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H: When can a mathematician have $3$ friends?
At a mathematics convention there are $n$ mathematicians. Each one has $3$ friends (friendship is symmetric). For what values of $n$ is this possible?
AI: For all even $n$ except $2$.
$n$ cannot be odd because if there are $k$ friendship-pairs, we must have $2k=3n$ (by cou... |
H: If $(a_{2n+1})$ and $(a_{2n})$ converge to $a$ then $(a_n)$ converges to $a$.
If $(a_{2n+1})$ and $(a_{2n})$ converge to $a$ then $(a_n)$ converges to $a$.
So far I realize that if $(a_{2n+1})$ and $(a_{2n})$ converge then for each $\epsilon>0$, there exists $N$ such that for all $n>N$, $|a_{2n+1}|, |a_{2n}| < \e... |
H: Describing a simple set
If I were asked to describe set A if:
$(∀x ∈ A)(∃n ∈ N) (2n = x)$
Would I be correct in saying that: for all values of x in A, there exists a natural number that is half of x.
Is there a way I can improve this?
AI: HINT: Every element of $A$ is an insert term here natural number. |
H: Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$
Is the following just a matter of showing the 3 properties that make up a metric??
Define d on $\Bbb R\times\Bbb R$ by $d(x,y)=\min \{1,|x-y|\}$. Show that $d$ is a metric on $\Bbb R$
$d(x,y)=0$ if $x=y$
$d(x,y)=d(y,x)$ for every $x,y \in X$
$d(x... |
H: How do I solve this recurrence equation using substitution?
f(1)=1, and f(n) = f(n-1)+2(n-1)
Using substitution, here are the first few steps:
f(n-1) = f((n-1)-1) + 2((n-1)-1)
f(n-1-1) = f((n-1-1)-1-1) + 2((n-1-1)-1-1)
And then eventually I see that f(n+(-1)*2^j) = f(n+(-1)*2^(j+1)) + 2n + 2(-1)*2^(j+1), w... |
H: What finite fields are quadratically closed?
A field is quadratically closed if each of its elements is a square.
The field $\mathbb{F}_2$ with two elements is obviously quadratically closed.
However, testing some more finite fields with this property, I didn't find any more. Hence my question is:
Which finite fie... |
H: Two differentiable functions
Prove or disprove the following statement: Let $f,g:[a,b]\to\mathbb{R}$ be differentiable with $g'\neq 0$, then there exists $c\in (a,b)$ s.t.
$$\frac{f(a)-f(c)}{g(c)-g(b)}=\frac{f'(c)}{g'(c)}.$$
I think this statement is true, it looks very similar to Cauchy's mean value theorem and I ... |
H: Prove that it's a subspace of $\mathbb{R}^3$
I have the following definition of $V_a$:
$$V_a := \{(x, y, z)^T \in \mathbb{R}^3 : y = 3x - az\}, \quad \text{for $a \in \mathbb{R}$}.$$
My first problem: I don't understand this definition. Which role does $a$ play in this formula? How do the vectors look like? Why the... |
H: If $\gcd(f(x), g(x)) = 1$, then $\gcd(h(x)f(x), g(x)) = \gcd(h(x), g(x))$
This is not homework, but I would just like a hint. The question asks
Let $f(x), g(x), h(x) \in F[x]$ (where $F$ is a field), and $\gcd(f(x), g(x)) = 1$. Show that $\gcd(f(x)h(x), g(x)) = \gcd(h(x), g(x))$.
Say $d(x) = \gcd(f(x)h(x), g(x)... |
H: For what values of a and b does the following limit equal 0?
I understand I need to make the sum of the individual limits equal 0 - but I'm a little lost. I computed the limit of the first term to be -4/3 via L'Hospitals Rule but Wolframalpha contradicts me (http://www.wolframalpha.com/input/?i=limit+x-%3E+0+%28si... |
H: Prove that $1^2 + 3^2 + 5^2+\cdots+(2n-1)^2 = (4n^3-n)/3$ for all $n \in \mathbb{N}$
Prove that $1^2 + 3^2 + 5^2+\cdots+(2n-1)^2 = (4n^3-n)/3$ for all $n \in \mathbb{N}$.
How can I solve this with induction? I've been working through a couple examples and for this one I can't relate the base case to the induction h... |
H: What Does $\cong$ (Congruence?) Mean in Linear Algebra
I've tried to look it up, but I don't know what it means to say that $V^* \cong V$. Also, what do I say, "$V$ dual is congruent to $V$"?
AI: In functional analysis there is a concept of the (topological) dual space $V^*$ of a normed vector space which is the se... |
H: Proving Cauchy Given Given Sequence Terms Arbitrarily Close
I need to show that $\{x_{n}\}$ is Cauchy given that there exists $0<C<1$ s.t. $|x_{n+1}-x_{n}|\leq C|x_{n}-x_{n-1}|$. Intuitively, that statement clearly implies $\{x_{n}\}$ is Cauchy, since it implies the sequence terms become arbitrarily close. But how... |
H: More series help, this time a telescopic sum.
$$\sum_{n=2}^\infty \frac{2}{n^2 - 1}.$$
I Tried setting it up as a telescoping sum as $$\frac{2}{n} - \frac{2}{n-1}.$$ but now i'm sure that cannot be correct. Mayhaps I need to complete the square? or pull out an N? I'm sorry for asking so many questions. I just real... |
H: Alternative way of expressing the limit of a function
When proving that the limit
$$ \lim_{h\to 0} \frac {f(x+h) - f(x-h)}{2h} = f'(x) $$
as $h \to 0$, am I correct in assuming that the limit
$$ \lim_{h\to 0}\frac {f(x)-f(x-h)}{h} $$
as $h \to 0$ is equal to $f'(x)$? If so, why?
Fixed: Put the wrong fraction in th... |
H: Why don't I need to use chain rule to find the derivative of arccos (x^2)?
I thought that I would need to use chain rule because of x^2 but apparently this is not the case, according to the work-through on calcchat.com.
It shows the derivative as:
[-1 / sqrt(1-x^4)] * 2x
Thanks in advance.
AI: That was the chain ru... |
H: Series question, telescoping.
Okay i'm starting to get a handle on this. my new question is $$\sum_{n=1}^\infty \frac{3}{n(n + 3)}.$$
I know i have to use Partial Fraction Decomposition, and what i came up with is
$$\sum_{n=1}^\infty (\frac{1}{n} - \frac{2}{n+3}).$$
Am I on the correct track? It doesn't seem to be... |
H: Differentiating $f(x)^{g(x)}$
Is there any general rule for what the derivative of $f(x)^{g(x)}$ (where $f(x),g(x)$ are differentiable functions) is in terms of $f(x),g(x),f'(x),g'(x)$.
In other words is there something analogous to product,chain and quotient rules for such expressions?
AI: Indeed there is, note th... |
H: Showing that intersections are not defined
I'm checking to see why intersections are not defined when looking at the class $A$ defined by:
$$ A = ON \cup [ON]^2\;,$$
where $ON$ is the class of ordinals and $[ON]^2$ the class of unordered pairs of distinct ordinals. Intersections are defined in $A$ if for any $x , y... |
H: Convex Function Inequality
Let $f: I \to \mathbb{R}$ where $I$ is an interval. We say that $f$ is convex if for every $a,b\in I$ and every $\lambda : 0<\lambda < 1.\\$
Prove that for any $x\in (a,b)$ $f(\lambda b + (1-\lambda)a) \leq \lambda f(b) + (1-\lambda )f(a) \implies f(x) \leq \frac{(x-a)}{(b-a)}f(b) + \frac... |
H: definition of limsup of a function
I already have some idea on what $\limsup_{x\rightarrow\infty} f(x)=\infty$ means but I would like to hear other ideas from the math stack exchange community. Maybe some intuition would be helpful.
AI: If you are talking about $\limsup,$ I assume you have a good bit of familiarity... |
H: $E$ an $n$-dimensional vector space. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$.
Let $E$ be a vector space of dimension $n$. Find all endomorphisms $f$ of $E$ which satisfy $f\circ f = \operatorname{Id}_E$.
Is trivial that $f = \operatorname{Id}_E$ is a solution, but I don't ... |
H: Determine all group homomorphisms from $\mathbb{Z}_{17}^{\times}$ into $\mathbb{Z}_7^{\times}$.
Determine all the group homomorphisms from $\mathbb{Z}_{17}^{\times}$ into $\mathbb{Z}_7^{\times}$.
I noticed that the group is cyclic and found that the generator of both groups is $\langle 3 \rangle$ but how can I fi... |
H: Real Analysis question in Second Fundamental Theory of Calc
Define
$$F(x)=\int_1^x \frac{1}{2\sqrt{t}-1} dt \quad \text{for all $x\ge 1$}.$$
Prove that if c>0, then there is a unique solution to the equation
$$F(x)=c, \quad x>1.$$
Attempt at a solution: I am not sure how to "prove" this. What I have so far is tha... |
H: N-binacci numbers and ratios generated by them
I was bored at home today and playing with "n-bonacci" numbers, numbers generated by $$x_0=0,x_1=1,...,x_k=k; x_n=\sum_{i=0}^{n-1}x_i$$
I made an assumption based upon the quadratic responsible for the golden ratio, that the ratio of sequential n-bonacci numbers could ... |
H: Evaluate $\oint_C\frac{dz}{z-2}$ around the square with vertices $3 \pm 3i, -3 \pm 3i$.
I'm having a tough time figuring out $\gamma(t)$ in $\oint_C f(z) dz = \int_a^b f(\gamma(t)) \gamma '(t) dt$. I think I have to split this into four parts, the four separate lines, but from there I'm trying to get things to canc... |
H: Is the Cantor set a subset of rational numbers, and is it countable or uncountable?
In Chapter 2 of Rudin's Priniciples of Mathematical Analysis, Rudin takes the Cantor set as an example of a perfect set in $\mathbb{R}^1$ which contains no segment. Here's the construction of the Cantor set and the proof:
2.44 The ... |
H: Show that $f:[0,1] \to [0,1]$ is continuous if $f(x) = x^{1/k}$ for any $k \in \mathbb N$
I'm very confused right now and I want to apply the theorem that says " A mapping f of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$". ... |
H: having trouble finding the inverse.
Let $f(x) = \frac{1}{4}x^3 + x -1$ What is the value of $f^{-1}(x)$ when $x=3$?
First, check to make sure the fuction is stricly monotonic.
$f^{\prime}(x) = \frac{3}{4}x^2 + 1$
$f^{\prime \prime}(x) = \frac{3}{2}x$ Which is a linear function thus $f(x)$ is strictly monotic
N... |
H: $0=a \cos (\phi + b) +c \cos (\phi +d)$
I am trying to find the maximum for a problem in continuum mechanics involving a 2D stress tensor. Now I arrived at a point where:
$0 = a \cos \left( \phi + b \right) + c \cos \left( \phi + d \right)$
My constants are $a$, $b$, $c$ and $d$. For which $\phi$ does the right sid... |
H: Regarding the question of compactness
My question is regarding Compactness and sequential compactness
How to show that the only compact subsets of $X$ are finite sets?
In case of specific sets like $\mathbb{N}$, we can choose cover containing open sets of the form $\mathbb N\setminus \{np: p$ is prime$, n\in \mathb... |
H: Monotone Sequence Limit Question
Let $\{a_n\}$ be a monotone increasing sequence that converges to a finite limit. If a monotone subsequence $\{a_{n_k}\}$ (with $n_{k+1}>n_{k}$, and $n_k\rightarrow\infty$) converges to a finite limit as well, are both limits necessarily equal?
AI: Yes. You do not even need the sequ... |
H: Big Oh Notation for a Recursive Algorithm
I have a question that I'm unsure of:
Express the complexity of the following method using big-O notation. You must explain how you arrived at your answer. What value is returned by the call fred(1,4)?
int fred(int x, int y)
{
if(y == 0)
return x;
else if((y... |
H: Use the definition of a limit to prove..
Use the definition of a limit to prove the following:
$$\lim_{x\to -2}(x-3x^2)=-14.$$
Our definition: Let $L$ be a number and let $f(x)$ be a function which is defined on an open interval containing $c$, expect possibly not at $c$ itself. If for every $\varepsilon>0$ there e... |
H: Definition of $ 1 + \frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{\ddots}}}}$
Is there a definition of $ 1 + \frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{\ddots}}}}$? I am somewhat familiar with continued fractions; that is, I am aware that their convergence depends on whether our input is rational or not. This is not a h... |
H: Calculating probability of difference of two distributions.
A has normal distribution of scores of students with $X \sim \mathcal N(625, 100)$
B has normal distribution of scores of students with $X \sim \mathcal N(600, 150)$
Now I have to calculate probablity of 2 A students's average marks to be greater than aver... |
H: RSA encryption without a calculator
I'm doing an RSA encryption and to get part of the solution I need to solve
$$C=18^{17} \pmod{55}$$
How would I solve this problem without a calculator
Thanks in advance
AI: Let's use successive squaring:
\begin{align*}
18^2 = 324 \equiv 49 &\equiv -6 \pmod{55} \\
18^4 \equiv (-6... |
H: Gcd of every other Fibonacci number
Let $f_n$ be Fibonacci Sequence.
$$gcd(f_{n},f_{n+2})=1,\quad \forall\,n\in\mathbb{N}.$$
Prove
Could you help me with this one? I have done the base case, I just can't figure out the inductive step. Thanks!
AI: Suppose that for a particular $k$ we have $\gcd(f_k,f_{k+2})=1$. We w... |
H: Let $I$ be an ideal of a commutative ring $R$. Let $1\in I$. Prove $I=R$.
Let $I$ be an ideal of a commutative ring $R$. Let $1\in I$. Prove $I=R$.
I think if $1\in I$, then the ring $R$ has unity, but I'm not sure where to go from there.
Any help/hints would be extremely helpful. ^_^
AI: By the definition of an id... |
H: Fourier transform of a real function is real
I was trying to find the Fourier transform of the function
$$x \mapsto \frac1{x^2 - 2x +2}$$
and I keep getting something with non-zero imaginary part. But the Fourier transform of a real function should be real, right? So I must be making a mistake?
What is a proof tha... |
H: Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly $4$ distinct real roots
Problem :
Let $f(x)$ be a 3rd degree polynomial such that $f(x^2)=0$ has exactly four distinct real roots, then which of the following options are correct :
(a) $f(x) =0$ has all three real roots
(b) $f(x) =0$ has exa... |
H: Is the set (0, $\infty$) open?
A set is open if it doesn't contain any of its boundary points. I think 0 is a boundary point here and I think it's the only one. So is the set open?
AI: You've hit the nail on the head. The only boundary point of $(0,\infty)$ is $0$, which is not in the set. Well done! |
H: The limit of $\sin(1/x)$ as $x\to 0$ does not exists
Prove that the following limit does not exist.
$$
\lim_{x\to 0} \sin\left(1 \over x\right)
$$
Our definition of a limit: Let $L$ be a number and let ${\rm f}\left(x\right)$ be a function which is defined on an open interval containing $c$, expect possibly not at ... |
H: How to prove $\frac{1}{m+n+1}-\frac{1}{(m+1)(n+1)} ≤ \frac{4}{45}$?
If $m$ and $n$ are any two positive integers then how do we show that the inequality $$\frac{1}{m+n+1}-\frac{1}{(m+1)(n+1)} ≤ \frac{4}{45}$$ always holds ?
AI: let
$$f(m,n)=\dfrac{1}{m+n+1}-\dfrac{1}{(m+1)(n+1)}$$
then we easy to
$$f(1,1)=f(1,2)=f... |
H: Two soccer team players sit at a round table question.
11 players of Team A and 11 players of Team B sit at a round table, the players from the two teams alternate, the goalkeepers sit together, but the captains do not sit together. In how many different ways can they be seated?
Here is my attempt. assume the goalk... |
H: For all $X \geq 1$,$\sum\limits_{1 \leq n \leq X} \mu(n) \left[\frac{X}{n}\right] = 1.$
I'm currently sitting with the following number theory problem:
Prove that for all $X \geq 1$, $$\sum_{1 \leq n \leq X} \mu(n) \left[\frac{X}{n}\right] = 1.$$
A few ideas I have tried: Using that $[x]=x - \{x\}$ which I couldn't... |
H: Limits, find a limit that exist in absolute value but not outside the absolute value.
Let I be an open interval that contains the point c and let f be a function that is defined on I except possibly at the point c. Suppose that lim |f(x)| as x->c exists. Give an example to show that lim f(x) as x->c may not exist. ... |
H: Can objects repeat in commutative diagrams?
Are objects allowed to repeat in commutative diagrams? This seems to be necessary when representing endomorphisms such as the morphism $f : X \to X$ in the category $\mathbf{Set}$, such as when $f$ is a constant function? Or in the category of binary relations, the morphi... |
H: Prove that $1<\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{3n+1}$
Prove that $1<\dfrac{1}{n+1}+\dfrac{1}{n+2}+...+\dfrac{1}{3n+1}$.
By using the Mathematical induction. Suppose the statement holds for $n=k$.
Then for $n=k+1$. We have $\dfrac{1}{k+2}+\dfrac{1}{k+3}+...+\dfrac{1}{3k+1}+\dfrac{1}{3k+2}+\dfrac{1}{3k+3}+\df... |
H: Prove this limit as $x\to\infty$
Let $f:(a,\infty)\to\mathbb{R}$ be a function. Suppose for each $b>a$, $f$ is bounded on $(a,b)$ and $\lim_{x\to\infty}f(x+1)-f(x)=A$. Prove
$$\lim_{x\to\infty}\frac{f(x)}{x}=A.$$
Here we assume $f$ to be arbitrary and no further conditions. $f$ could be continuous,discontinuous,as ... |
H: Proof of a limit theorem.
Prove the following theorem. Let $I$ be an open interval that contains the point $c$ and suppose that $f$ is a function that is defined on $I$ except possibly at the point $c$. If $m \le f(x) \le M$ for all $x$ in $I \setminus \{c\}$ and $\lim_{x\to c} f(x) = L$, then $m \le L \le M$.
Real... |
H: Isomorphism between $\mathbb{F}^{m \times n}$ and $\mathcal{L}(V, W)$
Let $\mathbb{F}^{m \times n}$ be the vector space of all $m \times n$ matrices and let $\mathcal{L}(V, W)$ be the vector space of all linear maps from a vector space $V$ to a vector space $W$ ($V$ and $W$ both over $\mathbb{F}$) with $\dim V = n$... |
H: Circles in Complex Planes
Points on the circle centre C and radius r are given by the equation $|Z-C|=r$ or $(Z-C)(\overline{Z}-\overline{C})=r^2$.
Where $Z = x + iy$.
When multiplied out, I understand that we have
$$Z\overline{Z}-C\overline{Z}-\overline{C}Z+C\overline{C}=r^2\;.$$
So the question I am stuck on stat... |
H: How to draw a set $\{(x,y): y^2 \leq x^2\}$
I want to draw a set $\{(x,y): y^2 \leq x^2\}$. I know what the result should look like (the blue region):
But I don't really see why is that so. I have
$$ y^2 \leq x^2 $$
I am dividing the domain in two: for $x<0$ and $x \geq 0$ and after square root I get:
$$\pm y \leq... |
H: A $2 \times 2$ matrix $A$ such that $A^n$ is the identity matrix
So basically determine a $2 \times 2$ matrix $A$ such that $A^n$ is an identity matrix, but none of $A^1, A^2,..., A^{n-1}$ are the identity matrix. (Hint: Think geometric mappings)
I don't understand this question at all, can someone help please?
AI... |
H: Maximum cardinality
Let $X$ be some set and $P$ be some subset of ${\frak P}(X)$. We can define the smallest cardinal $\frak k$ such that any $A \in P$ has cardinal $\leq \frak k$. Indeed we can consider the set of all cardinals of elements of $P$ and pick the supremum. On the other hand, it is not clear to me that... |
H: Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$
I want to prove this equation,
$$
\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}
$$
I thought of proving this equation by prove that you are using different ways to count the same set of balls and get the same result. But I'm stuck. Help me pl... |
H: Mean value property implies harmonicity
It is fairly easy to show that harmonic functions satisfy the mean value property, but it seems harder to show the converse. I've seen the following theorem without proof:
If $u \in C(\Omega)$ satisfies $$u(z) = \frac{1}{|\partial
B_r(z)|}\int_{\partial B_r(z)} u\,dS$$ for ... |
H: tank and pipes problem
A tank of 425 liters capacity has been filled with water through two pipes, the first pipe having been opened 5 hours longer than the second. If the first pipe were open as long as the second pipe, the first pipe deliver half the amount of water delivered by second pipe; if the two pipes were... |
H: Order of $(\mathbb{Z}\oplus \mathbb{Z})/\langle (4,2)\rangle$?
This is what I did:
any element is of the form $(a+4\mathbb{Z},b+2\mathbb{Z})$, where $a=0,1,2,3$ and $b=0,1$. So it the order should be $8$. But the answer is given to be $\infty$.
What is the wrong in the way i did?
AI: It is possibly that you just m... |
H: If $x^2+y^2=z^2$ has a solution then $5$ divides $xyz$
I tried to solve a question but I did not succeed yet...my question is about number theory. Here it is:
How can we show that if the equation $x^2+y^2=z^2$ has a solution then $5$ divides $xyz$?
Can we have a general method for solving questions like that?
I nee... |
H: Undefined limit $(0/0)$ for a function of two variables
I am trying to calculate the following limit:
$$ \lim_{(x_1,x_2)\to(1,1)} \frac{c_1 \cdot (x_1^2-1)}{\frac{x_1^{10}}{x_2^2}-1}$$ where $c_1$is a constant.
This is giving $\frac{0}{0}$ and I can't seem a way to escape it.
Can anyone help, please?
AI: If you m... |
H: What is the expected number of rounds that everyone get back their own ball?
Sppose there are $n$ people and each of them a unique ball. Suppose now they put the ball inside a box and the balls inside the box would then be mixed up. Then each of them will go out to draw a ball from the box.
Suppose that for each r... |
H: Represent a Toeplitz matrix in an array
I need to represent a $n \times n$ Toeplitz matrix in a $2n - 1$ array. I need to create a function that takes a pair $(i,j)$ and returns the value in the $2n - 1$ array.
I am having a difficult time trying to calculate the index. Can you please help me?
AI: Store the top row... |
H: Do you know how can I see any image of graph?
If exist homepage or method, please tell me.
I want see some image of graph like $y=x^2\sin\left(1/x\right)$ or $y=\frac{\ln(x)}{x}$.
Of course, I know that they are not complex, but I want to see it sketched lovely.
I know mathematica, but I don't have it.
AI: There ar... |
H: Field extension of quotient field
$1$. Let $ F $ be a field and $a,b$ be nonzero element in $L/F$.
How to show that a field extension $F(a,b)/F(a^{−1}b^{−1},a+b)$ is an algebraic extension?
I tried to show all element in $F(a,b)$ is an algeraic over $F(a^{−1}b^{−1},a+b)$ but
I failed....
$2$. Compute $[F(x):F(x^2/(... |
H: About proving an abelian group as a ring
This question is the following.
Consider the abelian group under addition of elements of the form $ar_0+br_1+cr_2$ where $a,b,c\in \mathbb Z$ and $r_0,r_1,r_2$ are variables.
1. Define a multiplication on this set that turns it into a ring with the property that $r_i*r_k=... |
H: Is each paracompact space with countable cellularity always Lindelöf?
Is each paracompact space with countable cellularity always Lindelöf?
We recall that the cellularity of a space is the minimal infinite cardinal $\kappa$ such that every family of pairwise disjoint open sets has cardinality less than or equal to... |
H: Game Theory Problem with Dice
I need solution to this game theory problem. It seems impossible to me.
Two players (1 and 2) play the following game. Player 1 must write the numbers from 1 to 18 on the sides of 3 dice without repeating the numbers. Then player two chooses one die and throws it. After that player 1 c... |
H: Proof that limits stay within the bounds
Prove the following theorem. Let $I$ be an open interval that contains the point $c$ and suppose that $f$ is a function that is defined on $I$ except possibly at the point $c$. If $m \le f(x) \le M$ for all $x \in I \setminus \{c\}$ and $\lim_{x \to c} f(x) = L$, then $m \le... |
H: When is a cyclotomic polynomial over a finite field a minimal polynomial?
When is the cyclotomic polynomial $f(x)$ over a finite field $\mathrm{F}_q$ also the minimal polynomial of some element $\alpha \in \mathrm{F}_q$?
AI: In general, a polynomial $f\in F[X]$ is the minimal polynomial of its roots over $F$ if and... |
H: Injective map from sigma algebras to partitions
I want to show that for every sigma-algebra $\mathfrak A$, you can define for all $x\in X$ the set of all $A_x:=\bigcap_{A\in \mathfrak A,x\in A}A$( a partition of the set $X$) and that this map is injective. Thus, for different sigma-algebras we get different partiti... |
H: Related rates: Find dA/dt of triangle, given d(theta)/dt -- Can't come to textbook answer
The question:
ABC is a triangle in which the lines $\overline {AB} = 20cm$, $\overline {AC} = 32cm$ and $\angle BAC = \theta$. If $\theta$ is increasing at the rate of 2° per minute, determine the rate at which the triangle's... |
H: Prove that lim sin (1/x) as x-> 0 does not exist.
Prove that lim sin (1/x) as x-> 0 does not exist.
Not really sure where to go with this, do I approach from both the right and left?
AI: Hint: Try to find two sequences $x_n\to 0$ and $y_n\to 0$ such that, for instance, $\sin(1/x_n)=1$ and $\sin(1/y_n)=0$. |
H: Quantum Hermiticity Bra-Ket notation please
If $A$ and $B$ are Hermitian operators, show that $$C~:=~i[A,B]$$ is Hermitian too.
My work:
$$\begin{gather}
C=i(AB-BA) \\
\langle\psi\rvert C\lvert\phi\rangle = i\langle\psi\rvert AB\lvert\phi\rangle-i\langle\psi\rvert BA\lvert\phi\rangle
\end{gather}$$
$A$ and $B$ ar... |
H: Lebesgue integration on real line
we know that if $A$ is a Lebesgue measurable set, $f$ and $g$ nonnegative, then
$$ f \leq g \implies \int_A f dm \leq \int_A g dm $$
Does the result still follow if we change $\leq$ with $<$ ??
AI: If $A$ is a null set, then we have of course
$$\int_A f\,dm = \int_A g\,dm = 0$$
re... |
H: Given a vector $\vec x$, are all vectors perpendicular to it constructible from skew-symmetrix matrices multiplied by $\vec x$?
As noted elsewhere, given a skew-symmetric matrix $S$ the vector $\vec x^T S$ is orthogonal to $\vec x$ since
$$\vec x^T S\vec x = -\vec x^T S^T\vec x = -(\vec x^T S\vec x)^T = -\vec x^T S... |
H: Discrete structures exercise
I have this exercise in my worksheet I am a beginner.
Prove or disprove that if $A,B$ and $C$ are sets such that $A\times B = A \times C$ then $B = C$.
AI: The statement is true if $A\neq \emptyset$. If $x \in A$ and $y \in B$, then $(x,y)\in A\times B=A\times C \Rightarrow y\in C$, so ... |
H: Given a vector $\vec x$, what is the maximum possible rank for a matrix $A$ such that $A\vec x=0$?
This is basically an inverse Eigenvector problem:
Given a vector $\vec x\in\mathbb R^n$ ($\mathbb C^n$), what is the highest rank $m$ a matrix $A\in\mathbb R^{m\times n}$ ($\mathbb C^{m\times n}$) can have such that ... |
H: Numbers of different ways to distribute $m$ balls into $n$ boxes?
So my question is this:
assuming I have $m$ balls how many ways there is to divide them into $n$ boxes (at least one ball for each box)?
For example if I have $7$ balls and I want to split them into $3$ boxes I can do:
$5, 1, 1 $
$4, 2, 1 $
$3, 3, 1 ... |
H: sum of the series $\sum e^{-n}\sin nz$
I need to Find the sum of the series $\sum e^{-n}\sin nz$ and indicate where the series converges. Make an appropriate statement about its uniform convergence.
I was doing calculation like below, but did not get any right way about the series.
$f_n(z)=e^{-n}\sin nz$, Clearly $... |
H: Proof something with multivariate normal distributions
Suppose that $X \sim N(\theta, \Sigma), X\in \mathbb{R}^p $, I need to prove
$$p(x) \propto \exp \{-\frac1 2 x^T\Sigma^{-1}x + x^T\Sigma^{-1}\theta\} $$
I can I do this?
I know that the density should be
$$ p(x) = \left(\frac{1}{2\pi}\right)^{n/2} \frac{1}{\sqr... |
H: Adjoint of resolvent of self-adjoint, densely-defined operator on a Hilbert space
Let $H$ be a Hilbert space, $T=T^*$ a densely-defined linear operator on $H$. Denote the resolvent set of $T$ as $\rho(T)=\{\lambda\in\mathbb{C}~|~T-\lambda$ has bounded, everywhere-defined inverse}, and define the resolvent of $T$ at... |
H: $f, g$ are continuous mapping from a connected set $S$ onto $\mathbb{C}^*$
$f, g$ are continuous mapping from a connected set $S$ onto $\mathbb{C}^*$
If $f^n=g^n$ for some positive integer $n$,
1.what is the relation between $f,g$?
2.If $f(x)=g(x)$ for some $x\in S$, what can be said about $f,g$ Then?
3.Show that... |
H: Stuck on order of integration problem
I need to set up a double integral for both orders of integration and use the more convenient order to evaluate the integral $\int\int_{R}\frac{y}{1+x^2}dA$, where $R$ is the region bounded by $y=0, y=\sqrt{x}, x=4$.
I have tried setting it up, but am getting different answers... |
H: Dimension reduction in the Lotka-Volterra model
I'm not sure if I can post this here, but check this out. This is some text from Boccara's Modeling Complex Systems. The thing which confuses me is that there is a dimension reduction from 4 parameters to just 1 (as stated in the text). This dazzles me, because I stil... |
H: How come independence does not imply linear independence? (Statistics)
$$
\begin{array}{c|lcr|c}
\text{X/Y} & \text{0} & \text{100} & \text{200} &\text{Pr(X)}\\
\hline
100 & 0.2 & 0.1 & 0.2 & 0.5\\
200 & 0.05 & 0.15 & 0.3 & 0.5\\
\hline
\text {Pr(Y)} & 0.25 & 0.25 & 0.5 &1
\end{array}
$$
Question: Are X and Y indep... |
H: Probability of two lamps failing within total 1200 hours when exponential density probability function is used for modeling failure
Question: Lamps are of a type with an average lifetime of 1200 hours. Assume that we can model the probability of failure of these bulbs by an exponential density function with mean μ ... |
H: Polynomial whose n no. of integrals are zero
Is it true that, say $y(x)$ be a polynomial of degree $\le$ n, such that $\int^1_0 x^i y(x) dx = 0$ for $0 \le i \le n$ then y(x) is zero polynomial ? Prove or disprove.
AI: Yes, by linearity, you get that $\int_0^1 y^2(x)dx = 0$. Since $y^2$ is continuous and non-negati... |
H: Why is the boolean closure of $F_{\sigma}$-sets not in $F_{\sigma}\cap G_{\delta}$?
In the Borel hierarchy, why is the boolean closure of $F_{\sigma}$ or $G_{\delta}$ equal to $F_{\sigma \delta} \cap G_{\delta \sigma}$? If I take the complement of an element in $F_{\sigma}$ I got an element of $G_{\delta}$, and fin... |
H: How is this result obtained?
I am reading a paper, and having a hard time determining how a result was obtained. The paper states that: Since the total number of linear-extensions is initially $n!$ and probing an edge reduces the number of linear-extensions by a $1-1/e \sqrt{n}$, then only $O(n^{3/2}log(n))$ probes... |
H: Solution properties of $au''(t) + bu'(t) + u(t) = 0 $
Let $au''(t) + bu'(t) + u(t) = 0 $.
Find the values of $a$ and $b$ so that the above ODE has a solution $u$ that:
$|u| \rightarrow \infty$ as $t \rightarrow \infty$
$u$ is perodic
$u \rightarrow 0$ as $t \rightarrow \infty$
Is there a way to determine this WIT... |
H: Find all entire functions that satisfy $f(2z) = (1-2z)f(z)$
This is for homework, and I could use a little help. The question asks
Find all entire functions that satisfy $f(2z) = (1-2z)f(z)$.
Here is what I have done so far. Since $f$ is entire, I wrote
$$ f(z) = \sum_{n=0}^{\infty} a_n z^n = a_0 + a_1z + a_2z^... |
H: Solve $x^2 = I_2$ where x is a 2 by 2 matrix
I tried a basic approach and wrote x as a matrix of four unknown elements $\begin{pmatrix} a && b \\ c && d \end{pmatrix}$ and squared it when I obtained $\begin{pmatrix} a^2 + bc && ab + bd \\ ca + dc && cd + d^2\end{pmatrix}$ and by making it equal with $I_2$ I got th... |
H: Reversing order of integration
I have to reverse the order of integration for the following problem, but the trig functions are tripping me up because I have to take domain and range restrictions into consideration.
Calculate
$$\int_0^1\!\int_0^{\cos^{-1}y}\!\!\sqrt{1+\sin x}\,dx\,dy$$
Does the region of integr... |
H: General linear group of a vector space
I am reading a text on Lie groups.
There is a whole chapter devoted to the group of invertible real or complex matrices of degree $n$, which are called the general linear groups (complex and real).
Further in the text, there is a reference to a general linear group of a vecto... |
H: Integral Of $\int \frac{y}{e^{3y^2}}dy$
I want to integrate the following :
$$\int \frac{y}{e^{3y^2}}dy$$
what I did so far is:
set $t=e^{3y^2}$ and $dt=6y\cdot e^{3y^2}dy$ from here I get get the expression:
$\frac{dt}{t}=6ydy$, there is another way to do it? any suggestions? thanks.
AI: $$ I = \int ye^{-3y^2} \ d... |
H: Convert a matrix so it can be multiplied on the right side, instead of the left
Sometimes I come across examples of geometric transformations that multiply the transformation matrix on the left side of the vector and sometimes on the right.
How do you modify a matrix so that when multiplied on the opposite side it... |
H: Prove by mathematical induction: $n < 2^n$
Step 1: prove for
$n = 1$
1 < 2
Step 2:
$n+1 < 2 \cdot 2^n$
$n < 2 \cdot 2^n - 1$
$n < 2^n + 2^n - 1$
The function $2^n + 2^n - 1$ is surely higher than $2^n - 1$ so if
$n < 2^n$ is true (induction step), $n < 2^n + 2^n - 1$ has to be true as well.
Is this valid argument... |
H: Series Reduction
I'm not too familiar with sequences and series and I would like to show
$ n = \dfrac{n}{2^{n-1}}(\sum\limits_{i=0}^{n-2} 2^i + 1) $
I've been playing around with it on paper and tried expanding and rearranging but I'm sure there's some property I'm not familiar with that I could exploit.
If someone... |
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