All Questions
1
vote
0answers
30 views
What are some definite integration techniques?
Are there any definite integration techniques which I could learn (calc AB student)? I mean techniques which don't require you to find the anti derivative. Thanks!
2
votes
1answer
49 views
what is $\lim_{x\to6}\frac{5}{(x-6)^2}$
$$\lim_{x\to6}\frac{5}{(x-6)^2}$$
Is it undefined or infinity and why ?
0
votes
1answer
14 views
Distribution for Response Times
I have samples from a response time population for a web transaction. I want to be able to use them to describe a distribution for the population but don't know a proper one to use. I have shied away ...
3
votes
1answer
66 views
$V=W_1\oplus\cdots\oplus W_k \iff \dim(V)=\sum{\dim(W_i)}$
If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
2
votes
2answers
31 views
Differentiable Operator Continuous?
Consider the space $C^{\infty}[a,b]$ with norm $||f|| = max_{[0,1]} |f(x)|$, with $f\in C^{\infty}[a,b]$. Is the differentiation operator $\frac{d}{dx}$ continuous on $C^{\infty}[a,b]$?
I'm very ...
1
vote
1answer
30 views
RSA encryption theory - modulo theory
I'm a bit mathematically challenged and have been working on the RSA cipher (good start). I can find the public and private keys and know how to work do modulo operations on a calculator. The problem ...
1
vote
1answer
55 views
Show that a function is continuous
Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$.
Show that the function $g$ defined on $\mathbb{R}$ by
$g(t) = ...
1
vote
1answer
35 views
Wealth indicator function for bidder agent logic
I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others).
Given:
Total number of competitors $n$
Amount of all ...
2
votes
1answer
35 views
Inverse image of prime ideal in noncommutative ring
We say an ideal $P$ in a ring $R$ is prime, if for any two ideals $I,J$ of $R$ the following implication holds: if $IJ\subseteq P$ then $I\subseteq P$ or $J\subseteq P$.
If $f\colon R \to S$ is a ...
0
votes
0answers
23 views
Conditional expectations calculation, check my work please.
Let $f_{X,Y}(x,y)=2(x+y)$, for $0<y<x<1$. Find $E[X|Y], E[Y|X]$.
This is purely a calculation, but that's my weakest spot, I always make some stupid mistake that loses me half the points! ...
1
vote
1answer
38 views
Proving that Bombieri's Theorem implies Linnik's theorem
I'm stuck on a line in the proof of Bombieri implies Linnik, where
Bombieri:
For primitive $\chi$ mod $q$ with $q \leq T$ we define $$N(\alpha, T; \chi)=\#\{\rho=\beta+i\gamma \;:\; ...
5
votes
2answers
51 views
Examples of non-cyclic group with a cyclic automorphism group
In introduction to algebra we got the exercise:
Let $G$ be a group. Show that when $\operatorname{Aut}(G)$ is cyclic $G$ is abelian.
This doesn't make that much trouble. Denote the center (all ...
0
votes
3answers
54 views
Generate all possible combinations of 3 digits without repetition
It's possible to generate all possible combinations of 3 digits by counting up from 000 to 999, but this produces some combinations of digits that contain duplicates of the same digit (for example, ...
2
votes
3answers
95 views
Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$
Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that
$$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$
?
I think it is but I can't prove it. Of course if $a_n ...
1
vote
2answers
34 views
Using mathematical induction to prove an identity related to combinatorics
Using Mathematical induction on $k$, prove that for any integer $k\geq 1$,
$$(1-x)^{-k}=\sum_{n\geq 0}\binom{n+k-1}{k-1}x^n$$
How should I proceed? The tutorial teacher attempted this question and ...
1
vote
1answer
25 views
Find the Cramer-Rao bound for an unbiased estimator of $b^2$
$X$ is a RV with pdf $f(x,b) = \frac{x}{b^2} \exp \{-\frac{x^2}{2b^2} \}$
I've got two different estimates: $\hat{b^2} =\frac{2}{\pi} (\frac{1}{n} \sum_{i=1}^n X_i)^2 $ using MME, and $\hat{b^2} = ...
1
vote
1answer
59 views
How are these inequalities simplified?
How does this:
(a > b && a > c && b <= c) ||
(a > b && a <= c && b < c)
simplify down to this:
...
0
votes
2answers
39 views
Find the relation between the dimension of the nullspace of $A$ and $A^t$
Let $A$ be a $n \times n$ matrix, what is the relation between the dimension of the nullspace of the homogeneous system of $A$ and the one of $A^t$?
2
votes
1answer
67 views
Analytically determine that $\arctan x$ is an odd function
Without producing the maclaurin series for $\arctan x$, how would determine whether it was odd or even?
1
vote
0answers
19 views
How to compress a linear operator and have the lossless composition property.
Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
1
vote
0answers
13 views
Partial cycles in projective resolutions of square-free algebra
Short version: Over a square-free algebra must every projective resolution of a simple module eventually terminate or contain a shift of itself as a direct summand?
I suspect not, but have not ...
1
vote
0answers
28 views
Problems sampling from a $pdf$ over $SO\left(3\right)$
I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series:
$$ f\left(\omega,\theta,\phi\right)=\sum ...
3
votes
2answers
37 views
Prove that semi-simple rings are Dedekind-finite
Just to be consistent with the terminology, let me define the words I'm using.
A module $M$ is simple if it has no proper non-zero submodules, and semi-simple if it can be written as a direct sum $M ...
0
votes
0answers
20 views
Prime and semiprime ideals of $A=T_3(D)$, the ring of $3\times 3$ upper triangular matrices over $D$
Let $D$ be a division ring. Could anyone tell me which are the prime and semiprime ideals of $A=T_{3}(D)$, where $A=T_{3}(D)$ the ring of $3\times 3$ upper triangular matrices with coefficients in ...
1
vote
0answers
17 views
Normalizing the Second Moment of $n$ Discs
Consider $n$ non-overlapping discs of diameter $d$ positioned (centred) at $P_1,\dots,P_n$ ($||P_i - P_j||\geq d, i\neq j$).
Graham and Sloane use the second moment as a measure of compactness for ...
2
votes
1answer
55 views
Does the splitting principle define chern classes for vector bundles if they are known for line bundles?
Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$.
Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is ...
1
vote
0answers
21 views
Construction binary tree
First let $\mu$ be the induced distribution of the random variable $X$ on $(\mathbb{R},\mathcal{B})$ and denote $EX=m$.
We also define for all $A\in G_{n+1}$ and $\omega\in X^{-1}(A)$
...
0
votes
0answers
20 views
Lagrange polynomial interpolation on matlab
i am trying to write a code that will input data sets and produce coefficients of the lagrange polynomial.
I have the code below but when i try and run it it tells me that xin is undefined. Can ...
0
votes
0answers
37 views
property of an increasing or decreasing function
For $x \in \mathbb{R}$, and $f(x)$ an increasing function, can we prove whether
$$ af(x)\lesseqgtr f(ax) $$
for $a >0$? If we have additional information that $f$ is homogeneous of some degree, ...
2
votes
1answer
47 views
Calculate the Riemann Stieltjes integral
This is not a homework question. It is a past exam question and I would appreciate some step by step help, as I never understood this concept in class.
Let $\alpha(t) = n^2$ for $t\in[n,n+1).$ ...
1
vote
1answer
45 views
Anonymous graphs and graph embeddedness
What are anonymous graphs, what is graph embeddedness, and how do they relate to each other? Very confused - I could not find short answer. Thanks.
3
votes
2answers
59 views
How to solve $at + b = 0 \pmod {(a-t)}$?
Is there any way (except trying for $t=0,1,2,\ldots,a-1)$ to solve the following equation for $t$ when $a$ and $b$ are known?
$$ at +b = 0 \pmod{(a-t)} \text{ with } a,b,t \in N $$
3
votes
2answers
141 views
Does this polynomial factorise further?
I just did a national exam and this question was in it, I am convinced this does not work:
Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorise this cubic fully.
My attempt
1 | ...
4
votes
2answers
64 views
How do I know when a form represents an integral cohomology class?
Suppose $M$ is an $n$-dimensional manifold, and $\omega \in \Omega^p(M)$ is a closed $p$-form. Moreover, assume that $d\omega = 0$, so that it represents a de Rham cohomology class.
I would like to ...
2
votes
3answers
93 views
Is the set of surjective functions from $\mathbb{N}$ to $\mathbb{N}$ uncountable?
I want to use Cantor's diagonalisation argument to prove that the set S of surjective functions of the form $\Bbb{N} \to \Bbb{N}$ is uncountable. The normal procedure is creating a matrix and filling ...
0
votes
1answer
39 views
Is a series (summation) of continuous functions automatically continuous?
I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a ...
-3
votes
0answers
129 views
Is there a flaw in Hao-cong Wu's Riemann hypothesis paper? [closed]
Hao-cong Wu published a paper in the EUROPEAN JOURNAL OF MATHEMATICAL SCIENCES, titled: Showing How to Imply Proving The Riemann Hypothesis
Is there a flaw in it?
5
votes
1answer
64 views
Why is the Derangement Probability so Close to $\frac{1}{e}$?
A derangement is a permutation of $\sigma$ of $\{1,2,3,\dots,n\}$ such that $\sigma(i) \neq i$ for every $i$. A common application of inclusion/exclusion in undergraduate combinatorics and ...
1
vote
0answers
15 views
Lipschitz constant for $F(t,y,z)=(z,f(y,z)\sin t)$
Let $f\in C^1(\mathbb R^2,\mathbb R)$.
Prove that all solutions for $x''=f(x,x')\sin t$ such that $x(0)=x(2\pi)$ and $x'(0)=x'(2\pi)$ have period $2\pi$.
I'm in the process of solving the above ...
0
votes
1answer
33 views
Can't establish a lower bound on a supremum
I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by
$$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$
for all $k\geq 1,1\leq j\leq k$.
This serves as an example of ...
0
votes
1answer
34 views
About continuous functions and aritmethic progression
I've try solve this question, but I haven't sucess...
The problem is the following:
A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show ...
0
votes
0answers
33 views
Bolza example like Question
I have to find $u$ minimizing $\int_0^1 F$ with $F(x,u(x),u'(x)) = (1-(u'(x))^2)^2+(u(x))^2$ with $u(0) = 0$ and $u(1) = 1$.
I'm relatively new to CoV and got told i should try ...
6
votes
2answers
88 views
A simple 2 grade equations system
If we have:
$$x^2 + xy + y^2 = 25 $$
$$x^2 + xz + z^2 = 49 $$
$$y^2 + yz + z^2 = 64 $$
How do we calculate $$x + y + z$$
3
votes
4answers
73 views
How to prove that $A\cap B\subseteq C$ and $A^c\cap B\subseteq C$ imply that $B\subseteq C$?
How do you solve this problem??
Suppose that $A\cap B\subseteq C$, and $A^c\cap B\subseteq C$. Prove that $B$ is a subset of $C$.
I don't know where even to begin
Can anyone help?
Thank you
2
votes
0answers
32 views
Solve the special integral
I want to solve a integral which contains a shift version
$$\int^{\infty}_{c}N [(1-e^{-1/t})]^{N-1} \frac{-1}{(t-c)^2}e^{-1/(t-c)}dt$$
This kind of integral has the form of normal integral
$$ \int ...
12
votes
1answer
353 views
Can you raise $\pi$ to a real power to make it rational?
We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:
Take $(\sqrt{2})^{\sqrt{2}}$
If it's rational, ...
0
votes
0answers
16 views
Variation of 3SAT is in NP-Complete
Consider the problem of "K-3SAT", a variation of 3SAT: Given a 3CNF formula O and an integer k, the machine determines whether the formula O has a satisfying assignment in which at most k variables ...
1
vote
3answers
34 views
Please help with this probalility problem.
Vince buys a box of candy that consists of six chocolate pieces, four fruit pieces and two mint pieces. He selects three pieces of candy at random without replacement.
Calculate the ...
1
vote
0answers
28 views
expanded geometric series?
I'm having some issues with the following series
$$
\sum_{n \geq 0} n^p r^n
$$
for a fixed positive integer $p$ and some real $r > 0$.
Is there any way to avoid going through linear combinations ...
2
votes
0answers
50 views
Proving identity involving sum
I'm stuck trying to prove the following identity, which is seemingly correct (from mathematica):
$$\displaystyle\sum_{i=j}^k\frac{(-1)^{i+j}{k+1 \choose i}{i \choose ...
