Information about the quantum states in a region of spacetime may be somehow coded on the boundary of the region, which has two dimensions less. This is like the way that a hologram carries a three dimensional image on a two dimensional surface.
http://www.encyclopedia4u.com/t/topological-space.html
Any metric space turns into a topological space if define the open sets to be generated by the set of all open balls. This includes useful infinite-dimensional spaces like Banach spaces and Hilbert spaces studied in functional analysis.
http://www.encyclopedia4u.com/m/metric-space.html
Metric space
A metric space is a space where a distance between points is defined. It is a topological space.
Formally, a metric space is a set of points M with an associated distance function (also called a metric) d : M × M -> R (where R is the set of real numbers). For all x, y, z in M, this function is required to satisfy the following conditions:
d(x, y) ≥ 0
d(x, x) = 0 (reflexivity)
if d(x, y) = 0 then x = y (identity of indiscernibles)
d(x, y) = d(y, x) (symmetry)
d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality).
These axioms express intuitive notions about the concept of "distance": distances between different spots are positive and the distance between x and y is the same as the distance between y and x. The triangle inequality means that if you go from x to z directly, that is no longer than going first from x to y, and then from y to z. In Euclidean geometry, this is easy to see. Metric spaces allow this concept to be extended to a more abstract setting.
A metric space in which every Cauchy sequence has a limit is said to be complete.
Metric spaces are paracompact Hausdorff spaces and hence normal (indeed they are perfectly normal). An important consequence is that every metric space admits partitions of unity and that every continuous real-valued function defined on a closed subset of a metric space can be extended to a continuous map on the whole space (Tietze extension theorem). It is also true that every real-valued Lipschitz-continuous map defined on a subset of a metric space can be extended to a Lipschitz-continuous map on the whole space.
Two metric spaces (M1, d1) and (M2, d2) are called isometrically isomorphic iff there exists a bijective function f : M1 → M2 with the property d2(f(x), f(y)) = d1(x, y) for all x, y in M1. In this case, the two spaces are essentially identical. An isometry is a function f with the stated property, which is then necessarily injective but may fail to be surjective.
Every metric space is isometrically isomorphic to a closed subset of some normed vector space. Every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
Distance between points and sets
If (M,d) is a metric space, S is a subset of M and x is a point of M, we define the distance from x to S as
d(x,S) = inf {d(x,s) : s ∈ S}
Then d(x, S) = 0 if and only if x belongs to the closure of S. Furthermore, we have the following generalization of the triangle inequality:
d(x,S) ≤ d(x,y) + d(y,S)
which in particular shows that the map x |-> d(x,S) is continuous.
Originally posted by phoenixthoth
so is it that the known universe is four dimensional with a 2 dimensional boundary? why are there 11 dimensions?
T duality
The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality, and comes about from the compactification of extra space dimensions in a ten dimensional superstring theory. Let's take the X9 direction in flat ten-dimensional spacetime, and compactify it into a circle of radius R, so that
A particle traveling around this circle will have its momentum quantized in integer multiples of 1/R, and a particle in the nth quantized momentum state will contribute to the total mass squared of the particle as
A string can travel around the circle, too, and the contribution to the string mass squared is the same as above.
But a closed string can also wrap around the circle, something a particle cannot do. The number of times the string winds around the circle is called the winding number, denoted as w below, and w is also quantized in integer units. Tension is energy per unit length, and the the wrapped string has energy from being stretched around the circular dimension. The winding contribution Ew to the string energy is equal to the string tension Tstring times the total length of the wrapped string, which is the circumference of the circle multiplied by the number of times w that the string is wrapped around the circle.
where
tells us the length scale Ls of string theory.
The total mass squared for each mode of the closed string is
The integers N and Ñ are the number of oscillation modes excited on a closed string in the right-moving and left-moving directions around the string.
The above formula is invariant under the exchange
In other words, we can exchange compactification radius R with radius a'/R if we exchange the winding modes with the quantized momentum modes.
This mode exchange is the basis of the duality known as T-duality. Notice that if the compactification radius R is much smaller than the string scale Ls, then the compactification radius after the winding and momentum modes are exchanged is much larger than the string scale Ls. So T-duality obscures the difference between compactified dimensions that are much bigger than the string scale, and those that are much smaller than the string scale.
T-duality relates type IIA superstring theory to type IIB superstring theory, and it relates heterotic SO(32) superstring theory to heterotic E8XE8 superstring theory. Notice that a duality relationship between IIA and IIB theory is very unexpected, because type IIA theory has massless fermions of both chiralities, making it a non-chiral theory, whereas type IIB theory is a chiral theory and has massless fermions with only a single chirality.
T-duality is something unique to string physics. It's something point particles cannot do, because they don't have winding modes. If string theory is a correct theory of Nature, then this implies that on some deep level, the separation between large vs. small distance scales in physics is not a fixed separation but a fluid one, dependent upon the type of probe we use to measure distance, and how we count the states of the probe.
This sounds like it goes against all traditional physics, but this is indeed a reasonable outcome for a quantum theory of gravity, because gravity comes from the metric tensor field that tells us the distances between events in spacetime.
Originally posted by Russell E. Rierson
Take two opposing mirrors as an overly simplistic example.
The number of self contained images increase, while the size of the images decrease.
[<-[->[<-->]<-]->]
A self including set.
T duality
The duality symmetry that obscures our ability to distinguish between large and small distance scales is called T-duality,
At that same time that Turing shows that any formalism for reasoning is incomplete, he exhibits a universal formalism for computing: the machine language of Turing machines. At the same time that he gives us a better proof of Gödel's incompleteness theorem, he gives us a way out. Hilbert's mistake was to advocate artificial languages for carrying out proofs. This doesn't work because of incompleteness, because of the fact that every formal axiomatic system is limited in power. But that's not the case with artificial languages for expressing algorithms. Because computational universality, the fact that almost any computer programming language can express all possible algorithms, is actually a very important form of completeness! It's the theoretical basis for the entire computer industry!
[...]
And I begin to get the idea that maybe I can borrow a mysterious idea from physics and use it in metamathematics, the idea of randomness! I begin to suspect that perhaps sometimes the reason that mathematicians can't figure out what's going on is because nothing is going on, because there is no structure, there is no mathematical pattern to be discovered. Randomness is where reason stops, it's a statement that things are accidental, meaningless, unpredictable, & happen for no reason.
Originally posted by matt grime
whenever people raise questions you can't answer or don't understand you post 'bored' smilies. Perhaps you are a genius. Perhaps you're an idiotic crank. I know where my money is
Originally posted by Russell E. Rierson
In simplest terms, a hypersurface is an [ n-1] dimensional space. For example, 3 dimensional space can be generalized to a 2-dimensional surface.
What the h**l is the relevance of R^17 ? 17 legs to the right triangle? All that is required is R^3 + 1 Then the [n-1] hypersurface is 2D.
x^n + y^n = z^n
as most mathematicians know (division algebra structures exist for 2,4,8,16 degree extensions of R and no others)
Originally posted by phoenixthoth
while it may be correct, can you give like some information about the isomorphism to further prove that the two realms are isomorphic? like in what context are they isomorphic (as sets, as groups, as vector spaces, etc.)?
As a way out of this conundrum, I have suggested that complete mathematical symmetry holds: that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. The elements of this multiverse do not reside in the same space but exist outside of space and time. Most of them are probably devoid of observers. This hypothesis can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato's realm of ideas or the "mindscape" of mathematician Rudy Rucker of San Jose State University exist in a physical sense. It is akin to what cosmologist John D. Barrow of the University of Cambridge refers to as "ð in the sky," what the late Harvard University philosopher Robert Nozick called the principle of fecundity and what the late Princeton philosopher David K. Lewis called modal realism. Level IV brings closure to the hierarchy of multiverses, because any self-consistent fundamental physical theory can be phrased as some kind of mathematical structure.
Originally posted by matt grime
non-formally differentiating functions of integers is it now?
Hi, this is easy to explain.
Notice that your equation is basically of the form
a^2 + (a+1)^2 + 1 = b^2
Never mind, for the moment, that you are considering consecutive ratios of b. First start by trying to understand which a's and b's can be solutions to this equation.
Well, this equation is the same as:
2a^2 + 2a + 1 - b^2 = 0.
Which is the same as
4a^2 + 4a + 2 - 2b^2 = 0
which is
(2a+1)^2 -2b^2 = -1
This is a special case of Pell's Equation
X^2 - 2Y^2 = -1 (i.e. where X=2a+1 and Y=b).
Now observe that (X,Y) = (1,1) is a solution.
It turns out that all the solutions
(X_k, Y_k) to this equation are given by
X_k + sqrt(2) Y_k = (1+sqrt(2))^k where k is odd.
Now your question relates to those X_k which are odd. Now compute some examples with k = 1,3,5,7,9,11,13... you will see a pattern. This explains your observations.