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H: Universal property of sum of ideals
If $A$ is a commutative algebra over the field $K$, and $I,J$ are ideals of $A$ (in particular, they are also algebras over $K$), is it possible to characterize the ideal they generate together $I + J = \langle I,J \rangle$ in terms of a universal property among $K$-algebras?
AI:... |
H: A very simple proof of a divergent sequence
I've already proved that: if $\,0 <x< 1$, then $(x^n)\rightarrow0$. Now I'm wondering about the opposite, when $x>1$ which clearly diverges. I'd like to know if the following approach is correct or maybe I need more sophisticated arguments.
Scratchwork: Let $x>1$ and sup... |
H: Let $f(z)=e^x+ie^{2y}$. For what values of $z\in\mathbb{C}$ does $f'(z)$ exist?
Let us define the function $f:\mathbb{C}\longrightarrow\mathbb{C}$ by
$$f(z)=e^x+ie^{2y}$$
where $z=x+iy$ for $x,y\in\mathbb{R}$.
Question: For what values of $z\in\mathbb{C}$ does $f'(z)$ exist?
Attempt: If $f'(z)$ exists at some point... |
H: Combinatorial Bijection?
I have the following problem, which seems pretty easy, but I'm not sure as to what exactly is meant by a combinatorial bijection. I know what a 'normal' bijection is. The problem and my work follows beneath.
Let $H$ denote the number of ones in a binary string. Give a combinatorial bijecti... |
H: A few questions relating to counting for midterm practise exam?
I'm doing some questions for my midterm practise exam (multiple choice) for discrete structures and would appreciate some help (My answer is bolded):
Using the 26-letter alphabet {a,b,c,...,z}, how many different 20-letter strings are there that start ... |
H: Another problem about irreducible polynomials over a (finite) field
I want to know whether it is true that over a finite field $K$ (with characteristic $p$, say), and for any positive integer $m$, does there always exist a prime (or equivalently, irreducible, since the polynomial ring over a field is UFD) polynomia... |
H: about restricting outer measures
If $\mu_0$ is an outer measure on an algebra, we can extend the premeasure to an outer measure $\mu^*$. By Caratheodory's theorem, the collection of $\mu^*$-measurable sets is a $\sigma$-algebra. Is an outer measure $\mu^*$ restricted to $\mu^*$-measurable sets a measure?
AI: Yes, i... |
H: Preforming Counting Permutations
Problem: A seven-person committee composed of Alice, Ben, Connie, Dolph, Egbert, Francisco, and Galvin is to select a chairperson, secretary, and treasurer. How many selections are there in which at least one office is held by Dolph or Egbert? Each person may only hold at most one o... |
H: Integral of $1/x$ about the asymptote with Cauchy versus complex
In short, the integral of $1/x$ from $-2$ to $1$ or some other such range is being confusing for me. By means of the Cauchy principal value it receives a value of $-\ln 2$. Whereas with complex integration it would seem I'd get $-\ln 2 + \pi i$. And i... |
H: Intuition Behind The Hyperreals
I know that there are an infinite number of hyperreals. But is it true that there are only two hyperreals with standard part equal to $0$ (the "finite" infinitesimal one and the "infinite" hyperreal)?
Put differently, is it wrong to view the hyperreals as a field "generated" by $\ma... |
H: Rule for Series
I apologize if this seems really stupid, but I've been stuck in finding the general pattern for the following series:
$$\sum_{n=1}^{\infty}\frac{2\cdot4\cdot6\cdots(2n)}{1\cdot3\cdot 5 \cdots(2n-1)}$$
The numerator is simple enough, it's just $2^nn!$. But what I'm really having trouble is finding th... |
H: inequality of $L^2$ functions
Let $f\in L^2(\mathbb{R})$. Suppose $g(x)=xf(x)\in L^2(\mathbb{R})$. Prove \begin{eqnarray*}
||f||_1\leq \sqrt{2}(||f||_2+||g||_2).
\end{eqnarray*}
How to prove this question?
AI: Hint: Similar to the argument in this post, note that
$$|f(x)|\le\frac{|f(x)|+|g(x)|}{1+|x|}.$$ |
H: Creating one Set from another using Set Builder Notation
I'm a little confused about set builder notation. If I have one set, how do I construct another set from the first set, supposing that I want to alter all the elements?
For example,
Let there be a set $A = \{1,2,3,4,5\}$
and I want to construct, from $A$, a s... |
H: Propositional logic-Predicates
I have this problem in my Discrete structures course Show why : ∀x P(x) ∨∀x Q(x) is not logically equivalent to ∀x(P(x)∨Q(x)) . Please help solve this
AI: Let $P(x)$ denote the proposition that $x$ is even ($x \in \mathbb{Z}$) and $Q(x)$ denote the proposition that $x$ is odd. ... |
H: Proof of my conjecture on closed form of $\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}$
Let $a$, $b\in \Bbb R^+$ and $m \in \Bbb N$ then My conjectural closed form is $$\int _{0}^{\infty}\frac{x^{a-1}e^{-mbx}}{1-e^{-bx}}\,{\rm d}x
=
\frac{\Gamma(a)}{b^a}\left\lbrack\zeta(a)-\sum^{m-1}_{k=1}\frac{1}{k^a}\righ... |
H: Can anyone explain to me this square root? step by step?
$$\begin{align}
v(p_1, p_2, w)
& = \sqrt{\frac w{p_1^2\left(\frac1{p_1}+\frac1{p_2}\right)}}
+ \sqrt{\frac w{p_2^2\left(\frac1{p_1}+\frac1{p_2}\right)}}
\\
& = \sqrt{\frac w{\left(\frac1{p_1}+\frac1{p_2}\right)}}
\left(\sqrt{\frac1{p_1^2}}+\sqrt{\frac1{... |
H: Invariance and Orthogonal Projection
Suppose $\mathcal{S}$ is a subspace of $\mathbb{R}^n$ and let $P$ be the orthogonal projection of $\mathbb{R}^n$ onto $\mathcal{S}$.
Show that $\mathcal{S}$ is invariant under a square matrix $A$ if and only if $AP=PAP$.
Show that $\mathcal{S}$ and $\mathcal{S}^{\perp}$ are inv... |
H: Decomposition of an ideal as a product of two ideals
How to show $$5\mathbb{Z}[\sqrt[3]{2}] = (5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1).$$
Firstly, I think that I can say that $$(5, \sqrt[3]{2}+2)(5, (\sqrt[3]{2})^2+3\sqrt[3]{2}-1)= (25,5(\sqrt[3]{2}+2),5((\sqrt[3]{2})^2+3\sqrt[3]{2}-1),5((\sqrt[3]{2})^... |
H: Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
Show that if $a$ and $b$ are positive integers with $(a,b)=1$ then $(a^n, b^n) = 1$ for all positive integers n
Hi everyone, for the proof to the above question,
Can I assume that since $(a, b) = 1$, then... |
H: Jacobi fields in polar coordinates.
This is from Sakai's Riemannian Geometry:
Let $(r, \theta)$ be polar coordinates of the plane. We define a Riemannian metric $g$ on the plane by $g(\frac{\partial}{\partial r}, \frac{\partial}{\partial r}) = 1$, $g(\frac{\partial}{\partial r}, \frac{\partial}{\partial \theta}) =... |
H: Anti-homomorphism of Hopf Algebra
I've got a quick question regarding the anti-homomorphism property. Specifically, what does it actually mean??
For a bit of context, I have the following question.
We define $U_q = U_q[o(3)]$ to be the associative algebra, with generators $L_0$, $L_+$ and $L_-$, with the defining r... |
H: Subspaces of $\Bbb R^3$
Given
$U=\{(1, -1, 3)^t\}$ and $V_a=\{(x, 3x-az,z)^t\}$ for any $a\in\mathbb{R}$, how to determine $a$, such that $U \cap V_a = \{o\} \land U+V_a=\mathbb{R}^3$?
I already found out that $V_a$ always describes a plane in this 3D space, because its base is $\{(1,3,0)^t,(0,-a,1)^t\}$ ($\dim(V)=... |
H: $\int^\infty_{-\infty} \frac{1}{\pi(1+x^2)} dx = 1$. How?
$$\int^\infty_{-\infty} \frac{1}{\pi(1+x^2)} dx = 1$$. How?
I can do
$$\int^\infty_{-\infty} \frac{1}{\pi(1+x^2)} dx = \frac{1}{\pi} \int^\infty_{-\infty} \frac{d}{dx} \tan^{-1}{(x)} \; dx$$ But how do I proceed? I remember the TA mentioning something about... |
H: Show that $\frac{(m+n)!}{m!n!}$ is an integer whenever $m$ and $n$ are positive integers using Legendre's Theorem
Show that $\frac{(m+n)!}{m!n!}$ is an integer whenever $m$ and $n$ are positive integers using Legendre's Theorem.
Hi everyone, I seen similar questions on this forum and none of them really talked abou... |
H: Generalized Bernoulli's inequality
I was able to prove Bernoulli's inequality, easily by simple induction.
However, I'm not sure how to prove the generalized inequality (generalized = for each sequence of numbers $i=1,\ldots,n$):
$$\prod\limits_{i = 1}^n {(1 + {x_i})} \ge 1 + \sum\limits_{i = 1}^n {{x_i}},\qquad... |
H: Uniform convergent and lipschitz continuous
I want to prove that if I have a sequence $ f_n\in C[0,1]$ that is uniform convergent to zero and all functions are lipschitz continuous, then the lipschitz constants form a zero sequence. Does anybody here know to show this?
AI: It is not true.
If I take $f_n(x) = \frac{... |
H: In a finite field $F$, how is $a^{|F|}=a$?
Let $F$ be a finite field with $|F|=q$. If $a\in F$, I have to prove $$a^q=a$$.
We know that if $a^r=1$, then $r|q$ (treating $F$ is a group under multiplication, and using Lagrange's theorem). Hence, $a^q=a^{r\frac{q}{r}}=1^{\frac{q}{r}}=1$. This is because $\frac{q}{r}$... |
H: Defining a map based on a group action on left cosets
If $H$ is subgroup of $G$ such that the index of $H$ in $G$ is $n$ and $\pi_H$ is the permutation representation of the action of $G$ on the left cosets of $H$, is $\pi_H$ a map from $H$ to $S_n$? I am a bit confused about how $\pi_H$ relates to the symmetric gr... |
H: Let $ (x_{1}, x_{2}) \sim (y_{1}, y_{2})$ iff $x_{2} =y_{2}$ on $ \mathbb{R} ^{2}$ . Then $\mathbb{R} ^2 /\sim$ is homeomorphic to $\mathbb{R}$
Let $ (x_{1}, x_{2}) \sim (y_{1}, y_{2})$ iff $x_{2} =y_{2}$ on $ \mathbb{R} ^{2}$ . Then $\mathbb{R} ^2/\sim $ is homeomorphic to $\mathbb{R}$
I am using Willard's book a... |
H: If a subset of the real numbers has no cluster points it is countable
Prove that if a subset of the real numbers has no cluster points it is countable.
I'd just like to see how this proof goes; preferably directly, if convenient.
AI: Here is a brief outline of the contrapositive. We show that an uncountable set h... |
H: Probability theory - Dice
Two guys are playing dice with each wagering $50. Player 1 chooses 2 as his lucky number, and Player 2 chooses 6. Every time their lucky number appears as a result, the player gets one point.
The player who gets 3 points first wins $100
Suddenly, the game has to be stopped. Player 1 chal... |
H: Find all $f: \mathbb{Q} \rightarrow \mathbb{R}$ such that $f(x+y) = f(x)+f(y)$
i have to find all functions $f: \mathbb{Q} \rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$.
So functions of the form $f(x) := ax, a \in \mathbb{R}$ satisfy the above condition:
$$
f(x+y)=a(x+y)=ax+ay=f(x)+f(y)
$$
But how do i proo... |
H: sum of divisors for given range of numbers from 1 to n
we are given a function F(n) for a number n which is defined as sum of the divisors of n (including 1 and n) ... now given an integer N we have to calculate
G(n) = F(1) + F(2) + F(3) + ..... + F(n)...
is there any formula for it???
i saw "Peter Gustav Lejeune D... |
H: Are side lengths enough to find the ratio of the diagonals of a quadrilateral?
Is it possible to find the ratio of two diagonals of a quadrilateral when the length of all sides are given??
AI: If you don't know some of the angles, then no. This is not to difficult to see by drawing some pictures (or even better, be... |
H: Solve the following Diffrential Equation $(x+y+1)dx+(2x+2y-1)dy=0$
I want to seperate variables in the following equation and need some advice:
$$(x+y+1)dx+(2x+2y-1)dy=0$$
what I tried to do so far is:
$$ydx+(x+1)dx+(2y-1)dy+2xdy=0$$
now how I should I continue? thanks.
AI: Hint:
Setting $$(x+y+1)=u\longrightarrow ... |
H: Prove if $f(x) = g(x)$ for each rational number x and $f$ and $g$ are continuous, then $f = g$
$f,g: \mathbb{R} \to \mathbb{R}$
I'd like to see a sketch for this proof.
[sorry for posting errors, I am on a cell phone]
AI: Hint. For every real number $r$ there is a rational sequence $(q_n)$ such that $q_n\to r$ as $... |
H: What does $\sigma(A_n)$ look like?
Let $\Omega=\mathbb{N}$, $A_n:=\left\{\left\{1\right\},\left\{2\right\},\ldots,\left\{n\right\}\right\}$.
What does then $\sigma(A_n)$ (generated $\sigma$-Algebra) look like?
Surely, $\mathbb{N}\in\sigma(A_n)$ and $A_n\in\sigma(A_n)$. But which sets are in $\sigma(A_n)$, t... |
H: Limit of the sequence: $\frac{k(k+1)}{(k+1)^2-qk^2}$
I am stuck on the proof of this limit:
given the sequence:
$$S(k,q)=\frac{k(k+1)}{(k+1)^2-qk^2}$$
$$\lim_{k\to\infty}S(k,q)=-\frac{1}{q-1}$$
How can I prove this limit?
AI: Hint: Expand the products and squares in the nominator and denominator and multiply both w... |
H: Fermat's Little Theorem and polynomials
We know that in $F_p[y]$, $y^p-y=y(y-1)(y-2)\cdots (y-(p-1))$. Let $g(y)\in F_p[y]$. Why is it valid to set $y=g(y)$ in the above equation to obtain $g(y)^p-g(y)=g(y)(g(y)-1)\cdots (g(y)-(p-1))$. This is done in Theorem 1 of Chapter 22 of A Concrete Introduction to Higher alg... |
H: Can we simplify $\sqrt{a}*\sqrt{a}$ to $a$ when $a \in \mathbb{R}$ and we do not know whether a is positive or negative?
Can we simplify $\sqrt{a}*\sqrt{a}$ to $a$ when $a \in \mathbb{R}$ and we do not know whether a is positive or negative? (Since $\sqrt{a}$ by itself is undefined in $\mathbb{R}$ when $a$ is negat... |
H: What is the maximum amount of eigenvectors that a square matrix n x n might have?
having the following arbitrary matrix:
$A=\begin{pmatrix}
a_{1,1} & \cdots & a_{1,n}\\
\vdots & \ddots & \vdots \\
a_{n,1} & \cdots & a_{n,n}
\end{pmatrix}$
What is the maximum amount of (linearly independent) eigenvectors and eigenva... |
H: Hilbert Spaces are Reflexive
I want to show that all Hilbert spaces are reflexive. I have found the following proof on StackExchange:
Hilbert Space is reflexive
However, I do not understand it. Essentially, we want to show that for all $g \in X^{**}$, (X is some Hilbert space) there exists a unique $x \in X$ such... |
H: Banach space with cardinality bigger than $\mathfrak{c}$.
By using the information contained in this post, we have that the cardinality of every Banach space is equal to its dimension, which in turn, is bigger or equal to $\mathfrak{c}$.
In my area of study, I have always beem studying spaces like $W^{1,p}$ for $p\... |
H: Probability of having lots of unique elements
If you sample $n$ integers from the range $1$ to $n$ inclusive it seems intuitive that you are likely to get a lot of numbers exactly once. Call $X_n$ the number of integers you get that occur exactly once in your sample. Is there a nice simple way of showing the foll... |
H: Why do I see i and k as the indices of summation?
I'm working on linear algebra and just wanted to clear up an uncertainty regarding whether there is a difference in the use of i and k as the dummy variables for the index of summation?
$\sum\limits_{i=1}^\infty {i^2} = \sum\limits_{k=1}^{\infty} {k^2}$ ?
I got co... |
H: Suppose $f$ is twice differentiable function such that ...
I am stuck with the following problem :
I did integration by parts which gives the result $\,\,f'(1)$.
Can someone explain? Thanks in advance for your time.
AI: Can you think of a reason that $f\,'(1)=f\,'(0)$? |
H: Changing variable
I've problem with formulating the following problem. I guess I need to express $v(d)$ in $v(t)$ but since $d=v*t$ I can't just replace $d$ with $v*t$ since I would get $v(t) = v...$, a recursive function.
A particle moves in a straight line. The velocity of the particle ($v$) depends on the trave... |
H: Evaluate $\binom{12}0+\binom{12}2+\ldots+\binom{12}{12}$ using binomial theorem
Solve the sum:
$$
{12 \choose 0}+
{12 \choose 2}+
{12 \choose 4}+
{12 \choose 6}+
{12 \choose 8}+
{12 \choose 10}+
{12 \choose 12}
$$
using the binomial theorem.
I know the binomial theorem:
$$ \left(a+b\right)^2 = \sum_{k=0}^{n} {... |
H: Prove that $f$ is a convex function if $f=d(x,C)$ and $C$ is convex.
Question: Suppose $V$ is a normed space and $f:V \rightarrow \mathbb R$ defined as $f(x)=d(x,C)=\inf_{c \in C}||x-c||_V$ where $C$ is a convex subset of $V$. Prove $f$ is a convex function.
Attempt at a solution: (I would like to know if it is cor... |
H: Roots of a Quadratic Problem
I'm struggling with this problem and was hoping I could get some advice. Here is the problem:
Let a and b be the roots of the quadratic equation $x^2−x−1/27=0$.
Without calculating the a and b show that $a^{1/3}+b^{1/3}$ is a root of the equation $x^3+x−1=0$.
Any help would be much appr... |
H: Complete ordered field is an Archimedean field that cannot be extended to an Archimedean field
As a bonus problem, our professor of real analysis asked us to prove that the real numbers (a complete ordered field) cannot be extended into an Archimedean field, with no definition of what he meant by extending.
I have ... |
H: How do you call functions integrable over any compact subset of their domain?
I'm quite sure there is a name for such class of functions but can not remember or figure out what terms to search for. A simple and very practical example would be "periodic Lebesgue" functions: functions which are periodic—and thus, exc... |
H: Showing that $f$ and $g$ are analytic continuations of themselves.
Let $$f(z)=\sum_{n=0}^{\infty} a^n z^n$$
$$g(z)=\sum_{n=0}^{\infty} (-1)^n \frac{(1-a)^n z^n}{(1-z)^{n+1}}$$
From this, I have:
$$f(z)=\frac{1}{1-az}, |z|<\frac{1}{|a|}$$
$$g(z)=\frac{1}{1-az}, |z||a-1|<|1-z|$$
I want to show that the two regions ha... |
H: simple math: finding what percentage B is performing when compared to A
Boy A's performance is $500$ and boy B's performance is $525$ for a particular task.
How can we calculate how much percentage B is performing better than A?
AI: The percentage at which B is performing better than A is given by:
$$\dfrac{(\text{... |
H: bilinear form decomposition
Let $V$ be a vector space over a field of $\text{char} \neq 2$. Then if $g: V \times V \to F$ is a bilinear form, and $U \subseteq V$ is a subspace, do we have $V = U + U^\perp$, where $U^\perp = \{x \in V | g(x,U) = 0\}$? Of course, this is the case when $g$ is an inner product. Is it t... |
H: Explaining and using the $N$-term Taylor series for $\sin x$
So I'm given the Taylor Series expansion of the sine function and I've been asked to prove it (Done) and then construct the following by my lecturer:
Explain why the Taylor series containing $N$ terms is:
$$\sin x = \sum_{k=0}^{N-1} \frac{(-1)^k}{(2k+1... |
H: How to solve this natural logarithms problem?
How do I take natural logarithm of the following?
$A - (Be^{-xy})$
AI: If you have the equation shown below on the left hand side of $\iff$, we can do the following: $$A - Be^{-xy} = 0 \iff Be^{-xy} = A$$ Now we can take the natural logarithm of each side:
$$\begin{alig... |
H: Laurent series for $\frac{e^z}{1 - z}$ for $|z| > 1$
I do this like $\dfrac{e^z}{1-z}$ = $-e \dfrac{e^{z-1}}{z-1}$ = $-e \sum_{k=0}^{\infty} \dfrac{(z-1)^k}{k!(z-1)}$ = $-e \sum_{k=0}^{\infty} \dfrac{(z-1)^{k-1}}{k!}$
However doesn't this give me the Laurent series around $|z-1| > 0$? Should I do a taylor expansion... |
H: What is an "incongruent" solution?
For example, "Solve the congruence (if possible), listing all the incongruent solutions:"
$$561x\equiv 3575\mod{1562}$$
I found $x\equiv 37+142t,\ 0\leq t\leq 10,\ t\in\mathbb{Z}$... There are 11 "incongruent solutions" because $(561,1562)=11$ and $11\mid 3575$... but what does "i... |
H: Geometrical interpretation of $P(x) + Q(y) = 0 $ when P,Q are polynomials of degree 2?
Special cases are circles ( $ (x-x_0)^2 + (y-y_0)^2 = R $ ) and ellipses.
Is there a geometric interpretation in the general case $ ( ax^2 + bx + c ) + ( dy^2 + ey + f ) = 0$?
AI: By completing the square, you find that
$$
(ax^2... |
H: Uniform convergence of $f'_n/f_n$
The following is the proof of Hurwit'z theorm in wiki
Let f be an analytic function on an open subset of the complex plane
with a zero of order m at z0, and suppose that {fn} is a sequence of
functions converging uniformly on compact subsets to f. Fix some ρ > 0
such that f(... |
H: Show: $\limsup$ does not change when changing finite many sets
Let $(A_n)_{n\in\mathbb{N}}$ be a series of sets. Define
$$
A^+:=\limsup\limits_{n\to\infty}A_n:=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k,~~~~~A^-:=\liminf\limits_{n\to\infty}A_n:=\bigcup_{n=1}^{\infty}\bigcap_{k=n}^{\infty}A_k
$$
Show ... |
H: Ordinary generating function question
Let $a_1\le a_2\le\cdots\le a_6$ and $b_1\le b_2\le\cdots\le b_6$ be positive integers, not necessarily distinct, such that $a_1+a_2+\cdots+a_6<b_1+b_2+\cdots+b_6$. When the $36$ sums $a_i+b_j$ are computed, it turns out that $k$ appears $k-1$ times for $k=2,3,\cdots,7$ and $1... |
H: Integral of product of two measurable functions
I have to show that for $\psi$ and $\phi$ two positive and measurable functions:
$$\int_X \phi\psi \, d\mu \le \sqrt{\int_X \phi^2 \, d\mu} \sqrt{\int_X \psi^2 \, d\mu}$$
I know that for $f,g$ two measurable functions it holds that:
$$\int_X fg \, d\mu \le \frac{1}{2}... |
H: How to prove that a series is equal to a recursive algorithm
I have the following sequence:
$$
y_n = \int_0^1 \frac{x^n}{x+5}\,dx, n = 0,1,\dots
$$
Now I have the following recursive algorithm:
$$
y_0 = \log{6} - \log{5}
$$
$$
y_n = \frac{1}{n} - 5y_{n-1}, n = 1,...
$$
I tried to prove that this algorithm is equal ... |
H: Grade calculation
My teacher said that $\frac{1}{3}$rd of the final grade will be based on Exam 1, $\frac{1}{12}$th each based on Exam 2-5, and $\frac{1}{3}$rd again based on Exam 6, how will my teacher calculate my final grade?
AI: If $x_1, x_2, \dots, x_6$ are your scores on Exams $1,2,\dots,6$, then your final g... |
H: Prove that $G/N$ is Abelian
Below is a homework problem I have gotten stuck on. I would really appreciate a hint, but please do not just give the answer away.
Let $G \subset \mathcal{M}_2(\mathbb{R})$ such that each $m \in G$ is
upper triangular with a nonzero determinant. Let $N = \left\{
\bigl(\begin{smallma... |
H: Finding Integers to Satisfy a Condition
Below is the question I have to answer. I'm not sure how to attack it. Any tips?
For how many integers
$n$ between $1$ and $6 \cdot 10^6$
does there exist at least
one pair of integers $(x, y)$ such that
$xn
+ 60y
= 1$?
AI: Hint: The equation
$$xn + 60 y = 1$$ has a ... |
H: Orthonormal bases for Hilbert spaces
In Reed and Simon (Functional Analysis) Theorem II.6 states that, given an orthonormal basis $\{ x_\alpha \}_{\alpha \in A}$ (not necessarily countable)for a Hilbert space $H$, every $y \in H$ can be written as a sum
$$
\sum_{\alpha \in A} (x_{\alpha}, y) x_\alpha
$$
where $(\... |
H: How you'd show that $f$ is not continuous?
How do you show that
$$f(x)=\begin{cases}2,&x < c\\1,&x\geq c\end{cases}$$ is not continuous at $c$ by using $\epsilon$-$\delta$ - formalism like here?
AI: Suppose $f$ were continuous. Let $\epsilon=0.1$, and $\delta$ be chosen so that $|f(x)-f(c)|<\epsilon$ for all $|x-... |
H: Is it ok to prove a subset of a group is an abelian group this way?
I'll admit from the start I'm being lazy, but all the same it makes thing's neater in my opinion - if it's valid.
Now it's known that if we have a group $G$ such that $g^2=e,\ \ \forall g \in G$ then $G$ is abelian. But what if we don't know G is a... |
H: How to solve this probability exercise?
We have a box and we have on it 6 balls with numbers from 0 to 5. We push out 3 balls in the way that after pushing out a ball we turn it back again in the box. What is the probability that the sum of the numbers will be equal to 5. Thanks in advance.
AI: Hint: how many choi... |
H: Number of points on $Y^2 = X^3 + A$ over $\mathbb{F}_p$
Let
$p\equiv 2\pmod{3}$ be prime and let $A\in\mathbb{F}^{∗}_p$
. Show that the number of points
(including the point at infinity) on the curve
$Y^2
=
X^
3
+
A$
over
$\mathbb{F}_
p$
is exactly
$p
+ 1$
I'm having trouble bring in the fact that $p
\equiv 2
\pmod... |
H: Dense curve on torus not an embedded submanifold
In reference to Showing a subset of the torus is dense, the responders helped show the poster that the image set $f(\mathbb{R})$ is dense in the torus. But, it's not immediately clear to me why the image set is not an embedded submanifold. If $f(\mathbb{R})$ is an em... |
H: $\sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x)$
Let $(X,d)$ be a metric space and $F : X \rightarrow [0, +\infty)$ a lower semicontinuous function. Then
$$ \sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x). $$
Is this true?
Intuitively it works since the increasing distance increases the sum of the two functions everyw... |
H: Is $\{(x,y)\in \Bbb R^2: x^2+4y^2+6x+4y+12=0\}$ the empty set?
I am attempting the following problem and can someone verify it ?
We see that $\,x^2+4y^2+6x+4y+12=0 \implies (x+3)^2+4(y+\frac 12)^2=-2$ and hence option (d) is the correct choice. Am I right?
AI: Yes, that is correct and a correct technique. |
H: Irreducibility of Polynomial in $\mathbb{Q}$
How can we show that, if $a>1$ is the product of distinct primes, then $x^n-a$ is irreducible in $\mathbb{Q}$ for all $n \geq 2$ and that it has no repeated roots in any extension of $\mathbb{Q}$?
Thoughts:
Perhaps we can do an inductive proof on $x^{k+2}-a$ as the $k=0$... |
H: Question regarding polygons
Can you prove, that if a equilateral lattice n-gon is constructible, then there will be such a polygon for which the sides have minimal length?
AI: The set of possible side lengths is exactly the set $S$ of all positive numbers of the form $\sqrt{a^2+b^2}$ for integers $a$ and $b$. In i... |
H: $A \oplus B = A \oplus C$ imply $B = C$?
I don't quite yet understand how $\oplus$ (xor) works yet. I know that fundamentally in terms of truth tables it means only 1 value(p or q) can be true, but not both.
But when it comes to solving problems with them or proving equalities I have no idea how to use $\oplus$.
Fo... |
H: Why the binary division of the two has the same quotent
I am trying to divide 110000 with 1101 in binary (long division). I use XOR each time. The divisions above should have quotent 100. However I might must misunderstood something. Could please anyone help me with that?
basically I am doing the following
--... |
H: partioning of $X$ such that $f_n$ converges uniformly to $f$
Let a measure be $\sigma$-finite and suppose $f_n \rightarrow f$ a.e.. Show that there exists $E_k$ and a null-set $F$ partioning $X$ such that $f_n$ converges uniformly to $f$ on each $E_k$.
I was trying to deduce it from Egorov's Theorem http://mathworl... |
H: Formula for the length of line that connects two sides of a triangle.
For the triangle in the picture, coordinates of $A$, $B$ and $C$ are known. Is there an explicit formula for length $XY$, as a function of height $h$? It's a function of other variables as well, but it's important that it depends explicitly on $h... |
H: show that torus is compact
I am having difficuties in showing a torus is compact. Initially I wanted to use Heine-Borel theorem, but after that I realise we are not working in $\mathbb{R}^n$ space. So a simple way to show torus is compact is by definition. But after defining an open cover for torus, I don't know ho... |
H: How to go about calculating this finite summation?
I have the summation
$$
\sum_{n = 0}^{10} \frac {1 + (-1)^n} {2^n}
$$
I have looked up how to work out sequences without manually finding each term and adding it up, but I have only found out how to work out problems like $n^2$, $n^3$, etc.
How would I solve this?... |
H: Proving a function is constant, under certain conditions?
The problem:
Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant.
I believe I have some intuition about why this is the case; i.e. if $t$ and $x$ are very close, ($|t - x| = \epsilon$),... |
H: Proof of orthogonal matrix property: $A^{-1} = A^t$
I have prooved this orthogonal property. Please correct it or show your version of the proof if I am wrong:
$A^{-1} = A^t$
$A^{-1} \times A = A^t \times A$
$I = I$
I appreciate your answer.
AI: There are two main definitions of orthogonality. Accepting one you can... |
H: Trigonometric inequality solving
How to solve this inequality $\left|\dfrac{\cos 2x + 3}{\cos x}\right|\geq 4$ ?
I tried to consider 2 cases:
1) When $\cos 2x \geq 0$ and $0<\cos x<1$
2) $\cos 2x\leq 0$ and $-1 < \cos x < 0$.
But I think that's wrong.
AI: As $\cos2x=2\cos^2x-1$
$$\frac{\cos2x+3}{\cos x}=\frac{2\co... |
H: Why does knowing where two adjacent vertices of regular $n$-gon move under rigid motion determines the motion?
I am reading the book Abstract Algebra by Dummit and Foote.
In the section about the group $D_{2n}$ (of order $2n$) the authors
claim that knowing where two adjacent vertices move to, completely
determine ... |
H: Norm space, linear operator exercise, help please!
$f \in L_2[a,b]$
$Uf(s):=\int_a^bk(s,t)f(t)dt$
$k(s,t):[a,b]^2\to R $ continuous.
show
1) $U:L_2[a,b]\to L_2[a,b]$, in other words, $Uf(s)\in L_2[a,b] \quad \forall f$
2) $U$ is linear and continuous
3) $||U||\leq ||k(s,t)||_2$
AI: Here are some hints:
Let $g_s... |
H: Category for measure spaces?
I know some things about measures/probabilities and I know some things about categories. Shortly I realized that uptil now I have never encountered something as a category of measure spaces. It seems quite likely to me that something like that can be constructed. I am an amateur however... |
H: Limit of $x_n/n$ for sequences of the form $x_{n+1}=x_n+1/x_n^p$
Given $x_1 = 1, x_{n+1} = x_n + \frac{1}{x_n} (n\ge1)$, Prove whether the limit as follow exist or not. If so, find it
$$\lim_{n\to\infty}\frac{x_n}{n}$$
Given $x_1 = 1, x_{n+1} = x_n + \frac{1}{\sqrt{x_n}} (n\ge1)$, Prove whether the limit as follow... |
H: Show that $f(X)$ has a dense subset.
Let $X$ be a topological space which has a dense countable subset $D$, and suppose $f\colon X \to Y$ a continuous function. Show that $f(D)$ is dense countable in $f(X)$.
AI: I see that you've decided that $f(D)$ is a good candidate. (Right on target with that one.) Since $f$ ma... |
H: $n\times n$ matrix determinant of rook configuration
We place $n$ rooks on an $n\times n$ chessboard in such a way that they don't threaten each other. To each such placement corresponds an $n\times n$ matrix in which there is a $1$ at the position of the rooks and $0$ at the other places. What is the determinant o... |
H: Is there a definition of cylinder that these equations satisfy
Our teacher is claiming that (in $\mathbb{R}^3$) the following surfaces are "cylinders":
$3x+y+\frac{7}{2}=0$
$y=x^2$
$z^2 = y$
$\frac{x^2}{4} + \frac{y^2}{4} = 1$
Is there any definition of cylinder that can justify this statement, and if so, which d... |
H: Mapping behavior of imaginary axis via $v=\frac{z-a}{z+a}$
I would like to know what the bilinear transform $v=\frac{z-a}{z+a}$ does to the imaginary axis, where $a$ is a real number.
I substituted $z=yi$ and calculated $|v|$ giving me $|v| =1$.
Is this enough proof to say that $v$ maps the imaginary axis to on the... |
H: laws of probability
Suppose that there is a $60\%$ probability that the product will be a success on the market (that means, the probability of failure is $40\%$). If the product is a success, you will get a profit of $\$200,000$, and if it is a failure, you will incur a loss of $\$100,000$. Should you develop this... |
H: How to find non-isomorphic trees?
"Draw all non-isomorphic trees with 5 vertices."
I have searched the web and found many examples of the non-isomorphic trees with 5 vertices, but I can't figure out how they have come to their answer. How exactly do you find how many non-isomorphic trees there are and what they lo... |
H: Laurent expansion of $f (z) = \frac1{z(z − 1)(z − 2)},$ (in powers of $z$) for $0 < |z| < 1,$ $1 < |z| < 2,$ and $|z| > 2.$
Find the Laurent expansion of $$f (z) = \frac{1}{z(z-1)(z-2)}$$
(in powers of $z$) for
a. $0 < |z| < 1$
b. $1 < |z| < 2$
c. $|z| > 2$
AI: The first thing you'll want to do, here, is what is kn... |
H: New ways to light the fire again
Recently I've been studying a lot of analytic geometry and this subject made my motivation drop. The thing is, the courses aren't stopping and I'm beginning to lose the passion I had before. I need ideas on how to "light the fire" again. Help would be much appreciated.
AI: "Shake it... |
H: Congruences with prime number and factorial
Prove that if $p\equiv 1 \pmod{4}$ is a prime number and $$x\equiv \pm \left(\frac{p-1}{2}\right)! \pmod{p}$$
then $x^2\equiv -1 \pmod{p}$
I think Wilson's theorem will come in handy here, used in some clever way, but I can't see it. I would be very grateful for any h... |
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