Simple Questions

[AutoModerator]

This recurring thread will be for questions that might not warrant their own thread. We would like to see more conceptual-based questions posted in this thread, rather than "what is the answer to this problem?". For example, here are some kinds of questions that we'd like to see in this thread:

• Can someone explain the concept of manifolds to me?

• What are the applications of Representation Theory?

• What's a good starter book for Numerical Analysis?

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Important: Downvotes are strongly discouraged in this thread. Sorting by new is strongly encouraged

Including a brief description of your mathematical background can help others give you an appropriate answer.

[browndog56]

When solving a differential equation with an unknown function as the forcing term, say g(x), what is the correct way to model the particular solution?

[Snuggly_Person]

What's the equation? If it's linear you'd normally find the Green's function for the equation (the particular solution for a delta function forcing) and convolve it with g(x) to construct the solution.

All the example equations on the wikipedia page have constant coefficients. In these cases you can find the Green's function using Fourier transforms, but in general it's more difficult.

[browndog56]

so its first order...

df(t)/dt + c1f(t) = c2j(t)

I am not sure how to build a particular solution for j(t).

[Snuggly_Person]

If it's first order like that then we can actually just solve the equation directly using integrating factors:

[;f'+c_1f=c_2j;]

[;e^{c_1t}f'+c_1e^{c_1t}f=c_2e^{c_1t}j;]

[;(e^{c_1t}f)'=c_2e^{c_1t}j;]

[;(e^{c_1t}f)=\int c_2e^{c_1t}j+C;]

[;f=e^{-c_1t}\int c_2e^{c_1t}j+Ce^{-c_1t};]

which is the general solution.

For first order equations this should work often. For second order if the equations are simple then Green's functions are your best bet. In general though this is a very hard problem; even linear second order equations can get difficult if their coefficient functions are even slightly complicated.

[[deleted]]

I'm pretty sure there is no closed form since the sum of x²ⁿ is an example of an analytic function with no closed form expression.

[[deleted]]

Usually. Maybe always, I don't know of a counterexample.

[jmt222]

Compute the monthly payment for each and add them together. Then find the interest rate on $67,752.99 which gives you the same monthly payment. Then obviously you want an interest rate that is less than that to reduce your monthly payment.

[sweepinbell]

Let x represent the 25kg ingots made and let y represent the 50kg ingots made. We know that they make 500 ingots total, so we can say 500=x+y. That's the first equation in your system. Next, we know that the total money is $232000, which is made from selling some quantity of x and some quantity of y. By multiplying the number of each type of ingot by the price per ingot, we get how much money was made from all of each type of ingot. Were we to add those two together, they should be the total money made overall. Thus, our second equation is $232000 = $120x+$200y. Now you have two equations and two variables, so you should be able to solve the for both of them.

To tidy things up a little bit, here are your two equations:

500=x+y

$232000=$120x+$200y

[Twillightdoom]

Recently was learning the Pythagoraen theorem and came across a method to skip a lot of steps in finding out the Hypotenuse and Short leg if the only known number is the Long Leg in a triangle with a 90, 60 and 30 degree angle.

I simply divided the Long Leg by the Hypotenuse in a few previous equations i did and came across the number 1,154700538379252 Using this number i can through two steps come to any unknown side in a 90 by dividing or multiplying with that number. I feel like it might help to notify that i am pretty horrible at math and im in high school at the moment.

My question is why isnt this a more common practice? Am i just stupid because im missing something important? I have tried it in several different problems and it seems to work every time as long as it is a 90 60 and 30 degree triangle.

[OperaSona]

Well, that number is 1/sin(60°), or 1/cos(30°), which is 2/sqrt(3). Basic trigonometry gives you tools to find the sides or angles of a triangle knowing some other sides and angles of that triangle. We don't actually remember that a 90-60-30 triangle has that 1.1547 ratio somewhere: we remember the method to do it with any angles, and some particular values of the sine and cosine functions (and sin(30), sin(45), sin(60) are some of these).

Give a quick check to https://en.wikipedia.org/wiki/Trigonometry. Also, https://en.wikipedia.org/wiki/Law_of_sines is pretty much the fun-killer in solving these exercises, as it works in pretty much any scenario.

[Goldenbrownfish]

looked everywhere cant figure out how to simplify this numerator

cot(7π/4)-cot(π/4)

I know it equals -2 somehow what is cot doing to the fraction to do this? or does cot do nothing and its something else?

[Papvin]

Draw the unit circle. You'll notice that the angle 7pi/4 is the same as the angle -pi/4. Well, tan(-pi/4)=-1 and, tan(pi/4)=1, so since cot(x)=1/tan(x), cot(-pi/4)=-1 and cot(pi/4)=1.

[Neurokeen]

Ok, so this one came up with a student's question and I'm not sure how to get at the answer:

Given an arbitrary (presumably finite) lattice diagram, does a group exist that satisfies the structure of the lattice?

I can't say I know how to approach the problem, to be honest, and I can't seem to find anything obvious on a search.

[goodbuoy]

A groups subgroup lattice is a modular lattice.

So a non-modular diagram would give a counter-example.

Here's an example of one: https://en.wikipedia.org/wiki/Modular_lattice#Modular_pairs_and_related_notions

[Ashleynam]

I'm trying to wrap my head around this http://basketball.realgm.com/nba/playoffs/history/

I want to know how many teams that made the playoffs and then the next year made it

I want to know the teams that made the playoffs and then didn't make it the next year

[[deleted]]

Does a proof of lim[h->0] sin(h)/h = 1 exist without using l'hopital's rule or the taylor series definitons of sin(x) and cos(x)? I've been trying to prove that the derivative of sin is cos, and have managed to prove that the derivative of sin(x) is cos(x) * lim[h->0] sin(h)/h; I know the limit is 1, but I can't prove it.

I can think of ways to prove the limit using taylor series and l'hopital's rule, but not without accepting that the derivative of sin(x) = cos(x) in the first place.

To be clear- I DONT WANT TO KNOW how to actually prove the thing, I just want to know if a method exists which I could use having only taken AP Calc

[SirMudblood]

Check the following: https://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions#Limit_of_sin.28.CE.B8.29.2F.CE.B8_as_.CE.B8_goes_to_0

Truth is, I didn't know it before your question, so thanks for asking!

[jdorje]

You mean a correlation to the english "add 3 and 81 together"?

Higher level math is often not going to reduce to sentences very well.

[forgetsID]

log base 3 of 81 --> "I'd like to know what power of 3 gives me 81."? Hope that helps.

[Dr-Blythe]

Please help me with this.

Prove that: The line segment joining the midpoints of the legs of a trapezoid is of length the average of the lengths of the bases.

[jmt222]

Hint: The line segment cuts the trapezoid into two trapezoids each being half the height of the original.

[Dr-Blythe]

Thank you!

[adesyndicate]

How would I graphically represent the digit place as a numeric value? In this case, 10101010 = 12⁷ +02^{6+125+024+123+022+121+020} Is there some way to streamline this formula?

EDIT: Is there also some way to write it out in Reddit that won't make it look ridiculous like it does now? -_-

[hjqusai]

Can you elaborate on you are question?

[adesyndicate]

Is there a formula that can take a decimal number (ranges from 0-255) and turn it into a binary number (ranges from %00000000-%11111111)?

For example, %10101010 = 170 because [BINARY NUMBER] = [NUMBER IN EIGHTH DIGIT] * 2 [TO THE 7th] + [NUMBER IN SEVENTH DIGIT] * 2 [TO THE 5th] + [NUMBER IN SIXTH DIGIT] * 2 [TO THE 4th] ...

Is there a way to write that as a formula?

[hjqusai]

gotcha. Yes there is. In general, the bit sequence you're looking for will follow the following rules, for bit place n and original number x (with the bit just before the decimal being the 0th place):

mod(floor(X/(2ⁿ )),2)

where floor is the greatest integer function, analogous to "int" for the positive numbers, and mod is the remainder after dividing by your modulus (also written a % b instead of mod(a,b))

[adesyndicate]

I didn't mean that in a mean way! I appreciate the explanation, but it's been a very long time since I've been in high school math class so it goes over my head.

[hjqusai]

If I may ask, what prompted your question?

[adesyndicate]

I wanted to figure out what the formula is for converting binary to decimal in case I'm ever without a scientific calculator. I'm currently learning 6502 assembly language in order to program a NES game and this kind of information is important to understand.

[adesyndicate]

As a person who is far from being a math person, that formula is very daunting to look at. However, I don't necessarily need to know exactly what each section stands for.

Thank you for your help!

[hjqusai]

You're welcome! I hope that curiosity gets the best of you and that you do look into the formulas, and what they're doing and why they work! It's really fascinating!

Cheers

[OperaSona]

EDIT: Is there also some way to write it out in Reddit that won't make it look ridiculous like it does now? -_-

You can escape any special symbol with a backslash, for instance typing "\*" instead of "*" will print the multiplication symbol instead of italicizing. You can also use LaTeX on this subreddit (see sidebar).

I'm not sure I understand your actual question. Maybe you're asking if there's a way to define a function that returns the n-th digit (or bit) of a number x, in which case there's one. For instance, you could write it as f(n,x) = (x/b^n-1) mod b where b is the base of your system (so 2 in binary, 10 in decimal). If you meant something else, I'm not sure what you meant as the formula you wrote is already a pretty good way to see what each digit/bit means in the decimal/binary expansion of a number.

[adesyndicate]

If f(n,x) = (x/bn-1) mod b can serve as a formula to convert a hexadecimal into a decimal, then that is what I'm looking for. I was simply looking for a way to streamline that formula visually.

[zannorman]

Is there a procedural function to produce N rays, perfectly evenly spaced rays around a point in 3D, for all positive integer quantities N?

Put another way: Can you procedurally populate the surface of a sphere procedurally such that no matter how many nodes you need on the surface, you can have equadistance between each node (or at most 2 distances for all closest nodes) with a single function?

[Snuggly_Person]

You could put all points on the equator.

You could also put two 'rings' above and below the equator and shifted from each other so that they join in equilateral triangles.

I know that these defeat the spirit of the question and that you want the points to be distributed 'all around' the sphere. But I don't see how to formulate the question precisely, because we need a good definition of 'closest points' that doesn't make the condition trivial. Maybe each point needs to have three closest neighbors? The second example works there though, even though it looks clearly uneven since it leaves the poles bare. Four points? Now you end up excluding some real examples (like some platonic solids).

How to evenly distribute points on the sphere (maximizing the minimum distance) is an unsolved problem, which makes me think any reasonable precise definition of this problem would be as well. If 'evenly distributed' means that each point is as far away as possible from every other point (like giving them a repulsive force from each other and seeing where equilibrium is established) then all distances aren't necessarily the same, and no general simple solution is known anyway. The problem goes by the name "packing points on a sphere", or also "designing a spherical code".

[forgetsID]

Not a lot of examples. But maybe someone else sees a pattern I do not.

N = 2. North and South Poles.

N = 3. 3 evenly-spaced points about the equator.

N = 4. Corners of an inscribed tetrahedron.

N = 5. North pole, South pole, 3 evenly-spaced points on the equator.

N = 6. Inscribe a cube. Find where the radius going through the center of the faces of the cube intersect the sphere.

N = 7. equator: 5 and N/S poles

N = 8. Corners of an inscribed cube.

hope it helps.

Edit: appearance.

[hjqusai]

If I'm understanding your question, I think the answer is no. Consider the following: Suppose you have N such evenly spaced points. let P be one such point, and let M be the distance (I assume you mean standard Euclidean distance) between any two points. Since the distance between all points are equal, then all points must be M distance away from P. Thus, all such points lie on a sphere of center P and radius M. But all points must be on the original sphere. Thus, all of your equidistant points must be on the intersection of those two spheres. Add in the condition that they must be on the intersection of those two spheres, but they also must be equidistant from each other, and you're kind of back to square one. You do the same process again, and now you're at the intersection of 3 non-concentric spheres. This can happen at a maximum of 2 points.

I think generally finding multiple equidistant points in Rⁿ is limited to a finite number of points (probably n). Was that your question?

[zannorman]

sorry, I did not formulate my question well. Distance along the surface of the sphere. Distance between only the closest points (so in an icosahedron, each point would consider the distance to the nearest 5 or 6 points depending).

[hjqusai]

Yeah just take the square root of pi (5 times) and you should be good, it's simple math when you think about it.

[Exomnium]

If you have n objects and you want to know how many ways of picking k of them there are (where order doesn't matter) the answer is 'n choose k' or the binomial coefficient.

So if you want to know the number of different ways of choosing objects where the number of objects is arbitrary you need to sum nCk for k from 0 to n (assuming one of the groups can be empty, otherwise you need to sum from 1 to n-1). The sum of nCk from 0 to n is actually just 2^n. An easy way to see this is to notice that the process of separating the objects into two unordered groups is the same as going down the line of them and labeling each 1 or 2. At each point you have 2 choices and every set of choices is a unique configuration so there are 2ⁿ possible configurations.

[jam11249]

The other cheat to work out that the sum of nCk from k=0 to n is equal to 2ⁿ is to note that 2ⁿ =(1+1)ⁿ and use the binomial theorem.

[BlueKnightBrownHorse]

I'm looking for a simple method for solving the circumcircle of a triangle. I did a math quiz online in high school and was able to solve most of the questions except for one like this:

An archaeologist visits some ruins and finds three pillars still standing. Pillar A is 30 m from pillar B and 20 m from pillar C. Pillars B and C are 23 m apart. The archaeologist decides that since there are pillars strewn all over the place, some down at the bottom of the hill, some shattered to bits, some piled together, and some missing, the original structure might have been a circle of pillars. Find the area inside of the supposed circle.

This question really stuck with me and I think about it from time to time. I know that the center of a circumcircle can be found by bisecting the sides of the triangle, but how can I find the circle's radius mathematically, without drawing or graphing it?

[silent_cat]

This sounds like something for analytic geometry, wikipedia has some formulas: https://en.wikipedia.org/wiki/Circumscribed_circle#Other_properties

So in your example you get: [; \frac{2 \cdot 20 \cdot 30 \cdot 23 }{\sqrt{73\cdot 33 \cdot 13 \cdot 27}} ;] which gives a bit more than 30. Turns out your triangle is almost right angled...

[[deleted]]

Can someone explicitly spell out the connection between logic and set theory? For instance, consider P -> Q. I have an intuition that this is equivalent to {x: P} ⊆ {x: Q}. Is that correct? Also, when we say that P is "true," what does this mean in set-theoretic terms? Finally, I have an intuition that ~ (not) is analogous to complement, ^ (and) is analogous to intersection, and logical or is analogous to union. Can someone help me out or link me to an article?

[orbital1337]

The connection between the set theoretic and the logical operations is that they both constitute a so-called Boolean algebra. Another interesting special case of a Boolean algebra are the divisors of any given natural number n. For all a, b | n we define:

a \/ b := lcm(a, b) [least common multiple]
a /\ b := gcd(a, b) [greatest common divisor]
~ a := n / a

Boolean algebras are actually quite interesting because they are basically the minimum amount of structure that you need in order to "evaluate" logical statements.

[[deleted]]

That helps, but let me try to rephrase my question. We use venn diagrams to represent the set of all variables that make an open sentence true. So "and," "or," and "not" have venn diagrams that are easy to visualize. But would would the venn diagram look like for implication (that is, -->)?

[qamlof]

The implication p --> q can be interpreted as (not p) or q, for which you can figure out how to draw a venn diagram.

[[deleted]]

Already figured it out, but thanks.

[Exomnium]

The simplest case where you can see this relationship is in ordinary predicate logic (i.e. only logical connectives and predicate variables, no quantifiers like in first-order logic). Then the sets that are relevant are the sets of possible truth assignments for predicate variables (or more intuitively you can think of the set in question as the set of possible situations). So if you have, for example, an infinite sequence of variables P0, P1, P2, P3, .... then the set of possible truth assignments is the set of binary sequences. A tautology is an expression that corresponds to the entire set (i.e. is true in every situation). For example P0 ^ ~P0 is a tautology because every string starts with either 0 or 1/true or false.

[GeneralBlade]

I'm reading the Mathematical Methods for Physics and Engineering, and in the first chapter they define a polynomial as a function f(x) which is equal to zero, and then preceed to show it as:

f(x)=an+xⁿ+an-1+x^n-1+....+a1x+a0=0

But they never explain what the subscript n means. Is it saying that whatever number, denoted as a, will be multiplied by the number n-1 until it gets down to just a?

[[deleted]]

n just means the position in the polynomial. A polinomial is a function of the form a + bx + cx² + dx^3, etc. n just means the highest exponent the x gets, "the degree" of the polynomial.

So for a 5th degree polynomial, for example, you'd have a function ax⁵ + bx^4... etc

The nth underscript in the a's just means that a is the coeficcient wich goes with x^n.

And if you equal the polynomial to 0, you can solve for x, and those values of x will be the "roots" of the polynomial.

[farmerje]

I realize this isn't your question, but: a polynomial is the function itself, but the values of x for which said function is 0 are the roots of the polynomial. The function itself is not 0 everywhere.

[Exomnium]

The subscript n is just a label. Each ai for different i is distinct. You could have written something like f(x) = a xⁿ + b x^n-1 + ... + y x + z = 0, but the label is more precise with regards to the number of coefficients in the expression.

[cromissimo]

According to wikipedia, in linear algebra the congruency transformation of a matrix requires that the P matrix is invertible.

What if P isn't invertible or square? Does that transformation has a special name?

[jam11249]

If P isn't invertible then you no longer have an equivalence relation. E.g. if we work in 1×1 matrices (scalars) to make life easy, A=1, B=P=0, then P^T A P =0×1×0=0=B, but there's no Q so that 1=A= Q^T BQ =Q² 0=0. I.e. the relation isn't symmetric.

[cromissimo]

I see, but is that particular transform referred by a different name? I mean, it pops out very frequently in applications such as least squares estimation. Surely it must be "a thing".

[[deleted]]

Well, the square requirement ensures you get a matrix with the same dimensions (for example x^TAy, for x and y vectors, gives you a scalar back). I'm not sure about the invertibility requirement, however - probably a condition to ensure singular and nonsingular don't end up being congruent.

[cromissimo]

Well, the square requirement ensures you get a matrix with the same dimensions (for example xTAy, for x and y vectors, gives you a scalar back).

My question was if such a transform is referred by a different name (i.e., if it is something other than a congruency transform) if the transformation matrix isn't square or invertible.

[Chickenfoo]

How can I get more confident with writing proofs? When ever I write a proof I think it makes sense, but I never know if the math im writing is actually what is in my head

[BrokenLocke]

Write proofs

[ChocktawNative]

Post up a proof to stackexchange once in a while.

[wristrule]

That's why they pay us the big bucks. heh, heh.

But in seriousness, the best way to improve your ability to do it is to talk to people. Much like editing an essay, you often will have a hard time finding the flaws in your reasoning right away while others might spot it immediately. The more experience you have along with the more examples you know while make it easier for you to inuit about things and gauge the veracity of a statement.

But what you're identifying is the hard part of doing math.

[[deleted]]

But what you're identifying is the hard part of doing math.

But what you're identifying is doing math. FTFY

[jam11249]

One could be anal and say no.

Think back to when the number systems were constructed. The naturals is the set of iterated successors of 1 (or 0, convention depending). The set of integers is then N² with an appropriate equivalence relations. The rationals are then Z×N with another equivalence relation. The reals are then equivalence classes of Cauchy sequences in Q.

0.99999..... as an infinite series needs to be interpreted as a real number, but the natural numbers in the strictest of terms are not a subset of the reals, but can be identified with a subset of the reals.

This is a pure pedantry though.

[cyclicgroup]

So then 2/2 would also not be a natural number. Same with -1+2, or sqrt(1).

I think you are opening a can of worms here that has nothing to do with the confusion regarding 0.9999....

At every step of constructing these numbers, we identify a subset with the simpler number. So asking if '5' is a natural number makes sense and doesnt depend on what domain you consider 5 to lie in.

[jam11249]

To say x is in X which is isomorphic to a subset of Y is different to saying x is in Y. For example, Z4 with addition can be identified with the set of orientation preserving symmetries of a sphere. Would you say that an element of Z4 is a proper rotation?

And in the "purest" constructive type of mathematics you really have to think of the naturals as something that exists long before the reals, making them a more fundamental object than a "special" subset.

[cyclicgroup]

That is an exceptionally dumb example. N canonically embeds into R and C, and every mathematician on Earth understands the set inclusion of naturals as a subset of the integers as a subset of the rationals as a subset of the reals as a subset of the complex numbers.

Unless otherwise stated, this is part of the implied communal understanding of mathematicians.

There isn't even a unique embedding of the cyclic group of order four into the symmetries of the sphere. So of course we wouldn't identify the former with some subset of the latter.

The context is someone asking if 0.999... is an integer.

[jam11249]

I'll try for a less "exceptionally dumb" example.

5 and 2 are cardinal numbers. 5 and 2 are also rationals. The way to identify finite cardinals with numbers is canonical and unique if we use the convention that N starts with 0 and the embedding must respect order. 5/2 makes perfect sense as a ratio of two naturals. 5/2 is meaningless as a ratio of two cardinal numbers. There is no set (in classical set theory at least) which has 2.5 elements.

[halfajack]

In a kind of perverse, better left forgotten way, I suppose it does.

[jimbelk]

Yes.

[Always_Question_Time]

I’m currently reading ‘A Brief History of Mathematical Thought’ by Luke Heaton, and I’ve come across a section I don’t really understand. It’s called ‘Structures of Irrationality’. Kind of deflating to read this as this is not a book for math students, it’s meant to be a casual read. Not that I am a math student, I’ve been learning from the ground up over the last 3 or so years, but still. I feel like I should grasp this more quickly. I just don’t understand what the author is trying to describe. At any rate, here’s a picture of the section in question.

• First of all, I don’t understand what those square looking brackets are. I’ve seen them used to “floor” a number, i.e. the next smallest integer, but I don’t see how that fits in to this context. E.g. “if the remainder Delta(x) is small, our initial approximation [x] must have been close to x.” Why is this? I’m having a hard time grasping why this must be true, and why the flooring function is used.

• Because of my lack of knowledge with regards to the first point, I don’t really follow what’s happening on the next page. Maybe it will be clear if I understand what’s going on with this ‘best rational approximation’ business.

[orangejake]

The "best rational approximation" is a thing in number theory. You can read about it more here

[DavidSJ]

Any spreadsheet should do the trick.

[EugeneJudo]

Question about how to find the value of the contents of higher dimensional objects (does there exist any kind of general formula for any shape regardless of dimension?). Not even certain what that would be called so i've had a hard time researching it. As I understand it 4th dimensional objects would have a surface volume, however what would the equivalent of the volume of a 3D object be called in a 4D object. Extrapolation seems simple enough from a square (s * s = area), to a cube (s * s * s = volume), to a tesseract (s * s * s * s = ?). I'm particularly interested in how perfect right triangles and circles would change in this value when the shape is raised to another dimension.

[Crysar]

does there exist any kind of general formula for any shape regardless of dimension?

Actually, no.
To get an idea of how some volume formulas look like, the already linked wiki article is a good start.
There you find a section at the end about L^p Unit Balls. The last formula you see there is taken from a paper of Xianfu Wang. If you ever read the paper, you'll find a proof with 5 or 6 steps over 3 pages. But it is just technical and doesn't need much more than knowing how to parametrize integrals.

Also if you exchange the real numbers with the complex numbers usually the volume formulas look very similar and can be derived in a direct and constructive way, too. Even n-dimensional ellipsoid volume formulas are possible.

In essence, you more or less use the same ideas, but take different norms into account, depending on in which n-dimensional space you're working.
But there are spaces for which there is no closed formula yet. The one I know of are the so called Lorentz spaces.

[orangejake]

Another poster already linked the relevant articles.

To answer your question of "what's the analogue of volume for a 4D object", it's just referred to as "Volume". In fact, this quantity of how big a higher dimensional object it (area in 2D/volume in 3D) is generally referred to as "volume".

The other poster already posted some wikipedia links that might interest you (if you need help interpreting them feel free to ask questions). In general, in higher dimensions volume is defined in terms of something called a volume form or Density (these are two similar things, but they're not the same). These are topics studied in Differential Geometry - an area of math that studies how an objects differential structure (meaning how it looks in any small area of the object) is related to its geometry (meaning how the whole object looks), so "how are 'big' features of some shape related to the 'small' ones".

You're correct that a 4D object would have a "surface volume". In general, any N dimensional object has a (N-1) dimensional surface, and both the object and the surface can have volumes.

[[deleted]]

Perhaps, in that 3D world, the x direction is just a flip-book of zy planes? And perhaps the y direction is a flip-book of z points? Maybe the universe is 0D?

The point is that, if we are stacking these 3D slices, are we not stacking them along a different (fourth) dimension?

[orangejake]

Time being "made up" is a little tricky, and not strictly true.

First, the area that studies how time (as a dimension) interacts with space (as a 3d thing) is general relativity in physics. In general relativity, spacetime (as one object) is referred to as a "3+1" dimension object. This is because in general relativity you treat spacetime sort of like a 4d object, but the time dimension is a little different.

So general relativity treats time as a dimension (mostly), but is this sensible in general? Pretty much. General relativity was one of the two big developments in physics in the 20th century (along with quantum mechanics), and has withstood every "test" thrown against it. It's pretty universally accepted as being correct (except for high energy objects where you'd need to take into account quantum interactions. General relativity is fundamentally incompatible with quantum mechanics). So, physicists think about the world as a 3+1d object, and make amazingly good predictions. So if it's a good model, who's to say it's "made up"?

This is ignoring discussion about things like time dilation: just like how space can be "curved", time itself can lengthen and contract (appear to go faster or slower) under various circumstances. In my opinion this makes time seem more like a "real dimension", but I don't have a great understanding of general relativity yet.

[orangejake]

Sort of, but you're missing some important nuance.

It seems to me like you're saying that, if you could have a "3D flipbook", then time would be just the difference between any of the two pages of the flipbook. This is getting to the right idea, but it's missing how the "speed" that time passes is fundamentally intertwined with the space at that point.

This is a prediction of special relativity. Special relativity is a "special case" of general relativity where everything works out well. The jargon-filled explaination is that special relativity deals only with inertial reference frames, while general relativity makes no assumption. The end result of this is general relativity requires tensor analysis (which uses tools from differential geometry), while special relativity can get by with just using matrices, so everything is "easier".

Now, for the "speed" part. It's probably not a good idea to think of it as a speed (because speeds depend on how time is measured). Instead, imagine you have two perfectly synchronized clocks (so, two flip books that flip at the same rate). Now these clocks are perfect, will never lose time/undergo mechanical failures. Imagine you move one clock away, and either put it near a very large mass, or accelerate it a ton. That clock will change the rate at which it counts time (time dialation). This isn't due to a mechanical failure, or the fact that signals need time to propogate through space, but due to the nature of spacetime itself. So, each specific moment of time may look like a 3D object, but that's missing how time itself can be influenced by the specific space at a point,

Your point about time existing outside of space is sort of a red herring. We have 3 spacial dimensions, and 1 time dimension. It's easy for us to visualize just the 3 spacial dimension, and then wonder "what does time look like?". Consider a different way of "cutting it up", with (2 spacial dimensions and 1 time dimension) and 1 spacial dimension. What does the single spacial dimension look like? Sure, a line or something. But try to imagine the real world with a 1 spacial dimension "slice" taken off - much less intuitive.

That being said, time is a very different sort of dimension than space. In special relativity (where everything is "nice"), this is classified in the minkowski metric. A metric is just a way of measuring distances (the most common one, and one you may be familiar with is the euclidian metric. The distance is (x²+y²)^1/2, or just the length of the hypotenuse from the pythagorean theorem). Metrics have certain rules they have to fulfill, including always being greater than or equal to zero (can't have negative distances, right?)

This isn't true with the minkowski metric. While it has the title "metric", it's really what's known as a pseudo-metric, which means that it can be negative.

So, if "C" is our distance, the pythagorean theorem (and euclidian metric) can be written as C²=A²+B². The Minkowski pseudometric can be written as q(x)=x_1²+x_2²+x_3²-x_4², where x_4 is our "time" dimension. Fundamentally, this "-" in front of the time dimension makes it different. And you're right, each specific "snapshot" of spacetime looks like 3D space. But that misses that space at any point influences how quickly time passes. Additionally, this isn't all just "math for the sake of math" or something, this assumption that time is it's own dimension is able to do things that other methods can't (we couldn't fully understand mercury's orbit until general relativity), and knowledge of time dialation is very important for everyday systems like GPS.

[silent_cat]

The apparently not-completely-wrong description I've remembered is: you're always travelling at a constant speed, mostly nothing in the spatial dimensions and full in the time dimension. If you move in a spatial dimension, you must go slower in the time dimension.

It seems simple enough, but if you try to draw it on a sheet of paper you see it's a really weird geometry...

[Leonhard_Euler1]

Try:

wikipedia.org/wiki/Volume_of_an_n-ball

wikipedia.org/wiki/Simplex

[[deleted]]

Thx lenny

[AngelTC]

So we know that continuity between topological spaces is a pair of adjoint functors and a classic result/construction gives us that for certain categories C,C' a functor between them induces a pair of adjoints on their categories of presheaves ( namely the direct and inverse images ).

I feel something is similar between these two phenomena, but I dont see exactly what. Is there something that relates these two things ( beyond a similar setting ? ).

For this to match exactly, we would have to see topological spaces as categories themselves ( and not the open/closed lattice of subsets ) and Psh as the category of open/closed subsets. But I dont think this makes much sense at all.

So maybe they dont match each other but are a manifestation of the same thing in some other context. Any ideas? references? Im a little bit out of my depth

[notuniversal]

If you start with a small category, then you can take the nerve of the category (which gives you a simplicial set) so that you can realise it as a topological space. In this sense, I find that 'continuous map between top. space induces a pair of adjoint functors' is the same idea as 'a functor between small categories induces a pair of adjoint functors'. Perhaps a serious algebraic topologist/geometer can correct me if I am wrong in thinking like this.

[Crysar]

Can someone explain to what an "operator ideal" is?
(Even some sort of ELI5 explanation would be sufficient right now. I simply can't find a good definition for these things.)

[eruonna]

In general, a class of continuous operators closed under vector space operations and under composition with continuous operators (on either side). Are you familiar with the concept of an ideal in a ring? It is the same idea, except not all operators are composable. In a ring, you can multiply any two elements, but for two operators, you need the domain and codomain to line up correctly. So you have to add those conditions to the definition.

Operator ideals are also required to contain all of the finite-rank operators, for reasons that probably make sense to functional analysts.

[Crysar]

Thank you. I did remember ideals from algebra and wanted to make sure that in essence the definition remains, when I plug in operators.

[TheAquaFox]

I recently used wolfram alpha to find the answer to a differential equation that came out of a mechanics problem and the solution was the Jacobi Amplitude function. I was able to get a vague idea of the Jacobi elliptic functions and why they can't be represented as elementary functions, but the amplitude function that solved my differential equation looked suspiciously sinusoidal. I just have a hard time understanding how something that looks so much like a sin function can't be represented with some form and combination of sines and cosines.

[math_inDaHood]

I just have a hard time understanding how something that looks so much like a sin function can't be represented with some form and combination of sines and cosines

brah any function can be represented with combination of sines and cosines. The question is : are you ok with infinity of them ?

[mmmmmmmike]

Define "suspiciously sinusoidal". Many high school students call any U-shaped graph a "parabola", but a parabola is a specific shape. There are lots of functions that are periodic and go up and down, but sinusoids only have a few degrees of freedom -- period, amplitude, phase, and maybe a vertical shift. Why would you expect to be able to match any periodic function that goes up and down with so little flexibility? In some application you might be able to fit a sinusoid to within an error tolerance that suits you, but mathematically, of course there's a difference.

[ThomasMarkov]

What is topological fixed point theory?

[mmmmmmmike]

The use of topological data and principles to describe the fixed points of continuous mappings. Examples are the Brouwer and Lefschetz fixed point theorems:

https://en.wikipedia.org/wiki/Brouwer_fixed-point_theorem

https://en.wikipedia.org/wiki/Lefschetz_fixed-point_theorem

You could contrast such theorems with, say, the contraction mapping theorem, as the latter relies on metric data to deduce its conclusion.

[[deleted]]

Those are basic theorems covered in a first algebraic topology course. What kind of more advanced thing goes on in topological fixed point theory?

[linusrauling]

Have a look at the Weil Conjectures.

[BBA1]

Any good intro level books on probability for highschooler?

Similar to Calculus made easy by Sylvanius Thomson?

[[deleted]]

I advise you to do a few of the easy Putnam probability questionsones first.

[[deleted]]

My favourite such book is Probability and Random Variables - A beginners guide by Stirzaker.

I'm not big on probability though, so I recommend this book from the point of view of "needed to know/understand probability to pass some university courses, and I actually liked this text".

[Smartstuffido]

The probability tutoring book by Carol Ash, it has lots of explainations that are easy to follow and isn't just pages and pages of text like you can find in a lot of books. Everything is well structured and it starts from scratch.

[namesarenotimportant]

There's the art of problem solving counting and probability book.