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H: Defining an ideal in the tensor algebra
In the wikipedia article about exterior algebra:
The exterior algebra $Λ(V)$ over a vector space $V$ over a field $K$ is defined as the Quotient algebra of the tensor algebra by the two-sided Ideal $I$ generated by all elements of the form $x \otimes x$ such that $x \in V$.... |
H: $f = 0 $ almost everywhere implies $\int_{\mathbb{R}} f = 0 $
my try: Suppose $\phi $ is imple function such that $ 0 \leq \phi \leq f$, then since $ f = 0 $ almost everywhere, then $\phi = 0 $ almost everywhere. Let $A = \{ x : \phi(x) = 0\}$. Therefore, $A^c = \{x : \phi(x) > 0 \}$ is a null set. Hence
$$ \int_{... |
H: Minimization problem with parameter
$a>0,\; b>0,\; S=$parameter $>0$.
$$a+b+\dfrac{S-2}{2(a+b)} \longrightarrow min$$
With condition that
$a\cdot b =1$
Using inequality of arithmetic and geometric means we get:
$$a+b+\dfrac{S-2}{2(a+b)} \geq 2+ \dfrac{S-2}{2(a+b)}$$
With equality when $a=b$. This is true, but the a... |
H: About the parity of the partition function.
I am reading this Kolberg's article, where he proofs that the partition function takes both even an odd values infinitely often.
http://www.mscand.dk/article.php?id=1555
Although I'm sure it's simple (the proof is very short), I can't understand how he gets the contradict... |
H: Proof using chain rule.
For a homework, I need to demonstrate that
$$ \left(\partial z \over \partial x\right)^2 - \left(\partial z \over\partial y\right)^2 = 4{\partial z \over \partial u}{\partial z \over \partial v}$$
where $z = f(x,y)$ and $u = x+y$ and $v = x-y$.
I tried by chain rule, but nothing.
Can anyone ... |
H: How do I find y' in this problem?
$$y^3 = cos(xy) $$
This is what I have so far:
$$(3y^2y'/y+xy') = -sin(xy)*y $$
$$ y' = -sin(xy) (3y^2/y)$$
I'm sure I did the last part wrong, but that was the best I could come up with.
AI: I do not see where your second line comes from, but the derivative of $y^3$ with resp... |
H: Combination Factorial formula
Is there a formula for working out combinations using the factorial of the 'choice' number?
E.g.: for a group of 6, what are the total possible combinations of up to 4 selections?
I have been doing this by adding the individual combinations, so (in the above example) I would add:
$$ \d... |
H: Uniqueness Proof for Division Algorithm using Contradiction
Let $a, b, \mathbb \in \mathbb {Z}$ and let there exist integers $q, q_1, r, r_1$ such that
the two pairs $(q,r)$ and $(q_1,r_1)$ satisfy the properties:
$$\ \ \ \ a = qb+r \quad \ \ \ \ \ \ \ ; 0 \le r \lt b \\
also, \ a = q_1 b + r_1 \quad ; 0 \le r_1 ... |
H: Solving $(3 {Q} {s^{2}})/(100h (h^{2} - 3hs + 3s^{2}))$ for $h$
$$Y = \frac{3 \times{Q} \times{s^{2}}}{100h \left(h^{2} - 3hs + 3s^{2}\right)}$$
where Q, s, and h are different variables.
The question I have is how to re-arrange this equation in terms of h?
So far, I have
$$Y = \frac{3 \times{Q} \times{s^{2}}}{100h... |
H: Let $L$ be the language defined by the regular expression $(a \vee b \vee c)(a \vee b \vee c)$
1) How does $|L| = 9$?
2) $|L^*| = \aleph_0$?
Thank you
AI: HINTS:
Either use the multiplication principle, or write out all words of the language in systematic fashion from the regular expression.
If $w\in L$, then $w,... |
H: Chain rule on matrix differentiation of trace in Kalman gain proof
I was reading the following derivation of the Kalman filter gain
http://www.robots.ox.ac.uk/~ian/Teaching/Estimation/LectureNotes2.pdf on pg 5-6.
From the following
$$L=\min_{K_{k+1}} trace(P_{k+1|k+1})$$
With
$$ P_{k+1|k+1} = \boldsymbol{ (I-K_{k+1... |
H: Finding $x$ in the domain of the function where the tangent line is horizontal
$$ y = x^3 - 4x^2 +4x + 2$$
I have to find all the values of the domain of the function where the tangent line at the point (x,f(x)) is horizontal.
I wasn't so sure on how to start this problem, so the first thing I did was derive the fu... |
H: Finding the value of $f_{x+y}$ for multivariate normal distributon
Given that bivariate normal distribution is
I need to find the value of $f_{X+Y}$ and the variable are standard normal.
If the variables are standard normal,
$$f_{X+Y}(z)=\int_{-\infty}^{+\infty}f_{X,Y}(x,z-x)\,\mathrm dx\\
=\int_{-\infty}^{+\inf... |
H: Find the limit -> Infinity with radicals
First guess to multiply by $x^{-1.4}$ so the radical in numerator with $x^7$ becomes 1 and other stuff becomes 0. But then denominator becomes $-\infty$. What is the right approach?
$$ \lim_{x \to \infty} \frac{\sqrt[5]{x^7+3} - \sqrt[4]{2x^3 - 1}}{\sqrt[8]{x^7 + x^2 + 1} - ... |
H: semigroup ideals
For the semigroup $$S_{3 \times 3} = \bigl\{(a_{ij}) \bigm| a_{ij} \in \mathbb Z_2 = \{0,1\}\bigr\}$$ (the set of all $3\times3$ matrices with entries from $\mathbb Z_2$) under multiplication. Find all the ideals of $S_{3\times3}$.
AI: Hint: For any $A \in S_{3\times 3}$ there are $B,C \... |
H: Almost the reverse triangle inequality
I'm wondering if one can get such an inequality (for $n$ a positive integer) :
$$\exists C>0\ \forall(x,y,u,v)\in \left(\mathbb{R}^n\right)^4,\quad \left| \left|x-y \right|^2-\left|u-v \right|^2 \right|\le C\left( \left|x-u \right|^2+\left|y-v \right|^2 \right) $$
I've tried u... |
H: Calculate the covariance matrix of $\hat{\beta}$
$y=X\beta+u$ where $u \sim N(0,\Sigma)$ and $\Sigma$ is symmetric & idempotent.
$X: n*k$, $y:n*1$, $\beta=k*1$, $u:n*1$ vector.
Suppose you apply LS(least square) to the model.
Calculate the covariance matrix of $\hat{\beta}$.
Since $\hat{\beta}=(X'X)^{-1}X'y$,
$E(... |
H: Characterizing trees where every diametral path shares an edge
In a graph $G$, a diametral path is a path of length $\text{diam}(G)$ joining two vertices that are at a distance $\text{diam}(G)$ from each other. Given a tree $T$, consider the set of all diametral paths of $T$. For example, if $T$ is say the star gra... |
H: showing that two surface is isometric
I tried to show that a parametrized surface $S$ in $\mathbb{R}^3$ given by $(u, v)$ ->$(u, v, u^2)$ is isometric to the flat plane.
At first, I found their first fundamental form, but they are different.
(I choose a surface patch defined by $(u, v)$ -> u$\mathbb{q}$ + v$\mathbb... |
H: Triangle inequality- complex
I am trying to prove the triangle inequality purely algebraically.
Let $z=x+iy$, $w=u+iv$.
Then,
$|z+w|^2$=$|(x+u)+i(y+v)|^2$=$(x+u)^2+(y+v)^2$=$x^2+2xu+u^2+y^2+2yv+v^2$
I tried the other way:
$(|z|+|w|)^2$=$(\sqrt{x^2+y^2}+\sqrt{u^2+v^2})^2$=$x^2+y^2+u^2+v^2+2 \sqrt{x^2+y^2} \sqrt{u... |
H: Change of an angle in a triangle
I've a triangle ABC. Where AC is the hypotenuse and the angle ABC is 90 degress.
AB is $15 km$ and changes with a speed of $600 km/h$.
BC is $5 km$ and changes with a speed of $0 km/h$.
At what speed changes the angle CAB in terms of $rad/h?$
I call AB for $x(t)$ and BC for $y(t)$. ... |
H: optimizing prime number algorithm
I am doing a function to return a list of prime number up to "n", one what to optimize the algorithm is the following:
"The next most obvious improvement would probably be limiting the testing process to only checking if the potential prime can be factored by those primes less than... |
H: Ring of $p$-adic integers $\mathbb Z_p$
There are a few ways to define the $p$-adic numbers.
If one defines the ring of $p$-adic integers $\mathbb Z_p$ as the inverse limit of the sequence $(A_n, \phi_n)$ with $A_n:=\mathbb Z/p^n \mathbb Z$ and $\phi_n: A_n \to A_{n-1}$ (like in Serre's book), how to prove that $\m... |
H: How do I solve a derivative that has an absolute value $x \times |x|$
I have a function x * |x|
To get the derivative I used the first principals:
$$ f'(x) = ( (x-h) * | x + h | - (x * |x|) )/ h $$
So if x is + I got
$$ x ^ 2 + xh - xh - h^2 - x^2 / h $$
$$ -h^2/h$$
$$ -h $$
$$ 0 $$
If x is negative:
$$ x^2 - 2xh ... |
H: Hausdorffness of a space
We know that if $X$ and $Y$ are two topological spaces with $Y$ Hausdorff, then for any two continuous functions $f:X\to Y$ and $g:X\to Y$, $\{x\in X: f(x)=g(x)\}$ is a closed set.
Is the converse true? Let us suppose that $Y$ be a topological space and for any topological space $X$ and fo... |
H: What does the result of derivating out all of the units from a "number" tell you?
I apologize for the terrible title, I'm not sure of the right way to say it.
Consider an equation which outputs in Watts $ \frac{kg \cdot m^2}{s^3} $. If you derivative out all the changes in kilograms, meters, and seconds what does ... |
H: Question About 3-Dimensional Integration
I'm having trouble with determining the volume of this solid.
$V := {(x,y,z) : 0 \le x \le 1, x^2 \le y \le \sqrt{x}, -x^{1/3} - y^{1/3} \leq z \leq x^{1/3} + y^{1/3}}$
I started off thinking $\mathrm{vol}(V) \leq \int_0^1 \int_{x^2}^{\sqrt{x}} x^{1/3} + y^{1/3} dy dx$ but I... |
H: Transformation (coordinates) - PDE
If I have $-\cos(x)u_x$ for $u\in C^2(\Omega), \Omega\subset\mathbb{R}^2$ and
$$
\xi:=-x-\cos(x)+y, \eta:=x-\cos(x)+y
$$
as transformation of the coordinates, what is then $-\cos(x)u_x$ after this transformation?
Do not know, how to solve this.
AI: We have
$$ x = \fr... |
H: definition of opposite catagory
I have been given two definitions of what an opposite category is. The first is one which is in my textbook and is follows:
If $\mathcal{C}$ is a catagory with objects $ob(\mathcal{C})$ and morphisms $C(U,V)$ for $U,V\in ob(\mathcal{C})$ we define the opposite catagory $\mathcal{C}^... |
H: How many ways can I express $X$ as a sum unique natural numbers to the power of $N$
I also posted the problem here on Stackoverflow: http://ejj.mobi/g5i5gh but that was concerned with the programming solution. Now I am much more interested in the theory behind this problem, or rather the fact that I don't understan... |
H: related rates of change
A balloon is rising vertically above a level, straight road at a constant rate of $1$ ft/sec. Just when the balloon is $65$ ft above the ground, a bicycle moving at a constant rate of $ 17$ ft/sec passes under it. How fast is the distance between the bicycle and the balloon is increasing $3$... |
H: Proof the Legendre polynomial $P_n$ has $n$ distinct real zeros
I need a proof to show that the inequality $m < n$ leads to a contradiction and $P_n$ has $n$ distinct real roots, all of which lie in the open interval $(-1, 1)$.
AI: By Rodrigues formula for Legendre polynomials,
$$\displaystyle P_n(x) = \frac{1}{2^n... |
H: compactness of the real projective plane
Let $\mathbb{P}_2(\mathbb{R})$ =( $\mathbb{R}^3$-{0})/~ where x~$r$x for any nonzero point x $\in \mathbb{R}^3$ and any nonzero $r \in \mathbb{R}$
I want to show that the $\mathbb{P}_2(\mathbb{R})$ is compact..
I used quotient map to solve, but it doesn't work. Would you i... |
H: Find $a$ with $a^2-a\equiv -1\pmod p$ when $p$ is a prime with $p\equiv 1\pmod 3$
I'm trying to prove that the primes with form $3k+1$ are not Eisenstein prime. This step: to find $a$ such that $a^2-a\equiv -1\pmod p$ when $p$ is a prime with $p\equiv 1\pmod 3$ is the only obstacle now, and I have thought of many m... |
H: Finding coordinates of a path
I am trying to find out whether to geometrical shapes overlap. This is based on an SVG-file.
The following is an example:
Group 1:
<g transform="matrix(1.0 0.0 0.0 1.0 257.0 124.95)">
<g transform="matrix(1.7880859 0.0 0.0 0.9476929 -0.05 0.0)">
<path d="M 18.9 8.15 L -18.... |
H: Picard group of a Affine scheme
How do we define a Picard group of an Affine scheme? Is there way to define as for commutative ring? Thanks
AI: The Picard group can be defined for any scheme $X$ as the group of isomorphism classes of invertible sheaves ( $\mathcal{O}_X$-modules that are locally free of rank $1$). G... |
H: Let G be a group, and let f(x)=x^(-1) for all x in G. Is f a function from G to G? If so, is it 1-1, onto?
Let G be a group, and let f(x)=x^(-1) for all x in G. Is f a function from G to G? If so, is it 1-1, onto?
Abstract algeba..
I know what it means to be 1-1 and onto..
1-1: f(a)=f(b)=>a=b
onto: there exists ... |
H: Calculate limit using Stolz-Cesàro theorem
Can someone help me calculate this limit using the Stolz-Cesàro theorem?
$\lim_{n\to \infty } \frac{1+\frac12+......+\frac1n}{\ln n}$
AI: $\ln n$ is unbounded and increasing and hence we can use the theorem:
$$
\displaystyle\lim_{n\to \infty } \frac{1+\frac12+......+\frac1... |
H: if $f : \mathbb{R} \rightarrow \mathbb{R} $ is continuous and $f(x) \to 0$ as $x \rightarrow \pm \infty $. Real Analysis (Continous function)
If if $f : \mathbb{R} \rightarrow \mathbb{R} $ is continuous and $f(x) \to 0$ as $x \rightarrow \pm \infty $. How to prove that $f(x)$ is bounded ?
AI: Since $\lim_{x\to \pm... |
H: Limit in form of 0/0
If I multiply by I get zero/2x anyway.
What manipulation needed to get 2/3?
AI: Remember the identity $$a^3 - b^3 = (a-b)(a^2 + ab + b^2).$$ If you think of $a$ as $(1+x)^{1/3}$ and $b$ as $(1-x)^{1/3}$, you find that multiplying by $$\frac{(1+x)^{2/3} + (1+x)^{1/3}(1-x)^{1/3} + (1-x)^{2/3}}{... |
H: A set of functions which is open in the space $C^1[0,1]$
Let $f:[0,1]\to [0,1]$ be a $C^1$ and increasing
function such that
$i)$ If $f(p)=p$ then $|f'(p)|\ne 1$
I want to prove that there exist an $\varepsilon>0$ such that if $g\in C^1$ and $||f-g||_{C^1}<\varepsilon$ then $g$ also have the two properties.
I don'... |
H: Can this polynomial transformation produce new symmetry?
I've got a polynomial transformation on $\mathbb{R}^6$, and I have a conjecture about it, but I'm having a hard time proving it. The transformation looks like this:
$
u:= abcde + abc + abe + ade + cde + a + c + e\\
v:= bcdef + bcd + bcf + bef + def + b + d + ... |
H: Marble ring: no two blacks are adjacent
Question: How many ways are there to arrange $20$ marbles, $6$ black and $14$ white, in a ring (circular arrangement) such that no two of the black marbles are next to each other?
Comments: It is easier to find the probability, as there are shortcuts one can make without hav... |
H: Disjunctive normal form expansion
I do not understand this at all.
Find the sum-of-products expansions of these Boolean functions.
$F(x, y, z) = x + y + z$
$F(x, y, z) = (x + z)y$
$F(x, y, z) = x$
$F(x, y, z) = x y$
How is $x$ not just $x$? This makes no sense to me at all and my book wants me to memorize a... |
H: Find number of ways of selection of one or more letters from the word: AAAABBCCCDEF
Hello the question is above , I can't understand such type of questions, please help.
the answer is 479 but how does it come?
AI: As $A$ occurs $4$ times, we can choose A in $4+1$ ways
So, the number of ways of selection of no or mo... |
H: Is ${\mathbb Z} \times {\mathbb Z}$ cyclic?
Not sure where to go with this, but I don't think it is cyclic..
AI: Hint: supose $\;m,n\in\Bbb Z\;$ are such that $\;\Bbb Z\times\Bbb Z=\langle (m,n)\rangle\;$, then among other things there must exist $\;x\in\Bbb Z\;$ s.t.
$$x(m,n)=(1,1)\implies xm=1=xn\implies\ldots ?$... |
H: Zariski's article on cohomology in algebraic variety
Zariski is supposed to have written an article on cohomology in algebraic geometry. It is supposed to be very good and cited as such for instance here. Can anyone supply a precise reference to this? I couldn't find such an article by search for author Zariski in ... |
H: Standard inductive problem
Question: Prove that $2^n \geq (n+1)^2$ for all $n \geq 6$.
I have tried to prove this below and I'm interested if my method was correct and if there is a simpler answer since my answer seems unnecessarily long for such a simple claim.
Inductive hypothesis $$2^n \geq (n+1)^2$$
We need to... |
H: Prove that $6$ divides $n(n + 1)(n + 2)$
I am stuck on this problem, and was wondering if anyone could help me out with this. The question is as follows:
Let $n$ be an integer such that $n ≥ 1$. Prove that $6$ divides $n(n + 1)(n + 2)$.
Note: An integer $a$ divides an integer $b$, written $a|b$, if there exists $q... |
H: Find the total work done by the force field F
Find the total work done in moving a particle in a force field given by:
$F=(y^2-x^2)i+(2-x+y)j$ along the curve $y=x^3$, from $(-1,-1)$ to $(1,1)$
Help is appreciated!
AI: In the case of a constant force and a linear motion, work is given by finding the component of ... |
H: Why output of FFT is same as input data size ?
What I understand from DFT formula below I can decide the N my self. I can try to use just 16 bins to describe a function or I may even use 4 , it won't be very accurate but I can do it right?
The confusing part is even most thrusted websites say next sentence taken ... |
H: How to find $\lim_{x\rightarrow\infty}\bigl(\frac{x+1}{x-2}\bigr)^{2x-1}$
$$\lim_{x\rightarrow\infty}\left(\frac{x+1}{x-2}\right)^{2x-1}$$
What are the steps to solve it? Probably the division should be multiplied by some expression.
AI: As $\displaystyle \frac{x+1}{x-2}=1+\frac3{x-2},$
$$\lim_{x\to\infty}\left(\fr... |
H: Existence of Algebra of anticommuting idempotents
Background and motivation: I'm wondering about the existence of an algebra which is in some ways similar to the exterior algebra, but is generated by idempotents rather than nilpotents.
Let $V$ be a finite-dimensional real vector space and let $\{e_i\}$ be a basis o... |
H: Random Variable with density and E(X)
$X$ is a random variable with value $0,1,2$ and $E(X)=1$, $E(X^{2})=3/2$. Find $f(x)$=the density of $X$ and find $E(X^{7})$=?
Here is what I did:
$E(X) = 0*f(0)+1*f(1)+2*f(2)$
$E(X^{2}) = 0^{2}*f(0)+1^{2}*f(1)+2^{2}*f(2)$
Then I got:
$f(1)+2*f(2)=1$
$f(1)+4f(2)=3/2$
Then I got... |
H: Proving logical equivalence: $P \Leftrightarrow P \vee (P \wedge Q)$
I'm a first year CS student about to write his first term test and this question is part of our practice package. I have not been successful in writing a sequence of equivalences to justify this proof.
$P \Leftrightarrow P \vee (P \wedge Q)$
I've... |
H: Degree of Johnson graphs and their subgraphs
What is the degree of Johnson graph $J(n,k)$ where $n>k$?
http://en.wikipedia.org/wiki/Johnson_graph
What are some good examples of subgraphs of Johnson graphs? The Johnson graph I am interested in is $J(n,2)$.
AI: From the Wikipedia page: Johnson graphs have ${n \choose... |
H: Calculate a ratio from part of a range.
I'm building a smartphone javascript application but my question today is really only math related. To give you a quick idea of what I'm doing, my code checks the smartphone's gyroscope to determine how much it is rotated. Based on that rotation, it's supposed to pan a backgr... |
H: To Prove That a Certain Set is a Manifold
Definitions and Notation:
Let us write $\underbrace{\mathbb R^n\times \cdots\times\mathbb R^n}_{m \text{ times}}$ as $(\mathbb R^n)^m$.
A rigid motion in $\mathbb R^n$ is a function $L:\mathbb R^n\to \mathbb R^n$ such that $||L(x)-L(y)||=||x-y||$ for all $x,y\in \mathbb R^n... |
H: Showing divergence
Suppose $a_n>0$, $s_n=a_1+ \dots+ a_n$, and $\sum a_n $diverges.
I need to prove that $\sum \frac{a_n}{1+a_n}$ diverges.
My attempt: We have $\forall n \in \mathbb{N}^*, a_n>0$
$\forall n \in \mathbb{N}^*, \frac{a_n}{1+a_n}-a_n\\= \frac{a_n^2}{1+a_n} \sim {a_n}^2 $
Since $\sum a_n$ diverges, the... |
H: What is the connectivity between Boltzmann's entropy expression and Shannon's entropy expression?
What is the connection between Boltzmann's entropy expression and Shannon's entropy expression?
Shannon's entropy expression:
$$ S= -K\sum_{i=1}^np_i\log (p_i) $$
AI: Mathematically they only differ in the constant mu... |
H: If the square of an integer is odd, then the integer is odd
The statement is: If the square of the integer x is odd, then x is odd.
My textbook says an indirect proof is applicable here but I came up with the following:
State that If $x^2$ is odd, then $x$ is odd.
Next, we assume that $x$ is odd.
We know if $x$ is... |
H: Did I answer this probability question correctly?
Hello I have the following question to answer:
Imagine three boxes, each of which has three slips of paper in it each with a number
marked on it. The numbers for box A are 2, 4 and 9, for box B 1, 6 and 8, and for box C
3, 5 and 7. One slip is drawn, independently a... |
H: How to construct the graph from an adjacency matrix?
I have the following adjacency matrix:
a b c d
a [0, 0, 1, 1]
b [0, 0, 1, 0]
c [1, 1, 0, 1]
d [1, 1, 1, 0]
How do I draw the graph, given its adjacency matrix above (I've added a,b,c,d to label vertices).
I don't understand how the vertex $d$ (e.g., the r... |
H: Norms on $\mathcal{M}_{n \times n}(\mathbb{R})$
On the space of all matrices $n \times n$ with real coefficients $\mathcal{M}_{n \times n}(\mathbb{R})$ we define two norms:
$||A||_1 := sup_{x\neq 0; x \in \mathbb{R}^n} \frac{||Ax||}{||x||} \\
||A||_2 := max |A_{ij}|.$
It's quite evident that these are indeed norms,... |
H: Continuously differentiable curves
How one can show that the unit cube $[0,1]^d$ cannot be covered by countably many continuously differentiable curves?
Thanks in advance
AI: By Sard's theorem, if $f:\mathbb R\to\mathbb R^d$ is continuously differentiable and $d>1$, then $f(\mathbb R)$ has measure zero. So, a count... |
H: Can there be a real solution to the square root of -1?
Basically what the title says:
Can there be a real solution to the square root of $-1$ (or any negative number in fact) or is it defined to be unreal?
Because of this: $$ \begin{align} \sqrt{-1} & = (-1)^\frac 12 \\[6pt]
& = (-1)^\frac 24 \\[6pt]
& = \sqrt[4]{(... |
H: hitting time of Brownian motion
I'm desperately trying to prove that for a standard BM $B_t$ the stopping time $T:=\inf\{t\geq0: B_t\geq\sqrt{1+t}\}$ is a.s. finite, i.e. $\mathbb{P}[T<\infty]=1$.
I actually tried to play around with $S_t=\sup_{s\in[0,t)}\frac{B_s}{\sqrt{1+s}}$ which should have the half-normal dis... |
H: Showing rational numbers are algebraic
A polynomial with integer coefficients is an expression of the form:
$f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$
where $a_n$, $a_{n-1}, \ldots, a_1, a_0$ are integers and $a_n$ is not equal to $0$.
a zero of the polynomial is a $c \in \mathbb{R}$ such that $f(c)=... |
H: Why does there seem to be so much error in the laws of sines and cosines?
I've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convention of naming the angles opposite these sides... |
H: How to do this integral analytically?
How to evaluate the following integral analytically $$f(x,y)=\int_{\Omega }d^2ke^{i(xk_x+yk_y)}$$, where the region $\Omega=\left \{ (k_x,k_y):k_x^2+k_y^2\leq K^2 \right \}$ with radius $K>0$.
What's the exact form of $f(x,y)$? Thank you very much.
AI: Use polar coordinates, an... |
H: Solving imaginary equation $z^3 = 5i + 5$
I need help solving the equation : $z^3 = 5i + 5$. I'm basically just starting to learn imaginary numbers and how to solve them.
Straight forward solution is a pain ( was trying to figure out with step-by-step solution from wolframalpha ). After that I was reading wikipedi... |
H: Prof that series converges
$(x_n)$, $x_1=\sqrt2$, $x_2=\sqrt{2}^\sqrt{2}$, $x_3=(\sqrt{2}^\sqrt{2})^\sqrt2$
$x_\left(n+1\right) = (\sqrt2)^\left(x_n\right)$
Does this series converge?
AI: Here are the steps you need to prove.
First prove that $1 < x_n < 2$ using induction.
Nest prove that $x_n$ is a monotone incre... |
H: Given real matrices $A,B$, exists $\tilde{B}$ so that $\langle Bx,x \rangle = \langle \tilde{B}x,x \rangle$
Given $A, B \in \mathbb{R}^{n \times n}$, show there exists $\tilde{B}\in \mathbb{R}^{n \times n}$ such that
$\langle Bx, x\rangle = \langle \tilde{B}x, x \rangle$ for all $x\in \mathbb{R}^n$ $\langle (A^2 ... |
H: Prove by mathematical induction that $\sum_{i=1}^{n}\frac{i}{2^i}\leq2$ for $n\ge 1$
I have this exercise by my professor that I have no idea how to solve. Any help would be greatly appreciated:
Using the method of mathematical induction show that for all $n \geq 1$, $n \in\mathbb{N}$
$$\sum_{i=1}^{n}\frac{i}{2^i}\... |
H: How to integrate $\int\limits_{|z| = R} \frac{|dz|}{|z-a|^2}$
I need to integrate, $\int\limits_{|z| = R} \frac{|dz|}{|z-a|^2}$ where $a$ is a complex number such that $|a|\ne R$.
So first I tried polar coordinates, which gives something I cannot continue.
Then I tried to write $|dz| = rd\theta = dz/ie^{i\theta}$ ... |
H: Combination Question [GRE PREP]
Three men (out of 7) and three women (out of 6) will be chosen to
serve on a 7 member committee. In how many ways can the committee be
formed?
I did 7C3 to get 35 men.
Then i did 6C3 to get 20 women.
Then i decide to add up 20 + 35 and get 55 but it is suggested i have to multi... |
H: Distributive properties of quanitifiers
What is the difference between
$\forall x~(~P(x) \to Q(x)~)$
and
$\forall x P(x) \to \forall x Q(x)$
To me they seem to be the same thing, what difference does it make where the quantifiers go?
AI: $\forall x (P(x) \rightarrow Q(x))$ means that for every $x$ that you care to ... |
H: probability without replacement and unknown number in urn
An urn contains N red chips, N blue chips, and N white chips all numbered 1 through N. Two chips are drawn at random and without replacement. Let A be the event that the two chips drawn are the same color; and let B be the event that they have the same numbe... |
H: Rational Zeros Theorem to show irrationality
Show that $\sqrt{3}+\sqrt{5}$ is irrational using the rational zeros theorem, you have to find a polynomial that has $\sqrt{3}+\sqrt{5}$ as a zero
AI: Let x = √3 + √5 Then x² = 8+2√15 and (x²-8)²=60 from which you can find your polynomial. With the Rational Zero test it ... |
H: What is the meaning of $\frac{0}{0}$?
I asked my teacher what is the real meaning of $\cfrac{0}{0}$, and the answer I got was "nobody knows".
I can't leave this subject "as is". I need a decent explanation, at least an explanation to why "nobody knows". I'm sure you'll come up with a few good answers.
AI: I really ... |
H: Finding Polynomials to Satisfy a Condition
I need to find polynomials $x(n), y(n)$ s.t. $x(n)(n^{2}+n+1)+y(n)(n^{2}+1)=1$, $\forall n \in \mathbb{Z}$.
I tried distributing it out:
$x(n)n^{2} + x(n)n + x(n) + y(n)n^{2}+y(n)=1$. I understand that for this to work, one of the terms has to equal one, and the other term... |
H: Rectangle divided into three triangles with two lines. One angle is given, what are all the others?
Let's suppose I have a rectangle divided into three triangles in the following way. No lengths of either the rectangle or triangles are known, only one angle is known.
I would like to know how to calculate all the ot... |
H: Characteristic $n$ and local rings
Prove that:
a) If A is a local ring then A has characteristic zero or a power prime.
Proof.
Suppose M is the unique maximal ideal of A then $A/M$ is Field in particular integral domain then $Char( A/M ) = 0$ or $p$ with some $p$ prime.
If $Char( A/M ) = 0$ then $\forall n \in \mat... |
H: How do I find $\lim_{x \to 0} \frac{\cos3x-\cos x}{\tan 2x^2}$?
Can't understand how to solve limit like this:
$$\lim_{x \to 0} \frac{\cos3x-\cos x}{\tan2x^2}$$
My attempt is:
$$\lim_{x \to 0} \frac{\cos3x-\cos x}{\tan2x^2}=\lim_{x \to 0} \frac{\cos3x}{\tan2x^2}- \frac{\cos x}{\tan2x^2}=\lim_{x \to 0}\frac{1}{\tan2... |
H: Show function $f(x)=\frac{r^2 \cdot a}{a^2+x^2}$ is strictly decreasing
I want to show, that
$$f(x)=\frac{r^2 \cdot a}{a^2+x^2}$$
with $x,a,r \in \mathbb{R}$ is strictly decreasing for $x>0$
A function is called strictly decreasing, if $x_1<x_2 \Rightarrow f(x_1)>f(x_2)$.
Perhaps someone can give me a hint how to s... |
H: Which sign does $\lim_{n\to\infty}(-2)^n$ have?
How can I express $\lim_{n\to\infty}(-2)^n$ using $\infty$? Which sign does it have, plus or minus?
AI: Neither. $\lim_{n \to \infty}(-2)^n$ does not exist. |
H: All subsets of $\mathbb{N}$ are open in $\mathbb{N}$?
Why precisely is this true? I've always sort of overlooked this and taken it for granted, but why exactly are all subsets of $\mathbb{N}$ open in $\mathbb{N}$?
AI: All subsets of $\Bbb N$ are open in $\Bbb N$ if and only if $\Bbb N$ is given the discrete topolog... |
H: Bounding integral of the form $\int_0^\infty |z+x|^{-n}dx$
I need to bound a sum of the form $\sum_{j=1}^m |z+j|^{-n}$ with $\Re z\ge1$ and $n \ge 2$. I am searching for a bound of the form
$$\sum_{j=1}^m |z+j|^{-n} \le C |z|^{-n+1}.$$
It is easy to see that
$$\sum_{j=1}^m |z+j|^{-n} \le \int_{0}^{m+1} |z+x|^{-n}d... |
H: Prove that $ℤ[i]^*= \{1,-1,i,-i\}$
Prove that $ℤ[i]^*= \{1,-1,i,-i\}$.
$\{1,-1,i,-i\} ⊂ ℤ[i]^*$ is trivial. But I'm not sure about the other inclusion.
Let $(a+bi) \in ℤ[i]^*$. Then there exist $c,d \in ℤ$ such that $(a+bi)(c+di)=1$.
Then $ac+bci+adi-bd=1$. Then $bc-ad=0$ and $ac=1+bd$. What can I conclude from thi... |
H: What is the definite integral $\int_{-1}^1 \frac{1}{x} dx$ equal to?
What is the definite integral $\displaystyle\int_{-1}^1 \frac{1}{x} dx$ equal to?
Is it $0$ like it would be if you just integrated and plugged in the bounds, or does the discontinuity at $x=0$ invalidate that line of reasoning and make this integ... |
H: What is the Y-intercept?
In class, my math teacher was teaching us how to draw functions on a coordinate plane, and he mentioned something about the Y-Intercept being an important step in creating/solving a function. But, what exactly is a Y-Intercept?
AI: The $y$-intercept of a function $f(x)$ is the point where t... |
H: Prove that there are no $x,y ∈ \mathbb N$ for which $x^2-y^2 = 10$
I began by factoring and got $(x+y)(x-y) = 10$
Then I tried cases and was able to prove the ones where $x$ and $y$ are equal-> because the equation will result to zero.
and also where $x < y$, because the answer will be negative.
How can I prove wh... |
H: Prove open set is not closed
The question might sound ridiculous, but I am not able to prove it with rigor. I tried proving it by the following definitions ONLY.
Open set: A set $U$ is open if for every $a$ belonging to $U$, there is
some $r = r(a) > 0$ such that the ball $B_r(a)$ is contained in $U$.
Closed se... |
H: Why can't this number be written as a sum of three squares of rationals?
This may be a very naive question and I apologize in advance.
Suppose that $n$ is a positive integer which cannot be written as a sum of three squares $a^2+b^2+c^2$ for integers $a,b,c\in\mathbb{Z}$.
Does it follow that $n$ cannot be written a... |
H: Is there a fast way to compute coefficient of some term of the product of some series'?
The example in wikipedia is
$$A=1-3x+5x^2-7x^3+9x^4-11x^5+\cdots$$
$$B=2x+4x^3+6x^5+\cdots$$
$$AB=2x-6x^2+14x^3-26x^4+44x^5+\cdots$$
And the term $x^5$ is given by
$$44x^5=(1\cdot6x^5)+(5x^2\cdot4x^3)+(9x^4\cdot2x)$$
So if you ... |
H: Derivative where x=a
If $f(x)=x^3+3x+2$
Find the number(s) a such that the tangent lines to the graphs of $f(x)$ and $f'(x)$ at $x=a$ are the same.
So far I have come up with:
$f'(x)=3(x^2+1)$
And when I graph both these functions out on my calculator the only point I am finding where $x=a$ is at $1$ (which could b... |
H: If the row-reduced form of matrix $A$ has a row of zeros, its columns do not span $\mathbb{R}^n$
Can someone explain why it is that, if the row-reduced form of an $n\times m$ matrix $A$ has a row of zeros, the columns of matrix $A$ do not span $\mathbb R^n$?
I'm not seeing the bigger picture here.
AI: Presuming the... |
H: Integral of the product of two Sine Functions Evaluated at Regular Intervals = 2 * Num. of Intervals Used?
Basically the I am considering the sum of the product of two sine functions:
$$\sum_{k=0}^{N-1} sin(2 \pi {1\over N} k) sin(2 \pi {1 \over N }k)$$
I am trying to make sense of the the case when you multiply th... |
H: show it is a random variable
Let X and Y be random variables and let A be an event. Show that
the function
$$Z(\omega)=\begin{cases}X(\omega) \quad \text{if} \; \omega \in A\\
Y(\omega) \quad \text{if} \; \omega \in A^c
\end{cases}$$
is a random variable.
I considered the case in which $Z(\omega)=I_A(\omega)$, wher... |
H: Prove that if $A$ is a set with $m$ elements, $B$ is a set with $n$ elements, and $A \cap B = \emptyset$, then $A \cup B$ has $m+n$ elements.
I'm working through a real analysis textbook, so I don't want the full answer. I'm only looking for a hint on this problem.
The book starts the proof like this: let $f$ be a ... |
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