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H: Probability inequality for sum of two random variables
Does anyone know if the following inequalities are right? If yes, what is the reference for them?
For random variable $x$ and $y$:
\begin{equation}
\mathbb{P}(x+y \ge x_0 + y_0) \le \mathbb{P}(\{x\ge x_0\} \text{or} \{y\ge y_0\}).
\end{equation}
AI: It should b... |
H: Find closed form for $1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$
Is there any closed form for the following?
$$1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$$
I tried to find one, but I failed.
I saw solution on Wolfram Alpha, but I didn't understand it:
Generating function: $$\mathcal G_n(a_n)(... |
H: Discrete dynamic models
We have the equation
$$x_{n+1} = ax_n(1-x_n) - v_n$$
Why are there only fixed points for $(a-1)^2 - 4av_0 \geq 0$?
Show that if $ 1<a<4$, there are 2 fixed points with $0<p_1 < p_2 <1$
For the first problem, I was able to calculate that the fixed points would be given by:
$$ p_{1,2} = \dfr... |
H: calculate riemann sum of sin to proof limit proposition
$$\lim_{n \to \infty}\frac1n\sum_{k=1}^n\sin(\frac{k\pi}{n})$$
I'm having trouble expressing $\sin(x)$ differently here in order to calculate the riemann sum.
I want to show that this converges to $\frac{2}{\pi}$ so it equals to $\int_0^1 \sin(x\pi)$.
Is there... |
H: What does $\sum_{n\in\mathbb N}\frac{n^x}{x^n}$ converge to when $x\in\mathbb R^+$ and $x>1$?
What does $\sum_{n\in\mathbb N}\frac{n^x}{x^n}$ converge to when $x\in\mathbb R^+$ and $x>1$?
I'm looking for a hint of how to tackle this problem.
AI: From the definition of the polylogarithm this is :
$$\sum_{n\in\math... |
H: Non symmetric urn and balls problem.
Consider an urn containing $N$ balls, of which m are white and $N - m$ are black. Balls are randomly selected from the urn according to the following rule:
If a black balls is selected, it is "observed" and the it is returned to the urn.
If a white balls is selected, it is "obs... |
H: Closed rectangle contained in set that does not have measure zero.
I was doing some analysis when it occurred to me that if I knew the answer to the following question many of the exercises would be simpler. However, I haven't been able to make much headway. Any help will be appreciated.
Question:
If I have a set ... |
H: (1)Questions about differentiable functions
1)The functions $f$ and $g$: $\mathbb{R} \rightarrow \mathbb{R} $ shall be 3-times differentiable.
Calculate $(f \cdot g)^{(3)}$.
1) $(f \cdot g)'=(f'g+fg')$
$(f'g+fg')'= (f''g+f'g')+(f'g'+fg'')= f''g+2f'g'+fg''$
$(f''g+2f'g'+fg'')'=(f'''g+f''g')+2(f''g'+f'g'')+(f'g''+f... |
H: Subset of Cantor set that isn't compact
How to prove that the Cantor set has a subset that is not compact? Actually, I want to prove that every infinite set $X\subset\mathbb{R}^n$ has a subset $Y$ that is not compact. If $X$ isn't bounded, then $X$ has a unbounded subset $Y$ that is not compact. If $X$ is bounded a... |
H: Showing that $\frac{2}{3\pi}\leq\int_{2\pi}^{3\pi}\frac{\sin x}{x} \, dx \leq \frac{1}{\pi}$
Please help showing that,
$$\frac{2}{3\pi}\leq\int_{2\pi}^{3\pi}\frac{\sin x}{x} \, dx \leq \frac{1}{\pi}$$
AI: Make use of the fact that $\dfrac1x$ is a decreasing function, to conclude what you want, i.e., for $0<a<b$ we ... |
H: Proving LHS and RHS
I just came across this problem from proving an equality for combinatorics. In general if I ask you to prove
Prove that
Expression X = Expression Y
Normally wouldn't one read from left to right and start showing the left is equal to the right?
Then I thought of Euclid's elements and the defini... |
H: Can I derive $i^2 \neq 1$ from a presentation $\langle i, j \mid i^4 = j^4 = 1, ij = j^3 i\rangle$ of Quaternion group $Q$?
(This question is related to the previous post I've posted few hours ago: (Dummit's AA, 1.5, P3) Are these presentations of the Quarternion group equivalent?)
I was trying to prove that the pr... |
H: Weakly closed implies sequentially closed
Another problem involving the weak topology:
Let $X$ be a normed space and $A \subset X$ weakly closed. Then $A$ is sequentially closed, that is: If $(x_n) \subset A$ and $x_n \xrightarrow{w}x$, then $x \in A$.
I know this characterisation is also used as definition of weak... |
H: Union of connected subsets is connected if intersection is nonempty
Let $\mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $\bigcap\mathscr{F}\ne\emptyset$. Prove that $\bigcup\mathscr{F}$ is connected.
If $\bigcup\mathscr{F}$ is not connected, then it can be partitioned into two d... |
H: problem with partial fraction decomposition
I want to do partial fraction decomposition on the following rational function:
$$\frac{1}{x^2(1+x^2)^3}$$
So I proceed as follows:
$$\begin{align}
\frac{1}{x^2(1+x^2)^3} &= \frac{A}{x} + \frac{B}{x^2} + \frac{Cx + D}{1 + x^2} + \frac{Ex + F}{(1 + x^2)^2} + \frac{Gx + H}{... |
H: A subspace of $\mathcal{B}(X;Y)$ isometric to $Y$
Let $X$ and $Y$ be normed spaces. If $\mathcal{B}(X,Y)$ is the space of all continuous linear functions of $^{X}Y$, I'm trying to prove that $\mathcal{B}(X,Y)$ has a closed subspace isometric to $Y$. Can someone give me only a clue.
AI: Hint: On a basis $e_n$ of $X$... |
H: Prove that $\{x_n\}_n$ is convergent.
Here is an exercise:
Let $x_n=1+2+\dots+\frac1n-\ln n$. Prove that $\{x_n\}_n$ is convergent.
(I believe that this can be found in the site, however I cannot find immediately, so I post it here.)
The hints are much appreciated. I don't want complete proof.
Thanks for your hel... |
H: Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$
Be a continuous function $f:[0,1] \rightarrow [0,1]$.
Show that there is a $\zeta \in [0,1]$ with $f(\zeta)=\zeta$ ($\zeta$ is called fixed point).
Consider the function $g:[0,1] \rightarrow [-1,1]$, $g(x):= f(x)-x$.
$g$ is continuous.
B... |
H: Jordan Canonical Form of a matrix
Determine the Jordan canonical form of the matrix: $\quad\begin{bmatrix} -7 & 9 \\ -4 & 5 \end{bmatrix}$
AI: Hints, to find the Jordan Normal Form, we can use the following approach (and others are possible too):
(1) Find the eigenvalues (we have a multiplicity $2$ eigenvalue $\la... |
H: Probabilty of A and D but not B and not C
I am trying to work a probability question and I am stumped.
Imagine there are 4 signs on the side of the road and each sign has a "chance of being seen" equal to the following
sign A: .75
sign B: .82
sign C: .87
sign D: .9
What are the chances of A AND D being seen, but NO... |
H: How should I handle the limit with the floor function?
I wanted to show that
$$ \lim_{n\to\infty} \bigg( \frac{n+\lfloor x \sqrt{n} \rfloor}{n-\lfloor x \sqrt{n} \rfloor}\bigg)^{\lfloor x \sqrt{n} \rfloor} =e^{2x^2} $$
After applying $x \sqrt{n} -1 \leq \lfloor x \sqrt{n} \rfloor \leq x \sqrt{n}$ and replacing $\s... |
H: Solving limits of trig functions.
a) $$\lim_{x\to\pi/2^+} \frac{\ln(x-\pi/2)}{\tan(x)}$$
b) $$\lim_{x\to 0^+} \frac{\ln(x)}{\arctan(x)}$$
How does one approach this question?
AI: Second Question: As $x\to 0^+$, $\arctan x$ approaches $0$ through positive values. In the meantime, $\ln x$ becomes very large negativ... |
H: Verification and hint for my answers to a basic statistics table
Sorry if this question seems very basic. I tried to find the answer by goolging. But because I do not know the correct keyword cannot find anything. This is a basic statistics problem. I attempted questions but I need them to get verified. I also do n... |
H: Can the semidirect product of two groups be abelian group?
while I was working through the examples of semidirect products of Dummit and Foote, I thought that it's possible to show that any semdirect product of two groups can't be abelian if the this semidirect product is not the direct product.
Here is my simple i... |
H: Let $\alpha$ be an ordinal then $\alpha=\cup\alpha$ or $\alpha=s(\cup\alpha)$.
Let $\alpha$ be an ordinal then $\alpha=\cup\alpha$ or $\alpha=s(\cup\alpha)$.
Attempt:
Since $\alpha$ is transitive $\alpha=\cup\alpha$ (here we are done) or $\alpha\supsetneq\cup\alpha$.
If $\alpha\supsetneq\cup\alpha$, since $\cup\a... |
H: Normalizing a continuous distribution
I work at a help/tutoring center at my university. Today a kid came in with this problem. I've only studied math and haven't drifted into physics, but he had this problem:
Let $P(x)=Ne^{-\frac{|x|}{a}}$. Then:
(a) Find $N$ such that $P(x)$ is properly normalized.
(b) Fi... |
H: Triple Scalar Product $=0\implies $Spanning a Plane? (and Checking for Colinearity)
If I have three vectors, $\vec{a},\vec{b},\vec{c}$, and their scalar triple product equals zero, that is $\vec{a}\times \vec{b} \cdot \vec{c}=0$, then I understand that it means the vectors are coplanar, and so they can not span $\m... |
H: Given governing equations, determine the condition for a solution and if they exist, solve the system.
Given the governing equations:
1a + 2b = 2c
2a + 4b = d
2a + 5b = e
3a + 9b = f
What conditions are required for a solution x = [a,b]^T to exist?
If these conditions exist, solve the system for a and b.
I set up A... |
H: Tensor Product is distributive.
Tensor Product is distributive.
I get stuck at the proof:
If $T$ is a $p$-tensor and $S, U$ a $q$ tensor. Then I need to show that
$$ T \otimes (S \cdot U) = (T \otimes U) \cdot (S \otimes U).$$
Denote $S(u_1, \ldots, u_q) \cdot U(w_1, \ldots, w_q) = S \cdot U (v_1, \ldots, v_q)$. ... |
H: Prove that $\exp(x)=3x$ has at least one solution for $x\in [0,1]$
Prove that $\exp(x)=3x$ has at least one solution $x \in [0,1]$.
$$e^x=3x$$
$$\Leftrightarrow e^x-3x=0$$
Let $$f(x) = e^x - 3x$$
$$f(0)=e^0 - 3 \cdot 0 = 1 > 0$$
$$f(1)= e^1-3 \cdot 1 = e - 3 < 0$$
Thus, since $f(1) < 0 < f(0) $, by the IVT:
$$\exi... |
H: Prove "casting out nines" of an integer is equivalent to that integer modulo 9
Let $s(x)$ be an abstraction for casting out nines of integer $x$. For all integers $x$, prove $s(x) \equiv x$ mod $9$
I'm not asking for an answer more of a way to attack this problem. Can't think of where to start
AI: Hint: Because $10... |
H: Why the two results are different?
I'm trying to evaluate $\int_C\dfrac{z+1}{z^2+9}$ where $C$ is a positively oriented simple closed contour containning $z=\pm3i.$
Method 1: Let $f(z)=\dfrac{z+1}{z^2+9}.$ Then the given integral is equal to $2\pi i\text{ Res}_{z=0}\dfrac{1}{z^2}f\left(\dfrac{1}{z}\right)$$=2\pi i... |
H: Simple yet tricky trigonometry
This might seem silly to ask, but how can I solve a trigonometry problem for the unknown $h$ in the form of:
$x + 45 = h/\tan 30$ and
$x = h/\tan 50$
AI: Since $h$ is the variable we are solving for, express both equations as functions of $h$:
$$x + 45 = \dfrac h{\tan (30^\circ)} \i... |
H: Proof of bipartite graph formula
I've come across a question that has got be stuck for hours.
I need to proof that:
Let $G$ be a graph $=(V,E)$, a bipartite graph with $n$ vertices and $e$ edges. Show that $$e\leqslant \left\lceil \frac{n}{2} \right\rceil \cdot \left\lfloor \frac{n}{2} \right\rfloor\;.$$
I can't ... |
H: Map values 0:90 to 2:0
I have values that range from 0 and 90, and I need to map their values to between 2 and 0. Examples would be 90 = 2, 0 = 0, 45 = 1. How can I go about getting these values converted down to the proper range?
Edit: I messed up on my examples.
Examples should be:
90 = 0, 0 = 2, 45 - 1
AI: Use a... |
H: Handshake problem. discrete math.
at a party, 25 guests mingle and shake hands. prove that at least one guest must have shaken hands with an even number of guests.
AI: HINT: Number the guests $1$ through $25$. Let $a_k$ be the number of guests with whom guest $k$ shook hands. Suppose that the numbers $a_1,a_2,\dots... |
H: Show that $(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$.
If $x>0,y>0,z>0$ and $x+y+z=1$, prove that $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z).$$
Trial: Here $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z) \\ \implies (1+x)(1+y)(1+z)\ge 8(y+z)(x+z)(x+y)$$ I am unable to solve the problem. Please help.
AI: $$(1+x)(1+y)(1+z) \ge 8(1-x)... |
H: how do you determine the derivative is true or false?
$$\frac{d}{dx} \cos(y^2)=-2y \sin(y^2)$$
I need to find out if this statement is true or false. My answer is false, but I'm not sure.It is false because the variables disagree.
AI: More formally, use the Chain Rule:
$$
\frac{d\cos(y^2)}{dx} = -\sin(y^2) \frac{d ... |
H: Proving a lemma about the union of sets, linear independence, spans
OK, here’s another lemma I’m being asked to prove and I am trying to see if I am in the right ballpark.
Let $L = \{y, y_1, y_2, y_3, \ldots, y_m\}$ be a linearly independent subset of $V$, a vector space over field $K$.
Let $S = \{x_1, x_2, \ldots,... |
H: Closed and bounded subsets of a complete metric space
Let $\{X_n\}$ be a sequence of closed and bounded subsets of a complete metric space such that $X_n\supset X_{n+1}$ for every positive integer $n$ and $\lim_{n\rightarrow\infty}(\text{diam }X_n)=0$. Prove that $\bigcap_{n=1}^{\infty}X_n$ contains exactly one po... |
H: Aquaintance problem in discrete math. induction proof.
I'm supposed to prove this by induction. I already proved it by contradiction, but I am lost on how to set it up for induction.
Prove that if at least two people are at a party, at least two of them know the same number of people. We assume that if person 1... |
H: changing summation limits
Let us say we have the following summation
$$\tag 0 \sum_{k=0}^{\infty }\sum_{l=0}^{\infty }g_{k}h_{l}\delta (t-(l+k)T)$$
Now, we let $n = l + k$. Then $l = n - k$ which becomes
$$\tag 1 \sum_{n=0}^{\infty }(\sum_{k=0}^{n }g_{k}h_{n-k})\delta (t-nT)$$
Note: (1) is the correct answer acco... |
H: Estimating a series $\sum_{k=1}^\infty e^{-k^2t}\leq \frac{1}{2}\sqrt{\frac{\pi}{t}}?$
Can we prove such an estimate $$\sum_{k=1}^\infty e^{-k^2t}\leq \frac{1}{2}\sqrt{\frac{\pi}{t}}?$$ I need it in my research....
AI: HINT
Note that for a monotonically decreasing function, we have that
$$\sum_{k=1}^{\infty} f(k) \... |
H: Calculate the probability of two teams have been drawns
If we know that team A had a $39\%$ chance of winning and team B $43\%$ chance of winning, how we can calculate the probability of the teams drawn?
My textbook mention the answer but I cannot understand the logic behind it. The answer is $18\%$. As working is ... |
H: Simple question about splitting fields
Let f(x) be a polynomial in F[x]. Let K be the splitting field of f over F.
Let a be an element of F. Is K also the splitting field of (x-a)f(x)? I think it should be.
I just want to make sure I'm not being stupid!
AI: Yes, you're right.
The splitting field of $f\in F[x]$ is ... |
H: Integration of complex function?
I have to find $$\int_C \frac{dz}{(z^2+9)^2} \,dz$$ if
C is the circle with the radius 3 and with the center at the point 2i.
Now,I know how to find the above integral. $$\int_C \frac{dz}{(z-a)^n} \,dz$$ But I have no clue how to solve and find this one.HELP :/
AI: Just consider ... |
H: Commutativity of reals using Cauchy seq.
I am trying to prove that real numbers are commutative using the definition of real numbers as the equivalence class of Cauchy sequences of rational numbers.
The following is what I know and what I plan.
RTP If $x,y \in \Bbb R$, then $x+y = y+x$ and $xy = yx$.
My understand... |
H: Is a normed topological space metrizable?
As stated in the title:
If there is a norm on a topological space, then we get a metric induced by the norm.
Is this true?
AI: Yes: if $\|\cdot\|$ is a norm on $X$, then $d(x,y)=\|x-y\|$ is a metric on $X$. See this Wikipedia article, especially this section to get starte... |
H: Normal subgroups of $\langle(123),(456),(23)(56)\rangle$
Let $G$ be a subgroup of the symmetric group $S_6$ given by
$G=\langle(123),(456),(23)(56)\rangle$. Show that $G$ has four normal
subgroups of order 3.
I may be missing something, but I can only find two of them: $\langle(123)\rangle$ and $\langle (456)... |
H: Are the subsets of homeomorphic spaces also homeomorphic?
Let $W$ be a subset of an $n$-dimensional complex topology vector space $Y$ such that $0\notin W$.We have known that $Y$ is homeomorphic to $C^n$ and let $S$ be the unit sphere of $C^n$.Can anyone show me that $W$ is homeomorphic to $S$?
AI: This is certai... |
H: Does ${\rm Res}_{z=z_0}\frac{p(z)}{q(z)}=\frac{p(z_0)}{q^{(m)}(z_0)}$ for all $m\ge 1?$
I know the following result involving pole:
If $p(z),q(z)$ be two functions analytic at $z=z_0$ and $p(z_0)\ne0$ and $q(z)$ has a zero of order $m$ at $z_0$ (i.e. $q(z)=(z-z_0)^mf(z)$ where $f(z)$ is analytic and nonzero at $z... |
H: Convergence of the sum of the subsequence $\sum_{k=1}^{\infty } 2^{k}x_{n_{k}}$
Let $\{x_{n}\}$ be a sequence such that $\lim_{n \to ∞}x_{n}=0$. Prove that there exists a sub sequence $\{x_{n_{k}}\}$ of $\{x_{n}\}$ such that $\sum_{k=1}^{\infty } 2^{k}x_{n_{k}}$
converges and $|\sum_{k=1}^{\infty } 2^{k}x_{n_{k}}|$... |
H: A first order theory whose finite models are exactly the $\Bbb F_p$
Is there a first order theory $T$ in the language of rings such that its finite models are exactly the fields $\Bbb F_p$ with $p$ prime (but no $\Bbb F_q$ with $q$ a proper prime power is a model of $T$)?
EDIT: Since this question turned out to be ... |
H: Evaluating the derivative of a Cantor-Vitali function
Let $\varphi \colon [0,1] \to \mathbb R$ a "Cantor-Vitali function", viz. take $x \in [0,1)$ and write it as
$$
x = \sum_{j=1}^{\infty} \frac{a_j}{3^j}, \quad a_j \in \{0,1,2\}
$$
with $a_j$ non definitely equal to 2. Then
$$
\varphi (x) := \begin{cases}
\sum_{... |
H: Fourier analysis on finite abelian groups
Can someone help me show if $f$ is a character of a finite abelian group then for all $a\in G$,
$$\sum_{[f]}f(a)\stackrel{}{=}
\begin{cases}
|G| & \text{if $ a$ is the identity} \\
0 & \text{otherwise}
\end{cases}$$
Where the sum runs over all characters of $G$,
I was ... |
H: A generalization of abelian categories including Grp
The category of groups shares various properties with abelian categories. For example, the Five lemma and Nine lemma hold in Grp. Is there a weakened notion of abelian category which also includes Grp such that the 5- and 9- lemmas are still provable by arrow cha... |
H: A sequence such that every rational is written infinitely
Since $\Bbb Q $ is a denumerable set $ \Bbb Q =\{ r_n: r \in \Bbb Q \}$ where $r_m \neq r_n $ if $ n \neq m $
Find a sequence $\langle x_n\rangle$ of real numbers such that for each $ n \in \Bbb N $, $\{m\in \Bbb N : x_m=r_n\}$ is infinite.
I found this diff... |
H: Inf of the measure of the neighbourhood of a compact set
Let $X$ bea compact metric space, and $\mu$ a Borel probability measure.
For $A\subset X$ de define $b_{\epsilon}(A):=\left\{ x\mid d(x,A)\leq
\epsilon\right\} .$
Suppose $A$ is compact and $\mu(A)=0,$ does this mean $\inf\mu(b_{\epsilon
}(A))=0?$
If this ... |
H: Unprovability of $i^2 =1$ from $\langle i \mid i^4 =1\rangle$ and similar problems
This question is related to Can I derive $i^2 \neq 1$ from a presentation $\langle i, j \mid i^4 = j^4 = 1, ij = j^3 i\rangle$ of Quaternion group $Q$?
I know I'm going too far but let me just ask...
1) Is indeed $\langle i \mid i^4 ... |
H: Consistency Trapezoidal rule
I want to prove that the consistency order of the trapezoidal rule is actually second order, that means that the error of the actual solution $x(t_{k+1})$ where we can restrict ourselves to the equation $x'(t)=\lambda x(t)$ and the approximation in each step $x_{k+1}=x(t_k)+\frac{\lambd... |
H: Taylor series of $ f = e^{x^2 + y^2}$ near $(0,0)$
I have to compute the second order Taylor series of the function
$ f = e^{x^2 + y^2}$ near $(0,0)$.
The Jacobian is:
$$ Df(x,y) = (2\ x\ e^{x^2 + y^2}, 2\ y\ e^{x^2 + y^2}) $$
and the Hessian:
$$ D^2f(x,y) = \left( \begin{array}{cc} 4\ x^2\ e^{x^2 + y^2} & 4\ x\ y\... |
H: Why does the limit of $ f(x) = \int_{0}^{x} e^{-t^2/2} dt$ exist?
For a function
$$ f(x) = \int_{0}^{x} e^{-t^2/2} dt$$
for $x \ge 0$ one has to argue whether $ \lim_{x \to \infty} f(x) $ exists or not.
I thought about the following:
$$ f'(x) = e ^ {-x^2/2} $$
$$ f''(x) = -x\ e ^ {-x^2/2} $$
Given $x \ge 0$, it's o... |
H: How many collections of four numbers, from 1 to 25, have a sum multiple of 5?
How many collections of four numbers, from 1 to 25, added together are a multiple of 5?
I.E the collection $${18, 23, 24, 25}$$ has a sum of 90, which is the maximum sum multiple of 5 that can be constructed with unique numbers from 1 to ... |
H: The difference between inside and outside
If I parameterise $\mathbb{R}^n$ with generalised polar coordinates $(r, \Theta)$ it is possible to partition $\mathbb{R}^n$ into three parts
$$A = \{x \in \mathbb{R}^n \mid r < 1\}$$
$$B = \{x \in \mathbb{R}^n \mid r = 1\}$$
$$C = \{x \in \mathbb{R}^n \mid r > 1\}$$
$A$ is... |
H: Find unitary matrix $U$ such that $U^*AU$ is diagonal, where $A = \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$
Given:
$$A = \begin{pmatrix} 1 & -i \\ i & 1 \end{pmatrix}$$
Find a unitary matrix $U$ such that $U^{*} A U = D$ where $D$ is diagonal.
Now the eigenvalues of $A$ are $0$ with eigenvector $(i,1)$ and $2$... |
H: Prove sequence is not convergent in a complete metric space
I'm looking over some previous analysis exams and I've come across this question:
Consider the set $\mathbb{R}$ of real numbers and the metric function
defined as:
$$d(x, y) = \begin{cases} 1 ~ \text{if} ~ x \ne y \\ 0 ~ \text{if} ~ x
= y \end{cases}$$
... |
H: Find a number $A$ so that $\lfloor A^{3^n} \rfloor$ are always odd
Find a number $A$ so that
(1) $\lfloor A^{3^n} \rfloor$ is always odd for $n\geq 1$;($\lfloor x \rfloor$ is the largest integer not greater than $x.$)
(2) $A>1$ and $A^{3^n}$ is never an integer for $n\geq 1$;(This is the place I have edited.)
(3... |
H: What is the difference between a population and a data set?
A data set is defined as a collection of data, while a population is defined as a collection of items under consideration.
What is the main difference between a data set and a population?
I have been seeing these two terms being used interchangeablely.
AI... |
H: Regularity of a solution of Laplace equation
Assume $\Omega$ is some open, bounded domain with smooth boundary - say $\Omega = B(0,1) \subset \mathbb{R}^3$. Let $v$ be a solution of the Laplace equation \begin{equation} \begin{cases} \Delta v =0 & \mbox{on } \Omega \\ v=f|_{\partial\Omega} & \mbox{on } \partial \Om... |
H: For a fixed and small $\epsilon$, finding the number of real roots of $x^{2}+e^{-\epsilon x}-2+\sin(\epsilon x)$
I saw the following question in an introduction to applied mathematics
exam (this is only the first part of the question):
Assume $0<\epsilon\ll1$ . Denote $$ f(x,\epsilon):=x^{2}+e^{-\epsilon
x}-2+\s... |
H: A small calculation .
How
$\sum_{k=0}^n (-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^k$
i got it
$\sum_{k=0}^n(-1)^n\times(-1)^{n-k}=\sum_{k=0}^n(-1)^{2n-k}$
And
is that $\mathbb E[\mathbb E(X)]=\mathbb E(X)$ ?
AI: Ad 1.: $(-1)^{2n-k} = (-1)^{2(n-k)+k} = ((-1)^2)^{n-k}(-1)^k = 1^{n-k}(-1)^k = (-1)^k$. So:
$$\sum_{k=0}^n... |
H: How to compute $\sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$?
I don't know how to compute $\sum_{n=0}^{\infty}\frac{(2n+1)}{n!}x^{2n+1}$,appreciate any help!
Is there any general rule for solving such problems?
AI: $$e^x=\sum_{n=0}^\infty\frac{x^n}{n!}$$
$$xe^{x^2}=\sum_{n=0}^\infty\frac{x^{2n+1}}{n!}$$
$$\frac{d}... |
H: Riesz-Representation theorem for a special class of functions
The original riesz representation theorem states
Let $X$ be locally compact hausdorff space. Then for any nonnegative functional $\Lambda$ on $C_c(X)$, there is a unique regular borel measure $\mu$ on $X$ such that $$\Lambda(f)=\int f\mu(dx)$$
for al... |
H: This set of ordinals is $\in$-transitive?
I was reading this problema about ordinals. I'm following the Z.F theory.
Let $A$ be a nonempty set of ordinals. Prove that if $\cup A \notin A$ then $\cup A$ it's a limit ordinal, i.e $\cup A $ is nontempty but that is directly, and also is not a sucesor of any ordinal.
I ... |
H: Finding a formula to a given $\sum$ using generating functions
Find a close formula to the sum $\sum_{k=0}^{n}k\cdot 5^k$
I tried using generating functions using the differences sequences with no luck.
AI: $$\sum_{k=0}^nx^k=\frac{x^{n+1}-1}{x-1}$$
$$\sum_{k=0}^n kx^{k-1}=\frac{d}{dx}\frac{x^{n+1}-1}{x-1}$$
$$\sum_... |
H: Silverman Adv. Topics example
I would like to refer you to Silverman's Advanced Topics in the Arithmetic of Elliptic Curves example 10.6:
Let $D$ be a nonzero integer, $E:y^2=x^3+D$ with complex multiplication by $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{-3})$. Let $\mathfrak{p}$ be a prime of $\mathcal{O}_K$ with ... |
H: Fourier Integral problem
Let $f(x)= \begin{cases} 3, & x \in [0,\pi]\\ 2x, & x \in(\pi,2\pi]
\\ 0, & x > 2 \pi \end{cases} $
Express $f$ as a Fourier integral.
I don't know which type of integral I'm supposed to use it. Normally I'd think they mean:
$$f(x) = \int_{0}^{\infty} a(u) \cos ux + b(u) \sin ux \ du$$
Wher... |
H: Where does the minimal polynomial lie?
Let $\theta = a + b \sqrt{D_1} + c \sqrt{D_2} + d \sqrt{D_1 D_2}$, where $a,b,c,d,D_1,D_2$ are integers. Is there any reason to believe the minimal polynomial for $\theta$ over $\mathbb{Q}$ should be an element of $\mathbb{Z}[x]$, without resorting to the notion of Galois conj... |
H: What is the maximum possible value of determinant of a matrix whose entries either 0 or 1?
My question is simply the title:
What is the maximum possible value of determinant of a matrix whose entries either 0 or 1 ?
AI: Quoting my question in another thread:
In fact, I don't even know how large the determinant of... |
H: Finding a function with properties
I am looking for a function $f(x)$ with the following properties:
Positive for $x\in(-\infty, 0)$ but tangent to the x-axis at $x=-1$
A root at $x=0$ and negative for $x\in(0, 2)$
A root at $x=2$ and positive for $x\in(2, \infty)$
I thought $f(x)=x(x-2)(x+1)^2$ would do the tric... |
H: Embedding torsion-free abelian groups into $\mathbb Q^n$?
Glass' Partially Ordered Groups states without proof:
Every torsion-free abelian group can be embedded into a rational vector space (as a group).
Can someone link me to a proof of this? It seems to me like it's probably false: $\mathbb Q^n$ is countable, ... |
H: Clearing up a property of orthonormal basis
I'm learning the concept of principal component analysis and in one explanation I found the following statement:
Let's take an arbitrary $n$-dimensional vector $\textbf{v} = (v_1, ..., v_n)$ and orthonormal basis $\textbf{e}_1, \textbf{e}_2, ..., \textbf{e}_n$. We can wri... |
H: What is a "foo" in category theory?
While browsing through several pages of nlab(mainly on n-Categories), I encountered the notion "foo" several times. However, there seems to be article on nlab about this notion. Is this some kind of category theorist slang? Please explain to me what this term means.
AI: It's sla... |
H: Find all generators of $ (\mathbb{Z}_{27})^{\times} $
"Find all generators of $ (\mathbb{Z}_{27})^{\times} $"
My attempt is below.
Since $ (\mathbb{Z}_{n})^{\times} $ is a cyclic if and only if $ n = 1, 2, 4, p^n, 2p^n $, $ (\mathbb{Z}_{27})^{\times} $ is cyclic.
And the order of $ (\mathbb{Z}_{27})^{\times} $ is $... |
H: Define parameter in linear system to get different number of solutions
For what values of the parameter $t$ does the following system of
equations have
no solution
more than one solution
exactly one solution
$$ (I):x+y+t\cdot z=-1$$ $$(II):3x+(t+1)y+(t-1)z=-1$$
$$(III):tx+2y+z=0$$
I'm not quite sure how to... |
H: Proving minima, maxima, and saddle points don't exist.
I've got the next function: $f(x,y)=x^3y^3$ where $x,y\in \mathbb R$
I need to determine whether there is minima, maxima or saddle point.
Easily enough, after doing the partial derivatives
$f_x'(x,y)=3x^2y^3$
$f_y'(x,y)=3y^2x^3$
I get the point $(0,0)$.
Now, ... |
H: Derivative of inverse function, why do I get this contradiction?
Consider two functions, $f(x)=\sin x$ $\;$ and $g(x)=\arcsin x$. Then, $f'(x)=\cos x$ $\;$ and $g'(x)=\frac{1}{\sqrt{1-x^2}}.$ We know that $g[f(x)]=x$, so $f'(x)=\frac {1}{g'[f(x)]}$ $\;$. $\:$ Substituting in, we have $\cos x=\frac{1}{\sqrt {1-\sin^... |
H: Point on an algebraic surface closest to another one
Given an algebraic surface $F(x,y,z)=0$, $F\in\mathbb{R}[x,y,z]$, and a point $P_0=(x_0,y_0,z_0)\in\mathbb{R}^3$, is there a possibility to (algorithmically) determine a point on $F(x,y,z)=0$ that is closest to $P_0$ with respect to the Euclidean distance?
AI: Ye... |
H: Checking if a function is injective
Let $\mathbb{C}$ be a small category, whose objects are thought of as "admissible worlds" and whose arrows as "temporal admissible developments". Let $X:\mathbb{C}^{\operatorname{op}}\rightarrow \operatorname{Sets}$ be a presheaf of sets defined on $\mathbb{C}$. Denote by $\sigma... |
H: Unique uniformly continuous function into complete space
Let $M_1,M_2$ be metric spaces such that $M_2$ is complete. Let $f$ be a uniformly continuous function from a subset $X$ of $M_1$ into $M_2$. Suppose that $\overline{X}=M_1$. Prove that $f$ has a unique uniformly continuous extension from $M_1$ into $M_2$ (t... |
H: Vorticity equation in index notation (curl of Navier-Stokes equation)
I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation:
$$
{\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = (\mathbf{u}\cdot\nabla)\pmb\omega - ( \pmb\omega \cdot\nabla)\mathbf... |
H: I haven't understood how does the line, $ H(Y)=-\int_D p(x)\log [\frac{1}{h}p(x)]dx$ appear?
Let $X$ has a pdf $p(x)$.
let $$H(X)=-\int_D p(x)\log p(x)dx$$
where $D=[x:p(x)>0]$
And,
$Y=a+hX$ ; $-\infty<a<\infty$ , $h>0$
so $Y$ has the pdf $g(x)=\frac{1}{h}p(\frac{x-a}{h})$
let $$H(Y)=-\int_{D_1} g(x)\log g(x)dx$$... |
H: The product of two natural numbers with their sum cannot be the third power of a natural number.
I wanted to know, how can i prove that the product of two natural numbers with their sum cannot be the third power of a natural number.
Any help appreciated.
Thanks.
AI: You're talking about the solvability of the Dioph... |
H: Can you prove this three-way linear map composition?
OK, this was an example that my prof gave when talking about surjective, injective and bijective functions. I also am curious if I am approaching this the right way. (Everyone here has been a really big help in pointing out to me where I am messing up).
It comes... |
H: Number of possible outcomes in a license plate
If a license plate consists of 3 letters followed by 3 digits and having at least one digit or letter repeated .. How many outcomes are there?
26 * 26 * 10* 10 * 10 .. Is that right?
AI: That is incorrect.
First, let's find the number of license tags, total, that can b... |
H: Convergence or divergence of $\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2}$
$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2}$$
The series is convergent or divergent?
Would you like to test without the full ...
I've thought of using the comparison test limit, but none worked, tried searching a number smaller or larger compared ... |
H: Boundedness and total boundedness
We say that a metric space $M$ is totally bounded if for every $\epsilon>0$, there exist $x_1,\ldots,x_n\in M$ such that $M=B_\epsilon(x_1)\cup\ldots\cup B_\epsilon(x_n)$.
Prove that if $M$ is a totally bounded metric space, then $M$ is bounded. Given an example to show that the c... |
H: How to determine the MU in economics?
I currently have a table like so:
Hours spent on Activity X | Total Utility
120
220
300
360
396
412
I know that Marginal Utility is calculated use slope formulate (delta Y / delta X).
I am asked to calculate the MU for 5 hours:
So in this case, $delta X = 5 - 1 = 4$ and $de... |
H: What is an example of monotonically decelerating/accelerating function?
I came across the term monotonically decelerating function. How can it be written as a mathematical function?
AI: Consider the simple case where an object moves along a $1$-dimensional axis.
Usually, the acceleration of an object whose positio... |
H: Solve the following equation algebraically for n. Show your working using factorial notation.?
a) $\dfrac{n!}{(n-2)!} = n(n-1)$
so the expression equivalent to $$\dfrac{n(n-1)}{6} = 12 \implies n(n-1)=72$$
I think its obvious the required solution is $8 \cdot 9 =72$, so $n=9$
b) $8Pn = \dfrac{8!}{(8-n)!} = 6720$
... |
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