A continuum is anything that gradually changes from one thing to another. But I want another word that describes something that changes gradually and whose opposite extremes are actually one and the same.   Have you ever seen one of those apple charts that describes the many different varieties of apples from sweet to tart? . They call it a spectrum, but a spectrum is not necessarily circular.  If the best answer to my question is "circular continuum" or "circular spectrum", so be it. But if anyone can think of a better word or phrase, I'd be interested to hear it.
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I'm drafting a text about music theory and, when looking for a way to describe the progress of steps between a note's octaves, the expression &quot;looping spectrum&quot; came to mind. I'm sure something like &quot;note wheel&quot; would be more elegant but, merely for the sake of curiosity: can a spectrum's ends come together? Note: an almost identical question was asked  but I'm afraid the answers were not very helpful.
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I am currently trying to learn some basic quantum mechanics and I am a bit confused. Wikipedia defines a  as a quantum of light, which it further explains as some kind of a .     What exactly is a quantum of light?   More precisely, is a quantum of light meant to be just a certain number of wavelengths of light (something like "1 quantum = a single period of a sine wave" perhaps?), or is the concept completely unrelated to wavelengths? In other words, how much is a single quantum?
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I can't think of the word to describe it. Something similar to "wanderer" or "roamer".   It's often used to describe people that don't stay in one place... not "migratory"...
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Fractals, when viewed as functions, are everywhere continuous and nowhere differentiable. Can this also be used as a definition for fractals?  i.e. Are all fractals everywhere continuous and nowhere differentiable? And also: Are all functions that are everywhere continuous and nowhere differentiable fractals?
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Parallel universe: in every universe, you are presented with a different decision. Does that mean there are an infinite amount of parallel universes? I.e., if in one universe I choose to go left, I am then presented with different choices vs if i choose right, and that creates more universes with more decisions to be made.  Or does it mean that in the end of one's life we end up in the same place, meaning that its just one parallel universe that eventually collides with the other.
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I'm looking for a more succinct word/small phrase that conveys the idea of "comparing and contrasting". Is there a word that hits two birds with one stone by representing both similarities and differences between two ideas/objects?
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Bats and rats are mammals. 1 and 2 are positive integers. Oaks and maples are trees. Carrots and cake are food.   I seek a word to describe this, the process of finding a concept that encompasses several given concepts. Is there such a word?
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I'm a huge fan of mathematical physics and I know what the formal definitions of those  two areas are, I've seen them. But I still get completely baffled when someone asks me to explain it simply. The difference is obvious to me, but I just can't seem to put it into words in a satisfying enough manner. So I'm asking you to help me...  What is the essential difference between  and ?  or if you prefer the rephrased version...  What was the motivation for introducing the name "mathematical physics" as a separate entity?  (It doesn't matter if you're one of those people who don't like labels, the thing is that we do have these two separated very often in academia.)
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In the context of topological spaces, I see the following major countability properties:  A space is:   "separable" iff it has a countable dense subset "second countable" iff if has a countable basis "first countable" iff the neighbourhood system of every point has a countable local basis.   (Definitions taken from Counterexamples in Topology by Steen and Seebach, 2nd ed. 1978 -- there may be differences in wording from other sources.)  The question I have is: "separable" into what, exactly? By which I mean to say: what is the thinking behind calling such a condition "separable"?  It appears not to be related to the concept of "separation axioms", which do immediately and obviously invoke an intuitive notation of separation, neither does it seem to have anything to do with "separated sets".  (Anyone using the spelling "seperable" or "seperated" will be immediately downvoted. :-) )
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The question pretty much sums up what I need to know. Why is it that light only gets dispersed into a spectrum when travelling through two non-parallel sides(like a prism) and not through something like a glass slide with parallel slides.  And how does this experiment(an experiment conducted by Newton) explain this? Because this is what is in my school textbook.
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The electromagnet spectrum looks to be continuous, yet energy is discrete. Shouldn't one see evidence of the electron's "quantum leap"? What's going on?     Moderator, please size my photo. Thank you.
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Can you give me an example of a function which is sequentially continuous but not continuous? (I know that in first-countable spaces this is equivalent, but what about in spaces without this condition?)  Thank you :)
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I am curious about the following diagram:     The image implies a circle of infinite radius is a line. Intuitively, I understand this, but I was wondering whether this problem could be stated and proven formally? Under what definition of 'circle' and 'line' does this hold?  Thanks!
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Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal  assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for non-tautological reasons, $\aleph_\alpha$ is provably larger than the cardinality of the continuum? I would hope not since it would seem pretty silly if something like $\alpha = \omega_0$ worked and we could say "well gee we can't prove that $c = \aleph_1$, but it's definitely one of $\aleph_1, \aleph_2, \ldots , \aleph_{73}, \ldots$". I (obviously) don't know jack squat about set theory, so this is really just idle curiosity. If a more precise question is desired I guess I would have to make it     For any countable ordinal $\alpha$ is the statement: $c &lt; \aleph_\alpha$ independent of ZFC in the same sense as the continuum hypothesis?   assuming that even makes sense. Thanks!
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As a start I assumed  which is the reciprocal of  as the frequency upper limit, but not so sure. On the other hand, is there a lower limit? Can we consider the reciprocal of our universe as a theoretical limit which is not yet limited, or just assume 0? I am just curious to know if you have any ideas with thanks.
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I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme but how do I get there?
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I thought the universe was finite, but then I read this:    And they seem to assume it is infinite. So which is it?