instruction stringlengths 12 30k |
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$v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let $K$ be a symmetric rank-2 tensor on the 3-manifold. Then apparently the following holds,
$\vert K_{ij} v^i v^j \vert \leq \... |
Modulus of a tensor? |
$v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let $K$ be a symmetric rank-2 tensor on the 3-manifold. Then apparently the following holds,
$$\vert K_{ij} v^i v^j \vert \leq ... |
How do I find a (lat, lon) point (any will do) on a circle where the only info I have is a (lat, lon) center point, and a radius measured in Feet (ft.)? |
I'm trying to understand how the performance of financial investments are measuring when the asset has multiple points of investment.
For example, say I invest 1000 in a stock. After 6 months, it's worth 1200. At this point, I invest an additional 2000, for a total value of 3200. If after an additional 6 months, the... |
$v^i$ be a vector on a Riemannian 3-manifold with metric $g_{ij}$ embedded inside a 3+1 space-time such that for some constant $N_M$ it satisfies the inequality $g_{ij}v^iv^j \leq N_M ^2$. Let $K$ be a symmetric rank-2 tensor on the 3-manifold. Then apparently the following holds,
$\vert K_{ij} v^i v^j \vert \leq \... |
How do I find a (latitude, longitude) point (any will do) on a circle where the only info I have is a (latitude, longitude) center point, and a radius measured in Feet (ft.)? |
How do I find a (longitude, latitude) point (any will do) on a circle where the only info I have is a (longitude, latitude) center point, and a radius measured in Feet (ft.)? |
If we have a covering $p:H\rightarrow G$ where $G$ is a topological group, then $H$ is also a topological group. The multiplication function can be defined as follows. Consider the map $f:H\times H \rightarrow G$ which is a composition of the map $p\times p$ and the multiplication function on $G$. Choose $h\in p^{-1}(e... |
I'm trying to understand how the performance of financial investments are measuring when the asset has multiple points of investment.
For example, say I invest `$1000` in a stock. After 6 months, it's worth `$1200`. At this point, I invest an additional `$2000`, for a total value of `$3200`. If after an additional 6... |
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized c. f. ? (bounty expires Sept. 26, 13h GMT) |
I would like to recover the probability mass function from the characteristic function of a <u>discrete</u> probability distribution using <i>Mathematica</i>.
<p> Ideally, I'd like to do calculations like this example (not necessarily this simple). To compute the pmf f<sub>X+X</sub>[x] of the sum of two iid discrete... |
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized c. f. ? (bounty expires Sept. 25, 13h GMT) |
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized c. f. ? (bounty expires Sept. 25, 7h GMT) |
How do I prove the partial denominators formula of the Bauer-Muir transformation of a generalized continued fraction ? |
The exercise is asking for a map from $A$ to the ordinals, not to itself. Why do you think the exercise is trivial? If $\lambda$ is regular and $A\cap\lambda$ is non-stationary (in $\lambda$) then there is a club subset of $\lambda$ disjoint from $A$. You can then pick $g(\lambda)$ in this club. You need to ensure, how... |
in constrained Lagrangian optimization what is a general way to figure out how the optimal point varies with respect to parameters of the constraints? For example maximize x*y when x + y < k, and I wanted to find dx*/dk and dy*/dk where (x*,y*) is the optimal point. How would I do that? |
I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. sqrt(x) can pick up where it left off, even if x changes a bit, because it uses Newton-Raphson. But exp(x) computed using a Maclaurin series or conti... |
I would like to recover the probability mass function (pmf) from the characteristic function (CF) of a discrete probability distribution using <i>Mathematica</i>.
<p> Ideally, I'd like to do calculations like this example (not necessarily this simple). To compute the pmf f<sub>X+X</sub>[x] of the sum of two iid disc... |
I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. $\sqrt{x}$ can pick up where it left off, even if x changes a bit, because it uses Newton-Raphson. But $\exp(x)$ computed using a Maclaurin series or... |
I think the hypothesis that f is a totally differentiable function is necessary.
A basic first result is that, f is holomorphic if and only if $\frac{\partial f}{\partial \overline{z}}=0$, and in that case $f' = \frac{\partial f}{\partial z}$, i.e "Holomorphic functions are independent of $\overline{z}$ and depend ... |
I think the hypothesis that f is a totally differentiable function is necessary.
A basic first result is that, f is holomorphic if and only if $\frac{\partial f}{\partial \overline{z}}=0$, and in that case $f' = \frac{\partial f}{\partial z}$, i.e "Holomorphic functions are independent of $\overline{z}$ and depend ... |
There is something in the definition of the [free product][1] of two groups that annoys me, and it's this "word" thing:
>If G and H are groups, a word in G and H is a product of the form
>
>$$
> s_1 s_2 \dots s_m,
>$$
>
>where each $s_i$ is either an element of G or an element of H.
So what is t... |
Without strenuous arithmetic.
Is there a programme I can download to do so?
What are the quadratic residues modulo 5^4 or 5^5?
Thanks! |
How can I find out whether a number is a quadratic residue in a large modulo? |
For getting geometric intuition about complex analysis Tristram Needham's book ["Visual Complex Analysis"][1] seems to be the best place you can turn to!
There, reading from page 216 on (and you can safely just jump there, if you already have arrived at questions like the one you are asking here), you will learn th... |
I want to point out that there are, topicwise, different paths from the most suggested standard one:
As noted by "jmoy", time is limited - the comments telling you to learn some physics or programming first or on the way seem to ignore this somewhat. Sure it is good and will pay off to have these things as backgrou... |
For getting geometric intuition about complex analysis Tristram Needham's book ["Visual Complex Analysis"][1] seems to be the best place you can turn to!
There, reading from page 216 on (and you can safely just jump there, if you already have arrived at questions like the one you are asking here), you will learn th... |
I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way:
1. They prove transitivity using induction.
2. They prove irreflexivity using induction and transitivity.
3. They prove asymme... |
I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. My $\mathrm{sqrt}(x)$ can pick up where it left off, even if $x$ changes a bit, because it uses Newton-Raphson. But $\exp(x)$ computed using a Maclau... |
Say I have a function $f(x) = ax^3 + bx^2 + cx + d$.
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $c, x_0 > 0$ exist, such that $0 \le f(x) \le cg(x)$ for all ... |
The well known formula for perfect numbers is
$P_n=2^{n-1}(2^{n}-1)$
this formula is obtained by observing some patterns on the sum of the perfect number's divisors. Take for example $496$
$496=1+2+4+8+16+31+62+124+248$
one can see that the first pattern is a sequence of powers of $2$ that stops at $16$, ... |
Say I have a function $f(x) = ax^3 + bx^2 + cx + d$.
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $k, x_0 > 0$ exist, such that $0 \le f(x) \le kg(x)$ for all ... |
Say I have a function $f(x) = ax^3 + bx^2 + cx + d$.
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $k, x_0 > 0$ exist, such that $0 \le f(x) \le kg(x)$ for all ... |
You've observed that $P_5 = 2^4(2^5-1) = 496$ can also be written as the sum of the first 9 powers of two minus the sum of the first four powers of two.
**Sums of powers of two**
Powers of two written in binary look like $1, 10, 100, 1000, \cdots$ but you can also write them like this $1, 1+1, 11+1, 111+1, \cdots... |
How to find a suitable contour to integrate round? |
> But I found that the pattern proceeds
> after 31, for example 31=25−20,
> 62=26−21, 124=27−22 and finally
> 248=28−23
> ...
> Well, is this something new? Have anyone seen this somewhere?
The prime factorization of P_n is $2^{n-1}(2^n - 1)$, where $(2^n - 1)$ is prime. Since $2^k$ for k = 0 to k=n-1, and the ... |
I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way:
1. They prove transitivity using induction.
2. They prove irreflexivity using induction and transitivity.
3. They prove asymme... |
The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if they are elementary but their proofs are very much sophisticated.
I would like to consider two famous questions: First... |
The motivation for this question is : http://math.stackexchange.com/questions/4066/rationals-of-the-form-fracpq-where-p-q-are-primes-in-a-b and some other problems in Mathematics which looks as if they are elementary but their proofs are very much sophisticated.
I would like to consider two famous questions: First t... |
Why do statements which appear elementary have complicated proofs? |
I'm trying to learn about the selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with.
I have some basic questions that someone might be able to help with:
Is the composition of two closed geodesics itself a closed geodesic?
Is composit... |
Can you advise some good books covering areas which are required for serious probability theory studies (e.g. measure theory, functional analysis)? Preferably this book should have some problem sets to work on. Thanks! |
A problem that is an example of this is x^2+16 . I have to know how to factor this over the set of complex numbers. How do you do that? I used to know its just been a long time. |
How do you factor a equation over the set of complex numbers? |
As the induction hypothesis is clearly not strong enough, how about strengthening your proof obligation? Just adding $x+1\not\in x$ will mean you'll also need $x+2\not\in x$, so I'd suggest $y\not\in x$ for all $y$ such that $x\le y$. |
I'm stuck trying to solve an inequality of the form $\left|\frac{f(x)}{g(x)}\right| \geq 1$; specifically, $\left|\frac{2x + 5}{x + 1}\right| \geq 1$
I tried the approach that was taught for solving inequalities in general, and solved first for x when the part enclosed in absolute value signs is $\geq 1$, and then ... |
How do I solve inequalities of the form $\left|\frac{f(x)}{g(x)}\right| \geq 1$? |
How `1+(x/2)+(x^2/3)+(x^3/4)+...` in denominator becomes `1-(x/2)-(x^2/12)+...` in numerator?
If I could consolidate the series in denominator in some form, then I could use binomial expansion in numerator by raising it to power -1, but I don't see the one in denominator in any known form other than converting it to... |
How is this series in denominator converted to a series in numerator? |
Say I have a function $f(x) = ax^3 + bx^2 + cx + d$ where $a > 0$.
It's clear that for a high enough value of $x$, the $x^3$ term will dominate and I can say $f(x) \in O(x^3)$, but this doesn't seem very formal.
The formal definition is $f(x) \in O(g(x))$ if constants $k, x_0 > 0$ exist, such that $0 \le f(x) \le... |
How do we go from $$\frac{1}{1+(x/2)+(x^2/3)+(x^3/4)+\dots}$$ to $$1-(x/2)-(x^2/12)-\dots$$?
If I could consolidate the series in denominator in some form, then I could use binomial expansion in numerator by raising it to power -1, but I don't see the one in denominator in any known form other than converting it to ... |
I'm stuck trying to solve an inequality of the form $\left|\frac{f(x)}{g(x)}\right| \geq 1$; specifically, $\left|\frac{2x + 5}{x + 1}\right| \geq 1$
I tried the approach that was taught for solving inequalities in general, and solved first for x when the part enclosed in absolute value signs is $\geq 1$, and then ... |
I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name? |
What is the proper geometric description of a the oval used for a horse racetrack? |
I was told to write the 3 cube roots of 27 in rectangular form how is this to be done? I know how to do cube roots on something like 5+0i in polar form, but how do you do it with just a number? |
How do you find the cube roots of just a number? |
Expansion of terms in set definition? |
I have attempted to google for this, but the searching is marred by the close relationship of Jordan Algebras and Quantum Mechanics. I have been passively thinking of this question for ages, tonight I finally post it. :)
My question is fairly simple. Similar to the universal enveloping algebra for Lie algebras, we s... |
Or, in other words, why are all functions not surjective? Isn't any subset of the codomain which isn't part of the image rather arbitrary? |
How is the codomain for a function defined? |
I am looking at Perturbation Theory by E.J. Hinch. The author introduces Van Dyke's matching rule:
(m term inner)(n term outer) = (n term outer)(m term inner)
To be used to match the terms in an inner solution to the the terms in the outer solution for some perturbed differential equation.
Hinch when comparin... |
Given a sequence of row sums $r_1, \ldots, r_m$ and column sums $c_1, \ldots, c_n$, all positive, I'd like to find a matrix $A_{m\times n}$ consistent with the given row and column sums that has the fewest nonzero entries. That is, we want to minimize $|\\{a_{ij} \ne 0\\}|$ under the constraint that $\sum_j a_{ij} = r_... |
The notation
$$\int \log(t)\\ d(\pi(t))$$
is an example of the [Stieltjes integral][1]. Some like this (I don't)
when discussing summation, as integrating with a step function $f(t)$
inside the $d$ notation is essentially a summation. Here $\pi(t)$
is constant, save at prime values of $t$ where it jumps by $1$. Th... |
In the same sense you could ask why all functions aren't total functions. The real question is perhaps "what is the use of writing $f:\ S\rightarrow T$" or even more general "what is the use of writing "$v \in S$"?
I'd expect that in most cases where it is mentioned explicitly it's a way to say "this object belongs ... |
I am looking at Perturbation Theory by E.J. Hinch. The author introduces Van Dyke's matching rule:
(m term inner)(n term outer) = (n term outer)(m term inner)
To be used to match the terms in an inner solution to the the terms in the outer solution for some perturbed differential equation.
Hinch when comparin... |
How can one build a 3x3 linear equation system (3 equations with 3 variables) if the variables are known, for example, a=1, b=2 and c=3?
Thank you in advance! |
How to build a linear equation system? |
Can we represent all algebraic structures in First-Order logic? |
Can we represent all algebraic structures in First-Order logic? |
In relation to this geometric construction:
![construction][1]
where D is the center of both circles, if the inner radius, the angle φ = ∠CAB, and the length Δg of line segment AB are given, what is the length Δx of line segment AC?
[1]: https://i.stack.imgur.com/2pXX7.png |
What is the difference in radii of two concentric circles given an angle and length of a triangle that is inscribed in the annulus? |
In relation to this geometric construction:
![construction][1]
where D is the center of both circles, if the inner radius, the angle φ = ∠CAB, and the length Δg of line segment AB are given, what is the length Δx of line segment AC?
[1]: https://i.stack.imgur.com/AoQDP.png |
Basically, if I have an [$L^p$ norm][1] $\left \| \mathbf{r} \right \|_p$ where $\mathbf{r} = \left \langle x, y \right \rangle$, how can I parametrize the circle $\left \| \mathbf{r} \right \|_p=k$ and then find the area?
Note: The equation basically turns into $|x|^p+|y|^p=k^p$. The problem with that is the absolu... |
How can one parametrize an $L^p$ norm circle? (in order to find the area) |
In relation to this geometric construction:
![construction][1]
where D is the center of both circles, if the inner radius (*x* = length of line segments DA and DE), the angle *φ* = ∠CAB, and the length Δ<i>g</i> of line segment AB are given, what is the length Δ<i>x</i> of line segment AC?
[1]: https://i.s... |
If you look at [this previous problem][1], I asked how to find the sum of all the areas between two taxicab geometry circles. However, upon learning about [$L^p$ norms][2], I thought it would be pretty interesting to extend the problem to all $L^p$ norm circles, not just $L^1$ (taxicab).
If $p=1$, then the result ha... |
A problem that is an example of this is $x^2+16$. I have to know how to factor this over the set of complex numbers. How do you do that? I used to know it's just been a long time. |
How do you factor an equation over the set of complex numbers? |
If you look at [this previous problem][1], I asked how to find the sum of all the areas between two taxicab geometry circles. However, upon learning about [$L^p$ norms][2], I thought it would be pretty interesting to extend the problem to all $L^p$ norm circles, not just $L^1$ (taxicab).
If $p=1$, then the result ha... |
I was maths student but now I'm a software engineer and almost all my knowledge about maths formulas is perished.
One of my client wants to calculate optimal price for perishable products. He has come across an article named **Optimal Pricing Policies for Perishable Products** by Miguel F. Anjos¤ Russell C.H. Cheng ... |
I have a series of problems that involve events that happen one after another, and where the usual formulas like P(A/B)=P(AB)/P(A) don't seem to help.
Example:
> In a horse race there are 3 horses. E12
> means horse 1 finishes before horse 2.
> E123 means horse 1 finishes before 2,
> which finishes before 3, e... |
How to solve this kind of Conditional Probability problems? |
If this is homework, it's a pretty strange one. :-? Anyway, do you like the following system?
$$
\begin{align}
x &= 1 \\\
y &= 2 \\\
z &= 3
\end{align}
$$
-Too much easy, I see.
So, try this: your system must fulfil the following equation:
$$
A
\begin{pmatrix}
1 \\\
2 \\\
3
\end{pma... |
What are some examples of cutting-edge research involving fractals or self-similar structures?
Who's actively contributing high-quality research in this field?
|
Where have fractals gone since Mandelbrot? |
If this is homework, it's a pretty strange one. :-? Anyway, do you like the following system?
$$
\begin{align}
x &= 1 \\\
y &= 2 \\\
z &= 3
\end{align}
$$
-Too much easy, I see.
So, try this: your system must fulfil the following equation:
$$
A
\begin{pmatrix}
1 \\\
2 \\\
3
\end{pma... |
It's said on Wikipedia page about [Post's lattice](http://en.wikipedia.org/wiki/Post's_lattice) that DM clone(all monotone self-dual functions) has majority function(xy + xz + yz) as one of its bases. What is the proof of this fact? |
This is mostly related to doing large modular exponentiation by hand. For example, a problem I was doing was to find the last 3 digits of $7^{9729}$; that is, find $7^{9729}\bmod{1000}$.
Using the simplest concept for Euler's theorem, I found that $7^{400}\equiv 1\pmod{1000}$, since $\varphi(1000)=400$. Using Carmic... |
How do I find the lowest $n$ for which $a^n \equiv 1 \pmod{b}$? |
This is not a solution, but a potential restatement of the problem.
Consider a single stick of length n(n+1)/2 that can be broken in any single place, leaving two pieces of length >=n. The resulting two sticks can be broken into 1...n easily, possibly in multiple ways. Can you show that for any position of that brea... |
Let random variable $X\sim\text{Binomial}\left(a+b,\frac{a}{a+b}\right)$, where $a$ and $b$ are positive integers.
I'm trying to prove that $\mathbb{P}[X\geq\mathbb{E}[X]]\leq\frac{3}{4}$, which appears to be true numerically.
Does anyone have a suggestion on how to proceed?
(The complication in this particular ... |
Let random variable $X\sim\text{Binomial}\left(a+b,\frac{a}{a+b}\right)$, where $a$ and $b$ are positive integers.
I'm trying to prove that $\mathbb{P}[X\geq\mathbb{E}[X]]\leq\frac{3}{4}$, which appears to be true numerically.
Does anyone have a suggestion on how to proceed?
The complication in this particular p... |
If this is homework, it's a pretty strange one. :-? Anyway, do you like the following system?
$$
\begin{align}
x &= 1 \\\
y &= 2 \\\
z &= 3
\end{align}
$$
-Too much easy, I see.
So, try this: your system must fulfil the following equation:
$$
A
\begin{pmatrix}
1 \\\
2 \\\
3
\end{pma... |
This question is motivated by the following homework problem. I'm trying to explicitly compute the homeomorphism $f:S^2 \rightarrow \mathbb{CP}^1$ by using stereographic projection and considering $\mathbb{CP}^1 = \mathbb{C}\cup \{\infty\}$. I'll want to prove that this is an isometry, where $S^2$ has the standard an... |
Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?) |
This is one of the question asked in a written test conducted by a company. The question sounded stupid to me. May be its not.
"Given the area of the coin to be 'A'. If the probability of getting a tail, head and the edge are same, what is the thickness of the coin? |
Sorry. I'll remove the ridiculous title soon. I just wanted to comment on something and realized I couldn't. I'm not sure the rep restriction on commenting makes sense, especially here where there are (at least for now) far fewer people than on MO.
----------------------------------
This question is motivated ... |
PLEASE UPVOTE ME SO I CAN COMMENT /// Does projectivizing always fix problems at infinity? (Or, am I making a mistake somewhere?) |
Generally one can see that there are infinite number of solutions for this equation $$a^{2}+b^{2}=c^{2}$$ by taking multiples of the solution $3,4$ and $5$.
Can i use this as a fact to prove, that $k^{2} + (k+1)^{2}$ is a perfect square for infinitely many $k \in \mathbb{N}$? Any hints, suggestions would be helpful... |
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