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
values |
|---|---|---|---|---|---|
21,947 | [
"https://ai.stackexchange.com/questions/21947",
"https://ai.stackexchange.com",
"https://ai.stackexchange.com/users/37831/"
] | During the first episode, it's 100% exploration, because all our Q values are 0. Suppose we have 1000 time steps, and it's terminated by meeting a reward. So, after the first episode, why can't we make it 100% exploitation? Why do we still need exploration?
| You can't guarantee that you have taken every action from every state, even with 1000 time steps. There would be multiple outcomes:
<ol>
<li>The episode terminates, either by success or failure before the 1000
time steps. The agent is trying to maximise reward, if this is achieved by taking less than 1000 steps then ... | BlueTurtle's answer is good, but I'd like to add something.
Your question realistically has nothing to do with Q Learning, in fact, you can ask the same thing about just about any RL algorithm. In fact, even in multi-armed bandits, you can easily see why your proposed method is suboptimal (please don't interpret this a... | https://ai.stackexchange.com |
411,876 | [
"https://physics.stackexchange.com/questions/411876",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/198391/"
] | In the calculation of the Feynman Amplitude for the muon neutrino-electron scattering (in the Charged Current way from W boson), or $e + \nu_\mu \rightarrow \nu_e + \mu$ (considering the 4-momentum conservation as $p_1 + p_2 = p_3 + p_4$, in the same sequency of the reaction described) the multiplication appears:
$64... | All the other terms in your answer are zero. I will demonstrate for one of the terms as the reasoning is similar
The second term and third term in your solution is $$-i \epsilon_{\mu \nu \kappa \tau}p_2^\kappa p_4^\tau p_1^\mu p_3^\nu - i \epsilon_{\mu \nu \kappa \tau}p_2^\kappa p_4^\tau p_1^\nu p_3^\mu$$
Since the ind... | Your result is correct; the contraction of a completely antisymmetric tensor, such as $\epsilon_{\mu\nu\alpha\beta}$, with a symmetric tensor, such are the products $p_1^{\mu}p_2^{\nu}p_3^{\alpha}p_4^{\beta}$ and $g^{\mu\nu}p_i^{\alpha}p_j^{\beta}$ in a QED or EW tree-level process (where any external momentum can be s... | https://physics.stackexchange.com |
122,796 | [
"https://electronics.stackexchange.com/questions/122796",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/8647/"
] | I have written the following function to calculate the current timestamp (since 00h Jan 1 1900). The output I get is:
<pre><code>Current Time & Date : 20:5:32 25/7/2014<\r>
2014 7 25 20 5 32<\r>
retval 1 3597523200<\r>
retval 2 3597523200<\r>
retval 3 3613161600<\r>
retval 4 361523520... | 20 * 60 * 60 = 72000
You need to perform the multiplication in 32 bits in order to avoid an overflow.
<pre><code>retval += (d.hr * 60UL * 60);
</code></pre>
| (3615235200-3613161600)/(24*60*60) = 24, so the change at 'retval 4' is correct and not an overflow, so your assertion that they are correct "til you get to retval 4" is false.
retval 5 however gives (3615241664-3615235200) = 60*60*20 - 2**16, so there is an overflow. Use a uint32_t on the right hand side of <code>re... | https://electronics.stackexchange.com |
5,671 | [
"https://electronics.stackexchange.com/questions/5671",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/1225/"
] | Is a function generator necessary for every day lab use, or is it special purpose equipment? That is, does it have similar utility to an oscilloscope, or multimeter - would you use it regularly enough to justify it's cost?
| If you're working with digital systems and square waves/pulse trains only, then it's probably not necessary. However, for analog amplifier design (ex. audio), it's an imperative.
If you haven't been stymied on a project because you lacked this tool, you probably don't need it. Spend your money on an oscilloscope a... | In the audio frequency range you can use your soundcard. Use google to find the software.
| https://electronics.stackexchange.com |
566,229 | [
"https://physics.stackexchange.com/questions/566229",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/156987/"
] | I read that homogenous space implies conservation of momentum (according to Noether's theorem) Conservation of momentum is <em>kinda</em> the statement that bodies continue moving with constant velocity unless a force is applied.
Suppose we instead lived in a universe where a bunch of frames could be described as absol... | <strong>After the comment by @ZeroTheHero I realized my answer is wrong. Here is the explanation of what went wrong. I will delete the answer later</strong>
Conservation of ordinary momentum is implied by homogeneity of space and time, not just space. The most easiest way to see it is by the fact that you can simulate ... | Let's say that anything moving relative to the "absolute rest" frame decelerates at a constant rate, <span class="math-container">$a = -x \ m/s^2$</span>, until it is at rest with respect to the "absolute rest" frame. Then, if you are in the absolute rest frame, you will see this deceleration.
If yo... | https://physics.stackexchange.com |
162,198 | [
"https://dba.stackexchange.com/questions/162198",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/115873/"
] | I have a database with the following:
<pre><code>|Customer number| |Sales date| |Cancel date|
</code></pre>
From this I want to count the number of active customers each year for lets say 10 years back.
If I want to do this for a single year I can do this (I think):
<pre><code>SELECT
customernumber
FROM
cust... | Thanks for all your answers. This is probably not the best solution, but this solutions does not require me to make a new table:
<pre><code>SELECT
customernumber,
'2015'
FROM
customers
WHERE
Datepart(year, salesdate)<2015
AND (Datepart(year, CancelDate)>2015 or Datepart(year, CancelDate) IS NULL)
UNION
SELECT
cu... | You need a tally table or calendar table that includes all years (that you want to report on). Call the table Allyears or something like that. As a field, you could put something like yearstart datetime and populate it with January 1 of each year (consider yearend as well). Then simply join the Allyears table to Tes... | https://dba.stackexchange.com |
512,003 | [
"https://physics.stackexchange.com/questions/512003",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/246513/"
] | I have the following assumption based on limited knowledge.
A black hole appears dark to us because any light that would be emitted from it, and any light that passes nearby, is caught by the gravity well and either captured or bent from its original course.
If this is true, from any point within the gravity well or t... | Anything that crosses the event horizon (including light) will swiftly proceed towards the "central" singularity (at least until it crosses the inner horizon; we will get back to that). In particular, it is impossible for an observer to hover inside the black hole. Hence any observer will be falling further down the bl... | The light is not exactly bent from it's original course, the blackhole curves spacetime and the light follows that curvature. The thing is that the time component of "spacetime", as well, is bent near a black hole, so that as the light approaches the event horizon, the time it takes (from an outside perspective) for th... | https://physics.stackexchange.com |
123,485 | [
"https://electronics.stackexchange.com/questions/123485",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/50412/"
] | I have a project that I've begun to work on, it's sort of an embedded PC type project, with a iMx6 arm processor. However, since this is a (potentially) commercial venture, we can't use prebuilt boards like the Pi etc. I have a thorough understanding of creating PCBs, however I've never made one with SMT ICs, and now I... | You've already accepted an answer, but I'll add this:
Keep in mind that all but the smallest (number of balls) and crudest (ball pitch) BGA layouts are going to require expensive multilayer boards- 6 or 8 layers, often not even the relatively inexpensive 4-layer type. Costly per board in small quantities and high up-f... | There are short-run electronics assembly services in most western countries, although they may be hard to find. I'd expect to pay $100-$200 per assembled board for this kind of thing, and possibly a few hundred $ in setup fees.
Note that buying the iMX6 components for short production runs may not be entirely simple.
| https://electronics.stackexchange.com |
289,460 | [
"https://softwareengineering.stackexchange.com/questions/289460",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/13552/"
] | What should you do if an API has changed the provided functionality? [In this case it would affect your project as a whole]
Some more context in my situation: Facebook eliminated the APIs ability to create, update, or delete event's via their API, they only provide read access. My whole application is based on creatin... | The first thing you should do is shout your favorite expletives while imagining the horrible tortures you would like to inflict on those who just killed your project. That's a great way to relieve stress and vent frustration.
After you calmed down, your next step would be to write an apology to your customers where yo... | In most cases it is possible to put a work around in place. the users can still create events right? just not through the api. so, worst case when they click a button you can have a webpage open.
You can probably find some halfway house where you parse the webpage and partially automate the Event creation
| https://softwareengineering.stackexchange.com |
90,676 | [
"https://softwareengineering.stackexchange.com/questions/90676",
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"https://softwareengineering.stackexchange.com/users/3542/"
] | I am part of a developer team that works with many other teams to maintain and improve an application that has been in use for at least 15 years. When it was first built and designed, TDD was unheard of.
The application is fairly stable, and we rarely encounter a show stopping bug, but we do average about one or two b... | Direct incorporation of full-blown TDD into a legacy code, maintenance project is a very hard sell. An approach I've seen work very well is this. For each bug that comes in, create an automated non-unit test that demonstrates the bug. By "non-unit", I mean something that can touch many parts of the system, hit datab... | At my company, I just went with the "only a grunt" method from JoelOnSoftware and began writing unit tests whenever I normally would have simply hacked together some sort of throw away console application. I started getting things done a lot faster with more stable releases, and got noticed for it. When asked what I wa... | https://softwareengineering.stackexchange.com |
348,989 | [
"https://softwareengineering.stackexchange.com/questions/348989",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/224262/"
] | I have a collection of Python objects, lets call them fruits. Some of these fruits might be faulty. To check for these faults, I have a number of functions that performs different checks, and returns a list of fruits that did not pass the check:
<pre><code>def not_delicious(fruits, limit):
return fruit for fruit i... | This looks like a job for polymorphism.
You want to bind checks and messages together without binding them to the <code>log_check</code> code. This is a good idea.
You're reluctant to use a class because the class would be stateless. To that I say: Feh! Stateless objects are perfectly fine.
An object is a bag of f... | If you e.g. don't want to rewrite a bunch of existing functions just to attach messages to them, you can do so externally:
<pre><code>CHECKS = {
(check_ripeness, 'This ain\'t no ripe enough'),
(check_taste, 'deliciousness does not conform to the standards'),
# etc
}
# somewhere in your method
for (predicate, me... | https://softwareengineering.stackexchange.com |
2,248,045 | [
"https://math.stackexchange.com/questions/2248045",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/380871/"
] | I am wondering if I'm thinking properly regarding this question:<img src="https://i.stack.imgur.com/eArlb.jpg" alt="enter image description here">
First, according to the pointwise limit, I did this informally:
As $n$ becomes very large, the interval $\left[\frac {1}{n}, \frac {2}{n}\right]$ will contain no $x$.
Cle... | Multiply by $R$ and integrate from $0$ to $1$:
$$ 0=\int_0^1 ((\rho R')'R + \lambda \rho R^2) \, d\rho. $$
Now integrate the first term by parts:
$$ \int_0^1 ((\rho R')'R \, d\rho = \left[ \rho R'R \right]_0^1 - \int_0^1 \rho R'^2 \, d\rho. $$
The first term vanishes if we assume that $R'$ is bounded as well, so
$$ 0 =... | One solution of Bessel's equation (without the right endpoint condition) is
$$
R_{\lambda}(r)=\sum_{k=0}^{\infty}\lambda^k\frac{(-1)^k\rho^{2k}}{2^{2k}(k!)^2}.
$$
A second linearly independent solution is obtained by variation of parameters, from which it can be shown that the second solution $S_{\lambda}$... | https://math.stackexchange.com |
555,655 | [
"https://stats.stackexchange.com/questions/555655",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/343076/"
] | I am running a multiple variable regression predicting GDP per capita for U.S. states with a bunch of independent variables. Currently I have included the District of Columbia in the data set which has a much higher GDP per capita. Should I still include D.C. in the dataset?
My reasoning to include it is that D.C. has ... | The criterion for making this decision should be based on
your purposes, and not influenced by the fact that the GDP per capita is for DC is about twice that of the leading state (NY) [which is in turn about twice that of the trailing state (MS)].
You say you are "... predicting DP per capita for U.S. states....$
... | As a general rule, you should not remove data from your dataset unless you think it was recorded incorrectly and so it unreliable due to measurement error. If a data point is correctly measured, but it merely has a value that is substantially different to other values in the data (making it an "outlier" in t... | https://stats.stackexchange.com |
52,924 | [
"https://security.stackexchange.com/questions/52924",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/16418/"
] | Imagine I have a website with user accounts. I want to authenticate users, so I need to know if the password they provide matches the one stored in my database. To protect the passwords from prying eyes, I want to obfuscate them, I decide to use AES in CBC mode to do so.
I know that I shouldn't use AES-CBC to do this,... | You are correct in that the salt and the IV are redundant: they both protect against the same type of attack (discovering that two accounts have the same password), and they do it in the same way (by making the encrypted representations of the password differ).
| <blockquote>
When encrypting the user password to store it in my database using AES-CBC, do I need to use a Key, an IV and a Salt? Whatever I need to use, how would they be used exactly?
</blockquote>
The IV is used to initialize the first 16 byte matrix of the AES-CBC encryption scheme. It is XORed with the first 1... | https://security.stackexchange.com |
219,385 | [
"https://softwareengineering.stackexchange.com/questions/219385",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/53095/"
] | As the question states: When implementing "SOA", is it a concept intended for communication between systems over a network or is it intended as a concept that operates within the language as a pattern?
| Service Oriented Architecture is an <em>architecture</em>, so the answer is neither.
It's not a design pattern within a language because it governs decisions far, far outside of the program design - notably, how all your business data is organized into services, which has a close relationship with your organizational ... | If you use it within a language as a pattern, you won't need some of the features of SOA:
<ul>
<li>loose coupling</li>
<li>mediated communication option</li>
<li>asynchronous communication (fire & forget messages)</li>
</ul>
So if you just want to have a clean contract between components, you can also do it witho... | https://softwareengineering.stackexchange.com |
110,837 | [
"https://physics.stackexchange.com/questions/110837",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/45670/"
] | I'm reading through Hartle's General Relativity and came across this question:
<blockquote>
Consider the following coordinate transformation from rectangular coordinates $(x,y)$, labeling points in the plane to a new set of coordinates $(m,n)$:
$$x = mn,$$
$$y = (1/2)(m^2 - n^2).$$
(c) Do the curves of consta... | Well, you can just calculate the metric:
$$\begin{aligned}
\mathrm ds^2 &= \mathrm dx^2 + \mathrm dy^2\\
&= \mathrm d(mn)^2 + \frac12\mathrm d(m^2 - n^2)\\
&= (m\,\mathrm dn + n\,\mathrm dm)^2 + \frac12(2m\,\mathrm dm - 2n\,\mathrm dn)^2\\
&= m^2\,\mathrm dn^2 + 2mn\,\mathrm dm\,\mathrm dn + n^2\,\ma... | $$dx= ndm + m dn\:,\quad dy = mdm -ndn$$
hence
$$dx^2 + dy^2 = n^2 dm^2 + m^2 dn^2 + 2nm dn dm + m^2 dm^2 + n^2 dn^2 - 2nm dn dm \:.$$ We conclude that:
$$ds^2 = (n^2 +m^2)(dm^2 + dn^2)$$
and there is no $dn\: dm$ term in the metric as stated.
| https://physics.stackexchange.com |
320,348 | [
"https://electronics.stackexchange.com/questions/320348",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/157973/"
] | my group-mate and I are currently trying to create a kind of a simple speaker which could vibrate up to 1 kHz. The speaker we designed was simply made up of small solenoid and a cylindrical magnet, we are using a function generator to generate the signal. However we noticed that the vibration can only be seen up to 400... | It's not clear to me whether this will preserve the orientation as you hope, but you can try this:
<ol>
<li>Shift-click all the objects to be moved (you can also shift-drag a
box around them). Don't include any vias.</li>
<li>Shift-doubleclick one of the selected items. The PCB Inspector dialog should pop up. Prope... | Probably just change the colour coding in Altium.
<ol>
<li>Mine has blue set as bottom layer and red as Top layer. </li>
<li>You can press 'L' to enter the configuration dialog box. </li>
<li>Double click on the color to change</li>
<li>Replace the bottom color layer with the top and voila!</li>
<li>Note that this sh... | https://electronics.stackexchange.com |
86,233 | [
"https://physics.stackexchange.com/questions/86233",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/16043/"
] | I know that under $SU(2) \times SU(2)$, the left-handed electron transforms under $ ( \frac{1}{2},0 ) $ representation and the vector gauge field $A_\mu$ under $ ( \frac{1}{2},\frac{1}{2}) $.
Since the electron transforms under $U(1)$, there must be a represenation under which it transforms. What is this representat... | The electron field transforms under the $\mathbf 1$ of $U(1)$, i.e., the generator is $i$ or $1$ depending on your convention/notation.
The gauge fields transform in the adjoint representation , but they transform as a connection, as @Adam mentioned. In other words, if $\psi \to g \psi$, then $D_\mu \psi \to g D_\mu ... | $U(1)$ is an Abelian group. Abelian groups only have 1-dimensional irreducible representation. Namely, transformation by a phase (in the case of the electron). The charge of fermion field is proportional to the coefficient of the phase. In particular, a field of charge $q$ transforms as $\Psi \to e^{i q \theta(x)} \Psi... | https://physics.stackexchange.com |
57,828 | [
"https://physics.stackexchange.com/questions/57828",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | Let's have the next case.
A rod (with mass $m$, length $L$ and a momentum of inertia $I$) at the initial time is located on a cylinder (with radius $R$) surface so that it's (rod's) center of mass lies on top of the surface. The сylinder is fixed. The rod moves along the surface without slip. There's need to find a fr... | If $\varphi$ is the angle of the rod then I get
$$ \ddot{\varphi} = \mbox{-} \frac{m r \varphi(g \cos\varphi+ 3 r \dot{\varphi}^2)}{I+m\,r^2 \varphi^2} $$
using vector calculus. Since $\ddot{\varphi} \propto -\varphi$ then the result is small angle linear harmonic motion. If <em>C</em> is the center of the rod then:
... | Your energy conservation is of the form:
$E= A\phi^2 + B\dot\phi^2 = const$
Take the time-derivative, then:
$2A\phi\dot\phi + 2B\dot\phi\ddot\phi = 0$
Follows SHO equation:
$\ddot\phi = - (A/B)\phi$,
and the angular frequency is $\omega=\sqrt{A/B}$
If you want to use the Lagrangian then
$L = T -V = B\dot\phi^2 - A\... | https://physics.stackexchange.com |
57,079 | [
"https://stats.stackexchange.com/questions/57079",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/21919/"
] | I have some data which I've analyzed using Kaplan Meier estimation. However, I have a gut feeling that this estimator is biased due to the high censoring rate in my data (nearly 50% censored at later times). What are some ways to address this in an analysis?
| The Kaplan-Meier estimator is <em>not</em> biased when a large proportion of individuals are censored. One of the problems we often observe is that the majority of power for the log-rank test is derived from early failure times which are difficult to observe in KM curves. It does mean that the median survival time is a... | K-M does not work well for censoring proportions >50%. If you can analyze the distribution of your data, it is better to use a parametric method such as MLE. In alternative, you can also use imputation methods.
| https://stats.stackexchange.com |
198,641 | [
"https://softwareengineering.stackexchange.com/questions/198641",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/4584/"
] | From the book <em>Think Like a Programmer</em> (emphasis mine)
<blockquote>
The Luhn formula is a widely used system for validating identification numbers. Using the original number, double the value of every other digit. Then add the values of the individual digits together (if a doubled value now has two digits, a... | It means that you're allowed to retain the value of one character in memory only until the next input operation happens. This is an idealization of the common practical constraint that an algorithm must be able to process arbitrary input without requiring arbitrary amounts of RAM. A simple practical test is: does your ... | The opposite would be for example reading the <em>whole</em> identification number into a string, and then making multiple passes over the digits of the identification number to calculate the Luhn checksum. The question is asking you to calculate the checksum without looking back to previous digits.
| https://softwareengineering.stackexchange.com |
176,984 | [
"https://physics.stackexchange.com/questions/176984",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/24371/"
] | For fermions
$$\bar{n}_{FD}=\frac{1}{e^{(\epsilon -\mu)/kT}+1}$$
and $\epsilon$ can be bigger or small than $\mu$.
However, for bosons:
$$\bar{n}_{BE}=\frac{1}{e^{(\epsilon -\mu)/kT}-1}$$
which implies that $\epsilon >\mu$ otherwise the occupancy is negative.
Is there another argument as to why $\epsilon >\mu$ ... | I found it:
$$\bar{n}_{BE}=-\frac{1}{\mathcal{Z}}\frac{\partial \mathcal{Z}}{\partial x}$$ where $x =(\epsilon -\mu)/kT$
$$\mathcal{Z} = \sum\limits_{n}^\infty e^{-n(\epsilon -\mu)/kT}$$
which converges if $\epsilon -\mu<0$
| The chemical potential is sort of the potential energy needed to add another particle from the surrounding reservoir to the system. Thus to add another particle to a particular single particle level requires energy if the chemical potential is larger than the energy of single-particle level.
If the chemical potential... | https://physics.stackexchange.com |
4,278 | [
"https://dba.stackexchange.com/questions/4278",
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] | One of my co-workers created a schema similar to the following. This is a simplified schema including only the parts necessary to address this question.
The system rules are as follows:
<ol>
<li>Departments can have 0 to many divisions. </li>
<li>A division must belong to only one department. </li>
<li>An artic... | Your self-reference could introduce a hierarchy of divisions within divisions. You need code (usually a trigger) to ensure that no divisions allow this.
The original schema using encoding is broken too. You have no enforceable FK
Personally, I'd consider this with a check constraint to ensure than only one of Departm... | Might I propose a new idea? The lookup table here is an auto-increment, in this way there is only ONE ID used between both "Division" and "Departments". This way also allows for "Departments" to be owned by multiple divisions (I think this is what you are after).
Whenever I have a set of unique tables that need to be ... | https://dba.stackexchange.com |
94,081 | [
"https://dba.stackexchange.com/questions/94081",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/60165/"
] | I am using <code>Postgres 9.3</code>.
When I update row in a table my flow looks like this:
<ol>
<li>Fetch old row (ex. <code>SELECT * FROM tbl WHERE id = 1</code>)</li>
<li>Validate new data in scope of old data (old row fetched in point 1) - this is done by my app outside of postgres.</li>
<li>Update row with new d... | Not sure why you are not updating directly with additional predicates in the where clause, but if that is not possible you can select for update as in:
<pre><code>SELECT * FROM tbl WHERE id = 1 FOR UPDATE
</code></pre>
This will lock the selected row and prevent updates of it while you do your validation.
| You could put a lock on the table while you do the validation.
<pre><code>BEGIN WORK;
LOCK TABLE tbl IN exclusive mode;
select * from tbl where id = 1;
--Do your validation
update tbl set .. where id = 1;
COMMIT WORK;
</code></pre>
| https://dba.stackexchange.com |
101,577 | [
"https://mathoverflow.net/questions/101577",
"https://mathoverflow.net",
"https://mathoverflow.net/users/21596/"
] | I am considering the Hecke operators $T_n$ acting on the space $M_k(\text{SL}_2(\mathbb{Z}))$ of weight $k$ modular forms of level 1. Are their eigenvalues always real?
I have read somewhere that the Fourier coefficients of a normalized eigenform are real. The coefficients are precisely the eigenvalues right? Is this ... | More simply, the eigenvalues are real because the Hecke operators are Hermitian for the Peterson scalar product (a fact which can be checked by a straightforward computation). See for example the introduction by Serre on modular forms in Cours d'arithmétique.
| In general, if $\pi$ is an automorphic representation of $\mathrm{GL}(n)$ with contragradient $\tilde\pi$, then $L(s,\tilde\pi)=\bar L(\bar s,\pi)$. In particular, the Dirichlet coefficients of $L(s,\pi)$ are real iff $\tilde\pi\cong\pi$. In the case of $n=2$ we have $\tilde\pi\cong\pi\otimes\omega^{-1}$, where $\omega... | https://mathoverflow.net |
1,519,682 | [
"https://math.stackexchange.com/questions/1519682",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/233658/"
] | How can one show that $\sqrt{x-1}+\sqrt{y-1}\leq \sqrt{xy}$
Assuming that :
$\sqrt{x-1}+\sqrt{y-1}\leq \sqrt{xy}$
So
$(\sqrt{x-1}+\sqrt{y-1})^2\leq xy$
$\sqrt{(x-1)(y-1)} \leq xy-x-y+2$
$ (y-1)(x-1)+3 \leq \sqrt{(x-1)(y-1)}$
Here I'm stuck !
| There are some errors in your calculation, e.g. a missing factor 2 in
$$
(\sqrt{x-1}+\sqrt{y-1})^2 = x - 1 + y - 1 + 2\sqrt{x-1}\sqrt{y-1}
$$
and in the last step the inequality sign is in the wrong direction and
the number $3$ is wrong.
<hr>
For $x \ge 1$, $y \ge 1$ you can square the inequality (since both
sides ar... | Starting with “Assuming that $X$” when $X$ is the thing to be proved is not the right thing to do. You can make your computations easier if you set
$$
t=\sqrt{x-1},\quad u=\sqrt{y-1}
$$
so $x=t^2+1$ and $y=u^2+1$. The inequality to be proved becomes
$$
t+u\le\sqrt{(t^2+1)(u^2+1)}
$$
that has a single radical. Since $t+... | https://math.stackexchange.com |
3,380,436 | [
"https://math.stackexchange.com/questions/3380436",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/711263/"
] | I am stuck on this Differentiation problem, any help would be great!
<blockquote>
If <span class="math-container">$A=xy$</span> and <span class="math-container">$x+5y=20$</span> find the maximum value of <span class="math-container">$A$</span> and the values of <span class="math-container">$x$</span> and <span class... | It may be useful to describe the problem as follows:
Maximize objective function <span class="math-container">$f(x,y) = xy$</span> subject to constraint <span class="math-container">$x + 5y = 20$</span>. Because the constraint is an equality, we may solve through simple substitution.
<span class="math-container">$x +... | Another way to handle constrained problems is the "Lagrange multiplier method. Write the function to be, in this case, maximized as <span class="math-container">$f(x,y)= xy$</span> and write the constraint as <span class="math-container">$g(x,y)= x+ 5y- 20$</span>. Then <span class="math-container">$\nabla f= y\vec{i... | https://math.stackexchange.com |
66,244 | [
"https://datascience.stackexchange.com/questions/66244",
"https://datascience.stackexchange.com",
"https://datascience.stackexchange.com/users/87707/"
] | I have a NumPy array of strings: 'n', 'y', wanna convert it into integer array of 0, 1, how to convert it?
<pre class="lang-py prettyprint-override"><code>imp = SimpleImputer(missing_values=np.nan, strategy='most_frequent')
X = imp.fit_transform(X)
X
array([['n', 'y', 'n', ..., 'y', 'n', 'y'],
['n', 'y', 'n',... | It is quite trivial
<pre><code>(X=='y').astype(int)
</code></pre>
Should do the trick. It simply converts your array to True or False according to your requirements and then astype will impose the required datatype. By default int will give you 1 for True and 0 for False.
| You could use the following code:
<pre><code>X[X=='y'] = 1
X[X=='n'] = 0
</code></pre>
This replaces the indexes of <code>'y'</code> with <code>1</code> and of <code>'n'</code> with <code>0</code>.
Generally the <code>X=='y'</code> returns a Boolean array which contains <code>True</code> where the <code>'y'</code> an... | https://datascience.stackexchange.com |
13,641 | [
"https://softwareengineering.stackexchange.com/questions/13641",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/5910/"
] | As IT manager what goals would you set for your programmers in your team. These goals would be used in the annual performance review. Any recommendations?
How would you create goals that actually measure programmers technical abilities.
| I'd try to dovetail this with the technical diretion of the company - are you trying to move all your GUI to WPF? Then learning WPF would be a good goal. I'd always emphasize learning the business - the more the programmers know about the business the better they can help automate.
| This can be really difficult, because good goals for employees should be specific and quantifiable. As we know, quantifying a programmer's productivity is definitely a conundrum.
In my experience managing programmers, I try to create goals based on the long term goals of the individual employee. For example, if the p... | https://softwareengineering.stackexchange.com |
128 | [
"https://mechanics.stackexchange.com/questions/128",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/105/"
] | I have a Honda Civic VTI, 1997 model, automatic transmission. When I start driving my car gives a feeling that one of the front wheel is wobbling and the steering wheel receives continuous jerks. This feeling goes away when I increase the speed.
Is it a wheel alignment or balancing problem?
| I would say that it is most likely a balancing issue. In order to find out, I would:
<ol>
<li>Use the car's jack to clear each front wheel from the ground. Give the wheel a spin and <strong>check to see that the spinning tire does not vibrate</strong> (sideways and/or up-and-down). This could indicate a damaged rim or... | I had this same problem. In my case it was a warped brake disc. You can see if it's the same in your case by seeing how hot your brake disc is after driving. Don't driving any more than you have to though, because you may damage the calipers. If it's the same problem then change the brake disc as soon as you can.
| https://mechanics.stackexchange.com |
65,297 | [
"https://chemistry.stackexchange.com/questions/65297",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/29436/"
] | $\ce{NH3}$ is a weak base so I would have expected $\ce{NH4+}$ to be a strong acid. I can't find a good explanation anywhere and am very confused. Since only a small proportion of $\ce{NH3}$ molecules turn into $\ce{NH4+}$ molecules, I would have expected a large amount of $\ce{NH4+}$ molecules to become $\ce{NH3}$ mol... | First, let’s get the definition of <em>weak</em> and <em>strong</em> acids or bases out of the way. The way I learnt it (and the way everybody seems to be using it) is:
<ul>
<li>$\displaystyle \mathrm{p}K_\mathrm{a} < 0$ for a strong acid<br>
$\displaystyle \mathrm{p}K_\mathrm{b} < 0$ for a strong base</li>
<li>... | First, let's understand the perspective of <em>weak acid</em> and <em>weak base</em>. This is in relation to pure water — like in most general chemistry courses.
Pure Water has a $\mathrm{p}K_\mathrm{a}$ of 14.
$\ce{NH3/NH4+}$ has a $\mathrm{p}K_\mathrm{a}$ of 9.25.
We know:
$$\ce{NH4+ + H2O <--> NH3 + H3O+}$$... | https://chemistry.stackexchange.com |
203,770 | [
"https://physics.stackexchange.com/questions/203770",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/83545/"
] | Generally, we say that conservation of energy is a <em>local</em> law; the change in energy in some small region of space is equal to the energy flux out of that region. However, in quantum mechanics, we can have superpositions of energy states. Then, when we measure them, they "instantly" achieve a certain energy. I'm... | This is not a problem for mainstream interpretations of quantum mechanics where there is no physical collapse of wavefunctions.
It's only a problem for fringe theories, such as spontaneous localization (aka objective collapse), where quantum mechanics is modified to induce a real collapse. From what I understand, vio... | As for all EPR-type situations, the answer is in the local statistics over an ensemble of identical copies. Say you do measure one pair of systems and find that system 1 is state $|0\rangle$ while its counterpart across the hall, system 2, is found in state $|E\rangle$. You can say that there is some "spooky-energy-tra... | https://physics.stackexchange.com |
30,016 | [
"https://electronics.stackexchange.com/questions/30016",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/3632/"
] | I know this might be a little bit funny question. But how can you determine the current that will used up as well as the power rating, when you a designing a power supply unit for a device. Because usually, I will always know the voltage, say 5V, but how can I calculate the current the device will use up.
Thanks.
| What current will be used is determined by the load, not the power supply. We've had a number of questions like this before, like "how do I limit the current of my 5A supply to 1A".
What your device will need is just the addition of the current for the different components on the board. You usually start with the po... | At a very simplistic level, if you know the voltage, and you know the resistance of your device, you will know the current needed: I=V/R
Alternatively, if you know what the power requirements are, again you can calculate: I=P/V
Your power supply needs to be able to deliver the required voltage at the greatest current... | https://electronics.stackexchange.com |
146,633 | [
"https://softwareengineering.stackexchange.com/questions/146633",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/9803/"
] | On my projects where the repository is shared between me and other programmers, I always write commit messages even if I'm the primary developer.
But on those projects where I'm the solo developer working on a project, and the repository is hosted on my personal laptop, and is not even hosted by the client, hence no o... | Well here is one reason: If you suddenly realize something has been broken for the last few hundred commits (possible if you commit at every minor edit, less feasible if you, like me, commit only "stable" snapshots), you can more easily find where you've inserted the bug if you've written clear commit messages, rather ... | I try to always. How many times do you look back and think, "Man what was I doing when I made this change." I do all the time. 30 seconds of writing a message can save you 20 minutes worth of work trying to remember.
| https://softwareengineering.stackexchange.com |
369,722 | [
"https://mathoverflow.net/questions/369722",
"https://mathoverflow.net",
"https://mathoverflow.net/users/116379/"
] | Let <span class="math-container">$\mathcal{H}$</span> denote a Hilbert space and <span class="math-container">$B(\mathcal{H})$</span> denote the algebra of all bounded operators on <span class="math-container">$\mathcal{H}$</span>. By recognizing the (Banach) dual of <span class="math-container">$B(\mathcal{H})$</span>... | Also the answer to the second question is <strong>yes</strong>, and the approximation may be chosen to converge in the point-ultrastrong<span class="math-container">$^*$</span> topology.
First, by choosing a net of finite rank orthogonal projections <span class="math-container">$p_i \in B(\mathcal{H})$</span> such that... | The answer to the first question is <strong>yes</strong>. This follows from the following more general result.
<strong>Terminology I: Ordered Banach spaces.</strong> By a <em>pre-ordered Banach space</em> I mean a pair <span class="math-container">$(X,X_+)$</span> where <span class="math-container">$X$</span> is a real... | https://mathoverflow.net |
120,925 | [
"https://softwareengineering.stackexchange.com/questions/120925",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/41217/"
] | Firstly I should explain the background. I am interested in high frequency trading programming roles. After looking at many job specs it is very clear that there is a big demand for programmers who have programmed Java and C++ on Unix as opposed to Windows.
My question is what are the differences a High Freq programme... | I worked at a HF shop and another services company for a short time. They care about speed. Speed of the software, speed of the network, and any other thing that could possibly have to do with speed.
They use Linux for stability, as other people have noted, but also because it allows you to tune, measure, and control... | Transaction servers are mostly(but not only) Unix: Thats because Unix is highly stable, and less resource hungry. A less important but not ignorable reason is legacy applications. You wouldnt want to move a stabe unix based transaction server to Windows.
But that doesnt necessarily mean that you have to program on U... | https://softwareengineering.stackexchange.com |
4,392 | [
"https://security.stackexchange.com/questions/4392",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/-1/"
] | I got shot down on ServerFault recently for suggesting that SSO can be <em>less</em> secure than having different credentials for each service. What are the arguments against SSO?
| Two views:
<strong>Corporate:</strong>
Whether you have control over all the machines, or at least the central auth server, you're better off with SSO. One place to hire employees, one place to terminate them, one set of credentials they have to worry about. Individual server compromises shouldn't leak credentials. U... | Imho the biggest concern with SSO is the sheer impact of a password breach, simply by factoring up all possible service accesses.
Otoh the younger history has shown that password reuse is more the rule than the exception. And SSO can also be used to enforce e.g. stricter password policies (use one password, but use i... | https://security.stackexchange.com |
366,512 | [
"https://mathoverflow.net/questions/366512",
"https://mathoverflow.net",
"https://mathoverflow.net/users/80052/"
] | In the book "Harmonic Measure" by Garnett and Marshall, we have the following result:
<strong>Lemma I.2.3</strong> Let <span class="math-container">$\mu$</span> be a positive Borel measure on <span class="math-container">$\partial{\mathbb{D}}$</span> and let <span class="math-container">$\{I_{j}\}$</span> be ... | The answer is no for <span class="math-container">$n \ge 2$</span>. Consider open cubes <span class="math-container">$Q(c,1)=\{c_i <x_i <c_i+1\}$</span> of side <span class="math-container">$1$</span>. Starting from <span class="math-container">$Q(0,1)$</span> and moving <span class="math-container">$c$</span> al... | It doesn't answer the question, but this is too long for a comment. I just want to point out that this is not possible with general rectangles even for Lebesgue measure on <span class="math-container">$\mathbb{R}^2$</span>. Since in one dimension we find that balls, cubes and rectangles are all the same this seems to i... | https://mathoverflow.net |
66,879 | [
"https://softwareengineering.stackexchange.com/questions/66879",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/17438/"
] | Whether an object physically exists or not, we can choose to model it in different ways. We could arbitarily use generalisation or composition in many cases. However, the GoF principle of "favour composition over generalisation [sic]" guides us to use composition. So, when we model, for example, a line then we create a... | It's not <strong>always</strong> the right choice. It is the <strong>favourable</strong> one in most cases. When a composite model requires change or extension, it is considerably more resistant to that because you can change the composition without fear of affecting other classes inadvertently.
How do we know this? F... | If you want a stronger rule of thumb beyond "favor composition over inheritance" then I might suggest something like this :
Of the two ways to specialize an object - Inheritance and Composition - you should
use inheritance only when you need your object to be polymorphic ( be substitutable ) for the base class you ar... | https://softwareengineering.stackexchange.com |
54,790 | [
"https://cs.stackexchange.com/questions/54790",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/48267/"
] | Consider the following algorithm for computing integer powers:
<pre><code>Procedure power(integer x, integer n)
power := 1
for i := 1 to n
power := power * x
return power
</code></pre>
Can we say that the loop invariant is $power \leq x^n$ ?<br>
Before the loop $power$ is initialized to $1$ ... | No, you can't say that $power \leq x^n$ is a loop invariant, since it is not maintained by the loop. For example, if $x > 1$ and at the current iteration $power = x^n$, then the invariant is satisfied at the beginning of the loop but not at the end of the loop.
Also, this loop invariant doesn't help you prove that ... | Let's rewrite your code using a while-loop so that everything is explicit.
<pre><code>{ 1 ≤ n }
i ≔ 0; p ≔ 1;
while i < n:
i ≔ i + 1; p ≔ p * x
{ p ≈ xⁿ }
</code></pre>
Note that the initialization sets i to 0 instead of 1!
Now after the loop is finished we will know that necessairly n < i and thattogether... | https://cs.stackexchange.com |
15,635 | [
"https://softwareengineering.stackexchange.com/questions/15635",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/5057/"
] | So I've had a series of really bad experiences and am wondering what I'm doing wrong.
I started out as a part time programmer at a major university. I developed a problem with my wrist and asked (nicely) for some help with my ergonomic accomodations.
My boss started screaming at me, then later claimed I screamed at ... | I think you need need to start doing more research on the company before taking a job and ask more pointed questions about the company in the interview. Since Health insurance is important to you, I would stay away from start-ups altogether as they can crash very suddenly leaving you in a similar situation (they usuall... | You sound like a reasonable guy. I think you may have just had monumentally bad luck.
There are good companies out there to work for; you just have to find them. Start wearing suits to your interviews. Be a professional. Talk to people who have a job with a good work environment, so that you know what that looks l... | https://softwareengineering.stackexchange.com |
48,573 | [
"https://math.stackexchange.com/questions/48573",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/6016/"
] | Given any positive integer, how can I think of a Markov Chain (states and transition probabilities) to have that integer as the period of two of its states?
Thanks.
| To be truly stochastic, assume that the state space of the Markov chain is $S\times C_d$ where $C_d=\mathbb{Z}/d\mathbb{Z}$ is the discrete circle of size $d$. And assume that the only transitions with positive probabilities go from some state $(x,k)$ to some state $(y,k+1)$ with $x$ and $y$ in $S$ and $k$ in $C_d$. Th... | Hint: No probabilities are needed. Just put them on a (directed) cycle.
| https://math.stackexchange.com |
246,223 | [
"https://electronics.stackexchange.com/questions/246223",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/17001/"
] | I'm creating a small assortment for my parts box, and I wonder about Zener currents. For example, a selection of 4.7V 500mW Zeners spans a wide range of Zener current values:
<ul>
<li>BZX79C4V7 = 5 mA Zener current (80 ohms Zener impedance)</li>
<li>1N5230B = 20 mA (19 ohms)</li>
<li>1N750A = 75 mA (19 ohms)</li>
</ul... | <blockquote>
As a rule, is a lower Zener current generally better?
</blockquote>
Not always. Zener voltage is affected by its drive current, such that if the drive current is too low the zener voltage will be less than the rated clamp voltage. For instance a 3.3 volt 5 watt zener may need 50 mA to reach its 3.3 volt... | The reverse current rating of a Zener diode is somewhat analogous to the forward current rating of a regular diode in that the current rating you would need in a particular application will depend upon the application.
In general, the Zener (reverse breakdown) voltage is dependent on the reverse current actually made ... | https://electronics.stackexchange.com |
7,159 | [
"https://chemistry.stackexchange.com/questions/7159",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/3828/"
] | I'm studying radioisotopes at the moment and balancing nuclear reactions isn't making sense in that more matter is coming out of the equation in negative β⁻ decay equations:
<span class="math-container">$$\ce{_6^{14}C -> _7^{14}N + e- + \bar{\nu_e}}$$</span>
Notice how the original element has decayed into a new e... | One neutron has changed into a proton, that's what has happened. Our convention in chemistry is to identify nuclear species by the proton number (or the "atomic number") and that is why a new nuclide with an <em>increased</em> atomic number is formed. Both the <span class="math-container">$\beta^+$</span> and <span cla... | What you are missing is that the atomic mass has not changed — mass is conserved — the top number on the isotope notation 14 for both carbon and nitrogen in first case (in fact, it goes down very slightly, but they rounded it up). The increasing atomic number (bottom number) just means a neutron became a proton.
| https://chemistry.stackexchange.com |
215,946 | [
"https://electronics.stackexchange.com/questions/215946",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/84456/"
] | I have read that if you don't unlug the charger from the socket after charging your phone, although it is not connected to the phone, it still consumes power and your electricity bill adds up.
On the other hand if I leave the charger plugged without the phone I assume the only power is reactive power and in residenti... | There are a couple of factors to consider.
1) Old-style wall-wart chargers use a transformer that runs at line frequency (50 or 60 Hz). High-quality line-frequency transformers <em>do</em> present a large reactive load to the AC Mains.
Unfortunately, most wall-wart chargers do <strong>NOT</strong> use high-quality ... | Actually the transformer inside the charger is not ideal and has some winding resistance and therefore will consume a small amount of active power even if the phone is unplugged.
| https://electronics.stackexchange.com |
136,171 | [
"https://electronics.stackexchange.com/questions/136171",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/57052/"
] | I am experimenting with VCFs for an analogue music synthesiser. Initially I tried a 3-stage low-pass filter based on the sinewave-generator circuit in the LM13700 datasheet but having removed the oscillator, as per schematic below:
<img src="https://i.stack.imgur.com/RgvPb.png" alt="3-pass LM13700 low-pass filter">
I... | The two filters have similar amplitude responses, but different phase responses. In other words, different harmonics of the original square wave are being delayed (phase shifted) by various amounts by the first filter, but they have a more nearly constant delay (linear phase shift) in the second filter.
While the two ... | As to why the first filter has asymmetries in response?
Look no further than the outputs being attached to darlington connected transistors or just the fact that there are transistors there with a weak pull down. The rate at which the output slews from positive to negative is dictated by the weak pull down. Essent... | https://electronics.stackexchange.com |
350,501 | [
"https://softwareengineering.stackexchange.com/questions/350501",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/200019/"
] | <pre><code>void Animation::playAnimation() const
{
static const int index = 0;
const std::string& animationFileName =
m_animationContainer.getAnimationName(index);
static const int zOrder = -1;
static bool isLooping = false;
AnimationBank::play(animationFileName,
zO... | Beyond @Christophe's very good answer, the code generated for the static is most likely worse than the one for the local variable, so if you're interested the under-the-hood benefit, statics are worse on modern processors.
The reason is that the statics must be located somewhere in memory that can be found by all th... | It is not a question of benefits, but a question of semantics:
<ul>
<li>A static variable in a function (even a member function), means that the variable is shared between all the calls of that function. So one call of that function has a side effect on the subsequent calls. </li>
<li>A non static variable is uniqu... | https://softwareengineering.stackexchange.com |
585,690 | [
"https://physics.stackexchange.com/questions/585690",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/274664/"
] | So, I have been watching some science videos regarding Einstein's theory on general relativity and until today the predictions based on his equations have been proven to stand.
My question would be: what happens in the scientific community if one experiment proves it wrong (not only Einstein, but even other laws and th... | <blockquote>
My question would be, what happens in the scientific community if one experiment proves it wrong
</blockquote>
We have already seen what happens in this circumstance by looking at what happened to Newtonian gravity.
First, well before the development of general relativity there were observations that did n... | Words like "proven" and "wrong" have to be used carefully in this context. It is more meaningful to talk about "accuracy" and "limits". If an experiment was conducted tomorrow that contradicted general relativity it would by no means make general relativity a useless theory, nor ... | https://physics.stackexchange.com |
149,478 | [
"https://softwareengineering.stackexchange.com/questions/149478",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/17466/"
] | Ok not really sure if I'm right.
I only recently learned that I needed to have contravariant interface to be able to pass that interface as a parameter in C# and this feature was only added in .NET 4.0.
So obviously there is some reason you can't do this Covariant or Invariant interfaces and it probably has something... | The need for covariance and contravariance can be best understood with an example.
Let's say you have a function that accepts a parameter of type <code>List<Base></code>, where <code>Base</code> is a base class that other classes inherit from. You would intuitively think that it would be just fine to pass a <co... | <blockquote>
I only recently learned that I needed to have contravariant interface to be able to pass that interface as a parameter in C# and this feature was only added in .NET 4.0.
</blockquote>
.NET 4 added co/contra-variance for generics. The variance concept though exists in any language with subtyping.
When d... | https://softwareengineering.stackexchange.com |
832,433 | [
"https://math.stackexchange.com/questions/832433",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/126222/"
] | Anyone knowns an example of a R-module finite genereted with a submodule not finite generated?
I find the following example: Taking the set of function $f:[0,1]\rightarrow\mathbb{R}$ seen as module of it self. This is finite generated. If we take the subset of functions $f$ such that $f(x)=0$ for all $x\in[0,1]$ exce... | Let $R$ be the ring of function $f:[0,1]\to\mathbb R$. Let $M$ be the $R$-module of functions $f$ such that $f(x)=0$ for all $x\in[0,1]$ except for a finite numbers of points.
Assume on contrary that $M$ is finitely generated with generators $g_1,\ldots,g_n$.
For each $a\in [0,1]$ let
$\chi_a(x)=
\begin{cases}
1&x... | Another example is $$A = \lbrace f: \mathbb{R} \to \mathbb{R} \rbrace$$ as module over itself, and the submodule generated by the functions $f$ with compact support, which can't be finitely generated.
| https://math.stackexchange.com |
270,784 | [
"https://electronics.stackexchange.com/questions/270784",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/130733/"
] | Hi I've recorded sound(field sound) using six microphones with sensitivity 5mv/pa by zoom recording device the results were obtained by dBFS unit and i need it in dBSPL, Is there any way to convert it?
because I have read several articles and come to the do not directly convert between the two units and need to cali... | You could try to go back and retroactively "calibrate" your original recordings. But that requires these things.
<ol>
<li><strong>FIXED RECORDING LEVELS</strong> Your original recording must have been done at some <strong>FIXED</strong> and unchanging recording level/gain from beginning to end. If you changed the rec... | Unless you calibrated the microphones AND the recorder at the time of the recording the absolute dB SPL reference is lost and unrecoverable. The reason is that you don't know what what the REAL sensitivity of the microphones are. "5mv/pa" is only the nominal design target. It does not necessarily represent the ACTUAL ... | https://electronics.stackexchange.com |
2,971 | [
"https://mathoverflow.net/questions/2971",
"https://mathoverflow.net",
"https://mathoverflow.net/users/1040/"
] | I am reading some papers which involve D-modules on a Lie algebra g, which are supported on the nilpotent cone n. They are equivariant for the action of G. (In particular, I consider g=sl_n).
It was explained to me (the statement, not the proof) that the category of such D-modules is semisimple, and that the simple ... | Watch out that there are more simple objects than it looks like at first glance, even for sl_n. Although the orbits are parameterized by partitions, they can carry nontrivial local systems whose intermediate extensions to the nilpotent cone will be new simple D-modules.
I have heard that the general problem of writ... | I think saying "constant sheaf" isn't quite right. You want to take the functions on each orbit, and do the intermediate extension of that D-module.
Now, that's a bit unexplicit, so let me try a different description. Recall that there is a Grothendieck simultaneous resolution, a map from <code>G x_B b</code> (the a... | https://mathoverflow.net |
406,247 | [
"https://physics.stackexchange.com/questions/406247",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/190526/"
] | I am trying to find the energy between the $n=2\leftrightarrow3$ transition for Muonic Hydrogen. My approach was to modify the Bohr model for standard hydrogen but taking the mass of the Muon $m_\mu\approx 207m_e$ instead of the mass of the electron $m_e$ and then substituting in the effective mass $m=\frac{m_\mu m_p}{... | $E_n = \frac{Z \alpha^2 \mu} {2 n^2}$ for hydrogen like atoms in the nonrelativistic approximation. For the hydrogen ground state ($Z=1$, $n=1$, $\mu=\mu_e$) this gives 1 Ry. Your answer should therefore be about $\mu_{\mu} /\mu_e$ times $1/4-1/9$ Ry, that is about 26 Ry ~ 400 eV ~ 6.4 $10^{-17}$ J.
| In order to obtain the energy levels of muonic hydrogen you can simply replace the reduced mass corresponding to the proton and electron with the reduced mass corresponding to the proton and the muon particle as you have already indicated.This should give you the correct Rydberg constant for the spectrum of this atom. ... | https://physics.stackexchange.com |
3,486,349 | [
"https://math.stackexchange.com/questions/3486349",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/691474/"
] | <span class="math-container">$\Omega$</span> is open in a topological space <span class="math-container">$X$</span> and <span class="math-container">$\{K_n\}$</span> are countable open sets such that <span class="math-container">$\bigcup K_n = \Omega$</span> and <span class="math-container">$K_n \subset \text{int} K_{n... | For <span class="math-container">$x<\frac{1}{2}, LHS>0>RHS.$</span>
For <span class="math-container">$\frac{1}{2}<x<1, LHS>\sqrt{2}>1>RHS$</span>.
For <span class="math-container">$1<x<2$</span>,
<span class="math-container">$$2^x = e^{x\ln 2}$$</span>
<span class="math-container">$$=\su... | Let's see if we can prove the result with resorting to logs or calculus.
We'll begin with a rewrite of <span class="math-container">$2^x\gt2x-1$</span>, letting <span class="math-container">$x=2u+{1\over2}$</span>, as
<span class="math-container">$$4^u\gt2\sqrt2u$$</span>
Since the inequality is obvious for <span cl... | https://math.stackexchange.com |
25,857 | [
"https://stats.stackexchange.com/questions/25857",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/5597/"
] | Basically, I want to manually compute what <code>predict(princomp(somedata))</code> produces.
In order to do that I need to have coefficients matrix so that
<pre><code>newdata %*% coefficients
</code></pre>
would produce the predicted values of each component.
Can anybody give me a clue to get the coefficients?
| If you read the documentation on the <code>princomp</code> function, via <code>?princomp</code> you'll see that there is a section titled "Value" that describes the components of the object returned.
The piece you are looking for is probably the loadings.
Additionally, if you type <code>stats:::predict.princomp</code... | The particular calculation done by <code>predict</code> in this case is <code>scale(newdata, object$center, object$scale) %*% object$loadings</code>, where <code>object</code> is the object returned by <code>princomp</code>. There's no way to reduce this to a single matrix multiplication, as in the general case <code>n... | https://stats.stackexchange.com |
100,509 | [
"https://physics.stackexchange.com/questions/100509",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/34032/"
] | We have
$$S=Nk_B[\ln(2 \cosh(x)) - x \tanh(x)]$$ where $$x = \frac{\mu B}{k_BT}$$
In need to show that at low temperatures entropy
$$S \approx Nk_B2xe^{-2x}$$
I wrote out the $\cosh(x)$ in terms of $e^{-x}$ and get that $\ln(2\cosh(x))\approx e^{-2x}$ at low temperatures, which would then mean I need to way to write ... | Let me expand on the previous answer. We expand each term separately. We have,
<span class="math-container">\begin{align}
\log \left( 2 \cosh x \right) & = \log \left( e ^x + e ^{ - x } \right) \\
& = \log \left[ e ^{ x } ( 1 + e ^{ - 2 x } ) \right] \\
& = x + \log \left( 1 + e ^{ - 2 x } \right) \\
&... | They probably mean to say "for large $x \gg 1$". The limit of low temperature is vague if they don't say how quantities are being fixed in the limit. Anyway, your approximations are wrong. For large $x$, $\ln(2 \cosh x)$ goes like $x$, and $\tanh x$ goes like $1 - 2 e^{-2x}$. Try graphing it if you don't believe me. :)... | https://physics.stackexchange.com |
311,348 | [
"https://dba.stackexchange.com/questions/311348",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/247004/"
] | We have a MariaDB 10.3 on a Redhat Server (VM) with 90 GB Ram and 20 VCPUs. I'm trying to optimize the Database. I installed Sysbench for Benchmarking and after a 50 seconds test I got this values (with default MariaDB variables):
<pre><code>#Benchmarking command:
sysbench oltp_read_write --threads=2 --report-inter... | I notice you have 20 CPU cores, but you're only running sysbench with 2 threads. You probably can get better results with more threads. The guideline often given is 2x the number of CPU cores, so try it with at least <code>--threads=40</code>.
I have also found sysbench itself has scalability limits (as any client app ... | To help other DBAs I wanna add some sentences from the book <em><strong>"O'Reilly High Performance mysql"</strong></em> to Bill's answer:
<strong>What Not to Do</strong>
You might be expected (or believe that you’re expected) to set up a benchmark suite
and “tune” your server by changing its configuration ite... | https://dba.stackexchange.com |
6,037 | [
"https://electronics.stackexchange.com/questions/6037",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/1609/"
] | I've been working in software for way too long, dislike it for a number of reasons, and would like to shift completely into electronics engineering work. I'd prefer analog, microwave, optoelectronics or maybe some specialty yet to be invented, but any electronics is better than any software. I have more talent and ... | In my experience the usefulness of the person decreases with the level of degree above Bachelors. The problem is that to get the higher degrees the candidate has to specialise in a narrower and narrower field and as a result loses their general spread of skills that will always be required in a real work environment. ... | If you can manage to get a job doing what you want to do without a degree, do it.
In the longer term, a Master's is a good investment, but you'll get more out of it if you have a year or two of experience first.
| https://electronics.stackexchange.com |
104,201 | [
"https://math.stackexchange.com/questions/104201",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/18802/"
] | I'm having some trouble with this question and can't really get how to prove this..
I have to prove $n^3+6n^2+11n+6$ is divisible by $3$ for all $n \geq 0$.
I have tried doing $\dfrac{m}{3}=n$ and then did $m=3n$
then I said $3n=n^3+6n^2+11n+6$ but now I am stuck.
| If you know what "mod 3" means then argue as follows:
$$n^3 + 6n^2 + 11n + 6 \equiv n^3 - n = (n-1)n(n+1) \equiv 0 \pmod 3 .$$
If you don't, then write this as:
$$ n^3 - n + 12n + 6n^2 + 6 = n(n+1)(n-1) + 3(2n^2 + 4n + 2), $$
and you're left with showing that both terms are divisible by $3$.
Now $n(n+1)(n-1)$ is alwa... | We have
$$
\begin{align}
n^3+6n^2+11n+6
&=6\binom{n}{3}+18\binom{n}{2}+18\binom{n}{1}+6\binom{n}{0}\\
&=6\left(\binom{n}{3}+3\binom{n}{2}+3\binom{n}{1}+\binom{n}{0}\right)
\end{align}
$$
so $6\mid(n^3+6n^2+11n+6)$ for all $n\in\mathbb{Z}$.
Of course, since $3\mid 6$, we have $3\mid(n^3+6n^2+11n+6)$, as request... | https://math.stackexchange.com |
481,037 | [
"https://physics.stackexchange.com/questions/481037",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/220110/"
] | If so, I have read that for <span class="math-container">$C_P$</span> and <span class="math-container">$C_V$</span> for an incompressible gas are identical (in <em>Heat Transfer</em> by Yunus A. Cengel). Then how will the relation between <span class="math-container">$C_P$</span> and <span class="math-container">$C_V$<... | All gases, including ideal gases, are compressible. That's because the molecules of gases are far apart and can readily be brought together by pressure. The specific heats of gases depend on the process. For an ideal gas, <span class="math-container">$C_p$</span> and <span class="math-container">$C_v$</span> are the sp... | Who said ideal gas is incompressible? Its (isothermal) compressibility is given by
<span class="math-container">$$\beta_{T}=-\dfrac{1}{V}\left(\dfrac{\partial V}{\partial P}\right)_{T}=-\dfrac{1}{V}\left(\dfrac{\partial}{\partial P}\dfrac{NK_{\rm B}T}{P}\right)_{T}=\dfrac{1}{V}\dfrac{NK_{\rm B}T}{P^{2}}=\dfrac{1}{P}$$... | https://physics.stackexchange.com |
68,530 | [
"https://physics.stackexchange.com/questions/68530",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/25999/"
] | How to find the commutator $[a, \sqrt{a^\dagger a}]$? Here $a$ is a usual bosonic annihilation operator, and $[a, a^\dagger] = 1$.
The first thing I tried is
$$
[x,A] = [x, \sqrt{A}]\sqrt{A} + \sqrt{A}[x, \sqrt{A}]
$$
which clearly shows either commutators similarity to derivatives and the difference between them. In ... | You have to use the eigenstates $|n\rangle $ of the operator $\hat{n} = a^\dagger a$.
You have, then, that $a \sqrt{\hat{n}} ~|n\rangle = a \sqrt{n} ~|n\rangle = \sqrt{n} ~ a |n\rangle = \sqrt{\hat{n}+1} ~ a |n\rangle ,$ where the last equality is because $a |n\rangle \sim |n-1\rangle$.
So, $\left[a,... | We are given
$$[\hat{a},\hat{a}^{\dagger}]~=~{\bf 1}.$$
Let
$$\hat{n}~:=~\hat{a}^{\dagger}\hat{a}.$$
Hints:
<ol>
<li>Prove that
$$\hat{a}\hat{n} = (\hat{n}+{\bf 1}) \hat{a}.$$</li>
<li>Prove that if $f:\Omega \subseteq \mathbb{C}\to \mathbb{C}$ is a sufficiently well-behaved function, then
$$\hat{a}f(\hat{n}... | https://physics.stackexchange.com |
483,632 | [
"https://stats.stackexchange.com/questions/483632",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/28715/"
] | I checked whether this question was answered before but because of notation, it's hard to see. I am a reading a paper that defines the following two RVs
<span class="math-container">$$
z \mid y \sim Binomial(\pi, y) \\
y \sim Poisson(\lambda)
$$</span>
then concludes (by integration and Bayes Rule) that
<span class="ma... | This follows from some fairly standard distribution theory. Define <span class="math-container">$Y_1 \sim \text{Poisson}(\pi \lambda)$</span> and <span class="math-container">$Y_2 \sim \text{Poisson}((1-\pi) \lambda)$</span> independently, and let <span class="math-container">$Y = Y_1 + Y_2$</span> and <span class="mat... | Its a bit of algebra, but here is my try
The expression of the density after you pull out the terms not involving <span class="math-container">$y$</span> are
<span class="math-container">$$ p(z) = \pi^z \exp(\lambda) \sum_{z \leq y} \binom{y}{z} (1-\pi)^{y-z} \dfrac{\lambda^y}{y!}$$</span>
The <span class="math-contain... | https://stats.stackexchange.com |
310,373 | [
"https://physics.stackexchange.com/questions/310373",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/144691/"
] | I have a couple of (noob) questions regarding Doppler Shift and light from a quantum physics perspective:
a) Since different observers will see the light at different frequencies depending on their reference frame / velocity thus resulting in Doppler Shift, does that mean that any light emitted exists in an infinite v... | A) Yes and No. Doppler shift is about apparent changes in frequency. Emphasis on apparent changes.The relative frequency of the light depends on the observer. Frequency is not, however, an inherent quality of the light. You may be conflating frequency and wavelength-a common error.
B) Perhaps. If you are firing them s... | I will try to answer your second question.
There is a principle in mathematics called the Pigeon Hole Principle. According to that principle, if there are 30 pigeons holes in which pigeons can be placed and there are 31 pigeons then there must be at least one pigeon hole that has more than 1 pigeon.
We can apply th... | https://physics.stackexchange.com |
45,029 | [
"https://physics.stackexchange.com/questions/45029",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/16251/"
] | We all know that space and time are the fundamental units, means no mathematical expression can express their relation to other variable fundamentally. But as we know that moving rod has a contracted length and dilated time which depend on the velocity of that rod w.r.t. an inertial frame in which the motion is happeni... | The units of space are meters (and square meters and cubic meters)<br>
The units of time are seconds<br>
The units of velocity are meters per second
So the units of velocity are not independent of the units of space and time.
<hr>
In classical mechanics, we can measure the position of an object using only three di... | As I have said space should include velocity in itself as extra dimension and same is true for time.Now as our saying if velocity is a form of space and also of time then the space and time both can be expressed in terms of velocity also. so what comes now that we will have two equations one expressing space-velocity r... | https://physics.stackexchange.com |
13,464 | [
"https://security.stackexchange.com/questions/13464",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/9317/"
] | I have read so many stories that a major DDoS attack is causing companies thousands of dollars to mitigate. I have two questions:
<ol>
<li>Is this because they try to keep the servers accessible to real site visitors while an attack is going on?</li>
<li>Does shutting the server down instead of trying to keep it up st... | It's because if you shut a server down, you can't use it for revenue. Imagine having a real shop: if you close it during the day, you can't sell anything and offer your services. It's as simple as that.
To improve my answer, it costs a lot to fight because it's pretty much impossible to defend yourself against such at... | Another aspect to the cost for a company, besides lost revenue, is the cost of investigating, verifying, and responding to such attacks. There are conversations with ISPs, combing through log files, re-writing firewall rules, communicating with customers, then postmortems.
But note that a company cannot 'stop' the att... | https://security.stackexchange.com |
311,910 | [
"https://electronics.stackexchange.com/questions/311910",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/114447/"
] | I'm looking for an extremely sensitive photosensor. I have a setup where a beam of light of some sort (weak laser, or something else) will be shining into the photosensor. Every now and then, a molecule with a diameter of about 1.0 - 1.3 nanometers will pass between the light source and the photosensor, and the photose... | No, that doesn't exist. Visible light has a wavelength much larger than your particles, so the waves will go right around them unimpeded
| Some very sensitive PIN diodes can be used as Geiger counters. The signal from the PIN diodes must be amplified a lot to detect radiation particles. Even with a weak light it should be able to detect dust particles flying by, but that is nowhere near the molucules that you want to detect.
| https://electronics.stackexchange.com |
81,661 | [
"https://physics.stackexchange.com/questions/81661",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/2653/"
] | I'm trying to help a child research a science project on refrigeration.
Refreshing my incredibly rusty thermodynamics skills....
The ideal gas law: $PV=nRT$. Let's take air at STP:
$P = 101\,kPa$
$V = 1\,L = 0.001\,m^3$
$R = 8.3\,J/mol\cdot K$
$T = 298\,K$
$n=PV/RT = (101000) (.001) / (8.3 \cdot 298) = 0.04\, moles$... | Why is the Copenhagen interpretation the most accepted one? I would say the answer is this:
<ul>
<li>it's the oldest more or less "complete" interpretation</li>
<li>hence you'll find it in many (all?) early text books, which is basically from where people writing modern text books copy from.</li>
<li>the overwhelming ... | Due to historical reasons, physicists who do not have a strong preference for a particular interpretation default to the Copenhagen one, despite some of its pseudo-mythical outgrowth - which you can just ignore if you are in the 'shut up and calculate' camp.
It doesn't help that every other interpretation (at least th... | https://physics.stackexchange.com |
1,819,649 | [
"https://math.stackexchange.com/questions/1819649",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/289959/"
] | $X$ and $Y$ are independent variables that follow the exponential distribution with the same parameter $\beta$. What is the probability $$P(X \geq k\cdot Y)$$ if $k>0$ ?
I don't really know where to start.
| $X$ and $Y$ follow an exponential distribution. Therefore, the density function is $f(t)= \beta \exp{(-\beta t)}$. First, we condition by $Y=k$ and integrates over the $k$.
$$P(X\geqslant kY) = \int_0^\infty \beta \exp{(-\beta t)} P(X\geqslant kt| Y=t)dt$$
Since $X$ and $Y$ are independent, $P(X\geqslant kt | Y=t)=P... | Required probability
$$
\iint_{x\ge ky\ge0}p_{X,Y}(x,y)dA=\int_{0}^\infty\int_{ky}^\infty p_{X,Y}(x,y)dxdy\\
=\int_{0}^\infty\int_{ky}^\infty p_X(x)p_Y(y)dxdy=\beta^2\int_{0}^\infty\int_{ky}^\infty e^{-\beta(x+y)}dxdy\\
=\beta\int_0^{\infty}-e^{-\beta(x+y)}\Big{|}_{ky}^\infty dy=\beta\int_0^{\infty}e^{-\beta(ky+y)} dy=... | https://math.stackexchange.com |
14,310 | [
"https://dsp.stackexchange.com/questions/14310",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/154/"
] | Is there any relationship between the FFT and linear predictive methods? Can an FFT result or the input to an FFT be modified to do non-circular prediction/extrapolation from the FFT results.
| At the least you should normalize it by the length of the sequence. But what you're trying to do is a pretty standard thing. I'm not sure what percentage difference really means. What you should probably be using is something like the root mean squared error:
$$RMSE=\sqrt{\frac{1}{N}\sum_{n=0}^{N-1}(x(n)-y(n))^2}$$
w... | You are calculating the relative error between two curves, when you curves have the opposite sign, the difference between the two curves may be even larger than the amplitude of the curve. That's probably the reason why your <code>mean(per)</code> reaches 200.
Like @Eric said, RMSE makes more sense, the Matlab impleme... | https://dsp.stackexchange.com |
308,402 | [
"https://mathoverflow.net/questions/308402",
"https://mathoverflow.net",
"https://mathoverflow.net/users/16183/"
] | <blockquote>
Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
$$\frac{1}{x+y^2+i0}?$$
</blockquote>
Remark. <strong>Equivalent question:</strong> consider the Schroedinger equation one the line
$$i\frac{\partial}{\partial t}\Psi(x,t)=\frac{\partial^2}{\partial x^2}\... | I think$^\ast$ the Fourier transform only vanishes for $u>0$, for $u<0$ instead
$$
\frac{1}{2\pi}\int_{-\infty}^\infty dx\int_{-\infty}^\infty dy\,\frac{e^{\mathrm{i}(ux+vy)}}{x+y^2+\mathrm{i}0^+}=-ie^{i\pi/4}\sqrt{\pi }(- u)^{-1/2}\exp\left({\frac{i v^2}{4 u}}\right)\;\;\text{for}\;\; u<0.
$$
You can check i... | The Fourier transform for $1/(x+y^2+\mathrm{i}\epsilon), \epsilon\in\mathbb{R}$ is formally compute as
\begin{align}
\int_{\mathbb{R}^2}\frac{e^{2\pi\mathrm{i}(ux+vy)}}{x+y^2+\mathrm{i}\epsilon}\,dx\,dy&=\int_{\mathbb{R}}e^{-2\pi\mathrm{i}(uy^2-vy)}\,dy\int_{\mathbb{R}}\frac{e^{2\pi\mathrm{i}ux}}{x+\mathrm{i}\epsi... | https://mathoverflow.net |
538,521 | [
"https://physics.stackexchange.com/questions/538521",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/90273/"
] | I am performing an experiment to measure the half-value thickness of different materials, and the radioactive source is <span class="math-container">$^{60}\text{Co}$</span>. To the extent of my knowledge, cobalt-60 is a <span class="math-container">$\beta$</span>- and <span class="math-container">$\gamma$</span>-emitte... | The <span class="math-container">$\beta$</span> rays won't escape from the source encapsulation (unless you've got an open source, which is unlikely for health and safety reasons). So you're just counting the <span class="math-container">$\gamma$</span> rays.
These will barely be affected by your aluminium foil. You n... | Test it with thicker aluminum, i have a hunch its breaking down the aluminum slightly. I know this isnt what your reqlly testing but try it and test the aluminums flexability. Im betting it becomes more brittle. Thank you
| https://physics.stackexchange.com |
3,171,798 | [
"https://math.stackexchange.com/questions/3171798",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/160881/"
] | I have the following ODE problem :
<span class="math-container">$8$</span> billion people in the world, increase each year by <span class="math-container">$2\%$</span>, how many people will be in <span class="math-container">$9$</span> months?
<strong>What I did :</strong>
The change each year is : <span class="math... | You don't have to assume the exponential form to begin with. In fact, if this is an exercise on ordinary differential equations, then you shouldn't in fact assume the form.
Instead, recognise the basic behaviour - this is a continuous growth model where the rate of increase of population <span class="math-container">$... | The function that you're after is an exponential one; to be more precise, it is of the type <span class="math-container">$f(t)=8e^{kt}$</span>. What is the value of <span class="math-container">$k$</span>? It musto be such that <span class="math-container">$f(1)=8\times1.02$</span>, which means that <span class="math-c... | https://math.stackexchange.com |
1,630,232 | [
"https://math.stackexchange.com/questions/1630232",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/76333/"
] | Suppose $R$ is a commutative ring and $D$ a multiplicatively closed subset. I'd like to show via universal properties that if $\mathfrak{a} \triangleleft R$ is an ideal, then $D^{-1}R/D^{-1}\mathfrak{a} \cong \bar{D}^{-1}(R/\mathfrak{a})$, where $\bar{D}$ is the image of $D$ in $R/\mathfrak{a}$. This is pretty close to... | Let us define a ring-homomorphism $u:R\to T$ by the universal property that a ring-homomorphism $f:R\to S$ factors (uniquely) through $u$ iff $f$ maps every element of $D$ to a unit and every element of $\mathfrak{a}$ to $0$. From the universal property of $D^{-1}R$, we see that any $f$ which factors through $u$ must ... | I brute-forced this just for the fun of it, in a slightly different flavour than the above answer. First, the composition <span class="math-container">$R\to R/\mathfrak{a} \to \overline{D}^{-1}(R/\mathfrak{a})$</span> sends <span class="math-container">$D$</span> to invertible elements and so we get a map <span class="... | https://math.stackexchange.com |
267,733 | [
"https://softwareengineering.stackexchange.com/questions/267733",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/157464/"
] | I don't know how to refactor a piece of code that differs from other in the number of conditions checked on an <code>if</code> clause. Let me show you a real world example I'm facing right now.
The only difference between the two methods is that on the inner <code>elseif</code>, it checks an additional condition on th... | PHP isn't a language I have much experience with, but as I understand it both loops are trying to filter the set of keys, so you can extract the structure of the loop into another method that accepts the conditions for filtering as an argument in the form of a function:
<pre class="lang-php prettyprint-override"><code... | So just for other PHP guys that may be struggling with functional programming I want to register here the steps I did following Jack's answer.
First I refactored each method to extract the condition so I could have an identical code base in both (note how the condition was extracted in both):
<pre><code>public functi... | https://softwareengineering.stackexchange.com |
117,189 | [
"https://cs.stackexchange.com/questions/117189",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/48180/"
] | Given a bipartite graph <span class="math-container">$G=(U,V,E)$</span>, find a maximum matching.
Algorithm G:
<ul>
<li>Let <span class="math-container">$S\gets\emptyset$</span></li>
<li>Mark all edges unmatched</li>
<li>For <span class="math-container">$i\gets1$</span> to <span class="math-container">$|E|$</span>
<... | Consider the path graph of length 3: <span class="math-container">$x-y-z-w$</span>.
Take <span class="math-container">$S$</span> to be <span class="math-container">$y-z$</span>, and <span class="math-container">$S'$</span> to be <span class="math-container">$x-y,z-w$</span>.
<hr>
A set of constraints for which your ... | There is a difference between a maximal matching and a maximum matching. A maximal matching is a matching that is not a proper subset of any other matching. Hence, it is not possible to add another edge to a maximal matching (we can say a maximal matching can't be augmented). A maximum matching is a matching of maxima... | https://cs.stackexchange.com |
20,331 | [
"https://physics.stackexchange.com/questions/20331",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/6743/"
] | I'm having trouble understanding the diagrams of elctromagnetic waves.
I have no problem with any concept in classical mechanics, and I think this can be answered without any relativity (which I don't know yet). I hope for an answer which is as intuitive as possible but any answer will be good.
So in the common diagr... | <blockquote>
<img src="https://i.stack.imgur.com/x2fQa.png" alt="Diagram of EM wave">
What exactly do the vectors represent? The wave is propagating though an eletromagnetic field. Are the lines just the field created by a particle?
</blockquote>
Each vector represents the value (magnitude and direction) of th... | So while the diagram might help it only shows you
<ul>
<li>one single propagation direction </li>
<li>only a static picture of the moving EM-wave </li>
<li>has constant amplitudes in k-direction </li>
<li>is only correct in the far-field zone</li>
</ul>
When you keep all this in mind it is actually a helpful picture... | https://physics.stackexchange.com |
2,725,952 | [
"https://math.stackexchange.com/questions/2725952",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/351436/"
] | I know this problem has been asked before by someone. However, my problem is a bit different. I found the following problem in Elementary number theory by Burton:
<blockquote>
Prove that the cube of any integer can be written as the difference of two squares. Notice that $n^3 = (1^3+2^3+3^3+.......+n^3) - (1^3+2^3+3... | Well, what happens when you try?
$(n+1)^3 = n^3 + 3n^2 + 3n + 1 =$
$(\frac {n(n+1)}2)^2 - (\frac {n(n-1)}2)^2 + 3n^2 + 3n + 1$
Meanwhile $(\frac{(n+1)(n+2)}2)^2 - (\frac {(n+1)n}2)^2 =$
$(\frac{(n+1)n}2 + \frac {2(n+1)}{2})^2 - (\frac {n(n-1)}2 + \frac {2n}{2})^2=$
$(\frac {(n+1)n}2)^2 + (n+1)^2n +(n+1)^2 - (\frac... | I think what you're supposed to pick up here is that $\sum_1^x n^3$ is equal to $\frac{x^4 + 2n^3 + n^2}{4}$. Subtract and see what you get.
| https://math.stackexchange.com |
2,107,350 | [
"https://math.stackexchange.com/questions/2107350",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/136523/"
] | <blockquote>
Suppose $a$ is a positive integer and another positive integer $b$ is obtained by jumbling up the digits of $a$. Suppose $a+b=10^{50}$. Then show that $10|a$.
</blockquote>
I first looked at what happens if $a+b=100$. Letting $a=10x+y$, if $b\neq a$ then $b=10y+x$ implying $11(x+y)=100$, a contradiction... | Assume $a \not \equiv 0 \pmod {10}$. Note that $a$ must be smaller than $10^{50}$, yet bigger than $10^{49}$.
Let $$a=\sum_{i=0}^{49}a_{i}10^{i}$$
Then let $(b_{0}, b_{1}, b_{2}, b_{3}, b_{4}, \dots, b_{49})$ be the digits of $b$ in order, and because of the definition of $b$ it follows that they are a rearrangement ... | (I'd like to post an answer using more intuition and less fancy technique. The proof may look a bit weird at first, but I think it is valid. )
Step 1: show that the last digit of $a$ can only be $0$ or $5$.
Step 2: show that the last digit of $a$ cannot be $5$.
Now step 1:
Suppose the last digit of $a$ is $2$. The... | https://math.stackexchange.com |
75,878 | [
"https://security.stackexchange.com/questions/75878",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/63425/"
] | I'm intrigued that many sites use seemingly random numbers with a random operator as a security check to validate that you're not a bot. Forgive my ignorance in captcha technology, but what is stopping the bot from pulling the simple math problem and calculating it on the fly?
Here's a sample I've pulled from a websit... | CAPTCHAs are one area of computer security where "roll-your-own" can be a good idea.
In order to break a CAPTCHA, a bot needs to be programmed to recognize and solve the CAPTCHA. For low-volume, low-value sites, the cost to program a bot to handle even a trivial CAPTCHA such as this is greater than the expected value... | It does stop the most rudimentary bots which repeatedly POST the same form. As you have mentioned, a more sophisticated bot can evaluate the result and append the unique nonce to every request, thus defeating this system.The most sophisticated bots can even go one step further by performing optical character recognitio... | https://security.stackexchange.com |
95,408 | [
"https://physics.stackexchange.com/questions/95408",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/28118/"
] | Say you have a <em>finite</em> rod of conducting charge. The field from one side of the rod looks like Figure 1:
<img src="https://i.stack.imgur.com/2L7X3.jpg" alt="enter image description here">
At the centre of the sheet, the field lines are straight because of symmetry. (for an infinitesimal length of of the rod) ... | Let us assume that the conductor has a dielectricity constant of $\varepsilon_{\rm rel}=1$. (This assumption is not too bad.) The charge in the rod collects as surface charge $\sigma>0$ at its surface $\Sigma\subset \mathbb{R}^3$. We assume furthermore, that the surface is smooth such that $\sigma$ is bounded, i.e.,... | Saying that the field lines are straight in the center of the rod is only an approximation when you are very close to the rod. As you start to move away the lines do diverge. You can actually see this in your Figure 1. As you have it drawn, the lines (even in the center) begin to move away from each other as you move f... | https://physics.stackexchange.com |
532,755 | [
"https://stats.stackexchange.com/questions/532755",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/255075/"
] | Suppose <span class="math-container">$X = \{(x_1,y_1),\cdots,(x_N,y_N)\}$</span> is our samples. Now if I currently copy paste <span class="math-container">$X$</span> one time, then our sample data will become <span class="math-container">$(X,X).$</span> And we know that the half of samples, the copy pasted part is mea... | Usually, you have to identify if the duplication is intended or at least an unintended artifact of a systematic procedure. Duplication can arise also naturally. One example would be data truncation. Suppose that you measure some lengths and truncate the result to centimeters. Human heights truncated to centimeters will... | To detect (near-)duplicates, you can compute nearest-neighbour distances. If you have a lot of variables this could be slow; one speed-up is to do it for a fixed or random subsample of variables first.
(A speed-up with better theoretical guarantees is to take a random low-dimensional projection and compute nearest-nei... | https://stats.stackexchange.com |
116,934 | [
"https://mathoverflow.net/questions/116934",
"https://mathoverflow.net",
"https://mathoverflow.net/users/27947/"
] | I intend to study mathematical logic , my purpose is to get to Godel's incompleteness theorems
I haven't study any mathematical logic before
so what is the good text which I can use for this purpose ?
I search for a book give me the right picture , and good explanations
I will use it as self-study
| There is a long exact sequence
<blockquote>
$$0 \to H^1(G,Hom(M,N)) \to Ext_{\mathbb{Z}G}^1(M,N) \to \cdots $$
$$\begin{array}{lll}
\cdots & \to & H^i(G,Hom(M,N)) \to Ext_{\mathbb{Z}G}^i(M,N) \newline
& \to & H^{i-1}(G,Ext_{\mathbb{Z}}^1(M,N))\to H^{i+1}(G,Hom(M,N)) \to \cdots
\end{array}$$
</blo... | As you've probalby noticed, we can't just take group cohoology. If $G$ is trivial, we are just taking exts in the category of abelian groups, which are nontrivial, but the group cohomology is trivial.
The spectral sequence idea is completely correct. The functor $Hom_{\mathbb ZG}(M,-)$ is the composition of the functo... | https://mathoverflow.net |
77,992 | [
"https://mathoverflow.net/questions/77992",
"https://mathoverflow.net",
"https://mathoverflow.net/users/14854/"
] | Is $\mathbb{C}[x,y]$ isomorphic to $\mathbb{C}[x]\otimes_{\mathbb{R}}\mathbb{C}[y]$ as rings?
Generally, in a category $\mathcal{C}$ with fibered product, morphisms $f: X\rightarrow Z$, $g:Y\rightarrow Z$, and $h: Z\rightarrow W$. when can say $X\times_Z Y\simeq X\times_W Y$?
| Well ${\bf C}[x,y]$ is isomorphic with $({\bf C}[x]) \otimes_{\bf C} ({\bf C}[y])$. But the tensor products <strong>over ${\bf R}$</strong> cannot coincide, because already ${\bf C} \otimes_{\bf R} {\bf C}$ has zero divisors. For example, $1 \otimes 1 \neq \pm \phantom. i \otimes i$ but $(1 \otimes 1)^2 = (i \otimes ... | Here's an answer to your second question. There is a cartesian square
$$X\underset{Z}{\times} Y \to X\underset{W}{\times} Y$$
$$\downarrow~~~~~~~~~~~~\downarrow$$
$$Z ~~\to ~~Z\underset{W}{\times} Z$$
where the bottom arrow is the diagonal map. Then the top arrow is an isomorphism if (but not only if) the bottom ar... | https://mathoverflow.net |
4,545 | [
"https://quantumcomputing.stackexchange.com/questions/4545",
"https://quantumcomputing.stackexchange.com",
"https://quantumcomputing.stackexchange.com/users/4206/"
] | I was wondering if anybody to help me to generate the following state.
It would be preferable if you use only Hadamard, CNOT and T-gates, on <span class="math-container">$\lceil\log_2(M+1)\rceil$</span> qubits:
<span class="math-container">$$|\psi\rangle = \frac{1}{\sqrt{2}}\biggl(|0\rangle + \frac{1}{\sqrt{M}}\sum_{j=... | Showing that <span class="math-container">$O=O^\dagger$</span> is equivalent to showing that <span class="math-container">$O^2=\mathbb{I}$</span>. In other words,
<span class="math-container">$$
O^2|x\rangle|y\rangle=|x\rangle|y\rangle
$$</span>
for all <span class="math-container">$x$</span> and <span class="math-cont... | Defining such oracles, you may visualize it as many controlled operations, especially <span class="math-container">$\text{CNOT}$</span>s which is an easy way to build oracles.
We know the effect of the <span class="math-container">$\text{CNOT}$</span> is if the control is a 1 then we add 1 into the target (you can se... | https://quantumcomputing.stackexchange.com |
103,143 | [
"https://dba.stackexchange.com/questions/103143",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/20589/"
] | I have run Paul Randals wait script and can see <code>SOS_SCHEDULER_YIELD</code> is at <code>60%</code> with an average signal wait of <code>10ms</code>. I have also run a script to find out the what percentage of my waits are signal waits and in total it is 47%.
I can seen in task manager that the CPU usage is high ... | If the two tables share ID space, surely then they should have a common parent table?
I would create a parent table and make the two other tables children of it with foreign keys. The parent has the autoincrement field, and you have to insert there first before inserting the child. Add a "type" column to the parent ta... | My suggestion would be to create a table like this
<pre><code>Create Table RelationTable
(
id INT UNSIGNED AUTO_INCREMENT PRIMARY KEY,
report_id INT UNSIGNED,
reportType TINYINT
);
</code></pre>
WHERE
<ul>
<li>id = unique id</li>
<li>report_id = unique id in table1 or table2</li>
<li>reportType = 1 for ... | https://dba.stackexchange.com |
310,316 | [
"https://mathoverflow.net/questions/310316",
"https://mathoverflow.net",
"https://mathoverflow.net/users/125359/"
] | (Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the example of $SO(2)$. All $SO(2)$ matrices $M$ can be written as ($\theta\in [0,2\pi[$)
$$
M=\begin{pmatrix}\cos\theta & \... | <strong>Elementary proof</strong>. The linear space $E$ spanned by $SO_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO_n$. Let $M=SR$ be a polar decomposition, where $S\in Sym_n^+$ and $R\in O_n$. This decomposition is unique with $S\in SPD_n$ if $M$ is non-s... | For $n>2$, the span of the matrices in $\mathrm{SO}(n)$ is the full space $M_n(\mathbb{R})$ of $n$-by-$n$ matrices with real entries.
One proof is using representation theory: If we let $S\subset M_n(\mathbb{R})$ denote the span of $\mathrm{SO}(n)$, then notice that this span is invariant under the action of $\mat... | https://mathoverflow.net |
110,724 | [
"https://mathoverflow.net/questions/110724",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4874/"
] | Hi there,
Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold?
For example, given a simple smooth curve on R^2, what's the radius of its tubular neighborhood? (One upper bound is given by the minimal curvature, but general it is not the radius)
Mayb... | As has already been pointed out, while the tubular radius bounds the curvature globally from below, the curvature information is not enough to correctly estimate this radius. Many examples can be considered, often silly ones when you allow your sub-manifold to be disconnected: consider the case of two parallel line seg... | The inverse of the curvature of a plane curve $C$ at a point $p$ is the radius of the osculator circle to the curve at the point $p$; see e.g. <em>Geometry and Imagination</em> by Hilbert and Cohn-Vossen. This suggests that the largest radius of a tube ought to be the inverse of the maximum of the curvature. Thi... | https://mathoverflow.net |
151,469 | [
"https://cs.stackexchange.com/questions/151469",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/147051/"
] | Let <span class="math-container">$α \in [0, 1)$</span> be a constant. For a rooted binary tree <span class="math-container">$T$</span> and a node <span class="math-container">$x$</span> in <span class="math-container">$T$</span>, we denote by
<span class="math-container">$|x|$</span> the number of nodes in the subtree ... | <span class="math-container">$$2|y| = (|y|-|z|) + (|y|+|z|)\le \alpha |x| + |x| - 1.$$</span>
So, <span class="math-container">$|x| \ge \frac2{1+\alpha}|y|$</span>.
Since <span class="math-container">$y$</span> is an arbitrary child of <span class="math-container">$x$</span>, if node <span class="math-container">$x$</s... | First, note that if <span class="math-container">$T$</span> is an <span class="math-container">$\alpha$</span>-good tree, then for any node <span class="math-container">$x$</span> with children <span class="math-container">$y$</span> and <span class="math-container">$z$</span>, without loss of generality, <span class="... | https://cs.stackexchange.com |
533,300 | [
"https://electronics.stackexchange.com/questions/533300",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/191832/"
] | I'm working on a project that requires rather high data transmission to a larger number of SPI slaves.
Will connecting multiple SPI slaves to the same master lower the througput alot? The master wont be able to communicate with all SPI slaves at the same time, so I'm guessing there will be a drop. But how do you determ... | Since SPI is a bus, the bus is shared between all connected components. With a single bus, the communication at any time is limited to a single pair of SPI master and slave. It's not possible to communicate with two slaves concurrently.
A single bus will have a lower transmission rate compared to two or more SPI buses ... | Let's analyze the worst case - all slaves require communication 100% of the time.
Obviously, the worst case rate would drop to 1/# of slaves.
If the duty cycle is less than 100% than the worst case rate would not be as bad.
| https://electronics.stackexchange.com |
163,626 | [
"https://softwareengineering.stackexchange.com/questions/163626",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/22776/"
] | Our product is a web based course managemant system. We have 10+ clients and in future we may get more clients. (Asp.net,SQL Server)
Currently if one of our customers need extra functionality or customised business logic, we will change the db schema and code to meet the needs.
(we only have one branch code base and ... | Parameterize as much as you can first, of course.
Avoid writing:
<pre><code>if(customer == 'foo') {
doBarCalculation();
}
</code></pre>
, but rather write:
<pre><code>if(customer.usesBarCalculation) {
doBarCalculation();
}
</code></pre>
Use the Strategy pattern:
<pre><code>customer.getCalculation().doCalcula... | Create a company table in the database. Every user gets associated with a company. Here is some pseudo-SQL for company-customizations for displaying dates, time-zone, fiscal offset, and contact-person:
<pre><code>table user:
id
company_id
... other fields ...
foreign key (company_id)
table company:
id
date_format
t... | https://softwareengineering.stackexchange.com |
665,505 | [
"https://physics.stackexchange.com/questions/665505",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/313241/"
] | Suppose, a trolley is moving along a frictionless surface with a constant velocity. After some time a mass is added to the trolley. I read somewhere that the theory of Conservation of motion will apply to the system. As in the momentum of the trolley will be the same even after adding the mass. But how's that possible?... | You can consider the addition of mass to be an inelastic collision where kinetic energy is not conserved by the momentum is conserved. The linear momentum in the horizontal direction of motion should remain the same before and after the collision.
<span class="math-container">$$mu = (m+M)v$$</span>
where <span class="m... | When you say "conservation of motion" you are referring to the conservation of momentum. <em>Quantity of motion</em> refers to momentum. Note that conservation of momentum not only works in horizontal and vertical directions, but in all three dimensions, <em>provided</em> net forces in all directions are zero... | https://physics.stackexchange.com |
549,002 | [
"https://math.stackexchange.com/questions/549002",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/99454/"
] | Could anybody provide a proof of the following identity identity:
$$ \sum_{n=0}^{N-1}\binom{N-1+n}{n}=\binom{2N-1}{N}$$
possibly using Symmetry property and Pascal's rule (or another easier way):
$$\binom{a}{b}=\binom{a-1}{b-1}+\binom{a-1}{b}$$
| $$\begin{eqnarray}
\sum_{k=0}^N {N+k \choose N} &=& \sum_{k=0}^N {N-1+k \choose N-1} + \sum_{k=-1}^{N-1} {N+k \choose N} \\
&=& \sum_{k=0}^{N-1} {N-1+k \choose N-1} + \sum_{k=0}^N {N+k \choose N} + {2N-1 \choose N-1} - {2N \choose N}
\end{eqnarray}$$
So
$$\begin{eqnarray}
0 &=& \sum_{k=0}^{N-1... | Note that $${2N-1 \choose N}$$ is the number of ways to choose $N$ distinct numbers from the set $\{1,2,\ldots, 2N-1\}$. Group the possibilities by the largest of these $N$ numbers. This largest number is at least $N$. For $k \in \{0, \ldots, N-1\}$ there are $${N - 1 + k \choose N-1}$$ cases where the largest number... | https://math.stackexchange.com |
302,445 | [
"https://physics.stackexchange.com/questions/302445",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/140084/"
] | I used the same argument in the proof of equation of continuity to flow of stones. Suppose I drop stones from the upper end of a vertical pipe. I am continuously dropping the stones so that at any instant the pipe is full of stones. The stones clearly have streamlines just like fluid flow because they have only straigh... | The answer to the first question is that the rate of flow involves not just the velocity and cross sectional area, but also density. With stones in a pipe, the area stays the same, so when the velocity rises, the density falls to compensate (the distance between stones get stretched vertically as their velocity rises)... | 1) About the rocks, you're ignoring the role of <em>air</em>. As they speed up, they separate, so air comes in to fill the spaces. With water, that can't happen (before drops form), so air pressure (and surface tension) squeezes the stream together.
2) About the water, you're ignoring the thickness $y_1$ of the slug o... | https://physics.stackexchange.com |
170,872 | [
"https://dba.stackexchange.com/questions/170872",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/122727/"
] | There is a debate on if a VARCHAR column breaks 3rd Normal Form. Specifically the column represents a NAME that can be derived from other metadata specific to the entity. Occasionally the column contains customized input from users.
Should the column be eliminated and the application concatenate the metadata together ... | If there are a lot of rows with customizations, I might keep the name column in the main table.
If on the other hand, only a small percentage of the rows in the main table ever contain a customized name column, I might consider making a related table to store the name column.
For instance:
<pre><code>IF OBJECT_ID(... | No, this does not break 3NF IMO. Why? Because although the value can usually be inferred by other records, it can not ALWAYS be inferred. Note 3NF as defined by Codd:
Every non-prime attribute of R is non-transitively dependent on every key of R
The key word here is EVERY. The way I would handle this is to have a sin... | https://dba.stackexchange.com |
384,160 | [
"https://softwareengineering.stackexchange.com/questions/384160",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/323770/"
] | I am confused about the two following statements and how do they both influence on which items be included into the Sprint backlog:
<ol>
<li>The Sprint Goal and Sprint plan is produced by the collaborative work of the entire Scrum Team.</li>
<li>The Product Owner builds the order of items in the Product backlog in ord... | The product owner knows what is important, and prioritizes the backlog. The team knows what is possible, and can estimate backlog items. Together, they can create a sprint goal that is both relevant and achievable. This is part of the Sprint Planning Meeting.
In the simplest case, this simply involves estimating the m... | This is a bit confusing, but in the Scrum Guide, there are two key responsibilities split between the Dev Team and Product Owner:
1) Product Owner owns the priority in the backlog.
2) Only the team can say how much work comes into the Sprint.
From a strictly Scrum Guide standpoint, this means that the conversation m... | https://softwareengineering.stackexchange.com |
1,938,767 | [
"https://math.stackexchange.com/questions/1938767",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/371462/"
] | <blockquote>
How many ternary words in seven letters do not contain the word $121$?
</blockquote>
My naive approach is:
<ul>
<li>If starts with 2 or 3, move to $T(n-1)$</li>
<li>If starts with 12, move to $T(n-3)$</li>
<li>If starts with 13, move to $T(n-2)$</li>
</ul>
But what if we start with $11$? I've seen som... | I think these cases cover the complement set:
<ul>
<li>Words containing $121$ exactly once</li>
<li>Words containing $121$ twice without overlap, but not $1212121$</li>
<li>Words containing $12121$, but not $1212121$</li>
<li>Words containing $1212121$ (this one's easy)</li>
</ul>
Can you take it from here?
| Let's do it recursively, for all lengths $n$.
We let $T_n$ be the number of "good" words of length $n$.
Let $A_n$ be the number that of good words of length $n$ that end in $12$.
Let $B_n$ be the number that of good words of length $n$ that end in $1$.
Let $C_n$ be the number that of good words of length $n$ that e... | https://math.stackexchange.com |
70,516 | [
"https://cs.stackexchange.com/questions/70516",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/66625/"
] | I am trying to find out how many times the "statement" is executed by finding its formula based on these loops:
<pre><code>int s = 0;
for(int k = n; k > 0; k /= 2)
{
for(int l = k; l < n; l++)
{
s++; // statement
}
}
</code></pre>
I have been stuck with this problem for a while since I couldn't really... | We can determine the <em>exact</em> value of $s$ as a function of $n$. For the sake of brevity, let $\lambda = \left \lfloor \lg n \right \rfloor$. We can replace the inner loop with:
<blockquote>
$s \leftarrow s+n-k$
</blockquote>
At the end of the program we'll have:
$$
s= \sum_{h=0}^{\lambda} n - \left \lfloor ... | The outer loop runs about log n times.
The inner loop runs 0 times during the first iteration when k = n. All the other times, 0 ≤ k ≤ n/2, so the inner loop runs between n/2 and n times, so we have O (n log n).
| https://cs.stackexchange.com |
505,298 | [
"https://physics.stackexchange.com/questions/505298",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/219182/"
] | Let's say I have some state vector <span class="math-container">$|\Psi(t)\rangle$</span>, and I express it as a linear combination of eigenstates of some operator, <span class="math-container">$\hat{Q}$</span>, with a discrete spectrum, which we will call <span class="math-container">$|q_n\rangle$</span>. Let's say tha... | Any separable solution to the Schrodinger equation is stationary. The mistaken step in your reasoning is "Each term of this sum is orthogonal, so they obey the Schrödinger Equation." That's not true.
If <span class="math-container">$Q$</span> commutes with the Hamiltonian (and there's no degeneracy), then <span class=... | The issue is either in the explicit time dependence you have assumed or in thinking your general <span class="math-container">$\Psi(x,t)$</span> is separable.
First, let's consider the case where your quantum state is an eigenstate of <span class="math-container">$Q$</span>:
<span class="math-container">$$|\Psi(t)\ran... | https://physics.stackexchange.com |
58,411 | [
"https://electronics.stackexchange.com/questions/58411",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/16910/"
] | I was playing around with my voltmeter and I came across something I don't know how to work the math of.
The circuit built was the one below, the right power supply was a 9V battery and the left was a six volt. Both resistors R1 and R2 are 1K ohms.
What is the math to determine how much current will go through (certa... | Diodes are non-linear devices so we need to make some assumptions (a.k.a. guess) about what operating state they are in.
With \$V_D = 9V\$ and \$V_A = 6V\$, I'm going to assume that D2 is forward biased and D1 is reversed biased. The arrows show the assumed direction of current flow. I'm calculating with \$R1 = R2 = 1... | I'm going to call V1 the voltage on the left in series with D1 and V2 the voltage on the right in series with D2.
The first thing to note is that the forward voltage drop for each diode is VF(D1) and Vf(D2). These values are usually 0.6V to 0.7V for a silicon diode.
Here, I drew it out for clarity:
<img src="https... | https://electronics.stackexchange.com |
73,561 | [
"https://softwareengineering.stackexchange.com/questions/73561",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/4702/"
] | Is it true that platform independency is more of a myth today?
Because programmers look for efficiency and power over portability which have to be compromised to some extent when achieving platform independency.
On the other hand architecture neutrality is welcomed because it provides a very powerful way to make a sof... | <strong>I rely on platform portability at the expense of power every day</strong>
I write my code on a Mac or Windows machine, and see it executed on a Linux machine. If I didn't have true platform portability this would be a nightmare.
Power is less of a compromise in my opinion because hardware is so cheap in compa... | Depending on the application somewhere between 80% and 100% of the code will have no impact on performance, as long as it isn't grossly inefficient. For this code, correctness, maintainability and portability should be your priorities; there is no need to sacrifice any of these for marginal performance improvements.
O... | https://softwareengineering.stackexchange.com |
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