asymptotics – Time to evaluate (1+x^i)

I am considering the asymptotic analysis required to convert a polynomial of the shape:

$P(x) = prod_{i = 1}^{n}(1 + x^{s_i})$

to its “full” representation, for example $P(x) = 1+x^3+x^5+x^8+x^9+x^{12}+x^{14}+x^{17}$.

Now, it would appear to me that we perform $n$ repetitions of performing $2^i$ multiplications, as for every expansion, we double the number of terms, as an example: $(1+x^2)(1+x^4)=1+x^2+x^4+x^6$. However, this smells a bit to me.

What is the asymptotic running time of expanding $P(x)$?

ag.algebraic geometry – Evaluate the following algebraic expressions. Show all of your steps

ag.algebraic geometry – Evaluate the following algebraic expressions. Show all of your steps – MathOverflow

calculus and analysis – How to use Mathematica to evaluate the limit of Exponential Functions?


What you wrote is

(E^-2 x + 3 x)/x // Simplify

(* 3 + 1/E^2 *)

Presumably you meant to write

expr = (E^(-2 x) + 3 x)/x;

First, check that l’Hopital’s rule is applicable

Limit(#, x -> Infinity) & /@ {Numerator(expr), Denominator(expr)}

(* {∞, ∞} *)

The derivatives are

D({Numerator(expr), Denominator(expr)}, x)

(* {3 - 2 E^(-2 x), 1} *)

The limits of the derivatives are

Limit(%, x -> Infinity)

(* {3, 1} *)

The ratio of the limits is the limit of expr

Divide @@ %

(* 3 *)


Limit(expr, x -> Infinity)

(* 3 *)

bash – Piping nc to sed and evaluate datetime for each row

Could someone provide any idea, how could i pipe data from nc to sed (or awk, or anything) and evaluate and write to file a timestamp after each row?
What i have:

# cat /etc/systemd/system/ncserver.service
Description=netcat listener

ExecStart=/usr/bin/bash -c '/usr/bin/nc -n -v -l -k -p 1313 >> /opt/data/$(hostname)-$(date -u +%%y%%m%%d-%%H).txt'


simple listener – receives text via raw tcp, sends into the file (restarts by cron once a day to generate new file)

I want to append current timestamp to each row of text coming (at the beginning or the end of the row, doesnt matter).
Tried sed, awk, but the problem is that i cannot evaluate date for each row – it evaluates only once, when service started.
Looks like this

109582 ?        Ss     0:00 /usr/bin/bash -c /usr/bin/nc -n -v -l -k -p 1313 | /usr/bin/sed 's/^/'$(date +%s)
 109583 ?        S      0:00 /usr/bin/nc -n -v -l -k -p 1313
 109584 ?        S      0:00 /usr/bin/sed s/^/1617555467;/g

Pipe may be not the answer at all, but i’m kinda limited by bash.
Any ideas?..

evaluation – What’s the difference between Evaluate and Replace command?

all, I’m new to Mathematica, I am very confused between Evaluate and Replace (/.) command. I feel they are very similar.

For example,

In(2):= y = x + 2;

y /. {x -> 3}

Out(3)= 5

However, if I use
In(1):= y = x + 2;

Evaluate(y, x = 3)

Out(2)= Sequence(5, 3)

For both cases, I can get y=5.
I think for the convenience, we can always use Replace instead of Evaluate for the simple substitution, correct? Or is there any case I should use Evaluate only instead of using replace If I want to substitute a symbol to another symbol or value?

Thank you

geometry – Evaluate $biggl(frac{V_{tetrahedron}}{V_{ellipsoid}}biggr)_{max}$ for the given scenario

What is the maximum ratio of the volumes of a tetrahedron $ABCD$ inscribed inside an ellipsoid of radii $a,b,c$, provided that the tetrahedron contain the center of the ellipsoid (within it, or on a face or edge thereof)?

Otherwise stated, evaluate $biggl(frac{V_{tetrahedron}}{V_{ellipsoid}}biggr)_{max}biggl|cup{abc}in{ABCD}$

plotting – Changes When Using Solve with Plot3D and Evaluate

I’m solving the following equation, which should be straightforward:

func1(A1_, A2_) := -((
   4 I + 4 I A2^2 - 12 I A1 (1 + A2 ((2 - I) + A2 - 2 I eig)) + 
    A2 ((1 + 2 I) + 2 eig) ((5 + 4 I) + 4 eig ((1 + 2 I) + eig)))/(
   8 A2));
sol1(A1_, A2_) := Solve(func1(A1, A2) == 0, eig);

I want to extract the analytic solution to this, but I’m not sure that what Mathematica is producing is correct. Here are the plots to explain what I mean:

Smooth plot using the unpacked solution

This is a smooth plot and something that I expect. I’m plotting this with the line:

Plot3D(Re(eig /. sol1(A1, A2)((1))), {A1, 0, 1}, {A2, 0, 1})

Now if I try to get the analytic expression for this solution and I plot it, I get something different. I can demonstrate this just by changing my Plot3D command:

Plot3D(Evaluate(Re(eig /. sol1(A1, A2)((1)))), {A1, 0, 1}, {A2, 0, 1})


enter image description here

As you can see, the plot is different; it’s jagged with a discontinuity. With this, I can’t be sure that the analytic solution I extract from Mathematica is right. I’d like to either replicate the first plot with an explicit expression that I can acquire somehow or find out what function exactly Mathematica is plotting in the first instance. Am I missing something in regard to rule solutions here?


differential equations – How to use Lyapunov Exponet to evaluate the performance of a chaotic map

Thanks for contributing an answer to Mathematica Stack Exchange!

  • Please be sure to answer the question. Provide details and share your research!

But avoid

  • Asking for help, clarification, or responding to other answers.
  • Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. MathJax reference.

To learn more, see our tips on writing great answers.

numerical value – N won’t evaluate expression numerically

Using N(expr) is not evaluating this expression numerically for me and I can’t figure out why. I originally thought it’s because I wasn’t doing NIntegrate but I’ve seen multiple examples of people using N(integral expr) to get a numerical result.

c1 = 1/(Integrate(e^(-0.04 x), {x, 5, 60}));
c2 = 1/(Integrate(e^(-0.16 x), {x, 5, 60}));
f1(x_) := c1*e^(-0.04 x);
f2(x_) := c2 * e^(-0.16 x);
P1 = N(f1(10)*f1(32)*f1(38)*f1(40))
P2 = N(f2(10)*f2(32)*f2(38)*f2(40))

out $$frac{log ^4(e)}{left(frac{25.}{e^{0.2}}-frac{25.}{e^{2.4}}right)^4 e^{4.8}}$$
out $$frac{log ^4(e)}{left(frac{6.25}{e^{0.8}}-frac{6.25}{e^{9.6}}right)^4 e^{19.2}}$$

I’ve tried evaluating it numerically in different stages like at c1, or in the function and still no numerical result. Any help would be appreciated

DreamProxies - Cheapest USA Elite Private Proxies 100 Private Proxies 200 Private Proxies 400 Private Proxies 1000 Private Proxies 2000 Private Proxies - Buy Cheap Private Proxies Buy 50 Private Proxies Buy 100 Private Proxies Buy 200 Private Proxies Buy 500 Private Proxies Buy 1000 Private Proxies Buy 2000 Private Proxies ProxiesLive New Proxy Lists Every Day Proxies123