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H: Vector Subspaces- Counterexample
ive been struggling to come up with a counter-example, ive been treating V as R^3. I would very much appreciate if someone could come up with a counter example and the right equality involving the equations below.
AI: Using your idea of $V=\mathbb{R}^3$:
a) Let $T=\{(t,t,0);t\in\ma... |
H: Measure theory problems.
Prove or disprove the following:
a)If $\mathscr A$ is a $σ$-Algebra on $Ω$ then {$Ω$ \ $A$ : $A$ element of $\mathscr A$} too.
b)A $σ$-Algebra with 3 elements exists.
c)A measure $μ$ on $P(\mathbb R)$ with $μ$({x}}=$1$ exists.
a) Isn't this equal to Ω \ $\mathscr A$? If so then this is the... |
H: Definition of partition of set regarding countability
In Stochastic Processes, we define a partition of a set as:
A countable collection of sets $U=\{A_1,A_2,A_3,...\}$(which may be a finite collection) which are pairwise disjoint is a partition of a set $D$ if
$\bigcup A_i=D$
My question laying here is that, doe... |
H: Mathematical series regarding complex (I think)
$\sum _{k=1}^{n-1} (n-k)\cos\frac{2k\pi}{n} $
I smell complex here...something regarding $n^{th}$ roots of unity... But I think there might be a catch...after all:
$n\sum_{k=1}^{n-1}\cos\frac{2k\pi}{n} -[ \sum_{k=1}^{n-1} cos\frac{2k\pi}{n}] -[\sum_{k=2}^{n-1} cos\fra... |
H: Matrix powers sequence bounded
Let $m\in\mathbb{N}^*$ and $A\in\mathcal{M}_m(\mathbb{C})$ such that the matrix sequence $(A^n)_{n\geq 0}$ is bounded. Is the sequence $(\|A\|^n)_{n \geq 0}$ bounded ?
AI: It does not need to be. Take
$$
A = \begin{bmatrix} \alpha & 1 \\ 0 & \alpha\end{bmatrix}, \qquad 0<\alpha<1.
$$
... |
H: How to prove that $\int_{-\pi}^{+\pi}\cos{(2x)}\cos{(3x)}\cos{(4x)}\cdots\cos{(2005x)}dx$ is positive
show that
$$I=\int_{-\pi}^{+\pi}\cos{(2x)}\cos{(3x)}\cos{(4x)}\cdots\cos{(2005x)}dx>0$$
This problem is my frend ask me,
My try:
$$I=2\int_{0}^{\pi}\cos{(2x)}\cos{(3x)}\cos{(4x)}\cdots\cos{(2005x)}dx$$
and I... |
H: big-O proof with power functions
I was wondering if anyone could show a proof for why $a^x$ is $\mathcal{O}(b^x)$ if $a$ and $b$ are constants and $a < b$. In other words, with power functions, does the function with the largest base always eventually overtake a function with a smaller base?
AI: I assume $0<a<b$. T... |
H: Conditional number: exercise
Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$?
Edit:
$\kappa(A)=||A||_2\,||A^{-1}||_2$
Now, there is a theorem that says $||A||_2=\sigma... |
H: Find coefficient of static friction if given initial velocity and distance?
I'm trying to work a physics problem about how to find the coefficient of static friction between two objects when given the initial velocity and distance. Specifically, the problem I am working is as follows:
A crate is carried in a picku... |
H: Derivative of a fraction with respect to another
I've found this derivative on a textbook
$\dfrac{d(c_{t+1}/c_t)}{d(\dfrac{\gamma}{c_t}/\dfrac{1-\gamma}{c_{t+1}})}=\dfrac{1-\gamma}{\gamma} \dfrac{d(c_{t+1}/c_t)}{d(c_{t+1}/c_t)}=\dfrac{1-\gamma}{\gamma}$
I would like to understand the first passage. Was $\dfrac{1-\g... |
H: Lagrange identity in integral form
I can prove Lagrange identity in discrete form, but I couldn't find any similarity to apply it for integral case. Here is what I mean,
$$
\Bigg(\int_a^bx(t)y(t)dt\Bigg)^2=\int_a^bx^2(t)dt\int_a^by^2(t)dt-\frac{1}{2}\int_a^b\int_a^b[x(s)y(t)-y(s)x(t)]^2dsdt
$$
Thanks for any help i... |
H: Computing a limit almost surely using the strong law of large numbers
Let $X_0=(1,0)$ and define $X_n \in \mathbb R^2$ recursively by declaring that $X_{n+1}$ is chosen at random uniform distribution from the ball $B(0,|X_n|)$ and $\frac{X_{n+1}}{|X_n|}$ is independent of $X_1,...,X_n$. Prove that $ \frac{\log |X_n... |
H: Updated: Constructing a bijection between $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\mathbb{R}$
I am supposed to construct a bijective function for the interval:
\begin{align}
I_2=\left(-\frac{\pi}{2} ,\frac{\pi}{2} \right] \longrightarrow \mathbb{R} \tag{Problem}
\end{align}
I first tried the easier case, ... |
H: A convergent / divergent sequence of positive numbers such that $\lim \frac{s_{n+1}}{s_n}=1$
I need to find both a convergent and divergent sequence of positive numbers such that $$\lim \frac{s_{n+1}}{s_n}=1$$
I think the question is asking me to play with the ratio test.
Just when I was about to write down the an... |
H: How do I continue to find the critical points of this function?
$\ f(\theta) = 6\sec \theta + 3 \tan \theta $
with the domain $\ 0 < \theta < 2π $
Here is what I get for the derivative:
$\ {dy \over d\theta} = 6\sec \theta \tan \theta + 3 \sec^2 \theta $
Then I set the derivative to 0:
$\ 6\sec \theta \tan \the... |
H: Probability of selecting 1 of i elements in a set of size n when choosing k elements
Given a set of $n$ elements, I will choose uniformly at random $k$ distinct elements.
Given a set $S$ of $i$ distinct elements from $n$, what is the probability that at least one of the $k$ elements will be in $S$?
AI: It's easier ... |
H: How can we calculate $\int \:\frac{1}{\sqrt{x^2-1}}dx$
Solve the following integral: $$\int \:\frac{1}{\sqrt{x^2-1}}dx$$
I attempted to solve it intergradation by parts by doing a
$$\int \:1\:\frac{1}{\sqrt{x^2-1}} \, dx$$
and set $u$ be $\frac{1}{\sqrt{x^2-1}}$ and $dv/dx$ be $1$:
but as I start doing, it gets... |
H: Counting Ballots?
How would I count the ballots in this scenario?
A ballot lists ten candidates for city council, eight candidates for the school board, and five bond issues. The ballot instructs voters to choose up to four people running for city council, rank up to three candidates for the school board, and appr... |
H: Indefinite integral with product rule
Assume the following integral:
$$\int \sin x \cos x\ \ dx $$
My understanding now is, that I can choose both for $f(x)$ and $g'(x)$ respectively, leading to different results:
$$f(x) = \sin x$$
$$g'(x) = \cos x$$
Leads to the result:
$$\frac{\sin^2x}{2} + c$$
Switching it aroun... |
H: How can we calculate $(\log_{x}{x})'$?
Related to this, I am looking for a solution for:
$(\log_{x}{x})'$ = ?
...where $x$ is not 1, but positive.
AI: Notice that $\log_x x=1.$ Is that enough? |
H: Probability of selecting 1 of n-i elements in a set of size n when choosing k elements
Given a set of $n$ elements, I will choose uniformly at random $k$ distinct elements.
First: Given a set $S$ of $i$ distinct elements from $n$, what is the probability that at least one of the $k$ elements will not be in $S$?
AI:... |
H: Find the area of a triangle that we don't know its base and height?
firstly find the area of m . And we don't have any clue just a picture ..
AI: I assume the areas of $BFE$ and $BFC$ are $4$, $10$ respectively.
$$\frac{A(BFE)}{EF}=\frac{A(BCF)}{FC}\Rightarrow EF=2k,\quad FC=5k$$
$$\frac{A(FCD)}{FD}=\frac{A(BCF... |
H: Matrix multiplication - Express a column as a linear combination
Let $A = \begin{bmatrix}
3 & -2 & 7\\
6 & 5 & 4\\
0 & 4 & 9
\end{bmatrix} $ and $B = \begin{bmatrix}
6 & -2 & 4\\
0 & 1 & 3\\
7 & 7 & 5
\end{bmatrix} $
Express the third column matrix of $AB$ as a linear combination of the column matrices of $A$
... |
H: "Convergence" of the sequence $a_k=2^{10^{\ k}}$
I've been observing final digits of each number in the sequence $$a_k=2^{10^{\ k}}$$
You get:
$\ a_0=2 \\ a_1=1024 \\ a_2= ...205376 \\a_3= ...069376\\a_4=...709376\\a_5=...9883109376\\a_6=...2747109376\\a_7=...1387109376$
And so on.
Obviously, the numbers are gettin... |
H: exchangeability vs. shift invariance
Let $X_n$ be a sequence of real RVs indexed by $n \geq 0$. Can someone provide an example of a sequence $X_n$ that is exchangeable (law invariant under finitely supported permutations on the natural numbers) but not shift invariant (law invariant under left shift)? What about ... |
H: Equation with a division in exponent
I have an equation: 10,9 * 2^(x/1,5) = 1000 and want to calculate the value of x.
x being in the exponent is my problem. How can I get to something like: x = ...
AI: You use the comma to denote the decimal so I'll use the same notation.
$$2^{x/1,5}=\frac{1000}{10,9}=\alpha\iff x... |
H: What is a real-valued random variable?
This question arose when someone (and surely not the least!) commented
that something like $\left(X\mid Y=y\right)$ , i.e. $X$ under condition
$Y=y$, where $X$ and $Y$ are real-valued random variables and $P\left\{ Y=y\right\}>0 $,
is not a well defined random variable. To see... |
H: Limit of $ (2^n (n!)^2)/(2n+1)!$
I want to show that
$$
\lim_{n \rightarrow \infty} \frac{2^n (n!)^2}{(2n+1)!} = 0,
$$
but it's been a long time since I took calculus, and I don't know how to do it. I've tried to squeeze it, but I didn't succeed...
Thanks in advance!
AI: Let $a_n = \dfrac{2^n(n!)^2}{(2n+1)!}$.
The... |
H: Need Help Solving Polynomial Equation
I'm working on an induction problem that basically boils down to this equation:
$$2(-1)^k+ 6(2^k)\left(-\frac{1}{2}\right)^{k+1} + (-1)^{k}=0$$
I'm fairly confident that the equation above is the solution to the problem, but I am unable to simplify it further in order to prove ... |
H: Why is the inner product not an element of the Hilbert space?
What I know about Hilbert space is that, elements in that space can be complex numbers.
But I was confused to read this statement from a book:
The inner product, being a complex number, is not an element of the Hilbert space.
Can someone elaborate this... |
H: Trying to understand an exercise using factorials with induction
Exercise:
Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1
Given Answer:
I will skip the basic step since I understand that part.
(n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand this line
But, I don't understand starting at this next... |
H: What's the limit of this sequence?
$\lim_{n \to \infty}\frac{1}{n}\bigg(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\cdots + 1 \bigg)$
My attempt:
$\lim_{n \to \infty}\frac{1}{n}\bigg(\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\cdots + 1 \bigg)=\lim_{n \to \infty}\bigg(\frac{\sqrt{1}}{\sqrt{n^3}}+\frac{\sqrt{2}}{\sqrt{n^3}}+\... |
H: Reducibility of the polynomial $x^4+1$
Show that $x^4+1$ is reducible in $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt2 \,i)$ .Is true that $x^{2n} +1 , n\ge3$ is reducible in $\mathbb{Q}(\sqrt2)$ and $\mathbb{Q}(\sqrt2\,i)$?
Is this correct to show that $x^4+1$ is reducible in $\mathbb{Q}(\sqrt2)$? Is there another... |
H: Mathematical Invariant
Start with the set {3, 4, 12}. You are allowed to perform a sequence of replacements, each time replacing two numbers a and b from your set with the new pair 0.6 a - 0.8b and 0.8 a + 0.6b. Can you transform the set into {4, 6, 12}? Look for an invariant.
I am having trouble determining what t... |
H: How to prove the following full rank condition?
Let $A$ be an $m \times n$ matrix with positive values.
Let $v$ be a vector of length $m$ with positive values.
Consider the following function of $x$, a vector of length $n$, such that $x_i \ge 0$ and $\sum_i x_i = 1$ (i.e. the following function is defined only for ... |
H: Topology of the completed upper-half plane
Define the topology on $\mathbb{H}^*
:
=
\mathbb{H}
∪
\mathbb{Q}
∪\infty$ by taking a basis of open sets around $\infty$ to be $S_{\epsilon} : =
\{
z
∈
H
: Im
(
z
)
>
1
/\epsilon
\}∪\infty$
, and taking
$Γ(
1
)$
-transforms to get bases of open sets around
the points in
$\... |
H: rref matrix equations - k2 7
This question is about reduced row echelon form, Gauss-Jordan, inverting matrices, and solving systems of equations.
I try to solve a system of equations with matrices. I know what operations are allowed, but I just seem to arrive at the wrong conclusion 50 % of the times. So here are t... |
H: Sums of squares have zero upper density
Define the upper density of a set $A \subseteq \mathbf{N}$ to be
$$\bar{d}(A) = \limsup_{n \to \infty} \frac{|A \cap [1,n]|}{n}.$$
Let $A$ be the set of sums of two squares, i.e. $A = \{x^2 + y^2 : x,y \in \mathbf{Z}\}$.
I know that any prime congruent to 1 modulo 4 is the s... |
H: rref matrix equations k2 - 5
This question is about reduced row echelon form, Gauss-Jordan, inverting matrices, and solving systems of equations.
I try to solve a system of equations with matrices. I know what operations are allowed, but I just seem to arrive at the wrong conclusion 50 % of the times. So here are t... |
H: How does a special case prove a surjection?
I have a problem understanding the following proof that claims a surjection. The text is translated from a german university textbook by Luise Unger (pardon any translation errors by me, please).
Given
$$f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$$
$$f((x... |
H: Finding the inverse of the sum of two symmetric matrices A+B
Consider calculating the inverse of matrix sum
$$A+B$$
where A is a symmetric dense matrix while B is a symmetric block diagonal matrix. I am interested in finding an efficient approach to update the inverse of the sum A+B if the values in one of the di... |
H: How can I apply Runge-Kutta to evaluate integral?
I would like to evaluate cumulative normal (0,1)-distribution values using Runge-Kutta method but the problem is that I don't know how to apply the method. Namely, if I have that $y'(x)=e^{-x^2/2}, y(0)=1/2$ and $y(3)$ is the value I would like to know, then how can... |
H: Understanding Counter example for "product of two quotient maps is a quotient map" from Ronald Brown
One can prove that the product of two open quotient maps is a quotient map.
Ronald Brown gives a counter example for the fact that this is in general not true for arbitrary quotient maps, in his book Topology and Gr... |
H: Minimum boolean lattice containing all poset of fixed size
I need help with the following:
What is the minimum $n$ such that the boolean lattice $2^{[n]}$ contains all posets of size $m$?
I noticed that it should contain a chain of length $m$, and the longest chain in $2^{[n]}$ has length $n+1$, so $n \ge m - 1$.
A... |
H: $P(X-np>n\varepsilon)\leq E\{e^{\lambda \cdot (X-np-n\varepsilon) }\}$
For $X \mathtt{\sim} \text{Bin}(n,p), \lambda > 0, \varepsilon > 0$, how do you show the following? $$P(X-np>n\varepsilon)\leq E\{e^{\lambda \cdot (X-np-n\varepsilon) }\}$$
Unless I made some error, it's pretty easy to see that:
$$E\{e^{\lambda ... |
H: Different types of Set Theory
I have just now learned that there are different types of Set Theory. I read Naive Set Theory by Paul R. Halmos, but other than that I have no other knowledge of other...set theories.
Could anyone explain the difference between Naive Set Theory and Axiomatic Set Theory? I thought the... |
H: Simplifying Boolean Algebra law
I've got a problem here that I could use help solving. I have simplified it to this point. Using Wolfram Alpha, I know it is still possible. My lecturer did it but I didn't catch all of it. It is frustrating me like mad and I didn't want to come here for just one question but I feel ... |
H: How can I use the distributive property to rewrite an algebraic fraction?
I have an expression:
$$N\left(\dfrac{N(N+1)(N-1)+3N}{3}\right)$$
Can can I use the distrubtive property to form:
$$N^2\left(\dfrac{(N+1)(N-1)+3}{3}\right)$$
If so, how? Could someone advise me on some material so O could strengthen my unders... |
H: Average number of predators and prey in Lotka–Volterra model?
Once again I wouldn't be surprised if this can be found maybe even on Wikipedia but I'm not a native English speaker and unfortunately couldn't find this myself.
So assuming standard Lotka–Volterra equations, exactly as written in Wikipedia, representing... |
H: What is the equal sign with 3 lines mean in Wilson's theorem?
I'm reading up on Wilson's Theorem, and see a symbol I don't know... what does an equal sign with three lines mean?
I'm looking at the example table and I still can't infer what they are trying to say about that relationship between equations.
AI: $$(n-1... |
H: Propositional Logic: Models/Counter-Models
Given the following task: (Given a single specification)
Use truth tables to check if the specification is consistent, and if it is - provide a model of the specification. If not, provide a counter model.
I know that a specification is consistent if there is at least one... |
H: Derivative of the area under $f(x)$ between $a(x)$ and $b(x)$
Consider the following, where $a(x)$, and $b(x)$ are both functions that are continuous in their domain:
$$
g(x) = \int\limits_{a(x)}^{b(x)}f(t)dt
$$
Is it the case that $g'(x)$ is always the following?
$$
g'(x) = f(b(x))\times b'(x) - f(a(x))\times a'(x... |
H: Find two cycles of length $r$ and $s$ such that the order of their product is not $\operatorname{lcm}(r,s)$.
I want to find two cycles of length $r$ and $s$ such that the order of their product is not $\operatorname{lcm}(r,s)$.
MY try: Take $ \pi = (123)$ and $\sigma = (12) $.
Then, $ \pi \sigma = (123)(12) = (13) ... |
H: Why and how are quaternions 'bilinear'?
What does it mean when we say that quaternion composition is 'bilinear'? I have observed that some authors write quaternion multiplication as:
While others specify:
Excuse the poor images, StackExchange did not seem to like my LaTeX.
Note that the sign on the vector cross p... |
H: Quotient $K[x,y]/(f)$, $f$ irreducible, which is not UFD
Does anyone know an irreducible polynomial $f \in K[x,y]$ such that the quotient $K[x,y]/(f)$ is not a UFD? Is it known when this quotient is a UFD?
Thanks.
AI: Yes, I think that the standard example is to take $f(x,y)=x^3-y^2$. Then when you look at it this ... |
H: What is the difference between positive presistent and null persistent state in a Markov Chain?
I'm not looking for the difference in the mathematical definition, but rather for an intuitive explanation of their differences and possible examples, so that I can have them in my head when solving/formulating problems.... |
H: $\frac{1}{z} \prod_{n=1}^{\infty} \frac{n^2}{n^2 - z^2} = \frac{1}{z} + 2z\sum_{n=1}^{\infty} \frac{(-1)^n}{z^2-n^2}$?
I am trying to show that
$$\frac{1}{z} \prod_{n=1}^{\infty} \frac{n^2}{n^2 - z^2} = \frac{1}{z} + 2z\sum_{n=1}^{\infty} \frac{(-1)^n}{z^2-n^2}$$
This question stems from the underlying homework p... |
H: Induction proof of inequality from linear recurrence
Consider the sequence:
$a_0 = 1; a_1 = 2; a_2 = 3; a_k = a_{k-1} + a_{k-2} + a_{k-3}; k \geq 3$
and the statement
$P(n) : a_n \leq 2^n$. Prove that $\forall n \in \mathbb{N}$, $P(n)$ holds.
I would like some help understanding this question. Also which is the bes... |
H: If $|x_{n+1}-x_n| < |x_n-x_{n-1}|$, then $(x_n)$ is a Cauchy sequence
Prove or disprove :
If $|x_{n+1}-x_n| < |x_n-x_{n-1}|$ for all $n\geq 2$, then $(x_n)$ is a Cauchy sequence
What I understand from this is if the difference between the $n$ and $n+1$ terms in the sequence is getting smaller and smaller, then, the... |
H: Using the Squeeze Theorem
Fix any $k∈N$, and let $a_1,...,a_k$ be $k$ positive numbers. Use the Squeeze Theorem to prove that as n aproaches infinity
$$\lim_{n\rightarrow +\infty} \left(\sum^k_{j=1}a^n_j\right)^{1/n}=\max(a_1,...,a_k) $$
AI: Suppose (it doesn't matter) that $a_1$ is $\ge a_j$ for all $j$. Then
$$a... |
H: Find moment generating function of Y = $e^X$
Let $X$ ~ $N(0,1)$ and $Y=e^X$. Find the moment generating function of Y.
I think I first need to find the cdf of Y. So I take:
$F_Y(y) = P(Y \le y) = P(e^X \le y) = P(X \le ln(y)) = F_X(ln(y))$
I think that part is correct. Now I get a little confused.
I think $M_Y(t) ... |
H: Showing that the mapping cone is a chain complex
Let $\alpha \colon \mathcal{A} \to \mathcal{D}$ be a morphism of chain complexes.
Let
$$
d^{C(\alpha)}_n
= \begin{bmatrix}
-d^{\mathcal{A}}_{n-1} & 0 \\
\phantom{-}\alpha_{n-1} & d^{\mathcal{D}}_{n-1}
\end{bmatrix}
$$
I want to show th... |
H: If $A$ is an $R$-module with some sort of ring structure, is it true that any $R$-submodule of $A$ is an ideal of $A$?
If $A$ is an $R$-module with some sort of ring structure, is it true that any $R$-submodule of $A$ is an ideal of $A$?
AI: No, think of the polynomial rings. For instance, set $A=R[x]$ (which is in... |
H: Divergence of $\sum_{k=1}^{\infty}\frac{k^k}{e^k} $
I'm trying to show the series $\sum_{k=1}^{\infty}\frac{k^k}{e^k} $ is divergent by the negation of the cauchy criterion. My thought was to break the sum into dyadic pieces that could be bounded from below to show the series divrges but I'm having trouble finding... |
H: Finding where the surface $z=2 + x + y^2$ intersects the $xy$ plane
I completely understand all you have to do is set $z = 0$ in order to find where the surface $z=2 + x + y^2$ intersects the $xy$ plane - I just do not understand how to solve this equation, wouldn't there be multiple solutions?
AI: If $z=0$ then yo... |
H: If $a,b \in \Bbb R$, prove that $|ab| \le (a^2+b^2)/2$
So far I have the first case when $a=b$:
\begin{align*}
|ab|
&= |b^2|\\
&=|b|^2\\
&=\frac{2|b|^2}2\\
&=\frac{b^2+b^2}2\\
&=\frac{a^2+b^2}2
\end{align*}
Case 2: $a>b$
Case 3 $a<b$
I've been stuck on this problem for a few hours now and don't know how to proceed.... |
H: Limit of the sequence $\{n^n/n!\}$, is this sequence bounded, convergent and eventually monotonic?
I am trying to check whether or not the sequence $$a_{n} =\left\{\frac{n^n}{n!}\right\}_{n=1}^{\infty}$$ is bounded, convergent and ultimately monotonic (there exists an $N$ such that for all $n\geq N$ the sequence is... |
H: Surgery on trivial knots
I know a theorem that any closed orientable 3 manifold can be obtained from the sphere $S^3$ by surgery along a framed knot.
I think I read or heard somewhere that as a surgery link, we can take unknots with integer framing.
I am confused whether if this means that we can find a surgery lin... |
H: For a normal random variable, $X \sim \mathcal{N}(0, 1)$, $F(-\sqrt{y}) = 1 - F(\sqrt{y})$?
For a normal random variable, $X \sim \mathcal{N}(0, 1)$, $F(-\sqrt{y}) = 1 - F(\sqrt{y})$?
How do I get that?
AI: Intuitive explanation. The probability density function drawn is $\mathcal N(0,1)$. Look at the two colored... |
H: Are algebraic properties consistent among ALL types of number groups?
I'm embarking on a self study course of group theory, modular arithmetic, and other mathematics relating to cryptography.
I notice that when studying modular arithmetic that they explicitly say that the relation is
compatible with the operation... |
H: sum in closed form $\frac{1}{3}+\frac{1\cdot 5}{3\cdot 7}+\frac{1\cdot 5\cdot 9}{3\cdot 7 \cdot 11}+...............$
Calculation of the given sum in closed form $\displaystyle \frac{1}{3}+\frac{1\cdot 5}{3\cdot 7}+\frac{1\cdot 5\cdot 9}{3\cdot 7 \cdot 11}+...............$
$\bf{My\; Try::}$ We can write the given i... |
H: Let $a_n$ and $b_n$ be sequences of real numbers. If $b_n$ is bounded and $\lim_{n \to \infty} a_n = 0$, then $\lim_{n\to\infty} a_{n}b_{n} = 0$
Prove the below statement:
Let $a_n$ and $b_n$ be sequences of real numbers. If $b_n$ is bounded and $\lim_{n \to \infty} a_n = 0$, then $\lim_{n \to \infty} a_n b_n=0$
Wh... |
H: Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$
Show that if $a$ is an even integer and $b$ is an odd integer then $(a, b) = (a/2, b)$
Hi everyone, I would like to know if my assumption is justified for answering the above question. Any constructive feedback is greatly appreci... |
H: Alternative proofs of the remainder theorem for polynomials
The theorem I've been tasked with proving is that for any polynomial function $f:\mathbb{R} \to \mathbb{R}$ and any $a \in \mathbb{R}$ there exists some polynomial $g:\mathbb{R} \to \mathbb{R}$ and some $b \in \mathbb{R}$ such that $$f(x)=(x-a)g(x)+b.$$
Us... |
H: Problem solving recurrences with generating functions
I'm trying to solve this linear recurrence with generating functions, but I keep getting stuck on the last few steps. I found the generating function, but after splitting it into partial fractions and putting it in sigma notation, I don't know how to simplify it... |
H: Limit of infinite measure union
Let $A_1,A_2,\ldots$ be Lebesgue measurable subsets of $\mathbb{R}$. It is certainly true, by monotonicity of $\mu$, that $$\mu\left(\bigcup_{i=1}^nA_i\right)\leq \mu\left(\bigcup_{i=1}^\infty A_i\right)$$
Is it true that $$\lim_{n\rightarrow\infty}\mu\left(\bigcup_{i=1}^nA_i\right)=... |
H: A connected subset of a metric space with at least two points has no isolated points
Show that if $E \subseteq X$ is connected and has at least two points, then $E$ has no isolated points. Thus a connected set with at least two points must contain infinitely many points.
I understand that if $E$ has no isolated p... |
H: Graph theory notation
Just a quick question on graphs: how do I read the notation in this question for the edge set? I can't find any explanation in my notes or online.
... specifically that first half of the union. I'm guessing the union just means that the {(4,0), (3,-2), (-2,-2)...} set of edges is also include... |
H: On the largest and smallest values of $ {D_{\mathbf{u}} f}(x,y) $, assuming that $ ∇f(x,y) ≠ 0 $.
I appreciate your time. If anyone can explain this problem, I would be most grateful. I need to understand this for a test, but I was not given any explanation.
Assume that $ ∇f(x,y) ≠ 0 $. Show that the value of $ {D... |
H: Infinite intersection and limits
I'm having difficulty understanding the relationship between a limit and an infinite intersection. Any help would be greatly appreciated.
Specifically, suppose we take any non-increasing sequence of numbers $x_n$ such that $x_n > a$ and $\lim_{n\rightarrow\infty}x_n=a$ then how is
$... |
H: Find a meromorphic function with given principal parts
I have the homework problem of finding an "elementary" meromorphic function $f(z)$ with the same principal parts as the sum $$\frac{1}{\pi z^2} + \frac{2}{\pi} \sum_{n=1}^\infty (-1)^n \frac{\cos n}{z^2 - n^2}$$ (and, for extra credit, computing the sum itself)... |
H: Reduction to LP
What will be the primal and dual of the following problem/
Given an undirected graph $G = (V,E)$, we want to assign non-negative weights to all the edges of $G$, denoted $\{ x_e \mid e\in E \}$ , such that the sum of all the weights is as large as possible,
subject to the constraint: for each verte... |
H: Measure theory singles out the countable cardinal. Why?
In some elementary analysis courses, we discussed what would fail without countable additivity, although it's not as if there would be some contradiction. It would merely be "not nice." We'd lose continuity under sequential monotone limits. Dunford and Schw... |
H: Normal subgroup and order
Let $G$ be a finite group and let $H$ be a normal subgroup. Prove that, for any $g\in G$, the order of the element $gH$ in $G/H$ must divide the order of $g$ in $G$.
All I have for this proof is; define the order of the element $gH$ as $n$ then this implies $g^nH = eH = H$.
How can I pr... |
H: Is this an acceptable congruency proof?
I have a past exam question that I proved as follows:
$$(\forall n\in \Bbb Z)\bigl((3n^2-5\equiv 2 \pmod 4)\lor(3n^2-5\equiv 3 \pmod 4)\bigr)$$
If odd:
$$3n^2 - 7 = k4,k\in \mathbb Z$$
$$3(2l+1)^2 - 7 = k4, l\in \mathbb Z$$
$$12l^2+12l-4 = 4k$$
$$3l^2+3l-1=k$$
If even
$$3n^2 ... |
H: Does $\lim_{x \to 2} \frac{2x+4}{x^2-4} $ exist?
So the lecturer's assistant is saying that the following limit exists and that it is $\infty$
So the equation
$$\lim_{x \to 2} \frac{2x+4}{x^2-4} $$
Now, if I go ahead and simplify the expression first I end up with
$$\lim_{x \to 2} \frac{2}{x-2} $$
and as far as I... |
H: Homomorphism and normal subgroups
Suppose that $\phi : G \to G'$ is a homomorphism between the groups $G$ and $G'$. Let $N'$ be a normal subgroup of $G'.$ Prove that the inverse image of $N'$ is a normal subgroup of $G$.
How can I prove this using the defintions?
A proof that was given confused me. The proof st... |
H: Calculating or estimating a sum of pairs of reciprocals with a constant sum
For given $M$, I would like to find
$$\sum_{\stackrel{i + j = M}{i < j}} \frac{1}{i}\frac{1}{j}.$$
I'm solving the problem programatically ATM, with a single for loop for any given $M$, and I want to solve it for all $M \in \{ 2, \ldots, N ... |
H: Number of arrangements for a varying number of Balls in N colors
Let $N\in\mathbb{N}$ be a number of colors. For each of these colors let $a_k$ be the number of indistinguishable Balls in the specific color. How many arrangements of balls can I find, when I am using all Balls given ($\sum_{k=1}^N a_k$)?
E.g.: Take ... |
H: For every $n$ large enough every zero of $1+z+z^2/2!+\dots z^n/n!$ is such that $|z|>r$
We need to show given $r>0$ there exists $n_0$ such that if $n\ge n_0$ then $1+z+z^2/2!+\dots z^n/n!$ has all $0$'s in $|z|>r$
I was thinking of using Rouche's Theorem but not able to construct $f(z)$, $g(z)$ with $|f(z)|>|g(z)... |
H: Is the projection from $\mathbb{Z}^2$ to $\mathbb{Z}^2$ $f(m, n) = (m + n,m − n)$ injective and how to prove it
I tried to solve it using linear span, but that seems to be a wrong approach.
Edit: and surjective
AI: Assume $f(a_1, b_1) = f(a_2, b_2)$. Then by definition of $f$ we have
$$
(a_1 + b_1, a_1 - b_1) = (a_... |
H: Generating via inverse transform problem.
Suppose we wish to generate $X \sim Binomial(2, \dfrac{1}{2})$ via inverse transform: $X = H(U)$ where $U \sim Uniform (0, 1)$. What is $H(y)$? ($H(y) = min\{x:y\leq F(x)\}$).
How can I do the discrete case? I have no idea how to solve this problem.
AI: Since $X\sim\mathrm{... |
H: Orthonormal Sets and the Gram-Schmidt Procedure
What my problem in understanding in the above procedure is , how they
constructed the successive vectors by substracting? Can you elaborate
please?
AI: Let $|w_1\rangle=|u_1\rangle$. Let us find $|w_2\rangle$ as a linear combination of $|w_1\rangle$ and $|u_2\r... |
H: Radius and Interval of Convergence Question
I have two problems in which I'm stuck finding the radius and interval of convergence:
1) $\sum\limits_{n=1}^\infty\frac{n^3(x+4)^n}{4^nn^{11/3}}$
Applying the ratio rule allows me to simply as such:
$(\frac{(n+1)^3(x+4)^{n+1}}{4^{n+1}(n+1)^{11/3}})(\frac{4^nn^{11/3}}{n^3... |
H: Showing an inequality
I wish to show $$|{(Re^{i \theta})^{-\frac{1}{2}}}\exp(\frac{-1}{Re^{i \theta}})| < \frac{M}{R^k}$$ for some M, k > 0
I've managed to reduce it to $$|R^{-\frac{1}{2}}| |\exp(\frac{-1}{Re^{i \theta}})|$$
but am unsure of where to go from here.
In context, I'm trying to find the inverse Laplace ... |
H: Matrix Equality
Can you help me to prove this equality ?
Let $A,B$ be $n\times n$ matrices. Let $[A,B]$ denote the usual matrix commutator and $e^{A}$ the usual matrix exponential. By hypothesis, let's say that $[A,[A,B]]=[B,[A,B]]=0$. We want to prove that, $\forall t\in \mathbb{R}$ we have that $e^{tA}e^{tB}=e^... |
H: Hopf Algebra - Adjoint Representation
I've been asked to prove the following;
$$ a \circ (bc) = \sum_{(a)} (a_{(1)} \circ b)(a_{(2)} \circ c)$$
Using the fact that the adjoint representation is as follows;
$$ a \circ b = \sum_{(a)} a_{(1)} b S(a_{(2)})$$
I've tried the expansion of the LHS as follows;
$$ a \circ (b... |
H: Maximising sequence of supremum
Suppose $\sup_{x \in X}f(x)$ is finite, where $X$ is some Banach space. Then there is a maximising sequence $x_n$ such that $f(x_n) \to \sup_{x \in X}f(x)$, right? Is this sequence countable?
AI: Yes. Let $n\in\mathbb{N}$. Then we can choose $x_n\in X$ so that
$$f(x_n) > \sup_{x\i... |
H: How to do this problem with vectors?
A plane flies in a direction NW at an airspeed of $141$ km/hr. If the wind at the plane's cruise altitude is blowing with a speed of $100$ km/hr directly from the north, what is the plane's actual speed and direction relative to the ground?
So I'm presuming you can do this wit... |
H: $\sin (x)$ for $x\in \mathbb{R}$
My confusion is how do we define : $\sin (x)$ for $x\in \mathbb{R}$.
I only know that $\sin(x)$ is defined for degrees and radians..
Suddenly, I have seen what is $\sin (2)$..
I have no idea how to interpret this when not much information is given what $2$ is...
does this mean $2$... |
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