On the roots of Bernoulli polynomials












2












$begingroup$


Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".




QUESTION: Or, does it? If so, what are these curves?




Request: Can someone post the complex plot here?






reference-request cv.complex-variables soft-question polynomials







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    4 hours ago










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    3 hours ago
















2












$begingroup$


Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".




QUESTION: Or, does it? If so, what are these curves?




Request: Can someone post the complex plot here?






reference-request cv.complex-variables soft-question polynomials







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    4 hours ago










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    3 hours ago














2












2








2


2



$begingroup$


Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".




QUESTION: Or, does it? If so, what are these curves?




Request: Can someone post the complex plot here?






reference-request cv.complex-variables soft-question polynomials







share|cite|improve this question









$endgroup$




Consider the Bernoulli polynomials denoted by $B_n(z)$. Now, start plotting the set of all (combined) complex roots $mathcal{A}_N$ of $B_n(z)$, say for $n=1,2,dots,N$ for some enough large $N$. It appears that $mathcal{A}_N$ branches into several "curves".




QUESTION: Or, does it? If so, what are these curves?




Request: Can someone post the complex plot here?






reference-request cv.complex-variables soft-question polynomials




reference-request cv.complex-variables soft-question polynomials






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 5 hours ago









T. AmdeberhanT. Amdeberhan

17.2k229126




17.2k229126








  • 1




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    4 hours ago










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    3 hours ago














  • 1




    $begingroup$
    One reference is arxiv.org/pdf/math/0703452.pdf.
    $endgroup$
    – Richard Stanley
    4 hours ago










  • $begingroup$
    @RichardStanley: thank you much for the quick reply with the reference.
    $endgroup$
    – T. Amdeberhan
    3 hours ago








1




1




$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
4 hours ago




$begingroup$
One reference is arxiv.org/pdf/math/0703452.pdf.
$endgroup$
– Richard Stanley
4 hours ago












$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
3 hours ago




$begingroup$
@RichardStanley: thank you much for the quick reply with the reference.
$endgroup$
– T. Amdeberhan
3 hours ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.



enter image description here



For the number of real roots, see OEIS sequence A094937 and references there.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Many thanks for the quick and generous response to my request for the plots.
    $endgroup$
    – T. Amdeberhan
    3 hours ago











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: "504"
};
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%2fmathoverflow.net%2fquestions%2f321507%2fon-the-roots-of-bernoulli-polynomials%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









2












$begingroup$

Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.



enter image description here



For the number of real roots, see OEIS sequence A094937 and references there.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Many thanks for the quick and generous response to my request for the plots.
    $endgroup$
    – T. Amdeberhan
    3 hours ago
















2












$begingroup$

Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.



enter image description here



For the number of real roots, see OEIS sequence A094937 and references there.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Many thanks for the quick and generous response to my request for the plots.
    $endgroup$
    – T. Amdeberhan
    3 hours ago














2












2








2





$begingroup$

Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.



enter image description here



For the number of real roots, see OEIS sequence A094937 and references there.






share|cite|improve this answer











$endgroup$



Here is an animation of the zeros of the first $100$ Bernoulli polynomials, produced using Maple.



enter image description here



For the number of real roots, see OEIS sequence A094937 and references there.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited 4 hours ago

























answered 4 hours ago









Robert IsraelRobert Israel

41.6k50119




41.6k50119












  • $begingroup$
    Many thanks for the quick and generous response to my request for the plots.
    $endgroup$
    – T. Amdeberhan
    3 hours ago


















  • $begingroup$
    Many thanks for the quick and generous response to my request for the plots.
    $endgroup$
    – T. Amdeberhan
    3 hours ago
















$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
3 hours ago




$begingroup$
Many thanks for the quick and generous response to my request for the plots.
$endgroup$
– T. Amdeberhan
3 hours ago


















draft saved

draft discarded




















































Thanks for contributing an answer to MathOverflow!


  • 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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f321507%2fon-the-roots-of-bernoulli-polynomials%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

Aikido

Image
Ĉi tiu artikolo citas neniun fonton . Vi povas plibonigi ĝin, aldonante fidindajn fontojn. Senfontaj asertoj povas esti forigitaj.  (Novembro 2012) Aikido Japana Nomo: Kanji 合気道 Hiragana あいきどう Spiradaj ekzercoj Aikido , aŭ Ajkido , estas moderna japana luktarto (Shin Budo) kompilita de Ueshiba Morihei dum la periodo inter la 1930-aj jaroj ĝis 1960-aj jaroj. Ĝi estas senperforta kaj multaj el ĝiaj teĥnikoj evoluis el Daito-Rjuo (speco de ĵuĵicuo) kaj el glava batalado. La filozofio de aikido devenis ĉefe de la religio Oomoto pro la kontaktoj inter O´Sensei ("granda majstro", esprimo uzata de la aikidistoj por nomi Ueshiba Morihei) kaj Deguchi Onisaburo Sensei el Oomoto. Oni konsideras, ke aikidisto havu iom da spiriteco, interalie pri la uzo de la Ĉi, KI, ĝi postulas nenuran teknikan trejnon. La nepo de la kreinto, Ueshiba Moriteru, nun estas "Doshu", tiu kiu montras la vojon. Enhavo 1 Aikido - Oomoto - Esperanto...
Read more

The minimum number of groups for any class cannot be less than 2 error

Image
0 $begingroup$ cv_split = StratifiedShuffleSplit(n_splits=10, test_size=0.3, random_state=0) # Performing cross-validation to estimate the performance of the models for idx, est in enumerate(estimator): cv_results = cross_validate(est[1](), x_value, y_value, cv=cv_split) I got following error from this code ValueError: The least populated class in y has only 1 member, which is too few. The minimum number of groups for any class cannot be less than 2. and my y_value looks like y_value = array([221900, 180000, 510000, ..., 360000, 400000, 325000]) I used this method before and y_value has the same shape as before. python data-mining cross-validation share ...
Read more

SMOTE: ValueError: Expected n_neighbors <= n_samples, but n_samples = 1, n_neighbors = 6

Image
0 $begingroup$ I am designing a multi class classifier for 11 labels. I am using SMOTE to tackle the sampling problem. However I face the following error:- error at SMOTE from imblearn.over_sampling import SMOTE sm = SMOTE(random_state=42) X_res, Y_res = sm.fit_sample(X_f, Y_f) error ~/.local/lib/python3.6/site-packages/sklearn/neighbors/base.py in kneighbors(self, X, n_neighbors, return_distance) 414 "Expected n_neighbors <= n_samples, " 415 " but n_samples = %d, n_neighbors = %d" % --> 416 (train_size, n_neighbors) 417 ) 418 n_samples, _ = X.shape ValueError: Expected n_neighbors <= n_samples, but n_samples = 1, n_neighbors = 6 Why does it say I have only 1 n_samples? When I tr...
Read more