Formulas on Habré

In 2014, British scientists conducted an experiment - they suggested to mathematicians to evaluate the aesthetic beauty of fifty different formulas, observing the reaction of their brain using functional magnetic resonance imaging (fMRI). During the observation, neuroscientists noticed that viewing some formulas caused a response in the prefrontal cortex, which is responsible for complex cognitive functions and emotions. It turned out that the perception of beauty formulas is very similar to the emotions that arise when viewing works of art or listening to music.

We suggest you look at a selection of beautiful (and not very) in the opinion of mathematicians formulas, and at the end of the publication - a small bonus.

The most "beautiful" flourish was the Euler identity, which is a consequence of the Euler formulas connecting the exponent of a complex number with trigonometric functions. What a beauty!

$1 + e ^ {i \ pi} = 0$

The second place in the charts went to the main trigonometric identity, connecting two main trigonometric functions:

$\ cos ^ 2 \ theta + \ sin ^ 2 \ theta = 1$

And how do you find the Gauss-Bonnet Formula? Bukovka to bukovka!
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$\ int_ {M} {KdA} + \ int _ {\ partial M} {k_g ds} = 2 \ pi \ chi (M)$

Well, or Gaussian integral (also known as Euler-Poisson integral). From the beauty already captures the spirit!

$\ int _ {- \ infty} ^ \ infty {e ^ {- x ^ 2} dx} = \ sqrt {\ pi}$

But the "ugly" according to scientists Formula Ramanujan. Well, scared, as if the cat piled!

$\ frac {1} {\ pi} = \ frac {2 \ sqrt {2}} {9801} \ sum ^ {\ infty} _ {k = 0} {\ frac {(4k)! (1103 + 26390k)} {(k!) ^ 4 396 ^ {4k}}}$

And in this line is created complete chaos:

$\ lim_ {8 \ rightarrow 9} \ sqrt {8} = 3$

Some more large formulas of varying degrees of attractiveness.

$\ sum_ {n = 1} ^ {\ infty} a_ {n} z ^ n \ and \ \ sum_ {n = 1} ^ {\ infty} \ frac {1} {a_ {n}} z ^ n$

$\ sum_ {n = 1} ^ {\ infty} a_ {n} \ and \ sum_ {n = 1} ^ {\ infty} \ frac {1} {a_ {n}} \ converges$

$\ lim {a_ {n}} = 0, \ lim {\ frac {1} {a_ {n}} = 0}$

$limsup \ sqrt [n] {| a_ {n} |} = l = \ frac {1} {R}$

$\ frac {1} {liminf \ sqrt [n] {| a_ {n} |}} = \ frac {1} {l '}$

$\ frac {1} {R} = \ limsup_ {n \ to \ infty} \ sqrt [n] {\ lvert a_n \ rvert} \ geqslant \ liminf_ {n \ to \ infty} \ sqrt [n] {\ lvert a_n \ rvert} = \ frac {1} {\ limsup_ {n \ to \ infty} \ sqrt [n] {1 / \ lvert a_n \ rvert}} = \ frac {1} {1 / R} = R.$

\ frac {1} {\ sqrt {a ^ 2 + ab + b ^ 2}} + \ frac {1} {\ sqrt {a ^ 2 + ac + c ^ 2}} + \ frac {1} {\ sqrt {b ^ 2 + bc + c ^ 2}} \ geq \ frac {2} {\ sqrt {ab + ac + bc} + \ sqrt {\ frac {a + b + c} {3 (a ^ 3 + b ^ 3 + c ^ 3)}}

$\ frac {1} {\ sqrt {a ^ 2 + ab + b ^ 2}} + \ frac {1} {\ sqrt {a ^ 2 + ac + c ^ 2}} + \ frac {1} {\ sqrt {b ^ 2 + bc + c ^ 2}} \ geq \ frac {2} {\ sqrt {ab + ac + bc} + \ sqrt {\ frac {a + b + c} {3 (a ^ 3 + b ^ 3 + c ^ 3)}}$

$\ left (\ sum \ limits_ {cyc} \ sqrt [3] {a ^ 2 + 4bc} \ right) ^ 3 \ sum_ {cyc} (a ^ 2 + 4bc) ^ 3 (ka + b + c) ^ 4 \ geq \ left (\ sum \ limits_ {cyc} (a ^ 2 + 4bc) (ka + b + c) \ right) ^ 4$

$\ left (\ sum \ limits_ {cyc} (a ^ 2 + 4bc) (ka + b + c) \ right) ^ 4 \ geq45 (ab + ac + bc) \ sum_ {cyc} (a ^ 2 + 4bc ) ^ 3 (ka + b + c) ^ 4,$

$a ^ 4 + b ^ 4 + c ^ 4 + d ^ 4 + a ^ 2b ^ 2 + b ^ 2c ^ 2 + c ^ 2d ^ 2 + d ^ 2a ^ 2 + 8 (1-a) (1- b) (1-c) (1-d) \ geq1$

And many of you have probably heard this story. At the beginning of the 70s of the last century, the company Parker issued an advertisement in which a hand was drawn, writing with a pen some formula:

$\ frac {3.5G + \ frac {V} {2}} {4 (H_2O) ^ 3} + 3 (360 ^ {\ circ}) = M$

The company management then received a lot of questions from chemists, physicists and other scientists with a request to clarify what was written, they say, what kind of a formula ?! It turned out that this is nothing but a comic martini recipe, which should be read as follows: take 3.5 parts of gin and 0.5 vermouth, add 4 ice cubes and shake with three movements.

Friends, as you probably already understood, we added support for mathematical formulas to the site - both beautiful and not so. To do this, we use the LaTex markup language (in the desktop version, the MathJax library is used to draw the formulas on the page, the mobile version, the mobile application, and the RSS formulas are displayed using SVG).

To add a formula to a publication, click the Σ icon in the toolbar. In the window that appears, select the lower case or block type of formula.

- the line formula is used to insert a formula in a paragraph of text;
- block formula is used to insert a formula from a new line.

After making the formula, click on the “Add Formula” button and it will appear in the text of the publication.

Formulas can be colored and made headings. Here, for example, is the Einstein-Pythagoras formula:

$E = m \ cdot c ^ 2 = m \ cdot (a ^ 2 + b ^ 2)$

Formulas work only in publications, there is no support for formulas in the comments yet.

Also do not forget that Habré has the opportunity to insert various oembed-objects, as we have already told . And, perhaps, someone missed the post about the design of publications .

Source: https://habr.com/ru/post/319600/

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Formulas on Habré

1+ei pi=0 1 + e ^ {i \ pi} = 0

cos2 theta+ sin2 theta=1 \ cos ^ 2 \ theta + \ sin ^ 2 \ theta = 1

intMKdA+ int partialMkgds=2 pi chi(M) \ int_ {M} {KdA} + \ int _ {\ partial M} {k_g ds} = 2 \ pi \ chi (M)

int − inftyinftye−x2dx= sqrt pi \ int _ {- \ infty} ^ \ infty {e ^ {- x ^ 2} dx} = \ sqrt {\ pi}

frac1 pi= frac2 sqrt29801 sum inftyk=0 frac(4k)!(1103+26390k)(k!)43964k \ frac {1} {\ pi} = \ frac {2 \ sqrt {2}} {9801} \ sum ^ {\ infty} _ {k = 0} {\ frac {(4k)! (1103 + 26390k)} {(k!) ^ 4 396 ^ {4k}}}

lim8 rightarrow9 sqrt8=3 \ lim_ {8 \ rightarrow 9} \ sqrt {8} = 3

frac3.5G+ fracV24(H2O)3+3(360 circ)=M \ frac {3.5G + \ frac {V} {2}} {4 (H_2O) ^ 3} + 3 (360 ^ {\ circ}) = M

E=m cdotc2=m cdot(a2+b2) E = m \ cdot c ^ 2 = m \ cdot (a ^ 2 + b ^ 2)

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