9
votes
3answers
1k views

why measure theory

I studied elementary probability theory. For that, density functions were enough. What is a practical necessity to develop measure theory? What is a problem that cannot be solved using elementary ...
6
votes
3answers
1k views

Importance of Neatness / Organization / Speed in Math?

Pretty simple question here but it does relate to math. I ask this as my writing is quite messy, possibly a cause of silly mistakes. How important is neatness in math? Does having messy writing put ...
6
votes
0answers
55 views

When is integration possible? [duplicate]

I'm not sure how to phrase this question. I'm sure I could write it in terms of operators between Frechet spaces, or something like that. Let me apologies to any analysts in advance for my lack of ...
2
votes
1answer
159 views

Problem with Discrete Parseval's Theorem

I think I must be missing something obvious, but I can't for the life of me see what it is. The discrete version of Parseval's theorem can be written like this: $$\sum_{n=0}^{N-1} |x[n]|^2 = ...
6
votes
5answers
103 views

Prove that $3^n>n^4$ if $n\geq8$

Proving that $3^n>n^4$ if $n\geq8$ I tried mathematical induction start from $n=8$ as the base case, but I'm stuck when I have to use the fact that the statement is true for $n=k$ to prove ...
3
votes
3answers
175 views

Why is $\det⁡(-A)=(-1)^n\det(A)$? [closed]

Why is $\det⁡(-A)=(-1)^n\det(A)$?
1
vote
2answers
137 views

Convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$

Is there a closed form formula for the convolution of $1/(1+x^2)$ and $\exp(-x^2/(4t))$, where $t>0$, i.e. the integral $$\int_{-\infty}^\infty ...
3
votes
1answer
279 views

Prove that for all positive integer n, the inequality $2n\choose n$ $<4^n$ holds

How do I prove that for all positive integer n, the inequality $2n\choose n$$<4^n$ holds? Thank you!
1
vote
4answers
181 views

Prove that $a^{(p-1)/2} $$\equiv$-1 (modp). Deduce that if $a, b$ are primitive roots modp, then $a\times b$ is NOT a primitive root mod p.

Let $a$ be a primitive root mod the odd prime p. Prove that $a^{(p-1)/2} $$\equiv$-1 (modp). Deduce that if $a, b$ are primitive roots modp, then $a\times b$ is NOT a primitive root mod p. Here ...
4
votes
3answers
102 views

area of triangle

In $\triangle ABC$ points $D,E,F$ are on the sides $AB,BC,CA$, respectively, with $AD=DB$, $CE=3BE$ and $AF=2CF$. If the area of $\triangle ABC$ is $480 cm^2$, how do we find the area of $\triangle ...
0
votes
2answers
57 views

An integral problem?

How do you integrate $e^{e^x}$? I was able to get it down to du/(ln u) but I wasn't able to go further. Thanks!
2
votes
1answer
128 views

Simplify difference of two arc tangents?

I have a problem, that I am trying to simplify, but there does not seem to be something obvious regarding it. Very simply, I am trying to figure out if there is a way to 'open' the following: $$ ...
0
votes
2answers
78 views

In how many ways can you choose $k$ out of $n$ people standing in line, So there's a space of at least 3 people between them

In how many ways can you choose $k$ out of $n$ people standing in line, So there's a space of at least $3$ people between them. Actually, I don't even know how to start on this one.
3
votes
1answer
247 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
4
votes
1answer
227 views

Conditions for a kernel of a bounded operator to be complemented

I am well aware of the problem of complementing subspaces in Banach spaces as it was discussed here and here . Nevertheless, I wonder whether there are conditions for existence of a complement $M$ ...
1
vote
1answer
35 views

Space of embedded surfaces with a common point

Consider the space of all embedded orientable surfaces in $ R^3 $ of constant mean curvature (the minimal case is included) passing through the origin. I'm asking if there exists a topology on this ...
1
vote
0answers
38 views

Invariant relation in ODE

It is well known that if function $g(x)$ is an invariant relation under ODE $\dot x = f(x)$ then $\frac{\displaystyle d}{\displaystyle dt}g = \lambda g$. More precisely. Let ...
1
vote
1answer
111 views

Set of left cosets is a group

Let $N$ be a normal subgroup of $G$. Let $G/N = \{gN : g \in G \}$. Show that the $(G/N, \circ, 1_{G/N})$ is a group, where: $\circ : G/N \times G/N \rightarrow G/N$ is defined by $(gN \circ hN) = ...
1
vote
1answer
29 views

System of Differential Equations Question Assistance

The following question has just left me confused with no real decent avenue of attack so any assistance on this would be appreciated. For the system of equations $t {\frac{d \vec x}{dt}} = A\vec x $ ...
11
votes
3answers
612 views

Irreducibility of $x^n-x-1$ over $\mathbb Q$

I want to prove that $p(x):=x^n-x-1 \in \mathbb Q[x]$ for $n\ge 2$ is irreducible. My attempt. GCD of coefficients is $1$, $\mathbb Q$ is the field of fractions of $\mathbb Z$, and $\mathbb Z$ ...
0
votes
1answer
489 views

How do I prove Poisson appraches Normal distribution

I want to prove why the mean and variance of a $\operatorname{Poisson}(\lambda)$, is different when the time index approaches infinite (it's approximated by the mean and variance of a Normal). For ...
0
votes
1answer
90 views

Confusing symbol in papers on hybrid logic

In literature about hybrid logic I'm reading for my thesis I've come across the following symbol: ::= Now, I've never seen this notation before. I can also not ...
1
vote
1answer
32 views

Decreasing from the horizontal asymptote

The function $f(x) = x^2/(x^2 - x -2)$ has the following graph. It has a horizontal asymptote $y=1$. For $x$ less than $-4$, the function is decreasing and its graph is under the asymptote. How is ...
1
vote
3answers
75 views

Taylor Polynomial for $x^{1/3}$

a. Compute the Taylor polynomial $T_3(x)$ for the function $(x)^{1/3}$ around the point $x=1$. b. Compute an error bound for the above approximation at $x = 1.3$. I'm having trouble figuring ...
0
votes
1answer
771 views

Mathematica vs Wolfram Alpha integration results

When I insert the following integration command in wolframalpha: ...
0
votes
2answers
95 views

Which of the following are subspaces of $M$?

Let $M$ be a vector space of all $3\times 3$ real matrices and let $$A=\begin{pmatrix}2&3&1\\0&2&0\\0&0&3\end{pmatrix}.$$ Which of the followings are subspaces of $M?$ ...
4
votes
1answer
210 views

Symmetrically splitting an octagon into quadrilaterals

I'm wondering whether it is possible to split an octagon into a finite number of quadrilaterals, such that the result is symmetric from all 8 directions (sides or points). There is one condition — any ...
1
vote
2answers
103 views

Trap Rule for sin(x)

Use the trapezoidal rule with $N=6$ to approximate the arc length of the curve $f(x) = \sin(x)$ from $x=0$ to $x=\pi$. So I found that $\Delta x = \frac{\pi}{6}$ which means that my interval ...
3
votes
1answer
90 views

How to prove the existence of infinitely many $n$ in $\mathbb{N}$,such that $(n^2+k)|n!$

Show there exist infinitely many $n$ $\in \mathbb{N}$,such that $(n^2+k)|n!$ and $k\in N$ I have a similar problem: Show that there are infinitely many $n \in \mathbb{N}$,such that ...
1
vote
2answers
64 views

What is the function given by $\sum_{n=0}^\infty \binom{b+2n}{b+n} x^n$, where $b\ge 0$, $|x| <1$

For a nonnegative integer $b$, and $|x|<1$, what is the function given by the power series $$ \sum_{n=0}^\infty \binom{b+2n}{b+n} x^n. $$ For $b=0$, this post shows $$ \sum_{n=0}^\infty ...
0
votes
0answers
28 views

Number of roots to system of Polynomials

If you have a system of k polynomials of dimension k and degree r is the number of solutions equal to: k^s? This appears to be the pattern and intuitively one could argue that each of the s systems ...
2
votes
2answers
90 views

For a diagonal matrix $M$, what is $e^M$?

For a diagonal matrix $$ M=\left(\begin{array}{ccc} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{array}\right) $$ show that $$ e^M=\left(\begin{array}{ccc} e^a & 0 & 0 \\ 0 ...
1
vote
1answer
42 views

Proof $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty {X(u)du} $

I see that we usually use the theorem to solve the Laplace transform, however i want to proof the theorem, who could give me some details!!! $L{\rm{[}}\frac{{x(t)}}{t}{\rm{] = }}\int_s^\infty ...
2
votes
1answer
132 views

Canonical log structure defined by a normal crossing divisor

Given a locally noetherian scheme $X$ and $D \subset X$ a normal crossing divisor. Let $j: U=X-D \hookrightarrow X$ the open complement immersion. Define the log structure $(M_X,\alpha_X)$ on $X$ as ...
0
votes
2answers
69 views

What is the dependent formula of the vectors

Are vectors $(1,0,3,4)$, $(1,0,1,1)$, $(4,1,2,3)$ and $(6, 1,8,11)$ linearly dependent? If yes, what is the dependence formula. I found out the determinant is $0$ isn't is supposed to be independent ...
0
votes
2answers
2k views

Similar cones - volumes and lateral areas

Two similar cones have volumes 9$\pi$ and 72$\pi$. If the lateral area of the larger cone is 32$\pi$, what is the lateral area of the smaller cone? I did the following... $\frac {(9\pi)^3} ...
4
votes
3answers
4k views

Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$

Let $f(x) = x^2 \sin{\frac{1}{x}}$ for $x\neq 0$ and $f(0) =0$. (a) Use the basic properties of the derivative, and the Chain Rule to show that $f$ is differentiable at each $a\neq 0$ and ...
0
votes
1answer
35 views

On finding adjoint of transformation.

Let $V$ be an inner product space and $v,w\in V$ be fixed vectors. Define $T(u)=(u,v)w$. How to find the adjoint mapping $T^*$?
0
votes
1answer
45 views

What is the proper way to determine error in my measurements?

How do I know what the error is in measurements I took using an oscilloscope? In the image below you will see on channel #1 of the oscilloscope (yellow) there is a pattern. I manually measured the ...
3
votes
0answers
47 views

find the dimension of $W.$

Let $W=\{p(B):p \text{ is a polynomial with real coefficients}\},$ where $B=\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}.$ Then find the dimension of $W.$ I have shown ...
4
votes
1answer
48 views

Apples and their volumes

An apple has a peel that is 1cm thick and a total diameter of 12cm. What percentage of volume of the apple is the peel? I tried $$\frac{\text{volume}(\text{radius of 6})-\text{volume}(\text{radius ...
5
votes
3answers
5k views

How many options are there for 15 student to divide into 3 equal sized groups?

How many options are there for 15 student to divide into 3 equal sized groups? Now i know the soultion is $\;\dfrac {15!}{5!5!5!3!}\;$ but i can't understand why. Can anyone please enlighten me?
3
votes
2answers
87 views

Proof that an embedding into $\ell^1$ is compact

Prove that any sequence $(x^{(n)})_{n\in\mathbb{N}}\subseteq\ell^1$ such that $\sum_{k=1}^\infty k\lvert x_k^{(n)}\lvert\leq1$ for all $n\in\mathbb{N}$ has a convergent subsequence. My thoughts ...
0
votes
1answer
36 views

A special subset of uniformly distributed numbers is still uniformly distributed?

Assume that I have a value range [1,1000]. I uniformly choose 10 numbers from [1,1000]. Assume that the chosen numbers are a1, a2, ..., a10. Besides, assume that they are ordered so that a1< ...
1
vote
1answer
49 views

Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?

Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius $t\in ...
0
votes
2answers
151 views

Show that $-Z$ is also a standard normal random variable.

Show that $-Z$ is also a standard normal random variable; that is, show that $P[-Z < x] = P[Z < x] \,\forall x.$
1
vote
3answers
38 views

Finding the limit of a particular function

Can not understand the following limit in a past paper. $$\dfrac{1}{[n(e^{\frac{1}{n}}-1)]}\rightarrow 1$$
1
vote
0answers
57 views

Fourier Transform of $(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$

I would like to calculate the FT of the following function: $$(\operatorname{sinc}(x))^2\cdot \exp(-ax^2)$$ Any hint is highly appreciated!
1
vote
4answers
95 views

Linear combination of vectors in $\mathbb{R}^3$

Show that any linear combination of $\pmatrix{1\\\frac{3}{2}\\0}$ and $\pmatrix{0\\3\\6}$ is also a linear combination of $\pmatrix{2\\3\\0}$ and $\pmatrix{0\\1\\2}$ I'm not sure how to do this. I ...
1
vote
1answer
81 views

Why is it okay to do this?

I am studying asymptotic recurrences for algorithms, and the book says: $$T(n) = 2T(n/2) + \Theta (n)$$ is technically $$T(n) = T(\lfloor n/2 \rfloor) + T(\lceil n/2 \rceil) + \Theta (n)$$ for an ...

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