1
vote
0answers
52 views

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 ...
0
votes
1answer
113 views

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 ...
1
vote
0answers
96 views

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 ...
2
votes
1answer
254 views

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 ...
1
vote
3answers
187 views

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.
1
vote
1answer
42 views

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 ...
0
votes
1answer
116 views

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 ...
1
vote
4answers
180 views

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 ...
1
vote
1answer
100 views

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 ...
1
vote
1answer
121 views

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". ...
2
votes
2answers
89 views

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 ...
7
votes
4answers
367 views

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 ...
1
vote
0answers
179 views

Linear Algebra - Find basis for $ImT$ and $KerT$.

$B=(u1,u2,u3) \in R^3$ $u1=(1,-1,0)$ $u2=(1,1,1)$ $u3=(1,2,3)$ $T : R^3 \rightarrow R^3$ This is expression matrix (not sure if that the right term in English) on basis B: $[T]_B ...
1
vote
1answer
130 views

Multiplying the results of a hypergeometric distribution to get a total probability

For a (trading) card game I would like to determine the probability of a specific hand from a deck of cards. I can determine the probability of a single card occurring any number of times in an ...
1
vote
2answers
112 views

What is the sum-capacity for a non-symmetric interference channel for information theorists?

This question is dedicated for people who are experts in information theory. An interesting result for a two user interference channel in information theory, is the sum-capacity to within one bit. It ...
1
vote
1answer
31 views

If $N$ is normal, $W^\ast(N)=\{N\}''$

I want to prove that if $N$ is normal, then $W^\ast(N)=\{N\}''$, where $W^\ast(N)$ is the Von Neumann algebra generated by $N$ and $\{N\}''$ is the bicommutant of $N$. for the inclusion $W^\ast(N) ...
2
votes
1answer
32 views

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 : ...
11
votes
1answer
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 ...
1
vote
0answers
40 views

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 ...
0
votes
1answer
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: ...
0
votes
1answer
115 views

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 ...
0
votes
1answer
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 ...
0
votes
1answer
45 views

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 ...
0
votes
3answers
54 views

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 ?
1
vote
0answers
32 views

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 ...
0
votes
2answers
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: $$ a_n=f(1)+f(2)+\cdots+f(n)-\!\int_1^n\!\! f(x)\,dx, $$ ...
-1
votes
1answer
117 views

The number Triangles in this picture [duplicate]

I want a method for find the number triangles in the under image.
1
vote
1answer
46 views

A problem on almost sure convergence of an average

I have the following exercise: Let $X_1, X_2 \ldots$ be such that $$ X_n = \left\{ \begin{array}{ll} n^2-1 & \mbox{with probability } n^{-2} \\ -1 & \mbox{with probability } ...
2
votes
1answer
63 views

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$.

List the elements of $\langle\frac{1}{2}\rangle$ in $(\mathbb{Q},+)$ and in $(\mathbb{Q}^*,\times)$. where $\mathbb{Q}^*:=\mathbb{Q}\setminus\{0\}$ My attempt: Well, I know that $\langle ...
3
votes
3answers
62 views

Group isomorphisms and a possible trivial statement?

I have the following set $G=\lbrace a,b,e \rbrace$ and I successfully computed the following Cayley-Table \begin{align} \begin{array}{|c|c|c|c|} \hline \circ& a & b & e \\ \hline a& ...
-1
votes
1answer
117 views

Borel Measures: Lebesgue Decomposition

Disclaimer Please, if you don't like self-answers just avoid this thread. (For more details see: Answer own Question?) Context Given a measure space $\Omega$ with sigma-finite measure $\lambda$. ...
2
votes
1answer
75 views

Showing a set is a subset of another set

I need to show that $(A \cup B) \subseteq (A \cup B \cup C)$ My Work So Far: What I really need to show is that $x \in (A \cup B)$ implies $x \in (A \cup B \cup C)$ So I translated my sets into ...
0
votes
1answer
52 views

Eigen values of a positive semidefinite matrix and its transpose

$A\in M_n(\mathbb{C})$ is positive semi-definite so there there exists unitary matrix $U$ such that $A=U^*DU$ where $D$ is the real diagonal matrix consisting of eigen values $(\ge 0)$ of $A$, now I ...
0
votes
1answer
184 views

Using differentials with volume of a cube

my question is The volume of a cube is increased from 1000 cubic centimeters to 1156 cubic centimeters. Use differentials to determine. the side length of the cube increases by? the surface area ...
6
votes
1answer
96 views

Is $\int f=f-1\iff f(\cdot)=e^{\cdot}$ proved this way correct?

I saw this on math overflow and made me wonder, why does it work, is it rigorous, can we really factor like this, and where can we use similar tricks; Let $\int$ denote $\int_0^x$ Then solve $$\int ...
1
vote
1answer
122 views

Proof of theorem 26.15 in “General Topology” by Willard

26.14 Definition. A simple chain connecting two points $a$ and $b$ of a space $X$ is a sequence $U_1,\ldots,U_n$ of open sets of $X$ such that $a\in U_1$ only, $b\in U_n$ only, and $U_i\cap ...
2
votes
0answers
542 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
5
votes
6answers
627 views

Problem in the second-derivative symbol.

The second derivative of this symbol according to the rules that we have learned the correct mathematical, I wish to know why this symbol is not used.
1
vote
1answer
41 views

What exactly does it mean for a maximal ideal to be unique in a principal ideal domain?

I'm currently reading about PIDs and have come across a question involving maximal ideals which at one point reads "Suppose that a Euclidean domain $R$ had a unique maxima ideal $P$". Does this mean ...
0
votes
1answer
31 views

How Can I define the derivative of matrix?

If I have a matrix: $$F(x) = \begin{pmatrix}f_1(x)& f_2(x) \\ g_1(x) & g_2(x) \end{pmatrix} $$ where $f_1,f_2,g_1$ and $g_2$ are differentiable functions. What would be the derivative of ...
2
votes
2answers
52 views

Prove that, given any positive integer n, some multiple of it must be of the form 99…900…0

Prove that, given any positive integer n, some multiple of it must be of the form 99...900...0 Give me a hand, please.
1
vote
1answer
73 views

If $X$ is a subset of $\omega_{\alpha}$ such that $|X| < \aleph_{\alpha}$, then $|\omega_{\alpha} - X| = \aleph_{\alpha}$

If $\alpha=0$, then $\omega_0=\mathbb{N}$ and $\aleph_0=$ countable. So $|\omega_{\alpha} - X| = \aleph_{\alpha}$ becomes $|\mathbb{N}-X|=|\mathbb{N}|$ which is true (the function $f:\mathbb{N} ...
1
vote
1answer
81 views

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: ...
0
votes
2answers
113 views

What is the number of digits of this number: $2^{333111160}$? [duplicate]

My question is: What is the number of digits of this number? : $$2^{333111160}$$
1
vote
1answer
186 views

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 ...
0
votes
1answer
35 views

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 ...
3
votes
6answers
141 views

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: ...
1
vote
0answers
39 views

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 ...
0
votes
4answers
86 views

A box contains 10 colored discs of which 2 are red

A box contains 10 colored discs of which 2 are red. A man pays 10 cent to play a game in which discs are pulled out on at a time, without replacement. If his first draw is a red disc, he will win 25 ...
1
vote
1answer
61 views

Expressing determinant as a linear combination of minors of fixed dimension

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 ...

15 30 50 per page