SFNQuestions

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Just for kicks and giggles, I'm going to define a function f ( r ) such that f ( r ) is the area of a circle with radius r.
And, I'm going to define it in terms of an integral
The details are here
http://www.analyzemath.com/calculus/Integrals/area_circle.html
(other such sites are also available)
It turns out that f ( r ) = pi r^2
Now, can I find an inverse for this function which is defined in terms of an integral (of course it's an integral its the area of a curve)
Well, yes I more or less can ( if we decide that either r in always >0 or that a negative radius is permitted as an alternative to a positive one)
I can calculate the radius ( r ) of a circle, given the area (a)
r=(a/pi) ^0.5
So, in this (trivial) case I can find an inverse fro a function that is defined in terms of an integral.
(It's entirely possible that someone better at maths than I cam could find more exciting examples but this one will do)
OK, That was easy, so I will now try something a bit more complex.
I will define a function, g, of two variables x and y such that g(x,y) = x+y,x+y
Sorry for my clumsy explanation what I mean is the equivalent of multiplication by the matrix
1,1
1,1
That matrix has the not very rare property of having no inverse.
Now imagine that I define a further function h(x,y) in such a way that it depends on an integral of g(x,y). It doesn't have to be anything clever.
The integral of g(x,y) dy,dx over some range will do nicely.
Now I contend that since the calculation of the intermediate i.e. g(x,y) is not invertible, the integral of that function also isn't invertible.
So, in this case it is not possible to invert a function defined in terms of an integral.
So, the answer to the question "Is there a way to invert a function defined by an integral?" is
Sometimes
You can if it's f(x), but you can't if it's h(x,y)
Incidentally, the choice of a function in two variables makes Fiveworld's solution a lot more interesting it would need a 2 D array, but not all 2 d arrays have an inverse.
If you had thought about what he said rather than discounting it rudely, you would have got to at least part of the answer.
(It's possible that someone is going to point out that I'm playing fast and loose with the definition of "function" here by sticking arrays into it. OK, fair cop, but the OP started it by introducing integrals)
Your failure to find in inverse comes purely from your choice to refrain from defining a branch, something that allows any function to be invertible and isn't limited to functional analysis but is applicable to linear algebra as well. Not only that, but this is not the example I provided. Anything that you can possibly identify in any way as any kind of continuous curve or line is invertible over that specific segment, it is only a matter of whether or not you apply the proper techniques to find it in terms of identified functions. In your own limited capabilities, y=x^2 shouldn't actually have an inverse, and yet every actual accredited mathematician in the world agrees that http://www.wolframalpha.com/input/?i=inverse+y%3Dx%5E2
And when it doesn't know the answer, it doesn't say "never going to be an answer, no one will ever know just give up on math" it says "no results in terms of *standard* mathematical functions" because accredited mathematicians know that a function is invertible if you apply the technique of choosing the proper domain and range, it is only the case that we may lack the techniques to explicitly identify it.
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But you asked an even broader question. If I asked "is momentum conserved?" a yes or no answer will be wrong. Because momentum conservation depends on the condition of whether or not there is a net external force. You asked about an arbitrary function and did not restrict it to polynomials. The answer is going to be "it depends on the function"
No the answer to momentum will depend on a vector analysis of the situation and whether the system is opened or closed. Like I said, every function can be inverted to some degree if the right domain is chosen, whether or not it happens over a particular domain isn't your problem, the topic you derailed is in reference to an example where we already know what the outcome should be.
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Your original question is too general to supply a specific answer and there is no universal answer available.
I garuntee you there is a technique that you can attempt to apply to the situation I provided that can also work in similar situations. If I choose to apply the technique to some random problem and I find the result is not expressible in currently defined functions, that's my problem, not yours.
Most functions defined by integrals have alternative definitions which may be more easily invertible.
From what I have seen, it is exceptionally rare to get an integral definition that somehow turns out to be nice and easy to work with in some alternate definition.
Bessel functions cannot be expressed in terms of simpler functions so must be inverted from tables or some other numeric procedure.
In terms of functions you know of, or, have cared to define. What the question and example asks for the technique to show how it's done to verify a result, or in other words, any amount of evidence to verify any of the "answers" thus far.
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That's simply not true.
It is true unless it is completely common knowledge that it's already been proven.
When the question is broad enough,
Nope, when the answer is wellknown enough, that's the rule, not broad enough. If you ask a broad question like "what's the inverse of a second degree polynomial?" "sometimes where x=0" isn't going to suffice, you need the quadratic formula at the very, most basic east.
I didn't say it was an arbitrary whim.
And you don't need to, it's very clear that it is.
You asked about "something that's clearly hard to work with like the gamma function or the exponential integral or the cosine integral or the error function or etc, "
I also specifically asked about a "function that is defined as an integral" and specifically gave a specific example to work through,
"In fact, you know what, let's take an example: arctangent. The arctangent function can be described as the integral of 1/sqrt(1+x^2) which after complex analysis or trig substitution we know actually takes the form of a complicated complexvalued logarithm containing a polynomial. But now, let's say I wanted to define the actual regular tangent function just starting out with the integral definition of the arctangent function (which for verification purposes we know takes the form of a sum/quotient of real/imaginary exponents)..."
would be "sometimes"; there is sometimes a way to invert such a function.
If it's a function that is continuous over some interval and you isolate a branch point, which is the only time you could ask for an inverse, the answer is never ever "sometimes" or "no."
If you are unhappy with the idea that maths sometimes says " It's impossible to tell" then I have bad news for you
Good thing it's not impossible to tell then. Even the gamma function with 100% certainty has an inverse over any branch point you choose, at the very least over the interval that it is monotonically increasing, it is simply the basic, common, average case that the inverse it is not currently defined in terms of elementary functions or possibly just known techniques, outside of a taylor series.
You are correct the other things you said  including the comments to fiveworlds were off topic.
This comment of yours is off topic.
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The correct answer to the question of the OP is either yes, no or sometimes.
Incorrect. In math an science, the answer isn't some arbitrary whim, an answer needs to include evidence and reasoning or it isn't credible and therefore is unusable.
None of your posts has included any actual answer to the problem just pointless grumbling about fiveworld's incomplete answer.
You might want to read the part of your post which I have quoted above, and then look in the mirror.
Off topic from mathematics, and I will continue to report that content until it is censored as per the rules of the site.
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I'm not here to teach you why oneway functions don't have an inverse. The only way to reverse a oneway function back to the state it was in originally is to remember what state it was in originally. For example the undo button uses arrays. There is no inverse for x^0 etc.
Not only is that a useless tautology which once again illustrates my point, but most known integral functions absolutely have an inverse over a large domain even if it's only known by it's taylor series, so now what you're saying is also completely irrelevant, it's as if you're just here to troll. If you're not going to take the time to answer the question accurately, don't waste the world's time with your post.
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We define x as an array of values under which we evaluate the function. Then we reverse the array.
This does not help anyone. The correct response would be to start looking at the inverse function theorem which definitely has something to do with the problem at hand.
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Then if you know better than those who try to guide you, you do not need to ask the question, right?
Just a reminder: the image of your first function is [math]h(x)[/math]. Then you claim it's preimage is [math]h^{1}(x)[/math]. So [math]h(x)=x[/math]. Agreed?
And so on.......
Why? Because the foundation of math and science isn't in automatically assuming every single thing you think of is automatically correct. I'm not just going to make a statement and assume it's going to work out, I'm going to test a lot of times, I only asked it here in case I didn't have luck finding out the right answer or to see if anyone else could confirm the results.
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If I have a given function that is defined as an integral, like of something that's clearly hard to work with like the gamma function or the exponential integral or the cosine integral or the error function or etc, is there a way to define an inverse of that function in a similar form that's NOT a copout dy integral? Not some dy integral that forces me to switch to integrating along the yaxis for no reason that solves absolutely nothing and never will, but something that starts as f(x) = integral(g(x))dx and hopefully takes the form of f1(x) = integral g^1(x)dx or something similar, something that is still defined on the domain of x.
In fact, you know what, let's take an example: arctangent. The arctangent function can be described as the integral of 1/sqrt(1+x^2) which after complex analysis or trig substitution we know actually takes the form of a complicated complexvalued logarithm containing a polynomial. But now, let's say I wanted to define the actual regular tangent function just starting out with the integral definition of the arctangent function (which for verification purposes we know takes the form of a sum/quotient of real/imaginary exponents)...
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I know there's an answer to this, because if there wasn't, every single person who ever said "cannot be put into terms of elementary functions" is automatically discretited, so I know they definitely must have done something to verify that.
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No, it is not valid (or only trivially).
First some terminology.....
The domain of a function is the set of all those elements that the function acts upon. Each element in the set is called an argument for the function
The codomain  or range  of a function is the set of all elements that are the "output" of the function. So for any particular argument the element in the codomain is called the image of the argument under the function.
So, Rule 1 for function...No element in the domain may have multiple images in the codomain.
Rule 2 for functions.....Functions are composed RighttoLeft
Rule 3 for functions.......Functions can be composed if and only if the codomain of a function is the domain of the function that follows it (i.e. as written, is "on the Left)
Rule 4 for functions....... For some image in the codomain, the preimage set is all those elements in the domain that "generate" this image. Notice that, although images are always single element, preimages are sets  although they may be sets with a single member in which case the function is said to have an inverse  not otherwise.
Look closely at what you wrote above. and check how many of these rules are violated.
Yeah I've found you're not entirely correct. You're obviously correct that not every function is invertable on every domain, but as a general technique it absolutely works because it's the same as doing a usubstitution and then back substituting it to find the inverse. I've done it tons of times now and it works every single time, long as I limit the domain.
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I am afraid I'm not that much of a believer.
Well you're in luck then because there's nothing to believe. Where does the statement 1+1=2 say anything about time or space? It's not a question of whether or not it physically exists, it's just a matter of whether or not a logical correlation is true, and every one is true regardless of any position of anything or any amount of time that passes. No matter where or when you are in the universe, 1+1=2 is a true statement within any set of axioms that allow that statement to be true. For something closer to home, take for instance a dimension: we can't pick it up, we can't see one, but it is nonetheless the consensus in physics that dimensions exist.
I can not exactly see what point you are trying to make. Can you state your thesis more clearly? Surely if you define the symbols differently you can make 2 and 3 mean the same thing. I don't see your core point here.
As you said, a lack of defined natural numbers would imply a lack of uniqueness. What I'm asking is that if you defined operators like succession but didn't define uniqueness, it seems like there would be nothing to verify that 2=3 is false, even in your point that you could disprove the statement with succession since that relies on mathematics that has already been established. So the symbols 2 and 3 could be equal if there's nothing explicitly to say that any given number x and y (or a and b) have to be unique values. This isn't philosophy this is more on the main topic.
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Personally I'm not convinced about that. Did the plays of Shakespeare exist before Shakespeare wrote them? Before there were humans on earth?
Logically speaking, logic itself is not dependent on any amount of time nor any amount or space. Even if the universe didn't exist, 1+1=2 would still be a true statement relative to some immaterial set of axioms. But Shakespeare's play requires a physical universe to exist based on its very definition.
I don't know what that means. Before Giuseppe Peano wrote down the axioms for the natural numbers, the world had no such axioms. If he'd written down different ones, history would be different. Many people believe the natural numbers have some sort of existence before there were humans. I'm not so sure. It's a matter of philosophy.
That's true to some extent, but what I am referring to is ruling out the possibility that one value is equal to another value, just because you write different symbols in this "symbolic game." It seems like without uniqueness, 2=3 could actually be a true statement. And as you can imagine, even if an element of a set is succeeded by another element of a set, it doesn't seem like you could garuntee the values are unique when dealing with complex or imaginary numbers, or even just angles. You can't even say that one complex imaginary number is greater than another.
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A proof is a logical deduction from some axioms. If you assume the Peano axioms you a "proof in PA" as they call it, and if you assume the axioms of set theory you get "proof in ZF".
I think in the present case the most fundamental proof is the one from set theory. The Peano axioms work but how do we know there is any model of them? So if you believe in the natural numbers then you can do a proof from Peano, but if you are a skeptic then you need the axiom of infinity to provide a model of them. That's a personal opinion. Most people accept PA as having some ontological referent, the "natural numbers of our intuition" or some such.
In the end it's just symbolic manipulation so you always have to start by taking some statements as given and not subject to proof.
So I guess you could say that [math]2 \neq 3[/math] is "obviously" true about the world. But in formal math, it's pretty arbitrary. The only reason [math]2[/math] and [math]3[/math] are different is that we define them that way. The symbol [math]3[/math] is defined as the successor of [math]2[/math], which is defined as the successor of [math]1[/math], which is the successor of [math]0[/math]. None of it means anything at all, it's just a symbolic game. That's the downside of formalization, you lose touch with reality. A child knows what it takes a logician years to prove.
Well symbols aside, logic is logic regardless of whether we discover it or not. All logical correlations are true all the time. What if a particular set of axioms lacked a specific statement of uniqueness? Could you still prove it?
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Yes that works too, directly from the order properties of the natural numbers or the reals. That's yet another proof.
So what actually distinguishes this from a proof and just something you can look at that's taken to be self evident? A proof...does what? Breaks something down until only the axioms that define the particular group remain?
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In set theory [math]2 = \{0,1\}[/math] and [math]3 = \{0,1,2\}[/math] and these are distinct sets by the axiom of extensionality. https://en.wikipedia.org/wiki/Axiom_of_extensionality
On the other hand in the Peano axioms, [math]2 \neq 3[/math] since two numbers are the same if and only if they are both successors of the same number; but [math]2 = S(1)[/math] and [math]3 = S(2)[/math].
https://en.wikipedia.org/wiki/Peano_axioms
Set theory and the Peano axioms are related by the Axiom of Infinity, which says (in effect) that there is a set that's a model of the Peano axioms. https://en.wikipedia.org/wiki/Axiom_of_infinity
Alright, that makes sense. What if their difference is not equal to zero? Does that qualify as a proof?
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Wow a site full of "mathematicians" and no one knows, that's...kind of amazing, this should be a millennium prize problem I guess.
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I could be wrong but I think the context is either within algebra or set theory. To me, you can look at this statement 2=3 and just see that it's wrong, but on the other hand, it's also often taken for granted that proving 1+1=2 takes pages of proof, so what's the actual proof behind proving it's wrong?
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I saw a discussion that got me wondering about problems related to abstract algebra and group theory. How you formally use group theory to disprove 2=3?
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From that other thread I wondered "hmm, how would I come up with that from scratch?" How would you go from a simple recursive series to that exponential formula?
http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibFormula.html#section1
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The noise probably comes from stickslip. Removing or sufficiently reducing static friction
Well that would defeat the purpose of the entire ball joint then, because it's suppose to function as a small armature component that holds its place.
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Oil? (I think that's why our joints are relatively smooth. While young, at least)
Are these better? https://en.wikipedia.org/wiki/Ball_joint#Spherical_rolling_joint
(I am not aware of the problem you refer to, so its just a guess)
No something mechanical that holds its place. The kind of ball joint I'm referring to is one that uses friction to maintain a stiff armature, but can move in any direction. It's similar to the one you presented but without the ball bearings.
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I notice that ball joints have that undesired creaking sound, whereas the bones in your body which are much bigger make nearly no noise when they transition from static to moving. Is there a way to keep the friction of a ball joint while making it quiet?
Or alternatively, is there some kind of gear equivalent to a ball joint that uses increased surface contact with helical spurs?
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I give up. Clearly you do not find your contradictory statements to be contradictory, so we'll ascribe it to me being thick and move on.
They're not contradictory, you're just refusing to acknowledge that your initial assumptions are wrong. As you ignored multiple times, the intention is the point out that I don't see any animals that fit the description, but in no logical way does my own observation rule out the possibility that such animals may have already existed and gone extinct or currently exist and are niche.
One of the major benefits of multicellularity is the ability of cells and organs to become specialized. I.e. as compared to a unicellular organism, in which every single cell is an individual which must perform every biological task necessary for the organism to live, the cells/organs in a multicellular organism can specialize in one or more specific roles. The leads to different optimal densities  e.g. a vertebrate has a rigid musculoskeletal system optimised to move around an enviroment, and a liquid blood and lymphatic system to transport compounds throughout the body.
Giving up variable density means giving up many of the benefits of multicellularity, and while some organisms do this (e.g. macroalgae, some fungi) they tend to be facultatively multicellular  exploiting some benefits of being both multi and uni cellular.
Right, no one's disputing that there are benefits to specialized cells, but I'm not saying they can't have blood vessels, I just mean the "skeleton" so to speak doesn't localize, that it extends homogenously from the center to the skin.
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Is there a way to invert a function defined by an integral?
in Analysis and Calculus
Posted · Edited by SFNQuestions
The x+y function that appears to just be a plane? Do you think that's not invertible in any way? When a=0, the integral gives you a constant line which when inverted gives you simply a vertical line, and to make it functional it only needs to be equivalent to a translated Dirac delta function or limit the range to a finite number. Literally anything that you can recognize as any kind of curve or shape can be flipped over the line y=x, that's really all you need to ask yourself, the rest is simply a matter of domain/range restrictions and whether or not it's in terms of known functions. Matrices are their own group, it has it's own rules. The definition for a bisection or an inversion or invertible does not necessarily have all the same applications and versatility as in functional analysis. Is there any particular reason you continue to completely evade the specific example I provided wherein the end result is already known but it's complex enough to illustrate the application? Honestly this site is so behind, stackexchange and physicsforums answer things so much quicker and with more detail compared to this site and without the cavalier attitude of the academically disappointing precedent set by the staff.
As an allegedly accredited staff member you should be able to view the edit history, and if you did you'd see I added the example only shortly after the post in order to give context and a more identifiable goal. Do you realize this looks like abuse and thus a lack of integrity?
Uh, no one's forcing you to be here, as a sentient being, your frustration is your own choice. If you don't have the expertise or experience for making progress with this particular technique, go ahead and do something else.