DrRocket

Your list is correct. But you don't have to actually calculate the numbers to see how many of them exist: 1. Between 11 and 1111 there are 53 numbers that are congruent to 12 mod 21. They are of the form 21k+12 , k=0,1,...,52 2. The question is how many of them are also congruent to 6 mod 9; i.e. for which k is there an m so that 21k+12=9m+6 3. Now 21k+12 = 3k+3 mod 9 = 3(k-1) + 6 mod 9 So 21k + 12 = 6 mod 9 if and only if (k-1) is a multiple of 3. 4. From the list in #1 those numbers are 1, 4, ..., 52. There are 18 of them.

You are certainly free to live in your own little world. But your personal definition is worthless in the world of mathematics, and the initial quetion was mathematical in nature.

And you would be wrong.

Had you read, and understood, the thread to which I provided a link, you would know that your "horse race" question was not ducked. You need to learn some physics. The heart of the matter is Stephen Hawking's "chronology protection conjecture". It can be proved under some reasonable, but not universally accepted assumptions. It may not be true at the quantum level. It remains a conjecture. Ontology has not contributed any understanding of the problem. Ontologists have contributed less. “…How can we understand the world in which we find ourselves ? How does the universe behave ? What is the nature of reality ? Where did this come from ? Did the universe need a creator ? Most of us do not spend most of our time worrying about these questions, but almost all of us worry about them some of the time. Traditionally these are questions for philosophers, but philosophy is dead. Philosophy has not kept up with modern developments in science, particularly physics. Scientists have become the bearers of the torch of discovery in our quest for knowledge.” -- Stephen Hawking in The Grand Design

"Between 11 and 1111" is clear and in this case it does not matter whether one means inclusive or exclusive. "and" is well-understood in mathematics. It means "and". When one means "or", one says "or". Pretty simple, no ?

You should be able to figure this out for yourself also. I post answers, not multiple questions. Big difference.

NG is a powerful vasodilator. It is used medicinally in small doses. Absorbed through the skin, it produces bad headaches. An NG headache is a recognized OSHA incident. NG can misbehave if you get it hot, or cold. NG decomposes in an autocatalytic reaction, so storage can be a problem. Best not mess with it unless you know what you are doing.

Figure it out for yourself. Multiple posting is not good form.

18 On how many bulletin boards have you posted the same question ? http://www.thescienceforum.com/Number-theory-problem-30858t.php

The answer lies not in philosophy and certainly not in ontology. It is a question of physics and a deep question at that. This issue has been recently discussed in the context of real physics -- much remains unknown. http://www.scienceforums.net/topic/55142-does-einsteins-theory-of-relativity-support-time-travel/

While the topic if this thread is not creationism, I think the analogy is clear: Debating creationists on the topic of evolution is rather like trying to play chess with a pigeon -- it knocks the pieces over, craps on the board, and flies back to its flock to claim victory." - Scott D. Weitzenhoffer

I understand. What is apparently not clear in what I said is that I don't think you can do that without knowing more about x, and "knowing more" means expressing x in the orthonormal basis of eigenvectors of A -- which entails solving for each vi. If [math] x= a_1v_1+...+a_nv_n[/math] for some choice of the [math]v_i[/math] then your various [math]2^n[/math] possibilities for [math]\textstyle \left( \sum_i \lambda_i x^T v_i \right)^2[/math] are just [math]\textstyle \left( \sum_i \pm \lambda_i a_i \right)^2[/math] Maximixation is a selection of + or - for each term. That seems to me to require knowledge of each [math]a_i[/math] (really just the sign of [math]a_i[/math] but that doesn't help) which requires solving for the [math]v_i[/math]. I don't think you can do what you would like. The only slight hope that I can see is this: quantitative finance has attracted the interest of some algebraic geometers. They might have developed an alternate approach to your fundamental problem (not this specific optimization problem but something deeper). You may want to talk to someone like that.

OK, I'm lost. If x is fixed then [math] \frac{x^T A x}{x^T x} [/math] is just a number. If x varies over the vector space then [math]\lambda_{\text min}[/math] is the greatest lower bound and [math]\lambda_{\text max}[/math] is the least upper bound of [math] \frac{x^T A x}{x^T x} [/math]. In order to make sense of your "all one's" vector it is necessary that the eigenvalues of A be distinct. So let's assume that. Since A is symmetric it follows that eigenvectors corresponding to disrinct eigenvalues are orthogonal. Thus your space has an orthonormal basis of eigenvectors of A say [math]v_1,...,v_n[/math] and this basis is determined up to [math]\pm v_1,..., \pm v_n[/math]. So now you are trying to maximize [math]|<x,\lambda_i v_i>|=\lambda_i |<x,v_i>|[/math] for some fixed x. I see no way to do this without expressing x in terms of the [math]v_i[/math] which is equivalent to knowing each [math]<x,v_i>[/math]. In fact one can concoct an x to make [math] \frac{x^T A x}{x^T x} [/math] anything that you like between [math]\lambda_{\text min}[/math] and [math]\lambda_{\text max}[/math] with a convex combination of the [math]v_i[/math]. Essentially you have an [math]x = a_1v_1+...+a_nv_n[/math] normalized so that [math] \sqrt {a_1^2+...+a_n^2} = 1[/math] and you are looking for the maximum of [math]|<x,v_i>|= |a_i| \lambda_i[/math]. This requires knowing all of the [math]a_i[/math]

If A is positive definite then [math]<x,y> = x^TAy[/math] is an inner product in the usual sense of linear algebra (a positive-definite bi-linear form). If you now define [math] ||x|| = \sqrt {<x,x>}[/math] then the Schwartz inequality is [math] |<x,y>| \le ||x|| \ ||y||[/math] with equality if and only if [math]x[/math] and [math]y[/math] are linearly dependent. So to maximize [math] \frac{(x^T A v)^2}{x^T x}[/math] for fixed [math]x[/math] one has to maximize [math](x^T A v)^2 = (<x,v>)^2[/math] which by the Schwartz inequality is bounded by [math](||x|| \ ||v||)^2[/math] with the bound achieved if and only if [math]v[/math] is a scalar multiple of x. Now you just have to decide what constraints your problem imposes on the scalar.

1. The question as to whether the universe is finite (compact) or infinite (non-compact) is a question of topology. Compactness is a topological property. Geometry and topology are different things. Sometimes they can be related and sometimes not. 2. The question of flatness is geometric, not topological. The relationship between curvature and topology in cosmology is based on questionable assumptions. If one assumes that the universe is homogeneous and isotropic then it can be shown that "space" is a Riemannian manifold of constant curvature and there are classification theorems that then relate curvature to topology. This is the source of the often seen correspondence: flat --- Euclidean 3-space; positive curvature --- 3-sphere ; negative curvature -- hyperbolic 3-space. This is legitimate only so long as one keeps in mind the assumptions that go with it. 3. There are flat spaces that are also compact. A flat torus is one such space. There are serious proposals that space could actually be a flat 3-torus. It is not isotropic in the technical sense, but a large flat torus could very well present itself observationally as being "the same in all directions". 4. "Flat, open, or closed" is not the set of options. "Closed" means compact without boundary in the terminology of manifold theory. "Open" means non-compact without boundary. "Flat" is a geometric property which is peripheral to the question. 5. Be careful with the physics literature. There are many misconceptions, particularly in older literature, regarding the connection between curvature and topology. Many physicists are unaware of the existence of compact manifolds having negative curvature (I know of at least one very well-known relativist who had not heard of them until I told him about them.) Thurston showed that there are lots of them. Not all are aware of flat compact manifolds either (The relativist knew about the flat torus.) 6. Even with the assumption of strict homogeneity and isotropy experimental determination of curvature is a problem. Topologically there is all the difference in the world among scalar curvature of 0 , -0.0000000000000000000000000000000001 and 0.000000000000000000000000000000000001 7. I doubt that any direct test of compactness will occur in the foreseeable future, but it may eventually be deducible from some aspect of a theory that is testable. No guarantees. BTW if you want to see some consideration of the flat torus see Brian Greene's new book The Hidden Reality. While I do not like the book, and it has some questionable mathematical reasoning, he does discuss the possibility of a "Pac Man space" and that is a flat torus.

Since [math](x^T A \alpha v)^2 = \alpha^2(x^TAv)^2[/math] for any scalar [math]\alpha[/math] there will be no maximium unless the quadratic form is degenerate and [math](x^TAv)=0 \ \ \forall v[/math] If [math]A[/math] is positive-definite and you require that both [math] x[/math] and [math]v[/math] be unit vectors in the norm associated to the inner product [math]<x,v>=x^TAv[/math] then the problem reduces to an application of the Schwartz inequality and a maximum is realized for [math]v = \pm x[/math] . If [math]A[/math] is not positive-definite then, off the top of my head, I'm not sure what can be said. You might take a look at the general subject of quadratic forms. What is the source of the question ?

The issue is not really relativistic effects, which are quite small until you are closer to c. Neither is it energy per se, though that can become an issue. The velocity of a rocket is a result of conservation of momentum of a body with variable mass (due to the expulsion of propulsive material. [math] V_{rocket}= V_{exhaust} \times ln( \dfrac {Mass_{initial}}{Mass_{final}})[/math] where [math] V_{exhaust}[/math] is the speed of the exhaust gas relative to the rocket. [math]V_{exhaust}[/math] is typically quoted in the industry, by gross abuse of units, in "seconds", the result of using English units and "cancelling" pounds-force by pounds-mass (yeah, it is ugly but traditional. Multiplying [math]I_{sp}[/math] in "seconds" by 32.2 gives you exhaust velocity in feet per second. A hydrogen-oxygen rocket would have an [math]I_{sp}[/math] of about 440 seconds. A solid rocket might have an [math]I_{sp}[/math] approaching 300 seconds (large space boosters are a bit less). Mass fraction, ratio of propellant mass to total mass are usually well below 90%, and a lot less for exotic systems where [math]I_{sp}[/math] might approach 10000 seconds (say ion propulsion). Do the math. 0.05 c is not in the cards. Stretching assumptions a lot you might get to 0.00075 c. Note that this does not address thrust which is dependent on mass flow rate. Since this involves consumption of power to accelerate propellant, if thrust is appreciable power consumption can be quite high (grows like the square of exhaust velocity). Therefore energy and power considerations dictate that exotic propulsion technologies are usually very low thrust.

Think about a regular yo-yo with a piece of weak elastic in place of the usual string. Several things happen at once -- the yo-yo starts to wrap up the string, but it also continues to stretch at the end of the initial "down" movement. .01% strain is still a lot of movement in a string that is a light-year in length. To make it easy think about the yo-yo moving slowly. If t is going fast then thev speed of the elastic wave (about sound speed) gets into the act and it gets more complicated.

No. Given a voltage source (like a cell) with an internal resistance [math] R_0[/math] maximim power dissipation in a load resistor [math] R_L[/math] occurs when [math]R_L=R_0[/math] . This results from the fact that the current is dependent on the resistance [math]E=I(R_0+R_L)[/math] and the power disipated in the load is [math]I^2R_L[/math]. A simple application of calculus shows that the maximum power dissipation occurs when [math]R_L=R_0[/math]. This is the resistive counterpart of "impedance matching" which maximizes the power to the speakers in your stereo.

The "observable universe" is time-dependent and location-dependent.. If the universe continues to expand at a non-decreasing rate then anything outside of our Earth-bound observable universe "now" will forever be causally disconnected from us. This scenario is dependent on what "dark energy" really is and how it behaves in the future. If the rate of expansion were to decrease, or if expansion were to reverse (and this is a possibility in some speculative theories) then objects outside of the current observable universe could re-enter the observable universe in the future. When you get down to it, as a practical matter nothing outside the galaxy is likely to affect us any time soon, and everything outside the solar system is out of our ability to exert any meaningful influence. Nevertheless, scientific curiosity dictates that we try to understand it. Science is not engineering. Many benefits have come from science, but the fundamental goal of science is understanding, not application. Attempts to understand the topology and geometry of space may yield insights affecting our understanding of the implications of general relativity or of a successor theory.

1. You got the year wrong. This is 2011. 2. There are quite a few other mistakes and inconsistencies. Pushing theories have been evaluated ad nauseum.. They don't work.

You, as usual, argue against a straw man. I have never said that curved spacetime is the only possible explanation of gravitation. It is the only currently available model that explains the body of empirical data. That is simply a fact. There are actually two models that match the data. They are general relativity and Einstein-Cartan theory. Both involve spacetime curvature. EC theory dispenses with the assumption that spacetime is torsion-free and is more difficult from a mathematical perspective. The two theories make predictions under most circumstances that are nearly the same and current technology does not permit experiments able to distinguish between the two. You might extend the list of contenders to Beckenstein's modification of GR to handle MOND ideas regarding the "dark matter" issue, but you still have curved spacetime. As to quantum gravity -- there is no current viable theory of quantum gravity. If and when it exists we will see. But any viable theory of gravitation, quantum or otherwise, will have to agree with the predictions of general relativity in most circumstances -- away from the center of a black hole and later than about 10^-33 sec after the big bang. What never ceases to amaze is your absolute confidence in assertions that you make that are factually and demonstrably just plain wrong. How can you discuss the ontology of a theory about which you have no clue ?

Morality need not be tied to religion. There are moral atheists. There are immoral clergy. How morality and democracy can possibly depend on "understanding the laws of nature" eludes me. Were the ancient Greeks uniformly immoral and was Athens not a democracy ? I am certainly glad to hear that either God will conform to the laws of nature or else (and that you can handle the "or else" part). You had better get to work on "the miracle of the five loaves and two fish". Let us know how you work that out with the Vatican and all Protestant religions. Your stance in dictating acceptable behavior for God rather reminds me of the local religious leaders who stated that God was only allowed to speak directly to their named prophet (and presumably He must first fill out the proper forms in triplicate and have them approved by the apostles). If you are a believer is this not a rather contradictory position ? [Einstein: "God does not roll dice." Bohr: "Einstein, don't tell God what to do."] Tying religion to politics is far more dangerous than tying it to science. I seem to recall that pursuit of freedom of religion resulted in the founding of a nation. We have fought wars to make sure that your philosophy would not be implemented. The Constitution of the U.S. was written to prevent just such an event.

You are missing the elasticity of the string. Special relativity precludes the existence of rigid bodies or infinitely stiff strings. The elastic stress wave propagates at about the speed of sound (the speed of sound strictly speaking applies to a vanishingly small stress), much less than c.

This is true, but has nothing to do with the given problem. Think about what was asked. Factor accordingly.