honzik

General bijection will not work because equal transformation is much stronger - it preserves lengths but bijection in general not. The length of every line segment has to be preserved so equal transformation of (0,1> into <0,1> must be identity or 1-x.

Equal transformation f in the 2-dimensional plane (E2) is transformation that satisfies: d(X,Y) = d(f(X),f(Y)) for every points X, Y from E2 (d is eukleidian distance - classical distance of two points). K is proper subset of L if and only if (iff) K is part of L but L is not part of K (ie. K <> L) Than the question is: Do exist sets A and B in E2 so that equal transformation exists so that f(A) = B (set equation) and A is proper subset of B?

Hi, I would like to know whether there exist two shapes A and B (in 2D plane) that are equal (ie. equal transformation between A and B exists) but A is proper subset of B. The definition of "being equal" (equal transformation preseves lengths - eg. rotation, traslation etc.) and "proper subset" (the sets are not equal) are I think commonly known. For infinite shapes these shapes A and B exist: eg. half-line, half-plane and so on but what about shapes that are bounded by say circle with the radius of 1? Does some examples exist (maybe some fractals) or can be proven that it is not possible? Thank you in any thoughts. Honzik

I have the key question to understand inflation: Could inflation as an event start in the whole (infinite) universe at the same time? If this is true how could the whole universe (all places in the universe) at the same time "decide" to start inflation? This seems to me unreal - How could two distant places at the same time know that they had to start some "action"? Thank you for answering / oppinions.

As I have read some articles about inflation - it is in fact possible that inflation started even in infinite universe, but as the "inflation bubble" spread through space it only affected finite space - and the rest of the space in the universe was not affected by this "inflation bubble"? Am I right? So what was at the border of this bubble? And - had this bubble some borer? In fact - this bubble still exists and we are in it - am i right? Thank you for your answers. PS: I am Czech and Honzik is my diminutive name - I was just looking for some login name.

Hi, I have just one "simple" question: Is cosmos(universe) finite or infinite? According to some articles I have read about inflation it seems that inflation theory works only with finite number of particles and finite space. Suppose that inflation "spread" in time - ie. that it started somewhere (in one place or in infinite places) and subsequently (at finite speed) the effect of inflation influenced surrounding places. Let's discuss the alternative that cosmos is not finite - because here we will (I think) get into some trouble: a) Let's suppose that inflation was "started" at bounded volume in (infinite) space. It then follows that: I) inflation still continues in some place in the universe (as it spreads more and more from the place where it started) - because space is infinite and the speed of inflation was (is) finite or II) inflation ended some time ago. But in this case there has to be some (infinite) space in the universe not influenced by inflation simply because the effect of inflation didn't reach this place (inflation took finite time and spread in finite speed) - but this can imply that space is not homogenous (in places which inflation didn't reach). And maybe it can imply some other hard to explain difficulties that a better knowledged cosmologist can explain. b) Let's suppose that inflation started in the whole infinite universe at approximatelly the same time. (In case it stated in the "half" of the universe - yet still infinite half - we will be at the same situation as in points aI) and aII).). But this point seems really strange - how could be inflation started at (approximatelly) the same time on the "opposite" (or better to say very very far - arbitrary far) places of the universe? How should these "opposite" places "know" that they had to start with inflation? And even if the inflation started in the whole infinite universe at the same time - there could be problem (as I nave read somewhere) with homogenity of the universe after inflation. c) The most interesting (and crazy) idea: the cosmos was finite before inflation and after inflation it was infinite - which seams to me really unbelievable. d) The whole inflation is just one quantum "change" (but I don't specify exactly the details) of one particle one one other quantum entity - that took place at one time all over the universe (quantum effects can show this behavior, i think). Can you please tell me which of these ideas could (theoretically and reasonably) happen and which ones are only pure speculations with no theory (known up to today) behind them or that are in contradiction in some reputable theory? Thank you very much for your answers. Honzik