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H: A short exact sequence of groups and their classifying spaces
Suppose that we have a short exact sequence of topological groups:
$$1 \to H \to G \to K \to 1.$$
I have found some papers mentioning that the above sequence induces a fibration:
$$BH \to BG \to BK.$$
Here $B$ assigns to each (topological) group its clas... |
H: Showing unique decomposition into parallel and orthogonal parts for any subspace
Given a general (possibly infinite dimensional) linear vector space $V$ with an inner product, how can you prove that for any subspace $S$, any vector v in $V$ can be uniquely expressed as
$$v = s + t$$
where $s \in S$ and $t$ is ort... |
H: Diagonalizability in $\mathbb{R}$ and $\mathbb{C}$
Give an example of a matrix $A\in M_{n\times n}(\mathbb{R})$ that is not diagonalizable, but A is diagonalizable viewed as a matrix over the field of complex numbers $\mathbb{C}.$
AI: The characteristic polynomial $x^2+1$ of $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ ... |
H: Conditional Probability Problem
An insurance company examines its pool of auto insurance customers and gathers the following information:
(i) All customers insure at least 1 car
(ii) 64% of all customers insure more than one car
(iii) 20% of the customers insure a sports car
(iv) Of those customers who insure ... |
H: Yitang Zhang: Prime Gaps
Has anybody read Yitang Zhang's paper on prime gaps? Wired reports "$70$ million" at most, but I was wondering if the number was actually more specific.
*EDIT*$^1$:
Are there any experts here who can explain the proof? Is the outline in the annals the preprint or the full accepted paper?
AI... |
H: What functions are solution to a homogeneous system of differential equations?
Given a vector $\vec{u} \in \mathbb{R}^n$. For what functions $\psi(t)$ can $\vec{x}(t) = \psi(t)\vec{u}$ be a solution of $\dot{\vec{x}} = A \vec{x}$ for some $n \times n$ matrix $A$?
I'm trying to prove that $\psi(t)$ has to be of the ... |
H: Convergence of $\int_{-\infty}^\infty \frac{1}{1+x^6}dx$
Okay, so I am asked to verify the convergence or divergence of the following improper integrals:
$$\int_{-\infty}^\infty \frac{1}{1+x^6}dx$$
and
$$\int_1^\infty \frac{x}{1-e^x}dx$$
Now, my first attempt was to use comparison criterion with $$\int \frac{1}{x^2... |
H: Convergence of $\int_{0}^{1} \frac{\ln^{2} x}{x^{2}+x-2} \ dx $
How do you show that $\displaystyle \int_{0}^{1} \frac{\ln^{2}x}{x^2+x-2} \ dx $ converges?
The singularity at $x=1$ is not an issue since it is removable.
But what about at $x=0$?
AI: You can show that
$$\int_0^1 dx\ \ln^n x$$
converges by changing va... |
H: Matrix Equation, Solving for Variables.
I'm going through my exercises, and came across a problem that wasn't covered in our lectures. Here's the question:
$
\begin{align}
\begin{bmatrix}
a-b & b+c\\
3d+c & 2a-4d
\end{bmatrix}
\end{align}
=
$
$
\begin{align}
\begin{bmatrix}
8 & 1\\
7 & 6
\end{bmatrix}
\end{align}
$... |
H: What is a "distinguished subset"?
I don't know if this is another word for something I already know or if it is something altogether different. I'm reading my textbook in CS about distributed algorithms, and this came up.
I googled for a definition but couldn't find one.
The state set $Q$ contains a distinguished ... |
H: Cardinality of intersection of sets
Consider the following problem: find $n(A \cap B)$ if $n(A)=10$, $n(B)=13$ and $n(A \cup B) = 15$.
I know if I want to find the union I use the Cardinal Number formula:
$$n(A\cup B) = n(A) + n(B) - n(A\cap B)$$
But how do I do it the other way: to find $n(A\cap B)$?
Would it be... |
H: What is the mathematical definition of index set?
I find some descriptions http://en.wikipedia.org/wiki/Index_set and http://mathworld.wolfram.com/IndexSet.html .
But can't find any definition.
AI: An index set is just the domain $I$ of some function $f:I\to X$. It's just a notational distinction between a function... |
H: number of zeros of function $\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$
$$f(z)=\prod_{n=1}^{\infty}\left(1-\frac{z^2}{n^2}\right)-1$$
How many zeros does the above function have in $\Bbb{C}$?
AI: As noted in a comment, this is essentially the problem of determining the number of solutions to $\sin(z)=z$.... |
H: Cardinality of Cartesian Product of finite sets.
If $a = \{1,2,3\}$ and $b = \{a,b,c\},\;$ FIND $\;n(a\times b)$
Or is it impossible to multiply these sets?
What will be the answer?
AI: Let's use capital letters for sets: so let $$A = \{1, 2, 3\},\;\;\text{ and} \;\; B = \{a, b, c\}$$ and $n(A) = |A| = 3,\;n(B) = |... |
H: What subjects should I study to learn about eigenfunctions? What good textbooks would you recommend for learning the subject?
I googled eigenfunction and look it up in wikipedia, but still I do not know where I should start to learn the subject. I have two questions, and allow me to repeat the title of this questio... |
H: Independent Spaces
What does it mean for spaces to be independent?
AI: Subspaces $W_1,...,W_k$ of $V$ are said to be independent if the only combination $$w_1+\cdots+w_k=0$$ with each $w_j\in W_j$ is $w_1=\cdots=w_k=0$. (This definition should be a bit earlier on the page.) |
H: Does this algorithm terminate in finite time?
I am trying to determine whether the following algorithm terminates:
int n;
int s;
s=3n;
while s>0
{
if s is even
{
s=floor(n/4);
}
else
{
s=2s;
}
}
So far, I have tried to see whether I can come up with a pattern for how the algorithm behaves fo... |
H: Find the modular residue of this product..
Please help me solve this and please tell me how to do it..
$12345234 \times 23123345 \pmod {31} = $?
edit: please show me how to do it on a calculator not a computer thanks:)
AI: We want to replace these big numbers by much smaller ones that have the same remainder on div... |
H: Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$
Compute the indefinite integral
$$
\int\frac{1}{1+x^8}\,dx
$$
My Attempt:
First we will factor $1+x^8$
$$
\begin{align}
1+x^8 &= 1^2+(x^4)^2+2x^4-2x^4\\
&= (1+x^4)^2-(\sqrt{2}x^2)^2\\
&= (x^4+\sqrt{2}x^2+1)(x^4-\sqrt{2}x^2+1)
\end{align}
$$
Then we can rewr... |
H: Mathematical Systems Question Help
Alright, this is another question for my math for teachers course. This question is not actually in the homework, but there are problems similar to it. I'd really like to learn how to do problems like this one, so it would be much appreciated if I was given a solid explanation/dem... |
H: Alternating functional Series Convergence SOS....
Does the following series converge?
$\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$
what is the radius of convergence?!!
AI: Note that $\sqrt{k}<k.$ Then
$\left| \frac{(-1)^k x^k \sqrt{k}}{k!} \right|<\frac{|x|^k k}{k!}=\frac{|x|^k}{(k-1)!}.$
For the series $\su... |
H: Basic probability questions
How is P(A , B) different from P(A $\cap$ B)? I'm genuinely curious as to why one might prefer one over the other.
Also as far as some probability P (A | B , C) goes, what is the order of operations i.e., is it P (A | (B , C)) or P ((A | B) , C)? Or are they both similar?
AI: The expres... |
H: Check whether the following polynomial is irreducible over $\mathbf Q$
I was trying this problem from my Abstract Algebra book exercise that says:
Show that the polynomial $x^2+\frac 13x-\frac 25$ is irreducible in $\mathbf Q[x]$.
What I tried: $x^2+\frac 13x-\frac 25 \equiv 15x^2+5x-6=f(x)$,say.
Now I compu... |
H: Two problems on analytic function and Mapping of elementary functions
Let $G$ be a region and let $f$ and $g$ be analytic functions on $G$ such that $f(z)g(z)=0$ for all $z \in G$. Show that either $f$ or $g$ is identically zero on $G$.
Here is how I do it: Assume $f$ is non zero on $B(a,R)$, then $fg=0$ implies $a... |
H: Find $u\in\mathbb{R}$ such that $\mathbb{Q}(u) = \mathbb{Q}(2^{1/2}, 5^{1/3})$.
I am having trouble finding such a $u$. My instincts at first told me to do the obvious thing and let $u = 2^{1/2}5^{1/3}$ but $u^{2} = \left(2^{1/2}5^{1/3}\right)^{2} = 2\cdot5^{2/3}$ but we want $\frac{1}{2}u^{2} = 5^{1/3}$ right?? ... |
H: Prove that $\sin^2{x}+\sin{x^2}$ isn't periodic by using uniform continuity
Before the problem is a proof that says a periodic function whose domain is $\mathbb{R}$ is uniformly continuous.So actually the problem is to prove $\sin^2{x}+\sin{x^2}$ isn't uniformly continuous.I hope to fellow the problem.Thanks for th... |
H: find the condition on A for the summation to be convergent
The summation is:
$$\sum_{n=1}^\infty \frac{ \sqrt { n + 1 } - \sqrt n }{n^A}$$
I don't know how to even begin. Hints??
AI: Hint: A standard beginning is to multiply top and bottom by $\sqrt{n+1}+\sqrt{n}$.
We end up with
$$\sum_{n=1}^\infty \frac{1}{n^A(... |
H: Prove that this set (involving fractional part of any rational number) is a partition of the set of rationals.
For any rational number $x$, we can writte $x=q+\,n/m$ where $q$ is an integer and $0\le n/m<1$. Call $n/m$ the fractional part of $x$. For each rational $r\in \{x : 0\le x<1\}$ ,let $A_r = \{ x\in \Bbb Q:... |
H: Show reflexive normed vector space is a Banach space
$X$ is a normed vector space. Assume $X$ is reflexive, then $X$ must be a Banach space.
I guess we only need to show any Cauchy sequence is convergent in $X$.
AI: Hint: (1) If $X$ is reflexive, $X$ is isomorphic to $X^{**}$. (2) Dual spaces are allways complete.
... |
H: Infinite imprimitive non abelian group?
My new question is
Is there an infinite, imprimitive and non abelian group?
Thank you for the further answers.
AI: Consider the subgroups $A={\rm Sym}(2{\bf Z})$ and $B={\rm Sym}(1+2{\bf Z})$ sitting inside $G={\rm Sym}({\bf Z})$. So $G$ is the set of bijections from the s... |
H: Is the truncated exponential series for matrices injective?
If $k$ is a field of characteristic $p$, we can define a map $\exp:\mathfrak{gl}_n(k)\to GL_n(k)$ by:
$$\exp(A)=\sum_{i=0}^{p-1}\frac{A^i}{i!}$$
In the answer to this question, we see that if $A^p=B^p=0$, and if $\exp(A)=\exp(B)$, then $A=B$. So if $p>n$,... |
H: Linear independence of $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$
Could you tell me why $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$ are lineraly independent?
I've tried $\alpha(x \sin x)^{\frac{n-1}{2}} + \beta (x\sin x)^{\frac{n+1}{2}} =0$
$(x\sin x)^{\frac{n-1}{2}}(\alpha + \beta... |
H: Counting Cosets of $\langle\tfrac12\rangle$, in $\Bbb{R}$ and in $\Bbb{R}^{\times}$
Describe the cosets of the subgroups described:
The subgroup $\langle\frac{1}{2}\rangle$ of $\mathbb{R}^{\times}$, where $\mathbb{R}^{\times}$ is the group of non-zero real numbers with multiplication.
The subgroup $\langle\frac{1}... |
H: Probability generating function and expectation
Let $X$ be Poisson random variable with parameter $Y$, where $Y$ is Poisson random variable, with parameter $\mu$. Prove that, $G_{X+Y}(s)=\exp\{\mu (s\exp^{s-1}-1)\}$
I know that, Poisson r.v. generating function is $G(s)=\exp\{\lambda(s-1)\}$. Do I need to calculate... |
H: If $E(|X+Y|^p)<\infty$, then $E(|X^p|)<\infty$ and $E(|Y^p|)<\infty$.
If $X$ and $Y$ are independent and for some $p>0$: $E(|X+Y|^p)<\infty$, then $E(|X^p|)<\infty$ and $E(|Y^p|)<\infty$.
How can I go from $E(|X+Y|^p)<\infty$ using independence to conclude something about $X$ all alone?
AI: By independence,
$$
E[|... |
H: Continuity in metric space
Let $F: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be defined by
$$ F(x,y) = \left( x^3 y,\ \ln(x^2 + y^2 + 1),\ \cos(x - y^2) \right) $$
When trying to show why $F$ is continuous where should I start?
AI: To give the answer of your question let me define first the vector valued functions.
A ... |
H: Is $e^z\sum_{k=0}^\infty\frac{k^3}{3^k}z^k$ analytic inside $|z|=3$?
Am I correct that the following function is analytic at least inside $|z|=3$? (I used the ratio test.) The solutions manual says that the function is analytic on and inside |z|=1, so I wonder if I'm having a conceptual misunderstanding. $$e^z\sum_... |
H: Is it possible for a function to be analytic anywhere outside the circle of convergence of its power series expansion?
Is it possible for a function to be analytic anywhere outside the circle of convergence of its power series expansion? I'm referring to analytic fuctions of course (i.e. those with power series exp... |
H: spherically symmetric configurations
$$\Delta S -S +S^3=0$$ How this Differential equation can be written in this form:
\begin{equation}
\frac{d^2S}{d\rho^2}+\frac{D-1}{\rho}\,\frac{dS}{d\rho}
-S+S^3=0
\end{equation}
Which is spherically symmetric configurations. D= dimensions.
Details in the paper equations (21, ... |
H: Notation of Planes
I have to finde the line $L := E_0 \cap E_1$, with
$E_0 = \left\langle \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \right\rangle ^\perp$
$E_1 = \left\langle \begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix} \right\rangle ^\perp + \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$
But I am unfamiliar with the nota... |
H: Velleman's Proof Designer problem 44
This is an exercise I didn't solve from the group of exercises present on Velleman's site that should be solved using Proof Designer (the little program that comes with the book).
Theorem:
If $\forall \mathcal{F} (\cup \mathcal{F} = A \rightarrow A \in \mathcal{F})$, then $\exis... |
H: Some questions about complexification of a real vector space
Could you tell me how to prove that if $f:U \rightarrow U$ is $\mathbb{R}$-linear, then:
1) $U^{\mathbb{C}}$ is a vector space over $\mathbb{R}$ (should I check all eight conditions for a vector space?)
2) $f^{\mathbb{C}}: U^{\mathbb{C}} \rightarrow U^{\m... |
H: On the equality of two generating functions related to plane partitions
I'd like to prove
$$\prod_{(i,j,k)\in\mathcal{B}(r,s,t)}\frac{1-q^{i+j+k-1}}{1-q^{i+j+k-2}}=\prod_{i=1}^r\prod_{j=1}^s\frac{1-q^{i+j+t-1}}{1-q^{i+j-1}},$$
where
$$\mathcal{B}(r,s,t)=\{(i,j,k):1\leq i\leq r,1\leq j\leq s,1\leq k\leq t\}.$$
I'm g... |
H: Minimal polynomials over the rationals and the reals
Find the minimal polynomial over $\mathbb Q$ and $\mathbb R$ for ...$\sqrt[3]{3}$, $1- i\sqrt{3}$, $2 + i$, $i\sqrt[3]{3}$
Sorry for my sqrt formulas .. I'm new here, hope to learn really fast to write a correct question in math symbols.
I have to find both mini... |
H: Question about eigenvalues
I have this :
i dont understand why they write $\lambda=m^2 , m\in \mathbb{N}\cup\lbrace0\rbrace$ ,
it's right that $\lambda=m^2$ is the eigenvalues of $(P_0)$ ,but $0$ is not an eigenvalue !.
after that why they write $\mathbb{N}\cup \lbrace0\rbrace$ 0 already belongs to $\mathbb{N}$... |
H: Which polygon tile grids allow convex polygons to be formed from multiple tiles?
If I have a grid made of equilateral triangles, I can easily form larger convex polygons as a set of tiles in that grid. I believe this holds for some (but not all) tilings of non-equilateral triangles.
The same for quadrilaterals - it... |
H: Taylor series of $f(x)=\frac {e^x-1}{x}$
I am asked to expand $f(x)=\frac {e^x-1}{x}$ centered at 0 using the known Talyor series of functions.
How to simplify the function so that it can be expanded more easily?
AI: If you expand $e^x$ and subtract 1, then you get something divisible by $x$. You should find
$$\dis... |
H: Applications of Double/Triple Integrals
This is the question that I need to solve using mathematica:
The concentration of an air pollutant at a point $(x,y,z)$ is given by: $$p(x,y,z) = x^2y^4z^3 \text{ particles}/m^3$$ We're interested in studying the air quality in a region in 3-space which satisfies $x\ge0$ and... |
H: How to resolve this algebra equation?
$$f = X^3 - 12X + 8$$ $a $- complex number, $a$ is a root for $f$
$b = a^2/2 - 4 $.
Show that $f(b) = 0$
This is one of my theme exercises ... Some explanations will be appreciated ! Thank you all for your time .
AI: \begin{align*}
f(b) &= f(\frac{a^2}{2} - 4) \\
&= (\frac{a^... |
H: proving that symplectic lie algebra is a subalgebra of GL
Suppose $S$ is n by n matrix over a field F. Define
$gl_S(n,F)=\{A \in gl(n,F): SA+A^TS= 0\}$
Show that this is a subalgebra of $gl(n,F)$
I get as far as:
$A \in gl_S(n,F)$ and $B \in gl_S(n,F)$
$S[AB] +[AB]^T S =SAB -SBA + (AB)^TS-(BA)^TS $
$=SAB + B^TA^TS... |
H: If $x^2\equiv 1 \pmod{n}$ and $x \not\equiv \pm 1 \pmod{n}$, then either $\gcd(x-1,n)$ or $\gcd(x+1,n)$ is a nontrivial factor of n
I'm reading elementary number theory and trying to understand the following problem: If $x^2\equiv 1 \pmod{n}$, $n=pq$, $p$ and $q$ are odd primes and $x \not\equiv \pm 1 \pmod{n}$, th... |
H: Unitary and transformation matrix
I have a question that I do not understand how to solve that.
Let $V$ be inner product space.
Let {$e_{1},...,e_{n}$} an orthonormal basis for $V$
Let {$z_{1},...,z_{n}$} an orthonormal basis for $V$
I have to show that the matrix represents
the transformation matrix between {$e_{... |
H: Parametric simultaneous equations
I stumbled on this one a few days ago and I'm probably missing something obvious...
I basically need to solve those parametric equations for the other coordinate $(x,y)$ other than the point $(t^2,t^3)$:
$$2y = 3tx - t^3$$
$$x = t^2$$
$$y = t^3$$
To have more context, the question ... |
H: $f(x)=|\cos x|+|\sin(2-x)|$ at which of the following point $f$ is not differentiable?
$f(x)=|\cos x|+|\sin(2-x)|$ at which of the following point $f$ is not differentiable?
1.$\{(2n+1){\pi\over2}\}$
2.$\{n\pi\}$
3.$\{{n\pi\over 2}\}$
4.$\{n\pi+2\}$
in all cases $n\in\mathbb{Z}$
well, is there any easy trick to sol... |
H: $X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then
$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then
$0$ is an isolated point
$(-2,0]$ is an open set
$0$ is a limit point of the subset $\left\{{1\over n}:n\in\mathbb N\right\}$
$(-2,0)$ is op... |
H: Applying Urysohn Lemma on $\mathbb{R^2}$
$A_1=\text{ Closed Unit Disk}$, $A_2=\{(1,y):y\in\mathbb{R}\}$, $A_3=\{(0,2)\}$.
Then there always exists a real-valued continuous function on $\mathbb{R}^2$ such that
$f(x)=a_j$ for $x\in A_j$, $j=1,2,3$
Iff at least two of the numbers $a_1,a_2,a_3$ are equal.
if $a_1=a_2=... |
H: For every monoid $M$ with zero is there a group $G$ such that $\mathrm{Grp}(G,G)\cong M$?
All monoids that I will consider here have identities. A monoid $M$ is said to have a zero iff $\exists z\in M \forall x\in M (zx=xz=z)$.
Let $M$ be any monoid with a zero. Must there exist a group $G$ such that $\mathrm{Grp}(... |
H: Nolinear system of equations.
$$
\left\{\begin{array}{l}
x^y = y^x \\
x-y\cdot\log_xy=(x+y)\cdot\log_xy
\end{array}
\right.
$$
Thanks for your time!
AI: With rules for logarithms the second equation translates to
$$
x - y\log_xy = \log_x y^x + y \log_x y
$$
and using the first equation this simplifies to
$$
x= y+... |
H: Vector Line Integral Question
I need to compute the line integral for the vector $\vec{F} = \langle x^2,xy\rangle$, for the curve specified: part of circle $x^2+y^2=9$ with $x \le0,y \ge 0$,oriented clockwise.
Once again, I'm stuck at the setup (this happens a lot with me). I know that I need to parameterize F, bu... |
H: is $(x+1)^4-x^4$ non-prime for all natural positive integers $x$
Looking at difference between two neighbouring positive integers raised to the power 4, I found that all differences for integer neighbours up to $(999,1000)$ are non-prime.
Does this goes for all positive integers?
And can someone please prove?
AI: H... |
H: What's the motivation of the ideal?
I'm reading a book on Algebra, it introduces the concept of ring after some examples, the concept of ideal.
Definition I.1.8. Let $(A,+,\cdot)$ be a ring and $I$ a non-empty subset of $A$. We say that $I$ is a ideal if:
$x+y\in I,\;\;\;\;\forall \; x,y \in I$
$ax\in I, \;\;\;\;... |
H: Does it make sense to talk about the concatenation of infinite series?
Does a series of numbers defined as the concatenation of two or more infinite series, for example all the positive integers followed by all the negative integers, make mathematical sense?
I came up with this problem while writing an implementati... |
H: Does the Weierstrass M-test show analyticity?
I'm trying to show (textbook exercise) that the riemann-zeta function is analytic. The solution is here:
Why does the proof say that the zeta series converges to an analytic function? Doesn't the M-test merely show uniform convergence? The zeta series (whose term is in... |
H: Prove that $f(f(x))=x$ has no roots .... $f$ having a general form
This problem gave me some headache, especially because $f$ have its own general form :
let $f(x) = ax^2 + bx + c$. Suppose that $f(x) = x$ has no real roots.
Show that equation $f(f(x))=x$ has also no real roots.
AI: Consider $g(x) = f(x)-x$. If $f... |
H: Improper Integral Question $ \int^{\infty}_0 \frac{\mathrm dx}{1+e^{2x}}$
I want to check if it's improper integral or not
$$ \int^{\infty}_0 \frac{\mathrm dx}{1+e^{2x}}.$$
What I did so far is :
set $t=e^{x} \rightarrow \mathrm dt=e^x\mathrm dx \rightarrow \frac{\mathrm dt}{t}=dx
$ so the new integral is:
$$ \i... |
H: Linear equation of 4 variables
I'm stuck on this Math problem :
How many solutions does the equation
$x_{1} + x_{2} + 3x_{3} + x_{4} = k$
have, where $k$ and the $x_{i}$ are non-negative integers such that $x_{1} \geq 1$, $x_{2} \leq 2$, $x_{3} \leq 1$
and $x_{4}$ is a multiple of 6.
I tried to write the possib... |
H: Determine conjugate function
Let $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=e^x$. Determine $f^*(y)$.
I try to use some inequalities to get supremum but it is impossible. Seemingly, I must consider some cases of $x$.
AI: $$f^*(y)=\textstyle\sup_x \langle y, x \rangle - e^x.$$
Since the supremum involves a smooth ... |
H: Intuition behind closed subsets of a metric space?
Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space.
Consider a metric space $$(X,d)$$
Then consider a subset of this space$$F$$
What the book is trying to convince me of is that F is closed iff a convergent seq... |
H: logs Challenge between two students >>be smart
two student were given the equation $2^{4x+6} = 3^{6x-3}$
1.steve rearranged to get $2^{4x+6} - 3^{6x-3} =0$
then wrote $\log (2^{4x+6} - 3^{6x-3}) = \log0$
are these legal steps ? if not explain what is wrong with them
2 Ali wrote $\log( 2^{4x+6}) =\log(3^{6x-3})$ ... |
H: A problem on square matrices
If $B,C$ are $n$ rowed square matrices and if $A=B+C, BC=CB, C^2=O$, then show that for every $n \in \mathbb N$, $$A^{n+1}=B^n(B+(n+1)C)$$
I tried to prove it using mathematical induction. But I could not get $P(1)$ to be true.
$$P(1): A^2 = B(B+2C)$$
I couldn't equate them.
Please of... |
H: A matrix problem :)
If $l_i,m_i,n_i$ ; $i=1,2,3$ denote the direction cosines of three mutually perpendicular vectors in space, provided that $AA^T=I$ ,where
$$A=\begin{bmatrix}
l_1 & m_1 & n_1 \\
l_2 & m_2 & n_2 \\
l_3 & m_3 & n_3 \\
\end{bmatrix} $$
I couldn't quite understand... |
H: Solving 3 simultaneous cubic equations
I have three equations of the form:
$$i_1^3L_1+i_1K+V_1+(i_2+i_3+C)Z_n=0$$
$$i_2^3L_2+i_2K+V_2+(i_1+i_3+C)Z_n=0$$
$$i_3^3L_3+i_3K+V_3+(i_1+i_2+C)Z_n=0$$
where $L_1,L_2,L_3,K,V_1,V_2,V_3,C$ and $Z_n$ are all known constants.
What methods can I use to obtain the values of $i_1,i... |
H: Can Green's theorem be used in a plane other than the xy-plane?
In the following 2D case, Green's theorem solves the following problem:
$$\vec{F}=\langle{xy+\ln{(\sin{e^{x})},x^2+e^{y^2}}}\rangle$$
$$\oint_C\vec{F}\cdot{d\vec{r}}=\iint_Dx\space{dA}$$
where C is the unit circle $x^2+y^2=1$, and D is the unit disk $x... |
H: Solving simple system of congruences
I have this example from wikipedia:
$$x \equiv 3 \pmod 4$$
$$x \equiv 4 \pmod 5$$
$$x = 4a + 3\\
4a + 3 \equiv 4 \pmod 5\\
4a \equiv 1 \equiv -4 \pmod 5\\
a \equiv -1 \pmod 5\\
x = 4(5b - 1) + 3 = 20b - 1
$$
But wikipedia shows $20b + 19$...what did I do wrong here?
AI: Your ans... |
H: Element by element formulae for 3x3 matrix inversion
Given a 3 x 3 matrix:
$$
A= \begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
$$
Can $A^{-1}$ be shown as as a 3x3 matrix with each element in terms of $a,b,c,d,e,f,g,h$ and $i$. Showing basic operators (... |
H: Definition of inner product
I'm studying the inner product part in linear algebra and it's a bit tricky to understand what inner product really means. It indicates the length of vectors? So if we calculate $<x,y>$ we have to make it as a scalar?
Here is an example. If $x$ and $y$ are column vector of $2*2$ matrix a... |
H: Does There Exist a Term for the Unique Nonpositive Square Root of a Nonnegative Real Number?
The term "principal square root" describes the unique nonnegative square root of a nonnegative real number.
Does there exist a term to describe the unique nonpositive square root of a nonnegative real number?
AI: I have hea... |
H: Solving modular equation
$$13863x \equiv 12282 \pmod {32394}$$
I need to solve this equation. If I'd found the inverse of 13863 and multiply the equation by this, I'd get the solution. So:
$$13863c \equiv 1 \pmod {32394}$$
And now - how can I find this inverse? The numbers are too big to just look for the multiplie... |
H: Big Greeks and commutation
Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering?
Clearly if $\mathbf{x}_i$ is a matrix then:
$$\prod_{i=0}^{n} \mathbf{x}_i$$
depends on the order of the multiplication. But, even if one accepts that it has a sequence, it is not clear if it should mean $\mathbf{x}_0\mat... |
H: Generating functions to solve recurrence relation
Use generating functions to solve the recurrence relation
$$ a_{n} = 3a_{nβ1} + 2 $$
with initial condition $a_{0} = 1$.
If I can bring it to $ a_{n}=k a_{n-1} $ I can solve it easily. Thank you
AI: Here is a start
$$ \sum_{n=0}^{\infty}a_{n+1}x^n = 3 \sum_{n=0}^{... |
H: Continuous random variable question
$ X $ is a non-negative continuous random variable with density function $f$ and distribution function $F$.
Use integration by parts to show that
$ \int_0^{\infty} ( 1- F(x)) dx =
\int_0^{\infty}xf(x)dx $
I'm quite puzzled on how to even integrate $F(X)$ to get $f(x)$ :S
AI... |
H: Proof of cauchy schwarz inequality in inner product space
$0 \le \lVert x-cy \rVert ^2= \langle x-cy,s-cy \rangle = \langle x,x\rangle -\bar{c}\langle x,y\rangle-c\langle y,x\rangle +c\bar{c}\langle y,y\rangle$.
If we set $c=\frac{\langle x,y\rangle}{\langle y,y\rangle}$ then the inequality becomes
$0 \le \langle x... |
H: How to solve this minimization problem?
I have a question which asks:
A cylinder shaped can holds $5000cm^3$ of water. Find the dimensions that will minimise the cost of metal in making the can.
What I did was express the height in terms of the radius as:
$$h=\frac{5000}{(\pi)(R^2)}$$
Then I differentiate it to:... |
H: Analysis: Integration (Riemann/Step functions)
Using the definition of the integral of a continuous function, and
that $\displaystyle\sum_{j=0}^{2^n-1} j = (2^n-1)2^{n-1}$ to show that
$\int_0^1x \ dx = \frac{1}{2}$
I'm having trouble even starting this question as I'm not sure how to take advantage of the su... |
H: Proving that the independent set problem is in NP-Complete
Consider the problem of "Independent set" in grahps. Given a graph $G$ and an integer $k$, the machine determines whether the graph $G$ contains an independent set of size $k$.
I need to prove that it's in NP-Complete by showing a reduction from another kno... |
H: Improper Integral $\int_{1/e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $
I need some advice on how to evaluate it.
$$\int\limits_\frac{1}{e}^1 \frac{dx}{x\sqrt{\ln{(x)}}} $$
Thanks!
AI: Here's a hint:
$$
\int_{1/e}^1 \frac{1}{\sqrt{\ln x}} {\huge(}\frac{dx}{x}{\huge)}.
$$
What that is hinting at is what you need to learn in o... |
H: How do I divide a set of data samples which follow a logarithmic distribution?
I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the interval as [min,max]. Now I need to... |
H: Is restriction of scalars well-defined on subspaces?
Let $K/k$ be an extension of fields and let $v_1,\ldots,v_r,u_1,\ldots,u_r\in k^n$. If the span of the $v$'s over $K$ equals the span of the $u$'s over $K$, must the two spans also be equal over $k$?
$$_K\langle v_1,\ldots,v_r\rangle=_K\langle u_1,\ldots,u_r\ran... |
H: Binomial coefficient series $\sum\limits_{k=1}^n (-1)^{k+1} k \binom nk=0$
I'm practicing for my maths term test mainly on binomial coefficients. I can't seem to find out how to prove the following identity. Any advice?
$$ \sum\limits_{k=1}^n (-1)^{k+1} k{{n}\choose k} = 0 $$
Thanks in advance.
AI: Hint: Note that ... |
H: Can you raise $\pi$ to a real power to make it rational?
We're all familair with this beautiful proof whether or not an irrational number to an irrational power can be rational. It goes something like this:
Take $(\sqrt{2})^{\sqrt{2}}$
If it's rational, then you proved it, if it's irrational, take $((\sqrt{2})^{\sq... |
H: Measuring distances on any coordinate system
I was reading the book The ABC of Relativity from Betrand Russell, and at some point, the author mentions a method for measuring the distance between 2 points on any coordinate system. He says that the formula was discovered by Gauss and is a generalization of the pythag... |
H: Proving combinatoric identity using the inclusion-exclusion principle
I need to prove the identity ${\displaystyle \sum_{k=0}^{n}(-1)^{k}{2n-k \choose k}2^{2n-2k}=2n+1}
$ using the inclusion-exclusion principle. It was hinted to think about the number of ways to color the numbers ${1,..,2n}$ in the colors red and... |
H: Need help with $\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx$
I need help with solving this integral:
$$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ dx,$$
where $\text{Li}_{s}(z)$ is the polylogarithm.
AI: $$\int_0^\infty x^{-\frac{3}{2}}\ \text{Li}_{\sqrt{2}}(-x)\ \mathrm dx=-2^{\sqrt{... |
H: Is a series (summation) of continuous functions automatically continuous?
I'm being asked to show that a given series (of rational functions) converges uniformly on a given disc, and then and asked to use this fact to show that integrating its limit function (i.e. a summation of rational functions) along a given co... |
H: About continuous functions and aritmethic progression
I've try solve this question, but I haven't sucess...
The problem is the following:
A continuous functions $f:[a,b]\rightarrow \mathbb{R}$ assume positive and negative values in its domain, show that there exists $a_1,a_2,\ldots,a_k$, k numbers that are arithmet... |
H: Does this polynomial factorize further?
I just did a national exam and this question was in it; I am convinced this does not work:
Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic fully.
My attempt
1 | 1 3 1 -5
| 1 4 5
|____________
1 4 5 0
$$(x - 1)(x^2 + 4x + 5)... |
H: Can't establish a lower bound on a supremum
I have a sequence of functions $f_{k,j}:[0,1]\to\mathbb{R}$ defined by
$$f_{k,j} = k^{\frac{1}{p}}\chi_{(\frac{j-1}{k},\frac{j}{k})},$$
for all $k\geq 1,1\leq j\leq k$.
This serves as an example of a sequence that converges to $0$ in measure, but not in $L^{p,\infty}$.
... |
H: Does the splitting principle define chern classes for vector bundles if they are known for line bundles?
Suppose we have definied chern classes $c_i$ for line bundles for all $i\geq 0$.
Let $E\to B$ be a rank $r$ vector bundle. By the splitting principle there exists a map $f:Y\to B$ such that $f^*E$ is a sum of li... |
H: Find the relation between the dimension of the nullspace of $A$ and $A^t$
Let $A$ be a $n \times n$ matrix, what is the relation between the dimension of the nullspace of the homogeneous system of $A$ and the one of $A^t$?
AI: Hint: For a given matrix $A$, the dimension of the left null space is equal to the dimens... |
H: How are these inequalities simplified?
How does this:
(a > b && a > c && b <= c) ||
(a > b && a <= c && b < c)
simplify down to this:
a > b && b <= c
Whereas this similar expression
(a <= c && c >= b && a <= b) ||
(a <= c && c < b && a < b)
does not simplify to
a <= c && a <= b
but instead to
(b <= c && a <= b... |
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