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H: Limit of $\lim_{x \to 0}\left (x\cdot \sin\left(\frac{1}{x}\right)\right)$ is $0$ or $1$?
WolframAlpha says $\lim_{x \to 0} x\sin\left(\dfrac{1}{x}\right)=0$ but I've found it $1$ as below:
$$
\lim_{x \to 0} \left(x\sin\left(\dfrac{1}{x}\right)\right) = \lim_{x \to 0} \left(\dfrac{1}{x}x\dfrac{\sin\left(\dfrac{1}... |
H: Some Results in $\mathbb{Z} [\sqrt{10}]$
This is a question from an old Oxford undergrad paper on calculations in $\mathbb{Z} [\sqrt{10}]$. We equip this ring with the Eucliden function $d(a+b\sqrt{10})=|a^2-10b^2|$. I want to prove the following results:
If $d(x)=1$, then $\frac{1}{x} \in \mathbb{Z} [\sqrt{10}]$
... |
H: Find $y$-Lipschitz constant
$$f(x,y)=x^3e^{-xy^2}, 0\leq x\leq a, y\in \mathbb R, a>0$$
I need to find $K>0$ such that $$|f(x,y_1)-f(x, y_2)|\leq K|y_1-y_2|$$ for all $0\leq x\leq a$ and $y_1,y_2\in \mathbb R$
I did this $$|x^3e^{-xy_1^2}-x^3e^{-xy_2^2}|=x^3|e^{-xy_1^2}-e^{-xy_2^2}|\leq a^3|e^{-xy_1^2}-e^{-xy_2^2}|... |
H: Subrings of $\mathbb{Q}$
Let $p$ be prime. Suppose $R$ is the set of all rational numbers of the form $\frac{m}{n}$ where $m,n$ are integers and $p$ does not divide $n$.
Clearly then $R$ is a subring of $\mathbb{Q}$.
I now want to show that if $\frac{m}{n}$ belongs to any proper ideal of $R$ then $p|m$.
Can someone... |
H: solve $y(x)=\cos \left(y'(x)\right) + y'(x)\sin (y'(x)), y(0)=1$
solve $$y(x)=\cos (y'(x)) + y'(x)\sin (y'(x)), y(0)=1$$
with wolfram alpha I got that a solution is $y(x)=x\arcsin x+\cos (\arcsin x)$
but I have no idea how to find it.
I tried transforming into an exact equation by letting $u=y'$ then I get $y=\frac... |
H: Smooth maps on a manifold lie group
$$
\operatorname{GL}_n(\mathbb R) = \{ A \in M_{n\times n} | \det A \ne 0 \} \\
\begin{align}
&n = 1, \operatorname{GL}_n(\mathbb R) = \mathbb R - \{0\} \\
&n = 2, \operatorname{GL}_n(\mathbb R) = \left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\Bigg| ad-bc \ne 0\right\}
\end{align}
$... |
H: Closed sets in a given topology
I came across a topology on $ \mathbb Z \times \mathbb Z $ whose basis is defined as follows:
$ B(m,n) = \lbrace (m,n) \rbrace $ if both m and n are odd
$ B(m,n) = \lbrace (m+a, n) | a = -1, 0 ,1 \rbrace $ when m is even and n is odd
$ B(m,n) = \lbrace (m, n+a) | a = -1, 0 ,1 \rbrace... |
H: How to determine a function of 2 variables from its derivative?
Please even the slightest advice would help!
If I have a function $V$ made of 2 variables $x_1$ and $x_2$,
and its derivative $$\frac{dV}{dt} = \frac{dV}{dx_1}\frac{dx_1}{dt} + \frac{dV}{dx_2}\frac{dx_2}{dt},$$
how do I find $V$? I don't know how t... |
H: How to show that there does not exist any integer $b$ with $f(b)=14.$
Let $f(x)$ be a polynomial with integer coefficients. Suppose that there exist distinct integers $a_1,a_2,a_3,a_4,$ such that $f(a_1)=f(a_2)=f(a_3)=f(a_4)=3.$ Then show that there does not exist any integer $b$ with $f(b)=14.$
AI: We have $f(x)=... |
H: A calculation of the norm of an ideal
Let $L$ be a number field of degree $n$ over $\mathbb{Q}$ and $\mathfrak{a}$ a non-zero ideal of the ring of integers $\mathcal{O}_L$. Suppose that $X=\{x_1,...,x_n\}$ is a $\mathbb{Z}$-basis of $\mathcal{O}_L$, and $Y=\{y_1,...y_n\}$ is a $\mathbb{Z}$-basis of $\mathfrak{a}$.
... |
H: Why is boundary information so significant? -- Stokes's theorem
Why is it that there are so many instances in analysis, both real and complex, in which the values of a function on the interior of some domain are completely determined by the values which it takes on the boundary?
I know that this has something to do... |
H: The relationship between plane curves and the derivative of the Wronskian
I have found a theorem but I did not understand the proof. I'm looking for a clarification of the proof or a different proof.
Let $f_1, f_2, f_3$ be the three components of a curve in $R^3$ parameterized by $t$. Let $W$ be the Wronskian matri... |
H: permutation/combination problem
There are 3 doors to a lecture room. In how many ways can a lecturer enter the room from one door and leave from another door?
I have done like this: They way of entering is 3 and exiting is also
3, therefore the total way will be 3*3=9.
AI: That’s the total number of ways in whic... |
H: Limit: $\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$
How can I find the following limit?
$$\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$$
It's a limit of type $\displaystyle 1^{\infty}$ and if I note with $\displaystyle f(x)=\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)$ then I mus... |
H: Characterization for compact sets in $\mathbb{R} $ with the topology generated by rays of the form $\left(-\infty,a\right) $
I'm trying to find a sufficient and necessary condition for a subset to be compact in $\mathbb{R} $ when the topology is generated by the basis $\left\{ \left(-\infty,a\right)\,|\, a\in\mat... |
H: How many friends for birthday party every weekend?
A friend claims that he is invited to a birthday party every weekend.
I know he needs to have at least 52 friends for that, but what is the "realistic" amount of friends that chances are 100% for beeing invited every weekend
AI: There is on $100\%$ guarantee, but y... |
H: possible combinations of 3-digit
How many possible combinations can a 3-digit safe code have?
Because there are 10 digits and we have to choice 3 digits from this,
then we may get $10^P3$ but A author used the formula $n^r$, why is that. What the problem in my calculations?
AI: The error is exactly the one that I ... |
H: Drawing balls with replacement, until I have one of each.
A urn has (n+1) types of balls, n of unique colors and the rest black. When picking a ball randomly from the urn, a colored (non black) ball has a probability of p of being picked. Each ball of color has equal probability of being picked, ie each has a (p/n)... |
H: Solve the recursion, $a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$
Bring the following recursion relation to an explicit expression:
$$a_n = 3a_{n-1}-3a_{n-2}+a_{n-3}+8$$
$a_{0} = 0$, $a_1 = 1$, $a_2 = 2$
All the examples I have seen were with maximum 2 steps back ($a_{n-2}$) and I thought I know how to solve those but I'm h... |
H: Square-Trangular Numbers Checking Answer
Problem: The first 2 numbers that are both squares and triangles are 1 and 36. Find the next one and if possible, the one after that.
Answer: 1225, 41616
Problem: Can you figure out an efficient way to find triangular-square numbers?
Answer: $s^2 = t(t+1)/2$, where $s,t \eps... |
H: Integrate ${\sec 4x}$
How do I go about doing this? I try doing it by parts, but it seems to work out wrong:
$\eqalign{
& \int {\sec 4xdx} \cr
& u = \sec 4x \cr
& {{du} \over {dx}} = 4\sec 4x\tan 4x \cr
& {{dv} \over {dx}} = 1 \cr
& v = x \cr
& \int {\sec 4xdx} = x\sec 4x - \int {4x\sec 4x\t... |
H: Elliptic Surfaces: a naive question
Ground field $\Bbb{C}$. Algebraic category. Smooth surfaces.
Let $S$ be a minimal elliptic surface $p:S\rightarrow C$ the elliptic fibration (general fiber = elliptic curve). Suppose the $m$-canonical system is non-empty and let $D\in \lvert m K \rvert$.
Why can we say that $D.F... |
H: Can someone clarify this implication
I'm reading a finance book, and I saw this implication that I don't understand. I mean where this g function come from? If someone can clarify this I would appreciate. Thanks.
If a have a function like $f(x,t)$ and the following equation
$ \frac{d f}{f}=\sigma dx \Rightarrow lnf... |
H: rounding up to nearest square
Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner?
ie.
$2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. What it does when x = 4 or 9 doesn't really matter.
I could write a function to do this... |
H: Can someone clarify Example I.I.2 from Hardy's Course of Pure Mathematics?
"If $\lambda, m,$ and $n$ are positive rational numbers, and $m > n$, then $\lambda(m^2 − n^2), 2\lambda mn$, and $\lambda(m^2 + n^2)$ are positive rational numbers. Hence show how to determine any number of right-angled triangles the length... |
H: equality of integrals without trigonometry
can someone show the equality of these two integrals without using any trigonometry;
$$ \int_{0}^{1} \frac{dt}{\sqrt{1-t^2}} = 2 \int_{0}^{1} \sqrt{1-t^2} \, dt $$
i'm working through a derivation of relationship between a circle's circumference and area and this equality ... |
H: Convergence of sequence
Does the following:
$$
\begin{align}
x_0 & = a \\
x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\
x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\
x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) \\
& {}\,\vdots \\
x_n & = x_{n-1} + \frac{1}{n}(b + x_{n-1}(c - 1))
\end{align}
$$
where $x_{0}$ is switche... |
H: Integral of $\cot^2 x$?
How do you find $\int \cot^2 x \, dx$? Please keep this at a calc AB level. Thanks!
AI: We use the identity: $$\cot x = \pm \sqrt{\csc^2 x - 1} \implies \cot^2 x = \csc^2 x - 1$$
So we can rewrite the integral as follows:
$$\int \cot^2 x \,dx = \int \left(\csc^2x - 1\right)\, dx$$
$$ \int \l... |
H: If $\mathbb f$ is analytic and bounded on the unit disc with zeros $a_n$ then $\sum_{n=1}^\infty \left(1-\lvert a_n\rvert\right) \lt \infty$
I'm going over old exam problems and I got stuck on this one.
Suppose that $\mathbb{f}\colon \mathbb{D} \to \mathbb{C}$ is analytic and bounded. Let $\{a_n\}_{n=1}^\infty$ be
... |
H: How to integrate $\cos\left(\sqrt{x^2 + y^2}\right)$
Could you help me solve this?
$$\iint_{M}\!\cos\left(\sqrt{x^2+y^2}\right)\,dxdy;$$
$M: \frac{\pi^2}{4}\leq x^2+y^2\leq 4\pi^2$
I know that the region would look like this and I need to solve it as difference of two regions. Lets say $C=A\cup B$, then $A = C - B$... |
H: $\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_2\cdot\aleph_1^{\aleph_0}$
I've seen this statement in multiple posts (e.g. here and here), but I can't seem to understand it. I can see why
$$\sum_{\alpha<\omega_2}|\alpha|^{\aleph_0}=\aleph_1^{\aleph_0},$$
by noting that every $\alpha<\omega_2$, so $|\alpha|\leq\a... |
H: $X$ topological space. $A$ open $A \cap Y = \emptyset \ \ \Longrightarrow A \cap \bar{Y} = \emptyset$?
I know this is an easy question, but I cannot demonstrate it properly.
Suppose by contradiction that $A \cap \bar{Y} \neq \emptyset$. Then $\exists \ x \in A \cap \bar{Y}$.
I need help formalizing this reasoning ... |
H: Simplify with fractional exponents and negative exponents
I am trying to simplify
$$ \left(\frac{3x ^{3/2}y^3}{x^2 y^{-1/2}}\right)^{-2} $$
It seems pretty simple at first. I know that a negative exponent means you flip a fraction. So I flip it.
$$ \left(\frac{x^2 y^{-1/2}}{3x^{3/2}y^3}\right)^2$$
Now I need to sq... |
H: Exponential integral question
How would I solve the following problem?
$$f(x)=\int\!\frac{4}{\sqrt{e^x}}\,dx$$
Using $u$ substitution I have set $u=e^x$ andd $du=e^x dx$
so would I have $$4\int\!\frac{du}{\sqrt{u}}$$
What would I do from here?
AI: Notice: You may use your method, but in doing so, recall that $\colo... |
H: Express $4+\sqrt{-2}$ as a product of irreducibles
This is part of an old Oxford Part A exam paper. (1992 A1)
Suppose we equip $R=\mathbb{Z}[\sqrt{-2}]$ with the Euclidean function $d$ defined by $$d(m+n\sqrt{-2})=|m+n\sqrt{-2}|^2$$
I want to determine the units of $R$ and express $4+\sqrt{-2}$ as a product of irre... |
H: Generalising the Chinese Remainder Theorem
We have that for $I,J$ ideals of some ring $R$ with $R=I+J$, $$\frac{R}{I\cap J} \cong \frac{R}{I} \times \frac{R}{J}$$
My question is whether the analogous expression for three ideals $I,J,K$ where $R=I+J+K$ is true?
I think I have found a counterexample with $R=\mathbb{... |
H: Dice probability when a number is disallowed in the first round
In the game Settlers of Catan a player starts each turn rolling 2 six sided dice. There's a variation of the game where if a 7 is rolled in the first round of a game (a 'round' is when each player has taken a turn) it doesn't count an needs to be re-ro... |
H: Calculating in quotient ring of $\mathbb{R}[X]$
Part of an old Oxford exam (1992 A1)
We want to find which elements of the quotient ring $\mathbb{R}[X]/(x^3-x^2+x-1)$ are equal to their own square.
Now, we note first that $x^3-x^2+x-1=(x-1)(x^2+1)$
Let $f(x)=(x-1)(x^2+1)$. Clearly we have $[1+(f(x))]^2=[1+(f(x))]$ ... |
H: How can I evaluate $\sum_{n=1}^{\infty} \frac{1+\sin^2 n}{n+n^{1.5}}$?
How can I evaluate this sum? My teacher asked but I can't get it.
$$\sum_{n=1}^{\infty} \frac{1+\sin^2 n}{n+n^{1.5}}$$
AI: You cannot expect a nice closed form answer to this. However, it is easy to show that it is bounded and thereby it conve... |
H: Counting Problem - N unique balls in K unique buckets w/o duplication $\mid$ at least one bucket remains empty and all balls are used
I am trying to figure out how many ways one can distribute $N$ unique balls in $K$ unique buckets without duplication such that all of the balls are used and at least one bucket rema... |
H: Topological extension property
Let $X$ and $Y$ be topological spaces. We say that the extension property holds if, whenver $S$ is a closed subset of $X$ and $f:S\rightarrow Y$ is continuous, $f$ can be extended to a continuous map on $X$.
A question asks whether the extension property holds for $X$ being any topo... |
H: Does $n n^{1/n} =O(n)$?
I was asked does $n n^{1/n} =O(n)$ ?
I can see that the left hand side is always bigger than $n$ but how would you prove the equality is false?
AI: The equality is true. Recall that $\mathcal{O}(n)$ means the function is eventually $\leq C n$, where $C$ is a positive constant.
In your case, ... |
H: $\pi$ is just a number, or also the circumference of a sub-unit circle?
A unit circle defined in the Cartesian plane has a radius of $1$ and a diameter of $2$. So making a full round is $2 \pi$. Now, $\pi$ is the ratio of the circumference over the diameter, so if I have a circle with diameter $1$ (radius $0\mathor... |
H: Please help finishing the calculation to find the Entropy of Pareto distribution.
Let $X$ follow Pareto distribution with parameters $\alpha, a, h$. That is, $X\sim Pa(\alpha,a,h)$, where $\alpha>0$ is the shape parameter, $-\infty < a < \infty$ is the location parameter, and $h>0$ is the scale parameter.
Then the ... |
H: How can I evaluate this sum?
How can I evaluate this sum ?
$$\sum_{n=1}^{m}n^p$$
I know when $p\in \Bbb N$ but when $p\in\Bbb R$ what do I do ?
please help and thanks for all
AI: If $p \in \mathbb{N}$, you have the Faulhaber's formula, i.e.,
$$\sum_{n=1}^m n^p = \dfrac1{p+1} \sum_{j=0}^p (-1)^j \dbinom{p+1}j B_j m^... |
H: Why do these two volume integrals express different values?
This is a question from a textbook I am reading:
Find the volume when the region R is revolved about the y-axis
when R is defined as the region bounded by $x = \sqrt{16-y^2}$ and
$x = 0$.
Because the function is symmetrical about the y-axis and being... |
H: How to deal with polynomial quotient rings
The question is quite general and looks to explore the properties of quotient rings of the form $$\mathbb{Z}_{m}[X] / (f(x)) \quad \text{and} \quad \mathbb{R}[X]/(f(x))$$
where $m \in \mathbb{N}$
Classic examples of how one can treat such rings is to find relationships l... |
H: Identity with Bernoulli numbers: $\sum\limits_{k=1}^{n}k^p=\frac{1}{p+1}\sum\limits_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j}$
How I can prove that
$$\sum_{k=1}^{n}k^p=\frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j}B_j n^{p+1-j},$$
where $B_j$ is the $j$th Bernoulli number?
I hope to find the answer. Thanks for help.
AI: This... |
H: Quadratic Equation with "0" coefficients
Let's say I have two objects $x$ and $y$ whose position at time $t$ is given by:
$$
x = a_xt^2+b_xt+c_x \\
y= a_yt^2+b_yt+c_y
$$
And I want to find which (if any) values of $t$ cause $x$ to equal $y$. That is
$$
(a_x-a_y)t^2+(b_x-b_y)t+(c_x-c_y) = 0
$$
This can easily be sol... |
H: find recursive solution $T(n)=2T(n/2)+n-1$
I want to solve this:
$$T(n) = 2 T\left(\frac{n}{2}\right) + n - 1 $$
I try :
\begin{align*}
n &= 2^m \\
T(2^m) &= 2T(2^{m-1}) + 2^m -1 \\
2 ^ m &= B \\
T(B) &= 2T(B-1) + B -1 && (1) \\
r - 2 &= 0 \implies r = 2 \implies T (B) = C*2^B ... |
H: Linear Transformation Orthogonality
True or False:
If $T$ is a linear transformation from $R^n$ to $R^n$
such that
$$T\left(\vec{e_1}\right), T\left(\vec{e_2}\right), \ldots, T\left(\vec{e_n}\right) $$
are all unit vectors
then $T$ must be an orthogonal transformation
The answer is ?
I know a linear transformat... |
H: Evaluating the integral: $\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx$
I am interested in evaluating the following integral:
$$
\int_{0}^{\infty} \frac{|2-2\cos(x)-x\sin(x)|}{x^4}~dx
$$
Using Matlab, Numerically it seems that the integral is convergent,
but I'm not sure about it. How can we prove that t... |
H: Linear Algebra determinant and rank relation
True or False?
If the determinant of a $4 \times 4$ matrix $A$ is $4$
then its rank must be $4$.
Is it false or true?
My guess is true, because the matrix $A$ is invertible.
But there is any counter-example?
Please help me.
AI: You're absolutely correct. The point ... |
H: Deducing a coefficient from a cubic polynomial?
I fully answered the question, and got that $k=-3$, but the answer says it's positive. Can anyone show me my mistake?
"Given that $x-2$ is a factor of the polynomial $x^3 - kx^2 - 24x + 28$, find $k$ and the roots of this polynomial."
Using factor theorem, I realised ... |
H: On Landau notations
How common it is to write e.g. $1-o(1)$ for a function that eventually approaches $1$ from below (or eventually equals $1$)? Would a better notation be $1-|o(1)|$ or what is meant is already obvious from $1-o(1)$ ? Obviously, precisely defining everything will work eventually, but I was wonderin... |
H: Matrix involving values of polynomials
I've been doing this problem but im stuck.
Be $f_1 f_2 f_3 \in \mathbb{R}_2$[$x $]. Proove that {$f_1$,$ f_2$,$ f_3$} form a base of $\mathbb{R}_2$[$x $] as $\mathbb{R}$ vector space, if and only if in $\mathbb{R}^{3x3}$ this matrix is invertible:
$ \begin{bmatrix} f_1(1) & ... |
H: A question on linear operators
This is a problem I’ve been working on as part of my studies for an upcoming comprehensive exam:
Let $F$ be a field, let $V\in F$-$\mathrm{Mod}$ be a finite-dimensional left $F$-vector space, and let $T\in\mathrm{End}_{F}\left(V\right)$ be an $F$-linear operator.
(a) Suppose that ... |
H: Is it possible that a vector space can be a finite union of proper subspaces?
Let $V$ be a vector space, and let $V_i$ for $i=1,\ldots, n$ be non-zero subspaces of $V$. Is it possible that $V=\cup_{i=1}^n V_i?$
The underlying field is assumed to be infinite.
AI: Let $V$ be a vector space over an infinite field $k$.... |
H: Set Notation (Axiom of Replacement)
This question is related to the one I asked yesterday here in that it's related to another one of the Zermelo-Fraenkel Axioms. After looking over the notation used to describe the axiom, that is:
$$ \forall \space x \space \forall \space y \space \forall \space z \space [\varphi... |
H: Find the factorization of the polynomial as a product of irreducible on rings R[x] and C[x]
Find the factorization of the polynomial $x^5-x^4+8x^3-8x^2+16x-16$ as a product of irreducible on rings $\Bbb R[x]$ and $\Bbb C[x]$
Testing with the simplest possible root in this case, $P(1) = 0$
Applying the schema of Ruf... |
H: Probability of senior citizens in a one million residence
In a city of over $1000000$ residents, $14\%$ of the residents are senior citizens. In a simple random sample of $1200$ residents, there is about a $95\%$ chance that the percent of senior citizens is in the interval [pick the best option; even if you can pr... |
H: Dimensions of vector subspaces in a direct sum are additive
$V = U_1\oplus U_2~\oplus~...~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + ... + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis of $V$]
Then it suffices to show $U_i\cap U_j-\{0\}=\emptys... |
H: Another Information Theory Riddle
The following nice riddle is a quote from the excellent, free-to-download book: Information Theory, Inference, and Learning Algorithms, written by David J.C. MacKay.
In a magic trick, there are three participants: the magician, an assistant, and a volunteer.
The assistant, who cl... |
H: Cubing a simple thing
I am trying to expand $\quad (x + 2)^3 $
I am actually not to sure what to do from here, the rules are confusing. To square something is simple, you just foil it. It is easy to memorize and execute. Here though I am not sure if I need to do it like multiplication where I take one $(x + 2)$ ter... |
H: Integration $\int \left(x-\frac{1}{2x} \right)^2\,dx $
Evaluate $$\int\!\left(x-\frac{1}{2x} \right)^2\,dx. $$
Using integrating by substitution, I got $u=x-\frac{1}{2x},\quad \dfrac{du}{dx} =1+ \frac{1}{2x^2}$ , and $dx= 1+2x^2 du$. In the end, I came up with the answer to the integral as :
$$\left(\frac{1}{3}+\fr... |
H: Integrate $\int {{{\left( {\cot x - \tan x} \right)}^2}dx} $
$\eqalign{
& \int {{{\left( {\cot x - \tan x} \right)}^2}dx} \cr
& = {\int {\left( {{{\cos x} \over {\sin x}} - {{\sin x} \over {\cos x}}} \right)} ^2}dx \cr
& = {\int {\left( {{{{{\cos }^2}x - {{\sin }^2}x} \over {\sin x\cos x}}} \right)} ^2... |
H: Find area of triangle ABC
BD Perpendicular AC , AB =BC=a
Find the area of triangle ABC
I have tried Googling , I used formula 1/2 (base X Height) . Used Pythagorean theorem. Anyone can suggest me solution.
AI: Let $y$ represent the distance $CD$. Then $y^2+x^2=a^2$, or $y=\sqrt{a^2-x^2}$. Now you have enough to f... |
H: A question about the degree of an element over a field extension.
Say $K$ is a field extension of field $F$. If element $b$ is algebraic with degree $n$ over $F$, we know that $[F(b):F]=n$.
Why is it that $[K(b):K]\leq n$?
AI: Recall the following facts: if $M/L$ is a field extension, then for any $a\in M$,
$$[L(... |
H: Why is $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] = p-1 $?
I know that $e^{2\pi i/p}$ is a root of $x^{p}-1$ and we can write:
$ x^{p}-1=\left(x-1\right)\left(\sum_{i=0}^{p-1}x^{i}\right) $
So $e^{2\pi i/p}$ is a root of $\sum_{i=0}^{p-1}x^{i}$, which means $[\mathbb{Q}(e^{2\pi i/p}):\mathbb{Q}] \leq p-1 $.
Can someone... |
H: Factoring the third degree polynomial $x^3 - 3x^2 - 4x + 12$ using long division
I am sure there is a better strategy that someone smarter than me would use, but I am not that person.
I am trying to factor$$x^3 - 3x^2 - 4x + 12 .$$
I do not know how, so I attempt to guess with long division. I cheat and look at the... |
H: How to minimize studying using mathematics?
A friend asked me this question earlier today, and it made me wonder how to come up with a general solution (where each variable is an integer):
I have a vocabulary test tomorrow at school. On it, there will be $W$
terms listed, and I will have to define $X$ of them. H... |
H: On convergence of problematic series.
Determine if the following series is converges or not
$$\sum_{n=2}^{\infty}\frac{1}{\ln^{\ln(\ln n)} (n)}$$
AI: Note that $$\ln\left(\frac{1}{\ln^{\ln(\ln n)}(n)}\right)=\ln1-\ln(\ln^{\ln(\ln n)}(n))=-\ln(\ln n)\ln(\ln n)=-\left[\ln(\ln n)\right]^2$$
Therefore
$$\sum_{k=2}^\inf... |
H: Closed form for $\sum^{\infty}_{{i=n}}ix^{i-1}$
How can I find a closed form for:
$$\sum^{\infty}_{{i=n}}ix^{i-1}$$
It looks like that's something to do with the derivative
AI: $$\sum^\infty_{r=n}rx^{r-1}=\dfrac {d\left(\sum^\infty_{r=n}x^r\right)}{dx}$$
From here, $\sum^\infty_{r=n}x^r=\dfrac{x^n}{1-x}$ if $|x|<1$... |
H: Power series of $f(x)=\sqrt{\frac{1+x}{1-x}}$
How do I find the power series form of $\,f(x)\,$:
$$\displaystyle f(x)=\sqrt{\frac{1+x}{1-x}}$$
I tried to multiply the fraction by $\,\dfrac{1+x}{1+x}\,$ but it didn't help...
AI: Try writing it as $ f(x) = \sqrt{ 1- x^2 } \times 1/(1-x) $ . Then write $ 1/(1-x) $ as... |
H: Algebra / Equation with 2 or more vaiables
A train moves past a telegraph post and a bridge 264 m long in 8 seconds and 20 seconds
respectively. What is the speed of the train?
I'm really confused if there are 2 or more variables. Can someone explain this in step by step? How x is removed until the end. Thanks.
O... |
H: solving for one variable in terms of others
A question from Steward's Precalculus textbook 5th, Pg 55,
the original formula is $$h=\frac{1}{2}gt^2+V_0t$$
the question asks to write the formula in terms of $t$, the answer is $$t=\dfrac{-V_0\pm\sqrt{v_0^2+2gh}}{g}$$
I don't know the steps on how to get there
AI: $$h... |
H: Finding the Taylor series of $f(z)=\frac{1}{z}$ around $z=z_{0}$
I was asked the following (homework) question:
For each (different but constant) $z_{0}\in G:=\{z\in\mathbb{C}:\,
z\neq0$} find a power series $\sum_{n=0}^{\infty}a_{n}(z-z_{0})^{n}$
whose sum is equal to $f(z)$ on some subset of $G$. Please speci... |
H: How to show that these integrals converge?
What test do I use to show that the following integral converges?
If you could provide me with the process that leads to the answer that would really help.
$\displaystyle \int_{0}^{1}\frac{x^n}{\sqrt{1-x^4}}\,dx$
$\displaystyle \int_{0}^{\pi /2}\frac{\ln(\sin(x))}{\sqrt{... |
H: Infinite products of a (finite) group
So I'm having a little trouble understanding the concept of infinite (cartesian) products of a group -- specifically, my notes (and, of course, homework questions) have concepts of, say $S_3^\mathbb{Z}$ and $S_3^\mathbb{R}$ (where $S_3$ is the permutation group of 3 elements, o... |
H: what is the sum of this?$\frac12+ \frac13+\frac14+\frac15+\frac16 +\dots\frac{1}{2012}+\frac{1}{2013} $
What is the sum of $$\frac12+ \frac13+\frac14+\frac15+\frac16 +\dots\frac{1}{2012}+\frac{1}{2013} $$
AI: Not really an interesting question, I fear ...
The exact answer is $\frac{A}{B}$, where $A$ is the 873-digi... |
H: How to form a cubic equation with the substitution method?
I had this question:
"Find the cubic equation whose roots are twice the roots of the equation $3x^3 - 2x^2 + 1 = 0$"
In my first attempt, I solved it through the use of simultaneous equations, where I let the cubic equation be: $x^3 + bx^2 + cx + d$. Howeve... |
H: How to prove two polynomials have no zeroes in common?
The question asked:
Divide the polynomial $P(x) = x^3 + 5x^2 - 22x - 6$ by $G(x) = x^2 - 3x + 2$. I did, and got the answer: $(x+8)(x^2-3x+2)-22$.
However, it now asks to: "Show that $P(x)$ and $G(x)$ have no zeros in common."
How do I prove this?
Thanks.
AI:... |
H: Find the closed solution of $s_{n} = 3s_{n-1} + 2^{n-2} - 1$
Find the closed solution of $s_{n} = 3s_{n-1} + 2^{n-2} - 1$ if $s_1 = 0, s_2 = 0, s_3 = 1$
I have attempted to use $p_n = c2^{n-2} - d$ [where $h_n = A(3)^n$, but to no avail] - i ended up with $c=-1$ and $d=-\frac{1}{2}$, which is incorrect.
Any help is... |
H: Conversion: dec to bin
We have:
$$\begin{align}
A&=\frac{19}{32} \\[0.3em]
B&=\frac{21}{32}\\[0.3em]
C&=\frac{19}{64}\\[0.3em]
D&=\frac{21}{64}.\\[0.3em]
\end{align}$$
In binary it is:
$$\begin{align}
A&=0,10011\strut \\
B&=0,10101\strut\\
C&=0,010011\strut\\
D&=0,010101\strut\\
\end{align}$$
Is it correct?
Does it... |
H: Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$
Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$
For the first part:
$\displaystyle\frac {4x... |
H: $V_\omega$ is countable
Is there an easy way to prove this? I found a book that suggests the injection $h:V_\omega\to\omega$ defined by
$$h(\{x_1,x_2,\dots,x_n\})=2^{h(x_1)}+2^{h(x_2)}+\cdots+2^{h(x_n)},$$
but I hit some snags while proving that the function is well-defined. Are there any other clever ways to prove... |
H: Is the ascending union of contractible spaces contractible
Let $\{Y_i\}_{i \in \mathbb{N}}$ be a collection of subspaces of $X$ such that each $Y_i$ is contractible and $Y_{i} \subseteq Y_{i+1}$. Is $\bigcup_{i\in \mathbb{N}}Y_i \subseteq X$ also contractible?
AI: No. Consider the half-open arcs
$$Y_n=\{e^{i\theta}... |
H: Finding the limit of a function?
The limit is actually easy: $\displaystyle \lim \limits_{t\to\infty}\dfrac{t^{k+1}}{e^t}$
One can use hopitals rule and say that ultimately the upper function will be reduced to a constant while the lower function will remain the same. Hence, the limit is zero. I was wondering if th... |
H: how to show that $f(x)$ can be expressed uniquely as follows: $f(x)=\prod_{i=1}^k[f_i(x)]^{n_i}$
Let $f(x)\in F[x],~F$ being a field, be monic. Then how to show that $f(x)$ can be expressed uniquely as follows:
$$f(x)=\prod_{i=1}^k[f_i(x)]^{n_i}$$
for some monic irreducible polynomial $f_i(x)\in F[x]$ and $k\in \ma... |
H: Does there exist $g$ s.t $g'=f$?
I have the following homework question:
Let G be the bounded open set shown in gray in this picture, whose
boundary consists of eight line segments. The endpoints of those
segments are, as shown, the points $-2,-1,-1+4i,1+4i,1,2,2+5i,-2+5i$.
Let $f:\, G\to\mathbb{C}$ be an arb... |
H: $\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find the integral function ... |
H: If $\sum a_n $ is a positive series that diverges, does $\sum \frac{a_n}{1+a_n}$ diverge?
Let $ \{a_n\}_{n=1}^\infty $ be a sequence such that $\displaystyle \sum a_n $ that is divergent to $+\infty$.
What can be said about the convergence of $\displaystyle \sum \frac{a_n}{1 + a_n} $?
Any hints, thoughts or leads w... |
H: If $x^p P(|X|>x|)=o(1)$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon
If $p>0$ and $x^p P(|X|>x|)=o(1)$ as $x\to\infty$, then $E(|X|^{p-\epsilon})<\infty$ for $0<\epsilon<p$.
It feels like the assumptions should lead to something like $\sum_n^\infty ((n+1)^{p-\epsilon}-n^{p-\epsilon})P(|X|>n)<\infty$ but I don... |
H: How do I work out what percent of my customers will be girls and what percent will be boys?
I know that 33.3333% of all girls questions would buy my product and that 80% of all boys questioned would buy it.
What i don't know is how to work out is statistically what percentage of our customers will be boys and what ... |
H: $\sum_{k=1}^{n} \binom{n}{k}k^{r}$
Find:$$\sum_{k=1}^{n} \binom{n}{k}k^{r}$$
For r=0 the sum is obviously $2^{n}$.
For r=1 the sum is $n2^{n-1}$.
For r=2 the sum is $n(n+1)2^{n-2}$.
Here's what I've tried:
$$\frac{d(1+x)^{n}}{dx}=n(1+x)^{n-1}=\binom{n}{1}+2\binom{n}{2}x+3\binom{n}{3}x^{2}+\cdots+n\binom{n}{n}x^... |
H: How to write in $2^x=5$ in logarithmic form?
How do I write:
$$2^x = 5$$
In a logarithmic form?
I've looked for a solution for some time now, so I decided to try here.
AI: $2^x = 5$
Now 'take log' on both sides:
$\log_2 2^x = \log_2 5$.
We can now use the property $\log a^b = b\times \log a$.
Thus,
$x \times \log_... |
H: Solve a polynomial involving geometric progression?
I have had trouble with this question:
"Solve the equation $8x^3 - 38x^2 + 57x -27 = 0$" if the roots are in geometric progression.
Any help would be appreciated.
AI: Let the roots be $a, a\cdot r, a\cdot r^2$
So using Vieta's formula $a+ a\cdot r+ a\cdot r^2=\fra... |
H: Which is true $A$ is subset of $B$ or $B$ is subset of $A$.
Consider the sets dened by the real solutions of the inequalities $$A=\{(x,y):x^2+y^4\le 1\}$$ and $$B=\{(x,y):x^4+y^6\le 1\}$$Then which is true $A$ is subset of $B$ or $B$ is subset of $A$.
This is Question No. $10.$ of this paper. Please help me.Than... |
H: How the inverse of this matrix be found?
How can the inverse of matrix
$A = \left( \begin{smallmatrix} 6&5\\5&4 \end{smallmatrix} \right)$ be $A^{-1} = \left( \begin{smallmatrix} -4&5\\ 5&-6 \end{smallmatrix} \right)$ where $\frac{1}{ad-bc} = \frac{1}{24-25} = \frac{1}{-1}$?
I thought that an inverse to this matri... |
H: When is the quotient algebra of a unital C* algebra helpful?
Let $\mathcal A$ be a unital C* algebra.
Which properties does $\mathcal B \subset \mathcal A$ has to have for it to make sense to form the quotient algebra $\mathcal A / \mathcal B$?
In cases where this construction makes sense, does $\mathcal A / \math... |
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