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End points? Who needs end points?
A few months back, I was looking at inner products over one argument function spaces (see here for a snapshot of my thinking at the time). The classic example are the Legendre polynomials.
You can build a function up by looking at its projection along very particular polynomials of increasing degree (constant, linear, quadratic, then degree 3, 4, etc). These polynomials can be thought out as perpendicular projections in the space of functions over the interval [-1, 1] with an inner product (the more abstract version of the vector dot product) being the product of the two polynomials (you can extend to some pretty arbitrary functions) integrated from -1 to 1.
Anyway, long story short, the polynomials for this case have really nice equations to generate the polynomials - the link describes at least four such relations.
So there's a more general case where one integrates the product of two functions of x times a weight function w(x) (which is always nonnegative), integrated again from -1 to 1. In this case, there are no known nice relations unlike the Legendre polynomials above.
You can still construct sequences of increasing degree polynomials just like before, using what's called Gram-Schmidt orthonormalization. You start with x^n as your degree n guess, and then subtract from that the projection of x^n into each of the lower degree polynomials. What's left over is not in the direction of any of the lower degree polynomials you already constructed. Then you divide by the square root of the integral of that residual polynomial squared times that weight function over [-1,1].
Well, I thought I had a trick for fixing that in the case where w(x) is strictly positive and bounded both from below and above by positive numbers (that is, 0 < M1 < w(x) < M2, for all x in [-1, 1]. (Incidentally, the Legendre system is a special case where w(x)=1 for all x in [-1,1].)
The idea is that I would map these polynomials into a new polynomial structure which happens to reduce to the Legendre polynomial system. Then I might be able to pull back those nice Legendre relations to the original non-Legendre situation.
So let's say that P_k(t) is the degree k Legendre polynomial over variable t. Q_l(x) is the degree l polynomial in the weight w system over variable x. I deliberately use two different variables because the polynomial systems will end up being over different multiplication systems which cause all kinds of complications.
Q_l is the usual polynomial over the polynomials ℝ[x], (ℝ being real numbers - this just means real valued polynomials of x) with addition + and multiplication *. I'll write it as (ℝ[x], +, *). P_k(t) will look similar with the same addition, but a completely different and unintuitive multiplication ☆, so we'll write that as (ℝ[t], +, ☆). I'll call the second "star polynomials". Powers of x will be written like x^3 (x to the power 3). Powers of t will be written like t^☆3 to indicate the funky multiplication being used.
The key mapping here between the two flavors of polynomials, is a sequence of polynomials {r_k(x)}, (which will be defined indirectly) where degree r_k = k, k >=0, mapped directly to t^☆k, which is also degree k. It's a basis (maximal set of linearly independent functions) mapped to another basis. This is enough to fully define a one-to-one and onto linear map between (ℝ[x], +, *) and (ℝ[t], +, ☆) with degree and the addition (+) operator being compatible, but not compatibility between the two multiplication operators.
Another important thing that is broken is evaluation of the variable. In the original polynomial series, you can evaluate x and then multiply, or multiply and then evaluate. They give you the same answer. That is (p*q)(3) = p(3)*q(3). It doesn't work for (ℝ[t], +, ☆). Let's say that x -> t, but x^2-3 -> t^☆2. If we naively try to set t = 0, then t^☆2 should be 0. But actually, t=0 implies x=0 and x^2-3 = -3. So t^☆2 = -3. Needless to say, this has ugly repercussions.
Anyway, r_k(x) is chosen so that Q_k(x) maps to P_k(t) for each k >=0. That is, in the ring based on x and normal multiplication, we have the Q_k basis over the weight w system. But we map it to a ring with Legendre polynomials over the t and the star multiplication.
So what can we say about the system. First, it's not elementary to deduce that star multiplication is commutative and associative. You have to define the multiplication so that a☆b = b☆a, and a☆(b☆c) = (a☆b)☆c. It turns out that the definition formula, (t^☆k)☆(t^☆l)=t^☆(k+l) is exactly what you need for both commutativity and associativity. That plus the fact that the star powers of t form a basis, means we get exactly what we need to move on. We dodge our first major problem.
Another feature that comes surprisingly easy is that continuous functions map to continuous functions. Just because we're mapping polynomials to (star) polynomials doesn't mean that we can map the limits of polynomials (such as continuous functions) to limits of polynomials. The somewhat peculiar definition of the weight function w(x) sandwiched between two numbers M1 and M2 is sufficient to get that polynomials that converge to continuous functions map to star polynomials that also converge to continuous functions.
We also can define a derivative, D over our star polynomials. It's linear and maps basis elements, D(t^☆k) = k t^☆(k-1), for k >=0 (equals zero when k=0). We get the usual derivative properties like D(p☆q) = Dp☆q + p☆Dq. I think chain rule works as well since it usually follows from the previous formula.
This allows us to construct an indefinite integral, I. I is what's called the generalized inverse of D. Linear functions need not have inverses, but they always have some sort of generalized inverse. Here, D (I(p))=p and I(D(p)) = p+C, where C is a constant. It's not quite an inverse because D maps constant functions to zero and thus, when you try to invert D with I, you'll be off by some unknown constant function C.
Here's where things went south. I tried to go from the above indefinite integral to a definite integral where I'm integrating from -1 to 1. The idea was that in my original polynomial space Q_k was the orthonormal basis (meaning perpendicular and length 1) of the weight w system. It is mapped directly to P_k, the orthonormal (well when scaled right) basis for the Legendre system. Two different bases and inner products but with the same outcome. (To show the second's inner product would have taken using one or more of the relations that Legendre polynomials have.) To then use the Legendre relations, one would first push the elements of the normal polynomials to the star polynomials, apply the desired Legendre relation, and pull them back to normal polynomial space.
The catch though is that one can't evaluate star polynomials consistently and hence, there is no good boundary conditions [the end points of the title] for which a definite integral can be defined. Perfectly good math ruined by a flaw in the structure.
A final remark is that this sort of thing is a simple version of renormalization from physics. The process is normally used to look at behavior of scale, particularly in attempts to remove infinite blowups that can show up as some aspect of the system is allowed to go to infinity (like perhaps the number of particles or energy of the system). My angle was that renormalization might also shift existing systems into a form that has a lot of nice properties and relations on it, just like my attempt to shift an arbitrary weight w system without known relations to the special Legendre system.
YouTube Fights Disinformation
YouTube recently took down a video of a bunch of guys having a chat wherein they say kids don't need to wear masks. Completely understandable being as the guys in question weren't proper politicians or talking heads but instead completely unqualified chuckleheads who were only medical experts from sketchy places like Harvard, Oxford, and Stanford.
University of Virginia Bans Critical Thinking
I shit you not, they actually suspended a student over asking problematic questions of a lecturer rather than simply Listening and Believing. And he's not allowed back to class until he undergoes psychiatric evaluation.
A Quick Reminder
Epstein did not kill himself.
Too Fucking Funny
MLB has decided that since Georgia passed racist Voter ID laws, they're moving their All-Star game over to Colorado. Which has had Voter ID laws for some time now. Maybe they don't see it as an issue since they're really good at keeping the black people run out of Colorado, so there's very few folks getting oppressed?
Testimony on the George Floyd killing
Speaker 1: (01:52)
Are you familiar with the use of force continuum?
Richard Zimmerman: (01:56)
Yes.
Speaker 1: (01:57)
Is that part of the Minneapolis Police Department use of force policy?
Richard Zimmerman: (02:01)
Yes, it is.
Speaker 1: (02:02)
Can you just describe in general what that means to the jurors
Richard Zimmerman: (02:05)
Yeah. Basically, the use of force continuum is guidelines, or it’s policy actually, that we have to follow. It’s when, for instance, when you arrive at a scene, no matter what the scene, the first level, the lowest level would be just your presence at a scene, in uniform. The next step up, maybe your verbal skills that you’ve learned to help diffuse a situation or learn information about whatever the situation is. The next step would be a soft technique, escorting the person by their arm, that type of thing. The next level would be a hard technique. That’s where you would use your, you maybe have to use your mace or handcuffs, that kind of thing. Finally, the top level on the continuum is deadly force.
[...]
Speaker 1: (04:11)
Have you ever, in all the years you’ve been working for the Minneapolis Police Department been trained to kneel on the neck of someone who is handcuffed behind their back, in a prone position?
Richard Zimmerman: (04:24)
No, I haven’t.
Speaker 1: (04:27)
Is that, if that were done, would that be considered force?
Richard Zimmerman: (04:30)
Absolutely.
Speaker 1: (04:32)
What level of force might that be?
Richard Zimmerman: (04:35)
That would be the top tier, the deadly force.
Speaker 1: (04:38)
Why?
Richard Zimmerman: (04:39)
Because of the fact that if your knee is on a person’s neck, that can kill them.
[...]
Speaker 1: (05:25)
Okay. Well, let me ask you this again. If you, as an officer, according to the training, you handcuff somebody behind the back, what’s your responsibility with regard to that person from that moment on?
Richard Zimmerman: (05:47)
That person is yours. He’s your responsibility. His safety is your responsibility, his wellbeing, and is your responsibility,
Speaker 1: (06:01)
Once you handcuff somebody, does that affect the amount of force that you should consider using?
Richard Zimmerman: (06:08)
Absolutely.
Speaker 1: (06:09)
How so?
Richard Zimmerman: (06:11)
Once a person is cuffed, the threat level goes down all the way. They’re cuffed. How can they really hurt you?
Speaker 1: (06:26)
Well, certainly there could be certain circumstances when a cuffed person could still be combative?
Richard Zimmerman: (06:32)
Oh, absolutely. Yeah. Yeah. But you getting injured is way down.
Speaker 1: (06:39)
What you mean by that?
Richard Zimmerman: (06:40)
Well, if you’re, you could have some guy try to kick you or something, but you can move out of the way. That person is handcuffed, and the threat level is just not there.
Speaker 1: (06:59)
So, by handcuffing somebody you’ve taken away some of their ability to harm you?
Richard Zimmerman: (07:04)
Absolutely.
Speaker 1: (07:08)
If somebody who is handcuffed becomes less combative, does that change the amount of force that an officer is to use under policy?
Richard Zimmerman: (07:19)
Yes.
Speaker 1: (07:21)
How so?
Richard Zimmerman: (07:23)
Well, if they become less combative, you may just have them sit down on the curb or, the idea is to calm the person down. if they are not a threat to you at that point, you try to help them, so that they’re not as upset as they may have been in the beginning.
Speaker 1: (08:01)
In your 30 years of training with the Minneapolis Police Department, and your experience, have you been trained on the prone position?
Richard Zimmerman: (08:15)
Yes.
Speaker 1: (08:17)
What has your training been, specific to the prone position?
Richard Zimmerman: (08:23)
Well, once you secure or handcuff a person, you need to get them out of the prone position as soon as possible, because it restricts their breathing.
Speaker 1: (08:38)
When you handcuff somebody behind their back … well, as part of training, have you been handcuffed behind the back?
Richard Zimmerman: (08:44)
Yes.
Speaker 1: (08:46)
Have you been trained on what happens to individuals when they’re handcuffed behind the back?
Richard Zimmerman: (08:52)
Yes.
Speaker 1: (08:52)
So, when somebody is handcuffed behind their back, how does it affect them physically?
Richard Zimmerman: (08:57)
It stretches the muscles back through your chest, and it makes it more difficult to breathe.
Speaker 1: (09:07)
If you put somebody in the prone position … Well, is it well-known this danger of putting somebody in the prone position?
Speaker 5: (09:16)
Sustained.
Speaker 1: (09:17)
How long have you had training on the dangers of the prone position, as part of a Minneapolis Police Officer?
Richard Zimmerman: (09:24)
For, since 1985.
Speaker 1: (09:30)
Is it part of your training regularly to learn about keeping somebody in the prone position?
Richard Zimmerman: (09:36)
Yes.
Speaker 1: (09:37)
What has the training band with regard to the prone position?
Richard Zimmerman: (09:41)
Once a person is cuffed, you need to turn them on their side or have them sit up. You need to get them off their chest.
Speaker 1: (09:51)
Why?
Richard Zimmerman: (09:52)
Because of the, as I had mentioned earlier, your muscles are pulling back when you’re handcuffed, and if you’re laying on your chest, that’s constricting your breathing even more.
Speaker 1: (10:14)
In your training as a Minneapolis Police Officer, are you provided with training on medical intervention?
Richard Zimmerman: (10:23)
Yes.
Speaker 1: (10:24)
I assume you’re not taught to be paramedics, but you receive some level of training?
Richard Zimmerman: (10:29)
Yeah. We’re first responders I think, is what our category would be.
Speaker 1: (10:33)
Does that include doing what we think of a CPR, chest compressions?
Richard Zimmerman: (10:37)
Yes.
Speaker 1: (10:38)
How often is that part of your training?
Richard Zimmerman: (10:42)
CPR? It’s every other year or so.
Speaker 1: (10:47)
As part of your training within the Minneapolis Police Department policies, is there an obligation to provide medical intervention when necessary?
Richard Zimmerman: (10:57)
Absolutely.
Speaker 1: (10:58)
What is the general teaching that you get with regard to medical intervention?
Richard Zimmerman: (11:04)
Well, again, it’s been that you need to provide medical care for a person that is in distress.
Speaker 1: (11:16)
Would that be true, even if you’ve called an ambulance to come to the scene?
Richard Zimmerman: (11:20)
Yeah, absolutely. The ambulance will get there in whatever amount of time, and in that time period, you need to provide medical assistance before they arrive.
TL;DR: The first quote establishes that there is a policy guideline, the "force continuum" for use and escalation of force. The second quote establishes that kneeling on a prone person is not official policy and it is considered the top level of deadly force in the "force continuum".
Finally, the meat of the testimony is in the final quote which establishes:
1. Police are responsible for people they handcuff and are required to give first aid medical care - even if an ambulance is called, should the person need it.
2. Handcuffed people are considered to be much lower threat because of the constraint and official policy is to use lower force on a handcuffed person.
3. Training is to move a handcuffed person from the prone position because of the breathing difficulties the position causes.
While it remains to be seen how these policies were enforced, I am reminded of people early on claiming this sort of thing is proper police procedure. Well, now we see it's not.
Say what now?
So let me get this straight:
Things it's racist to require ID for
Things it's not racist to require ID for
Voting
Gambling
Buying booze
Buying smokes
Buying pot
Drinking while in a bar
Not drinking while in a bar
Seeing naked people dance with a pole
Letting people see you dance naked with a pole
Consuming porn
Making porn
Buying a gun
Buying a fishing license
Buying a hunting license
Buying a house
Renting a house
Renting a hotel room
Buying a phone
Buying a car
Renting a car
Driving a car
Driving a boat
Adopting a dog or cat
Applying for foodstamps
Applying for welfare
Applying for social security
Applying for medicaid
Applying for medicare
Applying for unemployment
Applying for a job
Applying for a loan
Flying an airplane
Riding in an airplane
Opening a bank account
Cashing a check
Going to college
If this doesn't seem full of shit to you, you really need to take a hard look at why.
Hell Froze Over
A progressive agenda rag actually committed journalism for a change and fact-checked Biden! With actual facts! As unfavorably as their rating system allows!
That's where all this global warming is coming from. Someone slapped a Peltier between hell and earth.
Reality vs. The Narrative
For them wondering how big of a problem those scary-looking, scary-sounding "assault rifles" stack up as a preferred murder weapon: 2019's numbers are in.
Handguns: 6,368
Edged weapons: 1,476
Blunt weapons: 397
No weapons: 600
Rifles (all types): 364
Shotguns: 200
Other guns (???): 45
Handguns are, as usual, the clear winner. By a factor of 17x and change vs. every type of rifle combined.
Knives n swords n shit by a factor of 4.0x and change.
Going UFC on their ass by a factor of 1.6x and change.
And clubbing them like a baby seal by just under 1.1x.
Shotguns, you need to step up your game.
Yup, you're four times as likely to get taken out ninja-style than to get Rambo'd. But "assault rifles" are scary, so let's ban them and anything even kind of like them. No, we don't care if the previous "assault rifle" ban had literally no effect, it'll work this time!
P.S. If those 3,281 "Firearms, type not stated" get on your nerves, feel free to distribute them among the reported types in the ratio of those that were reported are distributed. Math says you'll get the exact same ratios I stated above when you do the new calculations.
Lightbulb Moment
Ya know, I've been mildly annoyed by progressives thinking they can just arbitrarily redefine a word so that it means whatever they hell they want it to for a good while now. And I was wrong. I was looking at things like those morons do for some reason. It's not a problem, it's an opportunity!
That realization had, here are a few words I used to not enjoy and their new meanings:
- *ist: a master in sexual techniques related to the root word.
- problematic: an adjective meaning somewhere between "badass" and "da shit".
- toxic: well endowed.
- trans*: a robot created for the purpose of activities relating to the root word. transsexual: sexbot. transgender: bot that accurately identifies male or female for those who can't. transphobic: bot that goes around scaring people. transcontinental: bot that serves you a crappy breakfast at a hotel.
- burn in hell: fuck me like grandpa used to then call the dogs in to lick me clean.
Okay, those are the ones I thought up while mowing. What have you guys got?
