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Let $f(x)=x^2$ be the parent function for the transformation $g(x)=(2x+12)^2 -1$. Follow the two points $(0,0)$ and $(2,4)$ from the parent function through the transformation.
Let's simplify $g(x)$ first. So, $g(x)=(2(x+6))^2 -1$. This form tells me the transformations of the graph.
I know that the $x$-values ... |
College Algebra Function Transformations Unique? |
Let $X$ and $Y$ be random variables taking values in the same abelian group $G$, so that we can define their sum $X+Y$. Let $(X_1, Y_1)$ and $(X_2, Y_2)$ be conditionally independent copies of $(X,Y)$ relative to $X+Y$; that is, for all $z\in G$, the random variables $(X_1, Y_1 | X+Y = z)$ and $(X_2, Y_2 | X+Y = z)$ ar... |
If the criteria for a subgroup of a group is to be non-empty and have inverse elements for every element in the subgroup, then why don't the subgroups of D<sub>4</sub> have inverse elements for each element in them?
[![Lattice of D<sub>4</sub>'s generators of subgroups of][1]][1]
[![Subgroup of D<sub>4</sub>][2]]... |
I was wondering if its true for some vector $y \in H$, where $H$ is a Hilbert space, that $\langle y, x \rangle = 0, \forall x \in H$ is enough to say $y = 0$.
I want to think this is true as $H$ is trivially topologically closed in itself, so we should be able to decompose $H = H \oplus H^\perp$, but this has that... |
>**Question:** Why $\| T^* x_2 \|^2_1= | \langle x_2，T T^* x_2 \rangle_2 |$?
Because, as just noted, $T$ is the adjoint of $T^*$, hence
$$\Bbb R_+\ni
\|T^*x_2\|_1^2=\langle T^*x_2,T^*x_2\rangle_1=\langle x_2,TT^*x_2\rangle_2.
$$ |
>**Question:** Why $\|T^*x_2\|^2_1=|\langle x_2，TT^*x_2\rangle_2|$?
Because, as just noted, $T$ is the adjoint of $T^*$, hence
$$\Bbb R_{\ge0}\ni
\|T^*x_2\|_1^2=\langle T^*x_2,T^*x_2\rangle_1=\langle x_2,TT^*x_2\rangle_2.
$$ |
Let $ \gamma\colon [a,b[\to \mathbb R^n $ be a curve (i.e. a continuous function). Studying differential geometry I came across two ways in which $ \gamma $ can be called "differentiable".
1. $ \gamma $ is differentiable if it is differentiable in the classical sense in the interior $ ]a,b[ $ (meaning $ \gamma(t) $... |
Suppose there are 25 indistinguishable balls which can go
into 7 urns in such a way that all configurations of balls are equally likely. For example,
(3, 0, 7, 9, 2, 1, 3) and (0, 18, 1, 1, 0, 0, 5) are equally likely, where the the ith number is the
number of balls in urn i. Given that all 7 urns have at least one ... |
For this question, I started with the dot product of an arbitrary vector u and v:
u•v
= ||u|| ||v|| cos(45)
= (1) (√2) (√2/2)
= 1
Then I used the u1v1 + ... + u3v3 = 1 property, and then subbed in u1 = 1 - u3 into the ||u|| formula, more subbing... and ended up with a vector all in terms of u3: ... |
I am trying to follow the proof given here: http://www.gautamkamath.com/writings/gaussian_max.pdf. In particular, I would like to understand the following crude bound mentioned in the paper:
[![enter image description here][1]][1]
**My Attempt**:
\begin{align*}
\mathbb{E}Y &= \mathbb{P}(|\{ i : X_i \geq 0 \}|... |
>**Question:** Why $\|T^*x_2\|^2_1=|\langle x_2，TT^*x_2\rangle_2|$?
Because, as just noted, $T$ is necessarily the adjoint of $T^*$, hence
$$\Bbb R_{\ge0}\ni
\|T^*x_2\|_1^2=\langle T^*x_2,T^*x_2\rangle_1=\langle x_2,TT^*x_2\rangle_2.
$$ |
Let us suppose that $F(x)=ax^{2}+bx+c$ and $G(x)=Ax^2+Bx+C$ where $a, b, c, A, B, $ and $C$ are real numbers with $a > 0$ and $A > 0$.
Let us suppose that the minimum value of the quadratic function defined by $F(x)$ is greater than or equal to the minimum value of the quadratic function defined by $G(x)$. Does it n... |
>**Question:** Why $\|T^*x_2\|^2_1=|\langle x_2，TT^*x_2\rangle_2|$?
Because, as just noted, $T$ is necessarily the adjoint of $T^*$, i.e.
$$\langle T^*x_2,x_1\rangle_1=
\langle x_2,Tx_1\rangle_2$$
for all $x_1\in H_1,x_2\in H_2$,
hence
$$\Bbb R_{\ge0}\ni
\|T^*x_2\|_1^2=\langle T^*x_2,T^*x_2\rangle_1=\langle x... |
> Let $A \in \mathbb R^{m \times n }, b \in \mathbb R^m$ and $0 \neq c \in \mathbb R^m$. Prove: Either the system $Ax = c$ or the system $A^T y = 0, c^T y = 1$ has a solution.
Looks a bit like Farkas Lemma but the $c^T y = 1$ confuses me. Can someone give me a hint?
I know two versions of Farkas Lemma:
$$ \... |
Sorry in advance if this is a bit messy, I am just trying to sort out some concepts.
Wikipedia essentially defines an admissible trading strategy as a stochastic process $H = (H_t)_{t\geq 0}$ such that the associated value process $\int H(u) d S(u)$ is lower bounded. As I understand it, this rules out doubling strat... |
Why are there different definitions of admissibility in the literature, and why do we need admissibility? |
Top to random shuffling is a method of shuffling a deck of N cards whereby the top card of the deck is removed and placed at random in the deck, and the procedure is repeated. I want to know the stationary distribution of the Markov chain where a state is a permutation of
the deck. I saw a proof using random walks on ... |
When self studying a text where it is not expected to be able to solve all (or most) of the problems, what are the appropriate criteria to use for advancement?
> A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all.... Many are introduced n... |
This is exercise 3.20, (ii) of Atiyah & Macdonald. $f^*$ is the induced map on $\operatorname{Spec}$ of $f: A \to B$, a ring homomorphism.
I have seen a counterexample on MathSE, stating that $k[t^2,t^3] \subset k[t]$ is one. But I cannot seem to understand why this is a counterexample. In this case $f$ must be the ... |
Let us assume A‘ $\subset$ A and $f: A \to A‘$ is bijective (so A is an infinite set according to Dedekind‘s definition). Let us further assume $h: A \to B$ is bijective. I wanna prove that B is also an infinite set.
There is a proof that goes like this:
We assume further $g: h \circ f \circ h^{-1} = B \to A \to ... |
[question 15][1]
[1]: https://i.stack.imgur.com/qhCz5.jpg
I've tried solving this but I get n = 4 instead of n = 5 which is the right answer.
I used the
> lagrange remainder =< (1/(n+1)!)f(n+1)(z)(x-c)^(n+1)
I plugged in 1 for the f(n)(z) and 1.2 for x and then solved until the inequality became true... |
Two cylindical shells of equal radius are inserted one into the other at various angle between the axes (I tried to give an example with the pic attached).
What is the maximum volume for the [cylindrical wedge][1] of the upper cylinder into the bottom one? For which angle between the axes it is reached?
[. And here's the thing about chain complex.
> **(Page 104 in [this file](https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf))**
> That being given, let ${ε}^q_{i,j}$ be a number which is equal t... |
Find the greatest possible number of back-to-back zeros at the end of the product of three
counting numbers if the sum of these three numbers equals 191.
(The number 202100 has exactly 2 back-to-back zeros at the end.)
I tried with examples such as 100,90,1 and 10,81,100, giving me a max amount of 3 trailing ze... |
[question 15][1]
[1]: https://i.stack.imgur.com/qhCz5.jpg
I've tried solving this but I get $n = 4$ instead of $n = 5$ which is the right answer.
I used the
> lagrange remainder $\leq \frac{1}{(n+1)!}f^{(n+1)}(z)(x-c)^{n+1}$
I plugged in 1 for the $f^{(n)}(z)$ and 1.2 for $x$ and then solved until th... |
I understand the solution given in my book, but I fail to understand why my solution is wrong. My thinking is that there are 5 positions in the committee, and since there must be at least one male and one female, call one slot the male slot and the other the female slot. There are 12 female to choose from for the femal... |
You have 12 female and 8 male employees. How many ways can you choose a committee of 5 people with at least one male and one female? |
Let us assume A‘ $\subset$ A and $f: A \to A‘$ is bijective (so A is an infinite set according to Dedekind‘s definition). Let us further assume $h: A \to B$ is bijective. I want to prove that B is also an infinite set.
There is a proof that goes like this:
We assume further $g: h \circ f \circ h^{-1} = B \to A \t... |
The question translates to whether
$$P(X_1 = x_1, Y_2 = y_2 | X+Y = z) = P(X_1 = x_1 | X+Y = z) \cdot P(Y_2 = y_2 | X+Y = z)$$
for all $z \in G$.
For example, in the specific instances where $X$ and $Y$ are to take values in an abelian group $G$ and when conditional on the sum $X+Y$, some kind of algebraic propert... |
Your question talks about one urn having at least $15$ balls while the solution you put word talks about exactly $15$. So what is it?
**Assuming "at least"**
One of the urns must have at least $15$ balls and the other six urns must have at least $1$ ball each. So let's start by placing $14$ balls in one of the ur... |
The following result is from Tao's note: https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/comment-page-1/#comment-682897.
>**Proposition**: Let ${\varepsilon > 0}$. Then, for sufficiently large ${n}$, a proportion of at least ${1-\varepsilon}$ of the cube ${[-1,1]^n}$ (... |
Two quadratic functions with min F(x) > min G(x). Does this imply that F(x) > G(x) for all x? |
Suppose there are 25 indistinguishable balls which can go
into 7 urns in such a way that all configurations of balls are equally likely. For example,
(3, 0, 7, 9, 2, 1, 3) and (0, 18, 1, 1, 0, 0, 5) are equally likely, where the the ith number is the
number of balls in urn i. Given that all 7 urns have at least one ... |
I understand the solution given in my book, but I fail to understand why my solution is wrong. My thinking is that there are 5 positions in the committee, and since there must be at least one male and one female, call one slot the male slot and the other the female slot. There are 12 female to choose from for the femal... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$I$ is holomorphic in its domain, ... |
How to differentiate $\frac{Z}{I-Z}(\frac{Z}{I-Z})^T$ if $Z$ is a matrix? |
In the book "Introduction to Linear Algebra" by Gilbert Strang, the author says the linear combination is cv + dw. I know that v and w are vectors. But are `c` and `d` continuous variables? i.e. can c or d be 1.1 or 1.2 etc? |
$E\left[ S_{t} \right] = E\left[ \lim\limits_{n \to \infty}\sum_{i=1}^{n} R_{i}(t) \right]=\lim\limits_{n \to \infty}\sum_{i=1}^{n}E\left[ R_{i}(t) \right]=0$
Each $R_{i}$ is an independent coin toss here, if the result is head, you win $\sqrt{t/n}$ dollar, otherwise you lose $\sqrt{t/n}$. I am curious why we coul... |
I'm working on a problem involving two linear unbiased estimators $T$ and $T'$ of a parameter $\theta$, defined from a sample $\{X_1, \dots, X_n\}$ with mean $\theta$ and finite variance. I aim to prove that $\text{Cov}_\theta(T, T') = \text{Var}_\theta(T)$, given that:
- $T = \sum_{i=1}^{n} \alpha_i X_i$ and $T'$ i... |
In you first approach you don’t actually ensure that this S which goes after vowels is the *second* S. Also, why are you considering $6!\over 3!2!$. The letter S goes last, it doesn’t participate in permutations of the six letters.
Instead, you should consider all the vowels and the letters S. There are $15\choose ... |
I want to prove that if $\partial(\Omega) \neq \emptyset$ then $\partial(\Omega)$ is closed, so we can prove that $\partial(\Omega) ^c$ is open, this means $int(\partial(\Omega)^c )=(\partial(\Omega))^c$
The first contention is clearly, but if we take $p \in (\partial(\Omega) )^c$ then $$p \notin \partial(\Omega)$$
... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$I$ is holomorphic in its domain, ... |
Suppose we have two inverse systems $(A_i)$ and $(B_i)$ of abelian groups and we have a homomorphism from one system to other, say $\lambda_i:A_i\rightarrow B_i$ for all $i$ (so that respective commutativity in diagram holds).
Even if each $\lambda_i$ is surjective, the map induced between inverse limits $\lambda: ... |
When self studying a text where it is not expected to be able to solve all (or most) of the problems, what are the appropriate criteria to use for advancement?
> A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all.... Many are introduced n... |
$$ I = \int_{0}^{\infty} \frac{\sqrt{2} \left( e^{-2vz - 2b\sinh(z)}\cos\left(\pi\left(\frac{1}{4} + v\right)\right) + e^{2vz - 2b\sinh(z)}\sin\left(\pi\left(\frac{1}{4} + v\right)\right) \right)}{\sqrt{e^z - e^{-z}}} \, \mathrm{d}z $$
Going through Desmos, the function does converge.$f(z)$, the integrand, is holomorp... |
In the book https://math.mit.edu/~gs/linearalgebra/ila5/indexila5.html, the author mentions that
> The vectors cv lie along a line. When w is not on that line, the combinations cv + dw fill the whole two-dimensional plane.
Is there a proof for this statement? Intuitively, my brain thinks that there might be a vecto... |
when two vectors do not lie on the same line, how does their combination fill the whole two-dimensional plane? |
I'm looking for advice on the **representation of the fundamental forms of surfaces in higher dimensional Euclidean space** and **verification of the embeddings**.
I have only studied differential geometry in **3D Euclidean space** so far, and **have not learned enough about manifolds**, but would like to study the ... |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
One way to prove the equality of row and column rank is to show that $\text{null}(T^*)=\text{annihilator}(\text{range}\; T)$ and then to invoke the rank-nullity theorem and a theorem about the dimension of annihilators. But this makes the result look like a numerical coincidence. I was wondering if there is a way to co... |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$.
A smooth vector field $X$ is conformal Killing if
$$\nabla_iX_j + \nabla_jX_i - \frac{2}{n} \text{div} X \, \,g_{ij} = 0$$
Proposition 3.2 in the paper "The Boost Problem in General Relativity" by Murchadha and Chistoudoulo claims t... |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
I understand the solution given in my book, but I fail to understand why my solution is wrong.\
My thinking is that there are $5$ positions in the committee, and since there must be at least one male and one female, call one slot the male slot and the other the female slot.\
There are $12$ females to choose from for ... |
You have $12$ female and $8$ male employees. How many ways can you choose a committee of $5$ people with at least one male and one female? |
Theoretically, we know that the range and domain of $f(x)=\cot^{-1}x$ is $(-\infty,\infty)$ and $(0,\pi)$, respectively. Its plot is a smooth continuous function.
Strangely, I find that Wolfram Mathematica plots it as an odd
discontinuous (at $x=0$) function in $(-\pi/2,\pi/2)$!
**Question 1:** Have you encount... |
In this question I use the term "nested Fourier series representation" to refer to an infinite series of one or more Fourier series versus a single Fourier series. Whereas a single Fourier series is periodic, a nested Fourier series representation is not necessarily periodic. When it is periodic it can be expressed as ... |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
What does circular inference (benign/virtuous or vicious) look like symbolically? Is there anything I can do with truth tables or some other means to better understand its structure and what relations among premises and conclusions are circular? |
According to [this source](https://www.math.uwaterloo.ca/~hwolkowi/matrixcookbook.pdf) (The Matrix Cookbook), we have that for $f(X) = \mathbf{a}^T X \mathbf{b}$ where $X \in \mathbb{R}^{n \times m}, \mathbf{a} \in \mathbb{R}^n, \mathbf{b} \in \mathbb{R}^m$, the derivative is $$ \frac{\partial f}{\partial X} = \mathbf{... |
My question is, how could I calculate the expected value of the number of digits of an octal representation of an $n$-digit decimal number?
How can this question be answered in general for two different bases? E.g., how many more digits would base-9 numbers have than base-12 numbers on average? |
On average, how many more digits do octal numbers have than decimal numbers? |
When studying a book (like Rudin ) where the problems are not intended to be fully solvable by a student, what criteria show you're ready to advance? |
I am wondering what the differences are between element-wise and jointly concave functions. How does an element-wise function look? I cannot imagine 2d function which is concave in terms of each variable but not concave jointly.
If we want to maximize a function with multiple variables, we should find the stationar... |
I want to prove that if $\partial(\Omega) \neq \emptyset$ then $\partial(\Omega)$ is closed, so we can prove that $\partial(\Omega) ^c$ is open, this means $int(\partial(\Omega)^c )=(\partial(\Omega))^c$
The first contention is clearly, but if we take $p \in (\partial(\Omega) )^c$ then $$p \notin \partial(\Omega)$$
... |
Let $\mathbf{x}$ be a $p\times 1$ multivariate Gaussian random vector with zero mean but not necessarily a diagonal covariance matrix(so that its component can be dependent). Let $\mathbf{a}$ be a deterministic vector, and $\mathbf{B}$ be a deterministic symmetric matrix.
Do we have $E[(\mathbf{a}^T\mathbf{x})^i(\ma... |
If $f(\mathbf{x}): R^p \rightarrow R $ is a multivariate odd-symmetric function in the sense that $f(\mathbf{x}) = -f(-\mathbf{x})$ for any $\mathbf{x}$ and it is absolutely integrable, does it integrate to 0 on $R^p$?
Any references will be appreciated. Thank you!
The multivariate odd symmetric definition was f... |
Given a matrix $\mathbf{A}$ and a vector $\mathbf{x}$, the MVP $\mathbf{Ax}$ gives the geometric interpretation of projecting $\mathbf{x}$ onto the coordinate system defined by the columns of $\mathbf{A}$. Does such an interpretation exist when a vector $\mathbf{y}$ is pre-multiplied to a matrix, i.e., $\mathbf{yA}$? O... |
How about doing the substiution $y=-x$,
$$\int_{\mathbb R^p} f(x) \mathrm dx= \int_{\mathbb R^p} f(-y) \mathrm dy = -\int_{\mathbb R^p} f(y)\mathrm dy$$
Can you finish the proof? |
I want to prove that if $\partial(\Omega) \neq \emptyset$ then $\partial(\Omega)$ is closed, so we can prove that $\partial(\Omega) ^c$ is open, this means $int(\partial(\Omega)^c )=(\partial(\Omega))^c$
The first contention is clearly, but if we take $p \in (\partial(\Omega) )^c$ then $$p \notin \partial(\Omega)$$
... |
It's well-known that a $T_1$ topological space is [fully normal][1] if and only if it is $T_2$ and paracompact.
It appears, looking at the proofs from Henno Brandsma's nice exposition [here][2] and [here][3], that we can drop the $T_1$ assumption for the implication that fully normal implies paracompactness (without... |
Given a matrix $\mathbf{A}$ and a vector $\mathbf{x}$, the MVP $\mathbf{Ax}$ gives the geometric interpretation of transforming $\mathbf{x}$ into the coordinate system with basis vectors defined by the columns of $\mathbf{A}$. Does such an interpretation exist when a vector $\mathbf{y}$ is pre-multiplied to a matrix, i... |
Tao (*Analysis I*, 4e, p. [270][1]):
> **If and only if** (iff). If $X$ is a statement, and $Y$ is a statement, we say that “$X$ is true if and only if $Y$ is true”, whenever $X$ is true, $Y$ has to be also, and whenever $Y$ is true, $X$ has to be also (i.e., $X$ and $Y$ are “equally true”). Other ways of saying the... |
Why does Tao say $x=3$ is logically equivalent to $2x=6$? |
> Let X be the smallest value obtained when k numbers are randomly chosen from the set 1,...,n. Find E[X] by interpreting X as a negative hypergeometric random variable.
This is Self Test Exercise 7.7 of Sheldon's A First Course in Probability.
The way I approached this is to first consider (for example) P(X = ... |
TLDR; can I do the second integral with the method I used for the first integral?
I was reading about some integrals when I saw the following:
$$\int_{0}^{\infty}\frac{dx}{x^4+2x^2\cosh(2t)+1} = \frac{\pi}{4\cosh(t)}$$
The way to do this is to write the denominator as $(x^2+e^{2t})(x^2+e^{-2t})$ and then this... |
Here is the game:
> $A$ plays against $B$. Each round, $A$ has probability $a$ of winning, $B$ has a probability $b$ of winning, and the probability of a draw is $d$. So $a+b+d=1$. Each round is independent. Winning criteria is the one who first reaches a streak of $N$. Let's say $N=10$. If a draw occurs, the stre... |
This Question asked on [math over flow][1]
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$
by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^k+1)^k dy$$
now lets define $a(n,k)$ ... |
I know that the Hairy ball theorem plus the following proposition
**Proposition**
For a smooth manifold $M$.
There exists a Lorentz metric on $M$ $\iff$ there is a continuous non-vanishing vector field on $M$
allow one to conclude that there is no Lorenzian metric on even dimensional n-spheres.
**I... |
$E\left[ S_{t} \right] = E\left[ \lim\limits_{n \to \infty}\sum_{i=1}^{n} \sqrt{t/n}~R_{i}(t) \right]=\lim\limits_{n \to \infty}\sum_{i=1}^{n}E\left[ \sqrt{t/n}~R_{i}(t) \right]=0$
Let $R_{1}(t),R_{2}(t),...,R_{n}(t)$ be a sequence of i.i.d random variables with mean 0 and variance 1. e.g ±1 fair coin flips.
I am ... |
I have found this interesting Matrix Multiplication cases where multiplying two 2 x 2 Matrices concatenates the entries. Does this have a name ? Has this been found before ?
\begin{gather*}
Some\ solutions\ to\ below\ eqn:\\
\\
\begin{pmatrix}
a & b\\
c & d
\end{pmatrix} \times \begin{pmatrix}
e & f\\
g & ... |
i have been attempting to do this problem as part of an exam review but having some trouble
Let {Xn} be a simple random walk on the integers with a probability of .4 of moving to the right.
a) If X0 = 3, find the probability that the walk ever reaches 8.
b) If X0 = 1, find the probability that the walk ever re... |
$E\left[ S_{t} \right] = E\left[ \lim\limits_{n \to \infty}\sum_{i=1}^{n} \sqrt{t/n}~R_{i}(t) \right]=\lim\limits_{n \to \infty}\sum_{i=1}^{n}E\left[ \sqrt{t/n}~R_{i}(t) \right]=0$
Let $R_{1}(t),R_{2}(t),...,R_{n}(t)$ be a sequence of i.i.d random variables with mean 0 and variance 1. e.g ±1 fair coin flips.
And $... |
Let us have a Riemmanian manifold $(M,g)$. Is there some “natural“/canonical way to lift the metric to the cotangent bundle $T^* M$, i.e. define a metric on $T^*M$?
I ask because I have read some articles on orthogonal separation of variables (on configuration space) heavily using Killing tensors and other notions f... |
I know every smooth manifold admits a Riemannian metric, and I think I can state that a Riemannian metric is a semi-Riemannian metric,right? The former being just a special case of the later, so the answer should be yes. **Is this reasoning correct? Otherwise how do I prove it?** |
Does every n-torus admit a semi-Riemannian metric? |
A deposit of 10,000 is done. During first year, the bank credits an annual effective interest rate of $\text {i}$. During the second year, the bank credits an annual effective interest rate of $\text {i-5%}$. After two years, he has 12,093.75
What would be the amount earned by an annual effective interest rate of $... |
I want to get the joint probability of two independent events, both conditional on another event.
What I know:
Probability of A is 0.2375.
Probability of B given A is .8.
Probability of C given A is .5.
Now, I know that B and C is a joint probability (B * C). I'm also fairly certain that the "un-conditio... |
How do you get joint probability of two events conditional on the same event? |
When I was studying my engineering degree, I learn from Linear Algebra about the quadratic form of ${x^T}Ax$. Therefore, I am just having simple academic curiosity regarding the presentation of order 3 and order 4 polynomials.
Does matrix presentation of cubic form and quartic form even exist and how do they look l... |
Matrix presentation of cubic form and quartic form? |
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