DrRocket

To apply the theory of probability you first need a probability space in which to formulate your problem and assertrions. You don't have one.l

Hint. None of these should take more than about five steps, and most quite a bit fewer.

no Go review what is meant by a generalinear combination. What is 2) ? This doesn't make sense.

It is a convention designed to make the usual algebra of exponents work [math] 1 = x/x = x^1 x^{-1} = x^{1-1} = x^0 [/math] Usually [math] 0^0 =1 [/math] by definition.

All the above are true. But for an everyday mundane application, balancing your ckeckbook uses only the abelian group operations of the integers.

The best available cosmological model, based on general relativity treats the universe as an intrinsic manifold without boundary. It is not known if space-like slices are finite (compact) or infinite (non-compact). In any case the universe is, by definition, the whole enchilada. There is no "elsewhere" into which it could expand. If there were, that would be part of the universe too. You should the thread on cosmology basics. http://www.scienceforums.net/topic/33180-cosmo-basics/

The Feynman Lectures on Physics. It can be read at many levels, from beginner to very advanced. After that there are lots of very good books on specific topics.

When you throw in quantum mechanics that procedure may not work. Not every function is a derivative. A derivative cannot have a "jump" discontinuity -- derivatives have the intermediate value property.

Note that in your question you use both the words "position" and "time". Your question may be more profound than you realize. In physics there are a few quantities that are taken as primitive. Time and position (aka space) are two of them. There are others, electric charge being one. There is currently no definition for either time or space that are any more fundamental than the operational definitions "Time is what clocks measure" and "Distance is what rulers measure". There are any number of crackpots with alternate viiews on the internet, but they cannot produce a viable rigorous theory. There are also researchers doing legitimate research that may someday present a different picture, but no such theory currently is available. General relativity, and also special relativity, unify the notions of space and time in a single entity, spacetime. Spacetime is both space and time and in a sense neither. Both space and time are observer dependent and only local concepts in general relativity. They are mixed together by curvature of the spacetime manifold. There is no global concept of either space or time. What there is is a metric that can be used to determine the "length' of the "world line" of an object's existence (what is called a timelike curve), and the units of that length are units of time -- what is called proper time. That same metric applied to a spacelike curve yields length in conventional units of length. All of this starts with the basic idea of the cartesian plane with a 3-D grid established with "rulers" and a time axis determined by a "clock". There is no more fundamental definition of what constitutes a clock or a ruler. The rest of the edifice is built using some relatively sophisticated mathematics -- Riemannian geometry. Any deep insight that produces a viable (as opposed to crackpot) more fundamental theory of space and time will be a truly major advance in physics, comparable to Einstein's invention of relativity.

If Ax=0 for all vectors x (nx1 matrices) then A represents the 0 linear function and is represented by the 0 matrix for any choice of basis vectors. Or you might consider what happens when you apply A to a basis vector and what that tells you about the matrix representation for A.

This is not at all clear and at the very least requires proof. You are making the unwarranted assumption that to each subset there must correspond a unique true statement expressed with a finite number of symbols from your countable set of symbols. Yes, but the proof doesn't quite work. There is a reason why Godel's proof is more subtle and complicated. However, he did use a numbering scheme so you are sort of on the right track.

It is somewhat simpler to simply note that [math] tA (\frac{1}{t} A^{-1}) = t \frac {1}{t} AA^{-1} = 1 \times I = I[/math] This works for linear operators in general and not just matrices on a finite-dimensional space.

Perhaps this can help those interested in delving into cosmology at some depth. The pillar of modern cosmology is one of the pillars of modern physics, general relativity. General relativity (GR) was formulated by Albert Einstein and announced in 1915. It has since received a great deal of attention, the mathematical foundations have been examined, the presentation refined, and a host of confirming experiments performed. General relativity, with its mathematical roots in Riemannian geometry is a formidable subject, and some of its predictions are contrary to everyday experience – i.e. “common sense” can be badly mistaken. That is no surprise as even special relativity, the precursor and “little brother’ of GR is surprising at first encounter. http://math.ucr.edu/home/baez/gr/gr.html http://en.wikipedia.org/wiki/Introduction_to_general_relativity http://en.wikipedia.org/wiki/General_relativity http://rspa.royalsocietypublishing.org/content/368/1732/5.full.pdf GR treats the universe over all time as a single entity – spacetime. This can also be done in Newtonian mechanics, so there is nothing really new about spacetime. What distinguishes GR is that spacetime is not just affine 4-space, but in fact is a Lorentzian 4-manifold of undetermined topology, with a curvature tensor that is also unknown but is determined by the distribution of mass/energy via a stress-energy tensor defined by a very complex set of partial differential equations. These equations, the Einstein field equations can only be explicitly solved in a few simple circumstances. Gravity is the result of curvature of spacetime. In general because of curvature neither space nor time have any global meaning. However, if one makes the assumption that spacetime is homogeneous and isotropic, then spacetime decomposes as a 1-parameter foliation by space-like 3-dimensional hyperplanes of constant curvature. The parameter serves as a surrogate for time and the hyperplanes as a surrogate for space. The hyperplanes inherit a true Riemannian metric from spacetime and expansion of space means that the distance between points increases as the value of the time-like parameter increases. Astronomical observations support the assumption that the universe is homogeneous and isotropic on the largest scales. Observations also support the expansion of space. http://scienceworld.wolfram.com/physics/HubbleConstant.html https://www.cfa.harvard.edu/~huchra/hubble/ http://map.gsfc.nasa.gov/universe/uni_expansion.html http://map.gsfc.nasa.gov/ http://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe http://aether.lbl.gov/www/science/cmb.html http://aether.lbl.gov/www/science/cmb.html Based on these assumptions and observations Hawking and Penrose in a series of papers used general relativity to conclude that, as a logical consequence, the universe began in an extremely compact form, and in fact predicted singular behavior (which is generally thought to indicate a limitation of general relativity to predict the first fraction of a second) http://web.archive.org/web/20080615011734/http://www.berkeley.edu/news/media/releases/2007/03/16_hawking_text.shtml/ http://rspa.royalsocietypublishing.org/content/294/1439/511.full.pdf+html?sid=0231aa70-32f1-40ac-bab7-91869da35ea6 http://rspa.royalsocietypublishing.org/content/295/1443/490.full.pdf+html?sid=0231aa70-32f1-40ac-bab7-91869da35ea6 http://rspa.royalsocietypublishing.org/content/300/1461/187.full.pdf+html?sid=0231aa70-32f1-40ac-bab7-91869da35ea6 http://rspa.royalsocietypublishing.org/content/314/1519/529.full.pdf+html So, while nobody knows what happened in the first fraction of a second, the big bang hypothesis in terms of subsequent expansion from an extremely compact state is on firm empirical and theoretical grounds. Inflation is not necessary to the big bang, but does use ideas from quantum field theory to explain why the universe is homogeneous on the large scale, yet exhibits anisotropy on smaller scales. It is not a fully verified, or even rigorously formulated, theory, yet. It is promising. It is supported by what has been seen in surveys of the cosmic background radiation. Attacking inflation as unproven is futile, because it is well-known to be just that. But interpreting “unproven” as fanciful or unlikely is simply a demonstration of ignorance. http://web.mit.edu/physics/news/physicsatmit/physicsatmit_02_cosmology.pdf Thus, modern cosmology rests on a solid foundation of empirical data and well-formulated theory. That does not make it immutable. Any physical theory is subject to refinement and extension. But any revision must meet equal standards of rigor. Anyone who rejects modern cosmology must meet the obligation of providing the basis for an alternative . That means providing a theory of gravity to replace GR, and the empirical data to support it. Further, that data must include ALL valid data, including that which currently provides evidence for the validity of GR itself. Addendum: useful references for the serious (these are NOT popularizations) Gravitation -- Misner, Thorne, Wheeler Gravitation and cosmology : principles and applications of the general theory of relativity -- Weinberg Cosmology -- Weinberg General Relativity -- Wald Principles of Physical Cosmology -- Peebles The large scale structure of space-time -- Hawking and Ellis General Relativity and the Einstein Equations -- Choquet-Bruhat

Hi. I'm a crusty old retired bastard with a strong interest in mathematics and physics and lots of academic and aerospace/defense experience in science and engineering. BS, MS EE. PhD mathematics. I take science, and not much else, seriously. I like to help neophytes. I have no use for cranks.