Show 12sqrtX is Not Uniformly Continuous on 01
Show that $x\ln(x)$ is not uniformly continuous on $(0,\infty)$
- Thread starter Matt
- Start date
- #1
Matt Asks: Show that $x\ln(x)$ is not uniformly continuous on $(0,\infty)$
I am trying to show that $f(x) = x\ln(x)$ is not uniformly continuous on the interval $(0,\infty)$. The solution given here Show $f(x)=x\ln x$ is not uniformly continuous does it by using $\epsilon-\delta$ but I want to do it by sequences if possible. The "x" term is messing things up because I cannot take $$x_n = \textrm e^{-n } $$ and $$y_n = \textrm e^{-n + 1}$$ because $|f(x) -f
|$ does not work out. Any ideas on sequences I can take?
- Jor_El
- Computer Science
- Replies: 0
Jor_El Asks: Additive vs Multiplicative model in Time Series Data
The above time series plot is a daily closing stock index of a company. I want to know which model between additive and multiplicative best suits the above data. I know what the two models are, but i haven't been able to figure out the correct model for the above data. Also, is there any way other than simple visualisation which can help me decide the correct model?
- Abdullah
- Chemistry
- Replies: 0
Abdullah Asks: Why is Ammonium Nitrate NH4NO3 ionic salt not molecular
As stated, I get the two are polyatomic ions but isn't NH4 like Nitrogen tetrahydride, which is two non-metals or covalent? And NO3 => Nitrogen trioxide also two non-metal or covalent.
So how do two covalent bonds become ionic? They all include only Non-metals; isn't it technically a covalent bond? I haven't any reasonable explanation on the internet; the only one I understood was both are polyatomic, and the first has a (+) charge, i.e. NH4 and the second a (-) NO3, its not like its a metal and a non-metal combining they both are from what i understand non-metals but why does that matter?
- Andrew James Combs
- Physics
- Replies: 0
Andrew James Combs Asks: 2006 Honda Accord "Stutters" above 2500 RPM, won't rev past 3000
I have a 2006 Honda Accord that won't rev past 3000 RPM. Somewhere between 2500 and 3000 the tachometer starts jumping back and forth rapidly between 2.5k and 3k and the car shakes pretty hard. If I let off the throttle so that it falls back below 2500 it goes back to running as normal.
This started on a ~200 mile round trip in the past few weeks. I added what oil I had and got it home. The coolant was really low so I refilled it. I took it to a mechanic who put a quart of oil in it. In the parking lot at the shop it would rev up, but on the drive home it started acting up again. It had codes for VTEC, evap, and a bad knock sensor. I replaced the knock sensor and gas cap and just like at the mechanic it ran fine for a few minutes then went back to the same problem behavior as before.
I would suspect the VTEC solenoid but it was just replaced in March. I've looked at a couple other descriptions of VTEC sensor failure but I haven't heard any of them describe these symptoms.
Should I go ahead and replace the VTEC solenoid and/or sensor? Or should I be looking elsewhere? Has anyone seen this kind of behavior before and know what I should investigate?
I should mention, there's also a crack in the air cleaner hose. It's pretty small but I haven't replaced it yet, I plan to soon.
The specific codes are: P0325 P2647 P0456
- QUANTUM WORLD
- Physics
- Replies: 0
QUANTUM WORLD Asks: Infinity potential well
For infinity potential well we take the anology of nucleus.Inside which proton and neutron is bounded in principle if it infinity potential well so particle should not come outside the nucleus.But what happened at the time of nuclear decay or radioactivity at that time particle come outside either through alpha or beta or Gama decay.how this could we possible??
- john mangual
- Physics
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john mangual Asks: How does this microwave kiln achieve such good insulation?
I am reading on Amazon about this microwave kiln, I am wondering how it achieves such high temperatures inside a microwave. What unusual material is this?
Possibly repeat of How do microwave kilns work?
- Thomas Ahle
- Mathematics
- Replies: 0
Thomas Ahle Asks: Integrating function over region is slow
I have a function
Code:
f[pt_] := Log[1/(1 - Abs[x])] + Log[1/(1 - Abs[y])] /. {x -> pt[[1]], y -> pt[[2]]} That looks like this:
I want to integrate it over a rotated rectangle:
Code:
Plot[NIntegrate[f[{x, y}], {x, y} \[Element] TransformedRegion[Rectangle[{-1/2, -1/2}, {1/2, 1/2}], RotationTransform[theta]]], {theta, 0, 2 Pi}] But it's very slow, and I keep getting errors like:
Code:
NIntegrate::slwcon: Numerical integration converging too slowly; suspect one of the following: singularity, value of the integration is 0, highly oscillatory integrand, or WorkingPrecision too small. I don't think this function should be that difficult to integrate.
I tried to rotate it myself like this:
Code:
r[t_] := {{Cos[t], -Sin[t]}, {Sin[t], Cos[t]}} Plot[NIntegrate[f[r[theta].{x, y}], {x, -1/2, 1/2}, {y, -1/2, 1/2}], {theta, 0, 2 Pi}] But it is still very slow. As in I had to kill it after multiple minutes of no results. I'm using Mathematica 12.1.1.0. Is there anything I can do to speed this up?
Bonus problem: I also want to find the maximum of the function. So I take
Code:
FindMaximum[{NIntegrate[f[r[theta].{x, y}], {x, y} \[Element] Rectangle[{-1/2, -1/2}, {1/2, 1/2}]], 0 <= theta <= 2 Pi}, {theta, .1}] However, this gives me a bunch of errors like
Code:
NIntegrate::inumr: The integrand g[x Cos[theta]-y Sin[theta], y Cos[theta]+x Sin[theta]] has evaluated to non-numerical values for all sampling points in the region with boundaries {{-(1/2),0.},{-(1/2),1/2}}. I tried to change the definition of r to
Code:
r[t_?NumericQ] := {{Cos[t], -Sin[t]}, {Sin[t], Cos[t]}} But it doesn't seem to improve things. It only changes the error to
Code:
NIntegrate::inumr: The integrand g[r[theta],{x,y}] has evaluated to non-numerical values for all sampling points in the region with boundaries {{-(1/2),1/2},{-(1/2),1/2}}. Note that the function is definitely defined at this location, and for all valid points in the region for any rotation.
Can anyone help me understand what I'm doing wrong?
- Sean
- Mathematics
- Replies: 0
Sean Asks: Support projection vs closed support projection of a normal state in enveloping von Neumann algebra
I preface this by saying that I am fairly new to the enveloping von Neumann algebra scene, so there may be some gaps in my understanding.
Given a $C^*$-algebra $A$ and a state $\phi$ on $A$, one may consider $\phi$ as a normal state on the universal enveloping algebra $A''$ of $A$. In this case, there should be a support projection $\text{supp}(\phi) \in A''$ for $\phi$, that is, a minimal projection such that $\phi(\text{supp}(\phi))=1$.
For a (concerning) example, if $A = C([0,1])$ and $\phi$ is the Lebesgue integral, this seems to give us a projection in $C([0,1])''$ which is smaller than the operator of multiplication by the indicator function of any set of full Lebesgue measure. This is a confronting possibility, so have I made a mistake somewhere or is this just evidence of how complicated $C([0,1])''$ is?
On the other hand, we could restrict to closed projections (in the sense of Akemann, The General Stone-Weierstrauss problem, 1969). Then for commutative $C^*$-algebras there is a smallest closed projection of full "measure", since (normal) states correspond to regular Borel probability measures and closed projections correspond to closed subsets of the spectrum. Furthermore, these "closed support projections" are much less pathological than the support projection in the enveloping von Neumann algebra.
I have a proof sketch via the universal representation that "closed support projections" do exist in the non-commutative case too, but nowhere do I use closedness, so I will not feel confident in its validity until I know what is going on with the support projection of the Lebesgue integral on $C([0,1])$.
I would also be interested to know if people have already thought about closed support projections, for example if it is known whether they are the closure of the support projection.
Thanks in advance.
Source: https://solveforum.com/forums/threads/show-that-x-ln-x-is-not-uniformly-continuous-on-0-infty.577252/
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