But where is this land of Four Dimensions?

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spring-of-mathematics:

isomorphismes:

Monotone and antitone functions
(not over ℝ just the domain you see = 0<x<1⊂ℝ)
These are examples of invertible functions.

Theorem on the inverse function of continuous strictly monotonic functions:
Suppose the function f:(a,b)→ℝ with -∞≤a<b≤+∞ is strictly increasing (resp., decreasing) and continuous. Letlim f(x)=α≥-∞ and     lim f(x)=β≤+∞ ,  if f is strictly increasing, resp.,x→a+                      x→b−lim f(x) = β≤+∞ and  limf(x ) = α≥-∞,  if f is strictly decreasing,x→a+                      x→b−Then f maps the interval (a,b) invertibly onto the interval (α,β). The inverse function f -1:(α,β)→(a,b) is also strictly increasing (resp., decreasing) and continuous, and we have:
lim(f -1)(y)=a    and   lim(f -1)(y)=b,    if f is strictly increasing, resp.,y→α+                     y→β− lim(f -1)(y)=b   and    lim(f -1 )(y)=a,    if f is strictly decreasing.y→α+                      y→β−
Analogous statements hold for semi-open or closed intervals [a,b].[Source]

Also, If (m,n) ⊂ (a,b), the function f:(a,b)→ℝ, that is also true.

spring-of-mathematics:

isomorphismes:

Monotone and antitone functions

(not over ℝ just the domain you see = 0<x<1⊂ℝ)

These are examples of invertible functions.

Theorem on the inverse function of continuous strictly monotonic functions:

Suppose the function f:(a,b)→ℝ with -∞≤a<b≤+∞ is strictly increasing (resp., decreasing) and continuous. Let
lim f(x)=α≥-∞ and     lim f(x)=β≤+∞ ,  if f is strictly increasing, resp.,
x→a+                      x→b−
lim f(x) = β≤+∞ and  limf(x ) = α≥-∞,  if f is strictly decreasing,
x→a+                      x→b−
Then f maps the interval (a,b) invertibly onto the interval (α,β). The inverse function f -1:(α,β)→(a,b) is also strictly increasing (resp., decreasing) and continuous, and we have:

lim(f -1)(y)=a    and   lim(f -1)(y)=b,    if f is strictly increasing, resp.,
y→α+                     y→β−
lim(f -1)(y)=b   and    lim(f -1 )(y)=a,    if f is strictly decreasing.
y→α+                      y→β−

Analogous statements hold for semi-open or closed intervals [a,b].[Source]

image

Also, If (m,n) ⊂ (a,b), the function f:(a,b)→ℝ, that is also true.

(Source: talizmatik, via geometric-aesthetic)

A history of 20th-century mathematics in ~500 words, with example pictures:

The inheritance of Cantor’s Set Theory allowed the 20th century to create the domain of “Functional Analysis”.
handwriting as functional dataa function that's neither concave nor convexweather data as functional dataTerence Tao's map of function spaceGoldman Sachs share price as functional data
This comes about as an extension of the classical Differential and Integral Calculus in which one considers not merely a particular function (like the exponential function or a trigonometric function), but the operations and transformations which can be performed on all functions of a certain type.
What is sine? What is cosine?
the point of calculus
The creation of a “new” theory of integration, by Émile Borel and above all Henri Lebesgue, at the beginning of the 20th century, followed by the invention of normed spaces by Maurice Fréchet, Norbert Wiener and especially Stefan Banach, yielded new tools for construction and proof in mathematics.

The theory is seductive by its generality, its simplicity and its harmony, and it is capable of resolving difficult problems with elegance. The price to pay is that it usually makes use of non-constructive methods (the Hahn-Banach theorem, Baire’s theorem and its consequences), which enable one to prove the existence of a mathematical object, but without giving an effective construction.

[I]n 1955 … [Nancy] … was the golden age of French mathematics, where, in the orbit of Bourbaki and impelled above all by Henri Cartan, Laurent Schwartz and Jean-Pierre Serre, mathematicians attacked the most difficult problems of geometry, group theory and topology.
a knotsome groups, drawn by Nathan Carter
John Baez's story of the reflection group of a pentagon, drawn by me
reflection group of the square in LaTeX
New tools appeared: sheaf theory and homological algebra … which were admirable for their generality and flexibility.
homological algebra
The apples of the garden of the Hesperides were the famous conjectures stated by André Weil in 1954: these conjectures appeared as a combinatorial problem … of a discouraging generality….

The fascinating aspect of these conjectures is that they … fuse … opposites: “discrete” and “continuous”, or “finite” and “infinite”.

Methods invented in topology to keep track of invariants under the continuous deformation of geometric objects, must be employed to enumerate a finite number of configurations.

Like Moses, André Weil caught sight of the Promised Land, but unlike Moses, he was unable to cross the Red Sea on dry land, nor did he have an adequate vessel. André Weil … was not … ignorant of these techniques …. But [he] was suspicious of “big machinery” ….

Homological algebra, conceived as a general tool reaching beyond all special cases, was invented by Cartan and Eilenberg (… in 1956). This book is a very precise exposition, but limited to the theory of modules over rings and the associated functors “Ext” and “Tor”. It was already a vast synthesis of known methods and results, but sheaves do not enter into this picture. Sheaves … were created together with their homology, but the homology theory is constructed in an ad hoc manner….

In the autumn of 1950, Eilenberg …undertook with Cartan to [axiomatise] sheaf homology; yet the construction … preserves its initial ad hoc character.

When Serre introduced sheaves into algebraic geometry, in 1953, the seemingly pathological nature of the “Zariski topology” forced him into some very indirect constructions.

………

Holy crap! I’ve been so busy with school and work that I’ve been neglecting my blog. I don’t even remember the last time I reblogged something.

ryanandmath:

Imagine you wanted to measure the coastline of Great Britain. You might remember from calculus that straight lines can make a pretty good approximation of curves, so you decide that you’re going to estimate the length of the coast using straight lines of the length of 100km (not a very good estimate, but it’s a start). You finish, and you come up with a total costal length of 2800km. And you’re pretty happy. Now, you have a friend who also for some reason wants to measure the length of the coast of Great Britain. And she goes out and measures, but this time using straight lines of the length 50km and comes up with a total costal length of 3400km. Hold up! How can she have gotten such a dramatically different number?

It turns out that due to the fractal-like nature of the coast of Great Britain, the smaller the measurement that is used, the larger the coastline length will be become. Empirically, if we started to make the measurements smaller and smaller, the coastal length will increase without limit. This is a problem! And this problem is known as the coastline paradox.

By how fractals are defined, straight lines actually do not provide as much information about them as they do with other “nicer” curves. What is interesting though is that while the length of the curve may be impossible to measure, the area it encloses does converge to some value, as demonstrated by the Sierpinski curve, pictured above. For this reason, while it is a difficult reason to talk about how long the coastline of a country may be, it is still possible to get a good estimate of the total land mass that the country occupies. This phenomena was studied in detail by Benoit Mandelbrot in his paper “How Long is the Coast of Britain" and motivated many of connections between nature and fractals in his later work.

(via mathmajik)

Famous Mathematicians

naturaj:

Thales of Miletus

Pythagoras of Samos

Euclid

Hypatia of Alexandria

Leonardo Fibonacci of Pisa

Leonardo da Vinci

Nicolas Copernicus

John Napier

Galileo Galilei

Johannes Kepler

Rene du Perron Descartes

Pierre de Fermat

Blaise Pascal

avenqcrs:

hey you

yes you

is today your first day of school? tomorrow? or maybe it’s your 50th day of school?

either way

youre looking great

you are going to rock this

i believe in you

youre gonna knock em out of the park, champ

best wishes, 

me

(via parasiticbackpack)

Sometimes when writing emails, I’ll write ‘p if(ffffffff)’ to indicate this is really a necessary condition, sucker!

—Logic professor (via philosophyprofessorquotes)

(via imathematicus)