# All Questions

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### Notation for gradients analogous to partial derivatives

Is there an equivalent of partial differentiation for functions taking multiple vectors as input? I mean the following. If we have a function $f(x,y)$, then a partial derivative is denoted as ...
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### continuous function from one metric space to another metric space

Is differentiation $f(x) \rightarrow f'(x)$ a continuous function from $C^1[a,b] \rightarrow C[a,b]$ ? Is integration $f(x) \rightarrow \int_a^x \! f(t) \, \mathrm{d}t$ a continuous function from ...
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### Mean of the Multivariate Wallenius Non-Central Hypergeometric Distribution

An urn contains $N$ balls where ball $i$ is of size $w_i$. We draw $n$ times without replacement. Let $x_i$ be the random variable indicating whether the ball $i$ has been drawn ($x_i=1$) or not ...
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### Maximum likelihood estimate of $N$ (trials) in Binomial

Suppose, we throw a biased coin $N$ times with $p(\text{head}) = \pi$, and we observe the number of heads as $k$ (could be any number, say $k=4$ for simplicity). We are interested in to find the most ...
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### how to solve $\int\frac{1}{1+x^4}dx$ [duplicate]

i want find the answer and metod of solve of $\int\frac{1}{1+x^4}dx$. I know $$\int\frac{1}{a^2+x^2}dx=\frac{1}{a}\arctan\frac{x}{a}+C$$, How I can use this to solve of that integration.
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### Is my method finding $\sup A$ and $\inf A$ fully correct?

$A=\left\{\dfrac{1}{n}+\dfrac{1}{n^2} \mathrel{\bigg|} n\in \mathbb N^*\right\}$ I have derived the function and I found $\dfrac{-n(n+2)}{n^4}$, so the function is strictly decreasing. Then I simply ...
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### Union of a finite number of open sets is open or not? Proper usage of this fact a proof

I received a homework assignment back and I was given full credit on the following proof: Let $S = \{ (x,y) \in \mathbb{R}^{2} | x \geq 1$ and $y \geq 1 \}$. Is $S$ closed? My proof is below ...
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### Linear Algebra - Prove Isomorphism.

Let $T : \Bbb R^n \rightarrow \Bbb R^n$ Linear transformation. Prove that there is a real number $\alpha$ that the transformation $\alpha I-T$ is isomorphism. isomorphism is only if $\ker T={0}$ or ...
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### Help solving: Normal Distribution problem without using the table OR with a given std

For a recent history test, scores follow the normal distribution with a mean of 70 points. 80% of the students scored below 88 points. What is the standard deviation of the scores? I have done a lot ...
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### Are the quaternions obsolete in pure mathematics?

I remember I read an article saying that "The quaternions $\Bbb{H}$ are obsolete in pure mathematics since the theory of vectors has been developed enough, however it is useful in computer science". ...
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### Let G be a finite cyclic group of order n. If d is a positive divisor of n , prove that x^d = e has exactly d distinct solutions in G

well i know that for a group to be cyclic then there must exist an element in G for example we call it g such that $G = \langle g\rangle$ and so $g^0 = e$ and $g^0 = g^n = e$ hence ...
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### How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$ [duplicate]

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
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### Finding expectation and variance for multiple items when given for 1 item

The problem I have here is that I know this is a normal distribution question, but I don't know how to find the Variance And Mean for 50 items, they have given for 1 item My book simply says : ...
180 views

### Simplifying polynomials

Suppose I have a (multivariate) polynomial with coefficients in $\mathbb Z$ or $\mathbb Q$, given in fully expanded form. How can I simplify this to reduce the number of elementary operations ...
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### Is it possible to find the norm fuction of a space from an inner product already defined for it?

I'm a noob on the subject of functional analysis. As the title of the question says: Is it possible to find the norm fuction of a space from an inner product already defined for it? e.gr.: Suppose ...
651 views

### Probability of Getting at least 2 correct answers out of 7 (3 choices are correct)?

Imagine there's a multiple choice question with 7 possible choices. 3 are correct and a student randomly selects 3 choices. What's the probability that he gets at least 2 correct? I thought it was: ...
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### Topology. Understanding what a base is intuitively

A collection $\{V_n\}$ is said to be base for $X$ if the following is true: For every $x$ that's an element of $X$ and every open set $G$ that is a subset of $X$ such that $x$ is an element of $G$, we ...
119 views

### Conic sections in standard form

I'm trying to convert the equation $$x^2 +2y^2 +4x-4y+4=0$$ into its standard form by choosing a new set of axes. Yet, when I go down the conventional route, there is no xy term so ...
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### Arguing that Graph, $G$, is 3 regular and pairwise edge-disjoint path can't share internal vertices

Let $G$ be a graph that is 3-regular. There are $n$ pairwise edge-disjoint $x,y$ paths. I want to show "since G is 3 regular, these paths cannot share internal vertices." I know the answer is supposed ...
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### Limit calculation, calculus

I am trying to find $$\lim_{x\to 0} \frac{e^{\alpha x} -e^{\beta x}}{\sin(\alpha x) + \sin(\beta x) }$$ but dont have a clue where to start, could someone give me a hint please ?
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### How would you approach this problem with Bayes: Coins with different biases

You have a sack of coins. Each coin can have a different bias. The biases are unknown. You flip each coin. If the coin comes up tails, you remove the coin from the sack. If it's heads, it remains ...
106 views

### Show that $\,a_n=f(1)+f(2)+\cdots+f(n)-\int_1^n f(x)\,dx\,\,$ converges

Let $\,f:[1,\infty)\to \mathbb R\,$ be a decreasing and lower bounded function. Show that the sequence $\{a_{n}\}_{n\in\mathbb N}$ defined as: $$a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx,$$ ...
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### The number Triangles in this picture [duplicate]

I want a method for find the number triangles in the under image.
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### octagonal number theorem $q$-Pochhammer symbol expression

Setting the exponents of this analogue of the series in Euler's Pentagonal Number theorem to be the octagonal numbers: $$U(q)= \sum_{n\in\mathbb{Z}} (-1)^{n}q^{n(6n-4)/2}$$ in mpmath: ...
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### What is the number of digits of this number: $2^{333111160}$? [duplicate]

My question is: What is the number of digits of this number? : $$2^{333111160}$$
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### Lyapunov-Schmidt reduction.

Use Lyapunov-Schmidt reduction to find an expression, or approximation, of the set of equilibria, as a function of the parameter $\lambda$, of the planar vector field ...
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### Mean of stochastic exponential

Suppose $X_t$ solves an SDE. Is it true to say that the identity, $$\mathbb{E}\left[e^{X_t}\right] = e^{\mathbb{E}[X_t]+\frac{1}{2}\text{Var}[X_t]}$$ holds only when the drift and volatility of ...
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### Proving AM-GM for the special case $n=3$

I know the AM-GM inequality and its proof which is relatively complex, though the case for $n=2$ is quite simple. However, I don't know of any special easier proof for the case $n=3$, specifically: ...
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### Existence of a A measurable function

Let $A$ be sigma algebra having subsets of $R$ only. We define a function from subset of $A$ to $R$ is said to be $A$ measurable iff every Borel set is pulled back to elements of $A$. Is there a ...
Suppose $k<n$. How does one express $\det\begin{pmatrix}a_1^1&\dots&a_n^1\\ \vdots&\ddots&\vdots\\ a^n_1&\dots&a^n_n\end{pmatrix}$ in terms of a linear combination of ...