#recurrence — Public Fediverse posts

"Le caractère essentiel du raisonnement par récurrence c’est qu’il contient, condensés pour ainsi dire en une formule unique, une infinité de syllogismes." – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

"Le caractère essentiel du raisonnement par récurrence c’est qu’il contient, condensés pour ainsi dire en une formule unique, une infinité de syllogismes." – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

Fatal Debris Avalanche On An Anthropogenically Disturbed, Earthquake-Perturbed Slope During Antecedent Rainfall [Turkey]
--
https://doi.org/10.1007/s10346-026-02713-0 <-- shared paper
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https://eos.org/thelandslideblog/gungoren-hillslope-1 <-- shared technical article
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https://geohazard.itu.edu.tr/en/news-detail/2025/01/23/could-the-g%C3%BCng%C3%B6ren-arhavi-%28artvin%29-landslide-be-predicted <-- shared technical article
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https://zenodo.org/records/14625940 <-- shared technical article/paper
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#rainfall #precipitation #pluvial #massmovement #monitoring #Turkey #Türkiye #Güngören #Artvin #remotesensing #earthobservation #GIS #spatial #mapping #LiDAR #UAV #InSAR #radar #spatialanalysis #spatiotemporal #massmovement #risk #hazard #geology #landslide #engineeringgeology #anthropogenic #humnacimpacts #manmade #quarry #highway #stability #earthquake #trigger #slopestability #naturalhazard #disaster #debrisflow #fatal #lossoflife #publicsafety #fieldwork #model #failure #geomorphology #kinematic #geomorphometry #creeping #extremeweather #multiple #recurrence #progressive #groundmotion #susceptibility

Fatal Debris Avalanche On An Anthropogenically Disturbed, Earthquake-Perturbed Slope During Antecedent Rainfall [Turkey]
--
https://doi.org/10.1007/s10346-026-02713-0 <-- shared paper
--
https://eos.org/thelandslideblog/gungoren-hillslope-1 <-- shared technical article
--
https://geohazard.itu.edu.tr/en/news-detail/2025/01/23/could-the-g%C3%BCng%C3%B6ren-arhavi-%28artvin%29-landslide-be-predicted <-- shared technical article
--
https://zenodo.org/records/14625940 <-- shared technical article/paper
--
#rainfall #precipitation #pluvial #massmovement #monitoring #Turkey #Türkiye #Güngören #Artvin #remotesensing #earthobservation #GIS #spatial #mapping #LiDAR #UAV #InSAR #radar #spatialanalysis #spatiotemporal #massmovement #risk #hazard #geology #landslide #engineeringgeology #anthropogenic #humnacimpacts #manmade #quarry #highway #stability #earthquake #trigger #slopestability #naturalhazard #disaster #debrisflow #fatal #lossoflife #publicsafety #fieldwork #model #failure #geomorphology #kinematic #geomorphometry #creeping #extremeweather #multiple #recurrence #progressive #groundmotion #susceptibility

RECURRENCE (Estats Units) presenta nou àlbum: "Art of Survival" #Recurrence #MelodicDeath #GrooveMetal #Metalcore #Novembre2025 #EstatsUnits #NouÀlbum #Metall #Metal #MúsicaMetal #MetalMusic

RECURRENCE (Estats Units) presenta nou àlbum: "Art of Survival" #Recurrence #MelodicDeath #GrooveMetal #Metalcore #Novembre2025 #EstatsUnits #NouÀlbum #Metall #Metal #MúsicaMetal #MetalMusic

RECURRENCE (Estats Units) presenta nou single: "Remember Who the Enemy Is" #Recurrence #MelodicDeath #GrooveMetal #Metalcore #Juny2025 #EstatsUnits #NouSingle #Metall #Metal #MúsicaMetal #MetalMusic

RECURRENCE (Estats Units) presenta nou single: "Remember Who the Enemy Is" #Recurrence #MelodicDeath #GrooveMetal #Metalcore #Juny2025 #EstatsUnits #NouSingle #Metall #Metal #MúsicaMetal #MetalMusic

1/3 Notes on #UAP Discussions: What is it when #Recurrence on a predominant but still speculative technical theme is woven into current discourse that is widely distributed ? If you have direct knowledge you can of course just state it. Independent researchers who do this are unfortunately at a disadvantage ; your going to be up against major players on #arXiv probably within 72 hours and the question then becomes was timing right for the greater good ?
There is another consideration though…

"[Le raisonnement par récurrence] est un instrument qui permet de passer du fini à l’infini. Cet instrument est toujours utile, puisque, nous faisant franchir d’un bond autant d’étapes que nous le voulons, il nous dispense de vérifications longues, fastidieuses et monotones qui deviendraient rapidement impraticables. Mais il devient indispensable dès qu’on vise au théorème général [...]" – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

"[Le raisonnement par récurrence] est un instrument qui permet de passer du fini à l’infini. Cet instrument est toujours utile, puisque, nous faisant franchir d’un bond autant d’étapes que nous le voulons, il nous dispense de vérifications longues, fastidieuses et monotones qui deviendraient rapidement impraticables. Mais il devient indispensable dès qu’on vise au théorème général [...]" – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

Surprising ADHD research finds greater life demands linked to reduced symptoms https://www.psypost.org/surprising-adhd-research-finds-greater-life-demands-linked-to-reduced-symptoms/?utm_source=dlvr.it&utm_medium=mastodon #ADHDresearch #mentalhealth #remission #recurrence #lifechallenges

"Le caractère essentiel du raisonnement par récurrence c’est qu’il contient, condensés pour ainsi dire en une formule unique, une infinité de syllogismes." – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

"Le caractère essentiel du raisonnement par récurrence c’est qu’il contient, condensés pour ainsi dire en une formule unique, une infinité de syllogismes." – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

New version of CRP Toolbox with a fast function to calculate #recurrence microstates
https://tocsy.pik-potsdam.de/CRPtoolbox/

Fernando Rosas (unfortunately not on Mastodon) asked on bsky:

"Has anyone figured out what exactly is the relation between the ideas of feedback, recurrence, and self-reference?"

A really interesting question.

He pointed to this paper for ideas: https://arxiv.org/abs/1711.02456
"Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos"

I did some desk research and found this cool paper:
https://arxiv.org/abs/1112.2141
"Resolving Gödel's Incompleteness Myth: Polynomial Equations and Dynamical Systems for Algebraic Logic"
that argues there is no essential incompleteness in formal reasoning systems if you look closely enough (using a more elaborate formalism based on polynomial equations to represent and evaluate logical proposition).

I wonder if analogous construction could be created for related theorems like the halting problem in computability theory.

#DynamicalSystems #IncompletenessTheorem #PolynomialEquations #HaltingProblem #Undecidability #SelfReference #Recurrence

Fernando Rosas (unfortunately not on Mastodon) asked on bsky:

"Has anyone figured out what exactly is the relation between the ideas of feedback, recurrence, and self-reference?"

A really interesting question.

He pointed to this paper for ideas: https://arxiv.org/abs/1711.02456
"Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos"

I did some desk research and found this cool paper:
https://arxiv.org/abs/1112.2141
"Resolving Gödel's Incompleteness Myth: Polynomial Equations and Dynamical Systems for Algebraic Logic"
that argues there is no essential incompleteness in formal reasoning systems if you look closely enough (using a more elaborate formalism based on polynomial equations to represent and evaluate logical proposition).

I wonder if analogous construction could be created for related theorems like the halting problem in computability theory.

#DynamicalSystems #IncompletenessTheorem #PolynomialEquations #HaltingProblem #Undecidability #SelfReference #Recurrence

Is there a good PHP recurrence package out there? I'm look for something that will handle RRULE stuff.

#PHP #RRULE #Recurrence

Is there a good PHP recurrence package out there? I'm look for something that will handle RRULE stuff.

#PHP #RRULE #Recurrence

"Le raisonnement par récurrence [...] est un instrument qui permet de passer du fini à l’infini [...]" – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence #maths #math

"Le caractère essentiel du raisonnement par récurrence c’est qu’il contient, condensés pour ainsi dire en une formule unique, une infinité de syllogismes." – Henri Poincaré (1854-1912)
#citation #mathématiques #récurrence

So, the first four pseudoprimes generated by the #recurrence sequence
a(n) = a(n-1) - a(n-2) - a(n-4), starting with (4, 1, -1, -2),
(that is, composite numbers passing the probable-prime test a(p) ≡ 1 mod p)
are, unless I have slipped up somewhere, 2706, 5205530, 42161779, 1146180898. That's a big gap before number four!

So, the first four pseudoprimes generated by the #recurrence sequence
a(n) = a(n-1) - a(n-2) - a(n-4), starting with (4, 1, -1, -2),
(that is, composite numbers passing the probable-prime test a(p) ≡ 1 mod p)
are, unless I have slipped up somewhere, 2706, 5205530, 42161779, 1146180898. That's a big gap before number four!

Here's a new one of these prime-detecting #recurrence sequences for you:

a(n) = a(n-1) - a(n-2) - a(n-4)
Start with 4, 1, -1, -2.
For prime p, a(p) ≡ 1 (mod p).

My search in 2013 passed over this because the first pseudoprime is just 2706, but the second one is 5205530 and the third is > 40 million.

The other good fourth-order one with coefficients in {+-1, 0} just seems to be basically the Perrin case again (characteristic polynomial is a multiple of the Perrin one).

But the analysis having to do with PV numbers doesn't apply to Lehmer's number, since it is a "Salem number": most of the other roots are ON the unit circle and will keep wreaking havoc with the moduli arbitrarily far out. So I understand that less. It is conjectured to be the smallest Salem number. #recurrence

But the analysis having to do with PV numbers doesn't apply to Lehmer's number, since it is a "Salem number": most of the other roots are ON the unit circle and will keep wreaking havoc with the moduli arbitrarily far out. So I understand that less. It is conjectured to be the smallest Salem number. #recurrence

Anyway, what got me started on all this again was that a post here that I can no longer find mentioned "Lehmer's number", the leading root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1. Naturally I wondered if the recurrence corresponding to this polynomial was a good prime detector. It is: the only pseudoprime less than, I think, 4 million is 2014. But it turns out that 10th-order recurrences that good are not all that hard to find, so it isn't super special in that respect. #recurrence

Anyway, what got me started on all this again was that a post here that I can no longer find mentioned "Lehmer's number", the leading root of x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1. Naturally I wondered if the recurrence corresponding to this polynomial was a good prime detector. It is: the only pseudoprime less than, I think, 4 million is 2014. But it turns out that 10th-order recurrences that good are not all that hard to find, so it isn't super special in that respect. #recurrence

So I guess all of this gives us a generalization of the Pisot-Vijayaraghavan numbers beyond the positive reals, and I'm sure this is well-known and someone has studied this already, but I haven't seen it. #recurrence

So I guess all of this gives us a generalization of the Pisot-Vijayaraghavan numbers beyond the positive reals, and I'm sure this is well-known and someone has studied this already, but I haven't seen it. #recurrence

I also found a lot where the leading roots are a conjugate pair of complex numbers, and there, if you add their powers, you get this sort of exponentially exploding sinusoidal oscillation--but (if the other roots have norm < 1) one where all the values get exponentially close to integers! That's real weird. #recurrence

I also found a lot where the leading roots are a conjugate pair of complex numbers, and there, if you add their powers, you get this sort of exponentially exploding sinusoidal oscillation--but (if the other roots have norm < 1) one where all the values get exponentially close to integers! That's real weird. #recurrence

It is, by the way, profoundly strange to see this exact expression where we raise a bunch of generally irrational and/or complex numbers to large powers, add them together, and it turns out to just be *rounding* the leading term to an integer.

But that's a remarkable property of these PV numbers--they get close to integers when you raise them to large powers. But the PV numbers are just the positive real ones--for my fifth-order recurrence, the leading root is negative. #recurrence

It is, by the way, profoundly strange to see this exact expression where we raise a bunch of generally irrational and/or complex numbers to large powers, add them together, and it turns out to just be *rounding* the leading term to an integer.

But that's a remarkable property of these PV numbers--they get close to integers when you raise them to large powers. But the PV numbers are just the positive real ones--for my fifth-order recurrence, the leading root is negative. #recurrence

Anyway, if you look at that Binet-like formula, you can see that the sum should pretty quickly be dominated by the root(s) of largest norm.

But we're doing modulo arithmetic on the sums, so the smaller roots will still be important--but *if* they all have norm less than 1, pretty soon all they're doing is rounding the sum to the nearest integer. But early on, they can be doing more than that. I think that is what causes some of the small-number weirdness. #recurrence

Anyway, if you look at that Binet-like formula, you can see that the sum should pretty quickly be dominated by the root(s) of largest norm.

But we're doing modulo arithmetic on the sums, so the smaller roots will still be important--but *if* they all have norm less than 1, pretty soon all they're doing is rounding the sum to the nearest integer. But early on, they can be doing more than that. I think that is what causes some of the small-number weirdness. #recurrence

But as yet, I have no good feel for what makes a "good" sequence whose probable-prime test passes few composites, preferably with a few small-integer coefficients in the recurrence.

The polynomials associated with the Lucas and Perrin numbers are famous: their leading roots are the golden ratio and the "plastic number" respectively, small Pisot-Vijayaraghavan numbers as I was saying. But this doesn't lead to a simple pattern for "good" sequences. #recurrence

But as yet, I have no good feel for what makes a "good" sequence whose probable-prime test passes few composites, preferably with a few small-integer coefficients in the recurrence.

The polynomials associated with the Lucas and Perrin numbers are famous: their leading roots are the golden ratio and the "plastic number" respectively, small Pisot-Vijayaraghavan numbers as I was saying. But this doesn't lead to a simple pattern for "good" sequences. #recurrence

Some other relevant facts: Sequences of this type all have a simple closed-form "Binet-like" exact expression for their elements:

\[ a_n = \sum_{j} c_j^n \] where the c_j are the eigenvalues of the matrix M. Those are also the roots of a monic polynomial that comes from det (Ix - M) = 0. Basically the leading coefficient is always 1 and the others are minus the recurrence coefficients. So these recurrences seem to have a lot to do with the study of algebraic integers. #recurrence

Some other relevant facts: Sequences of this type all have a simple closed-form "Binet-like" exact expression for their elements:

\[ a_n = \sum_{j} c_j^n \] where the c_j are the eigenvalues of the matrix M. Those are also the roots of a monic polynomial that comes from det (Ix - M) = 0. Basically the leading coefficient is always 1 and the others are minus the recurrence coefficients. So these recurrences seem to have a lot to do with the study of algebraic integers. #recurrence

What I was poking around with more recently was the question "are there more like that, where the first pseudoprime is very small but they're actually quite infrequent after that?" I only found *that* one because my search in 2013 was ignoring pseudoprimes that came out of the initial seed values! Otherwise I would have skipped right over it.

I refined my search and yes, that happens pretty often especially when you start adding a lot of terms. So I've been thinking about why.
#recurrence

What I was poking around with more recently was the question "are there more like that, where the first pseudoprime is very small but they're actually quite infrequent after that?" I only found *that* one because my search in 2013 was ignoring pseudoprimes that came out of the initial seed values! Otherwise I would have skipped right over it.

I refined my search and yes, that happens pretty often especially when you start adding a lot of terms. So I've been thinking about why.
#recurrence

Back then I found the first three pseudoprimes, then Rich Holmes found all the others up to a billion (there are not many), then Dana Jacobsen found the eighth one (which is over 3 billion) and determined that the next one is greater than 1.4 x 10^11.
#recurrence

Back then I found the first three pseudoprimes, then Rich Holmes found all the others up to a billion (there are not many), then Dana Jacobsen found the eighth one (which is over 3 billion) and determined that the next one is greater than 1.4 x 10^11.
#recurrence

So back in 2013, I tried that and found all kinds of interesting stuff. But the most fun one was fun enough to end up as https://oeis.org/A225984 . In that one, the matrix trace is -1, so a_p for prime p is congruent to -1 mod p, or equivalently to p-1.

There, the first composite that passes (listed as https://oeis.org/A225876) is 4, and it actually comes from one of the initial seed values. But the second one is 14791044, which is pretty amazing.
#recurrence

So back in 2013, I tried that and found all kinds of interesting stuff. But the most fun one was fun enough to end up as https://oeis.org/A225984 . In that one, the matrix trace is -1, so a_p for prime p is congruent to -1 mod p, or equivalently to p-1.

There, the first composite that passes (listed as https://oeis.org/A225876) is 4, and it actually comes from one of the initial seed values. But the second one is 14791044, which is pretty amazing.
#recurrence

Of course, that formula might pass some composite numbers too. The "pseudoprimes" for these sequences are a sort of extension of the idea of Fermat pseudoprimes. Kevin Brown did some messing around with this years before I did: https://www.mathpages.com/home/kmath346/kmath346.htm
But now we can do systematic searches for sequences of this type with infrequent pseudoprimes.
#recurrence

Of course, that formula might pass some composite numbers too. The "pseudoprimes" for these sequences are a sort of extension of the idea of Fermat pseudoprimes. Kevin Brown did some messing around with this years before I did: https://www.mathpages.com/home/kmath346/kmath346.htm
But now we can do systematic searches for sequences of this type with infrequent pseudoprimes.
#recurrence

If you write the recurrence that way, then the Lucas sequence and the Perrin sequence are both of the form a_n = Tr M^n. That's what determines the somewhat peculiar initial seed values. And now the "probable-prime" test looks like this:

for prime p, Tr M^p ≡ Tr M mod p.

That looks almost exactly like Fermat's Little Theorem, but with a matrix. At 1x1 it reduces to Fermat's Little Theorem.
#recurrence

If you write the recurrence that way, then the Lucas sequence and the Perrin sequence are both of the form a_n = Tr M^n. That's what determines the somewhat peculiar initial seed values. And now the "probable-prime" test looks like this:

for prime p, Tr M^p ≡ Tr M mod p.

That looks almost exactly like Fermat's Little Theorem, but with a matrix. At 1x1 it reduces to Fermat's Little Theorem.
#recurrence

So for instance, apply that matrix to the column vector of [3, 0, 2] and you get the column vector of [0, 2, 3]. The last two numbers 0, 2 shift out of the way and the bottom row of the matrix does the Perrin sequence addition to get the new number 3.
#recurrence

Recurrences of this type follow a general form. Suppose we write the recurrence as an nxn matrix M that acts on the last n elements of the series to produce the last n elements of the revised series. It will always have some off-diagonal 1s to shift the existing entries out of the way, then the coefficients in the recurrence appear in the bottom row. So for the Perrin numbers, it's \[ M = \left(\begin{array}{rrr}
0 & 1 & 0 \\
0 & 0 & 1 \\
1 & 1 & 0
\end{array}\right)
\]
#recurrence

Can we do better? Well, there are the Perrin numbers. Start with 3, 0, 2, then always add the two numbers BEFORE the most recent one:

3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29...

Count these starting with 0, then for any prime p, a_p ≡ 0 mod p. And the first composite one (the first "Perrin pseudoprime") is 271441. That's a lot better! Can we do better than that?
#recurrence