1
vote
3answers
179 views

Maximum order of a sum of functions

I'm being introduced to the Big-O notation via Susanna Epp's Discrete Mathematics with Appplications 3rd edition. The following defintion is stated on page 519: Let f and g be real-valued functions ...
1
vote
1answer
129 views

'Odd' Odd ratio question

Assuming Intervention group to be mothers having c-section and control group to be mothers having natural birth. Also, the numerical data is as follows: ...
3
votes
1answer
499 views

Tensor products

I'm trying to get my head round tensor products of vector spaces (I'm happy to see arguments in a more general setting, though). I am concerned principally with two statements: i) If $U,V,W$ are ...
1
vote
1answer
250 views

Show that if $f^{-1}((\alpha, \infty))$ is open for any $\alpha \in \mathbb{R}$, then $f$ is lower-semicontinuous.

I've tried looking for this question on this site, but I can't seem to find it. But if anyone can direct me to it, that would be great. But i'll pose my question in the mean time. As the title ...
4
votes
5answers
1k views

Analysis Problem: Prove $f$ is bounded on $I$

Let $I=[a,b]$ and let $f:I\to {\mathbb R}$ be a (not necessarily continuous) function with the property that for every $x∈I$, the function $f$ is bounded on a neighborhood $V_{d_x}(x)$ of $x$. Prove ...
-3
votes
1answer
138 views

Van Der Waerden Theorem [closed]

Can someone explain me what's the meaning of the term "l-equivalent" in the following paper: http://www.math.ucsd.edu/~ronspubs/74_01_van_der_waerden.pdf ? I saw the definition at the first lines, ...
4
votes
3answers
366 views

How do I find if $\frac{e^x}{x^3} = 2x + 1$ has an algebraic solution?

Is there some way of solving $$\frac{e^x}{x^3} = 2x + 1 $$ non-numerically? How would I go about proving if there exists a closed form solution? Similarly how would I go about proving if there exists ...
1
vote
1answer
71 views

Computing angle

See the drawing for the situation. Given lenghts a, b and c and also L, but k and angle alpha are unknown. How to compute this angle alpha? I know it is possible to compute if we first compute k in ...
3
votes
3answers
591 views

Gradient of a Mahalanobis distance

Short question: How can I calculate $\dfrac{\partial A}{\partial L}$ where $A = \|Lx\|^2_2= x^TL^TLx$? Is it $\dfrac{\partial A}{\partial L}=2Lx^tx$? Long question: I want to calculate the ...
0
votes
1answer
83 views

$ \mathbb{Z}_{3}\times\mathbb{Z}_{4} $ isomorphic a generated subgroup

I'm trying to prove that the group $\mathbb{Z}_{3} \times \mathbb{Z}_{4}$ is isomorphic to the generated subgroup by $(2,20)$ and $(9,10)$ of $\mathbb{Z}_{12}\times\mathbb{Z}_{40}$ . Here is my ...
0
votes
0answers
115 views

Very Basic Covering Space Q. - Two different covering maps over a same space?

Here's the situation I'm in. I have a map from the (closure of) Upper Half Plane ($\mathbb{U}$) into the punctured (closed) disk ($\mathbb{D}$) called $q$ that satisfies $q(0) = 1$, and $q(z+1) = ...
2
votes
1answer
1k views

Finding points of intersection which are coincident

I was practicing some basic math and I faced a question that I have no idea what does it requires me to do. Consider this simple function: $$f(x) = x^2-5x-12$$ Part a and b where pretty easy. I ...
1
vote
0answers
44 views

Adequate Trial Size Calculation.

Firstly, thanks for answering my last question. I appreciate the help and effort put in. I have another question to put forth. Again, it is related to Medical Statistics. There are one third of ...
3
votes
1answer
669 views

Distance between two gears surrounded by a known-length belt

This question is very similar (but not identical) to this one: Finding the distance between two gears (actually, we are trying to solve it on Bicycle Exchange: ...
1
vote
1answer
116 views

proving a fact about limits

$\text{Let}$$$\lim_{x\to d}h(x)=P,$$$$\lim_{x\to d}k(x)=Q\text{, and}$$$$h(x)\geq k(x)\text{ for all }x\text{ in an open interval containing }d.$$ $\text{Show}$$$P\geq Q$$ $\text{This is what I ...
1
vote
2answers
809 views

Proving $(0,1) $ is not countable

Recall that a countable set $S$ implies that there exists a bijection $\mathbb{N} \to S.$ Now, I consider $(0,1).$ I want to prove by contradiction that $(0,1)$ is not countable. First, I assume ...
4
votes
1answer
710 views

Prove that this sequence does not converge pointwise almost everywhere.

Suppose that $f_n$ is a sequence of real measurable functions on a set $X$ of finite measure, and suppose that there is some $\epsilon$ such that for all $n\geq1$: ...
0
votes
2answers
293 views

Functions involving infinite set -> infinite set and one-to-one and/or onto

Two true or false questions: $\mathbb{Q}^+$ means the positive rational numbers (no 0) $\mathbb{N}$ means all natural numbers Every function $f\colon \mathbb{Q}^+ \to \mathbb{N}$ is not one-to-one ...
0
votes
1answer
94 views

A complex function

I just need some help with the penultimate question of my coursework: Let $w=f(z)=\coth(z/2)$. Show that $w=f(z)=h(C) = (C+1)/(C-1)$ where $C=g(z)=e^z$. Find the image of the given points, ...
1
vote
1answer
214 views

Vandermonde matrix: if c is a solution to $Vx = 0$, then the coefficients $c_1, c_2, \dots, c_n$ must all be zero

Here is my problem: Given a vector x $\in \mathbb{R}^{n+1}$, the $(n + 1) \times (n + 1)$ matrix $V$ defined by $$v_{ij} = \begin{cases} 1 & \text{if } j = 1 \\ x_i^{j - 1} & ...
3
votes
0answers
188 views

Question on Lipschitz condition.

I want help in showing that $f$ is Lipschitz on $[0,1]$ $\implies$ that $f$ can written in the form $$f(x) = f(0) + \int_0^x h(x)~dt$$ for some bounded Lebesgue measurable function $h$ on $[0,1]$. ...
0
votes
2answers
254 views

Find the length (in cm.) of the hypotenuse?

A right angled triangle has sides of length X, Y and Z (all lengths in cm.). It is known that Z is the length of the longest side. The lengths of the other two sides satisfy the inequality ...
4
votes
1answer
270 views

If $f(x) = \frac{4^x}{4^x+2},$ find the value of $\sum \limits_{i=1}^{1999} f\left(\frac{i}{1999}\right) $

If $$f(x) = \frac{4^x}{4^x+2} $$ then find the value of $$f\left(\frac{1}{1999}\right) + f\left(\frac{2}{1999}\right) + f\left(\frac{3}{1999}\right) +\cdots+f\left(\frac{1999}{1999}\right).$$ ...
8
votes
1answer
2k views

A simple bijection between $\mathbb{R}$ and $\mathbb{R}^4$ or $\mathbb{R}^n$?

How to form a bijection from $(0,1]$ to $\mathbb{R}$: $$f(x) = \left\{\begin{array}{ll} 2-\frac{1}{x}&\text{if }x\in(0, .5]\\ \frac{2x-1}{1-x}&\text{if }x\in(.5, 1]. \end{array}\right.$$ ...
4
votes
3answers
497 views

limit of $\lim_{x \to 0}\left ( \frac{1}{x^{2}}-\cot x\right )$

Help me with that problem, please. $$\lim_{x \to 0}\left ( \frac{1}{x^{2}}-\cot x\right )$$
0
votes
2answers
196 views

Evaluate $n$th derivative at 0

I've found this task: evaluate $n$-th derivative at $x=0$ without finding the general formula: $f(x)=\dfrac{1+x+x^2}{1-x+x^2}$, $n=4$ $f(x)= \sqrt[3]{\sin(x^3)}$, $n=9$ It's interesting, how the ...
2
votes
2answers
140 views

Is this idea for a proof that $\mathbb{Q}$ is countable correct?

I first show that there exist a injection $f:\mathbb{Q}\rightarrow \mathbb{Z\times Z}$ and then we know that $\mathbb{Z \times Z}$ is a countable set so we deduce that $\mathbb{Q}$ is countable. And ...
2
votes
1answer
110 views

on generators of $k$-algebra

Let $(A,m,k)$ be a local ring, and $A$ is a finitely generated $k$-algebra, and the maximal ideal $m$ is nilpotent. Let $x_1,\ldots,x_n \in m$ and their canonical images in $m/m^2$ generate this ...
3
votes
2answers
199 views

Variation of the inscribed square problem

The inscribed square problem (summary here) is currently open: Does every Jordan curve admit an inscribed square? (It is not required that the vertices of the square appear along the curve in ...
6
votes
1answer
191 views

Existence of measure zero generating sets of additive real numbers.

Just getting around to posting thoughts I had regarding this question about the additive structure of the real numbers. I was interested in which sets generate $(\mathbb{R},+)$. First, is the ...
3
votes
3answers
169 views

Proving that a natural number is divisible by $3$

I am trying to show that $n^2 \bmod 3 = 0$ implies $n \bmod 3 = 0$. This is a part a calculus course and I don't know anything about numbers theory. Any ideas how it can be done? Thanks!
1
vote
1answer
90 views

Condition for Linear Map to be the Zero Map

Let $(V,\mathbb{K})$ be a vector space over some field $\mathbb{K}$ and let $T:\mathbb{K}\rightarrow V$ be a linear map such that $T(1) = 0$. I am trying to figure out whether it is necessarily the ...
1
vote
1answer
76 views

Does $|a^{-n} J_0(n)|$ converge?

Does the sequence $|a^{-n} J_0(n)|$ converge? I used the approximation $J_0(n) \approx \sqrt{\frac{2}{\pi n}} \cos(n - \pi/4)$ and assume that $a > 1$. Since the sample sequence of $J_0(n)$ is ...
3
votes
1answer
178 views

Is the kernel of a linear form $f : A^n \to A$ a projective module?

Let $R$ be a ring (not necessarily commutative, but if you have interesting things to say in the commutative case, please say them!) and $M$, $M''$ two finitely generated projective ...
2
votes
1answer
237 views

Set of intervals on the real line

I'm not sure how to approach this proof? any ideas Let $A$ be a set of intervals of the real line any two of which are disjoint - in other words, if $(a,b)$ and $(x,y)$ are distinct elements of $A$ ...
4
votes
1answer
476 views

On sums involving Euler's totient function

I've been struggling with the following claim without being able to prove it, so your help would be highly appreciated: Let $\varphi(n)$ be Euler's totient function. Show that there is a constant ...
2
votes
1answer
174 views

What are the relationships between combinatorics and randomness?

I was just reading the impressive paper by Tim Gowers The Two Cultures of Mathematics when I noticed the various connections between combinatorics and randomness. As a non-mathematician, it is not ...
4
votes
1answer
232 views

Lie algebra associated with the orthogonal group $\operatorname{SO}(2n)$?

How do I prove that the lie algebra associated with the even dimensional orthogonal group $\operatorname{SO}(2n)$ is given by matrices $B$ satisfying $B^\top K + KB = 0$, where $K = U^\top U$, $U$ ...
0
votes
1answer
108 views

Does a Jordan chain always start with an eigenvector?

Does a Jordan chain always start with an eigenvector? If so when computing a Jordan chain for a particular matrix do you have to start with an eigenvector, $v_0$ for particular eigenvalue then just go ...
4
votes
1answer
93 views

Asymptotics for almost all $x$

Theorem 2.2 in Shparlinski 2006 says: For all positive integers $n\le x$ except possibly $o(x)$ of them, the bound $$M(x)\ll\frac{x}{\log x}\exp\left((C+o(1))(\log\log\log x)^2\right)$$ holds. ...
7
votes
2answers
648 views

Linear functional on a Banach space is discontinuous then its nullspace is dense.

I need to prove that: If a nonzero linear functional $f$ on a Banach Space $X$ is discontinuous then the nullspace $N_f$ is dense in $X$. To prove that $N_f$ is dense, it suffices to show that ...
3
votes
1answer
810 views

Lie algebra associated with the symplectic group Sp(2n)?

The group Sp(2m) consists of the 2m×2m matrices A with the property At (A transpose)JA = J, where J is the 2mX2m standard skew-symmetric matrix. How do I prove that the lie algebra associated with has ...
1
vote
1answer
423 views

Natural basis for set of m x n matrices, visualized

I am told to consider the $m \times n$ matrices $E_{pq}$ described by: $[E_{pq}]_{ij} = \left\{ \begin{array}{ll} 1 & \textrm{if } i=p\textrm {, }j=q \\ 0 & \textrm{otherwise} \end{array} ...
0
votes
1answer
652 views

UMVUE of parameter $\frac{1}{1+\lambda}$

suppose $X_1,X_2,X_3$ are a random sample of exponential distribution with parameter $\lambda$. how can find UMVUE parameter $\frac{1}{1+\lambda}$. note: $(T=\displaystyle\sum_{i=1}^{3}X_i,\ ...
1
vote
2answers
5k views

Error propagation on weighted mean

I understand that, if errors are random and independent, the addition (or difference) of two measured quantities, say $x$ and $y$, is equal to the quadratic sum of the two errors. In other words, the ...
1
vote
1answer
93 views

Derivation of Borel measures w.r.t a “bigger” measure?

Given two Borel measures $\mu_1$ and $\mu_2$ on $\mathbb R$, is there always a Borel measure $\mu$ on $\mathbb R$ such that $$ d\mu_1=w_1 d\mu,\qquad d\mu_2=w_2 d\mu, $$ for some functions $w_1$ ...
-1
votes
1answer
153 views

Big Omega equation

I am struggling still with this equations...from my class materials.... This time we deal with lower bound -> BIG OMEGA: I know that: $$\Omega(g(n)) = \{f(n) : \exists c, n_0 > 0\,\forall n\ge ...
3
votes
3answers
182 views

Orthogonality relations of Characters

Could somebody please help me understand the jump from Proposition 10 to Proposition 11 in the following http://www.ms.uky.edu/~pkoester/research/charactersums.pdf Note: The orthogonality relations ...
3
votes
1answer
224 views

Two questions about generating functions

I'm very interested in the topic of generating functions, so I have two questions: I just realized that when I have exponential generating function for example $F(x)=e^{e^x-1}$, I can take n-th ...
11
votes
1answer
2k views

Why is every meromorphic function on $\hat{\mathbb{C}}$ a rational function?

I know that an analytic function on $\mathbb{C}$ with a nonessential singularity at $\infty$ is necessarily a polynomial. Now consider a meromorphic function $f$ on the extended complex plane ...

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