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H: 3xy + 14x + 17y + 71 = 0 need some advice
$$3xy + 14x + 17y + 71 = 0$$
Need to find both $x$ and $y$. If there was only one variable then this is easy problem.
Have tried:
$$\begin{align}3xy &= -14x - 17y - 71 \\
x &= \frac{-14x - 17y - 71}{3y}\end{align}$$
Then tried to put this expression everywhere instead of $x... |
H: Examples of 'almost'-vector spaces/modules, where distributivity fails?
I'm wondering what things go horribly wrong by not requiring distributivity in the defitinion of an $R$-module, or $k$-vector space. A 'concrete' example for $\mathbb{R}$ would be quite satisfying.
EDIT: Arturo Magidin points out there are two ... |
H: What are differences between semidirect product and direct product?
Given two groups $A, B$, we can construct direct product $A \times B$ whose elements are of the form $(a, b), a \in A, b\in B$. If $A, B$ are subgroups of a group $G$ and $A \cap B =\{1\}$, then we can construct semidirect product $A \rtimes B$ who... |
H: How do you solve this radical equation $\sqrt{2x+5} + 2\sqrt{x+6} = 5$?
I have a radical equation
$$
\sqrt{2x+5} + 2\sqrt{x+6} = 5
$$
and I am having trouble calculating an answer. I keep on getting weird numbers that are not correct as my answer. How do you solve this? A step-by-step procedure would be highly ap... |
H: Word Problem - Adding an amount after a certain limit
I cant solve this word problem:
Courier charges to a certain destination are $65$ cents for first $250$ grams and $10$ cents for each additional $100$ grams or part thereof. What could be the weight of package for which charge is $\$ 1.55$ ?
I am solving i... |
H: Prove $\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}$
How would I prove the following double angle identity?
$$\frac{1-\sin(2A)}{\cos(2A)}=\frac{1-\tan A}{1+\tan A}$$
My work thus far is
$$\frac{1-2\sin A\cos A}{\cos^2A-\sin^2A}$$
$$\frac{1-2\sin A\cos A}{(\cos A+\sin A)(\cos A-\sin A)}$$
Sadly I am stuck.
... |
H: Prove $\cot A\sin 2A=1+\cos 2A$
How would I prove the following two trigonometric identity.
$$\cot A\sin 2A=1+\cos 2A$$
This is my work so far
$$\frac{\cos A}{\sin A}(2\sin A \cos A)=1+\cos 2A$$
I am not sure what I would do next to make them equal.
AI: \begin{eqnarray*}
cotAsin2A=\frac{cosA}{sinA}(2sinAcosA)=cos... |
H: Is this correct for this expression?
I have seen the expression the identity product in somewhere and I try to express it again as
\begin{align}
%
\prod _{k=1}^K\left(1-x_{k}\right)=\sum _{k=1}^K
\frac{(-1)^k}{k!}\underbrace{\sum _{n_1=1}^K \ldots \sum
_{n_k=1}^K}_{n_1\neq n_2\neq ... |
H: Show matrix $A+5B$ has an inverse with integer entries given the following conditions
Let $A$ and $B$ be 2×2 matrices with integer entries such that
each of $A$, $A + B$, $A + 2B$, $A + 3B$, $A + 4B$ has an inverse with integer
entries. Show that the same is true for $A + 5B$.
AI: First note that a matrix $X$ with ... |
H: How to get a new point of a vector when rotated.
I want to obtain the new point of a vector that I rotate like this.
When I rotate them, I have the angle of rotation.
I want to know x and y, it rotates taking the reference point of 0,0
Thanks
AI: To give a general answer, you take your position vector $\vec{v}\in\... |
H: The count of functions from $X$ to $Y$?
Let $X$ be a set with $N$ elements, and $Y$ a set with $M$ elements. What is the count of possible functions from $X$ to $Y$?
If answer is $M^N$, then why can't it be $N \times M$? That is, why not count for each $x \in X$, the possible mappings from $x$ to each $y\in Y$?
A... |
H: Find the area of the region bounded by two curves
I need to find the area of the region that is bounded by $y=x^2-4$ and $y=2x-1$
I think I solved it, but I don't know what the right answer is so I'm not sure!
I got:
$$da = wl$$
$$=(x^2-4)-(2x-1)dy$$
$$=(x^2-2x-3)dy$$
$$\int{}da = \int{(x^2-2x-3)dy}$$
$$a = \frac{x... |
H: Integral of unimodal functions $f>0$.
Suppose $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is positive almost everywhere and integrable. We know that if $n=1$ and if $f$ is unimodal then the integral $F(x)=\int_{[-\infty,x]} f$ is convex for all $x<a$ and concave for all $x>a$, where $a$ is the point of unimodality. Supp... |
H: The relation between a metric space $(X,d)$ and the topological space that arises from it.
Consider the topological space $(\Bbb R,\mathfrak I)$ that arises from the metric space $(\Bbb R,d)$, with $d(x,y)=|x-y|$. I want to prove that $\partial(a,b)=\partial[a,b]=\{a,b\}$.
I have that
$$x\in \partial A \iff d(x,A... |
H: Cauchy Riemann Equations for $g(v(x,y))=u(x,y)$.
I have that $f(z)=u(x,y)+iv(x,y)$ for $z=x+iy$ is analytic on an open, connected set $U$. Suppose there is a function $g: \mathbb{R} \longrightarrow \mathbb{R}$ such that $g(v(x,y))=u(x,y)$. Prove that $f$ is a constant function.
I am having trouble applying the Cauc... |
H: Showing $\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n} $ does not converge uniformly.
Prove that the series $\sum_{n=1}^{\infty}\left\Vert x\right\Vert ^{n} $, $x\in\mathbb{R}^{n} $, does not converge uniformly on the unit ball $\left\{ x\in\mathbb{R}^{n}\mid\left\Vert x\right\Vert <1\right\} $.
I am not sure ... |
H: What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$?
What function satisfies $x^2 f(x) + f(1-x) = 2x-x^4$? I'm especially curious if there is both an algebraic and calculus-based derivation of the solution.
AI: We start from
$$x^2f(x)+f(1-x)=2x-x^4.$$
Replace $x$ by $1-x$. Then $1-x$ gets replaced by $x$.
So
$$(... |
H: Is composition of piecewise linear functions again a piecewise linear function?
I have some piecewise linear (not necessarily continuous) functions (also, in case it matters, in my specific case 'a' is larger than 0 in all functions). Is every the composition of those functions again a piecewise linear (not necessa... |
H: Finding the expectation of a truncating event given a particular outcome?
Approaching the following problem:
Gambles are independent, and each one results in the player being
equally likely to win or lose 1 unit. Let $W$ denote the net winnings
of a gambler whose strategy is to stop gambling immediately afte... |
H: Proving $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$, given that $\theta^3 + 11\theta - 4 = 0$
As the title says, given that $\theta^3 + 11\theta - 4 = 0$, I'm trying to prove that $\frac{-\theta + \theta^2}{2}$ is an algebraic integer in $K = \mathbb{Q}(\theta)$.
I know that ... |
H: A Couple of Normal Bundle Questions
We are working through old qualifying exams to study. There were two questions concerning normal bundles that have stumped us:
$1$. Let $f:\mathbb{R}^{n+1}\longrightarrow \mathbb{R}$ be smooth and have $0$ as a regular value. Let $M=f^{-1}(0)$.
(a) Show that $M$ has a non-vanis... |
H: The isomorphism of quotient groups
If $X$ is an abelian group and $A,B$ are its subgroup with $A\cong B$, is quotient group $X/A$ isomorphic to the quotient group $X/B$ ?
AI: Let X be the group of integers under addition. Let A be the group of even integers, B the group of integers divisible by 3. Then X, A, and B ... |
H: What is the average distance of a combination set?
I'm working on a genetic algorithm and would like to map each function to a set of "codons". So + -> 011. Given this, I would like to figure out how easy it would be for any given codon set to mutate into another codon set.
If you were to take two combinations, s... |
H: Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$
Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$.
Can anyone help me with this? Thank You!
AI: Method $1:$
We know $$ \tan^2A=\frac{1-\cos2A}{1+\cos2A} $$
Let us find the cubic equation whose roots are $\cos40... |
H: Proof of Gauss's Lemma (Riemannian Geometry version)
I was self-learning Do Carmo's Riemannian Geometry, there is a step in the proof of Gauss's Lemma what I can't quite figure out.
Since $d\,\exp_p$ is linear and, by the definition of $\exp_p$,
$$
\langle (d\,\exp_p)_v(v),(d\,\exp_p)_v(w_T)\rangle=\langle v,w_... |
H: Product rule for partial derivatives
I am going through the solution for a problem (1.7 from Goldstein's Classical Mechanics) where it says:
I don't understand why the right-hand side of the second line only contains 4 terms when there should be 5. The very last term on line 1 has been expanded into 1 term on line... |
H: Derivative iteration and factorials.
I'm in year 11 at high school and have just learn about calculus and the derivative of a function. I found that if I iterated the function $f(x) = x^n$ through the derivative process $k$ number of times where $k \leq n$ and $f(x)$ is $k=1$, $f'(x)$ is $k=2$ I would get: $[n(n-1)... |
H: How to write this in sigma notation?
Newton's formula for interpolation is
$$P(x)=c_1+c_2(x-x_1)+c_3(x-x_1)(x-x_2)+c_4(x-x_1)(x-x_2)(x-x_3)+\cdots$$
I prefer sigma notation, when it is possible. Can this be written in sigma notation?
AI: You can write $\displaystyle P(x)= \sum\limits_{i=1}^{+ \infty} c_i \prod\limi... |
H: Show $1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}}$ for $x \neq 0$
For $x \neq 0$, $$ 1 + 2 \sum_{n=1}^N \cos n x = \frac{ \sin (N + 1/2) x }{\sin \frac{x}{2}} $$
AI: Here is a well known trigonometric trick
$$
1+2\sum\limits_{n=1}^N\cos (nx)=
1+\frac{1}{\sin(x/2)}\sum\limits_{n=1}^N 2\... |
H: Martingale Problem and PDE's
Let $X$ be a RCLL Markov Process with generator $A$. Then I know that
$$ M^f = f(X)-f(X_0)-\int Af(X_s)ds $$
is a martingal for every $f\in \mathcal{D}_A$. If we suppose that $Af=0$, we see that $f(X)-f(X_0)$ is a martingale. Further I know that from the Markov property
$$E[f(X_{t+h})|... |
H: How to prove that $R/I \otimes_R M \cong M / IM$
Possible Duplicate:
Showing that if $R$ is local and $M$ an $R$-module, then $M \otimes_R (R/\mathfrak m) \cong M / \mathfrak m M$.
In one of the answers to one of my previous questions the following claim was mentioned:
$R/I \otimes_R M \cong M / IM$
So I tried t... |
H: Moving points along a curve on sphere.
I have two points on a unit sphere. I also have their coordinates.
theta=linspace(0,2*pi,20);
phi=linspace(0,pi,20);
[theta,phi]=meshgrid(theta,phi);
rho=1;
x=rho*sin(phi).*cos(theta);
y=rho*sin(phi).*sin(theta);
z=rho*cos(phi);
mesh(x,y,z)
xyz=randn(3,2);
xyz=bsxfun(@rdivide,... |
H: Evaluating $\int ^\frac{\pi}{2}_{0} \sin\left(2x+\frac{\pi}{4}\right)\ dx$
Find the exact value of the following definite integral:
$$\int ^\frac{\pi}{2}_{0} \sin\left(2x+\frac{\pi}{4}\right)\:dx=\left[-\frac{1}{2}(2x+\frac{\pi}{4})\right]^\frac{\pi}{2}_{0}$$
$$=-\frac{1}{2}\left(2\frac{\pi}{2}+\frac{\pi}{4}... |
H: Proving convexity of this set in $\ell^2$
This is a follow-up to the question I posted earlier this week.
Consider, for a fixed sequence $(a_n)_n\in\ell^2$ the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}\subset\ell^2.$$ Is this set convex in $\ell^2$?
According to the book [An i... |
H: Question about proof of $A[X] \otimes_A A[Y] \cong A[X, Y] $
As far as I understand universal properties, one can prove $A[X] \otimes_A A[Y] \cong A[X, Y] $ where $A$ is a commutative unital ring in two ways:
(i) by showing that $A[X,Y]$ satisfies the universal property of $A[X] \otimes_A A[Y] $
(ii) by using the u... |
H: What is the difference between "probability density function" and "probability distribution function"?
Whats the difference between probability density function and probability distribution function?
AI: The relation between the probability density funtion $f$ and the cumulative distribution function $F$ is...
if ... |
H: A circle on the plane
Possible Duplicate:
Parametric Equation of a Circle in 3D Space?
I know that, for example, if a circle is on a plane with counter-clockwise orientation, and with center $(a,b)$ and radius $R$, it has parametrization
$$r(t)=(a + R \cos{t};b + R \sin{t}) \quad 0 \leq t \leq 2\pi$$
and with c... |
H: Compute $ I_{n}=\int_{-\infty}^\infty \frac{1-\cos x \cos 2x \cdots \cos nx}{x^2}\,dx$
I'm very curious about the ways I may compute the following integral. I'd be very glad to know your approaching ways for this integral:
$$
I_{n} \equiv
\int_{-\infty}^\infty
{1-\cos\left(x\right)\cos\left(2x\right)\ldots\cos\left... |
H: Surface Area of the simple figure
There is a question which states
The solid brick figure shown is made of small bricks of side 1. When the large brick is disassembled into its component small bricks , the total surface area of all small bricks is how much greater than surface area of larger brick
Here is how ... |
H: A connected k-regular bipartite graph is 2-connected.
I've been struggling with this exercise; all ideas have been unfruitful, leading to dead ends. It is from Balakrishnan's A Textbook of Graph Theory, in the connectivity chapter:
Prove that a connected k-regular bipartite graph is 2-connected.
(That is, deletio... |
H: An infinite series of a product of three logarithms
I was told this interesting question today, but I haven't managed to get very far:
Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log \left(1+\frac{1}{2n}\right)\log \left(1+\frac{1}{2n+1}\right).$$
I am interested in seeing at least a few solution... |
H: Exponential function: Change base to exp
Why is the following true?
$$\left[\frac{N - it}{N}\right]^{j+1} = \exp\left(-\frac{ijt}{N}\right)$$
i,j - integers less than N.
Is there any theorem which allows me to get this result?
I tried with $$N=2^{32}, i=j=2^8, t=2^{16}$$ and its almost true.
AI: As has been made cl... |
H: Do such sequences exist?
I wish to know if there are real sequences $(a_k)$, $(b_k)$ (and if there are, how to construct such sequences) such that:
$b_k<0$ for each $k \in \mathbb{N}$ with $\lim\limits_{k \rightarrow \infty} b_k=-\infty,$
$$\sum_{k=1}^\infty |a_k| |b_k|^n< \infty, \space\forall n\in\mathbb{N}\cup\... |
H: How to prove that if $m = 10t+k $ and $67|t - 20k$ then 67|m?
m, t, k are Natural numbers.
How can I prove that if $m = 10t+k $ and $67|t - 20k$ then 67|m ?
AI: from first equation $k=m-10*t$,now put into equation we get that 67 divides $t-20*(m-10*t)$
or $67$ divides $201*t-20*m$, because $201/67=3$ ,it means ... |
H: Decay for the tail of a series.
Let $p>1$. I would like to have an estimate for the decay of the sequence $s_{n}=\sum_{k=n}^{\infty}k^{-p}$. Does anyone know of a bound of this type in the literature?
Thanks!
AI: Look at the proof of the integral test of convergence for a sequence; we identify $s_n$ as upper and... |
H: Partial Fractions Expansion of $\tanh(z)/z$
I have seen the following formula in papers (without citations) and in Mathematica's documentation about Tanh[]:
$$
\frac{\tanh(z)}{8z}=\sum_{k=1}^{\infty} \frac{1}{(2k-1)^2 \pi^2+4z^2}
$$
I have no idea how to prove it and I have also encountered in my research similar s... |
H: Explain this code to compute $\log(1+x)$
It's well known that you need to take care when writing a function to compute $\log(1+x)$ when $x$ is small. Because of floating point roundoff, $1+x$ may have less precision than $x$, which can translate to large relative error in computing $\log(1+x)$. In numerical librari... |
H: To show if $O(G)=p^{n}$ then $Z(G)\neq \{e\}$.
Possible Duplicate:
Normal and central subgroups of finite $p$-groups
I want to show that if $O(G)=p^{n}$ then $Z(G)\neq \{e\}$, where $p$ is a prime number and $Z(G)=\{a\in G | ax=xa, \forall x\in G\}$, which is also known as a center of the group $G$.
I think I ha... |
H: How to find estimated total time, based on time elapsed and bytes downloaded?
I'm doing C programming exercise for college, and couldn't figure out how to calculate the following value based on four values (2 of each from one type of variable). The exercise is to write a program for a progress bar for downloading a... |
H: Finding a point along a line a certain distance away from another point!
Let's say you have two points, $(x_0, y_0)$ and $(x_1, y_1)$.
The gradient of the line between them is:
$$m = (y_1 - y_0)/(x_1 - x_0)$$
And therefore the equation of the line between them is:
$$y = m (x - x_0) + y_0$$
Now, since I want another... |
H: Confusing double angle identity
How would I solve the following double angle identity.
$$\cos^4x=\frac{3}{8}+\frac{1}{2}\cos(2x)+\frac{1}{8}\cos(4x)$$
So far my work is
$$\frac{3}{8}+\frac{2\cos^x-1}{2}+\frac{1}{8}(2\cos^2x-1)$$
But how would I proceed.
AI: Notice that
\begin{eqnarray}
\cos(2x)&=& \cos^2 x - \si... |
H: How to show that $h(x^p) \equiv h(x)^p \pmod{p}$?
Possible Duplicate:
Why $g(x^{p})=(g(x))^{p}$ in the reduction mod $p$?
Let $h(x) \in \mathbb{Z}[x]$ and $p$ be a prime.
We know that for any integer $\alpha$ we have that $\alpha^p \equiv \alpha \pmod{p}$.
How can we use this to show that $h(x^p) \equiv h(x)^p \... |
H: Does the monoid of sets of numbers with addition have a name?
The monoid is the set of all sets of integers (but reals or complex numbers could work too). Addition between two elements is defined as $a+b = \{\ x+y\ |\ x \in a,\ y \in b\ \}$. As far as I can tell, the only category of algebraic structures this fits ... |
H: points of $X$ with non trivial stabilizers are discrete
so far I understand about the statement: let $p_i,i=1\dots,n$ has non trivial stabilizers
i.e $S_{p_1}=\{g:g.p=p, g\in G\}\neq\{e\}$, is non trivial subgroup of $G$ for $p_1$ and so forth upto $p_n$ we will get $S_{p_n}$,so we need to show $\{p_1,\dots,p_n\}$... |
H: Laplace transform of a product of Modified Bessel Functions
Working with a scalar field in 2 dimensions I've come to the following integral, from which I can extract the proper ultraviolet behavior ($a \ll 1$) of the theory:
$\int_0^\infty e^{-(4+a^2)x}\left[I_0(2x)\right]^2 ds$.
It is obvious to me that this is th... |
H: hints on solving $ \sin^2 x {d^2y \over dx^2} = 2 y$
How to solve this differentiation equation?
$$\sin^2 x {d^2y \over dx^2} = 2 y$$
I don't know how to begin. Can it be any simpler than this?
AI: Cleaning up Maple's solution, I get
$$
y \left( x \right) ={\frac {c_{{1}} \left( \cos \left( 2\,x \right) +1
\right)... |
H: Idempotents in $\mathbb Z_n$
An element $a$ of the ring $(P,+,\cdot)$ is called idempotent if $a^2=a$. An idempotent $a$ is called nontrivial if $a \neq 0$ and $a \neq 1$.
My question concerns idempotents in rings $\mathbb Z_n$, with addition and multiplication modulo $n$, where $n$ is natural number. Obviously wh... |
H: Spectrum of a field
Let's $F$ be a field. What is $\operatorname{Spec}(F)$? I know that $\operatorname{Spec}(R)$ for ring $R$ is the set of prime ideals of $R$. But field doesn't have any non-trivial ideals.
Thanks a lot!
AI: As you say $\mathrm{Spec}(R)$ is defined to be the set of all prime ideals of $R$. If $R$ ... |
H: "Simplifying" an extension of scalars
Let $A$ be a commutative $\mathbb Z$-algebra and $M$ be a $\mathbb Z\oplus \mathbb Z$-module.
Then $A\otimes_{\mathbb Z} M$ is an $A\oplus A$-module.
Is it true that $(A\oplus A)\otimes_{\mathbb Z\oplus \mathbb Z} M \cong A\otimes_{\mathbb Z} M$ as $A\oplus A$-modules? It seems... |
H: Independence of Rotation Matrix Definitions
I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix
$$
A = \left(
\begin{matrix}
a_{11}& a_{12}& a_{13}\\
a_{21}& a_{22}& a_{23}\\
a_{31}& a_{32}& a_{33}
\end{matrix}
\right)
$$
There are many prope... |
H: float result for two smallest integer division
I want to know the two integer number that division of them is this float. for example
x / y = 1.333333333....
$x$ and $y$ can be 8, 6 and 4, 3 ... i need x = 4 and y = 3.
For next example my number is 1.41 what are x and y? how can i find them ?
AI: As in J.D.'s edi... |
H: Finiteness of an integral w.r.t. a finite Borel measure.
Suppose $\mu$ is a finite Borel measure on $\mathbb{R}^3$. Define $h : \mathbb{R}^3 \rightarrow \mathbb{R}$ by
$$h(x) = \int_{\mathbb{R}^3} \dfrac{d\mu(y)}{\|x - y\|}.$$
Question 1: Must $h(x)$ be finite for almost every $x$, w.r.t. Lebesgue measure?
Question... |
H: Sampling a combination randomly
I want to sample a combination of $N$ elements (without replacement) from a list of $M$ elements where $M\gg N$. There are algorithms to do this when each element is picked with uniform probability. I want to do the same for the non-uniform case.
Let each specific element $i$ is ass... |
H: Almost a perfect cuboid
While reading a very old book on diophantine equations, I came across this exercise:
Find an infinite number of positive integer solutions of the equations
$$x^2 + y^2 = u^2$$
$$y^2 + z^2 = v^2$$
$$z^2 + x^2 = w^2$$
I have found a few solutions by hand, for example $x=240$, $y = 117$, $z = 4... |
H: How is exponent notation related to this example
I am not sure what is meant by exponent notation and therefore how to answer this question is baffling me.
Rewrite this in exponent notation:
$\sqrt[3]{x^2y(z-X)^5}$
AI: Oh so it is literally just a case of doing this?
$({x^2y(z-X)^5})^\frac{1}{3}$ |
H: About Poisson Equation.
I want to solve $ - \Delta u = f$ in $\Omega$ with $u = \phi $ on $ \partial \Omega$. But if I have the solutions of (1) and (2) below : $$ - \Delta u_1 = f \; \text{in } \Omega , \; u_1 = 0 \; \text{on } \partial \Omega \tag{1}$$ $$ - \Delta u_2 = 0 \; \text{in } \Omega , \; u_2 = \phi \;... |
H: A very simple Expected Value question
Can someone explain to me why is $P(Y = 1) = P(X = -1)+P(X = 1)$?
Why is P(Y = 1) the sum of P(X = -1) and P(X= 1)? I don't see how $Y = X^2$ comes into play? I am very new to this stuff.
AI: We have $Y=1$ iff $X^2=1$ iff $X=1$ or $X=-1$.
The only way that $X^2$ can be $1$ is ... |
H: Good book on evaluating difficult definite integrals (without elementary antiderivatives)?
I am very interested in evaluating difficult definite integrals without elementary antiderivatives by manipulating the integral somehow (e.g. contour integration, interchanging order of integration/summation, differentiation ... |
H: Are the matrix products $AB$ and $BA$ similar?
Given two matrices $A,B.$ On what conditions does $AB \sim BA$ hold?
AI: If $A$ is invertible, then $AB = A(BA)A^{-1}$ which shows that $AB$ and $BA$ are similar. Similar (no pun intended) proof if $B$ is invertible. |
H: How can I prove Stokes theorem using Green's formula?
$$ \int_{\partial \Omega} (u ~dx + v ~dy) = \iint_{\Omega} \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) ~dx ~dy $$ Then I want to prove that$$ \int_{\partial \Omega} w = \iint_{\Omega} ~dw, \;(w = u ~dx + v ~dy) $$ Would you give ... |
H: Continuous Dice rolls
A dice is being rolled continuously. Suppose you roll a 3 ( or any number ). What is the probability that you will get the next "3" exactly after n rolls?
AI: I will interpret "after $n$ rolls" as meaning that we want the probability that our next $n-1$ rolls do not yield a $3$, but the $n$-th... |
H: question about Skolem theories
Right now I am reading a proof of Downward Löwenheim-Skolem theorem in Hodges, but I am slightly confused about a proof Hodges makes. Let me write down some of the definitions.
Definition: Let $T$ be a first-order theory in a first-order language $\scr{L}$. >Then a skolemisation of ... |
H: Prime-base products: Plot
For each $n \in \mathbb{N}$, let $f(n)$ map $n$ to the product of the primes that divide $n$.
So for $n=112$, $n=2^4 \cdot 7^1$, $f(n)= 2 \cdot 7 = 14$.
For $n=1000 = 2^3 \cdot 3^3$, $f(1000)=6$.
Continuing in this manner, I arrive at the following plot:
Essentially: I would a... |
H: If $p$ is a factor of $m^2$ then $p$ is a factor of $m$
I'm a complete beginner and not sure where to go with this proof of Euclid's lemma. Any help would be greatly appreciated.
If $m$ is a positive integer and a prime number $p$ is a factor of $m^2,$ then $p$ is a factor of $m.$
So far I have:
Since we know th... |
H: Are minimal prime ideals in a graded ring graded?
Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minimal prime in $A$. Is $\mathfrak p$ a graded ideal?
Intuitively, this means the irreducible components of a projective variety are also projective varieties. When $A$ is Noetherian, I can give a proof... |
H: Does a disjoint set forest have multiple distinct "upwards closed" partitions?
The following is an excerpt from a powerpoint on the role of the inverse Ackermann function in determining the complexity of path compression.
Dissection of a disjoint set forest $F$ with node set $X$
Partition of $X$ into “top part” ... |
H: Evaluating $\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$
I've been trying to evaluate the following integral from the 2011 Harvard PhD Qualifying Exam. For all $n\in\mathbb{N}^+$ in general:
$$\int\limits_0^\infty \! \frac{x^{1/n}}{1+x^2} \ \mathrm{d}x$$
However, I'm not quite sure where to begin, ... |
H: Proof that if $s_n \leq t_n$ for $n \geq N$, then $\liminf_{n \rightarrow \infty} s_n \leq \liminf_{n \rightarrow \infty} t_n$
This is half of Theorem 3.19 from Baby Rudin. Rudin claims the proof is trivial. What I've come up with so far doesn't seem trivial, however, and is probably also wrong (my problem with it ... |
H: Identity related to binomial distribution?
While writing a (non-math) paper I came across the following apparent identity:
$N \cdot \mathop \sum \limits_{i = 1}^N \frac{1}{i}\left( {\begin{array}{*{20}{c}}
{N - 1}\\
{i - 1}
\end{array}} \right){p^{i - 1}}{\left( {1 - p} \right)^{N - i}} = \frac{{1 - {{\left( {1 - p... |
H: Has this operator $0$ as an eigenvalue / where is my error?
I know of a theorem that tells me, that every compact linear operator
on an infinitedimensional Hilbert space has to have the eigenvalue
$0$. On the other hand I have the operator
\begin{eqnarray*}
& T:\ell^{2}\rightarrow\ell^{2}\\
& \left(x_{1},x_{2},\... |
H: If both $a$ and $b$ $\not \equiv 0 \pmod{p}$ then $ab \not\equiv 0 \pmod{p}$
Any help with this proof would be great. Not even sure where to begin. I'm pretty much a total newbie.
If $a$ is not congruent to $0 \pmod{p}$ and $b$ is not congruent to $0 \pmod{p},$ where $p$ is a prime number, then $a*b$ is not cong... |
H: Under what conditions does the expression $(x+y)^4$ equal $x^4+y^4$?
In problem 16(c) of chapter 1 of Calculus, Spivak asks the reader to determine the conditions under which the expression $(x + y)^4$ equals $x^4 + y^4$. Clearly,
$$ (x + y)^4 = x^4 + y^4 \Leftrightarrow x = 0 \vee y = 0 \vee 4x^2 + 6xy + 4y^2 = 0 ... |
H: Count the number of n-bit strings with an even number of zeros.
I am currently self-studying introductory combinatorics by reading Introduction to combinatorial mathematics. I am currently in the first chapter, and I have a question regarding one of the examples. The question was asking to count the number of n-bit... |
H: Does this sequence of operators in Hilbert space, given by an algorithm, terminate
Let $H$ be an infinitedimensional Hilbert space and $T$ a compact
selfadjoint operator in it. Consider the following
Algorithm:
Let
$$
H_{1}=H,\ T_{1}=T
$$
and let $\lambda_{1}$ be that eigenvalue of $T_{1}$ whose absolute
value eq... |
H: Fibonacci Sequence Variants
I learnt about finding the $n$th Fibonacci number using matrix exponentiation in $\log n$ time. Then I tried finding similar formula for sequences of the form
$$S_{n} = S_{n-1} + S_{n-2} + a n + b$$
in terms of Fibonacci sequence.
But I could not find expression except for $a = 0$, in wh... |
H: Change of Variable for Lebesgue Integral on $\mathbb{R}^n$
Let $A=(a_{ij})$ be a real symmetric, positive definite $n \times n$ matrix and set
$$ F(x_1,x_2,\ldots, ,x_n)= \sum_{i,j}a_{ij}x_ix_j.$$
I am trying to show that for any non-negative measurable function $\alpha$ on the real line
$$ \int_{\mathbb{R}^n} \alp... |
H: Some Simple Algebra
\begin{align*}
x &= \frac 12 js + \frac 12 is \\\
y &= \frac 14 is - \frac 14 js
\end{align*}
How can I find a
\begin{align*}
i &= \\\
j &=
\end{align*}
conversion of this?
Edit:
I am not happy with the moderaters assumption on my syntax.
x = (j * s / 2) + (i * s / 2)
y = (i * ... |
H: When is there in a probability space no null sets?
I remember my lecturer saying that in some cases there will be no other null set than the trivial one (the empty set), but I can't remember exactly the condition.
I've been thinking and convinced my self that for finite sets, equipped with the power set and a proba... |
H: Wiener process question
When I look up the definition of 'Wiener process' at Wikipedia, it tells me:
$W(0) = 0$ and $W(t) - W(s) \sim N(0, t-s)$.
When I try to simulate this in matlab, I get different results when I define a vector $W1$ to be like:
$W1 = cumsum(dW)$, where $dW(j) \sim N(0, dt)$,
and a vector $W2$ t... |
H: Solving this pie-chart
Hi could anyone please guide me on how I would go about calculating the percentage of a specific sector from this Pi-chart. I think I am suppose to use data from the bar chart and apply it to the pie-chart but I don't really know how.
The question is
According to the data provided, in 2005 ... |
H: proving "$C^1([−1,1])$ is dense in the given space with given norm"
Define $$E = \left \{ f \in W^{1,2} (-1,1) \; | \; \| f \|_E := \left( \int_{-1}^1 (1-x^2 ) | f' (x) |^2 dx + \int_{-1}^1 | f(x) |^2 dx \right)^{\frac{1}{2}} < \infty \right \}.$$ Then how can I prove that $ C^1 ([-1,1]) $ is dense in $E$ ? Is th... |
H: Prove that this is a Banach space
Let $I=[0,1]$ and let $\displaystyle X:=\left\{f: I\times \mathbb R\to \mathbb R\colon \sup_{(t,x)}\frac{|f(t,x)|}{1+|x|}<\infty\right\}$.
Prove that $X$, equipped with the norm $\displaystyle \|f\|:=\sup_{(t,x)}\frac{|f(t,x)|}{1+|x|}$ is a Banach space.
My first attempt was to us... |
H: Minimize sum of smallest and largest among integers on the real line.
Suppose there are 3 non-negative integers $x$, $y$ and $z$ on the real line.
We are told that $x + y + z = 300$. Without loss of generality, assume
$x$ to be the smallest integer, and $z$ to be the largest.
How do I minimize $(x + z)$?
Attempt: ... |
H: Function $f(x)$ similar to exp(x) where $-f(x)$ is approximately $f(-x)$
I am wondering if there is a function $f(x)$ "similar" to the exponential function $\exp(x)$ such that:
$-f(x) \approx f(-x)$
I would also like $f(x)$ to have the following property:
$\frac{{f(a)}}{{f(b)}} = f(a - b)$
Or alternately,
$\frac{{f... |
H: Prove $\cos x= 2 \cos^2{\frac{x}{2}}-1=1-2\sin^2{\frac{x}{2}}$
How would I prove the following trig identity?
$$\cos x= 2 \cos^2{\frac{x}{2}}-1=1-2\sin^2{\frac{x}{2}}$$
I am not sure where to begin any help would be useful.
AI: I am sure you know the formula $\cos(a+b) = \cos a \cos b - \sin a \sin b$. Let $a=b = ... |
H: If $\lim_{n \to \infty }n \ln\left ( \frac{a_{n}}{a_{n+1}} \right )=g>1$, then $\sum_{n=1}^{\infty }a_{n}$ is convergent
Consider the series $\sum_{n=1}^{\infty }a_{n}$ where $a_{n}> 0$ for all $n\in \mathbb{N}$. Assume that: $\lim_{n \to \infty }n \ln\left ( \frac{a_{n}}{a_{n+1}} \right )=g$.
I need to prove that... |
H: Proof of convergence of a sum of mean-consistent estimators
After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these exercises.
Suppose $\theta_n$ and $\phi_n$ are mean consis... |
H: Word-Problem using a bar-chart
I can not manage to solve this problem. Any suggestions on how I can solve it.
If in 2006 Pharmacom spent the same dollar amount on administration (administration outgoings) as in 2005, but the total outgoings increased by ten percent, approximately what fraction of the total outgoin... |
H: Prove $\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}$
How would I prove the following identity?
$$\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}$$
I know $$\sin(a+b)=\sin a\cos b+\sin a \cos b.$$
So I did
$$\sin\frac{x}{2}\cos\frac{x}{2}+\cos\frac{x}{2}\sin\frac{x}{2}.$$
But what technique would I have to use to continue the prob... |
H: Show that $E(|X|)<\infty$ and $E(X_n)\rightarrow E(X)$
After a few weeks off I am back at my self-study of Measure-Theoretic probability. As always, I thank the community for any detail and answers they can provide as I try to work myself through these exercises.
Perhaps this is an application of Levi?
The question... |
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