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H: Proving all roots of a sequence of polynomials are real
Let the sequence of polyominoes $R_n(z)$ be defined as follows for $n\geqslant1$:
$$R_n(z)\;=
\;\sum_{r=0}^{\lfloor\frac{n-1}{2}\rfloor}
\tbinom{n}{2r+1}(4z)^r.$$
I would like to prove that all the roots of $R_n(z)$ are real (and thus negative since the coeffi... |
H: Usual topology and Heine Borel theorem
Heine-Borel theorem say: A subset in $\mathbb{R}^n$ is compact if and only if it is closed and bounded.
Is this theorem independent of the topology in $\mathbb{R}^n$?
If the answer is no, which is a counterexample?
I have seen a demonstration but uses the usual topology for ... |
H: Rotate a 3D vector on a plane
I have a 3D line vector with end points x0 and x1, which lies along the x-axis of a subsection of the plane, P.
However P has been translated, rotated and translated back from the global coordinate system by theta degrees along the global x-axis. The following image should illustrate m... |
H: About $|\operatorname{Sym}(\Omega)|$ when $\Omega$ is an infinite set.
Here is a problem:
Show that if $\Omega$ is an infinite set, then $|\operatorname{Sym}(\Omega)|=2^{|\Omega|}$.
I have worked on a problem related to a group that is $S=\bigcup_{n=1}^{\infty } S_n$. Does it make sense we speak about the relatio... |
H: Please explain this notation equation
I am confused by this equation as I rarely use math in my job but need this for a program that I am working on. What exactly does the full expression mean? Note that $m^*_i{_j}$ refers to a matrix whose values have already been obtained.
Define the transition matrix $M = ${$m_i... |
H: Max of Brownian motion with drift is finite almost surely
For $B_t$ Brownian Motion with drift $\mu<0$, I need to prove that the max value, $X = \max_{0<t<\infty}B_t$ is finite almost surely, ie $P(X<\infty)=1$.
Now, I know that because the mean is negative, it will go more and more negative, and it is also a super... |
H: An ignorant question about the incompleteness theorem
Let me preface this by saying that I have essentially no background in logic, an I apologize in advance if this question is unintelligent. Perhaps the correct answer to my question is "go look it up in a textbook"; the reasons I haven't done so are that I would... |
H: Further reading on the $p$-adic metric and related theory.
In his book Introduction to Topology, Bert Mendelson asks to prove that
$$(\Bbb Z,d_p)$$
is a metric space, where $p$ is a fixed prime and
$$d_p(m,n)=\begin{cases} 0 \;,\text{ if }m=n \cr {p^{-t}}\;,\text{ if } m\neq n\end{cases}$$
where $t$ is the multipl... |
H: Minimal diameter
I have a closed bounded convex shape in $\mathbb{R}^3$. I want to calculate what I would call the minimal diameter: find the plane (or, generally, the space of codimension 1) which minimizes the maximum distance from points in the shape to the plane, then take that maximum distance. I suppose one s... |
H: Turn fractions into $\mathbb Z_7$ elements
I had to perform a division between two polinomials $2x^2+3x+4$ and $3x+4$, my book suggests to do this operation without worrying about the modulo. So my result is $(3x+4)(\frac{2}{3}x+\frac{1}{9})+\frac{32}{9}$. Unfortunately my book fails to explain how should I perform... |
H: Strong markov property on max of brownian motion
For $B_t$ Brownian Motion with drift $\mu<0$, I have the max value, $X = \max_{0<t<\infty}B_t$ .
I need to prove with the Strong Markov Property that, $P(X>c+d)=P(X>c)P(X>d)$
a. It seems weird to me since one of the right hand terms is redundant...
b. I'll appreciate... |
H: $\mathcal{K}(\mathcal{H})$ is separable?
I am reading Arveson's Notes On Extensions of $C^*$-algebras, in proving the Corollary to Thm2, he seems to assume that $\mathcal{K}(\mathcal{H})$, the space of compact operators on a separable Hilbert space, is separable as a topological space, i.e., it contains a countable... |
H: Existence of a prime ideal in an integral domain of finite type over a field without Axiom of Choice
Let $A$ be an integral domain which is finitely generated over a field $k$.
Let $f \neq 0$ be a non-invertible element of $A$.
Can one prove that there exists a prime ideal of $A$ containing $f$ without Axiom of Cho... |
H: Tracing a point on a rotating circle
I have a large circle and a small circle as shown in the image. The distance between the centers of those circles. The Larger circle is rotating about it's center while traveling in direction D such that it travels its circumference in 1 revolution. There is a point on the smal... |
H: When is $(6a + b)(a + 6b)$ a power of two?
Find all positive integers $a$ and $b$ for which the product $(6a + b)(a + 6b)$ is a power of $2$.
I havnt been able to get this one yet, found it online, not homework!
any help is appreciated thanks!
AI: Hint: $(6a+b)(a+6b)$ is a power of $2$ iff $(a+6b)$ and $(6a+b)$ a... |
H: Inequality's root
Given $\sqrt x + \sqrt y < x+y$, prove that $x+y>1$.
Havnt been able to try this yet, found it online,
any help is appreciated thanks!
AI: If $\sqrt{x}+\sqrt{y}<x+y$, then after squaring both sides we have $x+2\sqrt{xy}+y<x^2+2xy+y^2$. Rearrange to isolate the square root:
$$2\sqrt{xy}<x^2+2xy+y^2... |
H: What means the boundary of a space
By definition topological spaces are clopen and then their boundaries are empty, but for example, is said that the boundary of the closed unit interval is it's two endpoints a so on.
whath is the meanining of "boundary" in this context?
AI: There are two notions of boundary in mat... |
H: Let $A\subseteq\mathbb{R}$ be open, with $A\cup (0,1)$ connected
Let $A\subseteq\mathbb{R}$ be open. If $A\cup (0,1)$ is connected then
A must be connected.
A must have one or two component.
$A\setminus(0,1)$ has at most two component.
$A$ must be a cantor set.
Take $A=(0,1/2)\cup(1/2,1)$. Then $A\cup (0,1)=(0,1)... |
H: the derivative for a Lipschitz function
For the solution of the ODE $dy/dt=f(t,y)$, $f$ has to be Lipschitz for all $t$. So, if $f$ is a function that is not differentiable with respect to $y$ but Lipschitz, what can I say about $f_y$? Can I estimate some norm of it? I can't say $\lVert f_y\rVert_C$ because $f$ is ... |
H: How complex exponential converges and "sum of exponents" rule holds
How is it the complex exponential converges for any value of $z$ in the complex plane? $$e^{z} = 1 + \frac{z}{1!} + \frac{z^2}{2!} \cdots\cdots$$
How is it the "sum of exponents" rule holds for complex exponential, that is $e^{w}e^{z} = e^{w+z} $?... |
H: Some questions on Laplace equation .
While i am revising for exam : I am facing some problems to understand things clearly . Here are my doubts :
a) If $u$ solves $\Delta u =0 , x\in \Omega; u=g , x\in \partial \Omega$ for non constant boundary data $g$ with $g\ge0$ and $g(x_0)\> 0$ for some $x_0 \in \partial \Omeg... |
H: Probability of a number being a multiple of another number
For a random whole number, n, between 0 and 4000 billion (don't know what that's called), is the probability that n is a multiple of 4096, 1/4096?
AI: If one end (either $0$ or $4000$ billion) is included, the probability is exactly $\frac 1{4096}$ as $4096... |
H: Notation for a function from all members of a tuple minus one.
Is there any compact, mainstream notation for a function from all members of a tuple minus one? What I have in mind is
$f\left(a_{1},\ldots,a_{n}\right)$ (except for $a_{i}$) $=a_{i}$
AI: If you mean an ordered tuple, then a common notation for
$$(a_1,a... |
H: Should one imagine diagrams/figures when working?
I'm working through Baby Rudin and find it exceedingly difficult to understand what's happening without drawing a small figure. For instance when proving properties of compactness, I would often draw figures like :
(Black is the set, red are the finite sub-covers)... |
H: Condition(s) that satisfy this equality
I am having difficulty understanding how my book came up with this answer.
Define $a \star b =ab+2b$, and suppose $x \star y = y \star x$. Then which of the following must be true?
A. $x+y=1$.
B. $y=0$.
C. $x=y$.
D. $x=-2$.
E. $xy=0$.
How did the text conclude that $x=y$ or... |
H: A question related to Novikov's condition
The well-known 'Novikov condition' says:
Let $ L = (L_t)_{t \geq 0} $ be a continuous local martingale null at 0 and $ Z = \exp(L - \frac{1}{2} \langle L \rangle) $ its stochastic exponential.
If
$ E[\exp(\frac{1}{2} \langle L \rangle_\infty)] \ < + \infty $,
then $ Z $ ... |
H: Integral of $e^{(x-x^3)/3n}$ from $0$ to $\infty$
How can you compute the following integral assuming $n>0$?
$$\int_{x=0}^{\infty}e^{\frac{x -x^3}{3n}}dx $$
Mathematica etc. fail to produce anything useful.
EDIT: I would be happy with an asymptotic result in $n$ if it is too hard to compute exactly. I don't know i... |
H: Question about the socle of a finite-dimensional algebra
Let $n\in \mathbb{N}$ and $k$ be an arbitrary field.
Is the socle of the algebra $k[x,y]/\langle x^2,y^{n+2}\rangle$ isomorphic to $k$?
Is $k[x,y]/\langle x^2,y^{n+2}\rangle$ a symmetric algebra or a Frobenius algebra or a self-injective algebra?
I would be... |
H: Using a single scalar equation to describe a line in space
A line in three-dimensional space may be described as an intesection of two planes, for example: $$\begin{align}x+y+z=0\tag{1}\\3x+7y=1\tag{2}\end{align}$$
This can be understood as two separate scalar equations or as a single matrix equation. (One may also... |
H: $G=\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to?
$G=\mathbb{Z}_2\times\mathbb{Z}_3$ is isomorphic to
$S_3$
A subgroup of $S_4$.
A proper subgroup of $S_5$
$G$ is not isomorphic to a subgroup of $S_n$ for all $n\ge 3$
What I know is Any finite group is isomorphic to a subgroup of $S_n$ for some suitable $n$(C... |
H: Lipschitz constant on a functional
Let $C$ be the space of continuous and nondecreasing functions defined on $[0,1]$ and endowed with the sup norm. Let $T:C\rightarrow C$ be a continuous mapping, and consider the following expression:
$$
U(Tz(x);z)=\int_{0}^{Tz(x)}\left[\int_{0}^{1}F(z(\xi))f(\xi)d\xi+\int_{s\in \G... |
H: Irrational distances, rational area triangles
Given any positive integer $n\ge3$ how to show that there are $n$ distinct points in the plane such that
1- the distance between any two points is irrational number and
2- each set of three points determines a non-degenerate triangle whose area is a rational number
AI:... |
H: Pythagorean Theorem for imaginary numbers
If we let one leg be real-valued and the other leg equal $bi$ then the Pythagorean Theorem changes to $a^2-b^2=c^2$ which results in some kooky numbers.
For what reason does this not make sense? Does the Theorem only work on real numbers? Why not imaginary?
AI: The Pythagor... |
H: limit question on Lebesgue functions
Let $f\in L^1(\mathbb{R})$. Compute $\lim_{|h|\rightarrow\infty}\int_{-\infty}^\infty |f(x+h)+f(x)|dx$
If $f\in C_c(\mathbb{R})$ I got the limit to be $\int_{-\infty}^\infty |f(x)|dx$. I am not sure if this is right.
AI: Let $f$ a continuous function with compact support, say ... |
H: Numerical methods book
I'm looking for an introductory book on numerical methods.
I'm beginning to learn to program (in Haskell, a functional language, if that would affect the recommendations). The reason I want such a book is to practice my programming skills by implementing easy math-related algorithms. For exam... |
H: Which of the following metric spaces are complete?
[NBHM_2006_PhD Screening test_Topology]
Which of the following metric spaces are complete?
$X_1=(0,1), d(x,y)=|\tan x-\tan y|$
$X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$
$X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$
$X_4=\mathbb{R}, d(x,y)=|e^x-e^y|$
$2$ is complete... |
H: How to construct a Bernstein set and what are their applications?
Bernstein Set: A subset of the real line that meets every uncountable closed subset of the real line but that contains none of them. It's from wiki.
My question is this: How to construct a Bernstein set? And what's its application in mathematics?
T... |
H: The inhomogenous Wave Equation
I've been trying to come up with a good way to get rid of an inhomogeneity in this PDE, but I have two different solutions. I am not sure if this is a question for the Math community. If not I'll ask the physics community about this
Consider this PDE which models the Wave Equation
$$... |
H: How many points are necessary to find a parallel ellipse, and how to do it?
So, I understand that to find an ellipse for sure you need at least five points. Why? The ellipse equation has only four variables ($x_0, y_0, a,\text{ and }b$). That's not actually my true question, just a curiosity. My question is, what i... |
H: Every ideal has an approximate identity?
Averson's 1970 paper on extensions of $C^*$-algebras seems to assume that every ideal has an approximate identity.
However, I am a little bit suspicious here, since he does not assume the closeness of these ideals-at certain steps, he proves something for the ideal of all f... |
H: Two proofs I'm having difficulty with
I've been given an assignment. Almost done except the last two are tripping me up. They are as follows:
1) if $2x^2-x=2y^2-y$ then $x=y$
2) if $x^3+x=y^3+y$ then $x=y$
I imagine they use a similar tactic as they both involve powers, but I've tried factoring,completing the squar... |
H: Trying to prove $\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\sqrt{1+(\sin(\frac{\pi}{t}) -\frac{\pi}{t}\cos(\frac{\pi}{t}))^2}dt$
I posted this incorrectly several hours ago and now I'm back! So this time it's correct. Im trying to show that for $n\geq 1$:
$$\frac{2}{n+\frac{1}{2}} \leq \int_{1/(n+1)}^{1/n}\... |
H: Convergent fraction for constant $e$?
I've just learned about e. I am very much the novice and my problem is that while trying to calculate the convergent fractions for e. For instance:
$${2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4}}}}}$$
I end up with 144/53?
I was wondering are there specific steps that... |
H: Combinatorics: how many unique albums...
I want to record a set of music albums so that each one is unique. Each album has ten tracks, and I've recorded 4 versions of each track. How many unique albums can I compile so that no two albums has the exact same set of tracks while still having only one version of track ... |
H: Overlaying Latin squares of order 4
Here are two latin squares overlayed upon each other to make one latin square, if you will. One "sub-latin" square is labeled with $1,2,3,4$ while the other is represented with $a,b,c,d$. There must be an $a$ corresponding to each of $1,2,3,4$ as you will see(and so on) and there... |
H: A question of a Buzyakova's 2005 paper
I came across a question of a Buzyakova's 2005 paper. This is a paragraphy of his paper, which is the last example in his paper. (I know it is a little complex to ask the complete question without the paper. )
I want to know in the last line, why $N-\cup_{\beta <\alpha}N_\bet... |
H: Power series identity
Possible Duplicate:
Summation of $\sum\limits_{n=1}^{\infty} \frac{x(x+1) \cdots (x+n-1)}{y(y+1) \cdots (y+n-1)}$
Through a numerical computation, I stumbled across the following identity. It takes place in the ring $(\mathbb{Z}[x])[[t]]$, which is complete with respect to the $t$-adic val... |
H: A misbehaved Power Series (contradiction with ln(2))
Possible Duplicate:
Explain why calculating this series could cause paradox?
Using the power series expansion $\ln(x+1)=\sum _{k=1}^\infty \frac{(-1)^{k+1}}{k}x^k$ we have $\ln(2)=(1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+\cdots+(\frac{1}{2k-1}-\frac{1}{2k})+\... |
H: $(a+b+c)^p-(a^p+b^p+c^p)$ is always divisible by...?
$(a + b + c)^p - (a^p + b^p + c^p)$
is always divisible by
(a) $p - 1\quad$ (b) $a + b + c\quad$ ( c ) $p\quad$ ( d ) $p^2 - 1$
$p$ is prime
I am able to solve this by substituting values and by euler theorem by assuming $( a + b + c )$ are co prime with ... |
H: Description of a matrix in first-order logic
Assume a 9 by 9 matrix with variable elements that are natural numbers ranging from 1 to 9 (like a Sudoku puzzle). I want to describe the entire matrix in first-order logic, but I'm having trouble thinking of a way to do so that isn't too horribly verbose.
Anyone have id... |
H: Partial Latin squares of even order
Can we show that $P$ is a partial latin square that is $n \times n$ where $n$ is even, where the upper quadrant $\frac{n}{2} \times \frac{n}{2}$ is filled and the rest is blank, then $P$ can be completed to a Latin square?
AI: Yes it can always be completed. This is a consequenc... |
H: Is the empty set partially ordered ? Also, is it totally ordered?
I am not sure on how to go about this. Please provide clear explanations.
AI: The question as phrased isn’t really meaningful, since you didn’t specify a relation on the empty set. However, there is only one, so I’ll assume that it’s the one that you... |
H: Write down the sum of sum of sum of digits of $4444^{4444}$
Let $A = 4444^{4444}$;
Then sum of digits of $A = B$;
Then sum of digits of $B = C$;
Then sum of digits of $C = D$;
Find $D$.
What should be the approach here?
AI: The approach is to use the fact that $4444 \equiv 7 \pmod 9,$
so that $4444^3 \equiv 1 \pmo... |
H: Continuous extensions of continuous functions on dense subspaces
I thought that if I have a function $f: \mathbb Q \to \mathbb R$ that is continuous then I can (uniquely) extend it to a continuous function $F: \mathbb R \to \mathbb R$ as follows: for $r \in \mathbb R \setminus \mathbb Q$ pick a sequence $q_n$ conve... |
H: Proving a theorem from topology
Theorem:
Suppose $Y \subset X$. A subset $E \subset Y$ is open relative to $Y$
if and only if $E = Y \cap G$ for some open subset G of X.
I don't understand what's happening neither can I follow the proof.
AI: For metric spaces you can argue as follows:
First we note, that ball... |
H: Solving an exponential distribution
In a simulation, I am trying to find the value of $d_i$ where:
$\displaystyle d_i \sim \frac{\epsilon_i}{\lambda_i}$ where $\epsilon_i$ is i.i.d. exponentially distributed with parameter = 1 and $i=1...n$.
Conditional on $\lambda_i$ the $d_i$ have an exponential distribution of $... |
H: Roots equal constant
\[
\sqrt{a-b} + \sqrt{b-c} + \sqrt{c-d} + \sqrt{d-a} = K
\]
for some real constant $K$ and some real numbers $a$, $b$, $c$ and $d$. Find $K$.
Again i apologize for the syntax and appreciate anyhelp thanks!
AI: Here $a, b,c,d,K$ are real numbers.
$\sqrt{a-b} , \sqrt{b-c} , \sqrt{c-d} , \sqrt{d... |
H: Determine if $(\mathbb N, \Sigma)$ is a poset, look for $\min(\mathbb N, \Sigma)$ and $\max(\mathbb N, \Sigma)$
Let $D(x) = \{y \in \mathbb N : y\text{ is a divisor of } x\}$ and let the relation $\Sigma$ be defined as follows:
$$\begin{aligned}x \Sigma y \Leftrightarrow D(x) \subseteq D(y) \end{aligned}$$
check $\... |
H: Reccurence relation: Lucas sequence
I need to solve the given recurrence relation:$$L_n = L_{n-1} + L_{n-2},$$ $n\geq3$ and $ L_1 = 1, L_2 =3$
I'm confused as to what $n\geq3$ is doing there, since $L_1$ and $L_2$ are given
I got $t = \frac{1\pm\sqrt 5}{2}$
Which got me the general solution, $ L_n = a $(golden... |
H: Prove by mathematical induction that $2n ≤ 2^n$, for all integer $n≥1$?
I need to prove $2n \leq 2^n$, for all integer $n≥1$ by mathematical induction?
This is how I prove this:
Prove:$2n ≤ 2^n$, for all integer $n≥1$
Proof: $2+4+6+...+2n=2^n$
$i.)$ Let $P(n)=1
P(1): 2(1)=2^1\implies 2=2$.
Hence, $P... |
H: Solve $k_1 a = k_2 b + c$
Find all $k_1, k_2$ that satisfy $k_1 a = k_2 b + c$ where everything are integers. It feels like there should be some easy way to describe this in terms of congruence and gcd.
AI: Let $d=\gcd(a,b)$. If $d$ does not divide $c$, there is no solution. So assume from now on that $d$ divides ... |
H: Prove the statement : $\log(k + 1) -\log k>\frac{ 3}{10k}$
Prove the statement : $\log(k + 1) - \log k > \frac{3}{10k}$
Approach :
$$\log(k+1)-\log{k} > \frac{3}{10k}$$
Clearly, $k\in\mathbb{Z}^{+}$
$$\log(k+1)-\log{k}=\log\bigg(1+\frac{1}{k}\bigg)$$
given base is $10$, so
$$\log\left(1+\frac{1}{k}\right) > \log\le... |
H: $f:X\rightarrow Y$ is a continuous surjection, $X$ and $Y$ are topological space
$f:X\rightarrow Y$ is a continuous surjection Then
If $V$ is open, does this imply $f(V)$ is open?
If $F$ is closed, does this imply $f(F)$ is closed?
If $A$ is an infinite subset, does this imply that $f(A)$ is so in $Y$?
Thank you ... |
H: How to do this summation: $\sum_{n=1}^\infty \log_{2^\frac{n}{2^n}}256$
How to do this summation?
$$\sum_{n=1}^\infty \log_{2^\frac{n}{2^n}}256=?$$
All I'm getting is $8(2 + 2 + \frac83 + \frac{16}{4} + \cdots )$ which is a diverging series.
AI: A single term of series is $$\log_{2^{\frac{n}{2^n}}} 256=\frac{\log_2... |
H: Properties of $xy^2/(x^2+y^4)$ near the origin
$f:\mathbb{R}^2\rightarrow \mathbb{R}$
Defined by $$f(x,y)= \frac{xy^2}{x^2+y^4}$$ if $x\neq 0,y\in\mathbb{R}$ and $$f(x,y)=0$$ if $x=0,y\in\mathbb{R}$
Then
it is continuous but not differentiable at origin
differentiable at origin
has all first order partial deriva... |
H: Fibonacci sequence, strings without 00, and binomial coefficient sums
Refer to the sequence $S$ where $S_n$ denotes the number of n-bit strings that do not contain the pattern 00.
By considering the number of n-bit strings with exactly i 0's, show that
$\displaystyle f_{n+2} = \sum_{i=0}^{\frac{n+1}{2}} \binom{n+1 ... |
H: What is The 3rd side length of Isosceles Triangle
I've a isosceles triangle which length is $10\;\mathrm{cm}$ , $10\;\mathrm{cm}$ and $x$.
If I want to make this triangle $120^\circ$ degree then what should be the $x$?
AI: Angles opposite to equal sides are equal, so one angle is $120^0$ while others are $30^0$ ea... |
H: A continuous map from $D$ unit disk, to $S^1$
$f:D\rightarrow S^1$ is a continuous then $\exists x\in S^1$ such that $f(x)=x$?
$f:S^1\rightarrow S^1$ then same as 1 holdd?
$f:E\rightarrow E$ then same as 1 hold? $E=\{(x,y):2x^2+3y^2\le 1\}$
by Fixed point Theorem I know 2,3 are correct, what about 1?
AI: In #1, ... |
H: Finding a well-defined solution to a matrix equation
I have the following problem:
Given two 2D real positive-definite symmetric matrices $M_1$ and $M_2$, find a matrix $T$ such that
$$ M_2=TM_1T^t$$
Clearly, the solution is not unique, but I don't care too much about that. All I need is a well-defined solutio... |
H: how to solve system of linear equations of XOR operation?
how can i solve this set of equations ? to get values of $x,y,z,w$ ?
$$\begin{aligned} 1=x \oplus y \oplus z \end{aligned}$$
$$\begin{aligned}1=x \oplus y \oplus w \end{aligned}$$
$$\begin{aligned}0=x \oplus w \oplus z \end{aligned}$$
$$\begin{aligned}1=w \o... |
H: Proving that a linear isometry on $\mathbb{R}^{n}$ is an orthogonal matrix
I wish to prove that if $T:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is defined by $T(v)=Av$ (where
$A\in M_{n}(\mathbb{R})$) is an isometry then $A$ is an orthogonal
matrix.
I am familiar with many equivalent definition for $A\in M_{n}(\mathbb{R})$
... |
H: If $H\unlhd G$ with $(|H|,[G:H])=1$ then $H$ is the unique such subgroup in $G$.
Here is a problem from "An introduction to the Theory of Groups" by J.J.Rotman:
Let $G$ be a finite group, and let $H$ be a normal subgroup with $(|H|,[G:H])=1$. Prove that $H$ is the unique such subgroup in $G$.
I assumed there was ... |
H: $GL_n(k)$ (General linear group over a algebraically closed field) as a affine variety?
In the context of linar algebraic groups, I read in my notes from the lecture that's already some while ago that $GL_n(k)$ is an algebraic variety because $GL_n=D(\det)$, $ \det \in k [ (X_{ij})_{i,j} ]$.
Now, $k$ is an algebrai... |
H: Determine monic and degree 3 polynomial in $\mathbb Z_p$
I stumbled upon this kind of problem and I really can't get the hang of it. Will anyone please outline the way to solve it?
Determine for which of the first $p > 0$ values the polynomial $f = 42x^4+21x^3-x+1 \in \mathbb Z_p$ is monic and has degree 3. Then fa... |
H: I have n flavors of icecream. I choose k scoops, where k can be larger or smaller than n. How to generate all possible sequences?
Say I have 4 flavors of icecream, a, b, c, and d, and I want to get 3 scoops.
So basically I want to generate all possible 4-tuples where the sum of all elements in each tuple adds up to... |
H: Proving that for every real $x$ there exists $y$ with $x+y^2\in\mathbb{Q}$
I'm having trouble with proving this question on my assignment:
For all real numbers $x$, there exists a real number $y$ so that $x + y^2$ is rational.
I'm not sure exactly how to prove or disprove this. I proved earlier that for all real ... |
H: show that integral converges even if it has a singularity
i am currently reading through a book on generalized functions, and there it is said that:
... $\int_{|x|\le r} |x|^{-t} dx$ converges for $t < n$ (in $n$ dimensions) and
diverges for $t \ge n$.
Why does it converge for $t < n$, its area/volume still go... |
H: Showing $\mathbb{Q}(\sqrt[4]{2},i)=\mathbb{Q}(\sqrt[4]{2}+i)$ using the Galois orbit of $\sqrt[4]{2} + i$
The following is a problem from I. Martin Isaac's Algebra. Let $E=\mathbb{Q}(\sqrt[4]{2}+i)$. I am trying to show $\mathbb{Q}(\sqrt[4]{2},i)=E$ with the following hint:
Find at least five different elements in... |
H: Changing the order of $\lim$ and $\sup$
Suppose that $f_n:X\to [0,1]$ where $X$ is some arbitrary set. Suppose that
$$
f_n(x)\geq f_{n+1}(x)
$$
for all $x\in X$ and all $n = 0,1,2,\dots$ so there exists $\lim_n f_n(x)$ point-wise, let's call it $f(x)$.
Define $f^*_n:=\sup\limits_{x\in X}f_n(x)$, $f^*:=\sup\limi... |
H: Question about proof of Arzelà-Ascoli
(Arzelà-Ascoli, $\Longleftarrow$) Let $K$ be a compact metric space. Let $S \subset (C(K), \|\cdot\|_\infty)$ be closed, bounded and equicontinuous. Then $S$ is compact, that is, for a sequence $f_n$ in $S$ we can find a convergent subsequence (conv. in $\|\cdot\|_\infty)$.
Pr... |
H: Birthday probability problem
There are four people in a room, namely P, Q, R and S.
Q's birthday is different from everyone else. What is the probability that P and R share the same birthday?
I'm getting $1/364$ as answer.
$(365*364*1*364)/(365*364^3) = 1/364$
AI: Two cases: If $P$ and $R$ share same birthday, th... |
H: Determine $a$ values allowing $x^2+ax+2$ to be divided by $x-3$ in $\mathbb Z_5$
Determine for which $a$ values $f = x^2+ax+2$ can be divided by $g= x-3$ in $\mathbb Z_5$.
I don't know if there are more effective (and certainly right) ways to solve this problem, I assume there definitely are, but as I am not aware... |
H: Prove that the dihedral group $D_4$ can not be written as a direct product of two groups
I like to know why the dihedral group $D_4$ can't be written as a direct product of two groups. It is a school assignment that I've been trying to solve all day and now I'm more confused then ever, even thinking that the teache... |
H: Value of a scaled Bessel function for negative argument
Is the function $\hat{i}_0(x) = e^{-|x|} \sqrt{\frac{\pi}{2x}} I_{\frac{1}{2}}(x)$ positive or negative for negative $x$?
$I_{\alpha}(x)$ above is a modified Bessel function.
Here are my arguments. Considering that $I_{\frac{1}{2}}(x) = \sqrt{\frac{2}{\pi x}} ... |
H: The trace norm cannot be increased by composing with a unitary operator
$\newcommand{\tr}{\operatorname{tr}}$
I was reading a proof for the statement $|\tr(US)|\leq |\tr(S)|$, for every endomorphism $S$ on a complex vector space $H$ and every unitary operator $U$ on the same space. Though the proof is short it uses... |
H: Memory of a neural network: is it forever?
As far as I remember a neural network cannot forget anything. Does this mean that no matters how evolved the network is, it's always going to throw me back the right output if I feed it an input?
And when I say "right output" I mean "precise output" without getting a bit w... |
H: Two notions of uniformizer
Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ideal of the local ring at a point $P \in X$?
AI: There is no connection because, as far as... |
H: Realization of graded algebras with Poincaré duality
Question: Given a finite dimensional positively graded algebra $A$
over some ring $R$ that satisfies Poincaré duality in some dimension
$n$, is there necessarily a topological space $X$ such that $H^*(X;R) \cong A$?
I recognise this is some sort of realizat... |
H: Fourier transform eigenvalues
Since I've studied the Fourier transform extension to the Hilbert space $L^2$, I wondered if there is a complete study relative to its eigenvalues. I know that its adjoint operator is the inverse transform, which means that I can't use the theory of self-adjoint operators to state some... |
H: Counting marbles
In how many ways can you distribute 9 marbles in 3 piles?
The marbles are all identical. I know its not a very deep question but I don't know how to solve it. There is a difference between (pile a getting 4 pile b getting 4 and pile c getting 1) and (pile a getting 1 pile b getting 4 and pile c ... |
H: Lipschitz continuity:$f_2 - f_1$ , $f_2$ constant $L_2$ and $f_1$ constant $L_1$
Suppose you have this two Lipschitz continuous functions:
$f_1$ ,with constant $L_1$ and $f_2$ with constant $L_2$.
I have to prove that $f_2 - f_1$ is Lipschitz continuous with constant $L_1+L_2$.
I did like this:
$|(f_2-f_1)(y)-(f_2-... |
H: Proof of the lemma used in proving that a finite-dimensional normed space is complete
I'm trying to understand the proof for the lemma:
$$\|\alpha _1 e_1 + \alpha _2 e_2 + \cdots + \alpha_n e_n\| \geq c (|\alpha_1|+|\alpha_2|+\cdots+|\alpha_n|)$$
where $c>0$ and the $e_i$s are finite and linearly independent.
The p... |
H: Permutation problem scheduling games
There are:
14 teams $t$ numbered from 0 to 13
13 different games $g$ numbered from 0 to 12
13 rounds $r$ numbered from 0 to 12
I want to make a planning such that:
each team plays against each other team
each team plays all 13 games
2 times the same game during one roun... |
H: which of the following metric spaces are separable?
which of the following metric spaces are separable?
$C[0,1]$ with usual 'sup norm' metric.
the space $l_1$ of all absolutely convergent real sequences, with the metric $$d_1(a_i,b_i)=\sum_{1}^{\infty}|a_i-b_i|$$
The space $l_{\infty}$ of all bounded real sequence... |
H: Can any Polynomial be factored into the product of Linear expressions?
Specifically I am wondering if...
Given a Polynomial of n degree in one variable with coefficients from the Reals.
Will every Polynomial of this form be able to be factored into a product of n linear (first degree) Polynomials, with the coeffici... |
H: Poisson random variable with mean going to infinity
I was looking for some facts on the probability theory and I found this exercise on Billingsley's "Probability and Measure" book (exercise 27.3, page 379). It doesn't look like a hard exercise but I'm having a hard time trying to prove the general case. Here it go... |
H: Dense subspace of $\ell^2$
Is the set \begin{align} A=\left\{a=(a_1,a_2,\dots)\in\ell^2 \ \ \lvert \ \ \sum_{k=1}^\infty \frac{a_n}{n}=0 \right\}\subset\ell^2
\end{align} dense in $\ell^2$
Is the following argument correct? Let $x=(1,0,0,\dots)$ and assume $\forall \epsilon>0\ \ \exists a\in A$ such that $\lVert ... |
H: Is $M(x)=O(x^σ)$ possible with $σ≤1$ even if the Riemann hypothesis is false?
The wiki page on Mertens conjecture and the Connection to the Riemann hypothesis says
Using the Mellin inversion theorem we now can express $M$ in terms of 1/ζ as
$$
M(x) = \frac{1}{2 \pi i} \int_{\sigma-i\infty}^{\sigma+i\infty} \fra... |
H: How to see that the shift $x \mapsto (x-c)$ is an automorphism of $R[x]$?
In the process of studying irreducibility of polynomials, I encountered the criterion that $p(x)$ is irreducible if and only if $p(x-c)$ is irreducible. When trying to determine what properties of the ring were preserved under this map $x \ma... |
H: Kernel of $T$ is closed iff $T$ is continuous
I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds:
\begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is closed}\end{align}
which probably holds for general operators $T:X\rightarrow Y$... |
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