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 |
|---|---|---|---|---|---|
199,081 | [
"https://softwareengineering.stackexchange.com/questions/199081",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/91799/"
] | I have a REST webservice that have a couple of resources, like Company and User. But now I need to make a modification to the API to return a special list combining company, user and maybe other type of resource. Like a list of recommendations with various type of resources.
Which is the best approach to solve this?
... | Map your Recommendation resources to <code>/recommendations</code> and then use this structure:
<pre><code>{
"users" : [
{ .. },
{ .. }, ..
],
"companies" : [
{ .. },
{ .. }, ..
]
}
</code></pre>
It allows the consumer to parse it properly based on the name... | I would define a separate Recommendation type with all the relevant information for recommendations in it.
In your model, this type does <em>not</em> map directly to a database table, but rather gets its contents from both the User and Company tables (possibly using an SQL UNION).
| https://softwareengineering.stackexchange.com |
496,642 | [
"https://math.stackexchange.com/questions/496642",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/73362/"
] | The following is not a subspace
$W_6=\{(a_1,a_2,a_3)\in \Bbb R\mid 5a^2_1- 3a^2_2+ 6a^2_3= 0\}$
because ($\sqrt{3},\sqrt{5},0)∈W_6$ and $(0,\sqrt{6},\sqrt{3})∈W₆$ but the sum ($\sqrt{3},\sqrt{5}+\sqrt{6},\sqrt{3})\notin W_6$.
If one of you guys could explain to me they arrive at this result I would really appreciat... | What you have written shows that there are two things in $W_6$ whose sum is not in $W_6$. This must happen if it is to be a subspace.
To answer "how does one arrive at this result?" the answer is probably "experimentation." Since it's obvious that the set contains the zero vector, and that $\lambda x\in W_6$ if $x\in ... | What you wrote says "$W_6$ is just those elements of $\mathbb{R}$ such that the equation $5\vec{a}_1^2 - 3\vec{a}_2^2 + 6\vec{a}_3^2 = 0$ holds." What you are considering is whether the set $W_6$ satisfies the definition of a linear subspace. This means that $\vec{0}$ must be an element of the subspace (trivially, it... | https://math.stackexchange.com |
2,749,833 | [
"https://math.stackexchange.com/questions/2749833",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/473627/"
] | <blockquote>
Define $C=\{\frac{n^2+12n+32}{n+5}\:|\:n\in \mathbb{N}\}$. Prove that $\inf C=\frac{45}{6}$
</blockquote>
<em>Attempt</em>:<br>
We'll show that $\frac{45}{6}$ is a lower bound of $A$. Let $c\in C$. There exists $n\in \mathbb{N}$ s.t. $c=\frac{n^2+12n+32}{n+5}$.<br>
$$\frac{45}{6}\leq\frac{n^2+12n+32}{... | Let $f(x)=\frac{x^2+12x+32}{x+5}=x+7-\frac3{x+5}$. Then $f'(x)=1+\frac3{(x+5)^2}>0$. So, $f$ is strictly increasing and therefore$$(\forall n\in\mathbb{N}):f(n)\geqslant f(1)=\frac{45}6.$$
| To answer to the question raised in a comment, it can be solved just using properties of inequalities: in the decomposition of
$$ f(n)=\frac{n^2+12n+32}{n+5}=n+7 -\frac 3{n+5}, $$
note $n+5$ is <em>increasing</em>, so as it is positive for all $n\in\mathbf N$, $\dfrac 3{n+5}$ is <em>decreasing</em>, and $-\dfrac 3{n+5... | https://math.stackexchange.com |
282,398 | [
"https://electronics.stackexchange.com/questions/282398",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/123812/"
] | I am looking at a portable inverter that claims to be able to output two modes through its Type B socket: 120V AC or 150V "HVDC". The documentation says the 150VDC mode can be used for resistive loads or for switched mode power supplies, avoiding conversion loss from the inverter. How can I tell what black box device n... | You seem to be getting confused between the efficiency of the LED and the efficiency of the circuit to drive the LED.
In terms of light output per unit of energy used by the LED they are an efficient way to generate light. In absolute terms they aren't great, they are around 10%<sup>[1]</sup> efficient in that respect... | The efficiency of a LED refers to how efficient the LED is. This has nothing to do with how efficient or not the driving circuit is.
In many cases, the overall circuit efficiency of LEDs is not much of a issue. If the LED is just being used as a indicator, it is low power in the first place. A typical green LED dro... | https://electronics.stackexchange.com |
714,098 | [
"https://physics.stackexchange.com/questions/714098",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/226687/"
] | In chapter 5 of the book "Statistical Mechanics" by Pathria it says
<blockquote>
Since the density matrix evolves in a unitary manner, the von Neumann entropy is time-independent
</blockquote>
Where the von Neumann entropy is defined as the trace
<span class="math-container">$$S[\rho(t)]=-\mathrm{Tr}\left(\rh... | Hint: Use the spectral decomposition to write
<span class="math-container">$$\rho(0) := \sum\limits_k \lambda_k \,|k\rangle \langle k| \tag{1} ,$$</span>
and then find an expression for <span class="math-container">$\rho(t)$</span> in terms of <span class="math-container">$\lambda_k$</span>. Especially note that <span... | Another neat method to prove this: write the von Neumann entropy as a limit of the Renyi entropies:
<span class="math-container">$$
S[\rho] = \lim_{n \to 1} S^{(n)}[\rho] = \lim_{n \to 1} \frac{1}{1-n} \log \text{Tr} \rho^n
$$</span>
Here, the identity is manifest: <span class="math-container">$U^{\dagger} U = 1$</span... | https://physics.stackexchange.com |
164,966 | [
"https://mathoverflow.net/questions/164966",
"https://mathoverflow.net",
"https://mathoverflow.net/users/26080/"
] | ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH).
<blockquote>
<strong>Question.</strong> For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{... | (A very partial answer only, but too long for a comment.)
In light of your second comment under Monroe's answer, I interpret your question as follows:
Given a formula $\varphi(x)$, write $\kappa_\varphi$ for the least cardinal satisfying $\varphi$. ($\kappa_\varphi$ could be undefined, of course. But you are only c... | Start with a model of GCH. Let $\kappa$ be singular. Add $\kappa^{++}$ Cohen subsets of $\mathrm{cf}(\kappa)^+$-- the forcing is $\mathrm{cf}(\kappa)^+$-closed and $\mathrm{cf}(\kappa)^{++}$-c.c. Then we'll have $\kappa^\kappa = \kappa^{++}$ but $\kappa^{\mathrm{cf}(\kappa)} = \kappa^+$ since we've added no $\mathrm... | https://mathoverflow.net |
55,954 | [
"https://cs.stackexchange.com/questions/55954",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/30102/"
] | Assume that we have a data structure which uses $O(\log n)$ space to store an integer $n$ and has a function $f$ which replaces the integer $n$ stored by $2^n$, i.e. $n=2^n$.
The time complexity of $f$ is $O(n_{_{old}})$ = $O(\log\ n_{_{new}})$,
is it admited to say that $f$ is a $O(\log n)$ function ?
| The running time of a function is usually measured with respect to the length of the input, which is represented by the parameter $n$. In your case, if the input is the integer $m$ then the input length is $n = \log m$ and the running time is $O(m) = O(2^n)$.
For functions whose output is much larger than the input, i... | No. The input for $f$ is $n_{\text{old}}$, which, as you mentioned, is stored using $O(\log n_{\text{old}})$ space. Thus the upper bound $O(n_{\text{old}})$ on the time complexity of $f$ is actually exponential in the input size, so all you can say is that $f$ is computable in $O(2^n)$.
| https://cs.stackexchange.com |
215,249 | [
"https://security.stackexchange.com/questions/215249",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/214539/"
] | I've changed ISP and am using their router. I wish to donate my old NETGEAR router still with its original Network Key (password). Would someone obtaining this router somehow get access to my computer files and data even though I am no longer using that network or ISP? Thanks.
| Using <code>eval</code> in this context doesn't create any vulnerability, as long as an attacker can't interfere with the arguments passed to <code>matchCondition</code>.
If you find it easier to read / program it this way, and you're confident that no untrusted input will ever go into your expression compiler, then g... | Today, everything is written by developers. Next month or next year, someone will say "hey, why not let the users write those themselves?" Bam.
Also, even if the rules are written by the developers only, do they or will they include any user-originated data? Something like titles, names, categories, for instance? This... | https://security.stackexchange.com |
146,090 | [
"https://security.stackexchange.com/questions/146090",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/35367/"
] | Nowadays there are a lot of hacked websites with stolen login information. In many cases the website states that no credit card data and/or payment information was stolen.
Why is that? What I assume is: That both, the database storing the payment data and the one storing user-credentials are separated from each other.... | <h2>PCI DSS</h2>
The major reason for this is a decade long effort by the payment cards industry to limit the extent of such breaches by requiring everyone who handles payment card data to either (a) conform to a set of security practices and (usually) audit requirements, or (b) stop handling payment card data themsel... | In the case of recently disclosed Yahoo data breach where 1bn user account information was stolen, it transpired that no credit card information was stolen because it was kept in a separate database in encrypted format.
Most organisations have rigid and robust methods to store credit card information, typically in a s... | https://security.stackexchange.com |
1,459,587 | [
"https://math.stackexchange.com/questions/1459587",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/79719/"
] | I have a given rooted tree where I need to reach all leaves atleast once (and stop at the last leaf I visit in the entire tour). We need to minimize the expected cost of visiting all leaves. Every hop of an edge costs 1 unit.
My approach : For each node, I calculate 3 parameters : expected value to visit all leaves in... | For <span class="math-container">$0\neq x\in N$</span> and some <span class="math-container">$\delta>0$</span> choose <span class="math-container">$\varepsilon=\frac{\delta}{\| x \|}$</span>. So <span class="math-container">$|f_n(x)-f_m(x)|< \|f_n-f_m\| \|x\|=\varepsilon\|x\|=\delta$</span> for sufficiently large... | observe that $||f_m-f_n||\leqslant \epsilon$ implies $\{f_n(x)\}_{1\leqslant n}$ is a cauchy sequence in $\mathbb{C}$ and hence convergent
so now define a function $f$ given by $f(x)={lim}_{n \to \infty}f_n (x)$ and show that $f \in N^* $ and $f_n \to f$
| https://math.stackexchange.com |
40,198 | [
"https://mathoverflow.net/questions/40198",
"https://mathoverflow.net",
"https://mathoverflow.net/users/9557/"
] | Hi
I have the following matrix
A=[a_11 a_12 a_13 1;
a_21 a_22 a_23 1;
.
.
.
a_n1 a_n2 a_n3 1]
I have seen that when some of a_ij are big for instance in the order of 200 , then condition
number is also big.
I would like to know is it possible to show it theoretically.
Is it po... | One possible way to proceed would be to get the singular value decomposition of your matrix and then look at the ratio of the largest singular value to the smallest singular value (a.k.a. the 2-norm condition number); largeness of this condition number implies largeness of the condition number with respect to the other... | Having large entries is not related at all to the condition number. For example, the matrix $10,000I$ (ten thousand times the identity) has condition number 1. One can easily generate matrices with arbitrarily large and small entries (even at the same time) and condition number 1.
In order to prove a bound, you need t... | https://mathoverflow.net |
442,366 | [
"https://physics.stackexchange.com/questions/442366",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/213689/"
] | I'm in the process of conducting research concerning free fall with air resistence. I'm dropping a beach ball filled with air from certain heigts, while measuring its speed and acceleration during the fall.
Theoretically, the initial (first few milliseconds) acceleration of the beach ball should be "g" (9,82 m/s^2).
... | (a) I assume that you've checked your means of measuring acceleration, say by using it to measure the initial acceleration of a dropped stone.
(b) I'd have thought that buoyancy (Archimedian upthrust from the surrounding air) might decrease the acceleration significantly. As you know, the upthrust is equal and opposit... | The initial acceleration is the effective weight divided by the inertia. The inertia is larger in air than in vacuum because of a fluid dynamic effect called the “added mass”. The added mass is one-half the mass of the air displaced by the ball (for an ideal fluid). The effective weight is the mass of the ball minus... | https://physics.stackexchange.com |
2,089,749 | [
"https://math.stackexchange.com/questions/2089749",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/253106/"
] | This is the last step in a proof I'm trying to do that the image of any one-dimensional representation of a finite group $G$ is cyclic. Because $g^n=id_G$ for some positive integer $n$ $\forall{g}\in{G}$, we know $\rho(g)^n=1$ as well. So $\textrm{im}(\rho)\subset{S^1}$, and is finite. Apparently this also means it mus... | It turns out that any finite subgroup of the multiplicative group of a field must be cyclic. In particular, since $S^1 \subset \mathbb{C}^*$ this implies the result you want.
To see this fact just note that if $G$ is abelian but not cyclic then $\exists n < |G|$ such that $g^n = e$ for all $g \in G$. But then if ... | Let $G$ be a (non-trivial) finite subgroup of $S^1$. Identifying $S^1$ with $\mathbb{R}/\mathbb{Z}$ (and choosing $[0,1)$ as the set of representatives), let $x$ be the minimal non-zero element of $G$.
If $G\neq \langle x\rangle$ then there is some $y\in G$ with $y\not\in\langle x\rangle$. Hence there is some natural ... | https://math.stackexchange.com |
296,520 | [
"https://mathoverflow.net/questions/296520",
"https://mathoverflow.net",
"https://mathoverflow.net/users/122571/"
] | While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
<ol>
<li>When defining Dolbeault cohomology, one uses $\bar{\partial}$ but not $\partial$. I wonder whether there happens any problem if one define a coh... | <ol>
<li>On differential forms, take complex conjugate to turn $\partial$ into $\bar\partial$, and holomorphic functions into conjugate holomorphic. </li>
<li>All of the proofs about differential forms then go through the complex conjugation effortlessly, including the Poincare lemma. We use $\bar\partial$ because we l... | If $X$ is a complex manifold and $E\to X$ is a holomorphic vector bundle, only $\bar\partial_E$ can be defined naturally, i.e., it depends only on the complex structures of $X$ and $E$. The $\partial_E$ operator cannot be defined intrinsically.
If $E$ has a flat connection, you can also define the $\partial$-operator.... | https://mathoverflow.net |
500,977 | [
"https://stats.stackexchange.com/questions/500977",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/254584/"
] | A common example of teaching binomial testing is viewing the results of some coin flips, seeing a total number of heads greater than the expected value, then formulating the null hypothesis that the probability of flipping heads is not biased and calculating the p-value from the binomial distribution. This null hypothe... | The generative story describes how each image sample is generated. The story is as follows - (a) Sample z ~ N(z | 0, I); (b) Sample x ~ N(x | f_mu(z), f_sig(z))
For any generative story, going forward tells us about test time, and going backwards helps us learn the parameters (note that last step has <code>x</code>, ou... | I think you're mixing up
<ol>
<li>How to compute or approximately compute <span class="math-container">$P(X)$</span></li>
<li>The "generative story"</li>
</ol>
Also, I might be wrong on this, but based on the last paragraph, it sounds like you're confusing <span class="math-container">$X$</span> with the enti... | https://stats.stackexchange.com |
677 | [
"https://mechanics.stackexchange.com/questions/677",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/428/"
] | I have a small leak at the top of my air conditioning condenser.
Last summer, I tried patching it with JB Weld. They do advertise as working with all metals, but I suspect it failed to hold because it doesn't bond well to aluminum. I could get enough refrigerant in to get some cold air, and then the patch would blow. ... | I think most pros will tell you not to bother trying to patch a condenser leak, just replace it. While that's not always an option, it's definitely an option you should consider first.<br>
Can you source cheap second-hand unit for your car? Unfortunately condensers and radiators are often damaged in car accidents so se... | I think that it would be best to replace it, they are not expensive for aftermarket parts. A good used one would be better than patching it. For best results, you will need to pull a good vacuum on the system to eliminate any air or moisture in the system before charging. It would be best to have it done professionall... | https://mechanics.stackexchange.com |
126,862 | [
"https://stats.stackexchange.com/questions/126862",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/21316/"
] | You can throw 3 dice 3 times. Every time a 3 comes, you save it and don't throw it any more. What are the odds you will end up with 3 3 3?
| Here is some hint:
Conditioning on the result of the first throw:
Let $A_0$ = {the first throw doesn't have 3}, $A_1$ = {the first throw has one 3},
$A_2$ = {the first throw has two 3s},
$A_3$ = {the first throw has three 3s} and $A$ = {ends up with 3, 3, 3}.
By law of total probability:
$$P(A) = P(A|A_0)P(A_0) + P(A|... | Here is some R code to check against the theoretical derivation:
<pre><code>perce=0; T=10^7
for (t in 1:T){
# first throw
res=sample(1:6,3,rep=TRUE)
val=sum(res==3)
# second throw
if (val<3){ res=sample(1:6,3-val,rep=TRUE)
val=val+sum(res==3)}
# third throw
if (val<3){ res=sample(1:6,3-val,rep... | https://stats.stackexchange.com |
397,187 | [
"https://math.stackexchange.com/questions/397187",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/21813/"
] | My textbook seems to suggest that the analytic functions are precisely the functions that can be written in terms of $z$ alone (no $x$ or $y$ or conjugate-$z$).
Am I inferring correctly?
Does this mean that $\sin (z+x)$ is not analytic? [where $z=x+iy$]
| More precisely, an analytic function can be expressed locally in terms of its power series. Here one must distinguish between two related notions: real analyticity and complex analyticity. Thus, the function you mentioned is real analytic (of two variables), but is not complex-analytic as a function of $z$.
| I think that $f$ "can be written in terms of $z$" alone means $\partial f/\partial \bar{z}=0$ here. It is easy to check that $\partial f/\partial \bar{z}$ is "equivalent" to the Cauchy-Riemann equation.
Note that you can obviously express a complex function in terms of $x$ and $y$, i.e. the real part and the imaginary... | https://math.stackexchange.com |
262,197 | [
"https://mathoverflow.net/questions/262197",
"https://mathoverflow.net",
"https://mathoverflow.net/users/76193/"
] | Margulis' normal subgroup theorem states that any normal subgroup of a higher rank irreducible lattice is either finite or of finite index.
What are the known counter-examples in rank $1$ ?
I am especially interested by $PU(2,1)$.
| Any cocompact lattice in a rank-one Lie group is <em>word-hyperbolic</em>. Olshanskii proved that such groups are SQ-universal, meaning in particular that they have uncountably many normal subgroups. Similar results are known for non-uniform lattices, which are <em>relatively hyperbolic</em>.
As Yemon Choi suggests ... | Also interesting in this context is the existence of compact complex surfaces whose universal cover is the ball and admitting an holomorphic map onto a Riemann surface with connected fibers. The kernel of the induced map is a finitely generated group of infinite index. This is similar to the existence (Rips counterexam... | https://mathoverflow.net |
3,455,248 | [
"https://math.stackexchange.com/questions/3455248",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/506709/"
] | <blockquote>
Define <span class="math-container">$\displaystyle F(s) = \sum_{n=0}^\infty
f_ns^n$</span>, where <span class="math-container">$f_n= P($</span> the first return to the origin at time <span class="math-container">$n)$</span>.
Suppose <span class="math-container">$F^{(n)}(s) = \displaystyle \sum_{k=0}^\... | Without using facts about generating functions (since it wasn't covered in 251).
<span class="math-container">$$
\textbf{Base case: n=1}
$$</span>
This follows directly from definition. So assume the proposition is true for n.
<span class="math-container">$$
\textbf{Induction step: n+1}
$$</span>
First note that th... | If <span class="math-container">$T_1,T_2,...$</span> are return times to the origin then (it is well known that) <span class="math-container">$T_1,T_2-T_1,...,T_{n+1}-T_n,...$</span> are independent and identically distributed. The sum of the first <span class="math-container">$n$</span> of these i.i.d. random variable... | https://math.stackexchange.com |
398,019 | [
"https://physics.stackexchange.com/questions/398019",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/99217/"
] | In the book of <em>The First Three Minutes</em> by Weinberg, at page 21, he talks about how do astronomers measure the speed of a luminous body along the line of sight by Doppler affect, i.e. the fractional change in the wavelength of the incoming light will be proportional to the speed of the body to $c$, but to use t... | As an addition to the correct answer of @flippiefanus, consider the element sodium.
When excited at low pressure by an electric arc, sodium vapour emits a complex spectrum of discrete wavelengths, <em>an atomic emission spectrum</em>, dominated by two intense emission lines with slightly different wavelengths: one a... | Stars emit lots of different wavelengths, but there are special wavelengths associated with the absorption spectra of the elements found in stars. These are known and they are the same everywhere, even in stars that are far away from us. So all the astronomers need to do is to identify the spectra of the elements in th... | https://physics.stackexchange.com |
624,547 | [
"https://electronics.stackexchange.com/questions/624547",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/314096/"
] | I'm looking at electronic speed control circuit design, and notice that most use a driver IC like TI's DRV series of chips. At the moment, sourcing these chips is extremely difficult, so I'm wondering why the MCU GPIOs can't switch the MOSFETs directly.
| Sometimes they can. But when they can't there are two main reasons- not enough voltage and/or not enough current.
The voltage is required to get many MOSFETs to turn on so they have low resistance. Often 10V is required. No common MCU can supply more than 5V and many these days are 3.3V on the GPIO pins.
The current is... | The MOSFET gate turn-on voltage may be higher than what an MCU I/O pin can provide. A driver circuit can translate the I/O logic output voltage into a strong 'on' gate voltage.
A MOSFET drive circuit must be able to charge and discharge the MOSFET's gate-source capacitance fast enough to produce the wanted output wavef... | https://electronics.stackexchange.com |
477,066 | [
"https://physics.stackexchange.com/questions/477066",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/177331/"
] | So, we started the study of thermodynamics by introducing <span class="math-container">$\text dU$</span> in a logical way:
<span class="math-container">$$
\text dU = T \text dS - P\text dV + \mu \text dN . \tag1
$$</span>
Later we started to see that all the properties of a system can be defined by a function <span c... | I recently opened my profile in this forum and faced this question from a year ago.
Although @CRDrost answer is not wrong, I think it would be nice to write here in a clear way how I have managed to understand the problem a year ago.
Here it goes:
When we first learn about partial derivatives, in the case of functio... | This is a very common problem induced by a choice we make in our notation for partial derivatives and it has to do with this idea that what you are keeping constant needs to be tracked very carefully in order to not run into contradictions.
To wit, you are trying to define a new function <span class="math-container">$... | https://physics.stackexchange.com |
124,789 | [
"https://mathoverflow.net/questions/124789",
"https://mathoverflow.net",
"https://mathoverflow.net/users/7443/"
] | Let $G$ be a finite group with normal subgroup $N$. Let $\chi$ be an irreducible complex character of $G$. Then Clifford's Theorem says that $\mathrm{res}^{G}_{N}\chi = e(\eta_1 + \cdots + \eta_r)$ where $e$ is a positive integer and each $\eta_i$ is a complex irreducible character of $N$ such that the $\eta_i$'s are t... | I don't think there's any clear cut relationship. In particular, the inequality $e \ge [\mathbb Q(\eta) : K]$ needn't hold. For example, take $G=S_5$, $N=A_5$ and let $\chi$ be the unique irreducible $S_5$-character of degree $6$. Then we have $\mathrm{res}^{G}_{N} \chi = \eta + \bar{\eta}$, where $\eta$ and $\bar{\eta... | The coefficient $e$ is always $1$ if $G/N$ is cyclic, but $|Q(\eta):Q(\chi_N)|$ can be arbitrarily large. For example, let $N$ be cyclic of prime order $p$, and let $G$ be the semidirect product of a cyclic group of order $p-1$ acting faithfully on $N$. Then $G$ has a rational valued character $\chi$ of degree $p-1$ an... | https://mathoverflow.net |
2,023,436 | [
"https://math.stackexchange.com/questions/2023436",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/-1/"
] | I am taking an ordinary differential equation class, and we are currently learning about power series. One thing that comes up is index shifting, for the most part, I can shift the index quite easily, but in the following case, I end up with a fraction index. I have an ODE of the form y'' + 2xy' + y = 0, with one of th... | The right-hand sum's lowest term is <span class="math-container">$x^3$</span>. We want <span class="math-container">$x^3$</span> to be of the form <span class="math-container">$x^{2n+1}$</span> so that we can combine the right-hand sum with the left-hand one. Clearly then, we can start the sum at n=1 and replace n with... | <strong>Hint</strong>
you can downshift or upshift.
$$\sum_{n=k}^N f(n)=$$
$$\sum_{n=k-1}^{N-1}f(n+1)=\sum_{n=k+1}^{N+1}f(n-1)$$
| https://math.stackexchange.com |
113,327 | [
"https://electronics.stackexchange.com/questions/113327",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/44136/"
] | I bought a car cigarette DC adapter yesterday with the following specs:
<pre><code>Input: DC 12V
Output: DC 17V
</code></pre>
I'm planning to connect it to a 12V battery I have at home, but I was wondering if the adapter input polarity matter or could damage the adapter if reversed?
| The adapter might have some sort of protection inside, but it will not work (and possibly it will fry) if you reverse the polarity. I bet you have this kind of plug:
<img src="https://i.stack.imgur.com/bugIC.jpg" alt="enter image description here">
The tip is the positive contact while the side plates should be conne... | Yes, it matters. Check it carefully or you probably will damage the device.
| https://electronics.stackexchange.com |
280,437 | [
"https://stats.stackexchange.com/questions/280437",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/42176/"
] | My specific case is this: There are a number of categories (Science, Health, Religion, etc.), each containing multiple documents. Assume for documents within each category, I perform topic modeling (LDA) with 10 topics ($k=10$). For example, for the Science category, each topic will correspond to Biology, Chemistry, et... | In math, generally there are more than one way to prove a equation. I think the previous Answers are correct, but differ from your approach. Your approach is correct also, but you need one more step.
At first, you missed hat on $\beta$ in your last equation. Using the fact that $\hat \beta =(X^TX)^{-1}X^Ty$, we have
... | The most compact way to see the equality is to use the orthogonal projection matrix $P = X(X'X)^{-1}X'$ and the residual-maker matrix $M = I-P$. Both these matrices are symmetric, $P'=P,\;\; M'=M$ and idempotent $PP=P,\;\; MM = M$.
We have
$$X\hat{\beta} = Py, \;\;y-X\hat{\beta}= My$$
Then
$$S_r = (y-X\hat{\beta})'... | https://stats.stackexchange.com |
3,752,741 | [
"https://math.stackexchange.com/questions/3752741",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/742435/"
] | For example A=λI ⇒ A²=λ²I where A is square, and λ∈ℝ.<br />
Or more generally AB=CD ⇒ ABAB=CDCD.<br />
Assuming all the necessary matrix products are possible, what other conditions would need to be fulfilled for this to hold?
| For my money, the answer you’re looking for has nothing whatever to do with matrices. For, in mathematics, when we write<span class="math-container">$$A=B$$</span> we’re saying that <span class="math-container">$A$</span> and <span class="math-container">$B$</span> are <em>the same thing</em>. In your case, you’re not ... | <span class="math-container">$AB=CD$</span>
<span class="math-container">$\Rightarrow ABAB=CDAB$</span>
<span class="math-container">$\Rightarrow ABAB=CDCD$</span>
In words: you are allowed to right multiply both sides by <span class="math-container">$AB$</span>, which is the same as <span class="math-container">$CD$</... | https://math.stackexchange.com |
168,879 | [
"https://physics.stackexchange.com/questions/168879",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/74685/"
] | How can gravitions exist without violating GR, since GR says that gravity is curvature in space-time.
| For energies below the scale where gravity becomes strongly coupled, the paradigm of QFT is applicable and teaches us that gravity is due to the exchange of massless spin-2 particles we call gravitons, and that the whole picture of curved space time is nothing but a nice way to equivalently represent the collective eff... | GR predicts gravitational waves, which hence exist without violating it. By the wave-particle duality, there must be a particle (or at least a quasi-particle) associated with (the quantization of) such waves. We arbitrarily name them gravitons. Whatever aspect of them may be found to violate GR simply indicates how we ... | https://physics.stackexchange.com |
74,513 | [
"https://security.stackexchange.com/questions/74513",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/62116/"
] | Assuming I currently use a safe way to salt-challenge-response-authentication-method, passwords are stored secured and all.
I now question myself - how do users set the password?
Meaning, they enter password as plain text in the textbox, and what happens next? should their client hash the password and send it to the s... | You should <strong>NEVER</strong> send passwords in the clear. As long as your communications are encrypted (TLS), your server can hash and salt the passwords when it receives them.
| For <strong>Web Apps</strong> you best go with
<ol>
<li>using TLS as mandatory. This includes mechanisms like HSTS, to prevent MITMs to redirect to fake http "counterpart".</li>
<li>sending password <strong>in the clear</strong> for both login and setup</li>
<li>salting and hashing of passwords on the server. Do somet... | https://security.stackexchange.com |
2,737,995 | [
"https://math.stackexchange.com/questions/2737995",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/520813/"
] | <blockquote>
Task: $\frac{du}{dt} = u(1-\frac uA)$ with $u(0) = u_{0}, 0 < u_{0} < A$
</blockquote>
I know that this is a variable separation type, however, I have no idea to solve it.
what should be the next step?
| We have
$$\frac{1-\frac{1}{1-\frac{1}{x-1}}}{1+\frac{1}{1+\frac{1}{x+1}}}
=\frac{1-\frac{1}{\frac{x-2}{x-1}}}{1+\frac{1}{\frac{x+2}{x+1}}}
=\frac{1-\frac{x-1}{x-2}}{1+\frac{x+1}{x+2}}
=\frac{\frac{-1}{x-2}}{\frac{2x+3}{x+2}}=-\frac{x+2}{(2x+3)(x-2)}$$
with $x\neq 1,-1,2,-2,-\frac32$
| We have:
$$1-\frac{1}{1-\frac{1}{x-1}} = 1-\frac{x-1}{x-2} = \frac{x-2-x+1}{x-2}=\frac{-1}{x-2}$$
and
$$1+\frac{1}{1+\frac{1}{x+1}} = 1+\frac{x+1}{x+2} = \frac{2x+3}{x+2}$$
so
$$\frac{1-\frac{1}{1-\frac{1}{x-1}}}{1+\frac{1}{1+\frac{1}{x+1}}} = \frac{\frac{-1}{x-2}}{\frac{2x+3}{x+2}} = -\frac{x+2}{(x-2)(2x+3)}$$
an... | https://math.stackexchange.com |
50,847 | [
"https://dsp.stackexchange.com/questions/50847",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/36797/"
] | Sorry for this question, I'm still new in signal processing area.
In OFDM system, let $x$ be the received signal, what do we call the process of multiplying the received signal $x$ by $e^{-j2\pi F_c t}$ , where $F_c$ is frequency carrier and $t$ is interval 0 : sampling rate: sampling rate * length of signal.
So th... | <blockquote>
I'm currently playing around with some compression algorithms and I'm asking myself if there is a type of data distribution / noise distribution that is easier to target with quantization (meaning less distortions at same rate). To my understanding i.d.d. Gaussian is the upper bound on "compression diffi... | It's very easy to make fundamental mistakes on the topic of information theory, unless you have some PhD or alike in the field. That being said as far as I remember the following could roughly be stated about your question.
Considering a discrete probability and source model of iid, the most efficient compression of a... | https://dsp.stackexchange.com |
554,995 | [
"https://physics.stackexchange.com/questions/554995",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/239775/"
] | Does a term for non-QM physics exist - in the sense of classical physics including relativity?
| The correct, modern, professional terminology for "non-quantum mechanics" is "classical mechanics".
Students are routinely confused by this because there is an old-fashioned tendency in texts that discuss special relativity to use "classical" to mean "non-relativistic". That usage only exists in educational genre mat... | The way I use it, "classical physics " includes classical mechanics, classical electrodynamics, special relativity and general relativity.
Otherwise I use classical mechanics, classical electrodynamics ....
Quantum physics for me includes quantum electrodynamics and quantum field theory.
I do not think there is gen... | https://physics.stackexchange.com |
545,879 | [
"https://math.stackexchange.com/questions/545879",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/100426/"
] | I know that $$\sum_{k=0}^n \binom nk k^2=2^{n-2}(n^2+n),$$ but I cannot find a way how to prove it. I tried induction but it did not work. On wiki they say that I should use differentiation but I do not know how to apply it to binomial coefficient.
<br>Thanks for any response.
(Presumptive) Source: Theoretical Exerci... | This is equivalent to proving $$\sum_{k=0}^n k^2 {n\choose k}=n(n+1)2^{n-2}.$$ Given $n$ people we can form a committee of $k$ people in ${n\choose k}$ ways. Once the committee is formed we can pick a committee leader and a committee planner. If we allow each person to hold both job titles there are $k$ ways for this t... | Dilip's comment points to the standard way. Here goes an alternative way:
If you are familiar with basic probability, you should recognize a canonical Binomial distribution in $$ \frac{1}{2^n} {n \choose k}$$
which count the number the successes in $n$ trials of an experiment with prob. $=1/2$ . This Binomial can als... | https://math.stackexchange.com |
347,102 | [
"https://softwareengineering.stackexchange.com/questions/347102",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/213138/"
] | I'm working with an external system which sends event-like data to us. An existing SDK from the third-party guys parses the JSON so in the end we get something like the following class:
<pre><code>public class Event
{
public string Type;
public dynamic Data;
}
</code></pre>
The requirements are as follows:
... | There's probably others, but the three obvious solutions I can think of are:
<ol>
<li>Use a <code>Dictionary<string, HanderDelegate></code> map,</li>
<li>Use a switch statement to map the "event" names to their handlers,</li>
<li>Use OO polymorphism (one class per event type) and some sort of factory.</li>
</ol>... | <strong>Your proposed design is good.</strong> I see no reason to object to it. There are possible alternative designs, but I can't think of anything decisively better.
If you have three types of events that are all handled by one handler, that information has to be encoded somewhere in any design. It might feel li... | https://softwareengineering.stackexchange.com |
281,146 | [
"https://stats.stackexchange.com/questions/281146",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/60537/"
] | Consider the weighted least square (WLS) regression estimate
$\beta_n=(X'\Omega^{-1}X)^{-1}X'\Omega^{-1}Y$
where $X,Y,\Omega$ has usual interpretation and $\Omega$ is known.
The under usual assumption $\sqrt{n}(\beta_n-\beta)\rightarrow N(0,1)$.
Now let us assume that $\Omega$ is unknown and estimated by a consist... | Although not an expert in Econometrics I would suggest the following. From simple reformulation, we can arrive at
$$\sqrt{n}(\hat{\beta}_n - \beta)=(n^{-1}X'\hat{\Omega}^{-1}X)^{-1}n^{-1}X'\hat{\Omega}^{-1}\sqrt{n}u .$$
If we can assume that $$(n^{-1}X'\hat{\Omega}^{-1}X)^{-1} $$ converges in probability, by applying... | $\hat{\beta}_n = \beta + [X'\hat{\Omega}^{-1}X]^{-1}X'\hat{\Omega}^{-1} \epsilon$, with $\epsilon = Y - X\beta$
To show that $\hat{\beta}_n$ and ${\beta}_n$ share the same asymptotic distribution, we need to show that
$[X'\hat{\Omega}^{-1}X]^{-1}X'\hat{\Omega}^{-1} \epsilon= [X'{\Omega}^{-1}X]^{-1}X'{\Omega}^{-1} \ep... | https://stats.stackexchange.com |
30,261 | [
"https://electronics.stackexchange.com/questions/30261",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/-1/"
] | Perhaps a Simple Question, but maybe Im misunderstanding "Voltage" drop in terms of LEDS.
So I have a Circuit wired up with a 9V Battery, one 270ohm Resistor and 3 Blue LEDS (that draw around....20ma and 2.0v forward voltage. (this is for testing purposes, I would always put a resistor with each LED)
I put in 1 Led a... | You're getting the expected result. What you see is the normal behavior of diodes in series and it's completely normal to have one resistor and a string of LEDs connected after it.
What's basically happening is this: When they told you that the forward voltage is 2 V, they lied. It actually depends on the current goi... | It's a little hard to tell from the photo, but it looks like you have the LEDs in parallel, rather than series. This is the difference.
<img src="https://i.stack.imgur.com/CTBeN.png" alt="Series or Parallel">
I'm sorry if I miss read your photo, and this is patronising. You should connect the LEDs in series to work ... | https://electronics.stackexchange.com |
352,060 | [
"https://physics.stackexchange.com/questions/352060",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/75734/"
] | Upon grabbing a glass with your fingers, the downward motion of the glass is halted by the frictional force, which is directed upward and related to the amount of force your fingers exert toward the glass (barely gripping a cold glass of water on a hot summer day will not be sufficient to prevent the glass from falling... | You'll probably be wanting to use the equation $$pV = nRT$$ where $p$ is the pressure, $V$ is the volume, $n$ is the number of moles $R$ is the molar gas
constant $(8.31)$ and $T$ is the temperature in Kelvin. You can easily rearrange this equation to get $$p = \frac{nRT}{V}$$ I'm a bit confused as to why you've broug... | Hydrogen has a critical temperature of 33K. So you can't get a true liquid simply by pressurizing room temperature hydrogen. It would be a supercritical fluid. You can squeeze more into a container with higher pressure, but it won't be a liquid.
Liquification can make transporting some materials easier (like propan... | https://physics.stackexchange.com |
9,672 | [
"https://chemistry.stackexchange.com/questions/9672",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/2595/"
] | If I test a pure solution of an acid chloride with a pH-meter, will I get a reading of pH 7?
When I add water to it, I know $\ce{HCl}$ gas will form, but this gas will dissolve in water to give an acidic solution.
My main confusion is that we call ethanoyl chloride an "acid" chloride, but is its pure solution acidic?... | An acid chloride is a covalent molecule. It has no pH. If you put a pH electrode into a reactive acid chloride you will dehydrate and acylate its hydrated glass surface ($\ce{#Si-OH}$) and damage its future response when properly used. pH electrodes are stored sitting in buffer or saline solution.
An acid chloride ... | pH-values only truely make sense in a solution whose solvent has acidic (as in ‘abstractable’) hydrogens. Apart from the obvious water this includes alcohols, enoliseable ketones and more. This means that a pure acid chloride cannot have a pH value, because there is no $\ce{[H+]}$ around.
However, all those solvents t... | https://chemistry.stackexchange.com |
44,854 | [
"https://dba.stackexchange.com/questions/44854",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/25142/"
] | I'm making a database that needs to have the following tables: Article, Video, Image, and Keyword. Keyword contains a fixed list of keywords. (For example when creating an article, you'd have to select from a list of keywords, not create a new one) Each Article, Image, and Video can have n number of keywords. I'm havin... | Something like:
<pre><code>Article ArticleKeyword Keyword
--------------- --------------- ------------
article_id (PK) <--- article_id (FK)
keyword_id (FK) ---> keyword_id (PK)
</code></pre>
There could be zero-to-many ArticleKeyword rows f... | Yes, in order to do a <code>many to many</code> relationship you have to have a join table. Basically your join table has two <code>one to many</code> relationships. One to each of the parent tables. This creates in essence a <code>many to many</code> relationship between the two parent tables.
<pre><code>Article... | https://dba.stackexchange.com |
15,619 | [
"https://dba.stackexchange.com/questions/15619",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/7684/"
] | College SQL class, using the book "SQL Fundamentals" by John J. Patrick. In the third chapter, he talks about using a "table of constants" to add columns to a select statement, where all rows have the same value.
For example, if you have table "characters", as so:
<pre><code>first_name last_name dept_code
--------... | Joe Celko mentions tables of constants in a couple of his books.
He suggests if using a table that a check constraint is added that ensures the table can contain no more than one row.
<pre><code>CREATE TABLE Constants
(
lock CHAR(1) DEFAULT 'X' NOT NULL PRIMARY KEY CHECK (lock = 'X'),
pi FLOAT DEFAULT 3.142592653 N... | In my experience, I have rarely developed code where a string constant is added to the result set. The proposed solution doesn't make sense to me. I would be highly suspicious of the code quality of a system that did this. Looking at this example I would code it as follows.
<pre><code>first_name last_name dept_co... | https://dba.stackexchange.com |
3,657,747 | [
"https://math.stackexchange.com/questions/3657747",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/574299/"
] | Question from a textbook.
Show that for non negative <span class="math-container">$x,y,z$</span> that
<span class="math-container">$$ (x+y+z) \sqrt{2} \leq \sqrt{x^2 + y^2} + \sqrt{ y^2 + z^2} + \sqrt{x^2 +z^2} $$</span>
and that for <span class="math-container">$ 0< x \leq y \leq z$</span>,
<span class="math... | Hint: <span class="math-container">$a)$</span> use <span class="math-container">$\sqrt{\frac{a^2+b^2}{2}} \ge \dfrac{a+b}{2}$</span> , add them up ( <span class="math-container">$3$</span> of them ). And
<span class="math-container">$b)$</span> the triangle inequality applies to <span class="math-container">$3$</span>... | <ol>
<li><span class="math-container">$(\sqrt{x^2 + y^2} + \sqrt{ y^2 + z^2} + \sqrt{x^2 +z^2} )^2= 2(x^2 + y^2+ z^2) + \;\;non-negative\;\; terms\geq 2(x^2 + y^2+ z^2)$</span> </li>
</ol>
which yields the first inequality.
<ol start="2">
<li><span class="math-container">$(y-x)^2 + (z-x)^2=y^2+z^2+2x^2-2x(y+z)\geq y^... | https://math.stackexchange.com |
41,740 | [
"https://mechanics.stackexchange.com/questions/41740",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/25749/"
] | I have a VW Polo 2003 1.2 and would like to add a speaker to each rear door. I am a complete newbie when it comes to electrical work in cars (actually in most things about cars other than the important simple stuff I suppose). Currently the car has 2 front speakers. I have taken apart one of the rear door trim panels a... | Because it wouldn't be very efficient.
The main advantage of internal combustion engines is that the energy density of their fuels (gasoline, diesel) is very good. You can go a long way on a relatively small, light tank. The drawback of them is that they aren't very efficient. Most of the energy in the fuel is lost to... | Although other answers provide good answers related to energy losses, there's an additional thing to consider.
How on earth are you planning to heat the air in a tiny cylinder at many kilowatts? This is required, because a typical four-cylinder car engine produces 100 kilowatts of power, and the efficiency is probably... | https://mechanics.stackexchange.com |
84,040 | [
"https://math.stackexchange.com/questions/84040",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/8955/"
] | <em>(I don't think that this is a good fit on cstheory, since I figure that this question already has a known answer. However, if you think that this would be a better fit there, please feel free to migrate it)</em>
It can easily be shown that REGULAR (the class of regular languages) is contained in DTIME(O(n)), beca... | For the benefit of the readers, here is an outline of the proof that an $o(n\log n)$-time TM accepts a regular language.
Consider the computation of the TM on some input $x$. Write $x$ as follows:
$$ |x_0|x_1|\cdots|x_n|. $$
Each $|$ signifies a boundary between two cells. The <em>crossing number</em> at a boundary on... | When you get below general polynomial time, you need to be careful which computation model we're working with -- a RAM can achieve much more in $O(n)$ time than a Turing machine can.
Anyway, with a "realistic" model, DTIME($O(n)$) can recognize much more than regular languages, such as $\{a^nb^n\mid n\in\mathbb N\}$ o... | https://math.stackexchange.com |
264,789 | [
"https://physics.stackexchange.com/questions/264789",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/100426/"
] | Can we derive Biot Savart law from Coulomb's law by setting a current source at an instant in place of one of the charges? Let's say $dl$ is the length of the conductor having $dQ$ charge which applies some force at the other charge at some instant of time. Taking derivates on both sides we get some current term. But h... | you are basically trying to undergo a transition from a law which is valid for static charges(or non relativistic speeds) to one which is valid for steady currents. That is why, simple differentiation is erroneous and does not include any magnetic field term in dE/dt.
| I am sorry that people always say no rather than I don't know. The comment of Ananyo is correct, like v^B produces E, v^E produces B, and you have on the left hand dl/dt which is v. So you are correct only needing some refinement in reasoning and what each variable means. A further indication that you are correct is th... | https://physics.stackexchange.com |
271,406 | [
"https://electronics.stackexchange.com/questions/271406",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/103685/"
] | I am building a simple voltmeter circuit that needs to measure up to about 35 volts. Looking around the web, I am finding all sorts of suggestions for voltage divider values, like 1 Mohm / 10 Kohm, and the one I am currently experimenting with, which is 10K/2K. I am having issues with allowing enough time between rea... | There are two things to consider:
<ol>
<li>The ratio of values</li>
<li>The input impedance of the ADC</li>
</ol>
Point 1 has hard and fast formulae. No rule of thumb here.
Point 2 is a little more "thumby". You want R2 (the lower resistor) to be low enough that the input impedance of the ADC doesn't skew the readin... | The ratio is whatever you need it to be to give the division you want.
The absolute values it depends on your application.
Your ADC input should have an effective resistance of 10+Mohms. You need the resistor in the lower half of your divider to be small in comparison to this otherwise the ADC input impedance is goin... | https://electronics.stackexchange.com |
267,477 | [
"https://physics.stackexchange.com/questions/267477",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/123308/"
] | One has, let's say, the inertia matrix of an object about it's center of gravity: $I_p$ in the coordinate system ($i, j, k$). What is the inertia matrix in another coordinate system, say ($i_1,j_1, k_1$) with $$ \begin{bmatrix} i \\ j\\ k\end{bmatrix} = A \cdot \begin{bmatrix} i_1 \\ j_1 \\ k_1\end{bmatrix}$$
| In this case it's probably best to be pragmatic. A pulse can be <em>described</em> as a superposition of sine waves that extend infinitely into space and time. But it's just that: a mathematical description that is useful for your purposes. There is not necessarily a physical meaning connected to it. Nevertheless, in q... | I just wanted to add to a previous (very accurate) answer: you can think of it as an Fourier expansion of the actual (physical) wave profile. It is not a real life process, it is a mathematical approximation. The wave pulse <em>can be thought of</em> as a superposition of plane waves, which happens to interfere destruc... | https://physics.stackexchange.com |
34,936 | [
"https://dsp.stackexchange.com/questions/34936",
"https://dsp.stackexchange.com",
"https://dsp.stackexchange.com/users/4391/"
] | Is it possible to calculate the maximum value of a time-domain signal from frequency-domain representation <em>without</em> performing an inverse transform?
| Suppose that Alice has a vector $\mathrm x \in \mathbb R^n$. She computes the DFT of $\mathrm x$
$$\mathrm y := \mathrm F \mathrm x \in \mathbb C^n$$
where $\mathrm F \in \mathbb C^{n \times n}$ is a Fourier matrix. Alice then tells Bob what $\mathrm y$ is. Since the inverse of the Fourier matrix is $\mathrm F^{-1}... | It's generally not possible to compute the exact maximum value, but you can compute a bound on the maximum value. Assuming your data are discrete-time, and you're using the discrete Fourier transform (DFT), you have the following relation between time domain and frequency domain:
$$x[n]=\frac{1}{N}\sum_{n=0}^{N-1}X[k]... | https://dsp.stackexchange.com |
81,690 | [
"https://mathoverflow.net/questions/81690",
"https://mathoverflow.net",
"https://mathoverflow.net/users/11100/"
] | Three points $z_1$, $z_2$, $z_3$ on the complex plane are given by the coefficients $a_k$'s of the cubic polynomial $f(z)=(z-z_1)(z-z_2)(z-z_3)=\sum_{k=0}^3 a_k z^k$. How does one express the (signed) area $V$ of the triangle with vertices $z_1$, $z_2$, $z_3$ in terms of $a_k$'s and $\overline{a}_k$'s? One is tempted ... | NB: Note that my $a_k$ have different signs from those defined in the question. For me,
$$
(z - z_1)(z-z_2)(z-z_3) = z^3 - a_1\ z^2 + a_2\ z - a_3,
$$
so that $a_k$ is the $k$-th elementary symmetric function of the $z_i$. This doesn't really affect the answer in any significant way.
While I don't think that the fi... | Not quite an answer, but a cute fact: if the three points (I use $x, y, z$ below, for typing convenience) are all on the unit circle, then the square of the area equals (using the OP's formula, and the observation that in this case $\overline{z} = 1/z$)
$-\dfrac{(x-y)^2 (x-z)^2 (y-z)^2}{16x^2y^2z^2}.$
The numerator i... | https://mathoverflow.net |
178,845 | [
"https://mathoverflow.net/questions/178845",
"https://mathoverflow.net",
"https://mathoverflow.net/users/36688/"
] | Let $H$ be a separable Hilbert space and $A$ is an invertible bounded operator on $H$. Can we approximate $A$ with an invertible operator $B$ such that $sp(B)$ is a countable set?
<strong>Motivation:</strong>
If the answer is yes, this would give's us an alternative proof of connected ness of $GL(H)$. Th... | I think the answer is "no"; here is a sketch of an argument though I will have to go back and check the details. First, there is nothing special about invertibility here; if every invertible operator is approximable by operators with countable spectrum, then every operator is, just by translation. I claim the unilatera... | The question has already been answered above (by Mike Jury). Here is another way of arguing: Any operator with countable spectrum is in the closure of the invertible operators. So are their limits. But again, the unilateral shift provides an example of an operator not in the closure of the invertibles. For if $\|S-X\|&... | https://mathoverflow.net |
418,936 | [
"https://electronics.stackexchange.com/questions/418936",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/210989/"
] | The Rds(on) of a MOSFET decreases when you apply higher gate voltage, but does it typically vary based on the source-drain voltage?
I will be using a MOSFET to switch very low voltages (10's to 1000's of µV) and was wondering if normal MOSFETs will work as long as the gate voltage is high or if the low source-drain vo... | The answer to your question <em>body</em> is "not significantly". The answer to your question <em>title</em> is "yes" -- but unless you have a really truly oddball FET, or you need insane accuracy, you don't need to worry.
RDS on is a parameter of a FET that's operating at low source to drain currents. It <em>specif... | The drain to source voltage if you are using the MOS as a switch would be very low, so it won't affect the Rds on. If you are only switching voltages in the range of 10 to 1000 uV then the body bias or back gate effect would be less, so the threshold of MOS won't vary much, hence the low Rds on.
| https://electronics.stackexchange.com |
207,764 | [
"https://electronics.stackexchange.com/questions/207764",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/6659/"
] | If a device "D" requires X watts power, and I have a power source Y that is less than the nominal voltage recommended by the device "D", can I obtain the same performance for the device by somehow boosting the current such that Y·(boosted current) = P?
I know this is a very simplified question that doesn't take alot o... | Device D needs X watts so providing source Y can provide a 10% more power than device D needs you can use a boost regulator to step up the voltage to suit the voltage required by device D.
The 10% more power accounts for the losses in a boost switching regulator. In other words if Y can supply a bit more power you can... | Along with P = V * I you also need Ohm's Law which states that V = I * R. In the case of a resistive load the voltage and current are proportional. The only way to increase the current is to increase the voltage.
The answer, therefore, is no.
Note that since V and I are proportional that power changes with <em>the sq... | https://electronics.stackexchange.com |
58,314 | [
"https://mechanics.stackexchange.com/questions/58314",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/25547/"
] | While I have a good knowledge on general service procedures for a vehicles engine, suspension, exhaust and other usual systems, I'm unsure of how to keep a vehicles electrical system in top-notch.
My question is, does the electrical system require routine maintenance, and if so, what tasks would it involve?
| Well, not much, but things to check include:
1) battery terminals - clean, tight and some protection ie petroleum jelly works well on lead/acid batteries
2) all cables (easily visible) are not rubbing or chafing anywhere
3) all bulbs / lights function.
Used to have distributors on the list and brush height & re... | Provided you have the required tools, I would add:
4) Load testing the battery<br>
5) Testing alternator output
and no tools required...
6) Making sure the top and sides of the battery itself are clean to prevent stray voltage loss
| https://mechanics.stackexchange.com |
152,621 | [
"https://dba.stackexchange.com/questions/152621",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/108361/"
] | Is it possible to compare permissions of two users on one SQL Server database?<br>
Is there any simple query to do it?
I am asking this because I want to find out if user B has different or same level of permissions as user A on a database.
| With help of colleague I got answer on my question.<br>
<em>EDIT:<br>
Forgot to mention, we had to check first if two or more users have server side permissions over database and than to check did anyone get permissions on database itself.</em>
<pre><code>--filter query
declare @filter nvarchar(200);
declare @DBLevel... | Here are two queries I have used to compare permissions between database users.
The first shows the database roles (if any) to which the database user has membership. The second shows individual GRANT and DENY permissions.
<pre><code>--Database user and role memberships (if any).
SELECT u.name, r.name
FROM sys.databa... | https://dba.stackexchange.com |
35,956 | [
"https://stats.stackexchange.com/questions/35956",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/11032/"
] | Taleb's book "The Black Swan" was a New York Times best seller when it came out several years ago. The book is now in its second edition. After meeting with statisticians at a JSM (an annual statistical conference), Taleb toned down his criticism of statistics somewhat. But the thrust of the book is that statistics ... | I read the Black Swan a couple of years ago. The Black Swan idea is good and the attack on the ludic fallacy (seeing things as though they are dice games, with knowable probabilities) is good but statistics is outrageously misrepresented, with the central problem being the wrong claim that all statistics falls apart i... | I've not read the book, but as stated the criticism seems pretty unreasonable to me. If extreme events are important, then statistics has appropriate tools in the toolbox, such as extreme value theory, and a good statistician will know how to use them (or at least find out how to use them and will be sufficiently enga... | https://stats.stackexchange.com |
3,005,589 | [
"https://math.stackexchange.com/questions/3005589",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/326865/"
] | Let <span class="math-container">$R$</span> be a <em>commutative</em> ring with <span class="math-container">$1_R$</span>.
I found the following theorem as an equivalent definition of <span class="math-container">$R-$</span>modules.
<blockquote>
<strong>Theorem.</strong> An abelian group <span class="math-container... | An abelian group <span class="math-container">$M$</span> can be given the structure of <span class="math-container">$R$</span>-module if and only if there exists a ring homomorphism
<span class="math-container">$$
\varphi\colon R\to\operatorname{End}_{\mathbb{Z}}(M)
$$</span>
If <span class="math-container">$M$</span> ... | <em>Let me give a try to answer and improve the theorem. Please feel free to edit my answer.</em>
<strong>Theorem.</strong> Let <span class="math-container">$R$</span> be a ring with <span class="math-container">$1_R$</span> and <span class="math-container">$(M,+)$</span> an abelian group (<span class="math-container"... | https://math.stackexchange.com |
116,082 | [
"https://mathoverflow.net/questions/116082",
"https://mathoverflow.net",
"https://mathoverflow.net/users/29853/"
] | I am reading "Category Theory" (2nd ed.) of Awodey, and I'm stuck at page 96 (proposition 5.12) when pullbacks are presented as functors:
The pullback under question corresponds to this square:
<span class="math-container">$$\begin{matrix}
C' \times_C A & \xrightarrow{h'} & A \\[1ex]
\downarrow \rlap... | Let us first consider a slightly simpler situation. The cartesian product of sets $A$ and $B$ is a set $C$ with two maps $p_1 : C \to A$ and $p_2 : C \to B$ such that ... (familiar condition inserted here). All cartesian products of $A$ and $B$ are canonically isomorphic, and among them there is a particular one, denot... | Actually, it looks to me that there's a conflation of two different issues. According to the boxed statement in the OP, we just have to exhibit a functor $h^\ast: \mathbf{C}/C \to \mathbf{C}/C'$ for a fixed morphism $h: C' \to C$. We are <b>not</b> being asked to prove that we can choose a strict functor (as opposed to... | https://mathoverflow.net |
417,810 | [
"https://electronics.stackexchange.com/questions/417810",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/210376/"
] | Pardon the very simple question, but I am a beginner looking to buy my first good solder for through hole work. I was looking at a 63/37 Kester 285 or AIM RMA, but one thing I’m curious about is the core size. Most of the products I’m looking at have anything from a 1.8 to 3.3% core, but I’m not sure what I should be... | <blockquote>
[From Question]: MOSFET transistor and normal transistor
</blockquote>
A MOSFET <strong>is</strong> a normal transistor - it's a Metal-Oxide-Semiconductor Field-Effect <em>Transistor</em>
A BJT is another normal transistor - a Bipolar Junction <em>Transistor</em>
The question itself doesn't make much ... | Bipolars operate by injecting small currents (electrons) into the base pin, and the emitter injects currents that attempt to cancel out the (opposite polarity) base currents. Because the BASE region is very thin, most of the emitter-injected charges miss their targets and end up gathered up in the Collector region of t... | https://electronics.stackexchange.com |
2,960,381 | [
"https://math.stackexchange.com/questions/2960381",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/268538/"
] | When I learned about BODMAS / PEMDAS rules in mathematics. I was told to follow certain rules.
For example if I have this expression: <span class="math-container">$3 − 2 + 4 − 1$</span> and to solve it <strong>I was told to always go from left to right</strong> because addition and subtraction in this case have the sa... | You may rearrange the terms how you wish before actually adding and subtracting without changing the value (assuming you do it the right way). That's not what "go from left to right" is about.
Subtraction is not an associative operation, and it doesn't associate with addition either. That means that given an expressio... | Depends on how it is taught, the relationship between minus and negative might not be obvious to beginners. The notion of negative usually comes much later after subtraction. That is <span class="math-container">$$a-b = a+(-b)$$</span>
Hence upon giving <span class="math-container">$a-b+c$</span>, some might misinterp... | https://math.stackexchange.com |
147,940 | [
"https://softwareengineering.stackexchange.com/questions/147940",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/10199/"
] | My boss wants to introduce new guidelines for the computers in our company. This means that the software environments will be completely standardized with a predefined set of tools. The goal is that all machines are completely reproducable and exchangable.
I'm wondering what the guidelines for developer's computers ar... | You're really asking several separate, but related, questions here.
<h2>Hardware</h2>
First: every company I've ever worked for had some kind of standardization of desktop machines. At a minimum, the hardware and OS were standardized across large groups (possibly in rolling categories: you get this year's machine when... | Even standardization for developers is crazy talk. I have never worked anywhere where everyone users the exact same set of tools. Is he going to buy 100 seats of Photoshop because the 2 guys on the graphics team need it?
What is going to happen is you will need some tool specific to your task on the project. They w... | https://softwareengineering.stackexchange.com |
563,287 | [
"https://physics.stackexchange.com/questions/563287",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/217974/"
] | First of all, I know that any math (or computation) a classical computer can do, we human can do with pen and paper at same type of efficiency. Obviously, that is because we tell a classical computer what to do steps by steps, and we are capable of completing each step.
My question is whether it is true with quantum co... | No, you actually can't.
<ol>
<li>First of all we have no idea whether something the size of a neuron can actually be in a superposition or interfere (essential for quantum computers). Even if they do, the effect is going to be very small, and it won't work for quantum computing. Qubits are made of electrons and photons... | Penrose notably thinks so, most don't.
Here is Max Tegmark's reasoning against it from Our Mathematical Universe pg. 207:
"an individual neuron must be able to be in a superposition of firing and not firing...so how long could a neuron keep secret whether it was firing or not?...10^-20 seconds"
He then uses a... | https://physics.stackexchange.com |
47,544 | [
"https://chemistry.stackexchange.com/questions/47544",
"https://chemistry.stackexchange.com",
"https://chemistry.stackexchange.com/users/-1/"
] | <blockquote>
How would Dalton's law be affected when there are two ideal gases in a
container at different temperatures?
</blockquote>
Let the gas with higher temperature be gas A and the gas with lower temperature be gas B. Then heat will be transferred from gas A to gas B due to which kinetic energy of the molec... | You cannot have two IDEAL gasses in a homogeneous mixture at different temperatures. As the collisions are stipulated to be non-elastic, all the kinetic energy of the particles (temperature) will be instantly transferred and spread out so that all the gas particles follow the Boltzmann distribution.
| Determine total number of moles of air in bulbs:
$$n=\frac{PV}{RT}=\frac{(1)(0.4+0.2)}{(0.082)(293)}=0.025$$
Consider final state of system. Let $P_F$ be the final equilibrium pressure, $n_1$ be the final number of moles in the larger bulb, and $n_2$ be the final number of moles in the smaller bulb:
$$n_1=\frac{(P_F)... | https://chemistry.stackexchange.com |
590,470 | [
"https://stats.stackexchange.com/questions/590470",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/355736/"
] | I'm reading a pdf document about simple linear regression, and it gave an expression of the least square estimator <span class="math-container">$\hat{\beta}_1 = \beta_1 + \frac{\sum^n_{i=0} \epsilon_i (x_i - \bar{x})}{\sum^n_{i=0} (x_i - \bar{x})^2}$</span>, but I'm confused on how they get to this answer.
I only know ... | This is essentially an algebraic exercise. Noting the <span class="math-container">$y_i = \beta_0 + \beta_1 x_i + \epsilon_i$</span> you then have:
<span class="math-container">$$\begin{align}
\sum_i y_i(x_i - \bar{x})
&= \sum_i (\beta_0 + \beta_1 x_i + \epsilon_i)(x_i - \bar{x}) \\[6pt]
&= \beta_0 \sum_i (x_i... | The least squares normal equations are
<span class="math-container">$$\mathbf X^\mathsf T\mathbf X\mathbf b = \mathbf X^\mathsf T\mathbf y \tag 1$$</span> and thus <span class="math-container">$$\text{(OLS)}: ~~\mathbf b =\left(\mathbf X^\mathsf T\mathbf X\right)^{-1}\mathbf X^\mathsf T\mathbf y. \tag 2\label 2$$</span... | https://stats.stackexchange.com |
267,873 | [
"https://security.stackexchange.com/questions/267873",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/288306/"
] | Given the hypothetical, you have digital evidence that state actors want to delete. Assuming two scenarios
<strong>Scenario #1</strong> - The state actors are not the agencies themselves, but bad actors within the agencies that have access to the tools. This eliminates the risk of warrants and a national security lette... | What about the good old DVD? Might have gone out of fashion nowadays but is has more than enough space to hold 2GB of data and it is downright impossible to delete data from it without physical access (a.k.a physically destroying the disc). Spread multiple discs over multiple locations to lower that risk.
Upload DVD co... | Answering my own question as a starting point
<ol>
<li>Buy a new laptop, 4 flash drives</li>
<li>Find a computer technician and request to create a new bootable drive from one of the flash drives</li>
<li>For the following steps, drive to a remote location without any wifi signal and keep computer fully offline.</li>
<... | https://security.stackexchange.com |
26,126 | [
"https://ai.stackexchange.com/questions/26126",
"https://ai.stackexchange.com",
"https://ai.stackexchange.com/users/44278/"
] | In Q-learning, all resources I've found seem to say that the algorithm to update the Q-table should start at some initial state, and pick actions (which are sometimes random) to explore the state space.
However, wouldn't it be better/faster/more thorough to simply iterate through all possible states? This would ensure ... | If your algorithm is executed multiple (or enough) times using an outer loop, it would converge to similar results as Q-learning would with <span class="math-container">$\gamma = 0$</span> (as you don't look what is the expected future reward).
In this case, the difference is that you would pass as much time to explore... | In short, yes, provided that you have a small number of states.
In pretty much any real system, the number of states is much higher than you could ever hope to explore exhaustively in any reasonable time. This is why you need to set some sort of exploration/exploitation policy to make sure that you mostly visit promisi... | https://ai.stackexchange.com |
2,763,523 | [
"https://math.stackexchange.com/questions/2763523",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/448693/"
] | Edit: I goofed up. Need to show that If $$(a^m+b^m)^{n}= (a^n+b^n)^m$$ then $m=n$ where $a,b$ are fixed positive reals and $m,n\in\mathbb{N}
$
| Factorizing by $a$ you obtain:
$$\left(1+\left(\frac{a}{b} \right)^n \right)^{m}=\left(1+\left(\frac{a}{b} \right)^m \right)^{n}$$
with $x=\left(\frac{a}{b} \right)^n$ this is the same as:
$$\left(1+x \right)^{m}=\left(1+x^\frac{m}{n}\right)^{n}$$
i.e
$$\left(1+x \right)^\frac{m}{n}=\left(1+x^\frac{m}{n}\right)$$
so th... | Counter example:
Take $a = b = m = 1$ and $n = 2$
Then
LHS is $4$ and RHS is $2$.
| https://math.stackexchange.com |
62,476 | [
"https://security.stackexchange.com/questions/62476",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/50410/"
] | The title is really bad, but I was able to find better. I have installed Debian on VMware, and set iptables to block any traffic except port 80 (inbound and outbound). When I run a nmap scan it shows that this port is filtered. Should it be this why or not?
| When nmap scans a TCP port (e.g. TCP/80 for HTTP traffic) a filtered response means that nmap did not get any response to the packet it sent. The other options for TCP ports are "closed" which means that in response to the SYN packet nmap sent, the host sent a RST packet (essentially indicating that there is no servic... | Filtered is a loose term when it comes to nmap, in most cases it refers to a firewall guarding it. In some cases these protections may be circumvented. In your case I think it's safe to say that if your config is right, you're safe (on ports other than 80)
| https://security.stackexchange.com |
589,620 | [
"https://electronics.stackexchange.com/questions/589620",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/297391/"
] | While preparing a measuring setup for a project, I've realized that the multimeter our lab can provide only supports resistance measurements up to 10^8 Ohm. Since the materials we want to measure go way beyond that, I intended to connect the material in parallel to another resistance and simply add one calculation step... | If the things you want to measure have a resistance that goes way beyond what you can measure with a DMM, then paralleling a shunt resistor to bring it into range will rapidly turn small measurement errors into big device under test errors.
You would do better to go back to first principles and devise your own measurem... | Lets do the math. A 1 GΩ resistor in parallel to a 100 MΩ resistor, the result is 90.90 MΩ. A 2 GΩ resistor instead is 95.238 MΩ.
A large change of the GΩ resistor will result in a small change of the parallel connection. A precise measurement of the GΩ resistors is impossible this way.
| https://electronics.stackexchange.com |
10,962 | [
"https://electronics.stackexchange.com/questions/10962",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/2028/"
] | What is the difference between "forward" and "reverse" voltages when working with diodes and LEDs?
I realize this question is answered elsewhere on the interwebs such as wikipedia, but I am looking for a short summary that is less of a technical discussion and more a useful tip to someone using diodes in a hobby circu... | The <strong>forward voltage</strong> is the voltage drop across the diode if the voltage at the anode is more positive than the voltage at the cathode (if you connect + to the anode).
You will be using this value to calculate the power dissipation of the diode and the voltage after the diode.
The <strong>reverse volt... | Forward-bias is when the anode (the pointy part of the symbol) is positive and the cathode (the bar) is negative. Reverse-bias is when the anode is negative and the cathode is positive. A lot of current flows when the diode is forward-biased, provided that the voltage is higher than 0.6V or so for a silicon diode or 0.... | https://electronics.stackexchange.com |
143,795 | [
"https://softwareengineering.stackexchange.com/questions/143795",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/33514/"
] | <h2>A quick introduction</h2>
My college semesters include a 8 week project working for an actual company with a software need in order to get some much needed practical experience. I have just started such a project with 5 other students. We're required to spend roughly 40 hours a week per student on this project. We... | For the specific project there's not much you can do until you find out what the client wants. However, there are some things you can do now so your team is ready to go.
<ol>
<li>How are you going to handle version control?</li>
<li>Will you do code reviews?</li>
<li>When will the daily standup meetings be held? What ... | <ol>
<li>Set up your version control system and document its configuration</li>
<li>Set up an automated build system</li>
<li>Set up automated Unit tests that integrate with the build system</li>
<li>Set up a web server to serve up static content like automatically generated docs and reports from your Continuous Integr... | https://softwareengineering.stackexchange.com |
71,781 | [
"https://cs.stackexchange.com/questions/71781",
"https://cs.stackexchange.com",
"https://cs.stackexchange.com/users/68046/"
] | I'm attempting to prove a problem is NPc, but I'm not sure which one would be optimal to use,
The problem is:
There are $n$ boars to be caged, and $m$ cages which each cage being able to hold $k$ boars.
Any boar can be put in in any cage, but certain pairs of boars can't be put together in the same cage.
I'm think... | First of all, this is a math question.
I think you are confused on how brackets are used. They indicate <em>precedence</em> of operations, and can be used <em>anywhere</em>, even in places where such indication is not necessary. For example,
$$3 \times 5 + 8$$
and
$$(3 \times 5) + 8$$
are both legitimate expressions a... | Apply the distributive law on <span class="math-container">$x\land(1\lor y)$</span> and see what you get.
| https://cs.stackexchange.com |
170,369 | [
"https://math.stackexchange.com/questions/170369",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/21373/"
] | Say $X$ ~ $Beta(\alpha,\beta)$. I want to prove the following,
$$4E(X)E(X^2)-2E(X^2)^2-4E(X)^2+2E(X^4)-4E(X^3)+4E(X^2)>0$$
Is there a simple way?
| <ul>
<li>$4E(X^2) - 4E(X)^2 = 4\cdot\text{var}(X) \geq 0$</li>
<li>$2E(X^4) -2E(X^2)^2 = 2\left[E((X^2)^2) - (E(X^2))^2\right]
= 2\cdot\text{var}(X^2) \geq 0$</li>
<li>$4E(X^3) - 4E(X)E(X^2) = 4\cdot\text{cov}(X,X^2)$</li>
</ul>
So, writing $Y = X^2$, your expression is
$$\begin{align*}
4\cdot\text{var}(X) + 2\cdot\t... | The $j$'th moment of the Beta distribution is
$$E[X^j] = \dfrac{\alpha(\alpha+1)\ldots(\alpha+j-1)}{(\alpha+\beta)(\alpha+\beta+1)\ldots(\alpha+\beta+j-1)}
$$
Expand out
$ 4E(X)E(X^2)-2E(X^2)^2-4E(X)^2+2E(X^4)-4E(X^3)+4E(X^2)$
and you get a rather complicated expression in $\alpha$ and $\beta$ whose numerator and den... | https://math.stackexchange.com |
297,959 | [
"https://dba.stackexchange.com/questions/297959",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/236915/"
] | Let's say I have a bunch of trains and each train has a schedule. A schedule is basically a list of stops and some metadata. So in JSON I would have something like
<pre><code>{
"train_no": 12345,
"schedule": [
{"stop_id": 54321, "station": "foo"},
... | If you are happy with this kind of implementation of schedule replacement, where the old schedule is gone forever, then I can see no problem with – nor, indeed, a better option other than – doing the replacement in two steps: first deleting the row with the old schedule and then inserting a new one with the new schedul... | I think ist os best to add to more fileds.
<ol>
<li>Delayed
here you can the minutes ot a time that is expected, that is btter because the people still can search for start and end and know that it is delayed and how much
</li>
<li>Canceledd
here also, rthe people can still search and know, that it is canceled
</li>
</... | https://dba.stackexchange.com |
129,963 | [
"https://stats.stackexchange.com/questions/129963",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/23946/"
] | I have a problem where I have sum of two random variables
1). Each distributed independently normally with different means ($\mu_1$, $\mu_2$) and sds ($\sigma_1$, $\sigma_2$). $Z=R_1+R_2$
2). Each distributed independently log-normally with different means ($u_1$, $u_2$) and sds ($s_1$, $s_2$). $LZ=LR_1+LR_2$
In th... | Since the sum of two independent normals is normal, you wouldn't get a uniform (as in your original title) by applying a logistic transformation on the sum.
<ul>
<li>If the means and variances are known (as your question implies), you could get a uniform result by using the normal cdf of the sum (mean is the sum of m... | If it's just a matter of doing the simulation, you can, in the normal case, do:
<pre><code>mu1 <- 0
mu2 <- 1
sd1 <- 1
sd2 <- 3
N <- 10^4
z <- rnorm(N, mean = mu1, sd = sd1) + rnorm(N, mean = mu2, sd = sd2)
x <- rbinom(N, size = 1, prob = exp(z)/(exp(z)+1))
</code></pre>
For the lognormal:
<p... | https://stats.stackexchange.com |
113,737 | [
"https://softwareengineering.stackexchange.com/questions/113737",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/37673/"
] | I have almost 5 years of experience as a web developer, and feel that I should be mid-level by now, but when I walk the walk I think I'm still pretty "junior" at it.
Here's what I think is the problem: For most of my career, I seldom had any real guidance by a more senior developer and zero experience with developer t... | Be prepared to learn from your team members and realize that the way you have been doing things for the last several years may not be "up to snuff" with the rest of your team members. Code reviews will likely reveal this if they are a part of the development process.
Communication is also important, so if you haven't ... | You'll need to have the coding skills, but you also need to make interviewers comfortable with you willingness to get along with others. You need more than just wanting to be a better programmer.
Start analyzing the way you currently do things. Do you return to code and find what you wrote confusing and difficult to c... | https://softwareengineering.stackexchange.com |
343,014 | [
"https://physics.stackexchange.com/questions/343014",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/160867/"
] | I've looked up and understand the reason behind the number of the acceleration of gravity $(9.8)$. But whenever people describe it they say it's $9.8 \text{m/s}^2$.
I don't understand why it's inverse time squared, why isn't it just a number like $9.8 \text{m/s}$?
| As any ordinary acceleration, the quantity $9.8\, \mathrm{m/s^2}$ means that the velocity increases by $9.8$ meters per second, each second. Hence $9.8$ meters per second, per second.
| Well, what is the definition of acceleration(due to gravity or otherwise)? It is
$$ \frac{\Delta v }{ \Delta t},$$
or, in words: it is the instantaneous change in velocity from one point to another divided by the associated time interval.
The unit for velocity is $m/s$. The unit for time is $s$. Hence, the units ... | https://physics.stackexchange.com |
1,927,726 | [
"https://math.stackexchange.com/questions/1927726",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/135919/"
] | I am having a little problem with my maths homework. The problem is as follows:
\begin{equation}
\int^a_0{\cfrac{dx}{x \ + \ \sqrt{a^2 \ - \ x^2}}}
\end{equation}
I tried to do the following but got stuck halfway:
Let $\ \ x \ = asin\theta, \ hence, \ dx = acos\theta \ d\theta $
$
\int^a_0{\cfrac{dx}{x \ + \ \sq... | $$I=\int^a_0{\cfrac{dx}{x \ + \ \sqrt{a^2 \ - \ x^2}}}$$
Let $x^2+y^2=a^2\implies dx=-\frac{y}{\sqrt{a^2-y^2}}dy$
Thus:
$$I=\int^a_0{\cfrac{dy}{y \ + \ \sqrt{a^2 \ - \ y^2}}}\frac{y}{\sqrt{a^2-y^2}}$$
Adding these together,
$$2I=\int^a_0{\cfrac{dx}{x \ + \ \sqrt{a^2 \ - \ x^2}}}\left(1+\frac{x}{\sqrt{a^2-x^2}}\rig... | Assuming $a>0$ and applying the substitutions $x=az$, $z=\sin\theta$:
$$ I(a)=\int_{0}^{a}\frac{dx}{x+\sqrt{a^2-x^2}}=\int_{0}^{1}\frac{dz}{z+\sqrt{1-z^2}}=\int_{0}^{\pi/2}\frac{\cos\theta}{\sin\theta+\cos\theta}\,d\theta $$
but due to the substitution $\theta=\frac{\pi}{2}-\varphi$ we also have $I(a)=\int_{0}^{\pi/... | https://math.stackexchange.com |
61,752 | [
"https://mathoverflow.net/questions/61752",
"https://mathoverflow.net",
"https://mathoverflow.net/users/14395/"
] | I know Vandermonde's convolution for binomial coefficients:
$$\sum_{j=0...k} \binom{n}{j} \binom{m}{k-j} = \binom{n+m}{k}$$
Is there a similar multiplicative convolution? More precisely, is there a simple formula for the coefficients $a(k_j,k'_j,k)$ in the following
identity?
$$\sum_{j} a(k_j,k'_j,k) \binom{n}{k_j}... | Riordan and Stein, in "Arrangements on Chessboards" (<em>Journal of Combinatorial Theory, Series A</em>, <strong>12</strong> 72-80, 1972) consider the numbers $A(r,s,k)$ defined by
$$\sum_{r,s} \binom{n}{r} \binom{m}{s} A(r,s,k) = \binom{nm}{k},$$ or, as others have pointed out, the number of $r \times s$ $(0,1)$-matr... | The interpretation darij gives in the comments has a fairly straightforward proof: ${nm \choose k}$ is precisely the number of $n \times m$ matrices of $0$s and $1$s with exactly $k$ $1$s. One can think of the terms ${n \choose k_j}$ and ${m \choose k_j'}$ as picking out the rows and columns that these $1$s inhabit, an... | https://mathoverflow.net |
25,087 | [
"https://mechanics.stackexchange.com/questions/25087",
"https://mechanics.stackexchange.com",
"https://mechanics.stackexchange.com/users/2899/"
] | I've heard of people starting engines in a build without a Radiator hooked up, to see if it fires.
If you have an old engine and want to see if it will run:
<ul>
<li>Can it be started without a radiator?</li>
<li>Will any damage occur? </li>
<li>Are there any advantages to doing this?</li>
</ul>
| It can <em>absolutely</em> be started without a radiator. You will not cause any damage as long as the engine does not overheat. If you don't run it long enough for the engine to get too hot, it's not an issue.
To give you an example of how it could be beneficial:
I used to own a 94 Camaro Z28. The engine in it was a... | An engine is a very large thermal mass. When the engine is cold it takes time to warm up. If you only run the engine for 15 to 30 seconds from cold there should be no problem. Running the engine any longer than that may cause the engine to overheat.
| https://mechanics.stackexchange.com |
1,124,655 | [
"https://math.stackexchange.com/questions/1124655",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/174765/"
] | Consider the inner product defined by polarizing the quadratic form $$q(x,y,z)=x^2-z^2+4xy-2yz$$ on $\mathbb{R}^3$.
Let $V=\langle(1,0,2),(3,-1,0)\rangle$. Could you show me how to find $V^\bot$ and $V\cap V^\bot$? I get $V^\bot=\langle(2,6,1)\rangle$, which doesn't feel right, and don't know how to calculate the inte... | The symmetric bilinear form that gives rise to your quadratic form $q$ is given by
$$
b((x_1,y_1,z_1),(x_2,y_2,z_2))=x_1x_2-z_1z_2+2x_1y_2+2y_1x_2-y_1z_2-z_1y_2
$$
Now the conditions of being orthogonal to $(1,0,2)$ and to $(3,-1,0)$ are respectively given by setting for instance $(x_2,y_2,z_2)$ equal to that vector... | you want to find $(x, y, z)$ so that it is orthogonal to $(1,0, 2)$ and $(3, -1, 0)$
that means $x + 2z = 0, 3x - y = 0$ if the basis of is more complex you will make a matrix made of these vectors as rows and row reduce them to solve for $x,y,z.$ in this example it is easier and we can set $z = -1$ and solve for $x = ... | https://math.stackexchange.com |
23,539 | [
"https://stats.stackexchange.com/questions/23539",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/9254/"
] | I'm sure this is a simple problem, but I'm not sure how to search and find the answer. I have a set of parameters that I've estimated using 'fitdist':
<pre><code>Fitting of the distribution ' gamma ' by matching moments
Parameters:
estimate
shape 0.2018062
rate 3.1255336
</code></pre>
Now I want to pull those ... | Using the <code>fitdistr</code> function from <code>MASS</code>, you can access the estimated parameters like I did it below. The same holds for <code>fitdistrplus:::fitdist</code>.
<pre><code>op <- options(digits = 3)
set.seed(123)
x <- rgamma(100, shape = 5, rate = 0.1)
ft <- fitdistr(x, "gamma")
ft$estimat... | In case anyone is still interested: you can just do this:
<pre><code>gamma.fit$estimate['shape']
gamma.fit$estimate['scale']
</code></pre>
| https://stats.stackexchange.com |
2,123,820 | [
"https://math.stackexchange.com/questions/2123820",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/411762/"
] | How do you rearrange
$$\frac{m_{1}}{\gamma_{2}} = \frac{\sqrt{\frac{m_1}{\gamma_2}}-m_1}{\gamma_1}$$
to get
$$\frac{\gamma_1}{\gamma_2} = \sqrt\frac{1}{m_1\gamma_2} - 1 $$
?
I understand this may be remedial but if someone here could show me the steps then that would be very helpful.
| Neither sequence has a limit because for both sequences, the claim
$$\exists L: \forall \epsilon > 0 \exists N\in\mathbb N\forall n\in\mathbb N: n>N\implies |L-a_n|<\epsilon$$
or, in writing:
<blockquote>
There exists an $L$ such that, for all $\epsilon>0$, there exists some $N$ such that $|L-a_n|<\... | The definition is: A point $l$ is said to be a limit point of a sequence $\{a_n\}$ if every neighborhood of $l$ contains infinite terms of the sequence. The limit points of first sequence are $1,2,3$ and for the 2nd sequence the limit points are $-1$ and $1$. As every neighborhood of $1$ contains infinite terms: namely... | https://math.stackexchange.com |
89,445 | [
"https://security.stackexchange.com/questions/89445",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/76707/"
] | <h1>Update:</h1>
Check the VyprVPN's app developer answer below, seems this is okay and an uninstaller is added to new versions to prevent such issues.
<h1>Original:</h1>
I was working with VyprVpn a few months ago, never worked with them again,
Today I installed LittleSnitch (An OS X Firewall and Network Monitor),... | <blockquote>
Is this kind of activity normal in VPN providers? they are legally
registered company, claim that respect privacy...
</blockquote>
That the domain names begin on dl. (short for "download") sounds like they are checking for updates, configuration files or whatever. That they do it every seconds seems l... | Hi I'm the developer who wrote VyprVPN for Mac. It looks like you've still got one of our software components installed. Macintosh applications that needs to alter your system, including VyprVPN, uses something called a "privileged helper tool” aka a system daemon. We install a helper tool with the user's permission, ... | https://security.stackexchange.com |
22,111 | [
"https://mathoverflow.net/questions/22111",
"https://mathoverflow.net",
"https://mathoverflow.net/users/4333/"
] | Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $F$ be a vector bundle on $X$ extending $E$, i.e., $F|_{U} = E$.
Is any extension of $E$ to $X$ isomorphic to $F$?
| This is true if $X$ satisfies Serre's condition $S_2$, i.e. $\mathcal O_X$ is $S_2$. Then a vector bundle is $S_2$ since locally it is isomorphic to $\mathcal O_X^n$.
More generally, a coherent sheaf $F$ on a Japanese scheme (for example: $X$ is of finite type over a field) which is $S_2$ has a unique extension from a... | This is false as stated; for example, if $X$ is obtained from a projective geometrically connected smooth surface over a field $k$ by gluing two points together and $U$ is the complement of the singular point, then the kernel of the restriction map from the Picard group of $X$ to the Picard group of $U$ is easily seen ... | https://mathoverflow.net |
2,255,850 | [
"https://math.stackexchange.com/questions/2255850",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/441249/"
] | I've tried substituting ${\frac{1}{x}}$ as $y$ and then I get $\lim_{y \to \ 0}\ (1)^{\frac{1}{y}}$ and that's infinity?
$$\lim_{x \to \infty}\ (1+ {\frac{1}{x}})^{x}$$
| Hint
$$\text{If }L=\lim_{x\rightarrow a}f(x)^{g(x)}=(\rightarrow1)^{(\rightarrow\infty)}$$
then
$$L=e^{\lim_{x\rightarrow a}g(x)(f(x)-1)}$$
| $\frac{d}{dx} \log x\mid_{x = 1} = \lim_{h \to 0} \frac{\log(1+h)-\log 1}{h} = \lim_{h \to 0} \log(1+h)^{1/h} = \lim_{h \to \infty} \log (1+\frac{1}{h})^h = $
$\log \lim_{h \to \infty} (1+\frac{1}{h})^h$.
But we know $\frac{d}{dx} \log x\mid_{x=1} = 1$, so it must be that $\lim_{h \to \infty} (1+\frac{1}{h})^h = e$.
| https://math.stackexchange.com |
124,434 | [
"https://softwareengineering.stackexchange.com/questions/124434",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/42710/"
] | I have used the web and Stack Overflow extensively during the past month or so in creating my final project for my C# class. I have used so much code that I didn't write myself that I feel I am being unethical by not giving proper credit to the people who helped me; or the websites that have provided excellent examples... | Think about it like this: It's a class. The purpose is for you to learn things. It does not really matter whether you learn them from the teacher or from some website, as long as you do learn. And the purpose of the assignment is twofold: having you learn actively (by <em>using</em> your knowledge, which tends to make ... | I think you should cite your sources, and bother most about <strong>learning more</strong>.
Homework is not useful per se, it is a mean for <strong>you</strong> to <strong>learn things</strong>. If you didn't learn enough, you are the loser (not your teacher).
| https://softwareengineering.stackexchange.com |
275,806 | [
"https://dba.stackexchange.com/questions/275806",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/215939/"
] | I'm currently designing a database schema for a small microservice which handles products.
The service contains a simple REST API which allows users to perform basic management of products and also some end points to query the full audit history about products. Schema already contains a snapshot-based audit table for p... | If you are going to follow the pattern of inserting into the audit table after each <em>insert</em>, no there is no need to retain the creation/alter timestamps as they are present in the audit table.
However, if the pattern is to only insert into the audit table after an update/delete (this is more common), then you w... | "Last updated" can be used to detect write conflicts, where a value is read, changed and written by different users concurrently. Having this value in the data table makes detection slightly simpler. A datetime value is not required for this use case, and can be problematic in certain corner cases.
| https://dba.stackexchange.com |
14,700 | [
"https://security.stackexchange.com/questions/14700",
"https://security.stackexchange.com",
"https://security.stackexchange.com/users/9378/"
] | Let me preface my question by saying that I understand that experience is generally more valuable than certifications when it comes to the IT profession.
Having said that, are vendor specific security certifications, such as Cisco's CCNA & CCNP Security routes, looked at similarly when compared to more general o... | Adding to the already great answers - which are all some variant of "it depends"... the "it depends" includes:
<ul>
<li>From Lucas Kauffman - it depends on your role - many certificationes depend on the role you're applying for. Some certifications apply to specific types of security roles.</li>
<li>From Eric - it's ... | This answer will vary widely depending on the organization. Obviously a company that is a pure Cisco shop is going to request that you have some Cisco certifications. While a company that is a Juniper shop for example will like that you have some networking certs and a good working knowledge of similar gear, but it wou... | https://security.stackexchange.com |
193,865 | [
"https://physics.stackexchange.com/questions/193865",
"https://physics.stackexchange.com",
"https://physics.stackexchange.com/users/-1/"
] | So here's the question:
<strong><em>A train window is 1 meter high and 10 meters long. A raindrop rolls down the window at a vertical speed of 5 meters per second(when stationary). If the train is moving at 30 m s–1 how far will it move in the time it takes the raindrop to roll down the window?</em></strong>
My pro... | If the raindrop's vertical velocity is constant as the train is both stationary and moving, the time taken for the raindrop to travel down the window would be:
$$t = \frac{1\ \text{m}}{5\ \text{m}/\text{s}} = 0.2\ \text{s}$$
Remember, the time $t$ would not depend on the speed of the train. The exercise also specifical... | This is one of those questions that can drive you crazy, since there is a great deal assumed and not stated. Let me try an alternate possibility.
Conceivably, the problem wants you to assume that, when moving, the overall speed of the raindrop remains fixed at 5 m/sec, but it travels in a straight line at an angle due... | https://physics.stackexchange.com |
261,335 | [
"https://softwareengineering.stackexchange.com/questions/261335",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/153680/"
] | This might be silly, but: let's say I want to delete a <code>Book</code> from the repo. Should I:
<ul>
<li><code>deleteBook(bookId)</code> - send the ID since in the most of the time, the ID is passed from the UI</li>
</ul>
or
<ul>
<li><code>deleteBook(Book)</code> - but then I need to fetch the book first? And this... | Repositories are fist-class citizens in the domain model, so they should operate in terms of ubiquitous language using entities and value objects. Therefore you should avoid naming repository methods after CRUD operations and using database table IDs (primary keys) to handle your entities. Instead you should focus on y... | I have always found it useful for the repository to track the entities it hydrates along with their primary key values in such a way that it stays with the context of the application. For instance, in your situation, I may have something like this on my BookRepository.
<pre><code>public class SqlBookRepository : IBoo... | https://softwareengineering.stackexchange.com |
172,855 | [
"https://mathoverflow.net/questions/172855",
"https://mathoverflow.net",
"https://mathoverflow.net/users/33550/"
] | What is the maximum (absolute) value of the binomial coefficient
$\begin{pmatrix}x \\ k\end{pmatrix} = \frac{1}{k!}x(x-1)(x-2)\dotsb(x-k+1)$
for real $x$ in the interval $0 \leq x \leq k-1$?
| It's easy to see that the extremum in $(0,1)$ has the same magnitude as the one in $(k-1,k-2)$ and is more extreme than any of the other extrema. The extremum in $(0,1)$ occurs at $x_0=(1+o(1))/\ln k$ (by looking at the derivative), but I'll only assume it is $\Theta(1/\ln k)$. Write $x_0=z/\ln k$ and substitute this ... | I'm not sure it adds anything beyond what's in Brendan McKay's answer, but I'll flesh out my comment. The absolute value of the binomial coefficient we are interested in may be written (for $x \in (0,1)$ and $k>1$ ) as
$\frac{x}{k}\prod_{i=1}^{k-1}(1 - \frac{x}{i}).$ Applying the AM-GM inequality to the displayed pr... | https://mathoverflow.net |
51,916 | [
"https://dba.stackexchange.com/questions/51916",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/29095/"
] | As part of a daily cron job, I need to run a query that processes a whole lot of data. This data is related to the visitors coming to a website, and updating the data with what we have captured previously.
The query relies on 2 derived tables (select queries in the <code>FROM</code> section), to do its work —
<pre><... | So the root problem as @Dmitriy mentioned, had to do with the derived queries. Basically, when operating with humongous data sets, derived tables can lead to a whole lot of pain as the underlying indexes from the tables comprising the queries are not available for the derived queries.
In short, if you're writing a <co... | First of all is goot way to help us to help you is show valid CREATE statements:
<pre><code>tbl_last_input_visit: #1072 - Key column 'idvisitor' doesn't exist in table
tbl_log_visit: #1072 - Key column 'site_id' doesn't exist in table
</code></pre>
Second: I will try to find out how to optimize it, but try to check d... | https://dba.stackexchange.com |
94,816 | [
"https://softwareengineering.stackexchange.com/questions/94816",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/24166/"
] | I may potentially have some work to implement new features to a client's existing web application. (I didn't write the original app).
I've never made an app that wasn't written from scratch, and I'm worried about integrating the new features without disrupting the service for too long (or at all).
I don't think the c... | Your basic work flow sounds entirely sensible. Particularly getting code under revision control and making sure you can recreate the app and database from that source.
I would suggest not updating the live site while it's running. Depending on how the site works you probably have two other options;
<ol>
<li>Set a n... | Automated integration testing is your FRIEND.
Go look up Selenium, which is an automated web site testing application. Set up a complete suite of tests that thoroughly exercises the CURRENT functionality of the site. Then, periodically as you develop, run your tests and satisfy yourself that your changes haven't broke... | https://softwareengineering.stackexchange.com |
139,925 | [
"https://dba.stackexchange.com/questions/139925",
"https://dba.stackexchange.com",
"https://dba.stackexchange.com/users/88728/"
] | I dont want to disturb the server values of xp_CMDShell & Show Advanced options. The criteria is to use SQL to run a batch file.
This is the script i am trying to solve it with:
<pre><code>DECLARE @advanced_options int
DECLARE @xp_cmdshell int
--CONVERT ( data_type [ ( length ) ] , expression [ , style ] )
SELE... | Error is at below select statements. If you do <code>ALT+F1</code> on sys.confiugrations you'll notice that <code>value_in_use</code> column is of type sql_variant and you are trying to assign it to int variables. To fix this, use below statements in your script and it should work fine.
<pre><code>SELECT @advanced_opt... | I normally put the original values in temp table and then once done with the script/task, I revert it back.
<pre><code>SET NOCOUNT ON;
IF OBJECT_ID('tempdb..#xp_cmdshelloption') IS NOT NULL
DROP TABLE #xp_cmdshelloption;
CREATE TABLE #xp_cmdshelloption (
configuration_id INT
,NAME NVARCHAR(35)
,origi... | https://dba.stackexchange.com |
506,227 | [
"https://math.stackexchange.com/questions/506227",
"https://math.stackexchange.com",
"https://math.stackexchange.com/users/97149/"
] | This function
$$f(x,y)=\frac{e^{xy}-\cos (x)+\sin(xy)}{x}$$
can be made continous for $f(0,y)$ by defining
$$f(0, y) = 2y .$$
My question is: how can i get to this conclusion ("$2y$ must be it") on my own?
I have tried using limits, substituting $u=xy$,... or polar, but no luck. Is there a list somewhere of things... | For fixed $y \ne 0$, we see from L'Hospital's rule that
$$\lim_{x \to 0}f(x; y) = \lim_{x \to 0} \frac{y e^{xy} - \sin{x} + y \cos{xy}}{1} = \frac{y - 0 + y}{1} = 2y$$
| Here is another method without using l'Hospital's rule:
$$f(x,y) = \frac{(e^{xy}- 1) +(1 -\cos x) + \sin xy}{x} = \frac{e^{xy}-1}{x} + \frac{1-\cos^2 x}{x(1 +\cos x)} + \frac{\sin xy}{x} = y\frac{e^{xy}-1}{xy}+\frac{x}{1+\cos x}\frac{\sin^2 x}{x^2}+y\frac{\sin xy}{xy}.$$
| https://math.stackexchange.com |
43,287 | [
"https://stats.stackexchange.com/questions/43287",
"https://stats.stackexchange.com",
"https://stats.stackexchange.com/users/-1/"
] | I'm currently writing a thesis and I have to interpret a coefficient estimate that's extremely negative compared to what I hypothesized.
How do I best communicate this? Describing it as <code>"very negative"</code> sounds a bit strange, but I can't think of anything better.
| If you expected the estimate to be negative, you could say something like "As expected, the coefficient estimate was negative, but its magnitude was much larger than expected".
| I would just write almost exactly what you did: "the coefficient was much lower than expected". Or, "based on .... [previous research? pilot studies?], we expected the coefficient to be in range X-Y. However, the estimated coefficient turned out to be much lower".
Clearly, this is not statistics, and there is no stati... | https://stats.stackexchange.com |
28,153 | [
"https://mathoverflow.net/questions/28153",
"https://mathoverflow.net",
"https://mathoverflow.net/users/6627/"
] | I have to simulate independent draws from a very complicated distribution. They only feasible way appears to be using MCMC. I was considering running thousands of chains in parallel, but that would slow things down from me considerably. So I am running one long MCMC chain. Assuming that MCMC chain run did converge nice... | From talking to statisticians, it seems like the standard thing to do is assume a yes. For a specified sequence or thinning, one can create Markov chains which exhibit 0 autocorrelation but a 'very large' amount of dependence despite the thinning, but the idea is that this should be pretty pathological, and so you are ... | Hi,
Thinning samples won't change your target distribution. It's probably a good idea since you'll end up with very unweildy data files otherwise (A brief mention of this is made in Bayesian Data Analysis by Gelman et al page 295).
Another unrelated point is that even though running parallel chains won't help re... | https://mathoverflow.net |
241,158 | [
"https://electronics.stackexchange.com/questions/241158",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/114086/"
] | Can anyone explain if it's good to have the Gnd pin of a Bypass Caps tied to IC's Gnd? I've observed some design using those methods. Just want to have a justified answer.
I'm not sure if the thickness of the board plays a huge role in here.
| <blockquote>
if it's good to have the Gnd pin of a Bypass Caps tied to IC's Gnd?
</blockquote>
Absolutely, if the capacitor ground isn't connected to the IC ground, it can't do its job of providing a low-impedance path between those two nets.
If you want to think of it in terms of ground loops, think about it this... | It's important to have any required high frequency decoupling capacitor as close to the IC pins as possible. That capacitor supplies short bursts of current to the chip (generalism alert) that would otherwise be provided from a capacitor a bit further away and fed down copper tracks. Keeping tracks short means keeping ... | https://electronics.stackexchange.com |
286,187 | [
"https://softwareengineering.stackexchange.com/questions/286187",
"https://softwareengineering.stackexchange.com",
"https://softwareengineering.stackexchange.com/users/182895/"
] | By Guard Statements I mean something similar to the first part of the function:
<pre><code>def doSomething(String something)
{
// Guard Statement
if(!something)
{
return false
}
// more stuff
}
</code></pre>
Say you might have several parameters, maybe you need to log that the method was ... | Don't think of short functions in terms of absolute LOCs number. It is irrelevant, because:
<pre class="lang-java prettyprint-override"><code>void demo(int a)
{
if (a < 0)
{
this.dealWithNegative();
}
for (int i = 0; i < a; i++)
{
yield this.doSomething(i);
}
}
</code></p... | A few options:
<ol>
<li>Don't use guards. Seriously, if a null parameter is going to blow up, just let it blow up. Also, while it's <em>possible</em> for some naughty programmer to call internal functions/services directly with bad data, you maybe don't need to guard everything.</li>
<li>Single line conditionals. This... | https://softwareengineering.stackexchange.com |
509,176 | [
"https://electronics.stackexchange.com/questions/509176",
"https://electronics.stackexchange.com",
"https://electronics.stackexchange.com/users/207355/"
] | I've recently started learning electronics/mechatronics. As part of my studies, I've been reading about devices such as RaspberryPi, Arduino, etc, and I'm interested in experimenting with such devices.
I have what would, I guess, be considered a "design" question. Let's say I wanted to undertake a small robot... | This question honestly is too broad. You have these criteria:
<ol>
<li>Suitability of peripherals</li>
<li>Suitability of processing power</li>
<li>Ease of software development</li>
<li>Ease of hardware development</li>
</ol>
3 and 4 completely dwarf the other two criteria so much when starting out. When starting, just... | <em>what I'm actually asking is a beginner technical question: Does such a robotic device, as described, require having some separate, additional computational device, such as an Arduino or Raspberry Pi, in order to operate (since this might be required to interface with the computer, or something), or are such devices... | https://electronics.stackexchange.com |
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