Log (0+) : EXPERIMENTO

Greetings humans, 

This is an experiment for me as much as it is an experiment for you, who are reading this. 

The ones of you who already knows me (or "knows" me), are already familiar with the fact that I create and put challenges of physics and mathematics on my instagram account (@theartoffugue [my username changes sometimes]). 

Travellers who randomly stepped into this blog, well... Just read the rows above to quickly get on the road. 

The aim of this blog, which I started as a sort of conversation with myself which ended up in silence, and hoping it will have a future, is now clear: to write here the challenges I put on IG with the solutions too, in order to create a record of funny useful interesting problems and, above all, to have them written down in a clear and more suitable way (through the help of HTML code instead of my hands with pen and paper). 

All the past challenges were once highlighted on IG, but I have recently removed them all so this is also a way to make a jump in the past in order to restore all them. I will indeed start with the oldest ones, and then back to the future! 

So, what else to say... Let's go! 

THE ROOTS OF UNITY

 Back to basics, I want to talk about polynomials and roots.

When studying polynomials, mathematicians are especially interested in their roots, the values of a variable that make the polynomial equal zero. A polynomial has as many roots as its degree (the value of the largest exponent), so that $x^3 + 3 = 0$ has three roots, whilst $x^9 - 3x^2 + 4 = 0$ has nine roots.

An important amusing field of study is the one in which we try to find a relation between a root of a polynomials and each others. For example: when graphed, the roots of some polynomials fall exactly on the vertices of regular polygons  (they stand apart by an exact geometric length). But there could be more sublet geometric relationships…

What kind of pattern can you get? Can you get ANY pattern? 

Let’s focus, for the moment, only onto an important class of polynomials: the cyclotomic polynomials (CP). Those are the ones that cannot be factored into smaller ones, but you can use them to build polynomials. The first two ones, and the simplest ones, are $(x+1)$ and $(x-1)$. You cannot factor them into smaller polynomials, but you can use them (for example by multiplying them) to get ($x^2 - 1$).

The tenth cyclotomic polynomials is $x^4 - x^3 + x^2 - x + 1$ (want to have fun? Prove it!).

The roots of CP follow a very special geometric pattern. To see it, we need to start with the complex plane in which the $x$-axis plots real numbers, and the $y$-axis plots imaginary ones. Then inscribe a circle of radius $R = 1$ (no matter if cm, metres or whatever) around the origin and you get the famous Unit Circle: the roots of CP ALL lie on this circle.

They bear an elegant name: the Roots of Unity. 

Seeking the Roots of Unity.

Polynomials consist of coefficients and variables raised to powers (see examples above). 

Roots are the values of x that make the polynomial equal zero. The roots of a CP all lie on a circle with radius 1 unit, centred around the origin of the complex plane. These are the “Roots of Unity”. 

You can admire the meaning of this just above. 

The problem is that the most of polynomials are non cyclotomic, and their roots are not the roots of unity. This is the case with almost any combination of coefficients, variables and exponents you could come up with. 

In $1965$, A. Schinzel and H. Zassenhaus predicted that the geometry of the roots of CP and non-CP differ in a very specific way: take ANY non CP whose first coefficient is $1$. Find his roots and graph them. Some may fall inside the unit circle, some right on it and others may fall outside it. 

Schinzel and Zassenhaus predicted hence that EVERY non CP MUST have at least ONE root that falls outside the Unit Circle and at least some minimum distance away. 

Another fascinating way to see this is in terms of repulsions: “the smallest roots of any non CP, which might fall within the Unit Circle, effectively push other roots outside the Unit Circle, like magnets pushing each other away 

It’s like to think about the roots as negative electric charged particles that repel each other with a force that decays when the distance increases. 

(Continues after the D.s. [Durante Scriptum])

D.s. I DARE thou to say that mathematics (and physics) is boring. "Shut up and calculate”. (R. Feynman).

Repulsive Roots.

So, according to Schinzel-Zassenhaus conjecture, every non CP must have at least one root that is at least some minimum distance outside the Unit Circle. That distance varies depending on the values of the largest power in the polynomial.

In the above example, the black dots are the roots of

$$x^{7} + 2x^{5} - 12x^{4} - 12x^{3} + 2x^{2} + 1 = 0$$

The conjecture’s main prediction has the feel of a physics equation. It says every non CP should have at least one root that is outside the Unit Circle by a distance equal to a constant number divided by the degree of the polynomial. 

If we had a non CP of degree $23$, the conjecture predicts that is should have a root at least $\dfrac{1}{23}$ of unit outside the circle.

It’s a powerful statement, but for decades only weaker forms of this conjecture has been managed to be proven. Indeed, Schinzel and Zassenhausem themselves managed only to prove that every non CP has a root at least $\left(\dfrac{1}{4}\right)^d$ [where $d$ = degree of the polynomial] outside the Unit Circle, a much smaller distance than the conjectured one.

Many improvements have been made, yet the conjecture still remains unsolved. Often, when a prominent math problem remains open for a long time, it’s because mathematicians simply lack the technique to solve it. Dream as you might of flying to the moon, you’re not getting there until someone invents a rocket. 

And then it comes V. Dimitrov: he transformed a question about the size of roots of polynomials into a question about the size of the values associated to a related but different type of mathematical object called power series (it’s like a polynomial, only with infinitely many terms).

Want to know more and really go deep? Have fun at ArXiv.org “A proof of the Schinzel-Zassenhausen conjecture on polynomials”. It’s really elegant and well understandable! 

Click here for the article

https://arxiv.org/pdf/1912.12545.pdf

VAULT

 Pages I need to store.

https://en.wikipedia.org/wiki/List_of_HTTP_status_codes

https://en.wikipedia.org/wiki/Separation_axiom

https://shop.arts-crafts.ca/products/jean-michel-blais-dans-ma-mains-sheet-music-book?variant=15074741026916

https://en.wikipedia.org/wiki/Pietro_Annigoni%27s_portraits_of_Elizabeth_II

https://it.wikipedia.org/wiki/Esistenzialismo

https://en.wikipedia.org/wiki/List_of_tests

https://it.wikipedia.org/wiki/Categoria:Figure_retoriche

https://it.wikipedia.org/wiki/Prima_lettera_ai_Corinzi

https://en.wikipedia.org/wiki/Norbert_Wiener

https://www.youtube.com/watch?v=9KnUGMHKec4

https://en.wikipedia.org/wiki/Jacques_de_Vaucanson

https://en.wikipedia.org/wiki/Machine_code

https://en.wikipedia.org/wiki/Synesthesia

https://edge.pokerlistings.it/assets/photos/cani-poker-Cassius-Coolidge.jpg?t=1407762730

https://www.arteworld.it/wp-content/uploads/2014/10/autoritratto-van-gogh-1887.png

https://it.wikipedia.org/wiki/Fenomenologia_dello_spirito

https://en.wikipedia.org/wiki/Symphony_No._7_(Beethoven)

https://en.wikipedia.org/wiki/EPA_list_of_extremely_hazardous_substances

AESTHETICALLY PLEASING INTEGRALS

I have decided to start a subsection (subset) of this blog by talking, and solving, the most beautiful (yes, aesthetically pleasant) integrals I have come across to during my "having fun time" periods, or generally during my studying periods. There are some of the which are really amazing, and they are amazing also because me myself and I have solved them. Sure, I don't claim to be the first one who solved them, since they are nothing special. But they are "hard enough" to make me compliment to myself for my courage and my will to attack them :)

THE ZEROTH API (Aesthetically Pleasing Integral)

$$J = \int_0^{+\infty} \dfrac{\sin(\pi x^2)}{\sinh^2(\pi x)}\ \text{d}x$$

Whose numerical result is $\dfrac{2 - \sqrt{2}}{4}$

PROOF

First of all we notice the integrand is even, hence we can write

$$\dfrac{1}{2}\int_{-\infty}^{+\infty}\dfrac{\sin(\pi x^2)}{\sinh^2(\pi x)}\ \text{d}x$$

Let's now define

$$f(z) = \dfrac{\cos(\pi z^2)}{\sinh(2\pi z) \sinh^2(\pi x)}$$

And not that because 

$$f(x\pm i) = \dfrac{-\cos(\pi x^2)\cosh(2\pi x) \pm i\sin(\pi x^2)\sinh(2\pi x)}{\sinh(2\pi x)\sinh^2(\pi x)}$$

we have

$$\int_{\gamma} f(z)\ \text{d}z = \int_{-\infty}^{+\infty} \left(f(x-i) + f(x+i)\right)\ \text{d}x$$

That is $2\pi i \times \text{Sum of the residues}$, which becomes $-\dfrac{\pi}{2} \times$ Sum of residues.

The poles are at $0$ and $\pm \dfrac{i}{2}$ and $\pm i$ hence we have: 

RESIDUES

If you are able enough to calculate the residues, which are rather easy, you'll find that the residue near zero is $-\dfrac{1}{2\pi}$

Instead we have $\dfrac{\sqrt{2}}{4\pi}$ near $\pm \dfrac{i}{2}$, whereas the residues around $\pm i$ are $-\dfrac{1}{2\pi}$.

Hence

$$\int_0^{+\infty} \dfrac{\sin(\pi x^2)}{\sinh^2(\pi x)}\ \text{d}x = \dfrac{\pi}{2}\left(-\dfrac{1}{2\pi} - \dfrac{1}{2\pi} + \dfrac{\sqrt{2}}{4\pi} + \dfrac{\sqrt{2}}{4\pi}\right) = \dfrac{2 - \sqrt{2}}{4}$$

As wanted.

LINES IN THE SAND

 Is there a way to predict sadness? Is there a way to predict pain? 

Or to avoid them. Predict and avoid. Or just avoid, if you cannot predict. 

I'm so tired. 

(This will be deleted)

CHARADE

Phosphorus trichloride and liquid trans-methyl propylaxine. 

Now, the third one that I haven't brought her yet is O-ethyl methyphonic acid. 

Combined, these chemicals make pesticide. They can also be adjusted. If you methylate the phosphorus trichloride, it forms methyl phosphorus dichloride... Which, when combined with the liquid trans-methyl propylaxine and the O-ethyl methyphonic acid, creates tiny particles. 

Very deadly, easily aerosolized, and completely different from pesticides.

It's nerve gas.

Beware to your significant other's wishes.

MILKSHAPE

Don't worry, it's the exact word. 

Have you ever wondered what milk really is? No? Well I did, I did some researches and now I share what I have studied and understood.

First of all, according to the animal species, milk has different components and it does considerably vary in its fats percentage (really high, for example, in sea mammals where in seals and cetaceans they can reach 50%. Also cloven-hoofed animals as moose, yaks and reindeers produce high calories milk). In any case: 

• Water is always the principal component; 

• Fats, especially sautéed ones, are the principal energetic source in milk, generally found under the form of phospholipids and glyceride esters of long and short chains of fatty acids like: 

Butanoic butyric acid; 

Hexanoic caproic acid; 

Octanoic caprylic acid; 

Decanoic capric acid; 

Docecanoic laurid acid; 

Tetradecanoic myristic acid; 

Hexadecanoic palmitic acid; 

Octadecanoic stearic acid; 

(saturated ones) and also

Caproleic acid cis-9 decanoic; 

Myristoleic acid cis-9 tetradecanoic; 

Cis-6 hexadecenoic palmitoleic acid; 

Cis-6 octadecenoic petroselinic acid;

Cis-9 octadecenoic oleic acid; 

Trans-9 octadecenoic elaidinic acid; 

Trans-11octadecenoic vaccenic acid; 

Linoleic acid, cis-cis-9,12 octadecadienoic acid;

(unsaturated ones). 

The present carbohydrates, the second energy source of milk, are made up in all the animal species almost entirely of (disaccharide) lactose. 

Lactose is the osmotically more active compound of the milk, with a relatively constant concentration in all the types of milk. 

Milk secretion is isotonic to blood, thanks to the recall of water from the blood to the breast due to lactose.

The amount of produced milk is directly proportional to the amount of synthesised lactose. 

There are other carbohydrates, mainly oligosaccharides composed of glucose, galactose, fucose, glucosamine, N-acetylglucosamine, galactosamine, N-acetylgalactosamine, and sialic acid generally conjugated to proteins (glycoproteins), such as k-casein, other than calcium sensitive ones for its solubility over a wide range of Ca++ concentrations and its low phosphorous content. 

2/3 (two thirds) of the proteins are represented by the phosphoproteins family generally called casein (k-casein for example).

Apparently the concentration of proteins in the milk of the different species is inversely proportional to the concentration of carbohydrates. 

The protein content of milk tends to be higher in species that are characterised by the rapid growth of puppies after birth.

The carbohydrate content, instead, is higher in milk of slow-growth species (like humans). 

Don't forget the possible presence of mineral substances, vitamins, aromatic substances (not chemicals, but meant as the responsible for the flavour and the smell), somatic cells (mostly macrophages) and bacteria.

Minerals or milk salt, are traditionally names for a variety of cations and anions with bovine milk. Calcium, phosphate, magnesium, sodium, potassium, citrate and chloride are all included as minerals and they typically occur at concentration of 5-40 mM (milli-molar). The milk salt strongly interact with casein, most notably calcium phosphate. It is present in excess and often much greater excess of solubility of solid calcium phosphate. 

In addition to calcium, milk is a good source of many other vitamins: A, B6, B12, C, D, K, E, thiamine, niacin, biotin, riboflavin, flolates and panthothenic acid are all present in milk.

The possible presence of pathogens, always possible in raw milk, with an average random incidence over time of around 20% of the milking, is linked to the state of health of the animal and to hygiene deficiencies and to the milking process and the environment status.

However, the risk is always present which is why adequate heating (boiling) is universally recommend by the Health Bodies. 

PH of the milk ranges from 6.4 to 6.8 and it changes over time. Milk from other bovines and non-bovine mammals varies in composition, but has a similar PH.

This should calm down your thirst, for a while. 

Here you can see a simplified representation of a lactose molecule, broken into glucose and galactose.