Solve a polynomial equation involving a gamma function

I would recommend using the new M12 function AsymptoticSolve for that. Your equation:

eqn = FD1((d-2)/2, ηs) + FD1((d-2)/2, ηs - vd) == 2 FD1((d-2)/2, η0);

We have to find the order of zero order approximation ηs when vd is small:

Simplify(Solve(eqn /. vd -> 0), (η0 | vd) ∈ Reals)

{{ηs -> η0}}

Now use AsymptoticSolve:

AsymptoticSolve(eqn, {ηs, η0}, {vd, 0, 5})

{{ηs ->
vd / 2 + ((-6 + d) (-2 + d) (-1 + d) vd ^ 4) / (
1536 η0 ^ 3) + ((2 – d) vd ^ 2) / (16 η0) + η0}

nt.number theory – Characterizations of prime Mersenne numbers involving the sum of the divisor function

In this article, we denote the sum of positive divisors based on an integer $ n geq 1 $ as $$ sigma (n) = sum_ {1 leq d mid n} d. $$

Then a bonus of the form $ 2 ^ p-1 $ it's called a Mersenne bonus. These are related to the unsolved problem related to even perfect numbers. In this article, I present two hypotheses, in the hope of knowing if it is possible to deduce that these are rights as characterizations of the prime numbers of Mersenne.

Conjecture 1. Yes $ m geq 1 $ is an integer that satisfies

$$ sigma left ( sigma left ( sigma left ( frac {m (1 + 1)} {2} right) right) right) = (2m + 1) sigma left ( 2m + 1 right), tag {1} $$
then $ m $ is a Mersenne bonus.

Conjecture 2. Let be $ k geq 1 $ a fixed integer. Yes $ m geq 1 $ is a satisfying integer

$$ sigma left ( left ( frac {m (m + 1)} {2} right) ^ k right) = left (2 left ( frac {m + 1} {2} right) ^ k-1 right) frac {m ^ {k + 1} -1} {m-1} tag {2} $$
then $ m $ is a Mersenne bonus.

Question. Are rights the previous conjectures? What work can be done on the veracity of these? Thank you so much.

Do not hesitate either, if you wish, to add comments on these equations.

examples – What can be an interesting problem of differential equations involving the definition of Gudermannian function?

In this post, I note the Gudermannian function as $$ operatorname {gd} (x) = int_0 ^ x frac {dt} { cosh t} $$
and its inverse as $ operatorname {gd} ^ {- 1} (x) $, see if you need the definitions, alternative definitions and identities, history and meaning in mathematics of Wikipedia with title Gudermannian function, or the article Gudermannian from the MathWorld encyclopedia.

A few days ago, I was wondering what could pose some interesting differential equation problems involving these special functions, I'm talking about ordinary differential equations and partial differential equations (where l & rsquo; It may be possible to provide initial values ​​or boundary conditions to obtain well-posed problems).

My first attempt was a pendulum type equation, because I know that the pendulum equation has a good mathematical / physical content. So we can try to solve $$ y & # 39; (x) + y & # 39; (x) + operatorname {gd} (x) = 0. tag {1} $$
And it can be integrated easily, see the code solve y''+y'+gd(x)=0 in Wolfram Alpha online calculator.

And using the method of variation of parameters one can calculate, easily, the general solution of
$$ y & # 39; (x) + y (x) + operatorname {gd} (x) = 0. tag {2} $$

Question. I wondered if it was possible to pose an interesting problem involving differential equations and the function of Gudermann or its inverse. If possible with good mathematical content (I say that perhaps it is possible to propose an interesting problem by invoking the particular meaning of some of these functions). If this is feasible, do not hesitate to add your answer by adding your problem proposal and / or your comments about it (if you are asked, you can work with valuable conditions and special fields). Thank you so much.

I do not add any geometry-related tags, but I know that partial differential equations are closely related to the geometry of varieties.

Number Theory – Is it possible to deduce from perfect odd numbers convolution sums involving divisor functions or other arithmetic functions?

Divide and use certain identities of (1) I have deduced the following facts, see also the remarks below. After these introductory paragraphs, to motivate our question, I ask if we can deduce useful information for perfect odd numbers $ n $ of the literature evaluating the sums of convolutions.

Made. A) if $ n $ is an odd perfect number of the form $ 36 M + $ 9 then
is satisfied with the following identity $$ 4 sigma left ( frac {2n} {3} right) = 3 sigma left ( frac {2n} {9} right) +2 (n + sigma (n)) tag {1} $$ and in addition also the congruence $ sigma_3 (n) equiv 18 text {mod} $ 36 is holding. B) If our odd perfect number $ n $ has the form $ 12 million + $ 1 then satisfied $$ 3 sigma left ( frac {n-1} {12} right) + sigma left ( frac {3 (n-1)} {4} right) = 4 sigma left ( frac { sigma (n) -2} {8} right), tag {2} $$
and also congruences $ sigma_3 (n) equiv 6 text {mod} $ 12 and $ sigma_5 (n) equiv 2 text {mod} $ 12 hold.

Notes on previous facts. My belief is that the opposite direction of $ (1) $and the other equation $ (2) $ are true, but I can not obtain that these prove that each equation is a characterization of an odd perfect number of the given form. I've tried working by invoking Euler's theorem to get perfect odd numbers $ n = 2 ^ { alpha} 3 ^ { beta} m $ such as $ (2, m) = (3, m) = 1 $ by case, to be deducted from the declaration $ (1) $ a contradiction, but if my calculations are correct, it's a failure: case 1 ($ beta = $ 2 and $ alpha = $ 0) returns $ sigma (m) = frac {18} {13} m $; case 2 ($ beta> $ 2 and $ alpha = $ 0) will be $ (3 ^ { beta +1} -1) sigma (m) = 4 cdot 3 ^ beta m $; case 3 ($ beta = $ 2 and $ alpha geq 1 $) was $ sigma (m) = frac {2 ^ { alpha + 1} cdot 9m} {13} $ and finally case 4 ($ beta> $ 2 and $ alpha geq 1 $) returns $ sigma (m) = frac {2 ^ { alpha + 2} cdot 3 ^ { beta}} {3 ^ { beta + 1} -1} $. And I can not find any counter-examples, I can not find any integers $ n equiv 0 text {mod} $ 9 satisfactory $ (1) $ and I can not find any integer $ n equiv 1 text {mod} $ 12 and as $ 8 mid (-2+ sigma (n)) $satisfactory $ (2) $. (If my congruences are rights) I have not used a lot of specific information about perfect odd numbers to infer these congruences.

Question. Is it possible to deduce assertions for perfect odd numbers, assuming they exist, using convolution sums with divisor functions $ sigma_k (n) $ or other arithmetic functions? I speak of congruences, of convolution sums analysis in the case of a specialization related to perfect odd numbers, of useful identities, or of congruences or specializations having applications in the study of perfect odd numbers. Thank you so much.

If you want to follow the approach in my fact, your answer is welcome: try to compute special equations for perfect odd numbers or to study similar congruences for different divider functions. $$ sigma_ {k} (n) equiv a text {mod} b. $$ It was my attempt / method of gathering information about odd perfect numbers, extracted from convolution sums involving arithmetic functions.

I add here the references that I used to deduce my facts, I used a specialization of an identity that is shown in the second paragraph of page 255 of (1), I also add the reference for the Touchard's theorem..

References:

1) James G. Huard, Zhiming M. Ou, Blair K. Spearman and Kenneth S. Williams, Basic evaluation of some convolution sums involving divider functions, Theory of numbers for the millennium II, A. K Peters (2002).

(2) J. Touchard, On prime numbers and perfect numbers, Scripta Math. flight 19pp. 35-39 (1953).

Identifying MITM Malware / Unknown Malware Involving SSL Connections

One of the services on my server is a Discord bot. That's what made me look for why.

In my syslogs, I noticed three points of growing concern:

do-agent(1066): 2019/08/25 08:50:21 
Sending metrics to DigitalOcean: Post https://sfo2.sonar.digitalocean.com/v1/metrics/droplet_id: 
x509: certificate is valid for *.com.com, com.com, not sfo2.sonar.digitalocean.com

discord-botd(26673): 2019/08/25 09:03:50
(DG0) wsapi.go:827:reconnect() error reconnecting to gateway, Get https://discordapp.com/api/v6/gateway: 
x509: certificate is valid for *.com.com, com.com, 
not discordapp.com

discord-botd(26673): 2019/08/25 09:04:59
(DG0) wsapi.go:827:reconnect() error reconnecting to gateway, 
x509: certificate is valid for www.chinanetcenter.com, oir.6rooms.com, upload.v.6.cn, pic.v.6.cn, uploadmp3.v.6.cn, *.1z123.com, ulink.6.cn, 
passport.6.cn, shrek.6.cn, www.huanpeng.com, img.huanpeng.com, mlog.chinanetcenter.com, mauth.chinanetcenter.com, i.g-fox.cn, s1.chunboimg.com, s2.chunboimg.com, s3.chunboimg.com, sstatic.chunboimg.com, s0.chunboimg.com, app.showcai.com.cn, auth.microfun.cn, ss.sysad.cn, ss.sysair.cn, sso.kongzhong.com, stc2.kongzhong.com, passport.kongzhong.com, auth-live.kongzhong.com, api.kongzhong.com, i.zhulang.com, m.zhulang.com, s.zhulang.com, app5.zhulang.com, www.cmyynet.com, start.crestdrop.net, load.ginamind.com, fast.sireech.com, play.homesava.net, 
qfcnc.calaprilia.net, mobcdn.znoopbag.net, start88.trackeast.com, play88.trackeast.net, amengsk.haitangbase.net, *.1zhe.com, res.samsungshop.com.cn, mobcdn.clerkin.net, h5cont.trueleffy.net, marsara.nidajudo.com, *.app.meitudata.com, *.meitu.com, *.meipai.com, *.meitubase.com, *.img4399.com, *.converse.com.cn, apk-ssl.tancdn.com, m.wywna.cn, media-qtil.licdn.com, media-exp1.licdn.com, media-exp2.licdn.com, media-exp3.licdn.com, media.licdn.com, platform-qtil.linkedin.com, platform.linkedin.com, static-qtil.licdn.com, static-exp1.licdn.com, static-exp2.licdn.com, static-exp3.licdn.com, static.licdn.com, m.staff.tcl.com, *.ourdvsss.com, cdn.zj96596.com, addons.cdn.mozilla.net, *.tcl.com, *.mall.tcl.com, www.17un.com, m.li0gx.cn, korhal8.clerkin.net, start88.nidajudo.com, usercenter-stage.ewfresh.com, pay-stage.ewfresh.com, mall-stage.ewfresh.com, order-stage.ewfresh.com, settle-stage.ewfresh.com, m.leinue.cn, m.aonanp.cn, m.nbuic.cn, m.xrhen.cn, m.zosue.cn, m.bustz.cn, m.yuwxe.cn, m.bxuwg.cn, m.ykdsbsc.cn, m.rushour.cn, m.nlpzzd.cn, *.xunsd.cn, m.mmgdfr.cn, m.kigoxhz.cn, m.xxqysj.cn, m.mmzdjq.cn, m.ybwbmk.cn, *.zhangyixun.cn, m.i2d1kc.cn, m.hr00.cn, m.ubmhu.cn, fsdext.fshares.com, fscant.fshares.com, www.fshares.io, fscan.fshares.io, fsdex.fshares.io, manage.fsdex.fshares.io, api.fshares.io, dex.api.fshares.io, manage.dex.api.fshares.io, chain.api.fshares.io, manage.chain.api.fshares.io, jpa.api.fshares.io, jpa.node.fshares.io, hka.api.fshares.io, hka.node.fshares.io, fs.fshares.io, guide.fshares.io, browser.api.fshares.io, wallet.api.fshares.io, wss.api.fshares.io, wss.dex.api.fshares.io, back.dex.api.fshares.io, back.chain.api.fshares.io, gate.dex.api.fshares.io, *.sg2046.cn, 
not gateway.discord.gg

After a restart, it disappeared.

openssl s_client -showcerts -connect did not show anything unusual (although I would have liked to have done it before a reboot)

Some antecedents:

  • Server is an updated Fedora 28 server.
  • The non-default services that I run are:
    • a Golang-based Web server (HTTP REST API)
    • a bot disc based on Golang
    • Digital Ocean Statistics Officer
  • SSH has no password and the firewall is limited to certain IP addresses.

I have never encountered this problem before, nor am I able to find similar results on Google.

Is it possible to identify how this happened or even if it is worrying?

Should I use the server?

Thank you!

co.combinatorics – A convolutional identity of $ 1 involving the Motzkin Triangle

the Motzkin Triangle $ T (n, k) $ is built according to the rules:

(1) $ T (n, 0) = $ 1;

(2) $ T (n, k) = 0 $ if $ k <0 $ or $ k> n $;

(3) $ T (n, k) = T (n-1, k-2) + T (n-1, k-1) + T (n-1, k) $.

After some quantified proof, I ask:

QUESTION. Can you provide a combinatorial proof for the identity below?
$$ sum_ {k = 0} ^ nT (n, k) T (n, k + 1) = sum_ {k = 0} ^ n binom {2n} {2k + 1} binom {2k + 1 } k frac1 {k + 2}. $$

Note. It's good to give other justifications to add variety to the discussion here, but I wish a combinatorial argument.

Differential Identity Involving a Logarithm – Mathematics Stack Exchange

By studying an effective string theory, I found the following identity:
$$ ln x = lim_ {s rightarrow0} frac {d x ^ {s}} {ds}. $$
I am, however, puzzled as to its derivation. Naively I would say that
$$ lim_ {s rightarrow0} frac {d x ^ {s}} {ds} = lim_ {s rightarrow0} s x ^ {s – 1} = 0, $$
which is obviously not the right approach. So my question is, how can I prove the first identity?

Plus, I would appreciate it a lot if you could help me give these questions the right labels.

Statistical inference – A basic question about a randomized test involving Type I error.

I have a fundamental question in the context of the verification of statistical assumptions, specifically randomized tests. Suposse that I have two actions (alternatives) on a certain unknown parameter $ theta in Theta $: the null ($ H_0 $) and alternative hypotheses ($ H_1 $).

In this case, the sample space is $ (0,15) subset mathbb {R} $. We know that the critical function is given by
$$ phi (x) = P (reject , , H_0 , , | , , x , , observed) $$

I do not know exactly if this definition really implies a conditional probability. Suposse I have the following critical function

$$
phi (x) =
begin {cases}
0, quad x in (0,2) \
p, quad x in (2,10) \
1, quad x in (10,15) \
end {cases}
$$

I can understand why

$$ P (reject , , H_0 , , , , H_0 , , is , , true) , = 0 times P (x in (0,2)) + p times P (x in (2,10)) + 1 times P (x in (10,15)) $$

The right side looks a lot like a wait. But I can not understand.

Number Theory – On the solutions of two identities respectively involving the Euler function and the Pochhammer symbol, or Stirling numbers of the second type

In this article, we refer to Stirling numbers of the second type by $ {n brace k} $and the Pochhammer symbol as $ (n) _k $. On the other hand, we note the total function of Euler as $ varphi (n) $.

We consider the following problems, where we only consider non-trivial solutions, for positive integers, see below in the definition of problems.

Problem 1 Find non-trivial solutions $ (n, k, x, y) $, which are the solutions for integers $ 1 leq k <n $ and for integers $ x> $ 1 and $ y> $ 1 of

$$ varphi left ((n) _k right) = x ^ y. $$

Problem 2 Find non-trivial solutions $ (n, k, x, y) $, which are the solutions for integers $ 1 leq k <n $ and for integers $ x> $ 1 and $ y> $ 1 of

$$ varphi left ({n brace k} right) = x ^ y. $$

Examples.

A) Problem 1 has, for example, non-trivial solutions $ (n, k, x, y) = (35,4,72,3) $ and $ (16,6,24,5) $.

B) Problem 2 has, for example, non-trivial solutions $ (n, k, x, y) = (30,27,1512,2) $.

I do not know if there is one aspect to highlight of these problems. Is any of these problems more potentially interesting than the other?

Question. I would like to know how to start studying the non-trivial solutions, the mentioned conditions, of Problem 1 or Problem 2. What can be a first professional statement on one of these problems? What I ask is what is the first professional statement that can be done to solve some of these problems. Thank you so much.

What I'm asking is that I know that asymptotics, heuristics, congruences and divisivility are important facts in the study of equations on integers, but I do not know which of these two problems is more interesting that the other (if it is), and I do not know how to glimpse what should be a first professional statement for some of these problems (the problem is more appropriate to obtain statements about its solutions).

low convergence in $ L ^ 2 $ and integral convergence involving test functions

Let $ Omega $ to be a bounded set of $ mathbb {R} ^ n $ and $ (f_n) _n subset L ^ 2 ( Omega) $ such as $ f_n to f in L ^ 2 ( Omega) $ weakly $ L ^ 2 ( Omega) $. Then for a given test function $ phi in C ^ infty_c ( Omega) $, do we have the following convergent property:
$$
int_ Omega | u_n | phi , dx to int_ Omega | u | phi , dx, quad textrm {as $ n to infty $.}
$$