text stringlengths 83 79.5k |
|---|
H: question on symmetric and nilpotent matrix
a) Let $A$ is of nilpotent of degree $k$ then according to the condition ${A}^{k}={B}^{2k}=0$ But I can't say anything about existence of such $B$
I have no idea on $b$, $c$, please help.
AI: (a) Can you show that if $\,A\,$ is a $\,n\times n\,$ nilpotent matrix , say$\,... |
H: A transform function from $(−∞,∞)$ to $(0,1)$?
I want to convert an integral from $(0, 1)$ range to $(-\infty, \infty)$ range by change of variable. What is the best transform function to do this - one that is simple, monotonic with $f(-\infty)=0$ and $f(\infty)=1$?
Thanks.
AI: $\newcommand{\logit}{\operatorname{lo... |
H: Non-isomorphic abelian groups of order $19^5$
I am trying to classify abelian groups of order $19^5$ up to isomorphism. Can anyone provide any approaches or hints?
AI: From the Fundamental Theorem for Fin. Gen. Abelian groups, it follows that we must take the partitions of 5 (all this can be googled easily):
$$\beg... |
H: $e^x(\ln x-c) =\sum \limits_{k=0}^\infty \frac{ x^{k} \Gamma'(k+1)}{ (k!)^2}$ Is it correct result?
$e^x=\sum \limits_{k=0}^\infty \frac{x^k}{k!}$
We can write $e^x=\sum \limits_{k=0}^\infty \frac{x^k}{ \Gamma(k+1)}$
Where $\Gamma(x)$ is Gamma function
$\Gamma(k+1)=k\Gamma(k)$
$\frac{\Gamma(k+1)}{\Gamma(k)}=k$
$... |
H: Check if two 3D vectors are linearly dependent
I would like to determine with code (c++ for example) if two 3D vectors are linearly dependent.
I know that if I could determine that the expression
$ v_1 = k · v_2 $is true then they are linearly dependent; they are linearly independent otherwise.
I've tried to constr... |
H: Low-Rank Approximations Book
I am looking for a source (book, online book, etc..) where I can find the theory behind low-rank approximations of matrices. In particular, I am interested in low-rank approximations used in optimization problems, such as minimizations of the Euclidean and Forbenius norms.
I have some ... |
H: Independent events and Dependent events
I have a question regarding these strikingly similar problems with contradicting solutions. This is somewhat long, so prepare
Probblem 1
Consider a bag of ten coins, nine are fair, but one is weighted with both sides heads. You randomly select a coin and toss it five times. L... |
H: Projective and injective modules; direct sums and products
I need two counterexamples.
First, a direct sum of $R$-modules is projective iff each one is projective.
But I need an example to show that, “an arbitrary direct product of projective modules need not be a projective module.”
If I let $R= \mathbb Z$ then ... |
H: A free group on the non-empty set $X$ is solvable iff $|X| =1$
Let $X$ be a non-empty set. Prove that $F_X$, the free group on $X$ is solvable if and only if $|X| = 1$.
We can see that if $|X| = 1$, then $F_X$ is abelian, and hence solvable. However, the other direction stumps me. Any suggestions?
AI: Yes: a quotie... |
H: Z-index of an arbitrary point on a flattened 3-dimensional triangle
I have a triangle in a 3-dimensional coordinate system that I want to draw to a screen. I'm able to flatten the triangle to 2 dimensions and determine whether an arbitrary point on the screen falls within the flattened object. What I don't know how... |
H: Elements of $\mathbb{F}_p$ having cube roots in $\mathbb{F}_p$
Let $p$ be a prime number, and let $\mathbb{F}_p$ be the field with $p$ elements. How many elements of $\mathbb{F}_p$ have cube roots in $\mathbb{F}_p$?
I had this question on an exam and after reviewing I am still not sure. Any help would be apprecia... |
H: Fastest numeric method for ODE
I am currently using Euler method and it is 4-8 times too slow. Which method will be fastest? I need it to compute Turing's reaction-diffusion system.
AI: Euler's method is the fastest possible single-step, explicit, non-adaptive method for a given fixed step size $h$.
If this is too ... |
H: How to compare big numbers that are outcome of different functions.
How is the best way to compare big numbers? They are result of two functions with different asymptotic growth. For example:
Googleplex which is $10^{{10}^{100}}$ to $1000!$
AI: $10^{googol}$ compared to $1000!$
$1000!=1000\times999\times998...<100... |
H: Is there a way to calculate the probabilities without tree diagram
Could someone tell me how on earth did they calculate $P(A_1\text{ and }A_3)$ and etc.. without drawing a tree diagram?
Thank you very much
AI: These are independent events and you want to know
$$\,P(A_1\cap A_3)=P(A_1)\cdot P(A_3)=\frac{1}{2}\cdo... |
H: What real numbers are in the Mandelbrot set?
The Mandelbrot set is defined over the complex numbers and is quite complicated. It's defined by the complex numbers $c$ that remain bounded under the recursion:
$$ z_{n+1} = z_n^2 + c,$$
where $z_1 = 0$.
If $c$ is real, then above recursion will remain real. So for what... |
H: Is there a better counter-example? (problem involving limit of composition of functions)
This question arises from one of the Berkeley Problems in Mathematics, fall 1981. It asks me to find a counterexample to the following claim:
Suppose $\lim\limits_{x\rightarrow x_{0}}f(x)=a$, $\lim\limits_{t\rightarrow a}g(t)=b... |
H: How can I bound this equation?
This is a problem I stuck in Berkeley Problems in Mathematics, fall 1983:
Let $x(t)=(x_{1}(t)...x_{n}(t))$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}^{n}$. It satisfies a differential equation of the form $$x'(t)=f(x(t))$$
where $f:\mathbb{R}^{n}\rightarrow \mathbb{... |
H: Scaling a square puzzle
Found this on the net today and lost.
On a table you have a square made of 4 coins at the corner at distance
1. So, the square is of size 1×1. In a valid move, you can choose any two coin let’s call them mirror and jumper. Now, you move the jumper
in a new position which is its mirror ... |
H: Hausdorff Measure
Given the Hausdorff Measure
Is it true that $H^1$(line)= Length of the line?
How can one prove it?
AI: On $\mathbb{R}^n$, thus in particular on $\mathbb{R}$, the $n-$ dimensional Hausdorff measure equals the $n-$ dimensional Lebesgue measure. This is not completely trivial for general $n$, not to... |
H: Finding second derivative
I am asked to find the second derivative of the function:
$$h(x)=\sqrt{x^2+1}$$
$$h(x)=(x^2+1)^\frac{1}{2}$$
$$h'=\frac{1}{2}(x^2+1)^\frac{-1}{2} 2x$$
$$h'=\frac{x}{\sqrt{x^2+1}}$$
$$h''=\frac{\sqrt{x^2+1} - x(\frac{1}{2}(x^2+1)^\frac{-1}{2}2x}{(\sqrt{x^2+1})^2}$$
And this is as far as I g... |
H: Asking solutions for the integral equations
This is from Berkeley Problems in Mathematics, Spring 86. It asks for $\lambda\in \mathbb{R}$, find all solutions of the following two equations:
$$\phi(x)=e^{x}+\lambda\int^{x}_{0}e^{x-y}\phi(y)dy; \psi(x)=e^{x}+\lambda\int^{1}_{0}e^{x-y}\psi(y)dy$$
My thought is to take... |
H: Request for a $\mathrm{Hom}$ functor example
Let $$ P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \xrightarrow{d_0} M \to 0$$
be an exact sequence of $R$-modules. Consider
$$ (*) \hspace{1 cm} P_2 \xrightarrow{d_2} P_1 \xrightarrow{d_1} P_0 \to 0$$
that is, the sequence with $M$ removed. Then this resulting sequ... |
H: floor division remainder and quotients may vary
I have question on behavior of floor division.
if i have,
-7/3 = -3 and remainder = 2
also -7/3 = -4 remainder = 5
it can have many results. When we make tally all are correct.Then what is the use of floor division is not stick to one solution as truncated divisio... |
H: Is it possible to extend an arbitrary smooth function on a closed subset of $R^n$ to a smooth function on $R^n$?
Assume that $K$ is a closed (or compact if necessary) subset in $\mathbb{R^n}$ and $f:K \rightarrow \mathbb{R}$ is a smoth function in the following sense:
for each $x \in K$ there exists a neighbourhood... |
H: Proving there are no subfields
I am trying to solve Q11 at pg. 582 from the book Abstract algebra
by Dummit and Foote, the question is:
Let $f\in\mathbb{Z}[x]$ be an irreducible quartic whose splitting
field has Galois group $S_{4}$over $\mathbb{Q}$. Let $\theta$ be
a root of $f$ and denote $K=\mathbb{Q}(\thet... |
H: Working with indices, where Einstein summation applies
Problem:
Let $$H:={1\over 2m}\vec p\cdot \vec p -{1\over r}$$, where $r=(\vec r\cdot \vec r)^{1\over 2}, \,\,\,\vec r= (x_1,x_2,x_3)^T$ and $$\vec R:= {1\over m}\vec p\times \vec L -\hat r$$. I wish to show that $${\partial H\over \partial p_i}{\partial R_j\ov... |
H: An example of finite groups
Is there any example of finite group $G$ with the following properties?
1) There is prime divisor $p$ of order $G$ such that the number of cyclic
subgroup of order $p$ is $p+1$.
2) The order of Sylow $p$-subgroup of $G$ is $p^{2}$.
3) $G$ is not $p$-group.
AI: Let $G$ be a group with all... |
H: Computing $10101011 \cdot 1025$
I'm trying to compute the following expression: $10101011 \cdot 1025$ in a simple, easy way without using a calculator, or an elementary school way. I realized that $1025=2^{10}+1$ so I need to compute now $10101011 \cdot 2^{10}+10101011$, How should I proceed?
AI: Forget binary and... |
H: Trace of a $227\times 227$ matrix over $\mathbb{Z}_{227}$
well, I know that the trace is the negative of coefficient of $x^{226}$ of the characteristic polynomial the matrix, but I dont know how the Char.Poly looks like in this case.please give me some hint.
Do I have to work in the splitting field of the chara... |
H: checking whether certain numbers have an integral square root
I was wondering if it is possible to find out all $n \geq 1$, such that $3n^2+2n$ has an integral square root, that is, there exists $a \in \mathbb{N}$ such that $3n^2+2n = a^2$
Also, similarly for $(n+1)(3n+1)$.
Thanks for any help!
AI: $3n^2+2n=a^2$, $... |
H: Countable union of countable sets(ZF)
Let ${{E_n}}_{n\in \mathbb{N}}$ be a sequence
such that every $E_n$ is countable.
Let $g_n : \mathbb{N} \to E_n$ be a bijection for every $n\in \mathbb{N}$.
Let $\alpha (n,k) = g_n(k)$
Let $A$ be the union of $E_n$'s.
Then $\alpha : \mathbb{N} × \mathbb{N} \to A$ is a surjectiv... |
H: Why is choosing elements in equivalence classes not a choice?
This is Asaf's answer from this link:
How do we know an $ \aleph_1 $ exists at all?
I don't understand this sentence that is;
From each equivalence class choose the representative which is an ordinal (which does not require any form of choice, as the e... |
H: Is this metric space complete: $d(f,g)=\max\limits_{t\in[0,1]} |f(t)-g(t)|$ for $C^1$-functions?
The space $\mathcal C^1[0,1]$ with the metric $$d(f,g)=\max_{t\in[0,1]} |f(t)-g(t)|$$
No, it is not complete metric space: by Stone-Weierstrass thm we know that $|x|$ can be uniformly approximated by sequence of polyn... |
H: Why does $ \| v_1 \cdots v_k \|_2 \leqslant \|v_k \|_2 \|v_1 \|_\infty \cdots \| v_{k-1} \|_\infty$ hold?
For $ v_i \in L^\infty \cap L^2 $, has a compact support, $v_i : \mathbb R^n \to \mathbb R$, $$ \| v_1 \cdots v_k \|_2 \leqslant \|v_k \|_2 \|v_1 \|_\infty \cdots \| v_{k-1} \|_\infty$$
holds? Then why?
AI: By ... |
H: Distance between a point and a m-dimensional space in n-dimensional space ($m
I am trying to find a method with a low computational cost to compute the distance of a point $P$ and a space $S$ that is defined by the origin $O$ and $m$ vectors $v_1, v_2, ..., v_m$ in an $n$-dimensional space ($m<n$). The vectors are ... |
H: Reduced frequency range FFT
Generally when one takes the FFT of a signal it "works" over the whole bandwidth dividing up the spectrum into chunks given by the resolution. If the bandwidth of the signal is 10khz and your resolution is 1000 then each "frequency" represents a chunk of 10hz(each bin is 10hz in size).
T... |
H: analytic on a disc with a hole
I am reading a proof of Cauchy's Integral Formula.In the proof,the author let $\phi (z,w)=[f(z)-f(w)]/(z-w)$ if $(z\neq w)$ and $f'(z)$ otherwise and leaves the readers to prove that $g(z)=\phi (z,w)$ is analytic for fixed $w$.I solve this problem by consider the power series around $... |
H: Functional and Linear Functional
may I know what is the distinction between functional analysis and linear functional analysis? I do a search online and came to a conclusion that linear functional analysis is not functional analysis and am getting confused by them. When I look at the book Linear Functional Analysis... |
H: About the convergence of $f_n:=\sin{\sqrt{t+4n^2\pi^2}}$
Let $f_n \in C([0,+\infty))$ be defined by
$f_n(t):=\sin{\sqrt{t+4n^2\pi^2}}$, for $n \in \mathbb N$ and $t \ge 0$.
Prove that $f_n$ converges pointwise to $f \in C([0,+\infty))$ and determine $f$;
study the uniform convergence of the sequence on bounded... |
H: Which statement is true regarding the centroid?
From a point $P$ on the circle $x^{2} + y^{2} = 4r^{2}$, tangents are drawn to $x^{2} + y^{2} = r^{2}$ at $Q$ and $R$. Then which of the following statement(s) are true regarding the centroid of $ΔPQR$
$A.$ is at a distance $r$ from chord $QR$.;
$B.$ lies on $x^{2} + ... |
H: The necessary and sufficient conditions for the solution of the equation $\frac{dy}{dx} = f(y)$ is locally unique.
$$\frac{\mathrm{d}y}{\mathrm{d}x} = f(y)$$
where $f(y)$ is continuous on $|y-a|\leq \epsilon$,and $f(y)=0$ iff $y=a$.
To Proof : For the initial value point on $y=a$,the equation has local unique solut... |
H: Question about factor rings
Assume $m_i$ are maximal ideals in a ring $R$.
Then I have $m_1 \cdot \dots m_{k}$ is an ideal in $m_1 \cdot \dots m_{k-1}$ hence I can quotient to get a factor ring $m_1 \cdot \dots m_{k-1} / m_1 \cdot \dots m_{k}$.
This ring is of course also an abelian group. Now I'd like to turn this... |
H: Boundary of product manifolds such as $S^2 \times \mathbb R$
Simple question but I am confused.
What is the boundary of $S^2\times\mathbb{R}$? Is it just $S^2$?
What would be the general way to evaluate the boundary of a product manifold?
Thanks for the replies!
AI: If you are interested in differentiable manifolds... |
H: Proof using Reductio ad absurdum (RAA)
Note: $\neg$ means 'not', $\rightarrow$ is 'conditional', $\land$ is 'and', $\lor$ means '(inclusive) or'.
Prove: $[\neg D \lor (A \land B)] \rightarrow[(J \rightarrow \neg A) \rightarrow (D \rightarrow \neg J)]$ using Reductio ad absurdum (RAA) or conditional proof (CP).
$\... |
H: What is the formula to find the count of (sets of) maps between sets that equal the identity map?
Given two finite sets $A$ and $B$, what is the formula to find the number of pairs of mappings $f, g$ such that $g \circ f = 1_A$?
AI: We look at the problem when $A$ and $B$ are finite. Suppose that $A$ has $a$ elemen... |
H: Limit of a subsequence
I studied a definition,and I didn´t find it in anyother book (but those I use).
It´s like a point of closure for sequences.We call $a$ "value of closure" of $(x_n)$ when $a$ is the limit of a subsequence of $(x_n)$.
The question is:
For a real number $a$ be a "value of closure" is necessary a... |
H: Continuous Actions and Homomorphisms
I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of $X$, have the compact open topology. I want to show that an action of $G$ on $... |
H: Schwarz inequality for unital completely positive maps
I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper:
Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between two $C^*$-algebras, then \begin{equation}\delta(A)^*\delta(A)\le \... |
H: Strange application of Cauchy's Integral Theorem
According to my book, Riemann's Zeta Function, Cauchy's Integral Formula is applicable to the following integral for all negative values of $s$:
$$-\frac{\Pi(-s)}{2\pi i}\int_{|z|=\epsilon}(-2\pi in - z)^{s-1}\frac{z}{e^z - 1}\frac{dz}{z} = -\Pi(-s)(-2\pi in)^{s-1}$$... |
H: The closure of $\overline{\{x\}}$ being irreducible and relating the generic point to its associated irreducible scheme
If $x$ is a point in $X$ where $X$ is a scheme, we write $\overline{\{ x\}}$ for the closure of $x$ in $X$.
$\mathbf{Question \;1}$: I am a bit confused why $\overline{\{ x\}}$ is irreducible. Ac... |
H: If a subset of $\mathbb{R}$ is closed and bounded with respect to a metric equivalent to the Euclidean metric, must it be compact?
Two different metrics $d$ and $\hat d$ in a space $X$ are said to be equivalent iff the topologies generated by them are the same, in other words $U\subseteq X$ is $d$-open iff it is ... |
H: Is $\pmb{\eta}\cdot\pmb{\omega_1} = (\pmb{\eta} + \pmb{1})\cdot\pmb{\omega_1}$?
$\pmb{\eta}$ - order type of $\mathbb{Q}$.
$\pmb{1}$ - order type of a singleton set.
$\pmb{\omega_0}$ - order type of $\mathbb{N}$.
$\pmb{\omega_1}$ - order type of the first uncountable ordinal.
It is easy to see that $\pmb{\eta}\c... |
H: Proving $\int_{1}^{\infty}\frac{\sin x}{\left(\log x\right)^{\frac{1}{2}}}dx$ converges
Prove that the following improper integral converges $$\int_{1}^{\infty}\frac{\sin x}{\left(\log x\right)^{\frac{1}{2}}}dx.$$
I see that you can show this using Dirichlet's Convergence Test, but how would you show it not usin... |
H: Research done by high-school students
I'm giving a talk soon to a group of high-school students about open problems in mathematics that high-school students could understand. To inspire them, I would like to give them examples of high-school students who have made original contributions in mathematics. One exampl... |
H: If the covariance matrix is $\Sigma$, the covariance after projecting in $u$ is $u^T \Sigma u$. Why?
I read in this answer that:
If covariance matrix is $\Sigma$, the covariance after projecting in
$u$ is $u^T \Sigma u$.
I fail to see this, how do I get the covariance of a set of points after projecting those p... |
H: Complex equation in maxima
I rested on this tutorial.
After issuing the command with "solve" function:
%i2 solve((a-b-sqrt(-c^2+2*c*y-y^2+r^2))^2+(d-y)^2=2*r^2*(1-cos(e)),y);
The output is:
Why there is unknown quantity "y" on the right side?
P.S.
There's no "Maxima" tag, what a pity! However, I was redirected he... |
H: How does this game work? (Number game: subtract prime)
Problem
Alice and Bob play the following game.They choose a number N to play
with.The runs are as follows :
1.Bob plays first and the two players alternate.
2.In his/her turn ,a player can subtract from $N$ any prime number less than $N$ or the n... |
H: Finding the divisors of the number $p^3q^6$
My text says that $p^3q^6$ has 28 divisors. Could anyone please explain to me how they got 28 here?
Edit:
$p$ and $q$ are distinct prime numbers Sorry for the late addition..
AI: There can be 0-3 factors of $p$, so there are 4 ways for that to occur. There are 0-6 factor... |
H: Farey sequences for polynomials?
Does a notion of Farey sequence (or something equivalent) exist for polynomials over finite fields?
AI: The terms in the Farey sequence exhaust the rationals, which form a field. The polynomials over a field don't form a field. The terms in a Farey sequence are listed in increasing ... |
H: Probability of Survival
Let $p(x)_{t}$ be the probability that a person aged $x$ survives until at least age $x+t$. Suppose we are given the following:
$p(x)_{1} = 0.99$
$p(x+1)_{1} = 0.985$
$p(x+1)_{3} = 0.95$
$q(x+3)_1 = 0.02$
Note that $q(x)_{t} = 1-p(x)_{t}$. What is $p(x+1)_{2}$?
So we want to find the pro... |
H: Are there "one way" integrals?
If we suppose that we can start with any function we like, can we work "backwards" and differentiate the function to create an integral that is hard to solve?
To define the question better, let's say we start with a function of our choosing, $f(x)$. We can then differentiate the func... |
H: What is the difference between normal and perpendicular?
What is the difference when a line is said to be normal to another and a line is said to be perpendicular to other?
AI: There are different kind of contexts you use term normal in mathematics. You often use perpendicular in case of two or three dimensional ge... |
H: Could someone explain this proof to me? Probability proof
If event $A$ and $B$ are events such that $P(A)$ and $P(B)$ are either $0$ or $1$ and $A$ is subset of $B$, then $A$ and $B$ are dependent events.
Proof: Since $A\subset B$, we have $A\cap B=A$ and so $P(A\cap B)=P(A)$.
$\therefore$ $P(A\cap B)-P(A)P(B)=P(A... |
H: Is there a name for the "most square" factorization of an integer?
For the definition that follows, I'm curious to know if there's a known name (to enable a literature search relating to algorithms).
Definition. Given an integer $n$, the maximally square factorization consists of the integers $\{a,b\}$ such that $n... |
H: What is the possible value of $x$ in the following case?
$$2^{y}+2(3^{y}) > 3(4^{y})$$ and $$y=3x^2+2x-2$$
Which of the following is a possible value of $x$?
$A -1.5 $
$B-2.5 $
$C -0.5 $
$D+0.7 $
$E+1.2$
$********$
I could just conclude from the $1st$ inequality that $$y∊(-∞, 0)$$
Should I put values of $y$ e.g. ... |
H: Is every set a subset?
Is every set a subset of a larger set? In other words, for an arbitrary set S, can one always construct a set S' such that S is a proper subset of S'?
Is this question even meaningful?
AI: Yes, one can: $S\cup\{S\}$ is a proper superset of $S$, since $S\in S\cup\{S\}$, but $S\notin S$. Thus, ... |
H: Definite integral with a complex number in Euler form
Well... I spent an hour trying to figure out how to go from lhs to rhs:
$$\frac { 1 }{ 2\pi } \int _{ -\infty }^{ +\infty } \phi _{ T }(u)\left( \int _{ k }^{ +\infty } e^{ -iux }dx \right) du=\frac { 1 }{ 2 } +\frac { 1 }{ \pi } \int _{ 0 }^{ +\infty } \R... |
H: 16 digit numbers divisible by 17
I wanted to know about the $16$ digit numbers those are divisible by $17$ and when this $16$ digit number is broken in groups of $4$ those groups of four are also divisible by $17$ and a check to verify their occurrence.
Emma.
AI: Use a divisibility rule:
Subtract 5 times the last... |
H: Need to find the recurrence equation for coloring a 1 by n chessboard
So the question asks me to find the number of ways H[n] to color a 1 by n chessboard with 3 colors - red, blue and white such that the number of red squares is even and number of blue squares is at least one. I am doing it in this way -
1.If the... |
H: Detail in definition of stochastic independence for families of events
In a probability theory script, I read a definition of independence where I don't understand one detail (Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space)
"A family of events $(A_i)_{i \in I}$, $A_i \in \mathcal{F}$ is called inde... |
H: How to find the inverse metrics?
I know one can calculate the inverse of metric tensor $g$ in coordinates as the inverse of it's matrix $g_{ij}$. However what I really liked about differential geometry is how one can actually avoid writing matrices, one just use expressions like
$$g = dx \otimes dx + dy \otimes dy ... |
H: A simple property of the norm of an cyclotomic integer
Let $l$ be an odd prime number and $\zeta$ be a primitive $l$-th root of unity in $\mathbb{C}$.
Let $\mathbb{Q}(\zeta)$ be the cyclotomic field.
Let $A$ be the ring of algebraic integers of $\mathbb{Q}(\zeta)$.
Let $\alpha \in A$.
Let $N(\alpha)$ be the norm ... |
H: Gauss' proof of the irreducibility of a cyclotomic polynomial
Let $l$ be an odd prime number.
Let $f(X) = 1 + X + ... + X^{l-1} \in \mathbb{Z}[X]$.
Probably Gauss was the first man who proved that $f(X)$ is irreducible.
I wonder how he proved it.
AI: The first proof presented here is a proof by Gauss. The original ... |
H: Finding the values of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.
i know that the values of $\cos n\pi=(-1)^{n}$ and $\sin n\pi=0$. Now i want to know that what is the general expressions of $\cos \frac{n\pi}{2}$ and $\sin \frac{n\pi}{2}$.
AI: There are two cases:
$n$ is even, write it as $2k$ and then you ha... |
H: Mahalanobis Distance using Eigen-Values of the Covariance Matrix
Given the formula of Mahalanobis Distance:
$D^2_M = (\mathbf{x} - \mathbf{\mu})^T \mathbf{S}^{-1} (\mathbf{x} - \mathbf{\mu})$
If I simplify the above expression using Eigen-value decomposition (EVD) of the Covariance Matrix:
$S = \mathbf{P} \Lambda \... |
H: How to get principal argument of complex number from complex plane?
I am just starting to learn calculus and the concepts of radians. Something that is confusing me is how my textbook is getting the principal argument ($\arg z$) from the complex plane. i.e. for the complex number $-2 + 2i$, how does it get $\frac{3... |
H: Ill posed PDE problem using Fourier transform.
I have got this question in my exam and i was not able to solve it . The hint that i had gotten was to use Fourier transform and solve it . But i couldn't .
.
Can anyone help me .
Thank you .
AI: Let $u$ and $\bar u$ be the solutions with initial values $f$ and $\b... |
H: Inequality problems
I have Maths test tomorrow and was just doing my revision when I came across these two questions. Would anyone please give me a nudge in the right direction?
$1)$ If $x$ is real and $$y=\frac{x^2+4x-17}{2(x-3)},$$ show that $|y-5|\geq2$
$2)$ If $a>0$, $b>0$, prove that $$\left(a+\frac1b\right)\... |
H: Proof of the divisibility rule of 17.
Rule: Subtract 5 times the last digit from the rest of the number, if the
result is divisible by 17 then the number is also divisible by 17.
How does this rule work? Please give the proof.
AI: Write your number $10a+b$.
Then because 10 and 17 are relatively prime,
$$17... |
H: Prove that $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$
$$\lim_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$$
Please,
Anyone could suggest me some way for this?.
Thanks.
AI: If indeed the limit was zero then every way we approach $(0,0)$ the limit would have to be $0$.
However, if ... |
H: Does the identity $\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2$ hold for $g \in U(n)$?
In a paper (corollary 1, p.14) the following identity is used:
Let g be a unitary matrix. Then:
$$\det(I+g^{-1})\det(I+g)=|\det(g-I)|^2 \text{ for }g \in U(n)$$
Now my question is why this holds
I calculated:
$$\det(I+g^{-1})\det(I+... |
H: 2x2 Matrices and Differences of Fractions
Consider the difference of two arbitrary fractions, $\frac{a}{b}$ and $\frac{c}{d}$.
$$\frac{a}{b}-\frac{c}{d}=\frac{ad-bc}{bd}$$
The numerator is the determinant of the 2x2 matrix $$ \left( \begin{array}{ccc}
a & c \\
b & d \\ \end{array} \right)$$
Is there any reason for ... |
H: generating function for $\sum\limits_k \frac{x^k}{k^2}$?
Does anyone know the generating function $f$ of
$$f(x) = \sum_{k=1}^{\infty} \frac{x^k}{k^2}$$
How can we get it? Thanks!
AI: If you're asking for a function which has the Taylor series in your question when expanded around $x=0$, the answer is the dilogarit... |
H: Arithmetic sequence
A sequence $(a_n)$ satisfies:
$|a_{m+n}-a_{m}-a_{n}|<\frac{1}{m+n}$ for all positive integers $m,n$
Show that $(a_n)$ is an arithmetic progression.
I thought to solve it like this way :
by induction $|a_{km}-ka_m|\le \frac 1m (\frac 12 +\frac 13 +...+\frac 1k)=\frac{H_k-1}{m}$.
Therefore $|\fra... |
H: Segment of $\mathbb{R}^2$?
I don't understand this sentence;
The segment $(a,b)$ can be regarded as both a subset of $\mathbb{R}^2$ and an open subset of $\mathbb{R}^1$.
If $(a,b)$ is a subset of $\mathbb{R}^2$, it is not open, but it is an open subset of $\mathbb{R}^1$.
What is 'segment $(a,b)$ in $\mathbb{R}^2$'?... |
H: Are the consequences of contradictions avoidable?
In common natural languages, there are two interpretations of the word "or".
Can you construct a formal logic based on the excluding notion of "or", such that from a contradictory ($A$ and $\mathbb{not}(A)$ is true simultaneously) it doesn't follow, that all formu... |
H: False proof of $R$ Noetherian, $I$ irreducible hence $I$ prime
Can you tell me what's wrong with my proof? Thanks.
Claim: If $R$ is a Noetherian ring and $I$ is an irreducible ideal in $R$ then $I$ is prime
Proof: Let $xy \in I$. We want to show that either $x\in I$ or $y \in I$. By contradiction assume neither $x\... |
H: Prove that an odd degree polynomial must cross any bounded continuous function in $\mathbb R$
Let $p (x)$ be an odd degree polynomial in one variable with coefficients from the set $\newcommand{\R}{\mathbb R} \R$ of real numbers. Let $g : \R → \R$ be a bounded continuous function. Prove that there exists an $x_0 ∈ ... |
H: Question about primary decomposition in Noetherian rings
I have a question about the following proof:
How do I get that $\mathfrak a$ is reducible? I thought perhaps one can argue that $\mathfrak a \cap \mathfrak a = \mathfrak a$ is a finite intersection hence $\mathfrak a$ can't be irreducible. But this feels stu... |
H: Composition of two polynomials
How's to make the composition of two polynomials? According to this page:
If $ P = (x^3 + x) $, $ Q = (x^2 + 1) $ then,
$ P\circ Q = P\circ (x^2 + 1) = (x^2 + 1)^3 + (x^2 + 1) = x^6 + 3 x^4 + 4 x^2 + 2 $
It seems that the $ (x^3 + x) $ becomes the $x^3$, then we have $( \space \space ... |
H: How can I prove that $V$ is irreducible as a representation of $O(V)$?
Let $V$ be a finite dimentional vector space over a field $k$. Let $g(\cdot,\cdot)$ be a nondegenerate symmetric bilinear form on $V$. Let $O(V)$ be the subgroup of $GL(V)$ that preserves $g$. Then $V$ can be viewed as a reprentation of $O(V)$. ... |
H: What do level curves signify?
Suppose I have a function $z=f(x,y)$, say like $z=\sqrt{x^2+y^2}$. By fixing some value for $z$ and varying all possible $x$ and $y$, we would get a level curve of $z=f(x,y)$. By changing values for $z$, one can get different level curves. For $z=\sqrt{x^2+y^2}$, the level curves would... |
H: What exactly is the difference between a derivative and a total derivative?
I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w / \partial r$ and $\partial w / \p... |
H: Evaluating $\int_{0}^{\infty}\frac{\arctan (a\,\sin^2x)}{x^2}dx$
This is the sequel of my previous question
$$I(a)=\int_{0}^{\infty}\frac{\arctan (a\,\sin^2x)}{x^2}dx$$ I want to use differentiation under the integral sign with respect to parameter "a" but so far without success.
Any hint?
AI: Thanks for the nic... |
H: Sequences, subsequences, and continuity of functions
It's been a few years since I studied point-set topology, and I'm a bit rusty on the basics. Would appreciate help with the following question.
Suppose $f:X\rightarrow Y$ is a map between two topological spaces, and I know that for any sequence $x_n\rightarrow x$... |
H: Subgroups written as products
Suppose a finite group $G$ is the product of two of its proper subgroups $G=AB$. Assume also that $A\lhd G$ and that $A,B$ have relatively prime orders. Isn't it true that any subgroup $H$ of $G$ can be written as $H=(H\cap A)(H\cap B)$?
AI: While Qiaochu's answer settles the original ... |
H: Expectation and median (Jensen’s inequality) of spacial functions
Let’s have a 1-Lipschitz function $f:S^n \to \mathbb{R}$, where $S^n$ is equipped with the geodesic distance $d$ and with the uniform measure $\mu$.
How can I show that such an $f$ satisfies Jensen’s inequality:
$(\int_{S^n} f d\mu)^2 \leq {\int_{S^n... |
H: Estimation Theory - Maximum Likelihood Estimation
The below homework question comes from Larsen and Marx, 4th edition.
Is the maximum likelihood estimator for $\sigma^{2}$ in a normal pdf, where both $\mu$ and >$\sigma^{2}$ are unknown, asymptotically unbiased?
I think I understand the notion that an estimator $\... |
H: Why does this equation converge to 1?
The following simple equation takes in an N-length (real) vector, and spits out a (real) number between 0 and 1. (I believe this means that it is a transformation mapping $\mathfrak{R}^N \rightarrow \mathfrak{R}^1$). This equation has the property that the answer will converge ... |
H: Examples of preorders in which meets and joins do not exist
Exercise 1.2.8 (Part 1), p.8, from Categories for Types by Roy L. Crole
Definition: Let $X$ be a preordered set and $A \subseteq X$. A join of $A$, if such exists, is a least element in the set of upper bounds for $A$. A meet of $A$, if such exists, is a g... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.