instruction stringlengths 12 30k |
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Here is an image:
[![enter image description here][1]][1]
I am given $\vec{n}$ and $\vec{v_A}$.
Apparently, $\vec{v_L}$ is perpendicular to $\vec{v_A}$ and $\vec{n}$ but I am struggling to see this even with the diagram? And then I am secondly struggling to see which direction I need to cross the vectors in t... |
Cross product - how do I know this landing direction is the cross product of the other two? |
The general element of $Z(A, v) + Z(A, w)$ is of the form $p(A)v + q(A)w$, where $p$ and $q$ are polynomials which matter up to multiples of $m_v$ and $m_w$ respectively. We wish to show that this is of the form $r(A)(v+w)$ for some polynomial $r$. Observe that $r(A)(v+w) = r(A)v + r(A)w$. Thus, we wish to find some po... |
Proof of Hutton's 1776 formula:
[![enter image description here][1]][1]
Why does the figure establish this identity?
$$
\frac{\pi}{4} = \arctan\left(\frac{1}{3}\right) + \arctan\left(\frac{1}{7}\right)
$$
[1]: https://i.stack.imgur.com/2ZMe3m.png |
A small ball is released from (-1,2,11) on z=x^3-3xy^2. Which vector does it start moving on?
I found the normal (-9,12,-1) and i know its not the answer. I really don't have any idea other then that. There no additional information given. |
Here is an image:
[![enter image description here][1]][1]
I am given $\vec{n}$ and $\vec{v_A}$.
Apparently, $\vec{v_L}$ is perpendicular to the cross product of $\vec{v_A}$ and $\vec{n}$ but I am struggling to see this even with the diagram? And then I am secondly struggling to see which direction I need to c... |
I tried to find the indefinite integral
$$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$
by using euler formula and put $x=\frac{\ln y}{2i}$ I got
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^p+1)^p dy$$
now lets define $a(n,k)$ as the coefficient of $x^k$ in the expression $\... |
Is a commutative and associative with neutral element operation and inverses on R^2 necessarily componentwise sum? |
I want to find the Fourier transform of $$ f(x) = \cos (kx)$$
using the Fourier transformation formula $$f(k)={1\over \sqrt{2\pi}}\int _{-\infty}^\infty f(x) e^{ikx}dk$$
How can I do that? |
Let's step back and look at the issue a bit more naively. If $X$ has finite mean $\mu$ and variance $\sigma^2$, then for any positive integer $n$, the iid sample total $n\bar X = \sum_{i=1}^n X_i$ will have mean $n \mu$ and variance $n \sigma^2$, which is simply a consequence of the linearity of expectation and the ex... |
From [1], we have the well known Lindeberg–Levy Central-Limit Theorem:
>[Lindeberg–Levy Central-Limit Theorem]
Suppose $X_1, X_2, \ldots, X_n$ is a sequence of independent and identically distributed random variables with $E[X_i] = \mu$ and $\operatorname{Var}[X_i] = \sigma^2 < \infty.$ Then, as $n$ approaches in... |
I am trying to do some calculations with involving the Ricci tensor, but I am off by a factor 2. Here's my problem: The manifold is $S^2$ with the standard metric $g=e^{2\phi}\delta_{ij}$, with $\phi=\ln2-\ln(1+x^2+y^2)$. Using formulas from this wikipedia [page](https://en.wikipedia.org/wiki/List_of_formulas_in_Rieman... |
A small ball is released from $(-1,2,11)$ on $z=x^3-3xy^2$. Which vector does it start moving on?
I found the normal $(-9,12,-1)$ and I know its not the answer. I really don't have any idea other then that. There no additional information given. |
Let's step back and look at the issue a bit more naively. If $X$ has finite mean $\mu$ and variance $\sigma^2$, then for any positive integer $n$, the iid sample total $n\bar X = \sum_{i=1}^n X_i$ will have mean $n \mu$ and variance $n \sigma^2$, which is simply a consequence of the linearity of expectation and the ex... |
I am wondering if anybody knows good exact bounds for the Median of a Hypergeometric distribution:
$$
X \sim \text{Hypergeometric}(N,K,n)
$$
One exact bound could be given by exploiting the [inequality][1] between Mean, Median and Standard Deviation.
Another direction might be the Binomial approximation of H... |
I am trying to do some calculations with involving the Ricci tensor, but I am off by a factor 2. Here's my problem: The manifold is $S^2$ with the standard metric $g=e^{2\phi}\delta_{ij}$, with $\phi=\ln2-\ln(1+x^2+y^2)$. Using formulas from this wikipedia [page](https://en.wikipedia.org/wiki/List_of_formulas_in_Rieman... |
A common schema in intuitionistic, constructive set theories is that of *$\it{\Delta_0}$ separation*:
$$
\forall x\exists y\forall z(z\in y\leftrightarrow z\in x\wedge\phi(x, z))\text{ provided $\phi$ is a $\Delta_0$ formula}
$$
So chosen as $\Delta_0$ sets are clearly predicatively defined, and full separation... |
Proof of Hutton's 1776 formula:
[![enter image description here][1]][1]
Why does the figure establish this identity?
$$
\frac{\pi}{4} = 2\arctan\left(\frac{1}{3}\right) + \arctan\left(\frac{1}{7}\right)
$$
[1]: https://i.stack.imgur.com/2ZMe3m.png |
When you ask what a matrix "really is", you are asking the wrong question. A matrix is, at its heart, just a big block of numbers arranged in a rectangular grid. That's it.
The important question is: what can you *do* with a matrix? That's a much more interesting story! In fact matrices can be used in many dif... |
I'm having trouble finding the Fourier transform of $g(t) = \cos^2{a x}$.
I know the answer has to be a sum of $3$ dirac delta functions, but I'm having trouble showing this. I'll show you where my work ran into a problem.
First, I used the identity $\cos^2 (x) = 1/2 \ (1 + \cos (2x))$. Then, I have:
$$\hat g... |
I'm sorry if this is a noob question but I got this statement asked on a test:
$$\forall\ x \exists y \,F(x,y) \rightarrow \exists y \forall x \, G(x,y)$$
where
$$M = \lbrace \text{Natural numbers > 0} \rbrace \\
F(x,y): \, x <y \\
G(x,y): \, x > y\\$$
The statement is supposed to be false? I get that the... |
How can I resolve this graphically derived identity? |
Let $\vec x=(x_1,-2,1,-1)$ and $\vec y=(-2,y_2,-1,-2)$, wich satisfies $\lVert \vec x\rVert=2\lVert \vec y\lVert$. Find the indeterminate values of $x_1$ and $y_1$.
So, assuming $\lVert _\dot{} \lVert =\lVert _\dot{}\lVert_2$, then:
$$\lVert \vec x\rVert:=\Bigg(\sum_{i=1}^{4} x_i^2\Bigg)^{1/2}=\sqrt {x^2_1+(-2)^2... |
Let $T: V \rightarrow W$ be a continuous/bounded linear map between Banach spaces. Let $J \subseteq \ker(T)$ be a closed subspace. Then I know that this induces a unique linear map on the quotient $\bar{T}:V/J \rightarrow W$ s.t. $\bar{T}\circ \pi=T$. I also know from topology that $\bar{T}$ is continuous w.r.t. the qu... |
Is the induced map on the quotient space bounded? |
So
- $e^{iz} = e^{ix-y} = \frac{e^{ix}}{e^y} = \frac{cos(x)+isin(x)}{e^y} = \frac{cos(x)}{e^y} +\frac{isin(x)}{e^y} = -1 + i$
Then
- $\frac{1}{e^y} cos(x) = -1$
- $\frac{1}{e^y} sin(x) = 1$
With some basic manipulation, we get $tan(x) = -1$ which gives us $x = \frac{3\pi}{4} + k$, for some integer $k$
... |
**So I would like to know what the fastest (asymptotically, worst case runtime) exact algorithm for the Euclidean Steiner tree problem is.** (More precisely, I would like to know the best known algorithm and runtime bound pair.)
Searching the Web, I found:
- A [2010 "survey"][1] on algorithms for the euclidean Ste... |
What is the asymptotically best algorithm for the euclidean Steiner tree problem? |
I define a neural network with $L$ layers ($L-1$ hidden layers). The forward pass is as follows:
$$
\mathbf{a}^{(l)} = f(\mathbf{W}^{(l)}\mathbf{a}^{(l-1)}+\mathbf{b}^{(l)})
$$
Where $l \in [0,L]$ and $\mathbf{a}^{(0)}=\mathbf{x}$.
We have a dataset $D = \{(x_i, y_i) \mid x_i \in \mathbf{x}, y_i \in \mathbf{y}... |
I define a neural network with $L$ layers ($L-1$ hidden layers). The forward pass is as follows:
$$
\mathbf{a}^{(l)} = f(\mathbf{W}^{(l)}\mathbf{a}^{(l-1)}+\mathbf{b}^{(l)})
$$
Where $l \in [0,L]$ and $\mathbf{a}^{(0)}=\mathbf{x}$.
We have a dataset $D = \{(\mathbf{X}_i, \mathbf{Y}_i) \mid \mathbf{X}_i = \math... |
Indeed, I partially understand defining the cartesian product as a set of functions, but I do not understand why we need to do that.
> Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$ as an ordered sequence such that $x_i\in X_i$.
> Similarly can we define $\pro... |
Cartesian products of a family of sets, why we need to define it as a set of functions? |
Problem :
Show by hand without any computer assistance :
>$$e^{e^2}>1000\phi$$
where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$.
I come across this limit showing :
$$\lim_{x\to 0}x!^{\frac{x!!^{\frac{2}{x!!-1}}}{x!-1}}=e^{e^2}$$
I cannot show it without knowing some decima... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
Although the constants are included in Structure definition and Interpretation definition, but Varibles are included only in Interpretation definition, what is the reason for that, is there any error when Variables mapping is included in the Structure definition? is there any logic book explains the answer? |
In Paul Zeitz's the Art and Craft of Problem Solving, he says on page 34 that one way to show that a matrix $\mathbf{C}$ is invertible is by showing that $\mathbf{C} b_i \neq 0$ for each basis vector $b_i$. However, this can't be true -- consider the example
$\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}$. Am I just tri... |
Indeed, I partially understand defining the cartesian product as a set of functions, but I do not understand why we need to do that.
> Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$ as an ordered sequence such that $x_i\in X_i$.
> Similarly can we define $\pro... |
Indeed, I partially understand defining the cartesian product as a set of functions, but I do not understand why we need to do that.
> Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$ as an ordered sequence such that $x_i\in X_i$.
Similarly can we define $\prod_... |
Indeed, I partially understand defining the cartesian product as a set of functions, but I do not understand why we need to do that.
> Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$ as an ordered sequence such that $x_i\in X_i$.
> Similarly can we define $\pro... |
Let $A,B\in M_2(\mathbb{R})$ be such that $A^2+B^2+2AB=0$ and $\det A= \det B$. Our goal is to compute $\det(A^2 - B^2)$. According to the chain of [comments][1] on *Art of Problem Solving*, the following statements are true:
1. $\det(A^2+B^2)+\det(A^2-B^2)=2(\det A^2+\det B^2)$. (Is this well known?)
2. (1) $\i... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
Although the constants are included in Structure definition and Interpretation definition, but Varibles are included only in Interpretation definition, what is the reason for that, is there any error when Variables mapping is included in the Structure definition? is there any logic book, which explains the answer? Can ... |
Indeed, I partially understand defining the cartesian product as a set of functions, but I do not understand why we need to do that.
> Let $X_1,X_2,\dots,X_n$ be sets. We have defined a point $$x=(x_1,\dots,x_n)\in \prod_{i=1}^n X_i$$ as an ordered sequence such that $x_i\in X_i$.
Similarly can we define $\prod... |
Consider the double sums :
$$A_3 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^3 + b^3}$$
$$A_4 = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{a^4 + b^4}$$
Is there a closed form for $A_3$ or $A_4$ ?
Can we prove they are irrational ?
***
Notice that double sums over rational functions can ... |
I was recently attempting a problem from Evans' PDE text and I solved it, but I'm not sure my answer is correct since it looks a bit fishy. I was hoping someone could verify it and correct me if I'm wrong.
The problem goes as follows:
Let U = $\{|x_1| < 1,|x_2| < 1\}$ denote the open unit square in $R^2$ centered... |
So, I got up to expanding $\frac{(\Re(z))^2}{|z|}$ to $\frac{z^2+2z\overline z + \overline z^2}{4|z|}$ but I don't see other sequences that converge to $0$ to spark inspiration on how to continue.
This one has just really thrown me off. Any tips?
Edit: Actually
$$\frac{(\Re(z))^2}{|z|} = \frac{r^2\cos^2(n)}{|... |
I was recently attempting a problem from Evans' PDE text and I solved it, but I'm not sure my answer is correct since it looks a bit fishy. I was hoping someone could verify it and correct me if I'm wrong.
The problem goes as follows:
Let U = $\{|x_1| < 1,|x_2| < 1\}$ denote the open unit square in $R^2$ centered... |
Given a univariate polynomial of degree $k$ raised to power $m$ with non-zero constant term $c_0$ and exponents $e_i \ge 1$ present (but not necessarily $k$ of them) it appears to me that $c_n/c_0^n$ (the coefficient of $x^n$ divided by the constant term of the polynomial raised to the power $m$) will be a polynomial o... |
>Let $S_n = \sum_{k=1}^n\frac{1}{\sqrt{n^2+k^2}}$. Calculate the following limit
$$\lim_{n \to \infty} n(n(\ln(1+\sqrt{2})-S_n)-\frac{1}{2\sqrt{2}(1+\sqrt{2})})$$
My intuition says that every from of $n$ times a parentheses in a form of $0 \cdot \infty$ otherwise the problem would be trivial.
So let's first calc... |
Probability of flips n to 2n-1 being all tails? |
Boundedness Theorem can be applied to a closed bounded interval. Now suppose on interval $(0, 1]$, continuous function $f(x)$ has a finite limit at $x\to 0+$ is also finite at $x=1$. It seems that $f(x)$ is bounded. But how to formally prove or if the assertion is not valid? |
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
$$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(n+1)(2n+1)}{12}-1} \prod_{k=1}^n (y^p+1)^p dy$$
now lets define $a(n,k)$ as the coefficient of $x^k$ in the expression ... |
How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$? |
It [seems like](https://math.stackexchange.com/questions/4476471) we should start with a probability distribution and infer a log likelihood based on that. Why is it a bad idea to start with regularizers and map this onto priors? For example, if I were walking around an ideal city grid, would distances not follow a Lap... |
Why is it bad form to translate from Lp norm to priors? |
It is a theorem that a representation $(\pi,V_\pi)$ of a compact group $K$ decomposes as a direct sum of irreducible representations.
My question is about Deitmar's treatment (Automorphic Forms page 198) of $K$-finite functions $G\rightarrow\mathbb{C}$, for $G$ a locally compact group and $K$ a compact subgroup. These... |
**Starting out with the wrong characterization of the formula you present seems to have set you on the wrong path.** Your first formula gives the *probability* that a random set of five distinct cards from a 52-card Poker deck will meet the criteria for "two pairs" but not for any better hand. That is, that there wil... |
Boundedness Theorem can be applied to a closed bounded interval. Now suppose on interval $(0, 1]$, continuous function $f(x)$ has a finite limit at $x\to 0+$, and is also finite at $x=1$. It seems that $f(x)$ is bounded. But how to formally prove or if the assertion is not valid? |
I have read the following theorem related to an $X \in L^2(\Omega)$ random variable's integral representation:
Let $w$ denotes a Wiener process with its natural filtration $\mathcal{F}$. Fix $T<\infty$. Let $h$ be an arbitrary $h\in L^{2}\left(\left[0,T\right]\right)$ deterministic function. The set $\mathcal{U}$ o... |
I got this problem as part of an interview...
> Problem Statment:
>
> You have two players, Frank and Jane, that take turns
> in rolling a fair k-sided die. Whoever rolls a k first wins the game.
> The Python program should output the probability that Frank wins the
> game for k=6 thru 99. That is, the output... |
**So I would like to know what the fastest (asymptotically, worst case runtime) exact algorithm for the Euclidean Steiner tree problem is.** (More precisely, I would like to know the best known algorithm and runtime bound pair.)
Searching the Web, I found:
- A [2010 "survey"][1] on algorithms for the euclidean Ste... |
Today I decide to study some topics of Algebra and then faced up with the definition of the direct sum of modules. To let us on the same page, the definition that I'm talking about is
>Given a ring $R$ and a family $(M_i)_{i\in I}$ of left $R$-modules, the direct sum of the family $M_i$ to be the set of all sequences ... |
Let $X: L^2(\mathbb{R^n}) \rightarrow L^2(\mathbb{R^n})$ be a multiplication operator, i.e. $f(x) \mapsto xf(x)$ for $f \in L^2$. Multiplication operators are known to be self-adjoint on some dense subset of $L^2$, and so by Stone's theorem on one-parameter unitary groups $X$ is the infinitesimal generator of some one-... |
I got this problem as part of an interview...
> Problem Statment:
>
> You have two players, Frank and Jane, that take turns
> in rolling a fair k-sided die. Whoever rolls a k first wins the game.
> The Python program should output the probability that Frank wins the
> game for k=6 thru 99. That is, the output... |
Today I decide to study some topics of Algebra and then faced up with the definition of the direct sum of modules. To let us on the same page, the definition that I'm talking about is
>Given a ring $R$ and a family $(M_i)_{i\in I}$ of left $R$-modules, the direct sum of the family $M_i$ to be the set of all sequences ... |
I have a silly question about the pullback of $ 1 $-forms. Take two smooth manifolds $ M $ and $ N $ and some smooth map $ F\colon M\to N $ between them. Take some $ 1 $-form $ \eta $ on $ N $. Suppose that with respect to some coordinates $ (y^1,\dots,y^m) $ on an open subset $ V $ of $ N $ we can write
$$
\eta{\res... |
>Let $P_n(x)=x^n-nx+1$ be a sequence of polynomials, where $P_n\colon[1, +\infty) \to \mathbb{R}$ and $n \ge 2$ <br/>
a) Show that for each $n$, $P_n(x)=0$ has exactly one solution, and for each $n$ let $x_n$ be that unique number such that $P_n(x_n)=0$. <br/>
b)Show that $\lim_{n \to \infty}x_n=1$ and study the conv... |
Considered an irreducible polynomial with coefficient over a field $\Bbb K$ $ (f(x)\in\Bbb K[x]$) and $deg(f(x))=n$ ; in my textbook is written that: $$ {\Bbb K[x]\over <f(x)>} = \{a_0+a_1x+...+a_{n-1}x^{n-1}+<f(x)>|a_i\in\Bbb K\} $$
Is it true? and if it's how can I prove this ?(I apologize if the question may seem s... |
It can be shown that taking y = 1/x and rotating about the origin can produce (part of) a hyperbola. Is there such a relation in 3D? Would it be possible to take the equation xyz = 1 and rotate it such that it forms a hyperboloid?
I tried figuring this out with a similar approach as one would in 2D: converting to an... |
Let $u: \mathbb{R} \to \mathbb{R}$ such that $u \in C^1 (\mathbb{R}) \cap \mathrm{BV} (\mathbb{R})$. Prove that
$$\frac{1}{\varepsilon} \int_{-\infty}^{\infty} |u(x+\varepsilon) - u (x)| dx \leq TV (u), $$
where $TV(u)$ is the total variation of u.
Any idea? |
The polynomials of degree less than that of $f$ give an irredundant collection of representatives for the quotient, yes. Maybe a nod to the division algorithm for polynomials in one variable with coefficients in a field? |
You are told you have a container containing N balls. There is only 1 white ball in the container.
S = The number of draws needed to obtain a white ball for the first time.
Prove that P(S = i) = 1/N if 1 <= i <= N
I do not understand how the probability of drawing a white ball can be 1/N on such a large range ... |
Considered an irreducible polynomial with coefficient over a field $\Bbb K$ $ (f(x)\in\Bbb K[x]$) and $deg(f(x))=n$ ; in my textbook is written that: $$ {\Bbb K[x]\over \langle f(x)\rangle} = \{a_0+a_1x+...+a_{n-1}x^{n-1}+\langle f(x)\rangle \mid a_i\in\Bbb K\} $$
Is it true? and if it's how can I prove this ?(I apolo... |
Does the moment generating function exist? |
I've recently become a little obsessed with all the ways limits can be composed - and the preconditions for this to take place. I took $A,B_{1},B_{2},C \subseteq \mathbb{R}$ and $f:A\to B_{1},\ g:B_{2} \to C$ and further assumed $f[A] \subseteq B_{2}$. And then I listed out all the ways to compose $f$ and $g$ with limi... |
Why does a projector being equal to its own central carrier imply that it is in the center of the commutant of its algebra? |
Let $R$ be an integral domain, and $a\in U(R)$ an invertible element of $R$ and $b\in R $. I will prove that the following are equivalent:
- $f(x)$ is irreducible in $R[x]$.
- $f(ax+b)$ is irreducible in $R[x]$.
First of all $f(x)$ and $ f(ax+b)$ have always the same degree $n$.
We suppose that $f(x)$ i... |
From [1], we have the well known Lindeberg–Levy Central-Limit Theorem:
>[Lindeberg–Levy Central-Limit Theorem]
Suppose $X_1, X_2, \ldots, X_n$ is a sequence of independent and identically distributed random variables with $E[X_i] = \mu$ and $\operatorname{Var}[X_i] = \sigma^2 < \infty.$ Then, as $n$ approaches in... |
You are told you have a container containing N balls. There is only 1 white ball in the container.
S = The number of draws needed to obtain a white ball for the first time.
Prove that P(S = i) = 1/N if 1 <= i <= N
I do not understand how the probability of drawing a white ball can be 1/N on such a large range ... |
It's easy to show (using the [Lindemann-Weierstrass theorem][1]) that, for $x\ne 0$, at least one of $x$ and $\sin(x)$ must be transcendental. But what about $x\sin(x)$? After all, the product of two transcendental numbers could be algebraic. Hence: Can the product $x\sin(x)$ be non-zero and algebraic? Might it even be... |
Can $x\sin(x)$ be algebraic when it is not $0$? |
From [1], we have the well known Lindeberg–Levy Central-Limit Theorem:
>[Lindeberg–Levy Central-Limit Theorem]
Suppose $X_1, X_2, \ldots, X_n$ is a sequence of independent and identically distributed random variables with $E[X_i] = \mu$ and $\operatorname{Var}[X_i] = \sigma^2 < \infty.$ Then, as $n$ approaches in... |
Can this Non-Linear Residual Function, $R(x)=((x+1)^3-1)^{1/3}-(x+1)$, be expressed exactly using a infinite series (using integer powers in $x$) or a finite polynomial function in $x$?
For example I have found one approximate series / polynomial function that is a reasonably accurate approximation above $x=2$
$$... |
I can see that there are $\binom{n+k-1}{k}$ cases of choosing k items of n types with repetition from <http://en.wikipedia.org/wiki/Multiset_coefficient#Counting_multisets>.
I wonder whether there is any formula about its sum for varying $k$.
In particular, I am interested in $\sum_{k=1}^{N} \binom{n+k-1}{k}$.
D... |
>A coin is tossed 100 times , Find the probability that tail occurs odd number of times!
I do not know the answer, but I tried this, that there are these $4$ possible outcomes in which tossing of a coin $100$ times can unfold.
1. head occurs odd times
2. head occurs even times
3. tail occurs odd times
... |
**Lemma** $\rm\,\ a\mid bc\Rightarrow a\mid (a,b)(a,c)\ \ $ [my notation swaps $\,\rm a,b\,$ vs yours]
**Proof** $\ \ \rm \color{#c00}{ad = bc} \ \Rightarrow\ (a,b)\,(a,c)\, =\ (aa,ab,ac,\color{#c00}{bc})\, =\, \color{#c00}a\,(a,b,c,\color{#c00}d)\ \ $ $\small\bf QED$
The OP has $\rm\,(a,b)=1\,\Rightarrow\,(a,... |
>Let $P_n(x)=x^n-nx+1$ be a sequence of polynomials, where $P_n\colon[1, +\infty) \to \mathbb{R}$ and $n \ge 2$ <br/>
a) Show that for each $n$, $P_n(x)=0$ has exactly one solution, and for each $n$ let $x_n$ be that unique number such that $P_n(x_n)=0$. <br/>
b)Show that $\lim_{n \to \infty}x_n=1$ and study the conv... |
In order to better understand the tricky $\exists L$ rule and its eigenvariable requirement from sequent calculus:
$$\Gamma, A[y/x] \vdash \Delta \over \Gamma, \exists xA \vdash \Delta$$
I decided to try proving the below proposition to build my intuition for this rule.
> If $A[y/x]\models\Delta$, provided tha... |
I've recently become a little obsessed with all the ways limits can be composed - and the preconditions for this to take place. I took $A,B_{1},B_{2},C \subseteq \mathbb{R}$ and $f:A\to B_{1},\ g:B_{2} \to C$ and further assumed $f[A] \subseteq B_{2}$. And then I listed out all the ways to compose $f$ and $g$ with limi... |
Let $D$ be a set of lines in the real projective plane such that for all $d\in D$, $\bigcap (D\setminus \left\{d\right\})=\emptyset$ and let $Dot(D)$ be the set of all dots that belong to at least two members of $D$.
$D$ is *complete* if for any pair of dot in $Dot(D)$ the line that contain both dots of the pair is... |
Does colinearity in construction with ruller has to occure by contruction? |
Considered an irreducible polynomial with coefficient over a field $\Bbb K$ $ (f(x)\in\Bbb K[x]$) and $\deg f(x)=n$ ; in my textbook is written that: $$ {\Bbb K[x]\over \langle f(x)\rangle} = \{a_0+a_1x+...+a_{n-1}x^{n-1}+\langle f(x)\rangle \mid a_i\in\Bbb K\} $$
Is it true? and if it's how can I prove this ?(I apolo... |
I want to prove Proposition 10.23 from Tomas Björk's ''Arbitrage theory in continuous time'' in the snippet below.
[![][1]][1]
[1]: https://i.stack.imgur.com/zqZQP.png
My attempt: For simplicity, assume everything is one-dimensional, with one risky asset $S$ and a risk-less asset $S^0$. Assume also that $r... |
Let $D$ be a set of lines in the real projective plane such that for all $d\in D$, $\bigcap (D\setminus \left\{d\right\})=\emptyset$ and let $Dot(D)$ be the set of all dots that belong to at least two members of $D$.
$D$ is *complete* if for any pair of dot in $Dot(D)$ the line that contain both dots of the pair is... |
Does colinearity in construction with ruller has to occure because we "decide" it in the contruction? |
That is exactly correct. Because the conditional variable $X \mid N = n$ is binomial with parameters $n$ and $p = 1/2$, you computed
$$\operatorname{E}[X \mid N] = N/2,$$
hence
$$\operatorname{E}[X] = \operatorname{E}[\operatorname{E}[X \mid N]] = \operatorname{E}[N/2] = \operatorname{E}[N]/2.$$
What are your rese... |
A teacher gave her students a paper square. The first student cut this
square into two shapes, using one straight cut not through any of the
paper’s corners. The second student cut one of the resulting shapes,
using one straight cut not through any of that shape’s corners, and so
on. After ten students had made the... |
Let the *multiplicative closure* operator $h$ be defined on $\mathcal{P}(\mathbb{N})$ by setting $hA$ equal to the smallest set containing $A$ that is closed under multiplication.
In L. F. Meyers' solution to Problem E $1941$ on pp. $408$-$9$ of the April $1968$ *American Mathematical Monthly* (https://www.jstor.org/s... |
how many natural numbers are needed to generate 14 distinct subsets under complement and multiplicative closure? |
I got this problem as part of an interview...
> Problem Statment:
>
> You have two players, Frank and Jane, that take turns
> in rolling a fair k-sided die. Whoever rolls a k first wins the game.
> The Python program should output the probability that Frank wins the
> game for k=6 thru 99. That is, the output... |
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