For example, when a river bifurcates into two separate streams of water, and then these two streams rejoin further along to become a single river once more.  Bifurcate itself is not generalized enough for the title purpose because (a) it implies a split into 2 parts, rather than n; and (b) it doesn't suggest anything about rejoining.  Example sentence can be as simple as "the river [term] as it travels along the flat valley", with the meaning as above, though the specific form might vary.  Perhaps branching could provide the n-parts concept, but I'm not sure how to incorporate rejoining in an elegant way.  Perhaps also there is some computer science or mathematical term for this idea, since it seems it might be useful in those fields.
121224
A  is river or stream that flows into a larger river.  A  a stream branching off a river.    Is there a word that combines both structures, the idea of a river that splits in two and later rejoins to become one river again?           /---------------\ ---------                 --------------------          \---------------/   My focus here is on the waterway, not the land. I’m looking for a word describing the behavior of the river splitting and later rejoining, without regard for the geography, and without regard for the cause or effect between land and water.
100529
Is it possible to have a list of items separated by commas where the last item in the list is not preceded by 'and'?     For example, "the river runs through dense bush, towns, cities, farms, dams, until it exits into the sea."     Or does it need to be "runs through dense bush, towns, cities, farms and dams, until it exits into the sea."
115585
A line may have infinite points becauase it may be expanded.But in case of a line segment it has 2 distinct points which are not movable.The distance between the end points in finite and known.  But still why do people(in my school) say that there are infinite points in a line segment.When I ask the teacher,she says you will learn at higher levels(what a genius way to get rid of question).
12708
In python, you can have a function return multiple values.  Here's a contrived example:  def divide(x, y):     quotient = x/y     remainder = x % y     return quotient, remainder    (q, r) = divide(22, 7)   This seems very useful, but it looks like it can also be abused ("Well..function X already computes what we need as an intermediate value.  Let's have X return that value also").  When should you draw the line and define a different method?
90983
The many-worlds interpretation of quantum mechanics has always been explained to me at a high level using examples of binary events (e.g. atom either did or did not decay at any given moment in time), which leads to a conceptually clean idea of "branching" into two distinct universes. But how does branching work if you have, say, a 70% probability of something happening (e.g. measuring the spin of an electron having gone through an SG apparatus oriented at an arbitrary angle)?  Do you say that 7 universes got to spin up and 3 that got spin down (to account for the 70% probability)? Does that mean those 7 universes are in-every-way identical copies of each other? But then what if you had something with a 71.87154819...% chance of happening? You would need an uncountably infinite number of branches to be able to represent arbitrarily precise probability ratios, and then subsets of those branches would contain uncountably infinite universes that are 100% degenerate and identical to each other. Is this what the standard many worlds interpretation assumes?  If not this, then what? You can't say that there's a 70% chance of universe A happening and a 30% chance of universe B happening if you're saying both happen. How does the many-worlds interpretation put a "weighting function" on different branches or outcomes?
8589
Sometimes there are many ways to define a mathematical concept, for example the natural base logarithm. How about sine and cosine?  Thanks.
36748
The symmetric derivative is always equal to the regular derivative when it exists, and still isn't defined for jump discontinuities. From what I can tell the only differences are that a symmetric derivative will give the 'expected slope' for removable discontinuities, and the average slope at cusps. These seem like extremely reasonable quantities to work with (especially the former), so I'm wondering why the 'typical' derivative isn't taken to be this one. What advantage is there to taking $\lim\limits_{h\to0}\frac{f(x+h)-f(x)} h$ as the main quantity of interest instead? Why would we want to use the one that's defined less often?
106849
(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.
22049
I think that the question is clear from the title:     Are there sets $A$ and $B$ such that $A \in B$ and $B \in A$?   My feeling is that the definition of any two such sets will be circular (we can't define $B$ till we define $A$ and we can't define $A$ till we define $B$), and hence will be excluded by some axiom.  On the other hand, I can't see an inherent contradiction, though I wouldn't be surprised if some form of Russell's paradox pops up, pushed behind one more set of braces.
76583
This is a really basic question, but it is one I never really thought about until now.   Let $\mathscr{G}$ be a tree. Then every pair of vertices in $\mathscr{G}$ is connected by a unique walk.   We are asked to prove the converse which is worded as:  If any two vertices of  $\mathscr{G}$ are joined by a unique walk, then $\mathscr{G}$ is a tree.   My question is whether the author means that if we can find a single pair of vertices that are joined by a unique walk in a graph (that may not be a tree), then we know $\mathscr{G}$ is a tree (which obviously is not true), or if he means something more a long the lines of "for all pairs of two vertices in $\mathscr{G}$".   Similarly, he asks us to prove:  If $\mathscr{G}$ is connected but becomes disconnected by the removal of any one edge, then $\mathscr{G}$ is a tree.   Again, in one case it is easy to find counterexamples of graphs with cycles (not trees) that can become disconnected by the removal of one of their edges, but if he wants us to prove that "if, for every edge, the removal of that edge will result in a disconnected graph, then the graph is a tree", the proof while also simple is very, very different.   I come from a more applied background and am not used to this kind of wording. Any insight would be much appreciated.
95149
A streamline cannot cross other one in steady state of a fluid but can they be tangent to each other ?  What I mean is consider fluid in steady state. if let us say particle A has its own path in fluid and at point B it has velocity x. But let us say another particle C has its own path and has to cross ppoint B it must also have same velocity and hence the two streamlines are tangent at B. Is this possible otlr is it only valid if particle A and C both have the exact same trajectory fro the beginning ?
120791
In some texts (mainly complex analysis or harmonic analysis) I sometimes see the following double subset symbol $\subset\subset$ for inclusion relation of two regions, e.g., $\Omega$ and $\Omega'$ are two regions in $\mathbb{C}$ such that $\Omega \subset\subset \Omega'$. I never figured out what it means exactly; I always interpreted it as the closure $\overline{\Omega}$ is contained in $\Omega'$ (so that some nasty boundary effects can be avoided). Is that right? Or does $\subset\subset$ mean some other kind of inclusion? Thanks.
28363
I am trying to prove that when an IVP in ODEs does not enjoy uniqueness, then it has infinitely many different solutions. I know that when the Lipschitz condition is satisfied, then there is a unique solution.    I guess, I am trying to show that if the IVP possesses two solutions, then we can create infinitely many solutions using these two. However, I am slightly lost as where to start the proof.
60872
I've tried solving it, but I'm not familiar with these kind of progressions and I couldn't get to a result. Also, I've found that the result is just $N/2$, but I don't get why would that be the answer, as it doesn't work for $N=1$ or $N=2$. Appreciate your help :)
10214
I'm having trouble understanding the motivation between wanting to transition between the two descriptions.  I figure that switching from  is useful if we want to recover a particle's trajectory maybe? Is there any other reason to do so? and then what are we measuring by switching from Lagrangian to Eulerian?
111928
I'm drawing maps in Illustrator. Our way of depicting divided highways is parallel lines with a fill. Here is an example:    The way I did this was a 1 point yellow stroke, command c, command f, send to back, explained stroke to 2 points  and make black.  These are a pain to make and edit. I'm wondering if there's a way to do this with a single stroke.
59112
For example, when some long prose passage ends, this appears:  * *  *  Then some new prose passage begins.  The three asterisks dividing the two prose sections are understood to divide the two sections into non-continuous temporal spaces.  I am interested in knowing if there is a specific name to describe symbolic or graphical markings that are meant to suggest temporal discontinuity.