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Suppose that we role $d$ dice. Let $X, Y$ be random variables, where
$X = \#$ rolled by the die with the highest value.
$Y = \#$ rolled by the die with the second highest value.
By convention, we allow for the possibility that $X=Y$ in case more than one die has the same value (e.g. in a scenario where two dice roll ... |
Let $N$ be a stopping time. i.e $\{N=n\} \in \mathbb{f}_n \forall n$. $\mathbb{f}_n$ is the filtration.
$\mathbb{f}_N=\{A\in \mathbb{f}, A\cap \{N=n\}\in \mathbb{f}_n \forall n\}$ is the sigma algebra of the stopping time N.
If a sequence of coins are flipped and N is the number of coins until the first head. (so we st... |
Let A and B be convex subsets of $\Bbb R^n$. The join of A and B is the set of all $\vec x$ such that $\vec x$ lies on a line segment with one endpoint in A and the other in B. I am wondering how to show that the join of A and B is a convex set.
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So to my knowledge a definite integral's significance is how it shows the "intensity" or area under the curve for a function.
However, I am confused then why the definite integral for
x from 0 to 1 = 1/2
is it representing the area under the curve with the implicitly assumed base/restriction being the x axis? becaus... |
Personal question: Is the ideal generated by the polynomial $x^2-2$ maximal in the ring $\mathbb{C}[x]$?
I know the ideal in $\mathbb{Q}[x]$ generated by $x^2-2$ is maximal, considering $\mathbb{Q}[x]/(x^2-2) \cong \mathbb{Q}\sqrt{2}$.
|
$5(-3x - 2) - (x - 3) = -4(4x + 5) + 13$ Solve for $x.$
Can anyone solve this equation and explain it to me? I don't understand this equation for some reason and I always keep getting different values.
|
Here is a plot of 17,723 Misiurewicz Points. The code below generates a set of polynomials u[m,n], the roots of which have periodicity (m-n) starting at iteration n. I stopped at 17,723 points because to get more by this method I had to generate and solve $2^{11}$-order polynomials. In other words, I hit the practica... |
The vector field given as $\vec{F}=\frac{\left \langle x,y,z \right \rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$
The region $D=\left \{ a^{2}\leq x^{2}+y^{2}+z^{2}\leq b^{2} \right \}$
I've some insights into this problem but would like to explore this further.
Evidently, $x^{2}+y^{2}+z^{2}=f\left ( x,y,z \right )$
But would ... |
Let $R=\mathbb{Z}[\sqrt{-5}]$ and $I=(2,1+\sqrt{-5})$ an ideal generated by $2$ and $1+\sqrt{-5}$. Show that $I$ is a maximal ideal.
So I tried to prove that if $a \notin I$ then $(I,a)$ must be the whole ring or the $I$ itself. That is, we can find a combination $ax+2y+(1+\sqrt{-5})z=1$ or $0$ for some $x,y,z \in \m... |
I think the easiest way to explain what I'm having trouble with is to give an example question:
A monkey is given 12 blocks: 3 Squares, 3 Rectangles, 3 Triangles, 3 Circles. Calculate the probability of it drawing three of each kind in order - say, 3 triangles, then 3 squares and so on.
I have done this question and go... |
Let $G$ be a group and $L^2(G) = \{f: G \rightarrow \mathbb{C} \}$. Now define an inner product on $L^2(G)$ by
$$\langle f, g \rangle = \sum_{x \in G}f(x)\overline{g(x)}$$
Where $\overline{g(x)}$ is the complex conjugate.
I know what it means for the inner product described above to be linear in $f$, i.e. $\langle f +... |
Let $X = -10Y + 10$. Let $r_1$ be the correlation between $X$ and $Z$ and $r_2$ be the correlation between $Y$ and $Z$. Then, which of the following is the best answer?
$r_1 = r_2$.
$r_1 = 10r_2$
$r_1 = -10r_2$
$r_1 = -r_2$.
None of the above is correct.
I think Choice No.3 is the best answer because of the coef. of ... |
Proving that differentiability implies continuity.
|
The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution.
Edit:-
Five cans of paint (numbered 1 through 5) were delivered to a professional painter.
Unknown to her, some of the cans (1 a... |
We roll a single die and the game stops as soon as the sum of two successive rolls is either 5 or 7.
We want to find the probability that the game stops at a sum of 5.
It seems like Markov chain with first-step analysis.
To find the transition matrix, I first need to define the states.
They way I define it is that ... |
I am going through some notes Physics, Topology, Logic and Computation: A Rosetta Stone, on category theory.
We first define the opposite category:
Given a category $C$, we define the opposite category $C^{op}$ as the category with the same objects as $C$, but a morphism $f:X\rightarrow Y$ in $C^{op}$ is a morphism $f... |
$W=\{f(x)\in P(\mathbb R) \colon f(x)=0 \text{ or } f(x)\text{ has degree }5\}$, $V=P(\mathbb R)$
I'm really stuck on proving this question. I know that the first axioms stating that $0$ must be an element of $W$ is held, however I'm not sure how to prove closure under addition or scalar.
I've tried using arbitrary p... |
Let$$F(x)=\begin{cases} 0,\ if\ x<0\\ \frac{x^2}{10},\ if\ 0\le x<1\\\frac{x+2}{8},\ if\ 1\le x<2\\ \frac{c(6x-x^{2}-1)}{2},\ if\ 2\le x\le3\\ 1,\ if\ x>3 \end{cases}$$ Find the value of $c$ for which $F(X) $is cumulative distribution function. Also evaluate $P(1\le x\le2)$.
I find out $c$ by $P(x\le3)=1\\ \implies \fr... |
Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that without loss of generality it can be assumed that $F(0) \ne 0$. Also, it is enough to find a polynomial where the prop... |
I have an exercise which is $\lim_{x \to 0^+} x^x$.
I have done it, and the answer is $1$, but I just need to know the easiest way to solve it without l'Hospital's rule, since in my country they don't allow us to use L'Hospital's rule.
|
Suppose joint probability density function is $f(x,y) = 1/(x+1)$ for $0<x<1$ and $0<y<x+1$. I try to calculate marginal density function $f_Y(y)$ by
$$f_Y(y) = \int_{y-1}^1 \frac{1}{x+1}dx = \ln(2)-\ln(y)$$
But this does not seem to be a right answer. Where did I go wrong?
|
After learning multiplicity in polynomials we were given the task of proving that:
if $ f(\alpha) = f'(\alpha) = f''(\alpha) = f'''(\alpha) =$ .... $f^{k-1}(\alpha) = 0$ and $f^{k}(\alpha) \not= 0$
then $(x-\alpha)$ is a root of $f(x)$ of multiplicity of $k$.
By logic we know this to be true however when trying to p... |
Let $A = \mathbb{Q}[x]$. Show that the ideal generated by $x^2-2$ is maximal.
I think it is sufficient to show that $\mathbb{Q}[x]/(x^2-2) \cong \mathbb{Q}\sqrt{2}$, where $\mathbb{Q}\sqrt{2} = \{a + b \sqrt{2} : a,b \in \mathbb{Q} \}$
I'm stuck on this problem.
Is anyone could help for this question?
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I've just started a logic and proof class, and I'm confused about what we learned.
Given a proof of the form $$(P \lor Q) \rightarrow R$$ why is it true that you only have to show $$P \rightarrow R$$ or $$Q \rightarrow R$$ to satisfy the original proof? Why even have a disjunction as a premise if you only need one of t... |
I have tried a lot of books to understand Koszul complex but eventually failed. I have tried Eisenbud's book, Weibel's homological algebra, but I just can not how to construct Koszul complex by tensor product and why this is equivalent to the construction by exterior algebra. Could some one recommend some more clearer ... |
The Calkin-Wilf tree is an infinite undirected graph (tree) which is constructed as follows: starting from the root at $\frac{1}{1}$, each node $\frac{a}{b}$ has two children:
a left child $\frac{a}{a+b}$
a right child $\frac{a+b}{b}$
This tree has the property that every rational appears in it exactly once, in lowe... |
To quote from Differential Forms in Algebraic Topology,
Set $x=c_1(S^*)$. Then $x$ is a cohomology class in $H^2(P(E))$. Since the restriction of the universal subbundle $S$ on $P(E)$ to a fiber $P(E_P)$ is the universal subbundle $\tilde{S}$ of the projective space $P(E_p)$, by the naturality property of the first Ch... |
The density of a random variable $X$ is
$f(x)$ proportional to $x^{-1/2}$ for $x \in [0,1]$$
and $f(x) = 0$ for $x \notin [0,1]$. Then, the mean of $X$ is
$\frac 12$
$\frac 1{\sqrt2}$
$\frac 13$
$\frac 14$
None of the above is correct.
By the formula $\int_{0}^1 x\times x^{-1/2} dx $ (the formula of the expectation... |
Let $A, B$ be $n \times n$ matrices with full rank. I'm interested in getting a bound on how the smallest eigenvalue of $S = (A^TA)(B^TB)^{-1} + (B^TB)(A^TA)^{-1}$ changes when I perturb $A$ and $B$. That is, given $Sx=\lambda x$ I want a bound on $\delta \lambda$ where $\left[(\tilde{A}^T\tilde{A})(\tilde{B}^T\tilde{B... |
I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps
$f_\alpha: A_\alpha \to X$
each of which is a cofibration. I also have weak equivalences
$g_\alpha: A_\alpha \to B_\alpha$.
And I know that $Y$ is obtained as the pushout ... |
The problem: ($\mathscr{F}(\mathbb{R})$ is the set of real valued functions)
Let $\phi:\mathscr{F}(\mathbb{R})\to\mathbb{R}\times\mathbb{R}$ be a function defined by $\phi(f)=(f(0),f(1))$ Prove that $\phi$ is a surjective homomorphism from $\mathscr{F}(\mathbb{R})$ to $\mathbb{R}\times\mathbb{R}$, and describe its ker... |
My attempt: I am getting $2^n(n!)^2$ .
First I paired $n$ boys and $n$ girls in $n!$ ways then these pairs can be arranged in $n!$ ways and in each of these pairs boy and girl can arrange themselves in $2!$ ways.
|
This is the given info:
$P(A)=2/5$ ,
$P(A \cup B)=3/5$ ,
$P(B|A)=1/4$ ,
$P(C|B)=1/3$ ,
$P(C|A \cap B)=1/2$ ,
Find $P(A|B \cap C)$
I know that $P(A|B \cap C)$ is equivalent to $P(A \cap B \cap C)$ divided by $P(B \cap C)$.
I was able to calculate $P(A \cap B \cap C)$ by doing $P(A)P(B|A)P(C|A \cap B)$ ... |
Let $T:V\to W$ be a linear map of vector spaces, and let $A\subset W$ be contractible. Then is $T^{-1}(A)$ also contractible?
|
Can you please give an example of a function with an infinite number of maxima and minima occurring in any finite time interval?
Edit: This question came to me as I was reading on the dirichlet conditions for the fourier transform of a function $g(t)$ to exist. Given that $g(t)$ is a non-periodic and deterministic sign... |
Given $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, with $X_1$ independent of $X_2$, as well as $Y = (X_1 + X_2)^2$, what is $\Pr(Y>\mathrm{E}[Y])$?
Misguided attempt:
$$
\begin{align}
&Y \sim \left(\mathcal{N}(\mu=1,\sigma^2=5^2)\right)^2\\
&\Pr(Y>\mathrm{E}[Y]) = \int_{\mathrm... |
Let $X \subseteq 2^{\omega}$ be a set of positive Lebesgue measure. Suppose that for every $\eta, \nu \in 2^{<\omega}$ of the same length, the measure of $X$ above $\eta$ is the same as the measure of $X$ above $\nu$.
Is it necessarily true that the outer measure of $X$ is $1$?
If the answer is positive, do we need cou... |
Why can't I use this proof to prove that the countable union of countable sets is countable without the axiom of countable choice?
Take the set of integers; some proper subset of it, call it $A$, can be mapped, with $f$, a 1-1 and onto function, to the set of even integers.
Some proper subset of the set of even integ... |
How do I solve:
$$
e^{4z}+e^{3z}+e^{2z}+e^z+1=0
$$
I'm getting lost on where to start. I tried using the definition
$$
e^z=e^x(\cos(y) +i\sin(y))
$$
But that doesn't seem to do me any good. I also tried using complex powers, but I seem to just be going in circles. Any suggestions?
|
Given:
$$\sum_{i=1}^\infty \frac2{7*i + 21} $$
The limit of the $nth$ term is 0, it means we aren't sure if it diverges.
On wolfram it says it diverges by comparison test, but how?
|
Show if $f$ is a norm:
For $\mathbb{R}^n$, Define $f: \mathbb{R}^n \rightarrow \mathbb{R} $ by $ f(x) = \|x\|^2$ =$\sum_{n} x_n^2 $
I tried to solve if $f$ satisfies the three properties of a norm: 1) zero vector 2)positive homogeneity 3)Triangle Inequality.
It satisfies the 1st property obviously.
For the 2nd property... |
Here's Theorem 2.43 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Let $P$ be a non-empty perfect set in $\mathbb{R}^k$. Then $P$ is uncountable.
Here's the definition of a perfect set:
Let $(X,d)$ be a metric space, and let $P \subset X$. Then $P$ is perfect if it is closed (i.e. i... |
In my notes, there is this theorem:
A sequence in $R^n$ converges (to a limit in $R^n$) iff it is Cauchy.
I understand that this theorem applies to all complete metric spaces, not just to $R^n$.
Now, here is my question:
The sequence of continuous functions $(f_n)$ in the space of continuous functions $C[0,1]$ equipped... |
(a + b)(b' + c')(a + b' + c)
where b' = b not and c' = c not.
I tried distributive because I'm not very good at applying the properties when multiplication is applied but I can with addition.
(a + b)(b' + c') = ab' + ac' + b'b + bc' = ab' + ac' + bc'
I think it can be simplified further but I'm not sure how. It would j... |
Suppose we have continuous random variable $X$ with distribution $f_X$. That is
$$ P\left(a \le X \le b \right) = \int_{a}^{b} f_X(x) \ dx $$
Now suppose I have a function $\phi: \Bbb{R} \rightarrow \Bbb{R}$
How does one compute the distribution of $\phi(x)$? That is I want to find a $g(x)$ such that
$$ P \left( a \... |
Background:
I came across the following problem in class and my teacher was unable to help. The problem was factorise $x^6 - 1$, if you used the difference of 2 squares then used the sum and difference of 2 cubes you came out with the following factorisation; $(x-1)(x+1)(x^2 +x + 1)(x^2 - x + 1)$. However if you used t... |
Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point P is reflected by the ellipse and arrivates at Q. The question is how to determine the reflection point on the ellipse. I mean is there a... |
Suppose there are $40$ balls with $10$ red, $10$ blue, $10$ green, and $10$ yellow. All balls with the same color are deemed identical. Now all balls are supposed to be put into $4$ identical baskets, such that each basket has $10$ balls. What is the number of ways to partition these balls?
I tried this problem, but it... |
I am trying to prove below equation with combinatorial argument but I have no idea how this works.
$$\sum_{i = 1}^{n} (i-1) = nC2$$
Can anyone give me a clue?
|
yesterday I asked a related question that was a specific case of my problem. It was solved here: Geometry/triangle problem determining leg-length but I would want to know if its possible to get a generic universal solution for the problem.
Given the next situation:
Problem scenario
We have the typical 2D coordinate sy... |
Let X be smooth and projective, D is an Cartier divisor of X. $\mathcal{N}_{D/X}$ is the normal bundle of D in X.
Let $q: X\longrightarrow Y$ be a finite quotient map of degree d. And q is totally ramified along D, with ramification index r. $q(D)$ is the image of D. Let $\mathcal{N}_{q(D)/Y}$ be the normal bundle of... |
Let $f$ be a sub-linear functional on a real vector space $X$ satisfying $f(x)+f(y)\ge f(x+y);f(\alpha x)=\alpha f(x);\alpha>0$
Show that it satisfies $f(0)=0 $
Thoughts:
Now $f(x+y)\leq f(x)+f(y)\implies f(0+0)\le f(0)+f(0)=2f(0)\implies f(0)\ge 0$
How to show that $f(0)\le 0$? Any help
|
Imagine two spaces:
An ‘input’ space with dimension $m$.
An ‘output’ space with dimension $n$.
$m \geq n$
There are points in each of these spaces defined such that some characteristic is defined. The characteristic is defined and valid in both spaces.
An example* might use RGB (Red, Green, Blue) values for the inpu... |
I dont think I really need to elaborate, do I? If you know what quaternions are then you know there are several imaginary-value options to choose from, or axes, along which the $\sqrt{-1}$ may exist.
By definition $i^2 = j^2 = k^2 = -1$ but no indication is given of going the other way.
I suppose we could just say that... |
Let $A,B$ be two families of sets. What is a term for the following families:
$$C = \{a\cup b|a\in A, b\in B\}$$
$$D = \{a\cap b|a\in A, b\in B\}$$
$$E = \{a\setminus b|a\in A, b\in B\}$$
Since they are similar in nature to Cartesian product, I thought of calling them "Cartesian union", "Cartesian intersection" and "C... |
I am trying to follow a guide for the method of characteristics; quoting the first example:
We use the method of characteristics to solve the problem
$ 2u_x - u_y = 0, \;\; u(x, 0) = f(x) $
(...) we can easily solve [the characteristic equations] to get:
$x = 2s + x_0 \;\;\; y= -s + y_0 \;\;\; u = u_0$
Up to here I h... |
I know that you can transform a posynomial function into an exponential function, which is convex. Does this imply that all posynomial functions are convex?
|
Let $\mathbb{Z}[\sqrt{2}]:= \lbrace a+b\sqrt{2}|a,b \in \mathbb{Z} \rbrace$. How many elements are there in $\mathbb{Z}[\sqrt{2}]/(1+3\sqrt{2})$?
I know that every equivalence class of $a+b\sqrt{2}$ contains an integer of the form $a-6b$, which is congruent to $a$ mod $6$. So does this mean that there are only 6 ele... |
I know that theoretically eigenvectors of real symmetric matrix are orthogonal to each other. So for each pair, dot product will be zero. But when I am calculating eigenvectors from real symmetric matrix using MATLAB as well as LAPACK, I am not getting dot product to zero. Can any one help me to clear about this issue... |
I understand that a smooth vector field on a manifold $M$, generates a "flow"/one parameter group action, lets say $\sigma(t,s): \mathbb{R} \times M \rightarrow M$, and $\sigma_t: M \rightarrow M$ gives a one parameter group of diffeomorphisms. My question is, do every diffeomorphism have to be an element of a such gro... |
I have a function in the form:
$$
\mathrm{e}^{-t\lambda} \cdot \left[t\lambda - {(t\lambda)^2 \over 2}\right]
$$
If one were to plot this for say $\lambda = \frac{2}{3}$ and $t$ from $0$ to $20$, this would produce a graph which has an 'unhappy parabola' shape till $t$ is around $3$, and then a 'happy parabola' shape w... |
I've this function to transform not using the general formula, but just substituting the known transform (i.e. $\text{rect}(t)\rightarrow \text{sinc}(f)$):
$\frac{\sin(6\pi t)}{t}$
I know the transform of the function $\frac{\sin(\pi t)}{\pi t}\rightarrow \text{sinc}(f)$, and i taught to apply the rescaling property, b... |
Let $n=\underbrace{11\dots1}_{1996\text{ figures}}$. Does there exist such a function $f(x)$ that for all real $x \ne 0, x \ne 1$ holds
$$\underbrace{f \bigg ( f \Big( \dots \big( f}_{n\text{ iteratoins}}(x) \big) \Big) \bigg)=\left (1-\frac {1}{\sqrt[n]{x}} \right)^n?$$
|
Give that $f$ is a decreasing continuous function and that
$$f(x+y) = f(x) + f(y) -f(x)f(y)$$
and $f'(0)=-1;$ Then it is to be found what is $\int_{0}^{1}f(x)dx$
I am at a loss on how to approach this sum. I have tried substituting $y=0$ & $x=x$ to no avail.
The given answers are -
A)$1$
B)$1-e$
C)$2-e$
|
About the sequence $d_n$ of the denominators of harmonic numbers, I know these facts:
It is unbounded, since $p\mid d_p$ for any prime $p$.
It contains only one $1$.
What more is known? Specially, is it true (as intuition suggests) that
$$\lim_{n\to\infty}d_n=\infty$$
? If not, is it known the inferior limit?
|
The function
$$z+\frac{1}{z}$$
seems to play a role in complex analysis.
However, does it have any applications besides what can be deduced in pure mathematics?
|
Prove that
$$\int_{0}^\pi x^{2k} \cos(h x) dx\geq 0$$
for all $k\in\mathbb N$ and $h$ even integer. I have tried with Induction Principle (for $h$) but without success.
|
Suppose the following integral
$$
\int\limits_{-\infty}^{\infty}\sin{x}dx
$$
In mathematical rigor, the following is the definition
$$
\int\limits_{-\infty}^{\infty}\sin{x}dx = \lim\limits_{n\to\infty}\lim\limits_{m\to-\infty}\int\limits_m^n\sin{x}dx
$$
Some places (especially in physics), say that since $\sin{x}$ is a... |
I'm trying to find an interpolating formula for a set of coefficients (I have $80$ at the moment).
I tried first to find an interpolating polynomial, but that was not useful: using the first $40$ coefficients only and extrapolate that formula to a guess for the other $40$ coefficients leads to catastrophic erro... |
The question is based on the example given in Intersection Theory under the heading Self-intersection. The example is as follows:
Consider a line $L$ in the projective plane $\mathbb{CP}^{2}$: it has self-intersection number $1$ since all other lines cross it once: one can push $L$ off to $L′$, and $L · L′ = 1$ (for an... |
So I have this is an assignment for algorithms. I've googled a lot, read the chapter in the book about big Oh notation, and I understand the concept. I do not however understand how to prove it. I know I need to work with a constant and put that out of the equation. And I know that the answer I've found for $f(n) = O(g... |
i have to calculate the Fourier transform in the title. My professor says the result is $\frac{1}{5-2\pi i f}$. I start from $H(t)e^{\alpha t}$, and i calculate the transform $H(t)e^{5t}\rightarrow \frac{1}{2\pi i f - 5}$. Now i must pass to $H(-t)...$ and i decide to apply the rescaling property $u(t/\lambda)=\lambda ... |
Let's say I have a field $\mathbb{F}$.
Now suppose I take the set $R = \{x \in \mathbb F^{\times}: \ y(x) \ge 0\} \cup \{0\}$ where $y$ is a function $y:\mathbb F^{\times} \rightarrow \mathbb{Z}$ satisfying $y(ab) = y(a) + y(b)$, $y$ is surjective, and $y(a + b) \ge \min\{y(a), y(b)\}$ for all $a,b \in\mathbb F^{\time... |
Suppose
${A_j},\,{\Delta _j} \in {\mathbb C^{n \times n}},\quad\big(\,j = 0,\,1,\,2,\,\ldots,\,m\,\big)$
${P_\Delta }\left(\lambda\right) = \left({A_m} + {\Delta _m}\right){\lambda ^m} + \, \cdots \, + \left({A_1} + {\Delta _1}\right){\lambda ^1} + \left({A_0} + {\Delta _0}\right)$ is a matrix polynomial, and $\lamb... |
Let $X$ be a topological space, $R$ is a commutative, unital ring. In a proof from lecture there is claimed that $H^*(S^1\times X,\{\text{pt}\}\times X;R)\cong H^*(D^1\times X,\partial D^1\times X;R)$ in singular cohomology and I'm not sure how to justify it.
It is $\partial(D^1)=S^0=\{-1\}\coprod \{+1\}$ and maybe it'... |
If $X$ is homogeneous and $p\in X$, then is $X\setminus \{p\}$ necessarily homogeneous?
This seems to work with all the simple examples I've tried. I would be interested in any counterexamples. Or if there is a theorem here, you can assume we are dealing with metric spaces.
EDIT: Greg's pointed out that $X=2\times S^1... |
How many $3$-digit positive integers can be represented as the sum of
exactly nine different powers of $2$?
What does this question mean? Is the sum of $9$ different powers of $2$ like $2^1+2^2+...2^9$? Then it must be $>1000$.
|
We already knew that if two independent chi-squared random variables, then their sum is also chi-squared with the degree of freedoms is the sum of theirs. How about the converse?
If $X\sim\chi^2(n)$ and $Y\sim\chi^2(m)$ and their sum $X+Y\sim\chi^2(n+m)$, can we conclude that $X$ and $Y$ are independent random variable... |
There is poisson random variable $X$
$$P(X=x)=\frac{\lambda^{x}}{x!}e^{-\lambda}$$
And define random variable $Z=\lceil \beta X \rceil$ ( $\beta$ is rational number which is lower than 1).
How can i find distribution of $Z$.?
|
For what values of $b$ does the sequence of functions:
for each $n\in\mathbb{N}$, let
$$f_n(x)=b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1]$$
converge uniformly in the space $C[0,1]$ equipped with the supremum metric $$d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$$
My working and my doubts:
I claim that the sequence $g_n(x)... |
In $S_{10}$ let us consider the permutation $\sigma = (123)(456)$ and the cyclic subgroup $ G = \langle \sigma \rangle$.
I can fairly easily find that $$\langle \alpha = (1,4,2,5,3,6)(7,8) \rangle \neq G$$ is a cyclic subgroup of the alternating group $A_{10}$ such that $\sigma \in \langle \alpha \rangle$ (and hence ... |
I need to understand the following
(i) $\{(x,y) \in R^2 |\;xy = 1\}\;U \{(x,y) \in R^2 |\;y = 0\}$ is not connected however
(ii) $ Y = \{(x,y) \in R^2 |\;x^2 + y^2 < 1\}\;U \{(x,y) \in R^2 |\;y = 1\}$ is path connected.
I attempt (i) along this line. For any $A = (a,0)$, I can find a ball of radius $1/2a$ such that ... |
Let $R = k[x_1,\dots,x_t,x_{t+1},\dots,x_n]$ and $>$ a monomial ordering on $R$. Then $>$ is an elimination ordering for $x_1,\dots,x_t \iff x_i >x_j^m$ for all $1\leq i \leq t, t+1 \leq j \leq n$ and all $m \geq 0$. (Ene and Herzog, Gröbner Bases in Commutative Algebra, Problem 3.2.)
Recall
Definition (Elimination O... |
Rudin says in page fourtheen in theorem 1.29 : If $a$ and $b$ are real, then $(a,b)=a+bi$. Proof he gives: $$a+bi=(a,0)+(b,0)(0,1)\\=(a,0)+(0,b)=(a,b)$$
of course this is correct (if we accept $a=(a,0)$), however i'm interested in how to justify that $a=(a,0)$, it seems entirely unrigorous to me, on one hand we have a... |
We have the matrix $A= \begin{pmatrix}
0 &2 &2 \\
2& 0 &2 \\
2& 2 & 0
\end{pmatrix}$, then one of the following is true:
$f_A(x)=m_A(x) $
The matrix $C=A^4-4A^2+5I$ isn't diagnolizable over $\mathbb R$
For the matrix $B=A^5+6A^3+A^2-I$ we have $f_B(x)\neq m_B(x)$
For the matrix $B=A^5+6A^3+A^2-I$ we have $f_B(x)... |
Determine lim f(x) as x approaches -2
Since there are two y values as x approaches -2 namely a gap at -1(from left and right) and a solid dot at -3, which of the following should be correct?
The limit DNE or the limit exists at y=-1?
Thank you.
|
I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too.
I would like to make a classical computation using maple. I would like define an abstract moving frame (e_1,e_2), then
1) get the dual frame omega_1, omega_2 (always abstract) omega_i(e_j)=d... |
Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant.
This problem can be solved as so:
If $deg(f(x))=n$, the upper equation implies that $n^2=2n-1$, or that $n=1$.
This implie... |
Given
$$G=\mathbb{Z}_2*\mathbb{Z}_2=P(a,b\mid a^2,b^2)$$
among other things I wanted to show that this group is infinite, what I did is consider the words of the form
$$abababa\ldots$$
they are all diferent from each other since they do not contain any of the relations described below, and also there is an infinite amo... |
Suppose that:
$0 \rightarrow M_{1} \rightarrow M \rightarrow M_{2} \rightarrow 0$
Is an exact sequence of $R$-modules (for $R$ commutative integral domain). Show that $\mathrm{rk}(M) = \mathrm{rk}(M_{1})+\mathrm{rk}(M_{2})$
I have shown the inequality $\mathrm{rk}(M) \geq \mathrm{rk}(M_{1})+\mathrm{rk}(M_{2})$, howeve... |
Let $G$ be a group. A regular representation is given as $V=\mathbb{C}[G]$, a vector space, where $l: G \to GL(V)$ be the action is given by $l(g)(\alpha)(h) = \alpha (g^{-1}h)$ for all $g,h\in G, \alpha \in V$.
The professor asked us to find a trivial sup-representation for the sub-representation $U=\{ \alpha \in V ... |
Let $A,B,C$ be pair-wise independent (like A and B are independent) such that$P(A^{c}\cap B)=0.1$ and $P(B\cap C)=0.2$ Show that $P(A^{c}\cup C)\ge \frac{7}{8}$.
$P(A^{c}\cup C)=1-P(C^{c}\cap A)\\=1-P(C^{c})\cdot P(A)$
what can i do next and how can I use the given information.
I know $P(A\cap B)=P(A)\cdot P(B)=0.1\te... |
What is the intuition and the proof behind the expression
$E[V | W] = E[E[V | U,W] | W]$
where $U,V,W$ are random variables. I know that $E[V | W]$ can be treated as a random variable which is a function of $W$ but I am having a hard time getting the intuition about the above expression of nested expectations.
|
Suppose $X$ is a commutative monoid and $f:X\to\mathbb R\cup\{\infty\}$ a function
and
$$g(x)=\inf\left\{\sum_{i=1}^nf(x_i)~\middle\vert~\sum_{i=1}^nx_i=x,n\in\mathbb N\right\}$$
$$h(x)=\inf\left\{\frac{g(mx)}m ~\middle\vert~ m\in\mathbb N\right\}$$
then how to prove that for all $k\in\mathbb N$, $h(kx)=kh(x)$ and $h(x... |
Suppose $\beta_1(t)$ and $\beta_2(t)$ are two parametric curves defined on $[0,1]$. Let $\beta_1^*(t)$ and $\beta_2^*(t)$ are two re-parametrized of the above curves.
Now, I looking for a re-parametrization, where the sum of two re-parametrized curve also have the same properties as the component curves (reparametrize... |
Let $f(x)$ be a polynomial in $x$ of degree greater than $1$.Degine
$g_i(x)$ by $g_1(x)=f(x)$,and $g_{k+1}(x)=f(g_{k}(x))$.Let $r_k$ be the
average of the roots of $g_k$.Determine $r_{89}$ if $r_{19}=89$.
My effort
Playing with the definitions I can see that
\begin{array}
gg_1(x)&=f(x) \\
g_2(x) &=f(g_1(x))=f(f(... |
By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish.
For example, in the case of the reccurence relation for the Bessel function
$$a_n = \frac{-1}{(n+\lambda)^2-m^2}a_{n-2}$$
Such is only sol... |
I am looking for a reference for the Schur Zassenhaus Theorem, saying that any normal Hall subgroup admits a complement.
An on-line search show that it is supposed to be in "The theory of groups" by Zassenhaus himself, but since I don't have the book I was hoping that someone can give a more exact Theorem or section in... |
The conditional statement is:
If today is February 1, then tomorrow is Ground Hog's Day.
I need to negate this but I am confused. Would it just be If today is not February 1, then tomorrow is not Ground Hog's Day? I think that is an inverse statement though. Please help me negate this conditional statement.
|
All birds are animals. All animals are four legged. Implications
a. All animals which are four legged are birds.
b. All birds are four legged
c. Some birds are four legged
d. Some birds are animals but not four legged.
Choices
a and b
b and c
only b
only d
What is the correct answer?
|
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