your $\Delta$.) Let $q_n$ be the ... | This is no answer but a description of an efficient way for computing the cardinality
of $S(n)$. I have to post it as an answer since it is too long for a comment.
Consider the subsets $U_a(n),C_a(n),L_a(n)$ of $S(n)$ defined as follows:
all coefficients of $U_a(n),C_a(n),L_a(n)$ are $\leq a$ and satisfy
the addition... | https://mathoverflow.net |
439,911 | [
"https://math.stackexchange.com/questions/439911",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/37139/"
] | Evaluate <span class="math-container">$$\int \limits_{0}^{\infty} \frac{x}{1+x^2} dx$$</span> by any method. In short I am interested in any method that overcomes the lack of convergence of this integral and gives an "number" to it.
<h3>EDIT</h3>
As I'm getting answers regarding convergence test, this should clear th... | Using the trivial measure, let $\int_{\mathcal{D}} f d\mu = 0$ for any function $f$ and any subset $\mathcal{D}$ of $\mathbb{R}$. Under this definition, your integral is convergent and its value is equal to $0$. But what are the applications of this stupid integral? None. To have a sensible integral, you want, for exam... | By definition
$$\int\limits_1^\infty\frac x{1+x^2}dx=\frac12\int\limits_1^\infty\frac{2x}{1+x^2}dx=\lim_{x\to\infty}\frac12\log(1+x^2)=\infty$$
Thus, the integral doesn't converge.
| https://math.stackexchange.com |
341 | [
"https://cardano.stackexchange.com/questions/341",
"https://cardano.stackexchange.com",
"https://cardano.stackexchange.com/users/128/"
] | <ol>
<li>What exactly is the 51% Attack?
</li>
<li>How hard would it be to accomplish this attack?
</li>
<li>How easy is it to overcome the attack should it actually occur?
</li>
</ol>
| The 51% attack is used by an attacker to take over the blockchain. Usually, the community of a blockchain has to reach consensus, which means that once someone mints a new block, other nodes validate it. The majority has to agree that the new block is valid and thus the blockchain has grown one block. All other nodes a... | The 51% number would imply the attacker controlled 51% of the STAKED ADA. Not necessarily 51% of all ADA. Of course the closer we get to 100% of all ADA staked the closer those two numbers get to each other.
| https://cardano.stackexchange.com |
164,808 | [
"https://mathoverflow.net/questions/164808",
"https://mathoverflow.net",
"https://mathoverflow.net/users/27932/"
] | Is there any routine technique to find a set of permutations which generate a Sylow $3$-subgroup of the symmetric group $S_{n}$?
| Such a Sylow $3$-subgroup is a direct product of "iterated wreath products" of the cyclic group of order $3$. It depends on the base $3$ expansion of $n.$ If $n = a_{0} + 3 a_{1} + \ldots + 3^{m-1}a_{m-1}$ where each $a_{i} \in \{0,2 \}$, then a Sylow $3$-subgroup of $S_{n}$ is the direct product over $i$ of a direct p... | <b>Answer by example:</b><br>
Let's take $n=16=9+3+3+1$.<br><br>
The 3-Sylow of $S_n$ is a subgroup of the symmety group of this graph:
$$
\bullet\!\!\!\stackrel{\textstyle/}{\phantom{\bullet}}\stackrel{\textstyle\bullet}{\stackrel{\textstyle |}\bullet}\stackrel{\textstyle\backslash}{\phantom{\bullet}}
\!\!\!\!\bullet
... | https://mathoverflow.net |
154,630 | [
"https://mathoverflow.net/questions/154630",
"https://mathoverflow.net",
"https://mathoverflow.net/users/43959/"
] | Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 2}(\partial\Omega)$ to be space of functions $u \in L^2(\partial\Omega)$ such that
$$|u|_{H^{\frac 12}(\partial\Omega)} = \... | Lions and Magenes define $H^\frac12$ by complex interpolation, hence your desired property holds by assumption; but they always assume $\Omega$ to have smooth boundary.
In the books of Adams and Grisvard you will find some related results, but - as far as I have (quickly) seen - not exactly what you look like.
| Here is a direct argument based on the definition by the Gagliardo norm
$$
\Vert u \Vert_{H^{1/2}}^2 = \int_{\mathbb{R}^N}\int_{\mathbb{R}^N} \frac{\vert u (x) - u (y)\vert^2}{ \vert x - y \vert^{N + 1}}\,dx\,dy + \int_{\mathbb{R}^N} \vert u \vert^2.
$$
In a first step you apply a weighted Hardy inequality on the b... | https://mathoverflow.net |
148,538 | [
"https://electronics.stackexchange.com/questions/148538",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/11636/"
] | I am using a TIP120 Darlington transistor in a through-hole (specifically TO-220) package, in a very simple application which does not generate much heat.
I remember reading that the metal tab at the top is only for connecting to an optional external heat sink.
Due to a design dimensional issue (unforeseen!), I need ... | I have done it, but I would never ship anything like that out.
I suggest clamping the "waste" end of the TO-220 in a vise firmly and sawing it with a fine tooth hacksaw blade, not allowing anything to touch the epoxy.
Shearing would be much faster and neater, but I fear it would distort the metal and cause the part ... | Removing the tab does not affect the functionality of the device, except for power dissipation. In fact, the SMT "DPAK" package is essentially that.
However, the <em>manner</em> in which you remove the tab may create excessive stresses on the device that will adversely affect its reliability. If you try to shear it of... | https://electronics.stackexchange.com |
16,215 | [
"https://chemistry.stackexchange.com/questions/16215",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/7365/"
] | In my textbook it is written that the order of basic strength of pnictogen hydrides is
$$\ce{NH3 > PH3 > AsH3 > SbH3 > BiH3}$$
I tried but could not find any explanation as to why this happens. What is the explanation?
| Each of these molecules has a pair of electrons in an orbital - this is termed a "lone pair" of electrons. It is the lone pair of electrons that makes these molecules nucleophilic or basic. As you move down the column from nitrogen to bismuth, you are placing your outermost shell of electrons, including the lone pair... | The hydrides of nitrogen family have one lone pair of electrons on their central atom. Therefore,they act as Lewis bases.As we go done the group, the basic character of these hydrides decreases. Nitrogen atom has the smallest size among the hydrides.Therefore the lone pair is concentrated on a small region and electron... | https://chemistry.stackexchange.com |
144,276 | [
"https://cs.stackexchange.com/questions/144276",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/143703/"
] | If so, I don't understand how, as a large part of back propogation is knowing what the 'real' answer is in comparison to 'predicted' answer. With 1 neuron in a hidden layer, we do not know the 'real answer' so theoretically it is impossible unless another technique is employed?
| What the backpropagation is computing, is the derivative (gradient) of that final output with respect to each neuron and each weight in the metwork.
This isn't a problem! You can think of it as computing the derivative of <span class="math-container">$f(g(x;w_1);w_2)$</span>, once with respect to <span class="math-cont... | Backpropagation computes the gradient of the loss function (with respect to the weights). Then, the weights are updated using this gradient. There are many tutorials on how back propagation works. Note that supervised learning assumes all training set instances are labelled, so we <em>do</em> know what the real answ... | https://cs.stackexchange.com |
267,844 | [
"https://security.stackexchange.com/questions/267844",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/204820/"
] | Since there are quite a few exploits of Intel ME firmware in the CPU (same applies to AMD), I would like to know what SIEM solutions are there for detecting these kinds of attacks.
To be more exact, I would like to know how to detect known exploits and known implants, but optionally would like to detect zero days and n... | Even if you don't share volumes, there are still potential security risks when running a remotely downloaded Docker image. For instance, the image may contain code that runs with elevated privileges inside the container, allowing it to perform actions such as accessing the host's network or consuming host resources.
Ad... | If the image is malicious, it <em>can</em> run malicious code on your system, even if you don't map any volumes for it.
Docker images are not intended to isolate malicious code from your host, it's intended to make easier the transition from development to testing to production. If you use a controlled and vetted image... | https://security.stackexchange.com |
185,160 | [
"https://softwareengineering.stackexchange.com/questions/185160",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/27417/"
] | I have plotted my team burn up chart and its velocity per iteration. To me it looks really bad (velocity fluctuates a lot). What should I be looking for to diagnose the root cause of this behaviour?
<img src="https://i.stack.imgur.com/68Duo.png" alt="enter image description here">
| It's perfectly normal to have a fluctuation at the first ten or so sprints, while the team is finding its rhythm. After that, it's perfectly normal for velocity to fluctuate around an average. Try plotting a running average of the last five sprints or so and you should see it level out. If not, some of the following... | Additional potential cause: during the later sprints, you are paying off technical debt from earlier sprints.
E.g. you have a management demo after sprint 3 and need to show happy-day scenario. To make it, you do the coding without error handling, without translation support, without unit testing. This is a valid deci... | https://softwareengineering.stackexchange.com |
128,894 | [
"https://stats.stackexchange.com/questions/128894",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/30869/"
] | My dataset consists of $n$ genes, each of them described by a vector of expression values, $5$ for "healthy" individuals, and $5$ for "unhealthy" individuals.
I am going to run $n$ t-tests (one for each gene) to identify which genes show a different behaviour between healthy population and unhealthy population.
Shoul... | You absolutely do want to apply a correction. The key idea is identifying significance by chance. As you increase the number of comparisons you increase the number of those that will be significant by chance.
For example, let's take the generic example of doing 100 comparisons using a significance threshold of 0.05.... | You say that no comparisons are being conducted because genes are not being compared to each other. However, each t-test is still a comparison. In fact, that's what a t-test is--a comparison of two means. In your case, each comparison is between the healthy group and the unhealthy group, rather than between gene A and ... | https://stats.stackexchange.com |
15,681 | [
"https://electronics.stackexchange.com/questions/15681",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/1571/"
] | I'm doing some work with a portable, cell-battery-based electronics and someone on the design team mentioned that pulsed current will more quickly deplete a battery than constant current, even if both have the same total energy consumed. Is this true? What is the reasoning behind it if it is true?
| P = I^2R.
So the power loss in the resistor goes up with the square of the current. Multiply this by time and you get energy (joules) lost to the resistor.
Example: Vbatt = 10 V and Rbatt = 1 ohm. 100 Joules required.
Pulsed load of 4 ohm:
<ul>
<li>Ipulse = 2A,</li>
<li>load_power = 16 W,</li>
<li>Time to deliv... | When you discharge battery, pulse current means that during these peaks current has to be higher, proportionally to your duty cycle.
Higher current means higher looses in battery itself due to internal resistance.
This is much more pronounced in usual prime batteries, and less in Lithium & NiMh due to their lower... | https://electronics.stackexchange.com |
310,559 | [
"https://softwareengineering.stackexchange.com/questions/310559",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/27217/"
] | I have never used a Continuous Integration system (CI) before. I primarily code in MATLAB, Python or PHP. Neither of these have a build step and I do not see how a CI could be used for my work. A friend on a large project in a large firm told me that language does not matter.
I do not see how CI would be of use to me ... | Continuous integration as a term refers to two distinct ideas.
The first is a workflow: instead of everyone in a team working on their own branch and then after a couple of weeks of programming try to merge their changes into the mainline, that changes are integrated (nearly) continuously. This allows problems to surf... | True, you do not have particular need of a CI system to perform builds and check that those builds are correct, but that is only part of what CI is about.
The purpose of CI is to detect errors as soon as possible, because generally speaking, the earlier an error is caught the cheaper it is to fix. To that end, in the ... | https://softwareengineering.stackexchange.com |
180,914 | [
"https://electronics.stackexchange.com/questions/180914",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/76479/"
] | As the title says, I have capacitors rated 100VDC 18pF. Can I use these on a board that uses 5V Max ?
I can't really find any direct answer to that question.
What I want to do is to put a crystal to a chip and hopefully use these caps on the crystal.
| The voltage rating on a capacitor is a maximum value; you can use it in a lower voltage circuit just fine.
| While Nick Johnson's answer is correct, there is a faint possibility that you might want to consider the following:
In the aerospace/hi-rel world, it is known that using tantalum electrolytic capacitors which are grossly overrated is not ideal. 50-volt capacitors used at 5 volts show a significantly higher failure rat... | https://electronics.stackexchange.com |
72,954 | [
"https://security.stackexchange.com/questions/72954",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/49511/"
] | I was curious how a DoS attack would affect a home router. In particular, I'm interested in how a SYN flood would affect a home router.
The reason I'm interested is due to a Cisco document I read. Within the document, it said SYN flood attacks can affect home routers. To me this seems odd because SYN floods must speci... | There are three main ways a SYN flood can work against a home router:
<ol>
<li>If the router is performing NAT and has a port forwarded to a server, a SYN flood can fill up the router's NAT table, causing it to drop connections.</li>
<li>The SYN flood can act as a simple bandwidth-starvation attack. A typical home ro... | Routers usually has running lightweight Web server that provides access to the configuration web interface. In most cases, it runs on port 80, 8080 or other common one (if isn't changed by the user).
The attacker (standing in the local subnet, if something like "Remote administration" is not enabled on the router, and... | https://security.stackexchange.com |
63,383 | [
"https://softwareengineering.stackexchange.com/questions/63383",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/7693/"
] | I have a friend that works for a company. They use a software product that is aged, and written in Delphi. There are concerns at this company about the life of the software company that provides the software (which they pay a monthly license to use). It's quite <em>specific</em> software.
My friend has spoken to hi... | Yes!
In fact if you don't suggest this - or at least bring it up I would hope that they would be a little concerned.
You need to help them read between their own lines. When they say that they want all the feature exactly as the previous application - surly what they are actually saying is "we don't want to change ou... | Find out what they have been getting for their license fee. There could have been updates provide in recent years. What may seem like a waste may be an insurance policy.
Have they determined this app does exactly what they want or are they reluctant to put in the effort to change it? They may not feel there is going t... | https://softwareengineering.stackexchange.com |
3,172,211 | [
"https://math.stackexchange.com/questions/3172211",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/620738/"
] | I am studying for an upcoming exam by doing practice problems, and this simplification was used in one of the solutions. However, I do not understand why or how these two terms are equal.
EDIT: had the wrong equation in the title at first; fixed now.
| <span class="math-container">$\begin{align}5(3^k - 2^k) - 6(3^{k-1} - 2^{k-1}) &= 5 \times 3^k - 6\times 3^{k-1} - 5 \times 2^k +6\times 2^{k-1}\\ &=3^{k-1}(5\times 3-6)-2^{k-1}(5\times 2-6)
\end{align}$</span>
In the first line, I am just grouping the like powers together(powers of <span class="math-container... | <strong>Key Idea</strong> <span class="math-container">$ $</span> Make <em>explicit</em> the <em>algebraic dependencies</em> among the exponentials by choosing a minimal generating set, here <span class="math-container">$\,x = 3^{\large k-1},\, y = 2^{\large k-1}\,$</span> so <span class="math-container">$\,3^{\large ... | https://math.stackexchange.com |
90,368 | [
"https://mathoverflow.net/questions/90368",
"https://mathoverflow.net",
"https://mathoverflow.net/users/14385/"
] | Let $X$ be a variety over some algebraically closed field $k$. In order to define the intersection product of the Chow ring one usually demands $X$ to be smooth. This is for example well explained by Fulton in his book Intersection Theory.
I was wondering whether it is possible to weaken this assumption. I would like ... | One thing that works under pretty general conditions is intersecting curves with $\mathbb Q$-Cartier divisors. This covers for instance the example mentioned by David.
Just in case:<br>
<strong>Definition</strong> Let $X$ be a normal variety and $D$ a Weil divisor on $X$. Then $D$ is called <em>$\mathbb Q$-Cartier</em... | Intersections of Weil divisors in normal surfaces wind up lying in $\mathbb{Q}$ rather than $\mathbb{Z}$. Here is the basic example of why:
Consider the projective closure of $\mathrm{Spec} \ k[x,y,z]/(xz-y^2)$. The group of Weil divisors is generated by the ideal class $D:=\langle x,y \rangle$. The group of Cartier d... | https://mathoverflow.net |
119,144 | [
"https://math.stackexchange.com/questions/119144",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/26725/"
] | This is part of a TopCoder.com algorithm practice question and I cannot wrap my head around it. I am given a lottery format and I need to calculate the odds. One particular format is that the numbers must be in non-descending order. The numbers do not have to be unique, so I can repeat the same number.
Example:
The "P... | Since each unordered pair of different numbers can be ordered in two different ways, the number of eliminated tickets is half the number of ordered pairs of different numbers. To count the ordered pairs of different numbers, note that to form such a pair you can first choose one of $n$ numbers, then you can choose one ... | If you are counting this over and over, you could do it with dynamic programming. Let $T[k][n] =$ how many times I can pick $k$ numbers in order out of $n$ total.
Of course $T[1][n] = n$ and $T[k][k] = 1$. Then to count $T[k][n]$, pick the largest number $m$ and count the rest using $T$, i.e. $$T[k][n] = \sum_m T[k-1][... | https://math.stackexchange.com |
181,649 | [
"https://dba.stackexchange.com/questions/181649",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/11296/"
] | I am pretty sure that this is a simple matter and that it's answered but I seems like not able to find it online, or to phrase it correctly.
I have a table (<code>ven_data</code>) that has a few columns: <code>id_v</code>, <code>lang</code>, more data columns
The unique constraint here is <code>(id_v, lang)</code> pai... | Normally when I have something like this, I have a table in one database and a VIEW pointing to that table in the other database. Can you just use a view on the other database which references the first one? If not then maybe a "materialized view/mirror table" of sorts which be a "slave" table that would get refresh... | Tables can be referred by full qualificators like <code>common.table_a</code>, <code>aux.table_b</code> etc from the working scheme <code>main</code>:
<pre><code>USE main;
SELECT *
FROM table AS w -- main.table referred
JOIN common.table_a AS z ON z.col_x = w.col_y
LEFT JOIN aux.table_b AS q ON q... | https://dba.stackexchange.com |
95,004 | [
"https://math.stackexchange.com/questions/95004",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/7325/"
] | Is this true: <strong>Every ideal of $K[x_1,\ldots,x_n]$ is generated by some subset with $\leq n$ elements?</strong>
It is true when $n=1$, since $K[x]$ is a PID.
I'm trying to prove it is not true for $n\geq2$, via the example $I:=\langle x^2,xy,y^2\rangle\unlhd K[x,y]$.
Does the SINGULAR code below confirm that $... | Expanding on my comment above:
Let $J=\langle x^3,x^2y,xy^2,y^3\rangle$. Since $J\subset I$, if $I$ is generated by two elements in $K[x,y]$ then its image, $I'\subset K[x,y]/J$, is generated by two elements as a $K[x,y]$-module.
But $I'$ in $K[x,y]/J$ is just $\{ax^2+bxy+cy^2: a,b,c\in K\}$. As a $K[x,y]$-... | Edit: I finally remembered how to do algebra, I hope. Below is a completely different answer than my original (and hopefully a correct one).
Your example works just fine. In fact, if we let $R = K[x_1,\ldots,x_n]$ and $\mathfrak{m}=\langle x_1,\ldots,x_n\rangle$ (which is maximal) then we get that $\mathfrak{m}^2$ req... | https://math.stackexchange.com |
242,707 | [
"https://security.stackexchange.com/questions/242707",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/248176/"
] | I recently purchased a wifi extender and I now have to set it up.
There are three options for how to set it up:
<ol>
<li>Through the app</li>
<li>By pressing the WPS button on the router and then pressing the WPS button on the extender</li>
<li>By connecting to the wifi network of the extender (which does not have a pa... | Using the WPS (Wi-Fi Protected Setup) button is generally considered a secure method. For two minutes after that, your access point will accept new devices onto the network, but not after, which means that the scope for attack is limited. If you are concerned, most access points allow you to find the connected device... | WPS can be easily hacked. So, make sure you disable WPS in settings.
First and foremost, update the firmware on the device, if possible. A firmware update can sometimes patch security vulnerabilities discovered by the manufacturer or 3rd parties after it left the factory.
Login to settings via the device's default IP a... | https://security.stackexchange.com |
173,427 | [
"https://math.stackexchange.com/questions/173427",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/24565/"
] | <blockquote>
What is $f(t)=X_{t+1}$, if $X_{t+1}=(1-p)(1-X_{t})+pX_{t}$ and $X_{0},p \in [0,1]$?
</blockquote>
And what are general methods for finding functions defined by such recurrent equations?
| I will assume that $t$ ranges over, say, the non-negative integers.
For your particular example, we have
$$\begin{align*}X_{t+2}&=(1-p)(1-X_{t+1})+pX_{t+1},\\
X_{t+1}&=(1-p)(1-X_t)+pX_t.\end{align*}$$
Subtract and simplify. We get
$$X_{t+2}-X_{t+1}=(2p-1)(X_{t+1}-X_t).$$
So if $Y_t=X_{t+1}-X_t$, we find that... | Well, you have $X_{t+1}$ equal to a weighted average between $X_{t}$ and $1 - X_{t}$. So there's a fixed point where these are equal, i.e., at $X=1/2$. To find the behavior away from that fixed point, define $Y_{t} = X_{t} - 1/2$: then $$
\begin{eqnarray}
Y_{t+1}&=&X_{t+1}-\frac{1}{2} \\ &=& \left(1-p... | https://math.stackexchange.com |
283,419 | [
"https://mathoverflow.net/questions/283419",
"https://mathoverflow.net",
"https://mathoverflow.net/users/43804/"
] | Given a finite CW or simplicial decomposition of a space $X$ and a ring homomorphism $\varphi:\mathbb{Z}[\pi_1(X)]\to F$ for a field $F$, if the $\varphi$-twisted homology is trivial, then the Reidemeister-Franz torsion $\tau^\varphi(X)\in F$ is an invariant of the twisted chain complex, well-defined up to multiplicati... | Yes, there is a very nice geometric interpretation over any manifold $X$ satisfying $\chi(X)=0$, for a version of Reidemeister torsion spelled out in Turaev's paper <strong>"Euler structures, nonsingular vector fields, and torsions of Reidemeister type"</strong>. Ian Agol's comment mentioned the Seiberg-Witten invarian... | Have you looked at the relation with Whitehead torsion and simple homotopy theory? I forget the details and am no expert, but there is a lot of nice low dimensional geometric topology hidden in that theory if that is any help. Look at Milnor's paper, Bull. Amer. Math. Soc., 72 (3):(1966) 358–426. but also look at the ... | https://mathoverflow.net |
15,583 | [
"https://dba.stackexchange.com/questions/15583",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/1756/"
] | In SQL Server 2008, I am using <code>RANK() OVER (PARTITION BY Col2 ORDER BY Col3 DESC)</code> to return data set with <code>RANK</code>. But I have hundreds of records for each partition, so I will get values from rank 1, 2, 3......999. But I want only up to 2 <code>RANKs</code> in each <code>PARTITION</code>.
Exampl... | You could put the original query using <code>rank()</code> into a subquery and wrap it with a query that filters the results.
| <pre><code>select * from (
SELECT Subject, Name, RANK() OVER (PARTITION BY Subject ORDER BY Score DESC) as RN
FROM Table
) a
where a.RN <= 2
</code></pre>
| https://dba.stackexchange.com |
128,754 | [
"https://electronics.stackexchange.com/questions/128754",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/50937/"
] | The configuration of the common emitter amplifier is one resistor at the emitter, on resistor at the collector, two bias resistors at the base as shown in this picture :
<img src="https://i.stack.imgur.com/EhVcU.jpg" alt="enter image description here" />
<ol>
<li>What is the benefit from this new configuration?</li>
<l... | For the function generator you must use the 50 Ohm output. Use the asym input on the amplifier (1/4 inch phone plug.) You will need a cable with a BNC connector on one end and a 1/4 inch phone plug on the other. Connect the shield on the BNC (outer connection) to the sleeve on the phone plug.
| The TTL / CMOS output on a function generator is <em>usually</em> a pulse output used to provide sync to other test equipment. The 50 Ohm output is the main output and will provide whatever signal shape you have selected.
Both outputs are unbalanced (what you are calling "asymmetric"). That is: they have a ground co... | https://electronics.stackexchange.com |
767 | [
"https://mechanics.stackexchange.com/questions/767",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/492/"
] | There are a number of different ODB2 scan tools out there, some PC based, some standalone units, some more for tuning and some more for diagnosing problems.
What are the different features that these devices provide?
What are the pros/cons among different models?
Is there one or two that "stand out from the crowd" as ... | If you're looking for personal use and already have access to a laptop computer, I'd highly recommend getting a PC (or Mac, if that's what you have)-based one. You buy the hardware, generally for $100 or less (I recommend the units from ScanTool.net - the less expensive ones are fine). The biggest advantage is that you... | I cannot answer all of your questions, as my experience is only with one OBD II scanner. I can give you the feedback that I've been using the very basic Actron CP9125 for years, and it has served my purpose of reading CEL codes (and resetting them) very well, on a number of different Japanese import cars.
The Actron C... | https://mechanics.stackexchange.com |
514,624 | [
"https://electronics.stackexchange.com/questions/514624",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/260094/"
] | Do beam-forming devices expect a particular antenna configuration? For instance, do they expect the antennas to be a certain distance from one another? Or parallel to one another? Etc.
Background Info:
I have a beam-forming capable WiFi router located in a utility room. I would like to detach the antennas and add exten... | Yes, it will be important to keep all antenna extensions equivalent to ensure the propagation delays are equal. You will want to keep any changes equivalent to each antenna to minimise efficiency loss. It’s probably best to keep the spacing the same, if you do increase the spacing, it's probably best to increase it bet... | There are two classes of beamformer. One type has the antenna configuration hard-coded into its mathematics, and attempts to construct geometrically beams towards wanted sources, and/or nulls towards unwanted sources. The other type uses general complex matrices to maximise SNRs of expected signals, and to the extent t... | https://electronics.stackexchange.com |
452,536 | [
"https://physics.stackexchange.com/questions/452536",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/93076/"
] | The action is defined as:
<span class="math-container">$$S = \int d^2\textbf{x}\,dt \left[\left(\frac{\partial h}{\partial t}\right)^2 + (\nu \,\nabla^2h)^2\right]$$</span>
The equation of motion is asked for, so use Euler-Lagrange:
<span class="math-container">$$\frac{\partial\mathcal{L}}{\partial h} = \frac{\partial... | The solution plays fast and loose with the calculus of variations, and uses a trick (or a rule of thumb) that you can usually get away with, but is not obvious to the beginner. Here's how it works:
The action is
<span class="math-container">$$
S = \int d^2\textbf{x}\,dt \left[\left(\frac{\partial h}{\partial t}\right... | <strong><em>Please read the edits/comments before continuing.</em></strong>
<hr>
When in doubt, write out the indices.
So we start with
<span class="math-container">$$ {\cal L} = \left(\frac{\partial h}{\partial t}\right)^2 + (\nu \partial_i \partial^i h)^2 = \left(\frac{\partial h}{\partial t}\right)^2 + (\nu \... | https://physics.stackexchange.com |
46,011 | [
"https://mathoverflow.net/questions/46011",
"https://mathoverflow.net",
"https://mathoverflow.net/users/-1/"
] | Let $K$ be a compact metric space, and $(E,d_E)$ a complete separable metric space.
Define $C:=C(K,E)$ to be the continuous functions from $K$ to $E$ equipped with
the metric $d(f,g)=\sup_{k\in K}\ d_E (f(k),g(k))$. Is the space $C$ separable?
The result is true when $E$ is the real line; this is Corollary 11.2.5 in D... | Yes, it appears e.g. as Theorem 4.19 in Chapter I of Kechris' <em>Classical Descriptive Set Theory</em>. (The relevant page is visible in Google Books if it's not in your library.)
| We have the following. Fix $X, (Y,d)$ polish spaces where $d$ is some bounded metric. Topologise $C^{0}(X,Y)$ by the metric $d(f,g)=sup_{x\in X}d(f(x),g(x))$. Then one can tweak Kechris' proof to show, that the subspace $S$ of uniformly continuous maps with bounded images, is Polish.
Is it possible to show that $C^{0}... | https://mathoverflow.net |
242,303 | [
"https://electronics.stackexchange.com/questions/242303",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/114720/"
] | I have overhead ceiling LED light fixtures.
When I turn on <strong>Main Switch</strong> of the house, I see the LED starts emitting very dim light ( have to look for a 3-4 second to perceive) even when LED's <strong>switch</strong> is <strong>off</strong>!
Is that normal or I need to check it out by an electrician ?
| By all means this seems like a bug of latest Altium version (16). I'll get in touch with Altium and start my PCB all over again.
| I just run into the same problem with AD 17.1 <em>and</em> 18.1.
I tried many different ways to resolve the issue, like deleting involved components in schematic and PCB, clearing net list in PCB, etc.
What eventually solved the issue was to delete the net labels for concerned nets in schematic, place new net labels... | https://electronics.stackexchange.com |
70,484 | [
"https://physics.stackexchange.com/questions/70484",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/26857/"
] | The question itself:
<blockquote>
A modern art sculptor anchors 1 meter of a cast iron pole in the ground and leaves 3 meters of the pole sticking out of the ground at a 60° angle. The owner of the metal shop told him that the pole is rated to withstand up to 9000 Nm of torque before it bends. If the sculptor decide... | The link you're looking for is hidden by the fact that Heisenberg kets $| \psi \rangle_H$ are not Schroedinger kets $|\psi \rangle _S$. The tranlation from the Heisenberg to the Schroedinger pictures is done via the time-evolution operator $U(t - t_0) = \exp\{-i H * (t - t_0)/\hbar\}$ (I do not explicity put hats on op... | In the Heisenberg representation, the "equivalent" of the Schrodinger equation is :
$$\hat H(t) = \frac{\hat P^2(t)}{2m} + V(\hat X(t))$$
with $[\hat P(t), \hat X(t)] = i\hbar$
If you are looking at eigenstates and eigenvalues of the hamiltonian, you will look for a constant Hamiltonian.
For instance, for the harmo... | https://physics.stackexchange.com |
877 | [
"https://datascience.stackexchange.com/questions/877",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/2798/"
] | Well this looks like the most suited place for this question.
Every website collect data of the user, some just for usability and personalization, but the majority like social networks track every move on the web, some free apps on your phone scan text messages, call history and so on.
All this data siphoning is j... | A couple of days ago developers from one product company asked me how they can understand why new users were leaving their website. My first question to them was what these users' profiles looked like and how they were different from those who stayed.
Advertising is only top of an iceberg. User profiles (either fille... | Most companies won't sell the data, not on any small scale anyways. Most will use it internally.
User tracking data is important for understanding a lot of things. There's basic A/B testing where you provide different experiences to see which is more effective. There is understanding how your UI is utilized. Categ... | https://datascience.stackexchange.com |
515,153 | [
"https://electronics.stackexchange.com/questions/515153",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/192175/"
] | These are terms that I have come across in multiple operating manuals for VFDs and usually, they just mention that it's used for variable torque applications. My doubt is "what exactly is it?" and is there a mathematical formula that I can use to derive this relation between voltage and frequency? I read some... | It is about efficient operation of your pump/fan. Affinity law says load torque is proportional to square of speed for fan or pump load. If you analyse a induction motor with equivalent circuit you will find that it is <code>slip</code> which determines efficiency. There is a slip value band from 0.05 to 0.15 where you... | Charles' comment is appropriate, but to simplify and give you some background:
AC machines are designed to operate at a certain flux level. Beyond that the machine can saturate which can cause damage, high currents, overtemperature, fuse/breaker trips, etc.
So to step back, consider just an inductor across the AC line... | https://electronics.stackexchange.com |
35,050 | [
"https://engineering.stackexchange.com/questions/35050",
"https://engineering.stackexchange.com",
"https://engineering.stackexchange.com/users/26462/"
] | I'm going to buy Thomas vacuum cleaner with Aqua filter and use it to sand walls a bit with grinder. And after finishing the decoration, use it as an ordinary cleaner. May construction plaster, putty sanding harm it? Or it is OK to use the household vacuum for small amount of grinding?
| Actually I've bought household Thomas Multi Clean X10 Parquet vacuum with Aqua filter and rent the industrial cleaner Hilti VC-40UM with Hilti DGH-130 concrete grinder. There was about 2 full buckets of dust, I cleaned the filter for Hilti about 10 times during work of single small 3x5 meters room. So I can say that ho... | I doubt it will last.
Professional units are designed for higher duty hours with more suction etc which means better bearings and greater power for example.
The home use equipment is not designed to be used at that type of duty hours - possibility of overheating etc.
A simple example is a toaster - buy one for a hom... | https://engineering.stackexchange.com |
2,042,632 | [
"https://math.stackexchange.com/questions/2042632",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/321627/"
] | Suppose $X_1, X_2$ are iid $N(0,1)$, why is it that $X_1X_2 \sim \frac{X_1+X_2}{\sqrt{2}}\frac{X_1-X_2}{\sqrt{2}}$? Here, "$\sim$" is used to denote the fact that they have the same distribution, or that their cdf's are the same. I understand that it is a result from probaility theory that $X_1+X_2$ and $X_1-X_2$ are i... | $X_1 X_2$ has some distribution (say, it has cdf $F$).
$\frac{1}{2} (X_1+X_2)(X_1-X_2)$ has some distribution (say, it has cdf $G$).
The statement is just saying $F=G$.
Note that you could just as well say $X_1X_2 \sim \frac{1}{2} (Z_1+Z_2)(Z_1-Z_2)$ where $Z_1,Z_2 \sim N(0,1)$ are i.i.d.
| $\frac{X_1\pm X_2}{\sqrt{2}}\sim N(0,1)$, and these two random variables are independent because they are jointly Gaussian (as linear transformations of the Gaussian vector $(X_1,X_2)$) and their covariance is zero. Indeed, $E [ \frac{X_1-X_2}{\sqrt{2}} \frac{X_1+X_2}{\sqrt{2}}]= \frac{1}{2} E [X_1^2 - X_2^2]=0$. Thus ... | https://math.stackexchange.com |
421,205 | [
"https://electronics.stackexchange.com/questions/421205",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/168663/"
] | On a custom board I have an ADC which measures -5V to +5V. And outputs in Hex via SPI like this:
<pre><code>0x8000 to 0x0000 = most negative to 0
0x0000 to 0x7FFF = 0 to most positive
</code></pre>
I just started studiyng ADCs so I am not sure what is going on. I beleive I have negative full scale of -5V and positive... | It looks like you have a 16-bit, 2's-complement ADC with a range of <span class="math-container">\$-5V \le V_{IN} < 5V\$</span>. If that's the case, then the digital values can be interpreted as signed integers from -32768 to +32767. The measured voltage is
<span class="math-container">\$ V_{IN} = \frac{DigitalValu... | If you count <em>most negative to 0</em> using integers, you count ..., -4, -3, -2, -1, 0
If you count <em>most negative to 0</em> using hexadecimals, you count (in your case) 0x8000, 0x8001 ..., 0xFFFE, 0xFFFF, 0
So, there is the 0xFFFF.
Now, your scale is 0X8000 to 0x7FFF, so, in decimal -32768 to 32767, the full s... | https://electronics.stackexchange.com |
31,063 | [
"https://softwareengineering.stackexchange.com/questions/31063",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/11893/"
] | This is semi-hypothetical, and as I've no experience in dealing with massive database tables, I have no idea if this is horrible for some reason. On to the situation:
Imagine a web based application - lets say accounting software - which has 20,000 clients and each client has 1000+ entries in a table. That's 20 millio... | Generally speaking, no, it makes no sense to have a table (I think you actually mean database here) per customer. 20 million rows is relatively small for a database table. Query speed against that should not be an issue so long as the database is properly tuned (indexed) and the queries are put together correctly. W... | Sounds like a bad idea.
Don't try to outsmart the database with exotic constructions like this. Database engines are designed with lots of optimizations to handle large data sets. For example, what you are describing sounds awfully close to an attempt for manually implementing indexes. Just use indexes provided by th... | https://softwareengineering.stackexchange.com |
586,533 | [
"https://electronics.stackexchange.com/questions/586533",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/291531/"
] | I was a developer of PV module.
so, I know the electrical characteristic of PV cells and MPPT concepts well.
I just wonder,
Can I use the buck converter for MPPT?
I saw that buck converter can be used for MPPT through the internet.
But, I thought,
Buck converter makes the PV current choped.
(Not kind of DC current with... | <blockquote>
<em>that's why Buck is not good for MPPT.</em>
</blockquote>
You appear to be making the assumption that a buck converter cannot be operated in CCM (continuous conduction mode). If you took a random sample of buck and boost converter circuits you'd probably find more buck converters running in CCM compared... | There are buck, boost, buck&boost converters. All of them make a "chopping" of the input current. To make the input current with low ripple, a capacitor bank is used.
There are also interleaved converters, like having multiple of them in parallel, having the synced switching frequency but delayed phase, l... | https://electronics.stackexchange.com |
425,653 | [
"https://physics.stackexchange.com/questions/425653",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/195695/"
] | It is known that electromagnetic waves with a high frequency possess a greater amount of energy than waves with lower frequencies. Why is this the case? Does it have anything to do with Planck's law?
| <blockquote>
It is known that electromagnetic waves with a high frequency possess a greater amount of energy than waves with lower frequencies.
</blockquote>
This isn't quite true. The energy carried by an electromagnetic wave is the product of two independent factors:
<ul>
<li>the energy of each individual photon... | Quantum mechanics tells you that the energy and frequency of an individual photon are related as $$E = h\nu$$ where $h$ is Planck's constant. This is part of the Quantum Hypothesis of Planck.
| https://physics.stackexchange.com |
302,517 | [
"https://physics.stackexchange.com/questions/302517",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/140323/"
] | In his famous paper on Special Relativity, Einstein derives the Lorentz Transformations. He considers a light beam emitted at time $t$ from the origin of the system of coordinates $k$ towards a point that moves with the origin of the system $K$ such that its coordinate on the $K$ system is $x'=x-vt$ and is then reflect... | It's also worth remembering that we don't strictly need a Taylor expansion in this case, because the purpose of a Taylor expansion is to linearize a function, which in this case we already know to be linear (as stated earlier in Einstein's paper, the transformation equations must be linear for space and time to be homo... | I have my attempt but it needs to be checked for logical faults.
<ul>
<li>$\tau$(0,0,0,t) = $\tau$(0,0,0,0) + [$\frac{\partial\tau}{\partial x'}$] * (0-0) + [$\frac{\partial\tau}{\partial y}$] * (0-0) + [$\frac{\partial\tau}{\partial z}$] * (0-0) + [$\frac{\partial\tau}{\partial t}$] * (t-0) = $\tau$(0,0,0,0) + t$\fra... | https://physics.stackexchange.com |
126,881 | [
"https://mathoverflow.net/questions/126881",
"https://mathoverflow.net",
"https://mathoverflow.net/users/6779/"
] | In Katz's article <em>p-adic properties of modular schemes and modular forms</em> in the Antwerp proceedings, the following definition of an elliptic curve over a base scheme $S$ is given:
<blockquote>
By an elliptic curve over a scheme $S$, we mean a proper smooth morphism $p: E \to S$, whose geometric fibres are c... | The argument that allows you to show that an elliptic curves defined as you say is a group scheme, and even a commutative one is the construction of a functorial and natural
isomorphism $E(T) \rightarrow Pic_{E/S}^0(T)$ for every $S$-scheme $T$. This allows
to see the functor $T \mapsto E(T)$ as a funtor in group, an... | This is N. Katz, B. Mazur, Arithmetic Moduli of Elliptic Curves, Ann. of Math. Studies 108, Princeton University Press, Theorem 2.1.2.
Note that the group scheme structure is unique by rigidity results.
| https://mathoverflow.net |
47,828 | [
"https://dsp.stackexchange.com/questions/47828",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/34286/"
] | I have a sinusoid in continuous time, with a frequency of 18kHz, it is sampled ideally with a continuous to discrete convertor, with a frequency of 27kHz. After that, we change the sampling speed in discrete time, using interpolator system and decimate system,so that result frequency equivalent in discrete time is 13.5... | It seems to me that the only way in which one of those solutions is possible is without aliasing. Probably there is a typo error, and the actual sampling rate is greater than 36 kHz. In such a case, for example, if the sampling rate was actually 37 kHz, we can interpolate with a factor I = 4, and then decimate with a f... | Assuming we are working with an ideal cosine that has a frequency response that is a perfect delta, then if the original signal is at $18 kHz$ and it is sampled at $27kHz$ then there will be aliasing (any input above $fs/2 = 13.5 kHz$ in this case will be aliased).
Working in digital domain $13.5 kHz$ would be $\pi$ ... | https://dsp.stackexchange.com |
106,422 | [
"https://security.stackexchange.com/questions/106422",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/92860/"
] | Ok, so been looking up public key encryption and using it a bit around some of my web applications. From some testing, I understand that the server contains two keys and sends the public key to the client.
With this kind of system, the client can send encrypted data.
My question is however, <em>how does the SERVER se... | Client generates a symmetric key, encrypts it with server public key and sends it to server. Server decrypts with private key and uses that as a network session key to speak securely with client. This is basically what TLS_RSA does.
| <strong>1. Handshake</strong>
It all starts first with a handshake. The client tells the server what version of SSL it supports and what cipher suites. The server answers with the information on which version of SSL and which cipher suite will be used.
<strong>2. Cerfiticate</strong>
The client now has to check if t... | https://security.stackexchange.com |
2,996,281 | [
"https://math.stackexchange.com/questions/2996281",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/590283/"
] | Here is what I have so far, and I am new to these kinds of proofs so please be detailed with me:
<span class="math-container">$| \frac{2x}{x-3}-(-4) | = |\frac{6(x-2)}{x-3}|$</span>
But I don't understand what I am supposed to do with this! I just don't understand. I know the answer, but I have no idea how to get ther... | In such a case it is often quite useful to use a substitution like
<ul>
<li><span class="math-container">$\boxed{x-2 = h} $</span> and check what happens to the expression as <span class="math-container">$\boxed{h \rightarrow 0}$</span></li>
<li>and note that <span class="math-container">$\boxed{|x-2| < \delta \Lef... | I know this isn't exactly a direct <span class="math-container">$\epsilon{-}\delta$</span> proof, but it is quick. The Algebraic Limit Theorem for functions says that, if <span class="math-container">$\lim_{x\to c}f(x)=a$</span> and <span class="math-container">$\lim_{x\to c}g(x)=b$</span>, then <span class="math-conta... | https://math.stackexchange.com |
2,166,075 | [
"https://math.stackexchange.com/questions/2166075",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/375741/"
] | <blockquote>
Prove $a_1+\cdots+a_n=\dfrac{(a_1+a_n)n}{2}$ inductively.
</blockquote>
Where $a_i=a_{i+1}-r$.
I tried to start proving it inductively, but any try lead to a bad conclusion, so I ended up proving it by making $a_n$ depend on $a_i$.
But I didn't know how to prove it inductively, so there is the problem... | If $n=1$, the result is trivial.
Suppose $$\sum_{i=1}^k a_i = \frac{(a_1+a_k)k}{2}$$
\begin{align}\sum_{i=1}^{k+1} a_i &= \frac{(a_1+a_k)k}{2}+a_{k+1}\\&=\frac{a_1k+a_{k+1}(k+1)-rk+a_{k+1}}{2}\\
&= \frac{a_1k+a_{k+1}(k+1)+a_1}{2}\\
&=\frac{(a_1+a_{k+1})(k+1)}{2}\end{align}
| No induction needed ... just use a simple trick famously used by Gauss when he was 10 years old:
Take two of these series, one going from $a_1$ to $a_n$, and the other one going back from $a_n$ to $a_1$, put them under each other, and add them up by entry (that is, add the first entries of the two series, then add the... | https://math.stackexchange.com |
368,373 | [
"https://mathoverflow.net/questions/368373",
"https://mathoverflow.net",
"https://mathoverflow.net/users/119875/"
] | I have a matrix
<span class="math-container">$$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$</span>
As you can see, the matrix is always self-adjoint for any <span class="math-container"... | For real <span class="math-container">$a,b,c$</span> and imaginary <span class="math-container">$d$</span> the matrix <span class="math-container">$A$</span> has <em>chiral symmetry</em>, meaning it anticommutes with a matrix <span class="math-container">$X$</span> that squares to the identity:
<span class="math-contai... | An equivalent trick : Let <span class="math-container">$J:= \operatorname{diag}(1,i,-1,-i)$</span>. Then <span class="math-container">$J^*AJ=iB$</span> where <span class="math-container">$B$</span> is real and skew-symmetric. Hence the spectrum of <span class="math-container">$iB$</span> (thus that of <span class="math... | https://mathoverflow.net |
283,286 | [
"https://dba.stackexchange.com/questions/283286",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/90813/"
] | I have an Azure SQL Database, one Table that has 3 Relevant Columns: Brand, ID and Month/Year and a Temporary Table that simply lists the Month/Year with an ID.
The table shows a list of all IDs that were active for that given month.
Month/Year is a datestamp column, with the date as the first of the Month. The ID Colu... | In general you should include the scripts that generate your Table schemas and populates them with data. (Also for performance questions, you should include the <strong>execution plan</strong> too.) I couldn't do a lot of testing without real data, but I believe this query should improve the performance you're seeing:
... | If <code>DAY(date)</code> is always 1 then you may use something like
<pre class="lang-sql prettyprint-override"><code>SELECT *,
CASE WHEN LAG([id]) OVER (ORDER BY [Month/Year] RANGE BETWEEN FOLLOWING 1 MONTH AND FOLLOWING 1 MONTH) IS NULL
THEN 'New'
WHEN LEAD([id]) OVER (ORDER BY [Mon... | https://dba.stackexchange.com |
82,139 | [
"https://electronics.stackexchange.com/questions/82139",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/27975/"
] | Can there be 3 select bits in a 4to1 MUX?
I needed it for a particular application and have no alternative.
| Yes, you can have that. You'll just have some selection combinations that make the same selection. For instance:
<pre><code>module mux_4to1(Y, A, B, C, D, sel);
output [15:0] Y;
input [15:0] A, B, C, D;
input [2:0] sel;
reg [15:0] Y;
always @(A or B or C or D or sel)
case ( sel )
3'b000: Y = A;
3'b001: Y = ... | If you have 2 MUXes with both enable and inhibit inputs, then you can connect the third bit to the enable input of one and the inhibit input of the other, expanding them into a 8-input MUX.
| https://electronics.stackexchange.com |
319,344 | [
"https://dba.stackexchange.com/questions/319344",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/263686/"
] | Running Postgres 13.8 on Ubuntu 20, I have a complicated-looking, yet relatively straightforward query. For some reason, the optimizer is wrong about the expected row count of the joins and is using a nested loop when it shouldn't. You can see that in the nested loop statements in the explain output, it's off by an ord... | You have a bunch of estimation problems here, some of which might be be easy to explain and some of which are hard to explain. But I would say the real problem is a missing index. You don't have an index on bundle_elements which starts with item_id. So instead, it is using an index which has item_id as it its 2nd co... | If the query plans are anything like mysql or sql server, they are estimates and are subject to change between running them. I ran into this issue when trying to get accurate rowcounts pre and post migration; the only way to get around it was to run a select count(*) from each table before and after. That served to w... | https://dba.stackexchange.com |
2,734,374 | [
"https://math.stackexchange.com/questions/2734374",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/253787/"
] | I don't think it is possible because that entails that only the <span class="math-container">$\mathbf 0$</span>-vector is in the eigenspace, but <span class="math-container">$\mathbf 0$</span> is not an eigenvector by definition.
However, my textbook says:
<blockquote>
For an <span class="math-container">$n\times ... | By definition to any eigenvalues correspond at least one eigenvector thus for a n-by-n matrix for each eigenvalue $\lambda_i$ we have $1\le$ dim(eigenspace)$\le n$.
| Following the definition, <span class="math-container">$\lambda$</span> is an eigenvalue of the matrix <span class="math-container">$A$</span> if there exists a non-zero vector <span class="math-container">$v$</span> such that:
<span class="math-container">$$Av = \lambda v.$$</span>
The definition itself assures that... | https://math.stackexchange.com |
46,815 | [
"https://mechanics.stackexchange.com/questions/46815",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/15494/"
] | I have a 1990 Econoline 250, it has a little leak in the valve cover gasket, it sits during the winter and the gasket contracts which makes it pretty bad the first time it starts up. After that not so bad. I know I have to get it fixed, but for the time being, is it safe to drive it with oil burning off the manifold? i... | It really depends, but most of the time you won't get enough oil off of the valve cover to start a fire. You're right, there will be a lot of smoke. Before you start it, you should try and clean off as much as possible. This will limit the amount of smoke which is produced. Also, check the oil to ensure there's enough ... | If it is a copious amount try to wipe off with rag (preferably when cold), otherwise will burn off pretty quick, no other issues.
| https://mechanics.stackexchange.com |
2,110,352 | [
"https://math.stackexchange.com/questions/2110352",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/367034/"
] | I'm struggling with applying my counting skills to probability problems. Specifically I'm grappling with how to enumerate the number of ways to draw colored balls from an urn in which we may have some balls different colors than others.
Here is an example.
Suppose we have $6$ red balls and $4$ blue balls and we want ... | $\binom{10}4$ counts the ways to select 4 distinct items from a set of 10 into <em>a set</em>. Order of the result doesn't matter. This may also be written as $^{10}\mathrm C_4$ .
Example: The probability of selecting two red balls among the four selected is: $$\mathsf P(R=2)=\dfrac{\binom 6 2\binom 4 2... | If things are same as we have red balls. Then after picking we don't need to take care about the arrangements of balls.
But if we are picking 4 balls R1, R2, R3, R4. We have to take care about their arrangements also.
| https://math.stackexchange.com |
112,660 | [
"https://mathoverflow.net/questions/112660",
"https://mathoverflow.net",
"https://mathoverflow.net/users/28176/"
] | Let (A,m) be a local ring and M be a finitely generated A-module contained in a free module F of rank r with length(F/M) < $\infty$. Then I have the following question : Is the statement "M doesn't have a non-trivial free summand if and only if M$\subset$mF " true? I was trying around Nakayama's lemma
| The maximal subgroups of $A_n$ are given by the O'Nan-Scott Theorem. They lie in one of the following classes:
1) $A_n \cap (S_{n-k} \times S_k)$, that is the stabiliser of a $k$-set.
2) $A_n \cap (S_a wr S_b)$ where $n=ab$, that is the stabiliser of a partition.
3) $A_n\cap AGL(d,p)$ where $n=p^d$ for some prime $p... | In general the question is too ambitious, but more can be said than might be expected. For example, it follows from results of J.G. Thompson that if $M$ is a maximal subgroup of a non-Abelian finite simple group $G$ and $M$ is nilpotent, then $M$ is a Sylow $2$-subgroup of $G$ and is non-Abelian (this does occur "in na... | https://mathoverflow.net |
193,726 | [
"https://softwareengineering.stackexchange.com/questions/193726",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/84905/"
] | As the number of bugs in a codebase increases, that number not only decreases the quality of the code, it also affects the mindset of the developers. Developer self-confidence falls when things are not going well. If self-confidence is not right, things more easily become a mess. How do you guys keep your self confi... | Learn about these:
<ul>
<li>Get to know the language better - starting with introductory books and then move on to advanced ones</li>
<li>Basic algorithm proofing techniques</li>
<li>Unit testing</li>
<li>Integration testing</li>
<li>Combinational testing</li>
</ul>
The purpose of knowing the language better and lear... | If you need help with keeping up your self-confidence, you probably need to talk with a personal councillor about that.
But it sounds like the problem here is that your project / team is in a dangerously disfunctional state. You / your team should be using on or more of the standard approaches to keeping code quality... | https://softwareengineering.stackexchange.com |
1,442,880 | [
"https://math.stackexchange.com/questions/1442880",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/45400/"
] | How is it that
$$\ln(2+\tan\theta) - \frac{1}{2}\ln\left(1+\tan^2\theta\right) = \ln \left(\frac{2 + \tan\theta}{\sec\theta}\right)$$
| $$\ln(2+\tan\theta) - \frac{1}{2}\ln\left(1+\tan^2\theta\right) = \ln \left(\frac{2 + \tan\theta}{\sec\theta}\right)$$
$$\ln(2+\tan\theta) - \ln\left(\sec \theta\right ) = \ln \left(\frac{2 + \tan\theta}{\sec\theta}\right)$$
| Hint: $$1+\tan^2\theta=\sec^2\theta$$ and
$$\ln x- \frac{1}{2}\ln y=\ln x-\ln y^{\frac{1}{2}}=\ln\frac{x}{y^\frac{1}{2}}.$$
I think you can finish from here.
| https://math.stackexchange.com |
371,540 | [
"https://math.stackexchange.com/questions/371540",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/28012/"
] | In a substep of a proof, I have a sentence $\varphi$ such that $\varphi\notin T$ and $(\neg\varphi)\notin T$. ($T$ is a consistent theory)
Information about $T$: $T$ is a consistent theory of a first order language $\mathcal{L}_\mathcal{A}$. Every model of $T$ is infinite. Assume $T$ is $\omega$-categorical.
<strong>... | Suppose that $T\cup \{\phi\}$ is inconsistent. Then $T\cup\{\phi\}\vdash\{\neg\phi\}$. Therefore, $T\vdash\{\neg\phi\}\Rightarrow \neg \phi\in T $
which is a contradiction.
| Suppose $T \cup {\varphi}$ is inconsistent. Then $T, \varphi \vdash \bot$, so [assuming $T$ has reductio] $T \vdash \neg\varphi$. So [assuming $T$ is closed under deducibility] $\neg\varphi \in T$. But by hypothesis, $\neg\varphi \notin T$. So $T \cup {\varphi}$ is consistent.
<ol>
<li>That depends on the bracketed as... | https://math.stackexchange.com |
127,220 | [
"https://electronics.stackexchange.com/questions/127220",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/52315/"
] | I was surprised to find that bridging neutral and ground in a socket at home tripped the RCD. Measuring with a multimeter, there is indeed 0.1 V between them. If the RCD trips at 30mA that would mean that the wiring in the house must have resistance lower than 3 ohms. (Sounds likely if household wiring is around 0.01 o... | 1) Why would you expect a potential difference not to appear? N and earth are tied together only in distribution/transformer boxes, that's a long way till your house, the N wire is usually carrying some current so its potential might slightly differ from ground.
2) If you call tripping the whole house an issue, well t... | With <strong>earth</strong> and <strong>neutral</strong> being tied at a distribution panel but the <strong>neutral</strong> being used to close the power circuit with <strong>live</strong>, there will be a potential difference but that will not be why the RCD tripped.
An RCD checks for a balanced current in the <stro... | https://electronics.stackexchange.com |
27,357 | [
"https://mathoverflow.net/questions/27357",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4087/"
] | When you are checking a conjecture or working through a proof, it is nice to have a collection of examples on hand.
There are many convenient examples of commutative rings, both finite and infinite, and there are many convenient examples of infinite non-commutative rings. But I don't have a good collection of finite ... | 2 families of examples that are sometimes useful to have in mind:
(1) The group ring of a non-abelian finite group over a finite commutative ring.
and
(2) the incidence algebra of a finite poset over a finite commutative ring (the ring of upper triangular matrices is a basic example of this).
Of course, both o... | From one point of view, the classification of finite non-commutative rings or even finite commutative rings is wild and tons of things can happen. From another point of view, finite rings are highly restricted and not all that much can happen.
By the Chinese remainder theorem, every finite ring is unique direct sum o... | https://mathoverflow.net |
6,226 | [
"https://mechanics.stackexchange.com/questions/6226",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/2601/"
] | We just recently got a Subaru Outback. It is a manual and has AWD. The dealer told us if we need it towed, it has to go on a trailer, and can't be towed with wheels down. I understand that this can cause transmission damage.
My question is, why would this cause transmission damage, when I can pop it into neutral an... | All wheel drive vehicles connect the front and rear axles via a transfer case or differential. While on a wheel lift tow one set is lifted off the ground and not spinning and the trailing wheels are spinning at road speed. This places a big load and resulting wear on the power transfer unit. Coasting allows all four wh... | You can dinghy tow an Outback or Forester behind a motorhome if these conditions are satisfied:
<ol>
<li>The Subaru is a Manual Transmission model; <strong>No automatic transmission subaru should ever be dinghy towed with wheels down!</strong></li>
<li>Use the gray "Valet" key to unlock the steering wheel. This is the... | https://mechanics.stackexchange.com |
2,015,307 | [
"https://math.stackexchange.com/questions/2015307",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/267981/"
] | Let $C^{0}[a,b]$ be the set of all continuous functions $f:[a,b] \rightarrow \mathbf{R}$ and $A \subset C^{0}[a,b]$. By
the extreme value theorem, $\|f\|_{\sup} < \infty$ for all $f \in C^{0}[a,b]$, so does this imply that
there exists an $M \in \mathbf{R}$ such that $M$ is a bound on $A$ under the sup norm?
| No, it does not. Every function is bound in its own right but unless you specify the set $A$ there may easily be no <em>common</em> bound on them. As a simple example, imagine the family of constant functions
$$f_q(x) = q.$$
This is continuous on $[a,b]$ for any $q$ with $\|f_g\|_{\rm sup} = |q|$. Given an $A$ that con... | If $f_n(x)=n$ the the sequence $\{f_n\}$ is a non-bounded subset of $C^{0}[a,b]$
| https://math.stackexchange.com |
37,497 | [
"https://stats.stackexchange.com/questions/37497",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/14174/"
] | Documentation states that R gbm with distribution = "adaboost" can be used for 0-1 classification problem. Consider the following code fragment:
<pre><code>gbm_algorithm <- gbm(y ~ ., data = train_dataset, distribution = "adaboost", n.trees = 5000)
gbm_predicted <- predict(gbm_algorithm, test_dataset, n.trees = ... | The adaboost method gives the predictions on logit scale.
You can convert it to the 0-1 output:
<pre><code>gbm_predicted<-plogis(2*gbm_predicted)
</code></pre>
note the 2* inside the logis
| You can also directly obtain the probabilities from the <code>predict.gbm</code> function;
<pre><code>predict(gbm_algorithm, test_dataset, n.trees = 5000, type = 'response')
</code></pre>
| https://stats.stackexchange.com |
105,388 | [
"https://mathoverflow.net/questions/105388",
"https://mathoverflow.net",
"https://mathoverflow.net/users/25921/"
] | Let $f:R\rightarrow R$. If there exists the finite limit $$\lim_{(x,y) \rightarrow a \atop x\neq y} \frac{f((y)-f(x)}{y-x}$$ then obviously there is a finite derivative $f'(a)$ and is equal this limit.
What about similar problem for higher order divided differences?
May is it true that existence of finite $$\lim_{(... | The answer is yes:
Assume we have a $\delta$ such that for $|x_i-a|<\delta$, we have $|[x_0,...,x_n;f]-\lim| < \epsilon$. Assume WLOG that $\lim = 0$ (by subtracting off a polynomial of degree $n$). Then if $y_0, ..., y_{n-1}$ and $z_0, ..., z_{n-1}$ are in the $\delta$-ball around $a$, we can show
$|[y_0,...,y... | In general the answer is no. For instance, if f is any odd function, then
$$\lim_{h\to 0}\frac{f(h)+f(-h)-2f(0)}{h^2}=0,$$
without any assumptions on differentiability of f. So it certainly does not follow that f''(0) exists.
| https://mathoverflow.net |
159,820 | [
"https://mathoverflow.net/questions/159820",
"https://mathoverflow.net",
"https://mathoverflow.net/users/45392/"
] | Let $F=\{z\in H: |z|\geq1, |Re(z)|\leq 1/2\}$. It is a fundamental domain of the modular group $\Gamma$ acting on the upper half plane $H$. (Strictly speaking, one should take part of the boundary from $F$ to make it a fundamental domain. For example, one can use $F_1=F-\{z\in F:Re(z)=1/2\}-\{z\in F:Re(z)>0 \text{ a... | As Misha says, this is false as stated. To see this: Apply the dilation $A \colon z \mapsto z/2$ to $F$. Note $i$ lies in the interior of $A(F)$. Since $i$ is fixed by a torsion element of $SL(2,\mathbb{Z})$, deduce that $A(F)$ is not a fundamental domain.
There are several different things you might be trying to... | As @SamNead notes, this is simply not true. It would be convenient if it <em>were</em> true, because then there'd be an easy proof of the self-adjointness of Hecke operators, yes. Indeed, this error is one of the standard, popular unfortunate-choices to try to prove that self-adjointness. The self-adjointness certainly... | https://mathoverflow.net |
3,804,380 | [
"https://math.stackexchange.com/questions/3804380",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/438651/"
] | <strong>Question.</strong> Suppose that <span class="math-container">$x,y,x',y'$</span> are positive integers satisfying <span class="math-container">$x^2-dy^2=\pm 1$</span> and <span class="math-container">$(x')^2-d(y')^2=\pm 1$</span> respectively. Assuming <span class="math-container">$x<x'$</span>, prove that <s... | You have
<span class="math-container">$$x^2-dy^2=\pm 1 \tag{1}\label{eq1A}$$</span>
<span class="math-container">$$(x')^2-d(y')^2=\pm 1 \tag{2}\label{eq2A}$$</span>
If <span class="math-container">$d \lt 0$</span>, there are no solutions in only positive integers and, if <span class="math-container">$d = 0$</span>, the... | Hm... isn't this trivial to prove?
First, it is easy to see that <span class="math-container">$d$</span> is positive.
Second, you can directly express <span class="math-container">$y$</span> and <span class="math-container">$y'$</span> as a function<br />
(it's the same function) of <span class="math-container">$x$</sp... | https://math.stackexchange.com |
3,714,997 | [
"https://math.stackexchange.com/questions/3714997",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/780973/"
] | What is different between field , Vector function and plane , they seem all shown with same equation:
<span class="math-container">$ \ r \left( t \right) = \left\langle { f \left( t \right),g \left ( t \right) , h \left( t \right)} \right\rangle$</span>
| For any inner product space, complete or not, let <span class="math-container">$v$</span> be any vector and consider the finite-dimensional subspace generated from <span class="math-container">$v,Av,\ldots,A^{k-1}v$</span>. Then this space is <span class="math-container">$A$</span>-invariant, so <span class="math-conta... | Since <span class="math-container">$A$</span> is self-adjoint, all eigenvalues of <span class="math-container">$A$</span> are real. Also, <span class="math-container">$\lambda$</span> is an eigenvalue of <span class="math-container">$A$</span> implies <span class="math-container">$Av=\lambda v$</span> for some non-zero... | https://math.stackexchange.com |
547,825 | [
"https://math.stackexchange.com/questions/547825",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | <blockquote>
Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.
</blockquote>
$\Bbb{R}$ must be Hausdorff. For $x_1, x_2 \in \Bbb{R}$ (where $x_1 \not= x_2$), if $d$ denotes the distance between these two points, then we can choose an open ball $U_1$ with radius ... | Your solution to Hausdorffness is not quite correct. Note what happens if your points are $\mathbf{x} = \langle 1 , 2, 3, \ldots \rangle$ and $\mathbf{x^\prime} = \langle 3 , 3 , 3 , \ldots \rangle$. (How do you choose the neighbourhoods $U_3$ and $U^\prime_3$?)
Recall that the usual product topology on $\mathbb{R}... | For not metrizable:let $A$ be the set of all sequences with elements in $\mathbb{R}$ with the property that the elements of all these sequences are positive .now the zero sequence(the sequence with zero elements) is a closure point for $A$ because every open set in $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ has a form $U_1... | https://math.stackexchange.com |
174,971 | [
"https://softwareengineering.stackexchange.com/questions/174971",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/57863/"
] | I've programmed in both C# and VB.NET for years, but primarily in VB. I'm making a career shift toward C# and, overall, I like C# better.
One issue I'm having, though, is curly brace soup. In VB, each structure keyword has a matching close keyword, for example:
<pre class="lang-vb prettyprint-override"><code>Namespac... | Put your starting curly brace in the same "rank" as your ending one, like this:
<pre><code>namespace ...
{
class ...
{
function ...
{
for ...
{
using ...
{
if ...
{
...
... | <ul>
<li>Depending on your IDE: put your cursor at the open/close brace and it'll highlight both that and the corresponding brace.</li>
<li>Collapse the block and it shows you where it opens/closes.</li>
<li>Write smaller code blocks. Seriously. Check out <code>Clean Code</code>, and never run into this problem again (... | https://softwareengineering.stackexchange.com |
81,252 | [
"https://dsp.stackexchange.com/questions/81252",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/61279/"
] | In my textbook, it is stated that for a discrete system, where the input and output are expressed by difference equations, to be causal, there needs to be initial rest. It is also stated that for the system to be linear, the initial conditions should all be zero. I understand why these cases are true, however (I have f... | <blockquote>
A linear system is causal if the following condition holds:
<span class="math-container">$$x(n) = 0\ \forall\ n \leq n_0\implies y(n) = 0\ \forall\ n \leq n_0 \tag 1$$</span>
where <span class="math-container">$n_0$</span> is an arbitrary point in time, and so we say that the system is initially at rest.
... | Linearity and causality are different things.
Causality means that the output is only a function of the present and past input samples (and output samples). <span class="math-container">$y[n] = x[n] + x[n-1]$</span> is causal and <span class="math-container">$y[n] = x[n]+x[n+1]$</span> isn't.
Linearity is simply define... | https://dsp.stackexchange.com |
200,769 | [
"https://mathoverflow.net/questions/200769",
"https://mathoverflow.net",
"https://mathoverflow.net/users/8628/"
] | We define an equivalence relation on $\mathcal{P}(\omega)$: for $x,y\in\mathcal{P}(\omega)$ we say $$x\simeq_{fin} y \text{ iff there is } n \in \omega \text{ such that }
x\setminus \{0,\ldots,n\} = y \setminus \{0,\ldots,n\}.$$
The set $\mathcal{P}(\omega)/\simeq_{fin}$ is usually written as $\mathcal{P}(\omega)/fin$... | Yes. This is an easy exercise:
For every $r\in\Bbb R$ fix some sequence of rational numbers $r_n$ such that $\lim r_n=r$. Now enumerate $\Bbb Q$ as $\{q_n\mid n\in\Bbb N\}$ and consider $A_r=\{k\mid\exists n:q_k=r_n\}$.
Then given $r\neq r'$, the sequences $r_n$ and $r'_n$ must be disjoint from some point onwards, be... | Consider the antichain $\{ [x]: x\in A \},$ where $A$ is an almost disjoint family of subsets of $\omega$ of size $2^{\aleph_0}.$
| https://mathoverflow.net |
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