Natural Density Puns

Natural is beautiful, yes. Embrace your natural hair, yes. All of that is important and there is absolutely nothing wrong with that. But I’m going to keep it real some of us unfortunately are not blessed in the natural hair department. The dream of thick luscious natural locks is not the reality for many naturals no matter how hard we try. I definitely go through waves of loving and hating my natural hair but it’s definitely been more of a hate relationship than anything else. I have 4c hair which I’ve learned to take care of in my 11 years being natural and trust me it has been quite the journey and I have had plenty of set backs.

In the natural hair community and society as a whole I still feel as though 4c hair does not get the love and recognition it deserves especially if it does not meet “special” criteria. What do I mean by “special criteria”? Here are a few examples:

1. Extremely high density
2. Thick coarse strands
3. Long

Many of us who decided to go natural had this idea in our heads of how our hair would be once our hair reached optimal health free of chemicals and relaxers. In our heads we had dreams of a thick, full, and luscious afro. Many of us starting our natural hair journey knew nothing about density, porosity or strand thickness, or the “real” science behind it all. Especially some of us naturals who started our journey a decade ago when going natural was still very new and there was very little information out there. The longer we have been in this journey and learned about our hair the more reality began to slap us in the face. The reality is long thick dense natural hair may not be in the cards for many of us and this due to one thing only, genetics. I’m just keeping it real. Many of us have tried every remedy imaginable, every hair supplement available to no avail. Why? Because those things cannot alter our genetic makeup. They can’t change the number of follicles on our head or the thickness of our strands.

The things that many women in the natural hair celebrate and embrace I cannot participate in. Twist outs on my fine hair are a joke, I absolutely HATE HATE HATE twist outs. I have tried every method imaginable, every product imaginable, I have not had one single twist out turn out right. Braid outs are a straggly mess and amplify my fine strands. Twists make me look like a wet balding poodle, my picked out afros are lopsided and see through. Wash and gos you say? Well that’s just asking for breakage and single strand kno

This is my hair like two years ago with texturizer after air drying for about an hour

https://imgur.com/vorv8yV

And this is my hair 3 months post going absolutely bald and 2 weeks trying to follow CGM, not too long after leaving the shower. The curls become less defined as my hair dries.

https://imgur.com/y7VpbOr

i discovered ithkuil literally less than a week ago , so i was wondering which natural language is the most "similar " to ithkuil , i searched a little and the closest i could find to ithkuil in conveying information was Latin . But i would like to hear from here.

(May be posting some questions here and there , because im interested in ithkuil)

I am planning on cutting my own bangs (I don't live in an area with good curly hairdressers) and am looking for some advice on how to section / prepare the hair for cutting. I have read some advice (cut dry, point cut, cut individual curls etc) but am still a little unsure how to section the hair for styling. My hair has a quite strong natural part on the side - should I part it in the centre on the wash day before cutting? Or style as normal and just brush forward the curls I want to cut? How far back should I cut the bangs? I already struggle with volume a bit on the non-dominant side so I don't want to risk it being permanently flat. Any help or advice is appreciated!

Current routine:

• Asiam coconut cowash
• Noughty to the rescue moisture boost conditioner
• Curlflo leave in conditioner and/or moisturising cream
• Giovanni LA hold styling gel

https://preview.redd.it/j16z1qd5pxb61.jpg?width=516&format=pjpg&auto=webp&s=dc71ef04034ac195479f88a5695a0ed3f41f6c93

https://preview.redd.it/mvoyyyyxoxb61.jpg?width=1582&format=pjpg&auto=webp&s=fe433f477f784b0535eb5fe840704dabf7181f6f

Not me in the Pic/Video

I am not a doctor, not do i play one on the internet!

I have very low density baby fine hair. 2c/3ab curls that do tend to lose volume and shape.  I had just discovered tha using the CM method..my cute-when-wet curls (that I hot perms for in the 80's and 90"s were achievable.

I did a DIY COVID extreme chop of my damaged hair and a bad cut (from one of the no appointment places jut before COVID-19 and it was NOT growing out well).  My new cut was a (ahem) bit shorter that I expected. It is not TOO bad (kind of a mid neck heavy bang shag).

Little did i know that wet cutting bangs at tip of the nose length would result in them.springing up an inch + above my brows!😳🤭

I am desperately trying to grow my hair out faster than .5 inch / month....and when doing research on growth promotion found studies on ways to hrow it faster AND have it grow in thicker and more dense.

I just started this process and thought I would share what Ive learned (but am just implementing so results are noy in).

I have some medical conditions that are exhausting & stopped blowdrying my hair straight & washing daily. BAM...curls!  Then I discoveted the CM method (which I follow most of so I could have better hair and less wash days & hopefully spend that limited energy elsewhere.  I'm enjoying experimenting with product and technique..but it is tiresome & I hope to perfect my routine soon.  I finally found products that work for me (AG curl line + their dry lift). The dry lift is a paste that you use at your roots to give lift. It also can absorb excess oil like a dry shampoo.it goes a loooong way.

If anyone knows a good hairspray that doesnt give Helmet Head please let me know! Drying alcohols not preferable but OK otherwise CG friendly.

NOW FOR THE HOOD STUFF ON HAIR GROWTH 🤓

Scalp Oil for hair growth, density and health:

Black seed oil, pumkin seed oil, fennegreek oil (cold pressed)  & Lavender,  tea tree & rosemary essential oils all promote hair growth and shaft health and/or prevent loss.  The first 3 oils are carrier oils and you can mix essential oils into them.  First 3 also have DHT blockers for male & female pattern balding amongst other hair/scalp growth promoting properties. The essential oils vary a little in what they do, but I believe  I picked the best four. Ratio of oils to carrier listed at bottom.  START WITH A LOW CONCENTRATION to make sure you don't react...then you can increase concemtrat

What I have heard from before is the repeated mentioning that Chebotarev density theorem for natural density only work for number field, and not function field of an algebraic curve. I didn't know how the proof goes before (it's really hard to find proof of this), but now that I have seen the proof for natural density, I'm surprised that it doesn't work for algebraic curve. So I probably misunderstood something about the proof, but I can't see where.

After making some modifications to adapt to the case of algebraic curve, I got the following sketch of the proof. Can someone point out to me where things gone wrong?

Let L/K be Galois extension of function field of algebraic curve, k=Fq be their constant field. Let g be an element of the Galois grop.

• The reduction to cyclic extension still work, with some modification. Let M be fixed field of g. We just need to reduce to the case where we only consider primes in M that are inert in L. Once we got reduction, by the same counting argument as in the number field case, each prime in K with Frobenius element g lift to exactly [M:K]/|C(g)| such inert prime in M, and the claim is true assuming it's true for L/M. The reduction to consider only inert primes is different in function field, but the result is the same. In number field, primes with norm being lowest prime power dominates; but in function field, primes with norm being highest power of q dominates. But in either way, the result is the same: when counting primes with Frobenius g, you can ignore primes with nontrivial degree over K.

• For the cyclic extension, nonvanishing of L-function still go through, but for even better reasons. We still have zetaL/zetaK is a product of L-function of nontrivial characters. But any zeroes and poles off the critical line of zetaL and zetaK is the same, by Weil conjecture. In particular, L-function is holomorphic and nonvanishing on Re(s)=1, but even better, it's still holomorphic and nonvanishing for all Re(s)>1/2.

• After L-function worked out, we can still feed its log derivative into Weiner-Ikehara with no real changes. The only differences is in the result, since the norms are all power of q, but this makes thing even easier and stronger. In particular, using the fact that highest degree primes dominates, this even gives us an asymptotic count for each degree.

So can someone help?

EDIT: I believed the issue had been solved. Claim is true for geometric extension, with essentially the same proof (Weiner-Ikehar