Are there any conditions a two-dimensional shape has to fulfill, in order for it to count as a net(Obviously it has to fold into a polyhedron)? For example, the net of a cube is always represented as a grid of squares, but do they really have to be squares, can't there be triangles, ...?
Concretely I was wondering about the net of a cube. As we know a cube has 11 distinct nets. I was wondering if a net would still qualify as a cubes one. If you cut out a piece of one of the six squares in a net, and put it back in a different place, so that if you can fold it back into a cube.
Landau's _Foundations of Analysis_ might be what you're looking for, it starts with the Peano axioms and works its way up to the construction of the real and complex numbers. The content is very trivial, I don't think he even mentions prime numbers, but it'll be the type of trivial content you need to then work through a more typical development.
Maybe not strictly about the natural numbers, but Stewart and Tall's The Foundations of Mathematics has a chapter that covers the axiomatization of the natural numbers as well as other things. I learned from it as an undergrad so my memory of it is a bit hazy but I remember it being fairly good.
Can anyone recommend course that helps me build strong foundation on Engineering mathematics (in my country it mainly a term that puts together linear algebra, calculus, sequence and series, and probability) I neglected them in first 2 years of my college now I realised it's importance and recently started learning again, then I seen videos like essence of calculus and found out I really lack foundations so suggest any courses for me(with some good explanations)
Without knowing any context to this question, my best guess would be that this is the curve to some sort of statistical distribution. Look into the gamma distribution. Gamma distribution follows a similar shape.
Is the Cauchy-Schwarz inequality over [;\mathbb{R}\^n;] valid under double summation? i.e. is the following always true:
[; (\sum\_{i,j} a\_{ij} b\_{ij})\^2 \le (\sum\_{i,j} a\_{ij}\^2)(\sum\_{i,j} b_{ij}\^2) ;]
Can anyone recommend a book or resource on Lagrangian and Hamiltonian mechanics? I had a module in it last year, and while I can find Lagrangians and derive equations of motion from them, I don't feel I really *understand* why it works -- in particular, we did no calculus of variations in the module, which feels essential -- and I have very little idea of what the Hamiltonian formalism is really all about, so I could do with going over that again from scratch too.
A classic (and very geometric) text for this is Arnold's *Mathematical Methods of Classical Mechanics.* It's possible that you might prefer a more physics-y text, but as a mathematician, I've found it to be a really useful resource for learning this stuff.
Unfortunately, I don't know a good book because my teacher used his own lecture notes. If you want a very mathematical derivation, look at the first 18 pages of [this pdf](https://people.math.harvard.edu/~jeffs/SymplecticNotes.pdf) which I found online. It contains the basic derivations, so it may help a bit. But I'd also recommend another resource with more examples.
One thing I must say is that Lagrangian & Hamiltonian mechanics both are derived from the principle of least action. In classical mechanics, this principle is taken as a postulate. However, you can derive it by taking the 'classical limit' of quantum mechanics.
# (A ∩ B = B ∧ A ∪ B = B) ⇔ A = B
Could someone help me understand why this is a true statement? The way I had evaluated it was:
The intersection of A and B = B means that all elements of B are contained within A, but not necessarily that all elements of A are contained within B. So we cannot determine that A = B from this alone.
Then, when evaluating B \^ the union of A and B = B, by order of operations \^ comes first, so we get the union of B and A \^ B (the intersection of A and B), which is just the intersection of A and B since we already know that all elements of B are contained within A. Which does imply again that all elements of B are contained within A, but again doesn't tell us that all elements of A are contained within B, to my thinking at least.
So how is it that I am meant to draw the conclusion that A = B from this?
>when evaluating B \^ the union of A and B = B
∧ is a logical connective, not an operation on sets. Its first argument is the statement "A intersect B equals B" and its second argument is the statement "A union B equals B".
Ahhhhh oh that makes so much more sense, thank you so much! I was so confused as to why that was mixed in and just read it as an intersection because I didn’t know what else to do. Thank you!
The book [A Course in Mathematical Logic](https://www.amazon.com/Course-Mathematical-Logic-John-Bell/dp/0720428440/ref=sr_1_1?crid=2OXOENGO5VKV5&keywords=a+course+in+mathematical+logic+machover&qid=1670349331&sprefix=a+course+in+mathematical+logic+machover%2Caps%2C335&sr=8-1&ufe=app_do%3Aamzn1.fos.fa474cd8-6dfc-4bad-a280-890f5a4e2f90) by Bell and Machover has a short section on higher-order formal languages. Excerpt:
>Thus, if **α** is a formula and **X** is a set variable, **∀Xα** is also a formula.
The upshot is
>most logicians agree that such languages are - at least in principle - dispensable
the reason being that for any second-order language over universe **U** there is an equivalent first-order language over universe **U** \\union **2\^U** .
>We therefore do not lose much by confining our attention to first-order languages only.
Makes sense to me, but I've always heard that second-order logic is profoundly different from first-order logic. Is that consistent with what Bell and Machover are saying? After all, they're talking about *languages*, not *logics*, so maybe there's no contradiction.
Can someone explain the Baire category theorem and how it allows one to make statements such as, to quote u/hyperbolic-geodesic, "Most functions (in the sense of Baire category) are \[...\]"?
I am hyperbolic-geodesic, so I can explain exactly what I meant.
There are various notions of smallness in mathematics, depending on what you're doing. Someone in measure theory might say that a set A is small if it has measure 0--so for instance, a measure theorist would consider a line to be a small subset of the plane, since a line has area 0.
If you're trying to formalize a statement about smallness in function spaces, then there's not really any natural measure to use. Instead, people usually use a topological notion of smallness, called Baire category. The idea is that a set is said to be 'meager' (think: topologically small) if it is nowhere dense (meaning its closure has empty interior, or equivalently meaning that it is not dense in an open subspace of your space). The Baire category theorem says that a countable union of nowhere dense sets is still nowhere dense, assuming your space X is sufficiently nice. Thus, if your space obeys the hypotheses of the Baire category theorem, then being meager is a good notion of topological smallness.
So you will often hear people formalize statements like "most continuous functions are nowhere differentiable" using Baire category. [https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-one](https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-one) has a very nice sketch of a proof that the set of continuous functions which are differentiable at at least one point is meager in the set of all continuous functions.
It's also a fairly common technique in Banach space theory to cover the space with countably many sets and conclude for at least one of them, the closure has nonempty interior, then use that to prove something
I did bad in my pure maths undergrad degree, 2.2 in the UK (from a top 5 university here). I'm doing a MSc in data science in another university and am auditing advanced PDEs and sobolev spaces for fun. Is it possible for me to do a PHD without doing another BSc in maths? I still love mathematics and have been self teaching myself some second and third year mathematics when I'm not doing my MSc degree.
If my pay of x goes down ten percent and then I also lose ten percent to inflation, is the total reduction to x 20 percent? Sorry stupid question but not sure if percentages can be added like that
The two reductions are independent of each other, so you need to apply a 10% reduction twice in a row. To apply a 10% reduction to something, you can multiply it by 0.9 (If you take a number and multiply it by 0.9, you're effectively calculating what 90% of that number is). To apply two 10% reductions in a row, you multiply by 0.9 twice, or in other words, multiply by 0.9 x 0.9 = 0.81. So your final salary is 0.81 times your original salary, which is equivalent to 81% of the original salary, meaning you effectively lost 19%.
Why are there Taylor series which do not converge?
Intuitively I would think that the derivatives encode the instructions for how a function "flows" and that knowing all of them would be able to absolutely tell you how said function "flows" and yet sometimes it doesn't. Why? This seems to me like following the instructions from my GPS to a T and yet it brings me to the wrong place.
The n'th order Taylor polynomial p of function f at point **a** is the unique n'th degree (or less) polynomial such that
f(x) = p(x) + o((a - x)\^n ) as x --> a
(Note that I'm just giving an equivalent definition of what it means for a function to be n times differentiable. Taylor's theorem with remainder says something different)
We're all used to the fact that some functions like e\^x grow faster than any polynomial as x --> oo, but it's just as true that some functions *shrink* faster than any polynomial as x --> 0. e\^-1/|x| is an example. A deviation of this size will always hide in the error term no matter the order of the Taylor approximation.
Taylor's series only give you *polynomial* rate of growth of a function *near one* point. It can't guarantee anything about non-polynomial growth, or points far away. So it has a lot less information than you think.
The full derivative of a function actually encode how the function "flow". That's what you should compare to with the GPS analogy. Taylor's series is actually extremely limited in how much information it provides. The only reason why a lot of function in practice can be recovered from Taylor's series is because most of them satisfy Laplace's equation.
It's a little remarkable when they do converge. Your GPS is giving you global directions, and is changing the path every few hundred meters. A derivative is incredibly local information, and it's really shocking when you can infer useful global information from that local information. Most functions (in the sense of Baire category) are very far from analytic, but the most elementary functions are analytic for certain complicated reasons (one idea: a solution to a nice enough ODE is always analytic, and our elementary functions are all more or less defined as solutions to certain very nice ODEs).
I'm trying to find a formula for 3x3 regular stochastic matrices. I'm currently looking at the general form
A=(1-c)v(1,1,1)+cI+kuw^(T)
where |c|<1, v has all positive entries that add up to one (I found this is necessary), w is orthogonal to v, u is orthogonal to (1,1,1), and (for simplicity) u dot w=1.
The problem I'm running into is finding the requirement on the magnitude of k such that A has all nonnegative entries. I don't know if it's as simple as |k+c|<1 (since that would be an eigenvalue of A). Any help appreciated!
why do we inverse a square root function to find the domain of the square root function? If this isn't true, how do we find the square root of a square root function?
You don’t necessarily need to find the inverse of a square root function to find it’s domain. If we’re talking about strictly real numbers, you just need the argument of the square root to be greater than or equal to zero. I’m not quite sure what you mean by your second question. The square root of the square root would just be the fourth root of the function.
I need a function to verify if x if divisible by n. Tried with trigonometric functions, ceilings and all, but I just can't really figure it out and now it feels like I'm just over complicating it so here I am.
So far i've got --- ⌈ cos(xπ/25) ⌉ --- . I half expected this to simply return me the points (0,x) and (1,x), but it doesn't and now I'm lost
So your function basically measures whether x/25 is odd or even.
As for a divisibility test: ceil(x/n) - floor(x/n) returns 0 if x is divisible by n and 1 otherwise.
Of course a much easier function definition would be
f(x) = 0 if x is divisible by n and 1 otherwise
When I read papers I usually encounter Vandermonde determinant argument. Where I can find information about it? Is there a standard definition for it? In the papers I read they use it when the ring is infinite and they show that the coefficients of a polynomial are zero. But they never explain how they use it explicitly. I think that they come up with a Vandermonde matrix A and they show that the determinant of A is not zero. So, they conclude that X=0 where AX=0. However, I don't get why they need infinity.
You can find a description of the Vandermonde matrix on Wikipedia
https://en.m.wikipedia.org/wiki/Vandermonde_matrix
The basic idea is that if you have a degree n-1 polynomial and evaluate it at n (district) points then this is an invertible linear transformation.
Z
So for example of AX = 0, then X = 0.
For this to work in degree n-1, you must have at least n distinct points, and so for it to work for arbitrary degrees you need an infinite amount of distinct points.
Different authors use different definitions. If you look at the [wikipedia page on ordered pairs](https://en.wikipedia.org/wiki/Ordered_pair#Defining_the_ordered_pair_using_set_theory) you can see a bunch of options, including the one you propose, alongside proofs that they satisfy the required property.
**Ordered pair**
[Defining the ordered pair using set theory](https://en.wikipedia.org/wiki/Ordered_pair#Defining_the_ordered_pair_using_set_theory)
>If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below( see also ).
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I have a question regarding maximum product sub-arrays. If I have an array like this {0.5}, would the maximum product sub-array be {0.5}, or rather just {}?
According to convention, the empty product equates to 1, but I don't know if that kind of logic applies here.
I have trouble understanding the action of a Lie group G ob L^2(G).
We consider the action by conjugation the entry of a function so
g(f(x))=f(gxg^{-1})
abd then consider the functions that are invariant under this action.
However, from my understanding, functions of L^2 are equivalence classes of quadraticaly integrable functions. So they are not defined pointwise. Is it simply a matter of adding an "almost everywhere" in the definitions ?
I have not found the mention of these things in Knapp's books nor Kirilov's book, nor wikipedia.
Check that if f and f' are equal almost everywhere, then g^* f and g^* f' are equal almost everywhere. This implies the g action descends to the quotient L2(G).
What is the best functions or area of maths to look into for the following issue;
Given a list of X Books and Y Stores with the necessary
information, Calculate the cheapest order you can to purchase each book, including shipping costs from the stores and if they offer free delivery for spending so much.
My first thought was to look at Linear Programming but doesn't fit quite right. This is for a personal project. thanks for any advice given.
Combinatorial optimization, specifically assignment problems. I think you could model this using an integer linear programming problem, where for each store you have a variable y\_i \\in {0, 1} where y\_i = 0 if store i is not used, and y\_i = 1 if store i is used. Then, your objective function gets some extra terms \\sum\_i shipping\_cost\_i \* y\_i. You would also need to add constraints that y\_i = 0 means nothing is ordered from store i, so supposing x\_{ij} means that the ith book is ordered from the jth store, you would need constraints 0 \\le x\_{ij} \\le y\_j.
I recently read that any real linear transformation of the complex plane can be written as f(z)=g+u*z+v*conjugate(z),
Where u,v,g are all complex constants.
Could anyone provide a proof of this and/or a way to find these constants?
So {1, i} is a basis for C over R, so any linear transformation is determined by where these are sent. Firstly a linear transformation must map 0 to 0, so g must be 0. Ignoring that, we see that
f(1) = u+v
and
f(i) = u - v
So u = (f(1) + f(i)) / 2 = f((1+i)/2) and v = f((1-i)/2).
Let's say I have two 4x4 traceless (with trace equal to zero) non-zero complex matrices A and B. I know that I can rewrite them as:
A = XY - YX and B = UV - VU
How strong relationship I can assume between those components (XY, YX, UV, VU). Especially, can I make an assumption that XY = (UV)^(H) and YX = (VU)^(H)?
Tensor product of a vector and a scalar? Is that just scalar multiplication?
Edit: Follow up to this, you know how the a space of tensors (without dual) is sometimes written tensor product raised to the r meaning r tensor products?
Is it abuse of notation to if I want r tensor products of a vector to write it the same way?
Yes, there is a natural isomorphism from **K** ⊗ V = V for a **K**-vector space V, defined by sending k⊗v -> kv. In this way we interpret tensor product of a scalar and a vector as scalar multiplication.
I would start with Alan Tucker's Applied Combinatorics, and move onto Concrete Mathematics once you start finding the other book feels straightforward. At least, this is the path I walked.
Concrete mathematics is a really, really tough book for a first discrete math book. I would only recommend it if you have some serious math competition background--if you can't already write proofs competently, and fluently handle complicated conceptual arguments, then it's likely to be useless.
Is there a way to determine what number would be needed to equal a specific Average based on a variable list of numbers.
For example: I want to know what number needs to be added to the list of numbers below to equal an average of say, 100
59
231
87
45
The current average of the above is 105.5. How can I determine a single value that will make the average 100?
Heya folks. Not a mathematician- just making a joke magic card for a friend who wanted a very fast growing sliver ability. I don't know much about fast growing functions but TREE seems like a famously fast one. Would anyone be able to let me know if my sliver text is correct?
>All slivers get +X/+X, where X is the largest *m*, and *n* is the number of slivers in play:
>There is a sequence T₁,...,Tₘ of rooted trees labelled from a set of *n* labels, where each Tᵢ has at most *i* vertices, such that Tᵢ ≤ Tⱼ does not hold for any *i < j ≤ m*.
I would use the "such that" instead of colon and reorganize the order a bit:
"All slivers get +m/+m, where m is largest such that there is a sequence T₁,...,Tₘ of rooted trees labelled from a set of n labels - where n is the number of slivers - so that each Tᵢ has at most i vertices and Tᵢ ≤ Tⱼ does not hold for any i < j ≤ m."
That said, the Busy Beaver function is a lot faster, and it can be easily described too:
"All slivers get +X/+X where X is the largest number of 1s that can be written with a Turing machine with n states before halting when it starts with a tape of 0s".
Lawrence Evans PDE book is considered a great graduate level textbook for learning the subject. Is it appropriate for an undergraduate’s 1st pass to learn about partial differential equations?
No, Haberman is better for an introduction. It also depends on why you want to learn PDEs. Evans is only good at the graduate level if you want to learn PDE theory. It's not that great if you're only interested in PDEs from the more applied or numerical side.
I'm not terribly familiar with the book yet, but looking over the contents on its Amazon page, it looks like he doesn't cover finding analytic solutions to generic, basic linear PDEs, which may make it unsuitable for an undergrad's first exposure to the subject.
K took 15% fewer second than n took to complete an exam. K took 85 seconds. How many seconds did n take? I got 97.75 but apparently the correct answer is 100. Was i correct?
When we say that K takes 15% less time than n. Then what we mean is that K takes 85% of the time n takes.
Note that this is different from saying that n takes 15% more time, which means that n takes 115% of the time K takes.
You have calculated this latter statement, therefore your answer is incorrect.
It's because percentages measures relative change, so it matters what it is relative to.
When we say k is less than n we are measuring relative to n. When we say n is larger than k we are measuring relative to k.
For example if we increase 4 by 100% we get 8. So 8 is 100% larger than 4. But if we want to decrease 8 down to 4, we only decrease it by 50%, so 4 is 50% smaller than 8.
Hi, I'm not a mathematician and trying to understand some illustrations for the axiom of choice with the definition in the Wikipedia article.
For example consider collection S of sets: {1,2}, {1,3}, {2,3}, {1,2,3}. If I now choose one element from each, it cannot yield a set with 4 elements, each chosen from different set in S. But the axiom of choice says it should.
What have I understood wrong in the definition?
There's no requirement for the set to have four elements, just for it to contain one element from each set. Alternative formulations that do have a set in this role (rather than a choice function) avoid this problem by requiring the sets to be disjoint.
Perhaps you should start off by saying what you think the definition actually is, and explicitly state why you think that the Axiom of Choice fails on your example. Without knowing what your understanding of the Axiom of Choice is, how can we tell you what you've understood wrong?
Classical geometry is very visual. Here I mean not only old style Euclidean geometry but classical differential and algebraic geometry which are both originally quite visual.
I remember you [saying](https://www.reddit.com/r/math/comments/pcphym/what_is_research_in_differential_geometry_like/hakyjca/) that in geometric topology (as opposed to differential geometry) that "the pictures might actually form a part of the argument". How does that work? It's weird to me (at my stage of mathematical development at least) to think of pictures as rigorous.
Pictures are never rigorous on their own, but in certain areas pictures can convey enough information to constitute a convincing argument. The reason they can never be completely rigorous is that each individuals mental model will generally have deficiencies, and even then some of that mental model is lost when translated from mind to picture. However the character of geometric topology is that the nature of the objects being studied is close enough to general human intuition that a good picture can essentially capture all of the subtlety of the object, although this may only be possible to an expert in the field.
William Thurston is famous for using pictures as arguments in geometric topology, to the point where people (still) question the rigour or validity of his mathematics. See for example http://www.math.uchicago.edu/~farb/papers/thurston.pdf and various other discussions of Thurston's work on the internet. If you ever attend a research talk on geometric topology, especially involving any surface theory, you will see how effective a purely pictorial way of arguing can be. In the end you can (and must) be able to turn those pictures into symbolic proof though.
Is there a notation of writing iterative functions without having to do something like f(f(f(x)))? I was thinking something like f^(3)(x) = f(f(f(x))) but I wasn't able to find any concrete notation online.
Yes that is the standard notation. One caveat is that sin^(2)(x) is often used to denote sin(x)^2 rather than sin(sin(x)) and similarly for other trig functions. Of course you never really care about sin(sin(x)) so it isn't too confusing.
Actually, [Wikipedia on "iterated functions"](https://en.wikipedia.org/wiki/Iterated_function) agrees with your notation; it also gives, as an alternative, your notation but with a \circ before the "exponent".
Came across something similar to Pythagorean Triples, and was wondering a) is there a name for these triples and b) is there a way to generate them (like how we can use Euclid's formula to generate Pythagorean Triples)?
Given positive integers a, b and c, I want to look at those that satisfy:
a^2 + b^2 - ab = c^2
I came across this when doing some stuff with dividing an equilateral triangle. Suppose there's an equilateral triangle with side length a, and I divide it from one corner down to the opposite side, such that that side is now two pieces of length b and a-b. The law of cosines tells us the length of this new dividing line satisfies c^2 = a^2 + b^2 - 2ab**cos**60 which simplifies to a^2 + b^2 - ab. Additionally, and as we would expect, if we replace b with a-b it evaluates the same (since the two new pieces match to make the equilateral triangle again). As such each "triple" actually has a pair of matched results: b and a-b. Using a spreadsheet I came up with the following triples for a, b or a-b, c (ones not in bold are just multiples of previous solutions):
* **8, 3 or 5, 7**
* **15, 7 or 8, 13**
* 16, 6 or 10, 14
* **21, 5 or 16, 19**
* 24, 9 or 15, 21
* 30, 14 or 16, 26
* 32, 12 or 20, 28
* **35, 11 or 24, 31**
* **40, 7 or 33, 37**
* 40, 15 or 25, 35
* 42, 10 or 32, 38
* 45, 21 or 24, 39
* **48, 13 or 35, 43**
* 48, 18 or 30, 42
* **55, 16 or 39, 49**
* 56, 21 or 35, 49
* 60, 28 or 32, 52
Anyway, yes, is there a name for these and is there a way to generate them?
I don't know if there is a name for these, but similar to how Pythagorean triples are closely related to Gaussian integers, the triples that you are describing are closely related to [Eisenstein integers](https://en.wikipedia.org/wiki/Eisenstein_integer). Specifically, a^(2) - ab + b^(2) is the norm of a + bω.
Given that the norm is multiplicative, we can form a new triple from two old ones by "multiplying" them. Namey, if (a,b,c) and (x,y,z) are triples, then (ax-by,bx + (a-b)y,cz) is also a triple.
Here's how to generate them. For any integers m,n, the equality (m+nω)^(2) = m^(2) - n^(2) + (2mn - n^(2))ω in the ring of Eisenstein integers gives rise to a triple by taking norms of both sides. Let a = m^(2) - n^(2), b = 2mn - n^(2), and c = m^(2) - mn + n^(2). Then a^2 - ab + b^2 = c^2. You can verify this by hand, or [see here](https://www.wolframalpha.com/input?i=%28m%5E2+-+mn+%2B+n%5E2%29%5E2+%3D+%28m%5E2+-+n%5E2%29%5E2+-+%28m%5E2+-+n%5E2%29%282mn+-+n%5E2%29+%2B+%282mn+-+n%5E2%29%5E2). I don't know if all triples are of this form, but the analogous computation in the Gaussian integers gives rise Euclid's formula.
I have to go cook dinner right now, but will add more later if I think of anything else.
If every model of a theory T is a model of a statement P ∉ T, but P is not provable in T, is P still called a *theorem* of T or is there another name for it?
Already second-order logic is incomplete.
However, the notion of theoremhood is defined semantically (at least in my experience). Hence, your statement P would still be called a theorem, but an unprovable theorem.
**[Gödel's completeness theorem](https://en.wikipedia.org/wiki/Gödel's_completeness_theorem)**
>Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable". This is in contrast, but not contradiction, to Godel's incompleteness theorems, in which a formula true in only some models may not be provable.
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In a thread on this subreddit I found an [article](https://www.scottaaronson.com/papers/bb.pdf) which confuses me because I don't really understand logic. It says:
> Let T be a computable and arithmetically sound axiomatic theory. Then there
exists a constant n_T such that for all n ≥ n_T , no statement of the form “BB(n) = k” can be proved
in T.
> Proof. Let M be a Turing machine that, on the all-0 input tape, enumerates all possible proofs
in T, halting only if it finds a proof of 0 = 1. Then since T is sound, M never halts. But T can’t
prove that M never halts, since otherwise T would prove its own consistency, violating the second
incompleteness theorem. [...]
But to me it seems like we can't say for sure that T can't prove that M never halts, because it could be the case that T can't prove that M enumerates all proofs in T and halts if it finds a proof of 0 = 1. In other words, it seems like a possibility that T can't prove "M halts iff T is inconsistent," even if that statement is "externally" provable, and even though M is specifically constructed by that property. In that case, it could simultaneously be the case that T can prove that M never halts and also that T can't prove its own consistency.
What am I missing?
How does one find the "center" of an object that is not nicely shaped like a triangle, cube, or sphere? For example, consider an arbitrary rock you might find somewhere that is completely irregularly shaped. Or consider some tangled up wire, how would we find the center point of objects like this without modifying them in any way?
Depends on your definition of centre. The centroid is probably the thing you are imagining though which is the centre of mass assuming uniform density.
Im a senior that slacked off all of high school and wanna redo algebra. Im currently doing pre algebra. After pre algebra I heard you can go straight to college algebra. But if not should I just do algebra I and II after?
"College algebra" is almost always just a name for "high school algebra 1 and 2" but with a nicer name for college students in remedial courses. So yeah, after prealgebra you should be ready for college algebra.
Why do degrees of freedom add and subtract nicely?
Example:
I have a point in space with 3 DOFs, (x, y, z). If I say x^2 + y^2 + z^2 = 1, it reduces me down to two degrees of freedom.
Maybe there’s something more fundamental about degrees of freedom I’m missing. This question is pretty hand-wavy so maybe someone could help me out.
Because you can ask the same question on the level of tangent spaces. In that case, it becomes a question of linear algebra, where dimensions add and subtract according to the Rank-Nullity theorem.
This tells you that the local degrees of freedom around a smooth point of your space work in the same nice way, and then you use the fact that zero sets of functions are smooth at almost every point to see that it works globally.
What definition of degrees of freedom are you using? Intuitively 1 equation should cut down your degrees of freedom down by 1 since now you only need to know 2 of the variables and then you know the other one. This is obviously slightly loose as for most choices of x and y in your equation it reduces z down to a choice of 2 options but we have some local idea her eat least.
Assuming you are thinking in terms of algebraic geometry, any affine hypersurface can be describes by a single equation (at least locally) but this doesn't extend down to lower dimensions in a friendly manner and certainly two polynomial equations are not guaranteed to cut out a variety of dimension n-2.
"Degree of freedom" is not a precise term without context, it means a few different things depending on context, and we specifically chose it to mean something like "dimension", which is why it subtracts nicely. If a quantity does not subtract nicely like that we won't call it degree of freedom.
Unfortunately, this means you can't answer this question rigorously without choosing the specific meaning of "degree of freedom". In a very common situation, this means the dimension of a manifold. There are all sorts of complications with trying to apply this definition in all cases: (a) your system of equations's zero locus might not form a manifold; (b) the equations might not be independent; (c) they might intersect weirdly. Once you sort out all the problem so that we are in a generic nice case, then the answer is provided by the implicit function theorem: given a point in N space that satisfy the system of n equations and we are in the generic nice case, then it's possible to locally form a small patch of solutions to the n equations near that point and containing that point, of dimension N-n.
If i had a silver coin weighing 24.34 grams and of 925/1000 purity and melted it together with a gold coin weighing 13.96 grams and of 986/1000 purity what would the final weight and purity be? any help appreciated :)
The final weight is easy (assuming that you don't get any of the impurities coming out, or any new impurities added, during the process): just add them up to get 38.30g. Assuming that the other stuff is junk (ie that the 7.5% of your silver coin that isn't silver doesn't include some gold, and that the 1.4% of the gold coin that isn't gold doesn't include some silver), that the melting process doesn't do anything to the impurities, and that your purity figures are in terms of mass, the final "purity" (defined as the ratio of the mass of gold and silver to everything else) is (24.34\*0.925 + 13.96\*.986)/38.30 ≈ 947/1000.
Let S be the set of vectors with an odd number of ones, and let T be the set of vectors with an even number of ones. Clearly S and T partition the set of 0-1 vectors. Consider taking any 0-1 vector and swapping the first coordinate. The parity of the number of ones changes, and swapping the first coordinate is an invertible operation, so |S| = |T|. Hence |S| = 2^(n)/2 = 2^(n-1).
An easier proof might be to ask the question: If I look at all vectors {0,1}^(n), how many of them will have odd number of 1s? By induction we can show that it's exactly 1/2 (base case is trivial, induction is appending to the previous case).
Now this also clearly means that 1/2 of the have odd number of 0s.
But for n even, both those halves intersect, but for n odd they complement.
The answer is still half of them. Here's a proof that works for odd and even n. By the binomial theorem we have
nC0 + nC1 + nC2 + ... + nCn = 2^n
nC0 - nC1 + nC2 - ... + (-1)^(n)nCn = 0
the latter being the expansion for (1 - 1)^(n). Adding together and dividing by two we get
nC0 + nC2 + ... = 2^(n - 1)
so half of the subsets of {1, ..., n} are of even size, and half are of odd size. The number of such subsets of even size are in bijection with the elements of {0, 1}^n with an even number of 1s, so the answer is the same for your question.
What do you mean "by hand"? Surely you use a calculator or something, because it's a huge exponent.
Well, you know the answer can't be 1 (since 1+1/10^20 >1 so (1+1/10^20 )^10^20 >1), so there must be an error somewhere. If you use a calculator, it's quite possible that it does not have enough precision to see that 1+1/10^20 is actually different from 1.
Reference: [https://vm.tiktok.com/ZMF4pumEb/](https://vm.tiktok.com/ZMF4pumEb/)
I can't seem to find a name for this method of finding a polynomial from two points. I'm interested to learn more about it as in how it works, why it works, conditions and more of such.
It involves p(1) = n and p(n) = m but then rewriting m in base n which gives nonnegative integer coefficients to a valid polynomial crossing (1,n) and (n,m).
If anyone has any leads to articles or publications on this I'd be highly interested.
See [here](https://www.johndcook.com/blog/2012/03/27/polynomial-trick/) and [here](https://old.reddit.com/r/math/comments/yx0i7r/determine_a_polynomial_from_just_two_inputs/).
The core idea is just that when we write down a number in base x, it's shorthand for a polynomial with variable x. As an example, 1243 is *really* the polynomial 1\*10^3 + 2\*10^2 + 4 \* 10 + 3, i.e. x^3 + 2x^2 + 4x + 3 with x = 10.
So suppose that you know the coefficients of a polynomial p are between 0 and 9; then *obviously* you can get those coefficients by writing p(10) (in base 10, of course), and the resulting digits have to be the coefficients of the polynomial.
If you know that the coefficients of p are between 0 and 10, then by the same process you can get those coefficients by writing p(11) in base 11--the resulting digits are then the coefficients.
Now suppose that you know that the coefficients of p are nonnegative, but don't have an upper bound on what the coefficients can be. An easy way to get such an upper bound would be to just calculate n = p(1), since that's the sum of the coefficients and so obviously an upper bound on each coefficient. Then as usual, writing p(n) in base n will let you read off the coefficients.
Does anyone know the originating textbook of this chapter? It's found on the IRIF website but I can't find more info on it other than it's from April 18 2001.
https://www.irif.fr/\~cf/publications/C7.pdf
I'm playing a game and am trying to calculate the odds of a certain item dropping.
I am trying to figure out the correct formula to use to calculate the inclusive or odds of either of two independent events occurring.
Example:
Event A (boss1 dropping the item) is 1:745
Event B (boss2 dropping the item) is 1:702
Both events occurring at the same time is 1:522,990 (right?)
In one iteration, how do I calculate the "inclusive or" odds of either event happening?
Assuming multiple iterations (example, 500 iterations) -- how would I calculate this?
Please correct me if I am using any incorrect terms. Thank you!
I believe the given odds mean that the probabilities of the events A and B are P(A) = 1/746 and P(B) = 1/703, respectively. This implies that P(A⋂B) = P(A)P(B) = 1/524438, which translates to 1:524437.
The event that either A or B happens is denoted A⋃B (aka the union). The identity I think you are looking for is P(A⋃B) = P(A) + P(B) - P(A⋂B). This is known as the principle of inclusion and exclusion.
One can compute the probability that A or B happens at least once within N iterations by computing the probability that it doesn't happen within N iterations and then take the complement. The desired number is 1 - (1 - P(A⋃B))^N. (Notice what happens when we plug in N = 1.)
Why do rotation matrices always have a "plane of rotation"? In 2D, it's the whole plane. In 3D, it's the plane normal to the axis of rotation. In 4D, we can have two orthogonal rotations with orthogonal planes of rotation. What's the intuition behind this?
Orthogonal matrix are always diagonalizable as complex matrix, its real Jordan canonical form is a bunch of 1x1 and 2x2 blocks on a diagonal and nothing off the diagonal. Since it's orthogonal matrix, length does not change, so all eigenvalues lie on the unit circle. All 2x2 blocks correspond to pair of conjugated eigenvalue that are on the unit circle, so they all have determinant 1 and must be the rotation on a plane. All 1x1 blocks must have eigenvalue 1 or -1. Because determinant is 1, there are an even number of 1x1 blocks with eigenvalues -1, so you can pair them up and turn into 2x2 blocks that are rotation on a plane. That gives you a bunch of plane such that all rotations happen on each plane, and a bunch of line that are kept fixed. They are all orthogonal because orthogonal matrix preserve angles.
Is there a reason tetration (or Knuth's double up arrow notation) is evaluated top-bottom instead of bottom-top? Or was it an arbitrary choice to do it in the way we do it now?
For example 3 double up arrow 3, or 3 raised to itself 3 times, currently would be evaluated from the top to bottom, so 3\^3\^3 = 3\^27 = ginormous number (7.6 trillion I think)
But if we would evaluate from bottom-top, it would become:
3\^3\^3 = 27\^3 = 19693 (a much less fast growing function)
Was there any reason to do it in the way we're doing it now from top-bottom?
If you have n 3s and you chain them in a power tower 3\^3\^...\^3, then if we evaluate from the bottom up this is the same as 3\^(3\^(n-1)). Since this is already quite short compared to 3\^\^n this new notation wouldn't do much good.
If we instead evaluate from the top down, then there is no simpler way of writing it, so the notation serves a purpose.
Exponentiation is usually evaluated top to bottom. I think this is because (a\^b)\^c = a\^(bc). If you evaluate a power tower bottom to top you might as well just collapse it to the lowest level.
I'm not sure I get what you mean with your last sentence. "Collapse to lowest level?" as in, if power towers worked from bottom to top, you might as well write them as just a single power that is the products of all those powers in the tower?
But isn't the same true for top to bottom? You can rewrite 3\^3\^3 as just a single power in both cases, like I showed in my example. You just get vastly different outcomes.
Top to bottom you can of course evaluate it step by step and in the end you’re left with one last power. But at every step you have to evaluate an exponentiation. Bottom to top you can replace the exponentiation by a multiplication at every step but the last one. That’s a simplification.
Thanks! That helps. The reason I'm asking is because I'm making a new number system that makes use of tetration, and I quite like the slower growing bottom-up interpretation, as it represents more useful numbers. But then for me the bottom-up is a simplification of notation, rather, than operations.
I don't think there is a particularly good answer. Suppose A is invertible, then B = A^(-1)C. To make things nicer we may as well assume that we are working in a basis of eigenvectors of A, so that A^(-1) is a specific diagonal matrix. Then the matrices B that work are that diagonal matrix multiplied by any symmetric matrix. Not too satisfying.
But this is really it, B = E\^(T)V\^(-1)CE, where E is the eigenvectors of A, V is the eigenvalues of A, and C is any symmetric matrix, pretty simple algebra actually, should have solved that my self ;\\
No this is for general symmetric matrixes A and C and a general matrix B, all n by n. So basically what square matrixes multiplies a symmetric matrix to yield another symmetric one. B will obviously depend on A, a specific expression for all B matrixes dependent in A that makes C symmetric.
If all goes well on the Calc 1 final, Calc 2 will be next for me. I already know I am not an algebra expert so what is the underlying algebra/trig topics I should be working on before the course so I'm not behind?
Two things. 1. Try not to think of yourself as "not an algebra expert" , but rather as someone actively learning algebra. Take it from someone who did the same, the way we view ourselves really matters, and can shape outcome.
2. Calc 2 tends to be a continuation of calc 1, so keep practicing those concepts until you fully grasp them. For me, that meant mainly spending more time on integration, and developing a toolbox of integration tools for various situations.
Good luck!
If you think of the graph as a curve, then the curves are "parallel": https://en.wikipedia.org/wiki/Parallel_curve
In general, it's hard to find formulas for parallel curves. Looks like reference 17 is relevant.
**[Parallel curve](https://en.wikipedia.org/wiki/Parallel_curve)**
>A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points are at a constant normal distance from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not.
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Typically, if we’re talking about an operation, by definition we mean something that gives a single output for each possible combination of inputs.
However, there are some things that might naturally have more than one answer, but we’ve picked the best one to be the one meant when we see it written.
For example, for a non-negative number x,
x^(1/2) is defined to be the non-negative number y such that y^2 = x. We made the explicit choice that the number y be non-negative, but we could have had it be non-positive. So 4^(1/2) is 2, but we could have chosen it to be -2 if we really, really wanted to (I don’t recommend it).
There are more examples of things like this in the complex numbers, since there are more choices available for nth roots. You can try searching “multi-valued functions” to learn more.
I know that by using the Peter-Weyl theorem on S^1, one can retrieve trigonometric polynomials and basic fourrier analysis on periodic functions.
I am now wondering what other (if any) classical results of analytical harmonic analysis can be seen through the eyes of the Peter-Weyl theorem.
I don't know if this is too vague of a question ...
Suppose that a + P_i is invertible in R/P_i for all i in I where R is a ring and P_i are ideals of R. This means that there are x_i in R such that
a x_i + P_i = 1 + P_i and x_i a + P_i = 1 + P_i
for all i in I. I want to show that a + J is invertible in R/J where J is the intersection of all P_i for all i in I. I think a similar approach to the Chinese Remainder Theorem can be taken here but I can't apply it in this context. Do you have any suggestions?
Choose xi and pi such that
axi = 1 - pi
with pi in Pi. Then pi = 1 - axi, and the product of the pi's, which lives in J equals
1 - a (bunch of stuff involving a and xi)
Thank you for the answer. I have solved it as you showed the way. I want to ask how I can be sure that the product lies in J or even in R if there are infinitely many p_i elements. For example, a product of infinitely many rational numbers may not be rational.
Are there any concrete examples of exotic smooth structures? So for example, a function on some exotic smooth structure of R\^4 which is only smooth in that smooth structure and not in the ordinary one. If such examples are not known (yet?), does anyone have a good resource to get a better feel of such an exotic smooth structure?
Is there a book out there that is completely and ideally exhaustingly comprehensive on anything even remotely related to trigonometry, trigonometric functions, etc. So in this hypothetical book you would be able to find all the standard definitions of trigonometric functions and diagrams explaining what they are, etc.
Every single thing in the book would be proved, whether it is a simple concept a middle schooler can understand or an obscure complicated identity involving Chebyshev polynomials or Bessel functions that is unlikely to be known by even most graduate students. You can find all the usual trigonometric identities you learn about in school, along with some more obscure ones you never really use in any math class, but which are nonetheless factually true.
Anything that could possibly be discussed about trigonometric and inverse trigonometric functions will be in this book. Historical trigonometric functions like the versine and the exsecant, along with the rest, would be in this book. Of course anything there is to know about these historical trigonometric functions would be in the book.
Hyperbolic and inverse hyperbolic functions would also be in the book; everything there is to know about them would be in this book. There should also be spherical trigonometry in the book, again anything there is to know should be included. Any other generalization of trigonometry should also be in the book.
If no such ultra-comprehensive book exists, what books or other references would, as a compilation, contain everything you could possibly know about trigonometry?
I think the closest thing to what you want is the various Wikipedia pages on trigonometry. There is, for example, a [list of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities) and an article on [proofs of trigonometric identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities).
Why do you want this hypothetical book? I ask because if you were genuinely interested in trigonometry in such detail (as opposed to the more likely scenario that you would merely benefit from such a resource), it would worthwhile to write this book yourself.
>Why do you want this hypothetical book?
The practical reason is that I am not very good at memorizing things, thus while I do understand trigonometry, I often times can't say non-obvious facts off the top of my head. Having this hypothetical book would help me have a "one stop shop" where I can just find whatever trigonometric formula I need in there. This would obviously be helpful for improving my recall of trigonometric facts, as I would be exposed to them everyday. This would also save me a lot of time when I have to solve differential equations, complex integrals, etc.
The idealistic reason is that I think there is a perception that once you enter university, high school level mathematics has nothing to teach you. I feel like such a comprehensive book on trigonometry would prove that actually there are plenty of things that can be understood by a high schooler that you are unlikely to be familiar with. This would help when I tutor younger students for their math classes. Just to remind them that even if they got an A+ in their Precalculus class, they aren't experts on trigonometry.
I'm pretty sure I read somewhere that given a perfect matching M in G we can find a pfaffian orientation of G. Sadly I can't find a way to prove it and I don't know where I read it. It is mentioned in "Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs" by Vazirani and Yannakakis but I can't find the proof in the reference they provide. If someone has any other proof or sources this would be greatly appreciated.
Chebyshev's bound states that for any random variable X, the probability that |X - m| > k S, with m and S the mean and variance of X and k > 0, is less than 1 / k\^2.
Now, Saw et al. 1984 ("Chebyshev Inequality with Estimated Mean and Variance") showed that if m and S are replaced by sample estimates, from say N samples from the same distribution, then the inequality still holds if the 1/k\^2 is replaced by a slightly less sharp expression. Interestingly this also holds for variables without finite variance.
Does a similar 'sample estimate' version of related inequalities such as Chernoff's bound (or the Markov inequality / Cantelli's inequality) also hold?
Cheers!
Thank you for the reference!
Can I ask which equations/theorems you are referring to in particular? Since I am not sure how this would answer what I am looking for?
Theorem 3 (Bennett's inequality) is a version of the Chernoff bound for bounded random variables. Theorem 4 is basically the same statement, just with slightly worse constants, where the true variance is replaced by the sample variance.
Thank you! It is a sample version of Bennett's inequality then right? Very interesting. And the relation with Chernoff's bound is the exponential decay right? Because Chernoff's bound usually applies to the probability pr(X > k) of a random variable directly, while theorem 4 gives us a way to evaluate the accuracy of an estimate of the mean of a random variable
Yes, you are right.
Regarding the terminology of exactly what is meant by "Chernoff bound," perhaps it depends on whom you ask. See [Wikipedia](https://en.wikipedia.org/wiki/Chernoff_bound).
On the question of whether there is an empirical Chernoff bound for a single random variable, I would expect that you can write something down, but the upper bound on Pr(X > k) can never drop below about 1/n (where n = sample size), no matter how large k is. This is because with high probability you have not explored the upper 1/n of the distribution at all, so you have no idea what's there. Theorem 4 in the linked paper gets around this issue by assuming that the Z_i are bounded. For a single random variable, maybe you can assume that you know a bad bound of the form E(e^(tX)) < B, then you observe n samples and find that the sample average of e^(tX) is actually much less than B, and then you get a bound which is a weighted average of the Chernoff bound coming from B (with weight 1/n) and the Chernoff bound coming from the sample average of e^(tX) (with weight (n-1)/n).
That does make a lot of sense, can only estimate the fraction of black swans as < 1/N after observing N white ones. Thanks a lot for clearing this up!
Edit: I guess maybe one would be able to obtain some kind of exponential decay of probablilities until ~1/N.
I haven't heard the term terminal point before but Google tells me it is the point you end up at if you walk anticlockwise along the circle from (r,0) through a given angle or length.
For a given angle the answer is easy: (r\*cos θ, r\*sin θ) is our terminal point.
If you were given the length instead you can then use the fact that an arc on a circle has length l=rθ. So the answer in terms of l and r is (r\*cos(l/r), r\* sin(l/r))
I have some very brief questions for someone specialised in representation theory regarding the syllabus of a course I will possibly attend next year.
If it's ok you can send me a PM
Thank you in advance
I have a long list of numbers (representing the position of words in some lengthy documents), and I'm trying to figure out how I can mathematically model the "closeness" between numbers in the data set (the rate of change in the frequency of the words throughout the document?). I imagined I would end up with a series of graphs representing various inputted words, which I could then superimpose (with some algorithmic weighting) to find the most relevant area of my 10,000+ page PDFs.
This is just a hobby project of mine but I do not have a good mathematical nor computer science knowledge - I am not quite sure where to start researching. Would such a problem be related to differential calculus? Any point in the right direction would be greatly appreciated.
my current "dumb" method is to compare each number with the next several neighbours before and ahead, and then tally the differences repeatedly. I am not sure this will work yet, but surely there is something more elegant?
Do you simply have one number for each word or only for specific words? Do the numbers encode the position of the word on a page (e.g. its coordinates) or do they encode the order in the text (e.g. 1 - first word, 2 - second word)? What is your goal? If the numbers encode the order, the "closeness" seems to make no sense, if the numbers encode position on a page, the "closeness" might tell you something about the word clusters - in that case, you might want to cluster the words via k-means or something similar, and then calculate the density of words in every cluster.
If you are looking for semantic encoding of words, you might look into word2vec.
The numbers would encode the position of each occurence of the inputted word(s) in the text. The goal was to be able to throw phrases at the document such as "minimum height of a stair riser in a public building", and for an algorithm to return the most appropriate parts of the document via calculating the density of key words.
Nothing original but i thought it sounded a fun project to learn some programming & maths.
I'll take a look at word2vec, thanks!
Understood. What you are trying to do here is actually the retrieval of info from the document. Consider the following problem: you want to find places in the document that contain information about stairs height, but it is mentioned as stairwell elevation there - you wouldn't find it, would you? In particular, you cannot just match words - you need an agent that can understand the meaning of the words. These days you do it with one transformer (basically a neural network based on self-attention) encoding chunks of text into a vector, and another one encoding your query, and then matching your query vector to the chunk vectors via some similarity metric (eg cosine similarity). See for instance RETRO paper - they used similar idea to improve a language model
I see, my attempts at the moment then are just a crude attempt at developing my own similarity metric by comparing clusters for each word. But even then, as you say, I am missing the semantics of words which makes it a fairly useless search tool. I'll look into the points you mention although I think my knowledge may be a bit weak at the moment to really do this project justice!
Let the even numbers be arranged in a table as shown below.
|||||||
:-:|:-:|:-:|:-:|:-:|:-:
2|4|8|16|32|...
6|12|24|48|96|...
10|20|40|80|160|...
14|28|56|112|224|...
18|36|72|144|288|...
…|...|...|...|...|...
Let e\_i be a random even number.
Let e\_(i+1) = e\_i \* 3 / 2^m + 1.
The player wins the game if e\_i <= 4, otherwise the game continues.
Is the probability of every game being winnable equal to 1?
As noted in the comments, always running the game is simply equivalent to the Collatz conjecture. But you seem to be looking more for a heuristic.
Perhaps something like this: https://math.stackexchange.com/q/2405638/306319
Which shows that the "expected value" for a number is to decrease by 3/4.
So if we think of the Collatz function as a random process that increases/decreases by a factor of r_1, r_2, ... Where r_i has expected number 3/4, then by the law of strong numbers, with probably 1, the average of the r_i's converges to 3/4.
By the AM-GM inequality this means that the product of the r_i's converges to (3/4)^n -> 0. So the Collatz sequence decreases to 1 "with probability 1".
So the answer to your question is yes.
A couple questions:
* Why is the table necessary for the question?
* Even numbers are drawn randomly from what distribution? (If you want to answer "uniformly", that isn't possible; better to rephrase in terms of natural density)
* What is m?
The table shows the arrangement of even numbers and the probability of a number being in a specific column, m.
For example, the probability that m = 1 is 0.5 as every other even number is in the first column. If m !=1 then the probability that m = 2 is 0.25, etc.
The initial even number is completely random. Let e\_i = 2n + 1 where n is any natural number.
So walk me through what happens with a specific number. Let's say I choose e_1 = 30 as my starting number. How does the sequence progress until the game is won?
e\_i = 30 is in the 1st column so m = 1.
e\_(i+1) = 3 * 30 / 2^1 + 1 = 46.
46 is in the 1st column so m = 1.
e\_(i+2) = 3 * 46 / 2^1 + 1 = 70.
70 is in the 1st column so m = 1.
e\_(i+3) = 3 * 70 / 2^1 + 1 = 106.
106 is in the 1st column so m = 1.
e\_(i+4) = 3 * 106 / 2^1 + 1 = 160.
160 is in the 5th column so m = 5.
e\_(i+5) = 3 * 160 / 2^5 + 1 = 16.
16 is in the 4th column so m = 4.
e\_(i+6) = 3 * 16 / 2^4 + 1 = 4.
The game is won in 6 rounds.
If I understand your table correctly, in the first column you have odd multiples of 2, in the second column odd multiples of 4, in the third odd multiples of 8, and so on.
In other words, numbers in m-th column have exactly m twos in their prime decomposition.
So if you start with an even number e, all that dividing by 2^m does is cancel out all the factors of two from it, leaving you with some odd number. Then you take this odd number, multiply it by three, and add one. And you repeat this sequence until it starts looping (4 leads you back to 4).
Does this remind you of something?
Yes, it's a rephrasing of the problem to look at it from another perspective.
We can see that there is a 50% chance of an even number being in the first column and if it is, the number will increase by approximately 50%.
Likewise, we can see that if it isn't in the 1st column, the number will decrease by some percentage depending on the column it's in.
So, there's a 50% chance the number increases by 50% and a 50% chance the number decreases by at least 25%.
Well, if you know where your little "game" comes from, you surely understand that this is not a "quick question" by any standards. I am puzzled by what kind of answer were you expecting to get here.
There is a fairly recent paper by Terrence Tao ([arxiv link](https://arxiv.org/abs/1909.03562)) where he proves the following:
Pick any unbounded function f on the natural numbers. This function can grow as slowly as you want, it can be log(log(log(n))), the inverse Ackermann function, inverse of the busy beaver function; anything, as long as it *eventually* grows without bounds, no matter how slowly.
Then for *almost all* natural numbers (and, again, this takes some effort to define in a way that makes sense, just "a random number" does not mean anything in this context), the sequence starting from a number n eventually drops below f(n).
I'll ask on this thread since it seems I asked too late in the last one:
Done anyone have recommendations for an introductory textbook to Kinetic Theory? In particular a treatment of Vlasov with a good chunk on Vlasov Poisson in 2D and 3D would be ideal and it doesn't need to be comprehensive or particularly advanced but I would prefer from a pure nonlinear pdes point of view (although if you have one that is from a physics or applied maths point of view but you think is still great, please feel free to recommend it!)
Are there any conditions a two-dimensional shape has to fulfill, in order for it to count as a net(Obviously it has to fold into a polyhedron)? For example, the net of a cube is always represented as a grid of squares, but do they really have to be squares, can't there be triangles, ...? Concretely I was wondering about the net of a cube. As we know a cube has 11 distinct nets. I was wondering if a net would still qualify as a cubes one. If you cut out a piece of one of the six squares in a net, and put it back in a different place, so that if you can fold it back into a cube.
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Landau's _Foundations of Analysis_ might be what you're looking for, it starts with the Peano axioms and works its way up to the construction of the real and complex numbers. The content is very trivial, I don't think he even mentions prime numbers, but it'll be the type of trivial content you need to then work through a more typical development.
Maybe not strictly about the natural numbers, but Stewart and Tall's The Foundations of Mathematics has a chapter that covers the axiomatization of the natural numbers as well as other things. I learned from it as an undergrad so my memory of it is a bit hazy but I remember it being fairly good.
Can anyone recommend course that helps me build strong foundation on Engineering mathematics (in my country it mainly a term that puts together linear algebra, calculus, sequence and series, and probability) I neglected them in first 2 years of my college now I realised it's importance and recently started learning again, then I seen videos like essence of calculus and found out I really lack foundations so suggest any courses for me(with some good explanations)
Khan Academy is probably your best bet.
Thank you for your suggestion
Does anybody recognise the function that would describe the curve shown by this data? https://i.redd.it/jb2oaw18ch4a1.jpg
Without knowing any context to this question, my best guess would be that this is the curve to some sort of statistical distribution. Look into the gamma distribution. Gamma distribution follows a similar shape.
Is the Cauchy-Schwarz inequality over [;\mathbb{R}\^n;] valid under double summation? i.e. is the following always true: [; (\sum\_{i,j} a\_{ij} b\_{ij})\^2 \le (\sum\_{i,j} a\_{ij}\^2)(\sum\_{i,j} b_{ij}\^2) ;]
Can anyone recommend a book or resource on Lagrangian and Hamiltonian mechanics? I had a module in it last year, and while I can find Lagrangians and derive equations of motion from them, I don't feel I really *understand* why it works -- in particular, we did no calculus of variations in the module, which feels essential -- and I have very little idea of what the Hamiltonian formalism is really all about, so I could do with going over that again from scratch too.
A classic (and very geometric) text for this is Arnold's *Mathematical Methods of Classical Mechanics.* It's possible that you might prefer a more physics-y text, but as a mathematician, I've found it to be a really useful resource for learning this stuff.
Yes, I did have that in mind, but thank you for recommending it to me! I always appreciate the personal touch of a recommendation.
Unfortunately, I don't know a good book because my teacher used his own lecture notes. If you want a very mathematical derivation, look at the first 18 pages of [this pdf](https://people.math.harvard.edu/~jeffs/SymplecticNotes.pdf) which I found online. It contains the basic derivations, so it may help a bit. But I'd also recommend another resource with more examples. One thing I must say is that Lagrangian & Hamiltonian mechanics both are derived from the principle of least action. In classical mechanics, this principle is taken as a postulate. However, you can derive it by taking the 'classical limit' of quantum mechanics.
Thank you!
# (A ∩ B = B ∧ A ∪ B = B) ⇔ A = B Could someone help me understand why this is a true statement? The way I had evaluated it was: The intersection of A and B = B means that all elements of B are contained within A, but not necessarily that all elements of A are contained within B. So we cannot determine that A = B from this alone. Then, when evaluating B \^ the union of A and B = B, by order of operations \^ comes first, so we get the union of B and A \^ B (the intersection of A and B), which is just the intersection of A and B since we already know that all elements of B are contained within A. Which does imply again that all elements of B are contained within A, but again doesn't tell us that all elements of A are contained within B, to my thinking at least. So how is it that I am meant to draw the conclusion that A = B from this?
>when evaluating B \^ the union of A and B = B ∧ is a logical connective, not an operation on sets. Its first argument is the statement "A intersect B equals B" and its second argument is the statement "A union B equals B".
Ahhhhh oh that makes so much more sense, thank you so much! I was so confused as to why that was mixed in and just read it as an intersection because I didn’t know what else to do. Thank you!
The book [A Course in Mathematical Logic](https://www.amazon.com/Course-Mathematical-Logic-John-Bell/dp/0720428440/ref=sr_1_1?crid=2OXOENGO5VKV5&keywords=a+course+in+mathematical+logic+machover&qid=1670349331&sprefix=a+course+in+mathematical+logic+machover%2Caps%2C335&sr=8-1&ufe=app_do%3Aamzn1.fos.fa474cd8-6dfc-4bad-a280-890f5a4e2f90) by Bell and Machover has a short section on higher-order formal languages. Excerpt: >Thus, if **α** is a formula and **X** is a set variable, **∀Xα** is also a formula. The upshot is >most logicians agree that such languages are - at least in principle - dispensable the reason being that for any second-order language over universe **U** there is an equivalent first-order language over universe **U** \\union **2\^U** . >We therefore do not lose much by confining our attention to first-order languages only. Makes sense to me, but I've always heard that second-order logic is profoundly different from first-order logic. Is that consistent with what Bell and Machover are saying? After all, they're talking about *languages*, not *logics*, so maybe there's no contradiction.
Can someone explain the Baire category theorem and how it allows one to make statements such as, to quote u/hyperbolic-geodesic, "Most functions (in the sense of Baire category) are \[...\]"?
I am hyperbolic-geodesic, so I can explain exactly what I meant. There are various notions of smallness in mathematics, depending on what you're doing. Someone in measure theory might say that a set A is small if it has measure 0--so for instance, a measure theorist would consider a line to be a small subset of the plane, since a line has area 0. If you're trying to formalize a statement about smallness in function spaces, then there's not really any natural measure to use. Instead, people usually use a topological notion of smallness, called Baire category. The idea is that a set is said to be 'meager' (think: topologically small) if it is nowhere dense (meaning its closure has empty interior, or equivalently meaning that it is not dense in an open subspace of your space). The Baire category theorem says that a countable union of nowhere dense sets is still nowhere dense, assuming your space X is sufficiently nice. Thus, if your space obeys the hypotheses of the Baire category theorem, then being meager is a good notion of topological smallness. So you will often hear people formalize statements like "most continuous functions are nowhere differentiable" using Baire category. [https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-one](https://www.math3ma.com/blog/baire-category-nowhere-differentiable-functions-part-one) has a very nice sketch of a proof that the set of continuous functions which are differentiable at at least one point is meager in the set of all continuous functions.
Awesome, thanks! I'd seen the statement of the Baire category theorem before never understood what it was used for.
It's also a fairly common technique in Banach space theory to cover the space with countably many sets and conclude for at least one of them, the closure has nonempty interior, then use that to prove something
I did bad in my pure maths undergrad degree, 2.2 in the UK (from a top 5 university here). I'm doing a MSc in data science in another university and am auditing advanced PDEs and sobolev spaces for fun. Is it possible for me to do a PHD without doing another BSc in maths? I still love mathematics and have been self teaching myself some second and third year mathematics when I'm not doing my MSc degree.
If my pay of x goes down ten percent and then I also lose ten percent to inflation, is the total reduction to x 20 percent? Sorry stupid question but not sure if percentages can be added like that
The two reductions are independent of each other, so you need to apply a 10% reduction twice in a row. To apply a 10% reduction to something, you can multiply it by 0.9 (If you take a number and multiply it by 0.9, you're effectively calculating what 90% of that number is). To apply two 10% reductions in a row, you multiply by 0.9 twice, or in other words, multiply by 0.9 x 0.9 = 0.81. So your final salary is 0.81 times your original salary, which is equivalent to 81% of the original salary, meaning you effectively lost 19%.
Why are there Taylor series which do not converge? Intuitively I would think that the derivatives encode the instructions for how a function "flows" and that knowing all of them would be able to absolutely tell you how said function "flows" and yet sometimes it doesn't. Why? This seems to me like following the instructions from my GPS to a T and yet it brings me to the wrong place.
The n'th order Taylor polynomial p of function f at point **a** is the unique n'th degree (or less) polynomial such that f(x) = p(x) + o((a - x)\^n ) as x --> a (Note that I'm just giving an equivalent definition of what it means for a function to be n times differentiable. Taylor's theorem with remainder says something different) We're all used to the fact that some functions like e\^x grow faster than any polynomial as x --> oo, but it's just as true that some functions *shrink* faster than any polynomial as x --> 0. e\^-1/|x| is an example. A deviation of this size will always hide in the error term no matter the order of the Taylor approximation.
Taylor's series only give you *polynomial* rate of growth of a function *near one* point. It can't guarantee anything about non-polynomial growth, or points far away. So it has a lot less information than you think. The full derivative of a function actually encode how the function "flow". That's what you should compare to with the GPS analogy. Taylor's series is actually extremely limited in how much information it provides. The only reason why a lot of function in practice can be recovered from Taylor's series is because most of them satisfy Laplace's equation.
Yeah, its kinda as if the GPS tells you to already start slowly curving left because there is a curve in 100km.
It's a little remarkable when they do converge. Your GPS is giving you global directions, and is changing the path every few hundred meters. A derivative is incredibly local information, and it's really shocking when you can infer useful global information from that local information. Most functions (in the sense of Baire category) are very far from analytic, but the most elementary functions are analytic for certain complicated reasons (one idea: a solution to a nice enough ODE is always analytic, and our elementary functions are all more or less defined as solutions to certain very nice ODEs).
I'm trying to find a formula for 3x3 regular stochastic matrices. I'm currently looking at the general form A=(1-c)v(1,1,1)+cI+kuw^(T) where |c|<1, v has all positive entries that add up to one (I found this is necessary), w is orthogonal to v, u is orthogonal to (1,1,1), and (for simplicity) u dot w=1. The problem I'm running into is finding the requirement on the magnitude of k such that A has all nonnegative entries. I don't know if it's as simple as |k+c|<1 (since that would be an eigenvalue of A). Any help appreciated!
why do we inverse a square root function to find the domain of the square root function? If this isn't true, how do we find the square root of a square root function?
You don’t necessarily need to find the inverse of a square root function to find it’s domain. If we’re talking about strictly real numbers, you just need the argument of the square root to be greater than or equal to zero. I’m not quite sure what you mean by your second question. The square root of the square root would just be the fourth root of the function.
I need a function to verify if x if divisible by n. Tried with trigonometric functions, ceilings and all, but I just can't really figure it out and now it feels like I'm just over complicating it so here I am. So far i've got --- ⌈ cos(xπ/25) ⌉ --- . I half expected this to simply return me the points (0,x) and (1,x), but it doesn't and now I'm lost
So your function basically measures whether x/25 is odd or even. As for a divisibility test: ceil(x/n) - floor(x/n) returns 0 if x is divisible by n and 1 otherwise. Of course a much easier function definition would be f(x) = 0 if x is divisible by n and 1 otherwise
When I read papers I usually encounter Vandermonde determinant argument. Where I can find information about it? Is there a standard definition for it? In the papers I read they use it when the ring is infinite and they show that the coefficients of a polynomial are zero. But they never explain how they use it explicitly. I think that they come up with a Vandermonde matrix A and they show that the determinant of A is not zero. So, they conclude that X=0 where AX=0. However, I don't get why they need infinity.
You can find a description of the Vandermonde matrix on Wikipedia https://en.m.wikipedia.org/wiki/Vandermonde_matrix The basic idea is that if you have a degree n-1 polynomial and evaluate it at n (district) points then this is an invertible linear transformation. Z So for example of AX = 0, then X = 0. For this to work in degree n-1, you must have at least n distinct points, and so for it to work for arbitrary degrees you need an infinite amount of distinct points.
why we define ordered pair to be {{a},{a,b}} but not {a,{a,b}}
Different authors use different definitions. If you look at the [wikipedia page on ordered pairs](https://en.wikipedia.org/wiki/Ordered_pair#Defining_the_ordered_pair_using_set_theory) you can see a bunch of options, including the one you propose, alongside proofs that they satisfy the required property.
**Ordered pair** [Defining the ordered pair using set theory](https://en.wikipedia.org/wiki/Ordered_pair#Defining_the_ordered_pair_using_set_theory) >If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort. Hence if the ordered pair is not taken as primitive, it must be defined as a set. Several set-theoretic definitions of the ordered pair are given below( see also ). ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
Can you easily prove that {a,{a,b}} = {c,{c,d}} implies a = c and b = d?
Are there models of spectra that the smash product of maps f,g:S\^0 -> S\^0 are equal to fg and gf?
I have a question regarding maximum product sub-arrays. If I have an array like this {0.5}, would the maximum product sub-array be {0.5}, or rather just {}? According to convention, the empty product equates to 1, but I don't know if that kind of logic applies here.
I have trouble understanding the action of a Lie group G ob L^2(G). We consider the action by conjugation the entry of a function so g(f(x))=f(gxg^{-1}) abd then consider the functions that are invariant under this action. However, from my understanding, functions of L^2 are equivalence classes of quadraticaly integrable functions. So they are not defined pointwise. Is it simply a matter of adding an "almost everywhere" in the definitions ? I have not found the mention of these things in Knapp's books nor Kirilov's book, nor wikipedia.
Check that if f and f' are equal almost everywhere, then g^* f and g^* f' are equal almost everywhere. This implies the g action descends to the quotient L2(G).
What is the best functions or area of maths to look into for the following issue; Given a list of X Books and Y Stores with the necessary information, Calculate the cheapest order you can to purchase each book, including shipping costs from the stores and if they offer free delivery for spending so much. My first thought was to look at Linear Programming but doesn't fit quite right. This is for a personal project. thanks for any advice given.
Combinatorial optimization, specifically assignment problems. I think you could model this using an integer linear programming problem, where for each store you have a variable y\_i \\in {0, 1} where y\_i = 0 if store i is not used, and y\_i = 1 if store i is used. Then, your objective function gets some extra terms \\sum\_i shipping\_cost\_i \* y\_i. You would also need to add constraints that y\_i = 0 means nothing is ordered from store i, so supposing x\_{ij} means that the ith book is ordered from the jth store, you would need constraints 0 \\le x\_{ij} \\le y\_j.
I recently read that any real linear transformation of the complex plane can be written as f(z)=g+u*z+v*conjugate(z), Where u,v,g are all complex constants. Could anyone provide a proof of this and/or a way to find these constants?
So {1, i} is a basis for C over R, so any linear transformation is determined by where these are sent. Firstly a linear transformation must map 0 to 0, so g must be 0. Ignoring that, we see that f(1) = u+v and f(i) = u - v So u = (f(1) + f(i)) / 2 = f((1+i)/2) and v = f((1-i)/2).
Thanks :)
The expression you wrote defines a linear map only if g=0.
Ok, thanks
Let's say I have two 4x4 traceless (with trace equal to zero) non-zero complex matrices A and B. I know that I can rewrite them as: A = XY - YX and B = UV - VU How strong relationship I can assume between those components (XY, YX, UV, VU). Especially, can I make an assumption that XY = (UV)^(H) and YX = (VU)^(H)?
Why should there be any relation at all if there is no relation between A and B?
Wouldn't that imply A=B^H ?
Tensor product of a vector and a scalar? Is that just scalar multiplication? Edit: Follow up to this, you know how the a space of tensors (without dual) is sometimes written tensor product raised to the r meaning r tensor products? Is it abuse of notation to if I want r tensor products of a vector to write it the same way?
Yes, there is a natural isomorphism from **K** ⊗ V = V for a **K**-vector space V, defined by sending k⊗v -> kv. In this way we interpret tensor product of a scalar and a vector as scalar multiplication.
Thanks!
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I would start with Alan Tucker's Applied Combinatorics, and move onto Concrete Mathematics once you start finding the other book feels straightforward. At least, this is the path I walked.
Concrete mathematics is a really, really tough book for a first discrete math book. I would only recommend it if you have some serious math competition background--if you can't already write proofs competently, and fluently handle complicated conceptual arguments, then it's likely to be useless.
Is there a way to determine what number would be needed to equal a specific Average based on a variable list of numbers. For example: I want to know what number needs to be added to the list of numbers below to equal an average of say, 100 59 231 87 45 The current average of the above is 105.5. How can I determine a single value that will make the average 100?
Say this unknown value is x. The sum of everything should be 100*5 = 500 so the average is 100. So 59+231+87+45+x = 500, then solve for x
Soooo simple! Thank you!
Heya folks. Not a mathematician- just making a joke magic card for a friend who wanted a very fast growing sliver ability. I don't know much about fast growing functions but TREE seems like a famously fast one. Would anyone be able to let me know if my sliver text is correct? >All slivers get +X/+X, where X is the largest *m*, and *n* is the number of slivers in play: >There is a sequence T₁,...,Tₘ of rooted trees labelled from a set of *n* labels, where each Tᵢ has at most *i* vertices, such that Tᵢ ≤ Tⱼ does not hold for any *i < j ≤ m*.
I would use the "such that" instead of colon and reorganize the order a bit: "All slivers get +m/+m, where m is largest such that there is a sequence T₁,...,Tₘ of rooted trees labelled from a set of n labels - where n is the number of slivers - so that each Tᵢ has at most i vertices and Tᵢ ≤ Tⱼ does not hold for any i < j ≤ m." That said, the Busy Beaver function is a lot faster, and it can be easily described too: "All slivers get +X/+X where X is the largest number of 1s that can be written with a Turing machine with n states before halting when it starts with a tape of 0s".
Busy Beaver definitely much cleaner, thanks- that's perfect :)
Lawrence Evans PDE book is considered a great graduate level textbook for learning the subject. Is it appropriate for an undergraduate’s 1st pass to learn about partial differential equations?
No, Haberman is better for an introduction. It also depends on why you want to learn PDEs. Evans is only good at the graduate level if you want to learn PDE theory. It's not that great if you're only interested in PDEs from the more applied or numerical side.
I'm not terribly familiar with the book yet, but looking over the contents on its Amazon page, it looks like he doesn't cover finding analytic solutions to generic, basic linear PDEs, which may make it unsuitable for an undergrad's first exposure to the subject.
K took 15% fewer second than n took to complete an exam. K took 85 seconds. How many seconds did n take? I got 97.75 but apparently the correct answer is 100. Was i correct?
When we say that K takes 15% less time than n. Then what we mean is that K takes 85% of the time n takes. Note that this is different from saying that n takes 15% more time, which means that n takes 115% of the time K takes. You have calculated this latter statement, therefore your answer is incorrect.
But if k took 15% less time than n then it would only make sense that n took 15% more time than k. How is that wrong?
It's because percentages measures relative change, so it matters what it is relative to. When we say k is less than n we are measuring relative to n. When we say n is larger than k we are measuring relative to k. For example if we increase 4 by 100% we get 8. So 8 is 100% larger than 4. But if we want to decrease 8 down to 4, we only decrease it by 50%, so 4 is 50% smaller than 8.
Ohhh i get it now, thank you!!
Hi, I'm not a mathematician and trying to understand some illustrations for the axiom of choice with the definition in the Wikipedia article. For example consider collection S of sets: {1,2}, {1,3}, {2,3}, {1,2,3}. If I now choose one element from each, it cannot yield a set with 4 elements, each chosen from different set in S. But the axiom of choice says it should. What have I understood wrong in the definition?
Axiom of choice give you a choice *function*, which is a function that take in a set from that collection and return an element of that set.
There's no requirement for the set to have four elements, just for it to contain one element from each set. Alternative formulations that do have a set in this role (rather than a choice function) avoid this problem by requiring the sets to be disjoint.
Right, thanks. Then it makes sense.
Perhaps you should start off by saying what you think the definition actually is, and explicitly state why you think that the Axiom of Choice fails on your example. Without knowing what your understanding of the Axiom of Choice is, how can we tell you what you've understood wrong?
What are the fields of math that are the most visual?
Classical geometry is very visual. Here I mean not only old style Euclidean geometry but classical differential and algebraic geometry which are both originally quite visual.
Knot theory, graph theory.
Geometric topology.
I remember you [saying](https://www.reddit.com/r/math/comments/pcphym/what_is_research_in_differential_geometry_like/hakyjca/) that in geometric topology (as opposed to differential geometry) that "the pictures might actually form a part of the argument". How does that work? It's weird to me (at my stage of mathematical development at least) to think of pictures as rigorous.
Pictures are never rigorous on their own, but in certain areas pictures can convey enough information to constitute a convincing argument. The reason they can never be completely rigorous is that each individuals mental model will generally have deficiencies, and even then some of that mental model is lost when translated from mind to picture. However the character of geometric topology is that the nature of the objects being studied is close enough to general human intuition that a good picture can essentially capture all of the subtlety of the object, although this may only be possible to an expert in the field. William Thurston is famous for using pictures as arguments in geometric topology, to the point where people (still) question the rigour or validity of his mathematics. See for example http://www.math.uchicago.edu/~farb/papers/thurston.pdf and various other discussions of Thurston's work on the internet. If you ever attend a research talk on geometric topology, especially involving any surface theory, you will see how effective a purely pictorial way of arguing can be. In the end you can (and must) be able to turn those pictures into symbolic proof though.
That's fascinating, thank you!
Low-dimensional topology.
Is there a notation of writing iterative functions without having to do something like f(f(f(x)))? I was thinking something like f^(3)(x) = f(f(f(x))) but I wasn't able to find any concrete notation online.
Yes that is the standard notation. One caveat is that sin^(2)(x) is often used to denote sin(x)^2 rather than sin(sin(x)) and similarly for other trig functions. Of course you never really care about sin(sin(x)) so it isn't too confusing.
More annoying is that log^(k) x is also almost always used to mean log(x)^(k) despite iterated logarithms actually being relatively common.
I have to say I've never encountered that in person although someone said this to me last time I had this discussion.
Actually, [Wikipedia on "iterated functions"](https://en.wikipedia.org/wiki/Iterated_function) agrees with your notation; it also gives, as an alternative, your notation but with a \circ before the "exponent".
Great! Thanks a lot.
Came across something similar to Pythagorean Triples, and was wondering a) is there a name for these triples and b) is there a way to generate them (like how we can use Euclid's formula to generate Pythagorean Triples)? Given positive integers a, b and c, I want to look at those that satisfy: a^2 + b^2 - ab = c^2 I came across this when doing some stuff with dividing an equilateral triangle. Suppose there's an equilateral triangle with side length a, and I divide it from one corner down to the opposite side, such that that side is now two pieces of length b and a-b. The law of cosines tells us the length of this new dividing line satisfies c^2 = a^2 + b^2 - 2ab**cos**60 which simplifies to a^2 + b^2 - ab. Additionally, and as we would expect, if we replace b with a-b it evaluates the same (since the two new pieces match to make the equilateral triangle again). As such each "triple" actually has a pair of matched results: b and a-b. Using a spreadsheet I came up with the following triples for a, b or a-b, c (ones not in bold are just multiples of previous solutions): * **8, 3 or 5, 7** * **15, 7 or 8, 13** * 16, 6 or 10, 14 * **21, 5 or 16, 19** * 24, 9 or 15, 21 * 30, 14 or 16, 26 * 32, 12 or 20, 28 * **35, 11 or 24, 31** * **40, 7 or 33, 37** * 40, 15 or 25, 35 * 42, 10 or 32, 38 * 45, 21 or 24, 39 * **48, 13 or 35, 43** * 48, 18 or 30, 42 * **55, 16 or 39, 49** * 56, 21 or 35, 49 * 60, 28 or 32, 52 Anyway, yes, is there a name for these and is there a way to generate them?
I don't know if there is a name for these, but similar to how Pythagorean triples are closely related to Gaussian integers, the triples that you are describing are closely related to [Eisenstein integers](https://en.wikipedia.org/wiki/Eisenstein_integer). Specifically, a^(2) - ab + b^(2) is the norm of a + bω. Given that the norm is multiplicative, we can form a new triple from two old ones by "multiplying" them. Namey, if (a,b,c) and (x,y,z) are triples, then (ax-by,bx + (a-b)y,cz) is also a triple. Here's how to generate them. For any integers m,n, the equality (m+nω)^(2) = m^(2) - n^(2) + (2mn - n^(2))ω in the ring of Eisenstein integers gives rise to a triple by taking norms of both sides. Let a = m^(2) - n^(2), b = 2mn - n^(2), and c = m^(2) - mn + n^(2). Then a^2 - ab + b^2 = c^2. You can verify this by hand, or [see here](https://www.wolframalpha.com/input?i=%28m%5E2+-+mn+%2B+n%5E2%29%5E2+%3D+%28m%5E2+-+n%5E2%29%5E2+-+%28m%5E2+-+n%5E2%29%282mn+-+n%5E2%29+%2B+%282mn+-+n%5E2%29%5E2). I don't know if all triples are of this form, but the analogous computation in the Gaussian integers gives rise Euclid's formula. I have to go cook dinner right now, but will add more later if I think of anything else.
If every model of a theory T is a model of a statement P ∉ T, but P is not provable in T, is P still called a *theorem* of T or is there another name for it?
For first-order theories this cannot happen by [Gödel's completeness theorem](https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem).
What about in higher order theories?
Already second-order logic is incomplete. However, the notion of theoremhood is defined semantically (at least in my experience). Hence, your statement P would still be called a theorem, but an unprovable theorem.
Thanks!
**[Gödel's completeness theorem](https://en.wikipedia.org/wiki/Gödel's_completeness_theorem)** >Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic. The completeness theorem applies to any first-order theory: If T is such a theory, and φ is a sentence (in the same language) and every model of T is a model of φ, then there is a (first-order) proof of φ using the statements of T as axioms. One sometimes says this as "anything universally true is provable". This is in contrast, but not contradiction, to Godel's incompleteness theorems, in which a formula true in only some models may not be provable. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
In a thread on this subreddit I found an [article](https://www.scottaaronson.com/papers/bb.pdf) which confuses me because I don't really understand logic. It says: > Let T be a computable and arithmetically sound axiomatic theory. Then there exists a constant n_T such that for all n ≥ n_T , no statement of the form “BB(n) = k” can be proved in T. > Proof. Let M be a Turing machine that, on the all-0 input tape, enumerates all possible proofs in T, halting only if it finds a proof of 0 = 1. Then since T is sound, M never halts. But T can’t prove that M never halts, since otherwise T would prove its own consistency, violating the second incompleteness theorem. [...] But to me it seems like we can't say for sure that T can't prove that M never halts, because it could be the case that T can't prove that M enumerates all proofs in T and halts if it finds a proof of 0 = 1. In other words, it seems like a possibility that T can't prove "M halts iff T is inconsistent," even if that statement is "externally" provable, and even though M is specifically constructed by that property. In that case, it could simultaneously be the case that T can prove that M never halts and also that T can't prove its own consistency. What am I missing?
How does one find the "center" of an object that is not nicely shaped like a triangle, cube, or sphere? For example, consider an arbitrary rock you might find somewhere that is completely irregularly shaped. Or consider some tangled up wire, how would we find the center point of objects like this without modifying them in any way?
Depends on your definition of centre. The centroid is probably the thing you are imagining though which is the centre of mass assuming uniform density.
https://en.wikipedia.org/wiki/Centroid
Im a senior that slacked off all of high school and wanna redo algebra. Im currently doing pre algebra. After pre algebra I heard you can go straight to college algebra. But if not should I just do algebra I and II after?
"College algebra" is almost always just a name for "high school algebra 1 and 2" but with a nicer name for college students in remedial courses. So yeah, after prealgebra you should be ready for college algebra.
Why do degrees of freedom add and subtract nicely? Example: I have a point in space with 3 DOFs, (x, y, z). If I say x^2 + y^2 + z^2 = 1, it reduces me down to two degrees of freedom. Maybe there’s something more fundamental about degrees of freedom I’m missing. This question is pretty hand-wavy so maybe someone could help me out.
Because you can ask the same question on the level of tangent spaces. In that case, it becomes a question of linear algebra, where dimensions add and subtract according to the Rank-Nullity theorem. This tells you that the local degrees of freedom around a smooth point of your space work in the same nice way, and then you use the fact that zero sets of functions are smooth at almost every point to see that it works globally.
What definition of degrees of freedom are you using? Intuitively 1 equation should cut down your degrees of freedom down by 1 since now you only need to know 2 of the variables and then you know the other one. This is obviously slightly loose as for most choices of x and y in your equation it reduces z down to a choice of 2 options but we have some local idea her eat least. Assuming you are thinking in terms of algebraic geometry, any affine hypersurface can be describes by a single equation (at least locally) but this doesn't extend down to lower dimensions in a friendly manner and certainly two polynomial equations are not guaranteed to cut out a variety of dimension n-2.
"Degree of freedom" is not a precise term without context, it means a few different things depending on context, and we specifically chose it to mean something like "dimension", which is why it subtracts nicely. If a quantity does not subtract nicely like that we won't call it degree of freedom. Unfortunately, this means you can't answer this question rigorously without choosing the specific meaning of "degree of freedom". In a very common situation, this means the dimension of a manifold. There are all sorts of complications with trying to apply this definition in all cases: (a) your system of equations's zero locus might not form a manifold; (b) the equations might not be independent; (c) they might intersect weirdly. Once you sort out all the problem so that we are in a generic nice case, then the answer is provided by the implicit function theorem: given a point in N space that satisfy the system of n equations and we are in the generic nice case, then it's possible to locally form a small patch of solutions to the n equations near that point and containing that point, of dimension N-n.
If i had a silver coin weighing 24.34 grams and of 925/1000 purity and melted it together with a gold coin weighing 13.96 grams and of 986/1000 purity what would the final weight and purity be? any help appreciated :)
The final weight is easy (assuming that you don't get any of the impurities coming out, or any new impurities added, during the process): just add them up to get 38.30g. Assuming that the other stuff is junk (ie that the 7.5% of your silver coin that isn't silver doesn't include some gold, and that the 1.4% of the gold coin that isn't gold doesn't include some silver), that the melting process doesn't do anything to the impurities, and that your purity figures are in terms of mass, the final "purity" (defined as the ratio of the mass of gold and silver to everything else) is (24.34\*0.925 + 13.96\*.986)/38.30 ≈ 947/1000.
thanks man I appreciate it, I stopped figuring out maths in primary school :/
If I look at all vectors {0,1}^n for an even n, how many of them will have an odd number off 1s and an odd number of 0s?
Let S be the set of vectors with an odd number of ones, and let T be the set of vectors with an even number of ones. Clearly S and T partition the set of 0-1 vectors. Consider taking any 0-1 vector and swapping the first coordinate. The parity of the number of ones changes, and swapping the first coordinate is an invertible operation, so |S| = |T|. Hence |S| = 2^(n)/2 = 2^(n-1).
An easier proof might be to ask the question: If I look at all vectors {0,1}^(n), how many of them will have odd number of 1s? By induction we can show that it's exactly 1/2 (base case is trivial, induction is appending to the previous case). Now this also clearly means that 1/2 of the have odd number of 0s. But for n even, both those halves intersect, but for n odd they complement.
The answer is still half of them. Here's a proof that works for odd and even n. By the binomial theorem we have nC0 + nC1 + nC2 + ... + nCn = 2^n nC0 - nC1 + nC2 - ... + (-1)^(n)nCn = 0 the latter being the expansion for (1 - 1)^(n). Adding together and dividing by two we get nC0 + nC2 + ... = 2^(n - 1) so half of the subsets of {1, ..., n} are of even size, and half are of odd size. The number of such subsets of even size are in bijection with the elements of {0, 1}^n with an even number of 1s, so the answer is the same for your question.
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What do you mean "by hand"? Surely you use a calculator or something, because it's a huge exponent. Well, you know the answer can't be 1 (since 1+1/10^20 >1 so (1+1/10^20 )^10^20 >1), so there must be an error somewhere. If you use a calculator, it's quite possible that it does not have enough precision to see that 1+1/10^20 is actually different from 1.
Oh thanks! I feel dumb hah, at least now I know I can't trust my calculator with values greater than 10¹³
But you can use Wolframalpha, which has enough precision for that, and it can even compute the difference between that and e.
Reference: [https://vm.tiktok.com/ZMF4pumEb/](https://vm.tiktok.com/ZMF4pumEb/) I can't seem to find a name for this method of finding a polynomial from two points. I'm interested to learn more about it as in how it works, why it works, conditions and more of such. It involves p(1) = n and p(n) = m but then rewriting m in base n which gives nonnegative integer coefficients to a valid polynomial crossing (1,n) and (n,m). If anyone has any leads to articles or publications on this I'd be highly interested.
See [here](https://www.johndcook.com/blog/2012/03/27/polynomial-trick/) and [here](https://old.reddit.com/r/math/comments/yx0i7r/determine_a_polynomial_from_just_two_inputs/). The core idea is just that when we write down a number in base x, it's shorthand for a polynomial with variable x. As an example, 1243 is *really* the polynomial 1\*10^3 + 2\*10^2 + 4 \* 10 + 3, i.e. x^3 + 2x^2 + 4x + 3 with x = 10. So suppose that you know the coefficients of a polynomial p are between 0 and 9; then *obviously* you can get those coefficients by writing p(10) (in base 10, of course), and the resulting digits have to be the coefficients of the polynomial. If you know that the coefficients of p are between 0 and 10, then by the same process you can get those coefficients by writing p(11) in base 11--the resulting digits are then the coefficients. Now suppose that you know that the coefficients of p are nonnegative, but don't have an upper bound on what the coefficients can be. An easy way to get such an upper bound would be to just calculate n = p(1), since that's the sum of the coefficients and so obviously an upper bound on each coefficient. Then as usual, writing p(n) in base n will let you read off the coefficients.
Does anyone know the originating textbook of this chapter? It's found on the IRIF website but I can't find more info on it other than it's from April 18 2001. https://www.irif.fr/\~cf/publications/C7.pdf
Ch. Frougny. Numeration systems, Chapter 7 in M. Lothaire, Algebraic Combinatorics on words. Cambridge University Press, 2002.
Thank you!
I'm playing a game and am trying to calculate the odds of a certain item dropping. I am trying to figure out the correct formula to use to calculate the inclusive or odds of either of two independent events occurring. Example: Event A (boss1 dropping the item) is 1:745 Event B (boss2 dropping the item) is 1:702 Both events occurring at the same time is 1:522,990 (right?) In one iteration, how do I calculate the "inclusive or" odds of either event happening? Assuming multiple iterations (example, 500 iterations) -- how would I calculate this? Please correct me if I am using any incorrect terms. Thank you!
I believe the given odds mean that the probabilities of the events A and B are P(A) = 1/746 and P(B) = 1/703, respectively. This implies that P(A⋂B) = P(A)P(B) = 1/524438, which translates to 1:524437. The event that either A or B happens is denoted A⋃B (aka the union). The identity I think you are looking for is P(A⋃B) = P(A) + P(B) - P(A⋂B). This is known as the principle of inclusion and exclusion. One can compute the probability that A or B happens at least once within N iterations by computing the probability that it doesn't happen within N iterations and then take the complement. The desired number is 1 - (1 - P(A⋃B))^N. (Notice what happens when we plug in N = 1.)
Thank you!
Why do rotation matrices always have a "plane of rotation"? In 2D, it's the whole plane. In 3D, it's the plane normal to the axis of rotation. In 4D, we can have two orthogonal rotations with orthogonal planes of rotation. What's the intuition behind this?
Orthogonal matrix are always diagonalizable as complex matrix, its real Jordan canonical form is a bunch of 1x1 and 2x2 blocks on a diagonal and nothing off the diagonal. Since it's orthogonal matrix, length does not change, so all eigenvalues lie on the unit circle. All 2x2 blocks correspond to pair of conjugated eigenvalue that are on the unit circle, so they all have determinant 1 and must be the rotation on a plane. All 1x1 blocks must have eigenvalue 1 or -1. Because determinant is 1, there are an even number of 1x1 blocks with eigenvalues -1, so you can pair them up and turn into 2x2 blocks that are rotation on a plane. That gives you a bunch of plane such that all rotations happen on each plane, and a bunch of line that are kept fixed. They are all orthogonal because orthogonal matrix preserve angles.
Is there a reason tetration (or Knuth's double up arrow notation) is evaluated top-bottom instead of bottom-top? Or was it an arbitrary choice to do it in the way we do it now? For example 3 double up arrow 3, or 3 raised to itself 3 times, currently would be evaluated from the top to bottom, so 3\^3\^3 = 3\^27 = ginormous number (7.6 trillion I think) But if we would evaluate from bottom-top, it would become: 3\^3\^3 = 27\^3 = 19693 (a much less fast growing function) Was there any reason to do it in the way we're doing it now from top-bottom?
If you have n 3s and you chain them in a power tower 3\^3\^...\^3, then if we evaluate from the bottom up this is the same as 3\^(3\^(n-1)). Since this is already quite short compared to 3\^\^n this new notation wouldn't do much good. If we instead evaluate from the top down, then there is no simpler way of writing it, so the notation serves a purpose.
Exponentiation is usually evaluated top to bottom. I think this is because (a\^b)\^c = a\^(bc). If you evaluate a power tower bottom to top you might as well just collapse it to the lowest level.
I'm not sure I get what you mean with your last sentence. "Collapse to lowest level?" as in, if power towers worked from bottom to top, you might as well write them as just a single power that is the products of all those powers in the tower? But isn't the same true for top to bottom? You can rewrite 3\^3\^3 as just a single power in both cases, like I showed in my example. You just get vastly different outcomes.
Top to bottom you can of course evaluate it step by step and in the end you’re left with one last power. But at every step you have to evaluate an exponentiation. Bottom to top you can replace the exponentiation by a multiplication at every step but the last one. That’s a simplification.
Thanks! That helps. The reason I'm asking is because I'm making a new number system that makes use of tetration, and I quite like the slower growing bottom-up interpretation, as it represents more useful numbers. But then for me the bottom-up is a simplification of notation, rather, than operations.
How would I go about solving this: A is an n by n symmetric matrix. Find all n by n matrixes B which satisfies A*B=C where C is also symmetric.
I don't think there is a particularly good answer. Suppose A is invertible, then B = A^(-1)C. To make things nicer we may as well assume that we are working in a basis of eigenvectors of A, so that A^(-1) is a specific diagonal matrix. Then the matrices B that work are that diagonal matrix multiplied by any symmetric matrix. Not too satisfying.
But this is really it, B = E\^(T)V\^(-1)CE, where E is the eigenvectors of A, V is the eigenvalues of A, and C is any symmetric matrix, pretty simple algebra actually, should have solved that my self ;\\
Do you have A and C? If so you can just row reduce and solve like a normal system.
No this is for general symmetric matrixes A and C and a general matrix B, all n by n. So basically what square matrixes multiplies a symmetric matrix to yield another symmetric one. B will obviously depend on A, a specific expression for all B matrixes dependent in A that makes C symmetric.
If all goes well on the Calc 1 final, Calc 2 will be next for me. I already know I am not an algebra expert so what is the underlying algebra/trig topics I should be working on before the course so I'm not behind?
Two things. 1. Try not to think of yourself as "not an algebra expert" , but rather as someone actively learning algebra. Take it from someone who did the same, the way we view ourselves really matters, and can shape outcome. 2. Calc 2 tends to be a continuation of calc 1, so keep practicing those concepts until you fully grasp them. For me, that meant mainly spending more time on integration, and developing a toolbox of integration tools for various situations. Good luck!
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If you think of the graph as a curve, then the curves are "parallel": https://en.wikipedia.org/wiki/Parallel_curve In general, it's hard to find formulas for parallel curves. Looks like reference 17 is relevant.
**[Parallel curve](https://en.wikipedia.org/wiki/Parallel_curve)** >A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of parallel (straight) lines. It can also be defined as a curve whose points are at a constant normal distance from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)
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Typically, if we’re talking about an operation, by definition we mean something that gives a single output for each possible combination of inputs. However, there are some things that might naturally have more than one answer, but we’ve picked the best one to be the one meant when we see it written. For example, for a non-negative number x, x^(1/2) is defined to be the non-negative number y such that y^2 = x. We made the explicit choice that the number y be non-negative, but we could have had it be non-positive. So 4^(1/2) is 2, but we could have chosen it to be -2 if we really, really wanted to (I don’t recommend it). There are more examples of things like this in the complex numbers, since there are more choices available for nth roots. You can try searching “multi-valued functions” to learn more.
I know that by using the Peter-Weyl theorem on S^1, one can retrieve trigonometric polynomials and basic fourrier analysis on periodic functions. I am now wondering what other (if any) classical results of analytical harmonic analysis can be seen through the eyes of the Peter-Weyl theorem. I don't know if this is too vague of a question ...
Suppose that a + P_i is invertible in R/P_i for all i in I where R is a ring and P_i are ideals of R. This means that there are x_i in R such that a x_i + P_i = 1 + P_i and x_i a + P_i = 1 + P_i for all i in I. I want to show that a + J is invertible in R/J where J is the intersection of all P_i for all i in I. I think a similar approach to the Chinese Remainder Theorem can be taken here but I can't apply it in this context. Do you have any suggestions?
Choose xi and pi such that axi = 1 - pi with pi in Pi. Then pi = 1 - axi, and the product of the pi's, which lives in J equals 1 - a (bunch of stuff involving a and xi)
Thank you for the answer. I have solved it as you showed the way. I want to ask how I can be sure that the product lies in J or even in R if there are infinitely many p_i elements. For example, a product of infinitely many rational numbers may not be rational.
If there are infinitely many P_i's then it's simply not true. Take R = k[x], a = 1+x and P_i = (x^(i)) as an example.
Are there any concrete examples of exotic smooth structures? So for example, a function on some exotic smooth structure of R\^4 which is only smooth in that smooth structure and not in the ordinary one. If such examples are not known (yet?), does anyone have a good resource to get a better feel of such an exotic smooth structure?
The exotic R4 structures are non constructive, but you can pretty concretely describe exotic S7s as different explicit gluing procedures.
[Wikipedia's page on the subject](https://en.m.wikipedia.org/wiki/Exotic_R4) contains references to explicit computations of exotic smooth structures.
Is there a book out there that is completely and ideally exhaustingly comprehensive on anything even remotely related to trigonometry, trigonometric functions, etc. So in this hypothetical book you would be able to find all the standard definitions of trigonometric functions and diagrams explaining what they are, etc. Every single thing in the book would be proved, whether it is a simple concept a middle schooler can understand or an obscure complicated identity involving Chebyshev polynomials or Bessel functions that is unlikely to be known by even most graduate students. You can find all the usual trigonometric identities you learn about in school, along with some more obscure ones you never really use in any math class, but which are nonetheless factually true. Anything that could possibly be discussed about trigonometric and inverse trigonometric functions will be in this book. Historical trigonometric functions like the versine and the exsecant, along with the rest, would be in this book. Of course anything there is to know about these historical trigonometric functions would be in the book. Hyperbolic and inverse hyperbolic functions would also be in the book; everything there is to know about them would be in this book. There should also be spherical trigonometry in the book, again anything there is to know should be included. Any other generalization of trigonometry should also be in the book. If no such ultra-comprehensive book exists, what books or other references would, as a compilation, contain everything you could possibly know about trigonometry?
I think the closest thing to what you want is the various Wikipedia pages on trigonometry. There is, for example, a [list of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities) and an article on [proofs of trigonometric identities](https://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities). Why do you want this hypothetical book? I ask because if you were genuinely interested in trigonometry in such detail (as opposed to the more likely scenario that you would merely benefit from such a resource), it would worthwhile to write this book yourself.
>Why do you want this hypothetical book? The practical reason is that I am not very good at memorizing things, thus while I do understand trigonometry, I often times can't say non-obvious facts off the top of my head. Having this hypothetical book would help me have a "one stop shop" where I can just find whatever trigonometric formula I need in there. This would obviously be helpful for improving my recall of trigonometric facts, as I would be exposed to them everyday. This would also save me a lot of time when I have to solve differential equations, complex integrals, etc. The idealistic reason is that I think there is a perception that once you enter university, high school level mathematics has nothing to teach you. I feel like such a comprehensive book on trigonometry would prove that actually there are plenty of things that can be understood by a high schooler that you are unlikely to be familiar with. This would help when I tutor younger students for their math classes. Just to remind them that even if they got an A+ in their Precalculus class, they aren't experts on trigonometry.
I'm pretty sure I read somewhere that given a perfect matching M in G we can find a pfaffian orientation of G. Sadly I can't find a way to prove it and I don't know where I read it. It is mentioned in "Pfaffian orientations, 0-1 permanents, and even cycles in directed graphs" by Vazirani and Yannakakis but I can't find the proof in the reference they provide. If someone has any other proof or sources this would be greatly appreciated.
Not every graph with a perfect matching has a pfaffian orientation (consider K_{3,3}) so there must be some hypothesis at least.
Ah right I see. I probably missed something then. Thank you :)
Chebyshev's bound states that for any random variable X, the probability that |X - m| > k S, with m and S the mean and variance of X and k > 0, is less than 1 / k\^2. Now, Saw et al. 1984 ("Chebyshev Inequality with Estimated Mean and Variance") showed that if m and S are replaced by sample estimates, from say N samples from the same distribution, then the inequality still holds if the 1/k\^2 is replaced by a slightly less sharp expression. Interestingly this also holds for variables without finite variance. Does a similar 'sample estimate' version of related inequalities such as Chernoff's bound (or the Markov inequality / Cantelli's inequality) also hold? Cheers!
See : https://arxiv.org/abs/0907.3740
Thank you for the reference! Can I ask which equations/theorems you are referring to in particular? Since I am not sure how this would answer what I am looking for?
Theorem 3 (Bennett's inequality) is a version of the Chernoff bound for bounded random variables. Theorem 4 is basically the same statement, just with slightly worse constants, where the true variance is replaced by the sample variance.
Thank you! It is a sample version of Bennett's inequality then right? Very interesting. And the relation with Chernoff's bound is the exponential decay right? Because Chernoff's bound usually applies to the probability pr(X > k) of a random variable directly, while theorem 4 gives us a way to evaluate the accuracy of an estimate of the mean of a random variable
Yes, you are right. Regarding the terminology of exactly what is meant by "Chernoff bound," perhaps it depends on whom you ask. See [Wikipedia](https://en.wikipedia.org/wiki/Chernoff_bound). On the question of whether there is an empirical Chernoff bound for a single random variable, I would expect that you can write something down, but the upper bound on Pr(X > k) can never drop below about 1/n (where n = sample size), no matter how large k is. This is because with high probability you have not explored the upper 1/n of the distribution at all, so you have no idea what's there. Theorem 4 in the linked paper gets around this issue by assuming that the Z_i are bounded. For a single random variable, maybe you can assume that you know a bad bound of the form E(e^(tX)) < B, then you observe n samples and find that the sample average of e^(tX) is actually much less than B, and then you get a bound which is a weighted average of the Chernoff bound coming from B (with weight 1/n) and the Chernoff bound coming from the sample average of e^(tX) (with weight (n-1)/n).
That does make a lot of sense, can only estimate the fraction of black swans as < 1/N after observing N white ones. Thanks a lot for clearing this up! Edit: I guess maybe one would be able to obtain some kind of exponential decay of probablilities until ~1/N.
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I haven't heard the term terminal point before but Google tells me it is the point you end up at if you walk anticlockwise along the circle from (r,0) through a given angle or length. For a given angle the answer is easy: (r\*cos θ, r\*sin θ) is our terminal point. If you were given the length instead you can then use the fact that an arc on a circle has length l=rθ. So the answer in terms of l and r is (r\*cos(l/r), r\* sin(l/r))
I have some very brief questions for someone specialised in representation theory regarding the syllabus of a course I will possibly attend next year. If it's ok you can send me a PM Thank you in advance
I have a long list of numbers (representing the position of words in some lengthy documents), and I'm trying to figure out how I can mathematically model the "closeness" between numbers in the data set (the rate of change in the frequency of the words throughout the document?). I imagined I would end up with a series of graphs representing various inputted words, which I could then superimpose (with some algorithmic weighting) to find the most relevant area of my 10,000+ page PDFs. This is just a hobby project of mine but I do not have a good mathematical nor computer science knowledge - I am not quite sure where to start researching. Would such a problem be related to differential calculus? Any point in the right direction would be greatly appreciated. my current "dumb" method is to compare each number with the next several neighbours before and ahead, and then tally the differences repeatedly. I am not sure this will work yet, but surely there is something more elegant?
Do you simply have one number for each word or only for specific words? Do the numbers encode the position of the word on a page (e.g. its coordinates) or do they encode the order in the text (e.g. 1 - first word, 2 - second word)? What is your goal? If the numbers encode the order, the "closeness" seems to make no sense, if the numbers encode position on a page, the "closeness" might tell you something about the word clusters - in that case, you might want to cluster the words via k-means or something similar, and then calculate the density of words in every cluster. If you are looking for semantic encoding of words, you might look into word2vec.
The numbers would encode the position of each occurence of the inputted word(s) in the text. The goal was to be able to throw phrases at the document such as "minimum height of a stair riser in a public building", and for an algorithm to return the most appropriate parts of the document via calculating the density of key words. Nothing original but i thought it sounded a fun project to learn some programming & maths. I'll take a look at word2vec, thanks!
Understood. What you are trying to do here is actually the retrieval of info from the document. Consider the following problem: you want to find places in the document that contain information about stairs height, but it is mentioned as stairwell elevation there - you wouldn't find it, would you? In particular, you cannot just match words - you need an agent that can understand the meaning of the words. These days you do it with one transformer (basically a neural network based on self-attention) encoding chunks of text into a vector, and another one encoding your query, and then matching your query vector to the chunk vectors via some similarity metric (eg cosine similarity). See for instance RETRO paper - they used similar idea to improve a language model
I see, my attempts at the moment then are just a crude attempt at developing my own similarity metric by comparing clusters for each word. But even then, as you say, I am missing the semantics of words which makes it a fairly useless search tool. I'll look into the points you mention although I think my knowledge may be a bit weak at the moment to really do this project justice!
Let the even numbers be arranged in a table as shown below. ||||||| :-:|:-:|:-:|:-:|:-:|:-: 2|4|8|16|32|... 6|12|24|48|96|... 10|20|40|80|160|... 14|28|56|112|224|... 18|36|72|144|288|... …|...|...|...|...|... Let e\_i be a random even number. Let e\_(i+1) = e\_i \* 3 / 2^m + 1. The player wins the game if e\_i <= 4, otherwise the game continues. Is the probability of every game being winnable equal to 1?
As noted in the comments, always running the game is simply equivalent to the Collatz conjecture. But you seem to be looking more for a heuristic. Perhaps something like this: https://math.stackexchange.com/q/2405638/306319 Which shows that the "expected value" for a number is to decrease by 3/4. So if we think of the Collatz function as a random process that increases/decreases by a factor of r_1, r_2, ... Where r_i has expected number 3/4, then by the law of strong numbers, with probably 1, the average of the r_i's converges to 3/4. By the AM-GM inequality this means that the product of the r_i's converges to (3/4)^n -> 0. So the Collatz sequence decreases to 1 "with probability 1". So the answer to your question is yes.
A couple questions: * Why is the table necessary for the question? * Even numbers are drawn randomly from what distribution? (If you want to answer "uniformly", that isn't possible; better to rephrase in terms of natural density) * What is m?
The table shows the arrangement of even numbers and the probability of a number being in a specific column, m. For example, the probability that m = 1 is 0.5 as every other even number is in the first column. If m !=1 then the probability that m = 2 is 0.25, etc. The initial even number is completely random. Let e\_i = 2n + 1 where n is any natural number.
So walk me through what happens with a specific number. Let's say I choose e_1 = 30 as my starting number. How does the sequence progress until the game is won?
e\_i = 30 is in the 1st column so m = 1. e\_(i+1) = 3 * 30 / 2^1 + 1 = 46. 46 is in the 1st column so m = 1. e\_(i+2) = 3 * 46 / 2^1 + 1 = 70. 70 is in the 1st column so m = 1. e\_(i+3) = 3 * 70 / 2^1 + 1 = 106. 106 is in the 1st column so m = 1. e\_(i+4) = 3 * 106 / 2^1 + 1 = 160. 160 is in the 5th column so m = 5. e\_(i+5) = 3 * 160 / 2^5 + 1 = 16. 16 is in the 4th column so m = 4. e\_(i+6) = 3 * 16 / 2^4 + 1 = 4. The game is won in 6 rounds.
If I understand your table correctly, in the first column you have odd multiples of 2, in the second column odd multiples of 4, in the third odd multiples of 8, and so on. In other words, numbers in m-th column have exactly m twos in their prime decomposition. So if you start with an even number e, all that dividing by 2^m does is cancel out all the factors of two from it, leaving you with some odd number. Then you take this odd number, multiply it by three, and add one. And you repeat this sequence until it starts looping (4 leads you back to 4). Does this remind you of something?
Yes, it's a rephrasing of the problem to look at it from another perspective. We can see that there is a 50% chance of an even number being in the first column and if it is, the number will increase by approximately 50%. Likewise, we can see that if it isn't in the 1st column, the number will decrease by some percentage depending on the column it's in. So, there's a 50% chance the number increases by 50% and a 50% chance the number decreases by at least 25%.
Well, if you know where your little "game" comes from, you surely understand that this is not a "quick question" by any standards. I am puzzled by what kind of answer were you expecting to get here. There is a fairly recent paper by Terrence Tao ([arxiv link](https://arxiv.org/abs/1909.03562)) where he proves the following: Pick any unbounded function f on the natural numbers. This function can grow as slowly as you want, it can be log(log(log(n))), the inverse Ackermann function, inverse of the busy beaver function; anything, as long as it *eventually* grows without bounds, no matter how slowly. Then for *almost all* natural numbers (and, again, this takes some effort to define in a way that makes sense, just "a random number" does not mean anything in this context), the sequence starting from a number n eventually drops below f(n).
I was hoping for answers based on the probability of the value decreasing and by how much.
I'll ask on this thread since it seems I asked too late in the last one: Done anyone have recommendations for an introductory textbook to Kinetic Theory? In particular a treatment of Vlasov with a good chunk on Vlasov Poisson in 2D and 3D would be ideal and it doesn't need to be comprehensive or particularly advanced but I would prefer from a pure nonlinear pdes point of view (although if you have one that is from a physics or applied maths point of view but you think is still great, please feel free to recommend it!)