ireducible polynomials with coefficients in ${0,-1}$












4














I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio







share|cite|improve this question
























  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28
















4














I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio







share|cite|improve this question
























  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28














4












4








4







I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio







share|cite|improve this question















I'm interested in the polynomials of the form $x^n -x^{n-1} - b_{n-2}x^{n-2} - cdots -b_1 x -1$ with the coefficients $b_k$ being either zero or one. The prototype is of course the Golden mean $x^2-x-1$. Is there a known name for such polynomials? I would search for info on them, but can't figure out how to form a reasonable search query.



To tighten things up: I'm interested in the ones that are irreducible with respect to one-another (are prime relative to one-another). As far as I can tell, there are the same number of these as there are irreducible polynomials over $mathbb{F}_2$ -- OEIS A001037 (should that be obvious?). As far as I can tell, they always have just one real root greater than one. I'm interested in the complex roots, too. Some of these show up in papers on "generalized golden means", typically the all-ones case $x^n -x^{n-1} - cdots -x -1=0$ or the all-zeros case $x^n -x^{n-1}-1=0$, but its not much enlightening. Another factoid: it seems the real roots are dense in [1,2], considering all such polynomials, taking $ntoinfty$. Is there a generating function? I'm guessing that texts in dynamical systems, chaos theory, fractals, subshifts, maybe number theory might discuss these; but under what name?



EDIT: The context for these polynomials is the iterated Beta Transform. I've written a 105-page tract providing a treatment of the Beta Transform, illustrating the usefulness and central natural of these polynomials. It can be found here: https://linas.org/math/solvit.pdf I still don't "understand them", and find this to be annoying.






dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio




dynamical-systems irreducible-polynomials fractals chaos-theory golden-ratio






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 26 at 9:41

























asked Feb 10 at 6:08









Linas

417210




417210












  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28


















  • In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
    – Andy Walls
    Feb 10 at 12:02












  • Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
    – Andy Walls
    Feb 10 at 12:37










  • Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
    – Linas
    Feb 11 at 8:28
















In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
– Andy Walls
Feb 10 at 12:02






In my playing around with polynomials like this with all the $b_n=0$, the roots are all close to the $n^{th}$ roots of unity. The roots are roughly in a circle in the complex plane since $x^{n-1}(x-1)=1$. IIRC, Even with $b_nne0$, the two highest order terms dominate as $n$ becomes large, so the roots are still roughly arranged in a circle in that case
– Andy Walls
Feb 10 at 12:02














Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
– Andy Walls
Feb 10 at 12:37




Glasser has an approach for finding the roots of the trinomial $x^n-x+t=0$, which you might be able to adapt to the trinomial $x^n-x^{n-1}-1=0$ but you might run into convergence problems with the hypergeometric functions. sciencedirect.com/science/article/pii/S0377042700002879
– Andy Walls
Feb 10 at 12:37












Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
– Linas
Feb 11 at 8:28




Ah, yes, thank you. I was seeing roots arranged in circles at various radii; (specifically, at a radius of 1 over the real root, in a related one-off problem in which these arise.) I see Glasser invokes the multiplication theorem, which is an old favorite of mine... and might provide the doorway I need.
– Linas
Feb 11 at 8:28










1 Answer
1






active

oldest

votes


















0














Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer





















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2644210%2fireducible-polynomials-with-coefficients-in-0-1%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0














Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer





















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04
















0














Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer





















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04














0












0








0






Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)





share|cite|improve this answer












Not an answer, but here's a MatLab/Octave script to visualize how the roots of
$$x^n-x^{n-1}-1 = 0$$
compare to the roots of unity
$$x^n-1 = 0$$



N = 10;

% x^n - 1 = 0

eta_poly = [1, zeros(1,N-1), -1];

% x^n - x^(n-1) - 1 = 0

r_poly   = [1, -1, zeros(1,N-2), -1];

eta = roots(eta_poly);

r = roots(r_poly);



% Plot the nth roots of unity and the other polynomial's roots

plot(real(eta),imag(eta),'.',real(r),imag(r),'+');

grid on;



% Sort the roots of unity by their arg() in [0, 2*pi)

ae = arg(eta)+2*pi*(arg(eta)<0);

[s,si] = sort(ae);

eta = eta(si);



% Match the closest roots of unity to the roots in r

% and permute r so that it lines up with the corresponding roots of unity

M = length(eta);

c = eta*ones(1,M)-ones(M,1)*r.';

[w,iw] = min(abs(real(c))+abs(imag(c)));

r = (abs(c) <= max(w))*r;



% What's the difference between the roots of unity and the roots of the

% other polynomial?

d = r-eta

r_d = abs(d)

phi_d = arg(d)






share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Feb 12 at 1:04









Andy Walls

1,539128




1,539128












  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04


















  • The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
    – Linas
    Feb 12 at 20:01










  • Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
    – Linas
    Feb 12 at 20:04
















The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
– Linas
Feb 12 at 20:01




The case of $x^n-sum_{k=0}^{n-1} x^k=0$ is interesting: the zeros approach the unit circle surprisingly slowly as $ntoinfty$.
– Linas
Feb 12 at 20:01












Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
– Linas
Feb 12 at 20:04




Also: for small n, the polynomials, roots are the "silver number", pisot number, tribonacci, tetranacci, etc. and have OEIS A092526 A058265 A109134 A086088 A060006 A293506 A103814 to describe them.
– Linas
Feb 12 at 20:04


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics 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.





Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


Please pay close attention to the following guidance:


  • 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.


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




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2644210%2fireducible-polynomials-with-coefficients-in-0-1%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Human spaceflight

Image
Human spaceflight From Wikipedia, the free encyclopedia Jump to navigation Jump to search "Space traveler" redirects here. For other uses, see Space traveler (disambiguation). @media all and (max-width:720px){.mw-parser-output .tmulti>.thumbinner{width:100%!important;max-width:none!important}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:none!important;width:100%!important;text-align:center}} Apollo 11 crewmember Buzz Aldrin on the Moon, 1969 International Space Station crewmember Tracy Caldwell Dyson views the Earth, 2010 Space Shuttle Discovery heads into space with a crew aboard, STS-121 in 2006 Inside a space suit on the Canadarm, 1993 Human spaceflight (also referred to as crewed spaceflight or manned spaceflight ) is space travel with a crew or passengers aboard the spacecraft. Spacecraft carrying people may be operated directly, by human crew, or it may be e
Read more

Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

Image
0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB]
Read more

File:DeusFollowingSea.jpg

Image
File:DeusFollowingSea.jpg From Wikipedia, the free encyclopedia Jump to navigation Jump to search File File history File usage No higher resolution available. DeusFollowingSea.jpg ‎ (238 × 211 pixels, file size: 12 KB, MIME type: image/jpeg ) Summary [ edit ] Media data and Non-free use rationale Description Cover art of the Deus CD Following Sea Author or copyright owner Deus Source (WP:NFCC#4) rocksucker.co.uk Date of publication 2012 Use in article (WP:NFCC#7) Following Sea Purpose of use in article (WP:NFCC#8) to serve as the primary means of visual identification at the top of the article dedicated to the work in question. Not replaceable with free media because (WP:NFCC#1) n.a. Minimal use (WP:NFCC#3) Low res image used in infobox of the article on the album Respect for commercial opportunities (WP:NFCC#2) n.a. Fair use Fair use of co
Read more