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H: Calculating expected value of unknown random variable
The question:
Micro Insurance Company issued insurance policies to $32$ independent risks. For each policy, the probability of a claim is $1/6$. The benefit amount given that there is a claim has probability density function
$$ f(y) =
\begin{cases}
2(1-y) &... |
H: What is the equation for a 3D line?
Just like we have the equation $y=mx+b$ for $\mathbb{R}^{2}$, what would be a equation for $\mathbb{R}^{3}$? Thanks.
AI: You can describe a line in space as the intersection of two planes. Thus,
$$\{(x,y,z)\in{\mathbb R}^3: a_1x+b_1y+c_1z=d_1 \text{ and } a_2x+b_2y+c_2z=d_2\}.$... |
H: What is "group of graph"?
I'm reading some old article and I have one small question: what in general is the group of a graph? By the article, definition should be in Harary's Graph Theory, but unfortunately I don't have any access to that book.
AI: The group of $G$ just means the automorphism group of $G$, i.e. th... |
H: Show that a local ring is equicharacteristic iff it contains a subfield
A local ring $(A,\mathfrak m)$ is equicharacteristic if $\operatorname{char} A=\operatorname{char} \kappa (m)$.
Need hints to solve the following question:
A local ring is equicharacteristic iff it contains a subfield.
AI: $\kappa (\mathfrak... |
H: Sobolev spaces - about smooth aproximation
Consider $\Omega $ a open and bounded set of $\mathbb R^n$ . Let $u \in H^{1}(\Omega)$ a bounded function. I know that there exists a sequence $u_m \in C^{\infty} (\Omega) $ where $u_m \rightarrow u$ in $H^{1}(\Omega)$ . I can afirm that exists $C>0$ where $|u_m (x)| \le... |
H: What does it say about a multivariate polynomial to be zero on a linear subspace?
If I univariate polynomial $f(x)$ that vanishes at a point $x_0$, we conclude that $x - x_0$ divides $f(x)$, and in particular that $f$ is reducible if $\deg f > 1$. Can anything of significance be said if a multivariate polynomial $... |
H: How is the area of a country calculated?
As countries' or states' borders are not straight lines but they are irregular in nature, I wonder how anyone can calculate the area of a country or state.
When do you think the area of a country or state was first calculated? Was it before satellites provided us accurate ... |
H: doubt about basic definition
I have a very basic doubt regarding my last post understanding the basic definiton
Can anybody clear the point what would be the result if I take $j=1$.
For ex. if I am taking $(a,x,1)$ and $(b,x,1)$ as the vertices, Will they be adjacent or not?
Here the value of $j$ is 1. I made the... |
H: Geometric similarities between points in an algebraic variety
If $f : \mathbb{R} \to \mathbb{R}$ is a univariate irreducible polynomial, Galois theory says that all roots are equivalent up to field automorphism (specifically, an automorphism of the field extension fixing the base field).
Can anything similar be sai... |
H: Finding the sum- $x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
If $S = x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
Find S.
Note:This is not a GP series.The powers are in GP.
My Attempts so far:
1)If $S(x)=x+x^{2}+x^{4}+x^{8}+x^{16}\cdots$
Then $$S(x)-S(x^{2})=x$$
2)I tried finding $S^{2}$ and higher powers of S to find some kind of r... |
H: Get a result and calculate back to it's Divisor
In the calculation 28 /7 = 4 the result is 4 and the divisor is 7. From the result i want to calculate back into the divisor. In other words, all I have to do in this case is is 4 + 3 and i get back to the divisor 7.
But that does not work when i do 28 / 2 = 14. In ... |
H: Slight generalisation of the Baire category theorem?
By the Baire category theorem, one cannot write a complete metric space $X$ as a countable union of closed nowhere dense subsets of $X$. Can this be generalised to say that there is no injection $f: X \hookrightarrow X$, $f(X) \subseteq \cup_{n \ge 1} X_n$, where... |
H: About mulit-variate Fourier series
If $f(x,y)$ is $2\pi$ periodic with respect to $x$ and $2\pi$ periodic with respect to $y$ respectively, then can I write $$ f(x,y) = \sum_{j,k \in \mathbb Z} c_{jk} e^{ijx} e^{iky}$$ where $$ c_{jk} = \frac{1}{4\pi^2}\int_0^{2 \pi} \int_0^{2 \pi} f(x,y) e^{-ijx} e^{-iky} dxdy \;?... |
H: Why does $(k-2)!-k \left\lfloor \frac{k!}{(k-1) k^2}\right\rfloor = 1,\;k\ge2\;\implies\;\text{isPrime}(k)$
Let $k$ be a integer such that $k\ge2$
Why does
$$(k-2)!-k \left\lfloor \frac{k!}{(k-1) k^2}\right\rfloor = 1$$
only when $k$ is prime?
Example:
$$\pi(n) = \sum _{k=4}^n \left((k-2)!-k \left\lfloor \frac{k!}{... |
H: Math equation a little hard for me
I got this equation to solve.
Im not very good at equation.
I got this equation
$( A\times \cos(B) ) \times C = D$
I want this form
$( A \times ? ) \times \cos(B \times ?) = D$
I want to do something like merge C in A and B for giving the result D
AI: If we have the equation: $(A\... |
H: Find an integrable $g(x,y) \ge |e^{-xy}\sin x|$
I want to use Fubini theorem on $$\int_0^{A} \int_0^{\infty} e^{-xy}\sin x dy dx=\int_0^{\infty} \int_0^{A}e^{-xy}\sin x dx dy$$
Must verify that $\int_M |f|d(\mu \times \nu) < \infty$. I'm using the Lebesgue theorem, so far I've come up with $g(x,y)=e^{-y}$ but am no... |
H: Extrema homework — maximizing the viewing angle of a picture on a wall
I have hit a problem in my homework and don't know how to solve it.
Here it is:
"A picture with height of 1.4 meters hangs on the wall, so that the bottom edge of the picture is 1.8 meters from the viewers eye. How far does the viewer have to s... |
H: If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.
If $a,b,c \in R$ are distinct, then $-a^3-b^3-c^3+3abc \neq 0$.
I think it is trivial because they are distinct.
So I wonder just saying "Since they are distinct" is enough to prove it?
Of course there could be several more detailed versions but I just... |
H: Axiom of choice and function with empty codomain
I'm having a little problem here, namely if the axiom of choice (Wikipedia) is
$$\forall X \left[ \emptyset \notin X \implies \exists f: X \to \bigcup X \quad \forall A \in X \, ( f(A) \in A ) \right]$$
and I choose the nonempty $X=\{\emptyset\}\neq \emptyset$ for wh... |
H: Verifying that $\mathbb Q=\bigcup_{n\ge 1} H_n$
Let $G=(\mathbb Q,+)$ and $r_1=p_1/q_1, r_2=p_2/q_2\in G$. I want to prove that:
$\langle r_1,r_2\rangle\subseteq \langle\frac{1}{q_1q_2}\rangle$
If $r_1,r_2,...,r_n\in G$ then $\langle r_1,r_2,...,r_n\rangle$ is cyclic.
If $H_n=\langle \frac{1}{n!}\rangle$ then e... |
H: Square Matrices Problem
Let A,B,C,D & E be five real square matrices of the same order such that ABCDE=I where I is the unit matrix . Then,
(a)$B^{-1}A^{-1}=EDC$
(b)$BA$ is a nonsingular matrix
(c)$ABC$ commutes with $DE $
(d)$ABCD=\frac{1}{det(E)}AdjE$
More than one option may be correct .
Also , taking... |
H: How to prove this function is surjective
I'm trying to solve this question:
In order to solve this question above, I found this function: $r/w\mapsto (r/s)/(w/s)$ such that $w/s\in T$, I almost proved this map is an isomorphism, I'm stuck just in the surjectivity part.
If we get an element $(r/s)/(w/s)$ of $T^{-1}... |
H: MCQ on a function
$f(x)=\left( \ln \left( \frac{\left( 7x-x^{2} \right)}{12} \right) \right)^{\frac{3}{2}}$
Choose correct options , more than one may be correct .
(a) $f$ is defined on $R^+$ and is strictly increasing.
(b)$f$ is defined on an interval of finite length and is strictly increasing
(c)range of functio... |
H: What does proving the Riemann Hypothesis accomplish?
I've recently been reading about the Millennium Prize problems, specifically the Riemann Hypothesis. I'm not near qualified to even fully grasp the problem, but seeing the hypothesis and the other problems I wonder: what practical use will a solution have?
Many ... |
H: Product of two functions converging in $L^1(X,\mu)$
Let $f_n\to f$ in $L^1(X,\mu)$, $\mu(X)<\infty$, and let $\{g_n\}$ be a sequence of measurable functions such that $|g_n|\le M<\infty\ \forall n$ with some constant $M$, and $g_n\to g$ almost everywhere. Prove that $g_nf_n\to gf$ in $L^1(X,\mu)$.
This is a quest... |
H: Finding the singularity type at $z=0$ of $\frac{1}{\cos(\frac{1}{z})}$
I have the following homework problem:
What kind of singular point does the function
$\frac{1}{\cos(\frac{1}{z})}$ have at $z=0$ ?
What I tried:
We note (visually) that $z_{0}$ is the same type of singularity for
both $f,f^{2}$ hence the ty... |
H: How to show that $\sqrt{1+\sqrt{2+\sqrt{3+\cdots\sqrt{2006}}}}<2$
$\sqrt{1+\sqrt{2+\sqrt{3+\cdots\sqrt{2006}}}}<2$.
I struggled on it, but I didn't find any pattern to solve it.
AI: $$\begin{aligned}\sqrt{1+\sqrt{2+\sqrt{3+\cdots \sqrt{n}}}}&<\sqrt{1+\sqrt{2+\sqrt{2^2+\cdots \sqrt{2^{2^{n-1}}}}}}\\&<\sqrt{1+\sqrt{2... |
H: If $b = c \times a$ and $c = a \times b$, and length $b$ = length $c$, $a$ is a unit vector.
If $\vec b = \vec c \times \hat a\,$ and $\,\vec c = \hat a \times \vec b\,$, and $|\vec b|$ = $|\vec c|$.
Assuming $\vec b \ne 0$.
I have managed to prove $\vec a$, $\vec b$ and $\vec c$ are orthogonal, but not much else... |
H: What are 'contexts' actually called?
Consider the following argument by contradiction.
\begin{array}{|l}
\mbox{We wish to deduce A.} \\
{\begin{array}{|l}
\mbox{Suppose not A.} \\
\hline \\
\mbox{Then B. Thus C. Therefore, contradiction. }
\end{array} } \\
\mbox{Thus, A.}
\end{array}
So to actually perform the arg... |
H: Chebyshev-Gauss quadrature with $\tan(x)$
With Chebyshev-Gauss quadrature, solve $\int_0^{\pi/4}x\cdot tan^2(x)$, for $n=3$.
Needs first to determine the change in the integral, to change the limits of integrals and then reduce in form integrate $$\int_{-1}^1{\dfrac{1}{(1-x^2)^{1/2}}}$$
http://upload.wikimedia.org/... |
H: ZF: Regularity axiom or axiom schema?
I have seen the axiom system ZF for set theory described including a single axiom of regularity (aka "foundation"), namely
$$\forall x\neq\emptyset \, \exists y\in x \ y\cap x = \emptyset$$
and also including regularity as an infinite axiom schema, with an axiom for every formu... |
H: Relationship between Mean Value Theorem and the maximum norm
I am seeking assistance with the following application of the Mean Value Theorem:
Let $x \in \Omega$ and construct an associated neighbourhood $N_x = (a, a+ \sqrt{\epsilon})$, such that $x \in N_x$ and $N_x \subset \Omega$. Then, by the Mean Value Theorem... |
H: solving an equilibrium equation
I have the following example:
$ pa_1=a_0\\ pa_0+qa_1+pa_2=a_1\\qa_0+qa_2=a_2 $
where p+q=1.
I can see how to get $a_1=(1/p)a_0 $ but from there they say from the third equation they produce $a_2 = (q/p)a_0$. Then the values are substituted into the normalising equation of $a_0 + a... |
H: Prove that if $p_1,\dots,p_k$ are distinct odd primes then 1 has $2^k$ square roots $\mod m$ where $m$ is the product of the primes.
I think I am most of the way through this proof but I am stuck. Here was my approach: I looked at the square roots of $1$ mod $105$, and noticed that each one corresponded to one less... |
H: Essential singularities of $\frac 1{e^z-1}$
How do I show that $\frac 1{e^z-1}$ has essential singularities (instead of say, poles) at $z=2n\pi i(n\in \mathbb Z)$?
I can't figure out how to show that the function does not go to infinity near $0$, or that it assumes every possible value near $0$. Exhibiting the laur... |
H: How to find the Laurent expansion for $1/\cos(z)$
How to find the Laurent series for $1/\cos(z)$ in terms of $(z-\frac{\pi}{2})$ for all $z$ such that $0<|z-\frac{\pi}{2}|<1$
AI: Note $\frac{1}{\cos z}=-\frac{1}{\sin (z-\frac{\pi}{2})}$.
Let $t:=z-\frac{\pi}{2}$. Then $0<|t|<1$, $\sin t=t-\frac{t^3}{3!}+\frac{t^5}{... |
H: Splitting of conjugacy class in alternating group
Browsing the web I came across this:
The conjugacy class of an element $g\in A_{n}$:
splits if the cycle decomposition of $g\in A_{n}$ comprises cycles of distinct odd length. Note that the fixed points are here treated as cycles of length $1$, so it cannot have m... |
H: Solve the following system of equations
Solve the following system of equations:
$\left\{\begin{matrix}
x^3(1-x)+y^3(1-y)=12xy+18\\
\left | 3x-2y+10 \right |+\left | 2x-3y \right |=10
\end{matrix}\right.$
AI: Perhaps asking Mathematica(WolframAlpha gives the answer as well) to solve it:
Solve[{x^3 (1 - x) + y^3 (1... |
H: $\Delta_{0}$ formulas
I am working through the Jech Set Theory book, and at the moment I am stuck at his definition of the $\Delta_{0}$ formulas:
A formula of set theory is a $\Delta_{0}$ formula if:
(i) it has no quantifiers, or
(ii) it is $\varphi \wedge \psi$, $\varphi \vee \psi$, $\neg \varphi$, $\varphi\righ... |
H: Error in book's definition of open sets in terms of neighborhoods?
The following is copied verbatim from a book (I. Protasov, Combinatorics of numbers, p. 14):
Suppose that to each point $x$ of a set $X$ a collection $\mathcal{B}(x)$ of subsets of $X$, which are called neighborhoods of $x$, is assigned so that the... |
H: Two questions about implications between $\mathsf{DC}, \mathsf{BPI}$ and $\mathsf{AC}_\omega$
Does the implication $\mathsf{DC} \implies \mathsf{BPI}$ hold?
And does the implication $\mathsf{BPI} \implies \mathsf{AC}_\omega$ hold?
I checked with Howard/Rubin's "Consequences of the Axiom of Choice" in part V where... |
H: How to get ${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$
I found this in my test book, any hints? Given $${n \choose 0}^2+{n \choose 1}^2+{n \choose 2}^2+\cdots+{n \choose n}^2 = {x \choose y}$$ Then find the value of x and y in n. According to the answer provided on last ... |
H: sum of coefficients of polynomial?
If $\sqrt{2+(\sqrt3 +\sqrt5)}$ is root of polynomial of eighth degree then, the sum of absolute values of coefficients of polynomial is?
I found this question on https://brilliant.org/assessment/s/algebra-and-number-theory/1974729/
I want to know is there any simple way to solve i... |
H: Binary search complexity
In sorted array of numbers binary search gives us comlexity of O(logN).
How will the complexity change if we split array into 3 parts instead of 2 during search?
AI: Same. You will get a running time only differing by constant factor ($\log_2 3=\frac{\ln 3}{\ln 2}$). |
H: From geometric sequence to function
I have this question:
Find the functions which equal the sums:
$$
x + x^3 + x^5 + ..
$$
Now, I can see in my result list, that its supposed to give
$$
\frac{x}{(1-x^2)}
$$
I can see why the numerator should be x, but I fail to see why the denominator should be
$$
(1-x^2)
$$
Can... |
H: Slope of tangent in $(x,y)$ on a circle $K=\{(x,y)\in\mathbb{R}^2|x^2+y^2=r^2\}$ with initial conditions
I came across an exercise with a sample solution that I unfortunately don't fully understand given that it's shortened.
Let $(x,y)$ be a point on a circle $K=\{(x,y)\in\mathbb{R}^2|x^2+y^2=r^2\}$ with $y\neq0$. ... |
H: Geodesic and Euler - Lagrange equation
If we have a metric $g_{\mu \nu}$, defined in a Riemannian manifold we can write the equation of the geodesic:
$$\frac{d^2x^\mu}{dt^2}+\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}$$ in which $\Gamma^\mu_{\alpha\beta}$ are the Christoffel symbols. The geodesi... |
H: Truncating a Time-Series Graph
I'm looking for a way to truncate outliers in a time-series graph.
Some context:
I'm plotting two different metrics on a single graph. It's important to understand what portion of "Metric A" takes up respective to "Metric B". "Metric B" can at times spike to a high value thus incr... |
H: Find point nearest to the origin
Find the points on the curve $5x^2 - 6xy + 5y^2 = 4$ that are nearest the origin.
The first method I've tried is I've taken the derivative of the equation to optimize (Pythagorean Theorem) and also the function of the curve using implicit differentiation and plugged stuff in (the wa... |
H: Number of solutions
I have a polynomial in integers $\psi (x)$ of degree $k$.
Consider the number of solutions
$$
\psi(z) \equiv u (\mod p^r)
$$
with
$$
(\psi'(z),p)=1.
$$
I was wondering how can I show that the number of solutions is $O(1)$?
Thank you!
AI: There are at most $\max(k,p)\leq p$ solutions to $\psi (... |
H: is $\hat{\theta}$ unbiased
consider a random sample of size n from a distribution with pdf $f(x;\theta)=\frac{1}{\theta}$ $0<x\leq \theta$ and zero otherwise. $0< \theta$
Now the first question was to find the MLE of $\hat{\theta}$ which I found to $X_{n:n}$ , now they want to find out if it is unbiased. My work... |
H: Why can a $2k$-regular graph be $k$-factored?
From Wikipedia:
If a connected graph is $2k$-regular it may be $k$-factored, by choosing
each of the two factors to be an alternating subset of the edges of an
Euler tour.
I don't understand why those alternating subsets form $k$-factors.
AI: In a connected $2k$-r... |
H: Def. of Homology Cell
I would like an explanation for the following definition.
"A metric space is an $\ homology \ cell $ if it is nonempty and homologically trivial (acyclic) in all dimensions."
What does "homologically trivial (acyclic)" mean?
What are intuitive examples of homology cells?
AI: From your questio... |
H: Problem on filters
$\mathcal{F}$ is filter on $\mathcal{I}$, but not ultrafilter.
Prove that $\exists$$\mathcal{X, Y}\notin\mathcal{F}$ | $\forall\mathcal{Z}\in\mathcal{F}$ $\mathcal{X}\cap\mathcal{Z}\neq$ $\mathcal{Y}\cap\mathcal{Z}$
Thank you for your time.
AI: Do you know that a filter $\mathcal F$ on a set $I$ ... |
H: How to formulate continuum hypothesis without the axiom of choice?
Please correct me if I'm wrong but here is what I understand from the theory of cardinal numbers :
1) The definition of $\aleph_1$ makes sense even without choice as $\aleph_1$ is an ordinal number (whose construction doesnt depend on the axiom of c... |
H: About Regulated Functions
Definition. Let $X$ be a Banach space. A mapping $f:[a,b]\to X$ is called regulated if it has one sided limits.
In the setting of a Hausdorff topological vector space $X$, can we still define regulated functions?
AI: As in a Hausdorff topological vector space the one sided limit of a funct... |
H: How to solve these equations with $x^2$, $xy$, and $y^2$?
Is there any good method to solve equations like this?
$$
\begin{cases} 2y-2xy-y^2=0\\2x-x^2-2xy=0\end{cases}
$$
This is what I did:
$$
\begin{cases} y(2-2x-y)=0\\x(2-x-2y)=0\end{cases}
$$
now I see, that:
$$
x=0
$$
$$
y=0
$$
and it's the first solution, now... |
H: Rudin Theorem 2.41 - Heine-Borel Theorem
When proving Theorem 2.41 in Principles of Mathematical Analysis:
Let $E \subset \mathbb{R}^k$. If every infinite subset of $E$ has a limit point in $E$, then $E$ is closed and bounded.
Rudin says,
"If $E$ is not bounded then $E$ contains points $x_n$ with
$$\vert x_n \ve... |
H: how can we prove that this function is homomorphism?
If $G_1 , ... , G_n$ are groups
let $G=G_1\times G_2\times\dotsb\times G_n$.
$E_p\colon G\to G_{p^{-1}(1)}\times G_{p^{-1}(2)}\times\dotsb\times G_{p^{-1}(n)}$
where $p \in S_n$ is defined as
$(g_1 , g_2 , \dotsc , g_n )\mapsto(g_{p^{-1}(1)} , g_{p^{-1}(2)} , \... |
H: Commutativity in rings of matrices
Let n be arbitrary positive integer and R arbitrary ring (perhaps, non-associative). Let's denote $M_n(R)$ the set of all $n х n$ matrices with entries from R.
As i know if R is non-trivial commutative ring without zero divisors then scalar matrices are the only matrices which com... |
H: Zorn's lemma implies the well-ordering principle
I am little confused about the proof given here http://euclid.colorado.edu/~monkd/m6730/gradsets05.pdf
On the second page, when defining $P$, the author says that $B\subset A$ and $(B,<)$ is a well-ordering structure. Isn't this exactly what we want to prove? How do ... |
H: Integral proof of logarithm of a product property
In one of my textbooks, the expansion of a logarithm product is proved using integrals.
$$\ln xy = \ln x + \ln y\iff \int_1^\left(xy\right)dt/t$$
$$\ = \int_1^xdt/t + \int_x^\left(xy\right)dt/t$$
Then, let $ u = t/x $ and substitute in the second integral:
$$= \int... |
H: How to solve equation like this?
Is there any good method to solve equations like this?
$$
\begin{cases} x^2=y^2\\-6xy+10y^2=16\end{cases}
$$
From the first equation, I see:
$$
1) x=1, y=1
$$
$$
2)x=1, y=-1
$$
$$
3)x=-1, y=1
$$
$$
4)x=-1, y=-1
$$
When I plug 2) and 3) to the second equation, I see it works.
So now ... |
H: Show the surface area of revolution of $e^{-x}$ is finite
I need to show that the surface area of revolution of $e^{-x}$ is finite when the region from $x=0$ to $x=\infty$ is rotated about the x-axis.
I tried using the surface area formula, but got stuck on the integration, so I thought that maybe there was a trick... |
H: Two closed subsets $A$ and $B$ in $\mathbb{R}$ with $d(A,B)=0$
I am looking for two closed subsets A and B (with $A\cap B = \emptyset$) of $\mathbb{R}$ with $d(A,B)=0$. I found a solution in $\mathbb{R}^2$, namely $A=\{(x,\frac{1}{x})\mid x>0\}$ and $B=\{(x,0)\mid x>0\}$. I know that those subsets have to be unboun... |
H: What is the limit of $\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ { k }^{ \left( \frac { 1 }{ 2k } \right) } } } $
When I was trying to solve this question
I came up to this: $$\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ { k }^{ \left( \frac { 1 }{ 2k } \right) } } } ={ 1 }^{ 1/2 }+{ 2 }^{ 1/4 ... |
H: $\operatorname{\mathcal{Jac}}\left( \mathbb{Q}[x] / (x^8-1) \right)$
$\DeclareMathOperator{\Jac}{\mathcal{Jac}}$
Using the fact that $R := \mathbb{Q}[x]/(x^8-1)$ is a Jacobson ring and thus its Jacobson radical is equal to its Nilradical, I already computed that $\Jac \left( \mathbb{Q}[x] / (x^8-1) \right) = \{0\}$... |
H: Evaluate the given limit by recognizing it as a Riemann sum.
Please note that this is homework. Please excuse my lack of $\LaTeX{}$ knowledge.
The Problem:
Evaluate the given limit by first recognizing the sum (possibly after taking the logarithm to transform the product into a sum) as a Riemann sum of appropriate... |
H: Dual vector space, proof of inclusion in a span
Question:
V is a vector space (of finite dimension) over F. We assume $\alpha, \beta \in V^*$ (The dual space) and they satisfy: $\forall v: (\alpha (v)=0 \Rightarrow \beta (v)=0$)
prove that $\beta \in sp(\alpha)$.
What I thought:
What I understand is that I need t... |
H: Holomorphic vs differentiable (in the real sense).
Why a holomorphic function is infinitely differentiable just because of satisfying the Cauchy Riemann equations, but on the other side, a two variable real function that is twice differentiable is not infinitely differentiable?
I'm asking this for two reasons:
1) $... |
H: A binomial inequality with factorial fractions: $\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$
Prove that $$\left(1+\frac{1}{n}\right)^n<\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+...+\frac{1}{n!}$$ for $n>1 , n \in \mathbb{N}$.
AI: We have by the binomial identity that
\begin{a... |
H: Convergence of infinite series $\sum (-1)^{n+1}\frac{1}{n!}$
I have this question: Do the series converge absolutely or conditionally?
$$
\sum (-1)^{n+1}\frac{1}{n!}
$$
I would say it does not converge absolutely, since I suggest, by using the ratio test, that
$$
\frac{a_{n+1}}{a_n}
$$
does not approach a limit
... |
H: Coordinate geometry: calculating the height of an equilateral triangle
If I have equilateral $\Delta ABC$ with A being $(-x,0)$ and B being $(x,0)$, how can I solve for the coordinates of C in terms of $x$?
I tried the following:
$2x^2 = x^2 + b^2 $ -- pythagorean thm, since we know that one side of the triangle is... |
H: Decreasing function
Given two increasing function $f(x)$ and $g(x)$ with all $x \ge 0$. Moreover, $f(x) > g(x)
$ for all $x \ge 0$ and $$\mathop {\lim }\limits_{x \to \infty } \left( {f(x) - g(x)} \right) = 0.$$
Is $d(x) = f(x)-g(x)$ a decreasing function ?
AI: It is not even eventually decreasing. Consider
$$
f... |
H: $f$ is continuous at $a$ iff for each subset $A$ of $X$ with $a\in \bar A$, $f(a)\in \overline{ f(A)}$.
Definiton.
$f$ is continuous at $a$ provided that for each open set $V$ in $Y$ containing $f(a)$ there is an open set $U$ in $X$ containing $a$ such that $f(U) \subset V$.
Problem.
$f$ is continuous at $a$ iff fo... |
H: definition of discriminant and traces of number field.
Let $K=\Bbb Q [x]$ be a number field, $A$ be the ring of integers of $K$.
Let $(x_1,\cdots,x_n)\in A^n$. In usual, what does it mean $D(x_1,\cdots,x_n)$? Either $\det(Tr_{\Bbb K/ \Bbb Q} (x_ix_j))$ or $\det(Tr_{A/ \Bbb Z} (x_ix_j))$? Or does it always same valu... |
H: For all $f: D_1(0) \to D_1(0)$ analytic with $f(\frac{i}{3}) = 0$, find $\displaystyle \sup_f\{\operatorname{Im} f(0) \}$
Let $\mathcal{F}$ denote the family of all analytic functions $f$ that map the unit disc onto itself with $f(\frac{i}{3}) = 0$. Find $M \equiv\sup\{\operatorname{Im} f(0) : f \in \mathcal{F}\}$.... |
H: Infinite series for arctan of x
this is a bit of a vague question so I won't be too surprised if I get vague responses.
$$\tan^{-1}(x) = x - (x^3 / 3) + (x^5 / 5) - (x^7 / 7) + \cdots $$ ad infinitum
I'm using this, where $x = (\sqrt{2} - 1)$ to calculate $\pi$, as $\pi = 8 \tan^{-1}(x)$
I have never really learn... |
H: On the notion of sheaf
Citing form Borceux, Handbook of categorical algebra, in the preface to volume 3:
The crucial idea behind the notion of a sheaf is to work not just with a "plain" set of elements, but with a whole system of elements at various levels. Of course, reasonable rules are imposed concerning the in... |
H: Prove positive definite matrix and determinant inequality
$A$ and $B$ are two real symmetric matrices and $A \succeq B$ (that means $A-B$ is a psd matrix), does that hold $|A|\ge|B|$? and why?
AI: Of course not. Consider, e.g. $A=I_2$ and $B=-2I_2$ with $\det A=1<4=\det B$.
Edit: However, the statement is true if $... |
H: secant method in maple
secant method in maple. Find a root of the statement $x^3-3x^2+4x-1=0$ with the initial value $x_0=0$ and $x_1=1$ with 5 digits point approximation.
AI: If this is homework and you're supposed to program it yourself then you could show what you've accomplished on your own already.
If you jus... |
H: Symbol for Mutual Inclusive events
is there any symbol for mutual inclusive(opposite of Mutually exclusive) events in probability.
I meant to say is for OR we have U symbol in Set Theory. Likewise is there any symbol for Mutual Inclusiveness.
AI: You want to say that two events $A$ and $B$ satisfy
If $A$, then $B$... |
H: Splitting open sets in perfect spaces
Suppose $V$ is a Hausdorff space which is perfect and $U\subset V$ is a non-empty open set. Can we find two disjoint, non-empty open sets $U_1, U_2\subset U$? Is there any natural class of spaces having this property?
AI: Sure: just pick distinct $x,y\in U$, use the fact that $... |
H: Probability example for homework
There's a highway between two towns. To reach the other, people must pay 10 dollars for cars and 25 dollars for bigger vehicles. How big income can we expect if the 60 percent of vehicles are cars and there are 25 incoming vehicles per hour?
What kind of distribution does it follow,... |
H: Integrating over a triangle
Let $\hat T$ be the triangle spanned by $(0,0)$, $(1,0)$ and $(0,1)$. Let
$$I(r,s) = \int_{\hat T} x^r y^s d(x,y)$$
with $r,s\in\mathbb N\cup\{0\}$.
Prove that
$$I(r,s) = \frac{r!s!}{(2+r+s)!}$$
To show this I rewrote the integral to
$$\int_0^1 \int_0^{-x+1} x^r y^s dy dx = \frac{1}{s... |
H: elementary substructure of $H(\kappa)$
$H(\kappa)=\{x:|TC(x)|<\kappa\}$, with $\kappa$ regular card. and $TC(*)$ the transitive closure.
Now we defined an elementary substructure:
$M\prec H(\kappa)$ if for every formula $\varphi$ and all $a_{1},\dots, a_{n}\in M$ holds: $\varphi^{M}(a_{1},\dots,a_{n})\leftrightar... |
H: A nonlinear differential equation
We are to solve
$$y=(y'-1)\cdot e^{y'}$$
Let $p=y'$, so $$y= (p-1)\cdot e^p$$
Differentiate: $$dy=(e^p + pe^p-e^p)dp=pe^p\, dp$$
From $$dx=\dfrac{dy}{p}=e^p\,dp$$
I find $$x=\int(e^p)dp=e^p$$.but the answer on my book is $$y=[\ln(x-c)-1]\cdot(x-c)$$
Help?
AI: $$x=e^p \implies p=\ma... |
H: Linear Independence of $v$, $Av$, and $A^2v$
Let A be a 3x3 matrix and $ v \in \Re^3 $ with $ A^3v = 0 $ but $ A^2v \neq 0 $.
Show that the vectors $ v, Av $ and $ A^2v $ are linearly independent.
(From Alan MacDonald's Linear and Geometric Algebra, Problem 3.1.5).
AI: First, you might try to see what would happen ... |
H: Counting number of solutions with restrictions
I want to count the number of non-negative integer solutions to an equation such as
$$x+5y+8z=n$$
I can do this using generating functions; for example, the answer here is
$$[x^n]\frac{1}{(1-x)(1-x^5)(1-x^8)}$$
where $[x^n]$ is the coefficient of $x^n$.
But what if I ... |
H: Drawing a triangle in a unit circle
This is a question that I derived for a long time ago. It asks if we draw a triangle in a unit circle does all arc lengths $(\alpha ,\beta ,\theta)$ and sides of triangle $(a,b,c)$ can be rational numbers? Intuitively I believe that the rationality does not hold but I can't deriv... |
H: How to isolate j?
can anyone explain me how to isolate the j variable please?
$$q = \frac{1 - (1 + j)^{-n}}{j} p $$
TIA
AI: This can be solved exactly ("isolating $j$") only under very special circumstances. If $n > 4$ it is probably hopeless (and for $n = 3$ or 4 the exact solution will turn out to be a horrible m... |
H: A property of radical ideals
Let $A$ be a commutative ring with $1 \neq 0$.
Theorem (Atiyah-MacDonald 1.13 (v)). Let $\mathfrak{a, b} \subseteq A$ be ideals. Then $\sqrt{\mathfrak{a + b}} = \mathfrak{\sqrt{\sqrt{a} + \sqrt{b}}}$.
Question. Is the following generalization true?
For any finite collection of ideals ... |
H: Number of abelian groups Vs Number of non-abelian groups
I would like to see a table that shows the number of non-abelian group for every order n. It is a preferable if the table contains the number of abelian groups of order n (this is not necessary though). If anyone could provide me with such a table , I would b... |
H: Principal ideal domain not euclidean
Can anyone give an example of a principal ideal domain that is not Euclidean and is not isomorphic to $\mathbb{Z}[\frac{1+\sqrt{-a}}{2}]$, $a = 19,43,67,163$?
I believe it is conjectured that no other integer rings of number fields have this property. What about other rings?
AI:... |
H: Supposed counterexample to Liouville's theorem
I'm trying to understand Liouville's theorem, and I don't see why $f(z)=e^{-|z|^2}$ isn't a counterexample. It's bounded ($0 < f(z) \leq 1$), so it must be somehow that it's not holomorphic. Isn't it differentiable everywhere?
AI: No, it's nowhere differentiable (exc... |
H: Need help simplifying complicated rational expression.
Studying for my final and I can't figure this out.
Simplify:
$$\large\frac{\frac{3}{x^3y} + \frac{5}{xy^4}}{\frac{5}{x^3y} -\frac{4}{xy}}$$
AI: The shortest route is to note that the least common multiple of all four denominators of the ‘small’ fractions is $x^... |
H: Comparing $2^n$ to $n!$
Comparing $\;2^n\;$ and $\;n!\;$
I need to rewrite one or the other so they can be more easily compared (i.e. have similar form). Because of the factorial, I'm a little lost as to how to compare the two functions. Normally, I would take the logarithm when trying to re-express a power like $2... |
H: Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$
Is $\sum_{n=1}^{\infty} {x^2 e^{-nx}}$ uniformly convergent in $[0,\infty)$?
So I started by saying that by the geometric series test where $a=x^2$ and $|r| = |\frac{1}{e^x}| \leq 1$, the series converges pointwise.
But how do I exactly p... |
H: $\lVert A\lVert<1$ implies $1-A$ invertible only true in complete spaces?
It is a well known fact that
if in a Banach space $X$ a bounded linear operator $A:X\to X$ satisfies $\lVert A\lVert<1$, then $1-A$ has a bounded inverse.
I was wondering wether completeness is actually a necessary condition. I think the f... |
H: Coloring dots in a circle with no two consecutive dots being the same color
I ran into this question, it is not homework. :)
I have a simple circle with $n$ dots, $n\geqslant 3$. the dots are numbered from $1\ldots n$.
Each dot needs to be coloured red, blue or green. No dot can be couloured the same as his neighbo... |
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