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H: Solving systems of equations using matrices
I'm teaching myself how to solve systems of equations using matrices and row operations. I get the basic mechanics of it (legal operations, etc.), but it seems like it's kind-of a crapshoot deciding where to begin, and choosing the "wrong" operation to start with can lead... |
H: continiouty of maping from set back into itself.
Let $f: [a,b] \to [a,b]$ be continuous. Show that the equation $f(x) = x$ has at least one solution in $[a,b]$.
Firstly im going to assume $x \in [a,b]$ thus a is the min and b is that max or vice versa assume the first. thus x >a and $x <b$ so we can divide the int... |
H: Partial fraction decomposition of $\frac{x-4}{(x-2)(x-3)}$
I'm trying to do the partial fraction decomposition of the following rational expression:$$\frac{x-4}{(x-2)(x-3)}$$
Here are the steps I preformed:
\begin{align*}
x-4 & = \frac{A}{x-3} + \frac{B}{x-2}\\
x-4 & = Ax - 3A + Bx - 2A\\
x-4 & = x(A+B) - (3A + 2B)... |
H: Finding the limit for functions with two variables
I know that when we have a limit of a function with $2$ variables, the limit must be the same, regardless of the path we take. So this is useful for proving that a limit doesn not exist. But when you've tried this method for different "paths" (e.g., $(x,0),(0,y),(x... |
H: Is there a formula to calculate the minimum height of an n-nary tree with L leaves?
I'm trying to figure out if there is a way to calculate the minimum height of an n-nary tree with L leaves. Is there such a formula?
AI: I found it:
$$ \text{minimum height}= \lceil \log_n L \rceil$$ |
H: Solving ODE using frobenius method. 3 coefficients
I'm trying to learn frobenius method by solving some problems (ODEs).
For example:
$$xy''+(2x+1)y'+(x+1)y=0$$
Let $y=\sum\limits_{n=0}^\infty a_nx^{n+r}$. Then, I took derivatives and put into the equation:
$$\sum\limits_{n=0}^\infty a_n(n+r)^2x^{n+r-1}+2\sum\limit... |
H: Small question about derivative
how to derive $\int_0^1 G(t,s) e(s)ds$ with respect to $t$
Where $G(t,s)$ is a Green function and $e:(0,1)\rightarrow \mathbb{R}$ continuous and $e\in L(0,1)$
Please help me
Thank you
AI: You can commute the derivative and the integral operators in this case. See here. |
H: Geometric proof
Let the three sides of a triangle be $a,b$ and $c$. If the equation
$$a^2+b^2+c^2=ab +bc+ac$$
holds true, then the triangle is an equilateral triangle.
How do we prove this? An answer or even the slightest hint will be appreciated.
AI: Note that
$$a^2 + b^2 + c^2 = ab + bc + ca \implies (a-b)^2 +... |
H: What's a better way to integrate this?
$$ \int \frac{1}{x^2 + z^2} dx $$
I tried substitution and also by parts. By parts is getting messy and I am not sure if I am getting the right answer. I am trying to figure out an easier way or the proper way to integrate this. Could someone please show me?
AI: The right subs... |
H: Definite integration of a trigonometric function
How to integrate $$\int_0^{\pi/2}\!\dfrac{2a \sin^2 x}{a^2
\sin^2 x +b^2 \cos^2 x}\,dx $$
my first step is $$\frac{2}{a} \int_0^{\pi/2}\!\dfrac{a^2 \sin^2 x}{a^2 +(b^2 - a^2) \cos^2 x}\, dx $$
I would kind of want to do some sort of $u=\cos x$ substitution, to get ... |
H: double integrals and iterated integrals
Give an example (if any) for a non-integrable function $f:\mathbb{R\times R}$ $\to$
$\mathbb{R}$ with domain in $[0,1]^2$ such that both iterated integrals exists(i.e. in both order of integration).
Here is what I have got:
$$
f(x,y) =
\begin{cases}
e^{-xy}\sin x \sin y, & \... |
H: What is the general equation of a cubic polynomial?
I had this question:
"Find the cubic equation whose roots are the the squares of that of $x^3 + 2x + 1 = 0$"
and I kind of solved it. In that my answer was $x^3 - 4x^2 + 4x + 1$, but it was actually $x^3 + 4x^2 + 4x - 1 = 0$.
I took the general equation of a cubic... |
H: $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$
If $E(X^2)=1$ and $E(|X|)\ge a >0$, then $P(|X|\ge\lambda a)\ge (1-\lambda)^2a^2$ for $0\le \lambda \le 1$.
I can see from the well known inequality $E(|X|) \le E(|X|^2)^{1/2}$ that it must be the case that $a\le 1$. But what to do next I'm not sure.... |
H: A definite multiple integral
$$\int_0^1\int_\sqrt[3]{x}^1 4\cos(y^4)\,\mathrm dy\,\mathrm dx$$
What I got was
$\sin(1)x+\cos(x^2) dx$ and now I am stuck.
I suddenly froze. Could someone help me? Haven't done calculus for a long time.
AI: Note that this is a definite integral, and hence the result should be a numb... |
H: When are two vectors parallel if the vectors are $5e_1-3e_2+\alpha e_3$ and $\beta e_1 + 2e_2 + 3e_3$
When are two vectors parallel if the vectors are $$5e_1-3e_2+\alpha e_3$$ and $$\beta e_1 + 2e_2 + 3e_3$$
The alpha and beta are parameters.
AI: Two vector $\vec {n_1}$ and $\vec{n_2}$ are parallel when we can wri... |
H: Understanding weighted linear least square problem
I am having difficulty in understanding about weighted linear least squares.
Could anybody explain me instead of minimizing the residual sum of squares why we need to minimize the weighted sum of squares? Further, I want to know about the term weighted?
Although I ... |
H: When is $T$-Alg monoidal closed?
Given a category $\mathcal{V}$ and a monad $(T,\eta,\mu)$, what would be the sufficient conditions on $\mathcal{V}$ and $T$, for the category of $T$ algebras to be monoidal closed?
(I'm pretty sure that Kock proved that, if $T$ has strength and is commutative, then $T$-alg is close... |
H: Closed subset of an affine variety... is it affine?
Preliminaries
So, first of all let me give you the definitions I'm dealing with. Let $k$ be an algebraically closed field, and $\mathbb{A}^n = k^n$.
An affine variety is a closed and irreducible subset of $\mathbb{A}^n$.
Here we endow $\mathbb{A}^n$ with the Zaris... |
H: Zeros of a cubic polynomial with rational coefficients
While discussing a related problem, one of my friends came out with a question as follows:
Is it possible that a cubic polynomial $p(x) \in \Bbb{Q}[x]$ has all of its zeros to be both real and irrational? That is, can $p(x)$ be factored into the following form?... |
H: Are these ODEs equivalent?
I have the following set of ordinary differential equations:
\begin{equation}
\left\{
\begin{array}{l}
\dot{a} = f_1(a, b, c, d) \\
\dot{b} = f_2(a, b, c, d) \\
\dot{c} = f_1(c, d, a, b) \\
\dot{d} = f_2(c, d, a, b)
\end{array}
\right.
\end{equation}
where $f_1$ and $f_2$ are two functio... |
H: Do these definitions of congruences on categories have the same result in this context?
Let $\mathcal{D}$ be a small category and let $A=A\left(\mathcal{D}\right)$
be its set of arrows. Define $P$ on $A$ by: $fPg\Leftrightarrow\left[f\text{ and }g\text{ are parallel}\right]$
and let $R\subseteq P$.
Now have a look... |
H: Class Group of $\mathbb Q(\sqrt{-35})$
As an exercise I am trying to compute the class group of $\mathbb Q(\sqrt{-35})$.
We have $-35\equiv 1$ mod $4$, so the Minkowski bound is $\frac{4}{\pi}\frac12 \sqrt{35}<\frac23\cdot 6=4$. So we only need to look at the prime numbers $2$ and $3$.
$-35\equiv 5$ mod $8$, so $2$... |
H: Symplectic Form Preserved by Orthogonal Transformation
I'm trying to prove that the symplectic form
$$\omega = d(\cos\theta) \wedge d\phi$$
is preserved by the action of $SO(3)$ on $S^2$ where $\phi$ and $\theta$ are spherical polars. Now $SO(3)$ simply acts by
$$\theta \mapsto \theta + \epsilon, \ \phi \mapsto \ph... |
H: How to solve this integral for a hyperbolic bowl?
$$\iint_{s} z dS $$ where S is the surface given by $$z^2=1+x^2+y^2$$ and $1 \leq(z)\leq\sqrt5$ (hyperbolic bowl)
AI: A related problem. Note that,
$$ z=\sqrt{ 1+x^2+y^2 } \implies z_x=\frac{x}{\sqrt{ 1+x^2+y^2 }},\quad z_y=\frac{y}{\sqrt{ 1+x^2+y^2 }} $$
$$ \iint_... |
H: Application of Open Mapping Theorem
This was stated without proof in the complex analysis text I am reading (Complex Made Simple by Ulrich, page 107). I'm sure it's easy, but I'm tired and need a little help.
Let $f$ be nonconstant and holomorphic in some region $V$ and assume $f'$ is nonconstant. Define
$$\Omega =... |
H: How to graph an absolute value equation?
How would you graph:
$|x+y|=1$ ?
I can do the normal $y=|x+1|$ and all that. But how would you do a question with two of these unknowns in the absolute value?
Any help would be greatly appreciated, thanks.
AI: It doesn't seem as though $|x+y|=1$ is a function, since for $x=... |
H: value of fraction
Can we find out the value of
$$\frac{1+i}{1-i}$$
I have tried to solve it by multiplying $(1+i)$ to both sides and in the end see that the result is still $i$. Am I correct,or is there a different solution?
AI: You divide by complex numbers, by multiplying by its conjugate, that is, replace in th... |
H: How can I practice Jean-Robert Argand idea of the rotation of a square root of -1
I am studying complex numbers and I really need an intuition on how they work.
I found the following video of Mathematician named Adrien Douady
https://www.youtube.com/watch?v=2kbM96Jr4nk
He explains complex numbers in a algebraic ... |
H: Find vectors vertical to given vectors with certain length
Given the vectors $\mathbf{u,v}$ in R³, determine all vectors that are
vertical to $\mathbf{u}$ and $\mathbf{v}$ with length = 1
Every vector $\mathbf{x'}$ that is to be found must meet these two conditions:
$\mathbf{x'} \cdot \mathbf{u} = \mathbf{x'} ... |
H: Are the area of a circle inscribed in a square and the area of the "spandrels" (the four corners that remain) commensurable?
And how would you demonstrate that most simply?
See the beginning of my blog post for a little more:
http://seekecho.blogspot.fr/2013/02/different-ilks.html
AI: So you want to show that the a... |
H: Given $x_1=1,x_2=2,x_{n+2}=3x_{n+1}-x_n\forall n\in\mathbb N$. Find $x_n$.
Given $x_1=1,x_2=2,x_{n+2}=3x_{n+1}-x_n\forall n\in\mathbb N$. Find $x_n$.
I tried to find ways to telescope, but failed. Please help. Thank you.
AI: There is a general way to solve those type of recurrences, through the characteristic pol... |
H: How do I calculate typical group size?
If I have a set of groups of individuals (e.g. people), and I want to calculate the typical group size (as observed by individuals), how do I do this?
Wikipedia refers to this as "mean crowding" or "Typical Group Size" but doesn't give a formula: http://en.wikipedia.org/wiki/G... |
H: get the distribution function
I have the following density function:
$ f(x) = \left\{
\begin{array}{l l}
cx^2+|x| & \quad \text{if -1/2<x<1/2}\\
0 & \quad \text{otherwise}
\end{array} \right.$
we know that $\int_{-\infty}^{\infty}{f(x)dx = 1}=\int_{-1/2}^{1/2}{cx^2+|x|dx = 1}$ and I get $c=9$.
The dis... |
H: Show that $\frac {a_1^2}{a_2}+\frac {a_2^2}{a_3}+...+\frac {a_n^2}{a_1}\geq a_1+a_2+...+a_n$ using AM-GM.
Given $a_1,a_2,...,a_n$ be positive reals. Show that $\displaystyle\frac {a_1^2}{a_2}+\frac {a_2^2}{a_3}+...+\frac {a_n^2}{a_1}\geq a_1+a_2+...+a_n$ using AM-GM.
I know how to slve it using rearrangement ineq... |
H: sum of ten squares
You are given an unlimited supply of $1\times 1,2\times 2,3\times 3,4\times 4,5\times 5,6\times 6$ squares.Find a set of ten squares whose areas add up to $48$.If not the whole solution,even a little prod in the right direction would help.
AI: One solution is obvious:
8 squares of 1x1
1 square of... |
H: Calculate gas-station probabilities
I would like to calculate probabilities for the next exercise:
Knowing the average amount of cars that drive per minute into a gas-station is 3.
** How can I calculate the probability of arriving at least 12 cars into the station in a period of 5 minutes?
** And the probability ... |
H: A problem related to an integral equation
I am stuck on the following problem that is as follows:
The integral equation $\quad \varphi(x)-\lambda \displaystyle\int_{-1}^{1}\cos[\pi(x-t)]\varphi(t) dt= g(x)$ has
1.a unique solution for $ \lambda \ne 1$ when $g(x)=x$
2.no solution for $\lambda \ne 1$ when $g(x)=1$
... |
H: Intersection between sphere and cylinder
I have a sphere and a cylinder.
I have the center and the radius of each of them.
the sphere:
radius = $r_1$
center = $(x_1,y_1,z_1)$
the cylinder:
radius = $r_2$
height = $h_2$
center = $(x_2,y_2,z_2)$
how do I know if there is an intersection?
I read this one: http://en.wi... |
H: Simplifying fractions - Ending up with wrong sign
I've been trying to simplify this
$$
1-\frac{1}{n+2}+\frac{1}{(n+2) (n+3)}
$$
to get it to that
$$
1-\frac{(n+3)-1}{(n+2)(n+3)}
$$
but I always end up with this
$$
1-\frac{(n+3)+1}{(n+2)(n+3)}
$$
Any ideas of where I'm going wrong?
Wolfram Alpha gets it to correct f... |
H: Power series of $\frac{\sqrt{1-\cos x}}{\sin x}$
When I'm trying to find the limit of $\frac{\sqrt{1-\cos x}}{\sin x}$ when x approaches 0, using power series with "epsilon function" notation, it goes :
$\dfrac{\sqrt{1-\cos x}}{\sin x} = \dfrac{\sqrt{\frac{x^2}{2}+x^2\epsilon_1(x)}}{x+x\epsilon_2(x)} = \dfrac{... |
H: Raise a number to the "y" power without using exponents.
This is kind of a spinoff on my question Divide by a number without dividing.
Can anyone think of some clever ways to raise any given number to any given power without using an exponent anywhere in your equation/formula?
$$x^{y}=z$$
AI: You can always use the... |
H: Non-vanishing 2-form on quartic surface.
Let $S\subset \mathbb P^3$ be a quartic surface defined by a homogeneous degree 4 polynomial $F\in k[x_0,x_1,x_2,x_3]$. $S$ is a K3 surface, so it has a unique non-vanishing $(2,0)$-form $\omega$ up to scalar.
How can this $\omega$ be computed?
AI: If $S$ is smooth, $\omega... |
H: A simple question about a bounded function
Let $f$ be a function defined on $[0,\infty)$. If $|f(x)| \leq M$ for all $ x \in [0, \infty)$, then can I say $$ \exists C,R >0 : |f(x)| \leq \frac{C}{(1+x)^2}\;\;(x \geq R) $$
is equivalent to $$ \exists C >0 : |f(x)| \leq \frac{C}{(1+x)^2} \;\; (\forall x \in [0,\inft... |
H: Linear combination of vectors in $\mathbb{R}^3$
Show that any linear combination of $\pmatrix{1\\\frac{3}{2}\\0}$ and $\pmatrix{0\\3\\6}$ is also a linear combination of $\pmatrix{2\\3\\0}$ and $\pmatrix{0\\1\\2}$
I'm not sure how to do this. I have a proof sketch for showing that any 2-dimensional vector is a line... |
H: Apples and their volumes
An apple has a peel that is 1cm thick and a total diameter of 12cm. What percentage of volume of the apple is the peel?
I tried
$$\frac{\text{volume}(\text{radius of 6})-\text{volume}(\text{radius of 5})}{\text{volume}(\text{radius of 6})}$$
and got 42%. Is this correct?
AI: $$
\frac{\frac... |
H: How many options are there for 15 student to divide into 3 equal sized groups?
How many options are there for 15 students to divide into 3 equal-sized groups?
Now I know the solution is $\;\dfrac {15!}{5!5!5!3!}\;$ but I can't understand why.
Can anyone please enlighten me?
AI: Consider the following points:
Suppo... |
H: On finding adjoint of transformation.
Let $V$ be an inner product space and $v,w\in V$ be fixed vectors. Define $T(u)=(u,v)w$. How to find the adjoint mapping $T^*$?
AI: From the definition of adjoint:
$(Tu, z) = (u, T^*z)$
So, for all $u, z \in V$:
$(Tu, z) = ( (u,v)w, z ) = (u,v) \cdot (w,z) = (u, \overline{(w,z... |
H: Does $u\in L^p(B)$ implies $u_{|\partial B_t}\in L^p(\partial B_t)$ for almost $t\in (0,1]$?
Let $B$ be the unit ball in $\mathbb{R}^N$ with center in origin and consider the space $L^p(B)$ with Lebesgue measure ($1<p<\infty$). Let $B_t\subset B$ be a concentric ball of radius $t\in (0,1]$. For fixed $t$ we can def... |
H: Which of the following are subspaces of $M$?
Let $M$ be a vector space of all $3\times 3$ real matrices and let $$A=\begin{pmatrix}2&3&1\\0&2&0\\0&0&3\end{pmatrix}.$$ Which of the followings are subspaces of $M?$
$\{X\in M:XA=AX\}$
$\{X\in M:X+A=A+X\}$
$\{X\in M:\text{trace}(XA)=0\}$
$\{X\in M:\det(XA)=0\}$
AI: Hi... |
H: Differentiability of $f(x) = x^2 \sin{\frac{1}{x}}$ and $f'$
Let $f(x) = x^2 \sin{\frac{1}{x}}$ for $x\neq 0$ and $f(0) =0$.
(a) Use the basic properties of the derivative, and the Chain Rule to show that $f$ is differentiable at each $a\neq 0$ and calculate $f'(a)$.
You may use without proof that $\sin$ is differ... |
H: Proof that an embedding into $\ell^1$ is compact
Prove that any sequence $(x^{(n)})_{n\in\mathbb{N}}\subseteq\ell^1$ such that $\sum_{k=1}^\infty k\lvert x_k^{(n)}\lvert\leq1$ for all $n\in\mathbb{N}$ has a convergent subsequence.
My thoughts on this: Clearly $\lvert x^{(n)}_k\lvert\leq\frac{1}{k}$ uniform in $n$... |
H: Similar cones - volumes and lateral areas
Two similar cones have volumes 9$\pi$ and 72$\pi$. If the lateral area of the larger cone is 32$\pi$, what is the lateral area of the smaller cone?
I did the following...
$\frac {(9\pi)^3} {(32\pi)^2} = \frac {x}{(32\pi)^2}$
resulting in a lateral area of $4\pi$. Is this ri... |
H: A special subset of uniformly distributed numbers is still uniformly distributed?
Assume that I have a value range [1,1000].
I uniformly choose 10 numbers from [1,1000].
Assume that the chosen numbers are a1, a2, ..., a10.
Besides, assume that they are ordered so that a1< a2< ...< a10.
Here comes my question.
If I ... |
H: How do I prove Poisson appraches Normal distribution
I want to prove why the mean and variance of a $\operatorname{Poisson}(\lambda)$, is different when the time index approaches infinite (it's approximated by the mean and variance of a Normal).
For example:
$$
N_k = N_1 + (N_2 - N_1) + (N_3 - N_2) + ... + (N_k - ... |
H: Set of left cosets is a group
Let $N$ be a normal subgroup of $G$. Let $G/N = \{gN : g \in G \}$. Show that the $(G/N, \circ, 1_{G/N})$ is a group, where:
$\circ : G/N \times G/N \rightarrow G/N$ is defined by $(gN \circ hN) = ghN$
and $1_{G/N} = 1*N$.
Thanks for your help.
AI: First verify that the group operation... |
H: Decreasing from the horizontal asymptote
The function $f(x) = x^2/(x^2 - x -2)$ has the following graph. It has a horizontal asymptote $y=1$. For $x$ less than $-4$, the function is decreasing and its graph is under the asymptote. How is this possible when $\lim_{n \to -\infty} f(x) = 1$? Can a function decrease aw... |
H: How to prove the existence of infinitely many $n$ in $\mathbb{N}$,such that $(n^2+k)|n!$
Show there exist infinitely many $n$ $\in \mathbb{N}$,such that
$(n^2+k)|n!$ and $k\in N$
I have a similar problem:
Show that there are infinitely many $n \in \mathbb{N}$,such that
$$(n^2+1)|n!$$
Solution: We consid... |
H: An integral problem?
How do you integrate $e^{e^x}$? I was able to get it down to du/(ln u) but I wasn't able to go further. Thanks!
AI: No, not Calculus AB level. This antiderivative is "not elementary" in the technical meaning of that term. https://en.wikipedia.org/wiki/Elementary_function
added
Maple says
$$
... |
H: In how many ways can you choose $k$ out of $n$ people standing in line, So there's a space of at least 3 people between them
In how many ways can you choose $k$ out of $n$ people standing in line, So there's a space of at least $3$ people between them.
Actually, I don't even know how to start on this one.
AI: HINT:... |
H: Prove that for all positive integer n, the inequality $2n\choose n$ $<4^n$ holds
How do I prove that for all positive integer n, the inequality $2n\choose n$$<4^n$ holds?
Thank you!
AI: Hint: The LHS is the number of $n$-element subsets of $[2n]$, while the RHS is the number of all subsets of $[2n]$. |
H: Defining a subset
The question I have to answer is as following in Swedish:
Hur många mängder X uppfyller {a, b, c} ⊆ X ⊆ {a, b, c, d, e}?
Loosely translated (I do not know mathematical terminology well in English):
How many sets X satisfy {a, b, c} ⊆ X ⊆ {a, b, c, d, e}?
I do not know how to solve this question.... |
H: Help in a proof of a result in Hungerford's book
I need help to proof the last part of this corollary:
I didn't understand the part (IV) because the author proves just the canonical projection case and the statement says "every nonzero homomorphism of rings $R\to S$ is a monomorphism".
Thanks in advance.
AI: If e... |
H: Do there exist $29$ consecutive integers so that every of them has exactly $2$ distinct prime factors?
Do there exist $29$ consecutive integers, denote $a,a+1,\cdots,a+28$, so that every of them has exactly $2$ distinct prime factors?
For example, $25$ has only one distinct prime factor, and $30$ has $3$ distinct... |
H: Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$
I'm stuck at this algebra problem, it seems to me that's what's provided doesn't even at all.
Provided: $$a+b+c=0$$
Find the value of: $$\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$$
Like I'm not sure where to start, a... |
H: looking for reference for integral inequality
Math people:
I would like a reference for the following fact (?), which I proved myself (I am 99% sure the proof is valid) but which has probably been done before. My proof was a little messy. If no one can supply a reference, I would appreciate an elegant proof. Her... |
H: Calculating permutations if the sequences have to be in ascending order?
How would you go about calculating the number of permutations in ascending order.
Obviously if you had (a set of) 3 numbers you have $ 3! $ permutations:
(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1)
But only one of these is in asce... |
H: For what values of a the function $y=x^6+ax^3-2x^3-2x^2+1$ is even
I want to know for what valuyes this function is even
I know that $f(x)=f(-x)$ to proove that function is even. how its helps me?$$y=x^6+ax^3-2x^3-2x^2+1$$
Thanks!
AI: $$f(x)=f(-x)\\
x^6+ax^3-2x^3-2x^2+1=x^6-ax^3+2x^3-2x^2+1\\
(a-2)x^3=(2-a)x^3
$$... |
H: Partial fraction decomposition with a nonrepeated irreducible quadratic factor
I'm trying to do a partial fraction decomposition on the following rational eqn with a nonrepeated irreducible quadratic factor:
$$\dfrac{-28x^2-92}{(x-4)^2(x^2+1)}$$
I've broken it down into an identiy: $-28x^2 -92 = A((x-4)(x^2-1))+B(x... |
H: Differential equations basic problem
I know this is a basic Physics problems but somehow I can't solve it.
We have the differential equation: $2x''x^2 - 4 x^2x' - 2 x^3 = 0$
We have to conclude that the system:
$x' = y $
$y' = 2y + x$
..is equivalent to the differential equation. How can I do it?
Thanks in advanc... |
H: Calculate the angle between two curves $f(x)=x^2$ and $g(x)=\frac{1}{\sqrt{x}}$
I want to Calculate the angle between two curves on their intersect $f(x)=x^2$ and $g(x)=\frac{1}{\sqrt{x}}$, what I did so far is:
$$x^2=\frac{1}{\sqrt{x}} \rightarrow x=1$$then :
$$\tan(a)=\left |\frac{f'(a)-g'(a)}{1+f'(a)*g'(a)}\righ... |
H: Convergence of random variable to a negative constant
Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$
I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost surely).
The almost surely case is obvious from the definition of the co... |
H: Construct a linear programming problem for which both the primal and the dual problem has no feasible solution
Construct (that is, find its coefficients) a linear programming problem with at most two variables and two restrictions, for which both the primal and the dual problem has no feasible solution.
For a line... |
H: Prime decomposition in ring extensions
Let $R\subseteq R'$ be Dedekind domains, let $\mathfrak{p}$ be a nonzero prime ideal of $R$. Then $\mathfrak{p}R'$ is an ideal of $R'$ and it has a factorization
$$\mathfrak{p}R'=\mathfrak{P}_1^{e_1}\cdots\mathfrak{P}_g^{e_g}$$
in which $\mathfrak{P}_1,\ldots\mathfrak{P}_g$ ar... |
H: Does that series converge or diverge?
Does the series $$\sum \limits _{n=3}^\infty \frac{(-1)^{[\log n]}}{\sqrt{n}}$$ converge or diverge? As usually, $[x]$ denotes the integer part of $x.$
AI: Calculate the sum, $n=\lceil e^k\rceil$ to $\lfloor e^{k+1}\rfloor$. There are about $(e-1)e^k$ terms. Each has the same s... |
H: Dual of holomorphic functions (with the $L^1$ topology)
Let $\Omega$ be a connected domain of the complex plane, and let $E$ be the vector space of integrable holomorphic functions on $\Omega$. Then it can be checked that $E$ is a closed subspace of $L^1(\Omega)$. My question (which I'm sure has a well-known answer... |
H: Math school online?
As a hobbyist programmer with little to no math skills (no science skills either, thank you religious schools) I am finding math encroaching on my life more and more often. I have always laughed and said that I'm just not good at math because my brain doesn't work like that.
My younger brother c... |
H: Need help in finding counterexample
I need to find example that this isn't correct:
Let $R_1,R_2,R_3$ be binary relations on set $A$. Prove that this is not correct: $(R_1\cup R_2)\circ R_3 \supseteq(R_1\circ R_3)\cup(R_2\circ R_3)$
AI: According to one convention for writing the composition of two relations, $\lan... |
H: Age is fraction of year man dies
My friend sent me a question from an olympiad and im not sure that we have followed the right method, we both did the same thing:
The age of a man was 2/61 of the year in which he died. How old would he have been if he
lived until 1992?
Surly then he dies in 1992 and then his age is... |
H: notation question (bilinear form)
So I have to proof the following:
for a given Isomorphism $\phi : V\rightarrow V^*$ where $V^*$ is the dual space of $V$ show that $s_{\phi}(v,w)=\phi(v)(w)$ defines a non degenerate bilinear form.
My question : Does $\phi(v)(w)$ denote the map from $v$ to a linear function $w$? (... |
H: Contour Integral: $\int^{\infty}_{0}(1+z^n)^{-1}dz$
I'm working through Priestley's Complex Analysis (really good book by the way) and this Ex 20.2:
Evaluate $\int^{\infty}_{0}(1+z^n)^{-1}dz$ round a suitable sector of angle $\frac{2\pi}{n}$ for $n=1,2,3,...$
Can someone advise what the contour may be? If we use a ... |
H: Cramer-Rao Lower Bound
Assume that $X_1,X_2,\ldots,X_N\sim N(\mu,2^2)$ and $Y_1,Y_2,\ldots,Y_M\sim N(0,\sigma^2)$.
a)Find the Cramer-Rao Lower Bound (CRLB) for the variance of the unbiased estimators of $\mu$.
b)Find the CRLB for the variances of the unbiased estimators of $\mu^2$.
c)Is the MLE, $\hat{\mu}$, a unif... |
H: special case of Nagata's Lemma (Matsumura p.86)
Let $K$ be a field and $R$ a valuation ring of $K$ with maximal ideal $m_R$. Let $a \in R$ such that $1-a \in m_R$.
Statement: For any $s$ that is not a multiple of the characteristic of $R/m_R$, the element $(1+a+a^2+\cdots+a^{s-1})^{-1}$ is inside $R$.
How do we pr... |
H: inner product space and fixed vector
My professor gave this question:
$V$ is an inner product space, $v$ and $w$ are fixed vectors in $V$. Let $T:V\to V$ be a linear map such that $T(u)=(u,v)w$. Find $T^*$.
Now I need to find the rank of $T$ and to find the matrix that represents $T$ by the standard basis of $\ma... |
H: Linear Algebra - Vector Spaces of Vector Spaces
If we have a Vector Space such as $\Bbb R$, we can make Vector Spaces out of it. For example, let $v_1,v_2,v_3\in \Bbb R$. We know that $(v_1,v_2,v_3)$ is a Vector Space - it is $\Bbb R_3$ - and its dimension is $3\times \dim {(\Bbb R)} = 3$. Thus we can see that $\Bb... |
H: Integrate by parts: $\int \ln (2x + 1) \, dx$
$$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x + 1) \, dx = x\ln (2x + 1) - \int {2x \over 2x + 1} \cr
& = x\ln (2x + 1) - \int 1 - {1 \... |
H: Combination calculation with reducing set size
My statistics aren't too great, so I'm struggle to work out the result of the following situation.
Say you have 5 sets of 5 possible options (25 options total); and you select 1 option from each set. Each time you select 1 option from a set, that set is removed from th... |
H: probability that a random line segment parallel to the hypot. of a triangle with legs 3 and 4 will inclose an area of at least half
Sorry for the unclear title. It was difficult to explain the problem in a concise way in 150 characters.
A right triangle has the legs 3 and 4 units, respectively. Find the probability... |
H: Derivative of conjugate transpose of matrix
Building off of my previous question, I am trying to derive the normal equations for the least squares problem:
$$
\min_W \|WX - Y\|_2 \\
W \in \mathbb{C}^{N \times N} \quad X, Y \in \mathbb{C}^{N \times M}
$$
The intuitive way of viewing this problem is that I am trying ... |
H: Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$
If $x>1$ and $x^2+\dfrac {1}{x^2}=83$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$
a) $764$
b) $750$
c) $756$
d) $760$
In this question from given I tried to approximate the value of $x$ which should just above to 9 then I tried to calculate ... |
H: Linear Algebra - Another way of Proving a Basis?
If we have a Vector Space $V$, can we prove that a set $S$ of vectors $\in V$ is a basis of $V$ given that:
$S$ contains the same number of vectors as $\dim{(V)}$.
Every vector in a basis of $V$ can be written as a linear combination of the vectors in S
Example: L... |
H: Book Suggestions for an Introduction to Measure Theory
Couldn't find this question asked anywhere on the site, so here it is! Do you guys have any recommendations for someone being introduced to measure theory and lebesgue integrals?
A mentor has suggested a book that's in french, but unfortunately I don't know fre... |
H: Why isometric isomorphic between Banach spaces means we can identify them?
Is the "isometric" part really necessary? For what reason is that?
Eg. we prove that there is an isometric isomorphism between $(L^p)'$ and $L^q$ ($(p,q)$ conjugate) and then we identify them together as the same space. If they were isomorph... |
H: prove by induction that $P\left(\bigcup\limits_{i=1}^{n} E_i\right) = 1-\prod\limits_{i=1}^{n}(1-P(E_i))$, $E_1,E_2,\ldots , E_i$ independent
Suppose $E_1,E_2,\ldots , E_i$ are independent events. prove by induction that $$P\left(\bigcup\limits_{i=1}^{n} E_i\right) = 1-\prod\limits_{i=1}^{n}(1-P(E_i))$$
The first s... |
H: 4 equations with set of numbers
I must make 1 addition (x+y=z), 1 subtraction (x-y=z), 1 multiplication (x*y=z), and 1 division (x/y=z) equation with the following numbers. All the numbers must be used to fill x, y, and z of each equation. x, y, and z can consist be 1, 2, or 3-digit numbers.
0-4 available
1-9 avail... |
H: Minimizing Mean Squared Error for Exponential Function
I have a function that I'm trying to model using an exponential function and I'm trying to determine the constants for the exponential. I know I could optimize it using trial-and-error in R or another language, but I'd like to learn an analytic solution.
I figu... |
H: Find the value of $\frac{\tan\theta}{1-\cot\theta}+\frac{\cot\theta}{1-\tan\theta}$
I want to know an objective approach to solve these type of expression in a quick time
Which of the expression equals to
$$\dfrac{\tan\theta}{1-\cot\theta}+\dfrac{\cot\theta}{1-\tan\theta}$$
a)$1-\tan\theta-\cot\theta$
b)$1+\tan\th... |
H: Property of natural numbers involving the sum of digits
How can you prove that every natural number $M$ or $M+1$ can be written as $k + \operatorname{Sum}(k)$, where $\operatorname{Sum}(k)$ represents the sum of the digits of some number k.
Example:
$$
248 = 241 + \operatorname{Sum}(241) = 241 + 2 + 4 + 1$$
AI: Hin... |
H: Blackjack card counting, with one whole deck should the "count" end on zero?
When playing blackjack if you are card counting for a single deck, should the count always come to zero at the end of the deck? Wouldn't it depend on strategy and a corresponding betting table?
Would the optimum card counting strategy end ... |
H: A tricky probability question.
I have been asked the following question, and unfortunately I have no idea how to proceed. Here is the question:
Suppose we have 99 empty papers and we wrote numbers from 1 to 99(using each number) on one side of the papers randomly. Then we mixed all the papers randomly and started t... |
H: Finite abelian $2$-group
If $G$ is a finite abelian group such that $o(x)=2$ for all $x \neq e$ and $|G|=2^n$ for some $n\in\mathbb N$, prove that $G \cong \mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$ ($n$ factors).
Any help is appreciated. I asked this question before and was suggested to try induction but I have... |
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