text stringlengths 83 79.5k |
|---|
H: How do I find the base angles without a vertex angle in a isosceles triangle?
How can I find the base angles in a isosceles triangle if the vertex angle is missing?
Normally, I would go: 2x + vertex angle = 180, but now even the vertex is missing, the only thing I have is a line in the middle, with 90 degree angle.... |
H: Unary minus on squared number
According to algebra as I know it, $-2^2 = 4$, but most calculators expand this to $-2 * 2 = -4$, which yields a different answer. This is because of the order of precedence. In traditional math, I've seen that the unary operator is applied before exponents. This is not the case for... |
H: Integer division
I think I found a mistake in the princeton review "Cracking the GRE" 2014 edition on page 408. The problem is as follows:
If $\frac{13!}{2^x}$ is an integer, which of the following represents all possible values of $x$?
a.) $0\le x\le10$
b.) $0\lt x\lt9$
c.) $0\le x\lt10$
d.) $1\le x\le10$
e.) $1\l... |
H: conditional probability of dependent events
I posted another question earlier but i can't modify it so please bare with me.
I have a problem where i need to find it's conditional probability: probability of Pc(B)=(B|C) where the event of B={no two people are born in the same month} and event C={exactly three people... |
H: When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$
When is $f_{xy}(x,y)\neq f_{yx}(x,y)?$, where $f_{xy}$ and $f_{yx}$ denote the mixed (second) partial derivatives of a multivariable function $z=f(x,y)$.
AI: It's worth studying the following results:
Schwartz Theorem or Young Theorem or Clairault Theorem.
And see examples wh... |
H: Dual of this primal optimization problem?
How would one find the dual of the following problem?
$min_x 1/2 ||y-x||_2^2 + \lambda||x||_1 $
Can someone please explain to me how to do this since there are no specific constraints?
AI: You could reformulate your problem as
\begin{align*}
\operatorname*{minimize}_{x,u} ... |
H: From a Hilbert space to another Hilber space, is a norm preserving invertible transformation also unitary?
Given two Hilber spaces - $H_1, H_2$ and a transformation $T:H_1 \to H_2$ that is norm preserving and invertable, does this imply that $T$ is also unitary transformation, namely that it preserves the inner pr... |
H: Prove uniform converging for $\{f_n(x)\} = x^n$,
The definition says that:
Suppose $D \subseteq R$.
$\{f_n(x)\}$ uniform converging to $f(x)$ in $D$ if any $\epsilon > 0$ , $\exists N(\epsilon)$, such that for any $n > N(\epsilon)$ and any $x \in D$:
$|f_n(x) - f(x)| < \epsilon$
I need to prove uniform convergen... |
H: The empty function and constants
On Wikipedia and also after searching this forum there is stated that for each $A$ there exists a unique function $f : \emptyset \to A$ called the empty function for $A$.
In the theory of Algebraic Data Types (ADT's) and in functional programming languages (like Haskell) where one p... |
H: How to show that a given set is a vector space?
I am having some issues with this problem in my Linear Algebra textbook. The goal is to either show that the given set, $W$, is a vector space, or to find a specific example to the contrary:
\begin{Bmatrix}
\begin{bmatrix}
a\\
b\\
c\\
d
\end{bmatrix} :
\begin{matri... |
H: What could be the lead to prove $||X||_2 \leq ||X||_F \leq \sqrt{rank(X)}||X||_2$?
In the above statement, $||X||_2$ = $L_2$ norm of X and $||X||_F$ = $Frobenius$ norm of X.
It appears to me that the $L2$ norm of X and $Frobenius$ norm of X are the same. How should i proceed to prove the above statement?
AI: Consid... |
H: Spectrum of a ring is irreducible if and only if nilradical is prime (Atiyah-Macdonald, Exercise 1.19)
Can anyone help me with this exercise, please?
A topological space $X$ is said to be irreducible if $X\neq\emptyset$ and if every pair of non-empty open sets in $X$ intersect, or equivalently, if every non-empty ... |
H: Can the inverse of a function be the same as the original function?
I was wondering if the inverse of a function can be the same function.
For example when I try to invert
$g(x) = 2 - x$
The inverse seems to be the same function. Am I doing something wrong here?
AI: You're correct. A function that's its own inverse... |
H: Computing a formula for $\partial^2 f/\partial{v}\partial{w}$
Let $f(x,y,z)$ be of class $C^2$. Putting $x = u + v - w$, $y = 2u - 3v$, $z = v + 2w$ makes f into a function of u, v, and w. Compute a formula for $$\frac{\partial^2f}{\partial{v}\partial{w}}$$ in terms of the partial derivatives of f with respect to x... |
H: Convergence of "sliced" power series
Let $\phi(t)=\sum_{k=1}^\infty a_k t^k$, $x=t^m \in \mathbb{C}$ for some fixed $m\in \mathbb{N}$ be a convergent power series. I guess that $a_0=0$. For $r=0,\ldots,m-1$ and $k=mq+r$, why are the power series
$$\tilde\phi_r(x) = \sum_{q=0}^\infty a_{mq+r}x^q$$
convergent?
The is... |
H: What is the easiest way to solve the following linear system
What is the easiest way to solve the following linear system for $a,b,c,d,e$ in terms of $h$?
I want to do this quickly.
Are matrices the way to go?
$a+b+c+d+e=0, (-a+c+2d+3e)h=1, a+c+4d+9e=0, -a+c+8d+27e=0, a+c+16d+81e=0$
Thanks!
AI: Yeah: Gaussian elim... |
H: Establishing an Upper Bound for a Curious Function
Suppose I have a sequence of positive real numbers $a_1, \, a_2, \, \dots \,a_n$ such that the following is satisfied:
$\sum \limits_{i=1}^{n}a_i = 1$
I am trying to find the smallest value of $L$ such that the following is also true, for any arbitrary sequence:
$L... |
H: find infimum of $(n+1)^2 / 2^n$ (using Bernoulli's inequality)
What is the infimum for:
$$(n+1)^2 / 2^n$$
I've tried to simplify $2^n$ with Bernoulli's inequality $((1+1)^n \ge (1+n))$
but it didn't work out..
AI: Hint: $$\frac{(n+2)^2}{2^{n+1}}\cdot\frac{2^n}{(n+1)^2} \to \frac12 < 1.$$ |
H: Residue of a complex function at some pole.
How can one visualize residue of a complex valued function at some given pole? I know how to find it. but I want to know its significance and its geometric nature. Why do we study it? thank you.
AI: If you take a complex function $f(z)$, we call $\bar{f(z)}$ the polya ve... |
H: When does $f(a),f(f(a)),f(f(f(a)))...$ produce better and better approximations to $x=f(x)$?
I tried to approximate the solution to $x=f(x)$ for some given $f$, by guessing $x=a$, then I observed that $x=f(a)$ was an even better approximation, and $x=f(f(a))$ and so on was even better, so why does this method work ... |
H: How can I prove that this function is uniformly continuous?
How to show that $f(x)=\frac{x}{1+|x|}$ is uniformly continuous?
Thank you. Also, how do I become good at writing these proofs?
AI: You can show that it is differentiable, and has bounded derivative. Then use Lagrange's mean value theorem. |
H: Subgroup $\{(1),(12)\}$ in $S_3$ is not kernel of any homomorphism
How do I prove the following statement
Subgroup $\{(1),(12)\}$ in $S_3$ is not kernel of any homomorphism.
AI: $(1\,2\,3)^{-1}\circ (1\,2)\circ (1\,2\,3)=(1\,3)$ is not in this subgroup, hence it is not normal. |
H: Pointwise lower bound for holomorphic functions on the unit disk
I am looking at Problem 8C in August 2012 complex analysis qualifying exam from University of Wisconsin-Madison (direct link to the file).
Let $f$ be a holomorphic function on the unit disc $\mathbb{D}$. Fix
$z_0\in\mathbb{D}$. Suppose that $f(0) ... |
H: Limit of sequences.
Could someone tell me how one could work out the limit of the following:
1) Ax= (1.001)^x) / (x^1000) as x->∞
2.) Bx=( 1^100 +2^100+......n^100) ^1/n as n->∞ .
I know the final answers but I can't really show how do you get there step by step. Any help?
AI: Hint:
$$1\le\left(1^{100}+2^{100}+\ldo... |
H: Sampling with replacement events vs. probability of coverage
I have a deck of $N$ cards, when $k \leq N$ of the cards bear a mark. I sample from the deck uniformly and with replacement until I find a marked card. I then erase the mark, and place the card back in the deck. The expectation for the number of sampli... |
H: polynomial ring over finite field
Can someone provide a proof for the following? I wrote a one page proof. I must be doing something wrong. There has to be a quicker way to prove this. Thank you ahead of time. I will highly rate for your help.
AI: I'm curious what approach you took in your proof, as this would help... |
H: Is $\{w \in \Sigma ^* : |w|_2 mod 4 = 2\}$ a regular language?
Given alphabet $\Sigma = \{1,2,3\}$, is $\{w \in \Sigma^* : |w|_2 \bmod 4 = 2\}$ a regular language?
I tried so hard on finding a regular expression but couldn't...
AI: /([13]*2[13]*2[13]*2[13]*2)*[13]*2[13]*2[13]*/ |
H: Residue of $\frac{1}{(e^z-e)^3}$ at $z = 1$
I'm trying to calculate the residue of $\dfrac{1}{(e^z-e)^3}$ at $z = 1$. The answer is $\dfrac{1}{e^3}$, but having trouble seeing how one would arrive at that. Any hints?
AI: If $z=1+u$ and $u\to0$, then $f(z)=(\mathrm e^z-\mathrm e)^{-3}=\mathrm e^{-3}(\mathrm e^u-1)^{... |
H: Find the coordinates of intersection of a line and a circle
There is a circle with a radius of $25$ ft and origin at $(0, 0)$ and a line segment from (0, -31) to (-37, 8). Find the intersections of the line and circle.
I am asking for somebody to analyze what I am doing wrong in calculating the answer, given the... |
H: Using Wolfram Alpha to solve a system of linear equations
How do I input the below system of equations in Wolfram Alpha in order to solve for the unknowns?
I'm wondering if there's some kind of code that can be written in order to make wolfram alpha understand what I'm talking about.
$$\left.\begin{matrix}
a+b+c+... |
H: Where have you seen this topological ring used?
Let $R$ be a ring and define the topology on $R$ to be that where the open sets are unions of ideals. Then this forms a topological ring - let me know if you need proof. Have you seen it used anywhere or what can you do with it now that you know it's a topological r... |
H: Which method is the most practical for this volume problem?
Find the volume of the solid generated by revolving the region bounded by
$$ y = \frac{3}{\sqrt{x^2 + 9}}, x = 4, y-axis $$
Rotated about the y-axis.
After drawing the graph, I wanted to simply take the integral in regards to y and do the disk method but ... |
H: Multiple integral over General Domains
This is a rather general question but I am having trouble conceptualizing the integral.
Say that we for the function $f(x,y)$ we integrate $y$ over $v(x)$ and $u(x)$ and $x$ over an interval of constants $\left[a,b \right]$.
Then what exactly does the integral measure? Since... |
H: What is the order of this pole at $\pi/2$?
Consider:
$$f(z)=\frac{\cos(z)}{(z-\pi/2)^4}$$ at $z = \pi/2$.
What is the order of the pole? How can I see this? I keep getting $2$, but my textbook says that the answer is $3$.
AI: $$
\frac{\cos z}{(z-\pi/2)^4}=\frac{-(z-\pi/2)+\frac1{3!}(z-\pi/2)^3+O((z-\pi/2)^5)}{(z-... |
H: When is the formula for the infinite geometric series valid
When is the formula $$S_{\infty} = \dfrac{a}{1-r}$$ valid?
Does |$r| <1$?
AI: Since I found this very useful when learning geometric series, I'll show you this to enhance your understanding of the formula:
Given a finite geometric sequence, let's say exem... |
H: Probability of finding empty seats
Supposed we are using a round table with 10 seats, there are 7 people who pick their seats randomly. you and your friend are late and you two want to have 2 seats next to each other. Another person is also late and he will arrive before you.
One friend call, he assures that there... |
H: Convergence of an infinite series of conjugated complex numbers
Take the closed form of this Hadamard/Weierstrass product:
$$\displaystyle \frac{\sinh(\pi s)}{\pi s} := \prod_{n=1}^\infty \left(1- \frac{s}{0 + n i} \right) \left(1- \frac{s}{{0 - n i}} \right)$$
Easy to see that the complex zeros occur in conjugated... |
H: How to show $\Bbb R$ is Archimedean?
Suppose $X$ is a real number such that $X > 0$. We want to show there exists and $n \in \mathbb{N}$ such that $X \geq \frac{1}{n} $.
MY attempt: If $X < \frac{1}{n} \; \; \; \forall n $ then $X \leq 0 $ by passing to the limit. Contradiction. Can someone show me a way to show th... |
H: If this warped coin is flipped 5 times, what is the probability that more heads than tails occur?
A warped coin has probability of 0.5 of landing Heads, probability of 0.4 of landing Tails, and probability 0.1 of landing on its Edge. It is flipped 5 times. What is the probability that more Heads occur than Tails?
A... |
H: finding determinant as an function in given matrix
Calculate the determinant of the following matrix as an explicit
function of $x$. (It is a polynomial in $x$. You are asked to find
all the coefficients.)
\begin{bmatrix}1 & x & x^{2} & x^{3} & x^{4}\\
x^{5} & x^{6} & x^{7} & x^{8} & x^{9}\\
0 & 0 & 0 & x^{10} & x^... |
H: Can check my arithmetic/work on this arc length problem?
Find the length of the curve $y^2 = x^3$ from (0,0) to $(\frac{1}{4}, \frac{1}{8})$.
I know that the arc length formula for rectangular problems is:
$$ \int_a^b \sqrt{1 + (\frac{dy}{dx})^2} $$
So, to get a, I subtract $\frac{1}{4}$ from 0. To get b, I subtrac... |
H: How to deduce that $\gcd(a, b) = \gcd(a, a+b)$ given that the common divisors of $a$ and $b$ are the same as the common divisors of $a$ and $a+b$?
This homework question asks to first prove that the common divisors of $a$ and $b$ are the same as the common divisors of $a$ and $a+b$, which I have done, and then dedu... |
H: who has the higher probability to buy pizza
Supposed that A, B, C are given a bunch of homework (N). They decided to share the homework and work individually on each part. A will take 20%, B take 30% and C take 50% of the homework (N).
Because A is a good student so he will make 1% error (doing it wrong) in the ho... |
H: Basis of a subspace of all continuous real-valued functions
This is a homework problem I have and I have the right answer, but I don't know how to do it in proof form . . .
$W$ is the subspace of all continuous real functions spanned by $\{\cos^2(t),\ \sin^2(t),\ \cos(2t)\}$. Find a basis for $W$. I know the given... |
H: Use the definition of continuity to prove a function is defined at every nonnegative real number
Use the definition of continuity to prove that the function f defined by f(x) = (x)^(½) is continuous at every nonnegative real number.
Our definition of continuity: Let I be an interval, let f:I-R, and let c(element of... |
H: $\int_A f dm \leq 0 $ for all $A$ lebesgue measurable implies $f \leq 0 $ a.e
$$ \textbf{Problem} $$ $\int_A f dm \leq 0 $ for all $A$ lebesgue
measurable set implies $f \leq 0 $ a.e
$$ \textbf{Solution (Attempt)} $$
We want to show $X = \{ x : f > 0 \} $ is a null set, that is $m(X) = 0 $. Consider, $X_n = \{... |
H: Order of modular group
Prove $|(\mathbb{Z} / p^e \mathbb{Z} )^{\times}| = p^e - p^{e-1}$
I know it has something to do with the fact that we have $p^e$ elements and we're substracting $p^{e-1}$ multiples of $p$, but I'd like to know how to formally prove this.
AI: Let $a \in \mathbb{Z}$. First show that in $\mathbb... |
H: About counting number of n-tuples
Let n-tuples be $(x_1,x_2,x_3,...x_n)$ and $0\le x_i<q$ ($x_i$ is integers) for $i=1,2,3,...,n$.
First part of the question was about the number of n-tuples. I got this part right, (number of n-tuples)$=q^n$
But for the second part, it is asking the number of n-tuples considering t... |
H: Vector in a linearly dependent set is a linear combination of other vectors in that set?
I have a true/false HW problem:
If $S$ = {$v_{1}$, $v_{2}$, . . . $v_{n}$} is linearly dependent then any $v_{k}$, inf ≥ k ≥ 1, is a combination of the other vectors in $S$.
The setup I did is:
$v_{k}$ = -($a_1/a_k$)$v_1$ - ($a... |
H: Taylor polynomial about the origin
Find the 3rd degree Taylor polynomial about the origin of $$f(x,y)=\sin (x)\ln(1+y)$$
So I used this formula to calculate it $$p=f(0,0)+(f_x(0,0)x+f_y(0,0)y)+(\frac{1}{2}f_{xx}(0,0)x^2+f_{xy}(0,0)xy+\frac{1}{2}f_{yy}(0,0)y^2)+(\frac{1}{6}f_{xxx}(0,0)x^3+\frac{1}{2}f_{xxy}(0,0)x^2... |
H: Help needed with first-order logic representation
I'm very new to first-order logic. I've been working on some tasks below, and would appreciate if somone could check if I have understood and solved the questions correctly
Task:
Assume that $B$, $F$ and $K$ are relational symbols so that
$Bx$ interpreted as "$x$ ... |
H: Subgroups containing kernel of group morphism to an abelian group are normal.
Let $\varphi:G\rightarrow H$ be a group homomorphism from group $G$ to group $H$. Show that, if $H$ is abelian, all subgroups of $G$ that contain $\mathrm{ker} (\varphi)$ are normal in $G$.
AI: Hint: $G / \ker \varphi$ is abelian. Hence $... |
H: $\lim_{(x,y)\to(0,0)} \frac{xy}{\sqrt{x^2+y^2}}$
Let me be honest, when some teacher ask me to calculate such limit, I always try to found two paths with different results to show that such limit don't exist.
But, I try four paths all fall at zero... and wolframalpha (internet boss (after math.stackexchange.com)) t... |
H: Is the intersection of a sequence of nested subspaces nonempty?
Say $X$ is a topological space (compact). if $\{ A_n \} $ is a collection of nonempty closed subsets of $X$ such that $A_{n+1} \subseteq A_n $ for all $n$, then does it follow that $ \bigcap_n A_n $ is non-empty??
My try: Since $X$ is compact topologic... |
H: $(0 \leq f' \leq f$ on $\mathbb R$ and $f(a)=0)$ $\implies f=0$?
Given that $0 \leq f' \leq f$ on $\mathbb R$ and $f(a)=0\in\mathbb R$ for some $a\in\mathbb R$, how do I prove that $f$ is identically zero? Using the mean value theorem naively didn't really get me anywhere.
AI: Since $f'\ge 0$, then
$$
f(x)=f(a)... |
H: Show that there is no invertible continuous function $f:[0,1) \to (0,1)$
This is a problem I came across recently and have been trying to figure out, but I'm not having a lot of luck!
By considering the restriction of $f$ to $(0,1)$ or otherwise, show that there is no invertible continuous function $f:[0,1) \to (0,... |
H: What can be the value of $m$ in following equation
During calculations I got this step
$$(e^m/((m+1)^{m+1}) )^{3n/4} = 1/2^n$$
I want the value of m here??
AI: Well, you can eliminate $n$ by raising both sides to $4/(3n)$:
$$\frac{e^m}{(m+1)^{m+1}} = \frac{1}{2^{4/3}}.$$
I don't think there's much hope of a closed-... |
H: Using the Squeeze Theorem in Sequences
My textbook has an example that says "Show that the sequence {${c_n}$} $= (-1)^n \frac{1}{n!} $ " converges, and find its limit.
It tells me that I must "find two convergent sequences that can be related to the given sequence" which the textbook states that the two possibilit... |
H: Show that $C[a,b]$ is a complete space under the metric $d(f,g)=\sup_{t\in [a,b]}|f(t)-g(t)|$.
$C[a,b]$ is a normed vector space of all continuous complex valued functions on $[a,b]$, with supremum norm
$$\|f\|_\infty=\sup_{t\in [a,b]}|f(t)|.$$
The metric induced by the norm is
$$d(f,g)=\|f-g\|_\infty = \sup_{t\i... |
H: $A$ is dense iff there is not open subset nonempty in $X \setminus A$
$A$ is dense iff there is not open subset nonempty in $X \setminus A$. Let $X$ be topological space.
my try: If $A$ is dense then $X = Cl(A) $. we can assume there is an open $O$ such that $O \subseteq X \setminus A $. My question is : Can we a... |
H: Non-trivial open dense subset of $\mathbb{R}$.
I recently found the following exercise real analysis:
Let $A\subseteq\mathbb{R}$ be open and dense. Show that
$$\mathbb{R}=\{x+y:x,y\in A\}$$
I think it is not too hard to prove. But do we have a non-trivial example of such a set? So my question is:
Can we find a... |
H: Convergence of $\sum_{n=1}^\infty 1/(n - c)^2$
I'm curious as to whether or not the following series converges,
$$\sum_{n=1}^\infty \frac{1}{(n - c)^2},$$
where $c$ is some positive constant, $c \notin \mathbb{Z}_{>0}$.
Initially my intuition lead me to believe that it did converge since, for large enough $n$, o... |
H: $|G| = pqr$ with $p$, $q$ and $r$ distinct primes. Show G is not simple.
$|G| = pqr$ with $p$, $q$ and $r$ distinct primes. Show G is not simple. I know this might have been asked and answered before. I just wanted someone to tell me if my argument is OK:
Let $|G| = pqr$, and assume $p < q < r$. We have at least on... |
H: spanning over vectors, but this span has only 1 vector
I have seen a problem that asks the following:
Is the following a subspace:
$$H = \text{span}\left\{\left[\begin{array}{c}2\\0\\3\\4\end{array}\right]\right\}$$
I know the answer is yes it is a subspace, because a SPAN is a subspace by a corollary and I have ev... |
H: Jordan Normal Form and eigenvalue 0
I understand the processes of putting a matrix into Jordan normal form and forming the transformation matrix associated to "diagonalizing" the matrix. So here's my question:
Why is it that when you have an eigenvalue x=0 with algebraic multiplicity greater than 1, that you don't ... |
H: The size of $\sigma$-field generated by finite sets
If we have a sequence of sets $F_i$ $i=1,2,...,k$ and $F_i \subset F_{i+1}$. what is the size of the $\sigma$-field, $\sigma(F_i, i=1,2,..,k)$? I guess the size is $2^{2^{k}-1}$ but I can't prove it. How can I prove this result?
AI: Assuming inclusions are proper,... |
H: Critical Values of a Function
I need to find the critical values of $h(t) = t^{3/4} - 2t^{1/4}.$
So I began by finding the derivative of the function and simplifying:
\begin{align*}
h'(t) &= (3/4)t^{-1/4} - (2/4)t^{-3/4} \\
&= \frac{3}{4t^{1/4}} - \frac{2}{4t^{3/4}}\\
&= \frac{3t^{3/4} - 2t^{1/4}}{4t}
\end{align*}
... |
H: Differentiability of Linear Maps
I am wondering whether all linear mappings have first-order partial derivatives (or stronger properties such as being continuously differentiable at all orders). Formally, suppose $A$ is an $m \times n$ matrix and define the mapping $F: \mathbb{R}^{n} \to \mathbb{R}^{m}$ by $$F(x) =... |
H: Chance of marrying a girl
My girlfriend's father has a magic - fair - coin, he agrees to let me marry his daughter if I play his game: I have to toss the coin couple times until I see the head comes up. Then if the number of times I have tossed is divisible by three, I cannot marry his daughter, otherwise I can mar... |
H: Limit point compactness implies sequential compactness
I am trying to go through the proof of: Suppose $ X $ is metrizable space. If $X$ is limit point compact, then $X$ is sequentially compact.
Proof:
Let $(x_n)$ where $n\in \mathbb{N_0}$ be a sequence of points in $X$. We need to find a convergent subsequence. ... |
H: Please help on this matrix transformation problem
Let A be the matrix below and define a transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ by $T(U) = AU.$ For each of the vectors $B$ below, find a vector $U$ such that $T$ maps $U$ to $B$, if possible. Otherwise state that there is no such $U$.
$$
\begin{pma... |
H: Show the sequence ${n^m + 1}\over {n^{m+1} + 1}$ with $m \in \mathbb{R}$ is decreasing
Consider the sequence $\{a_n\}_1^\infty$ such that $ a_n = $ ${n^m + 1}\over {n^{m+1} + 1}$ and $m \in \mathbb{R}$
EDIT: This is incorrect for $m < -1$, Then add the condition $m \geq -1$
I want to show this sequence is monotoni... |
H: $f : X \to Y $ continuous and surjective. $A $ dense in $X$ $\implies$ $f(A)$ dense in $Y$
$$ \textbf{PROBLEM} $$
$f : X \to Y $ continuous and surjective. $A $ dense in $X$ $\implies$
$f(A)$ dense in $Y$
$$ \textbf{SOLUTION(ATTEMPT)} $$
Suppose $A$ is dense in $X$. Then we must have by definition that $Cl(A) =... |
H: legendre's formula in number theory
Prove that $(k!)^{(k-1)!}$ divides $(k!)!$ (k is a non-zero positive integer). I Know Legendre's formula for counting p's power in $n!$ for any prime p,but here we have $(k!)!$ and can't establish a useful formula for that to help prove this problem.
I would be grateful for your... |
H: Why is Wolfram giving me a different answer for standard deviation?
I have the following set of data:
Raw Scores x-x̄ (x-x̄)²
-----------------------------
7 -6 36
8 -5 25
10 -3 9
14 1 1
26 13 169
-... |
H: A function can be extended to a uniformly continuous function
Let $S$ be a subset of the metric space $E$ with the property that
each point of $S^c$ is a cluster point of $S.$ Let $E'$ be a complete
metric space and $f: S\to E'$ a uniformly continuous function. Prove
that $f$ can be extended to a continuous... |
H: we need to find the points where $|f(z)|$ has maximum and minimum value
$f(z)=(z+1)^2$ and $R$ be the triangle with vertices $(0,0),(0,1),(2,0)$, we need to find the points where $|f(z)|$ has maximum and minimum value
so here $z=2$ is the point of maximum and $z=0$ is minimmum (intuitively), as $f$ is analytic so ... |
H: Show a subring contains certain elements.
Show that the set of all real numbers of the form
$a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$
with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$.
Proof. We show that the set of all real numbers of the form
$a_0 + a_1\pi + a_2\pi... |
H: Which of the following subsets of $R[x]$ are subrings of $R[x]$?
Which of the following subsets of $R[x]$ are subrings of $R[x]$? Prove or disprove.
All polynomials with constant term $0_R$.
The elements of this set have the form $r_nx^n + r_{n-1}x^{(n-1)} + \dots + r_{1}x$. This is closed, since $R[x]$ does not ... |
H: What line are inverse functions on the complex plane reflected over?
On the real plane (xy plane) inverse functions are reflections of their original functions over y=x. Is there such line for complex functions and their inverses?
AI: Since we have more dimensions to deal with when working with functions of a compl... |
H: "Problem of points (POP)" explanation
Quite recently, there was a question related to "Problem of Points" at MSE. I did some literature survey on POP in the internet and found explanations using an example of coin toss. I have been trying to get an explanation in terms of a rolling a die.
Two players keep rolling... |
H: $f_n \rightarrow 0$ in $L^{1}$ implies $\exists N$ such that $\lim_{ k\rightarrow 0 }\frac{1}{2k}\int_{-1/k}^{1/k} f_N(x) dx = 0$?
Suppose $f_n \geq 0$ and $f_n \in L^1(\mathbb{R})$ for $n=1,2,\ldots.$
If $f_n \rightarrow 0$ in $L^{1}(\mathbb{R})$, must there be an $N$ such that
$$
\lim_{ k\rightarrow 0 }\frac{1}{2... |
H: Determine truth value
If $ x \in \bigcup\{A:A \in \mathscr{A}\} $, then $x \in A$ for some $A \in \mathscr{A}$. $\mathscr{A}$ is a nonempty collection of sets and I have to determine the truth value of the above statement. I know about truth tables. $p\text{ and }q$ is true if both $p$ and $q$ are true, stuff like ... |
H: We need to find the limit of $\sum_{n=0}^{\infty} (n+1)z^n$
We need to find the limit of $\sum_{n=0}^{\infty} (n+1)z^n$
for what values of $z$ does the series converges?
for convergence we need $\limsup|(n+1)z^n|^{1\over n}<1$
i.e $|z|<{1\over (n+1)^{1\over n}}$
Thank you for help.
AI: HINT: Your series is the deri... |
H: How do mathematics define a point?
I have a serious doubt. How do mathematicians define a 'point' in a space or a plot? If we have a clear explanation for a 'point' , I think my doubt on infinitesimals and infinity will be clarified.
AI: Your serious doubt has been shared by many great thinkers throughout history! ... |
H: Pairwise independence implies intependence of unions
Is the following statement true?
Let $\mathcal{A}$, $\mathcal{B}_1$, $\mathcal{B}_2$ be $\sigma$-algebras such that $\mathcal{A}$ is independent from $\mathcal{B_1}$ and $\mathcal{B}_2$. Then $\mathcal{A}$ is independent form $\sigma\{\mathcal{B}_1\cup\mathcal{B... |
H: Connectedness of the the punctured plane and the right open half-plane
Show that any set obtained by removing a single point from $\mathbb{R}^2$ is still connected, where $\mathbb{R}$ is the real numbers.
Then show that $\Bbb H = \{(x,y) : x>0\}$ is connected. By considering the function $$f(x, y)/x,$$ or otherwis... |
H: Finding modulus and argument of $\,z³ - 4\sqrt3 + 4i = 0$.
I think I am messing up somewhere as the principle argument should be a nice number from the standard triangles such as $\frac{\pi}{4}$, $\frac{\pi}{3}$ or $\frac{\pi}{6}$ or something close. (That's what we have mainly been working with)
I have made $z^3 =... |
H: $f : [a, ∞) → R$ is a continuous function. If $\lim_{x→∞} f (x) = L$, prove that $f$ is uniformly continuous on $[a, ∞)$.
Suppose that $f : [a, ∞) → R$ is a continuous function. If $\lim\limits_{x→∞} f (x) =
L$, prove that $f$ is uniformly continuous on $[a, ∞)$.
My attempt at the proof:
Well since I have to use bo... |
H: On the five-point set $X=\lbrace a,b,c,d,e \rbrace$, construct two topologies, one that is Hausdorff and one that is not Hausdorff
On the five-point set $X=\lbrace a,b,c,d,e \rbrace$, construct two topologies, one that is Hausdorff (other than the discrete topology) and one that is not Hausdorff (other than the tr... |
H: Prove that $m=(x+y^2, y+x^2+2xy^2+y^4)$ is a maximal ideal of $\mathbb{C}[x,y]$.
Prove that $m=(x+y^2, y+x^2+2xy^2+y^4)$ is a maximal ideal of $\mathbb{C}[x,y]$.
I can show that the ideal $(x,y)$ of $\mathbb{C}[x,y]$ contains $m$ and $(x,y)$ is a maximal ideal. Therefore to show that $m$ is itself a maximal ideal... |
H: A closed set in $\mathbb A^2_k\times\mathbb P^1_k$
Let $k$ be an algebraically closed field and consider the Zariski topology on $\mathbb A^2_k$ and on $\mathbb P^1_k$.
If
$$X:=\left\{((x_0,x_1),(y_0:y_1))\in\mathbb A^2_k\times \mathbb P^1_k\,\bigg| x_0y_1=x_1y_0\,\right\}\subseteq \mathbb A^2_k\times \mathbb P^1_k... |
H: 31,331,3331, 33331,333331,3333331,33333331 are prime
31,331,3331, 33331,333331,3333331,33333331 are prime. This law can continue it? Will there emerge a composite number?
Without using a computer how to judge.
AI: 333333331 is not prime; it is divisible by 17. This does not require a computer. Euler did calculation... |
H: Calculate double integral of ...
I was doing a homework problem but now I'm stuck. The problem says:
Calculate $\iint_{S} \frac{dx dy}{\sqrt{2a - x}}$ where S is a circle of radius $a$ which is tangent to to both coordinate axes and is in the first quadrant
The cartesian equation for a circle of radius a, and cente... |
H: collection $\mathcal{B}$ of subsets $V = \{ x + yk : k \in \mathbb{Z} \} $ for $x,y \in \mathbb{Z}$ form a basis for some topology of $\mathbb{Z}$
Problem:
The collection $\mathcal{B}$ of subsets of the form $V = \{ x + yk : k
> \in \mathbb{Z} \} $ for $x,y \in \mathbb{Z}$ is a basis for some
topology of $\mathb... |
H: diagonalizability and finding a basis for $\mathbb R^3$
My question is about matrices:
let $T$ be a linear operator on the vector space $\mathbb R^3$ which is represented in the standard basis by the matrix
$$
\begin{bmatrix}
-9 & 4 & 4 \\
-8 & 3 & 4 \\
-16 & 8 & 7 \\
\end{bm... |
H: Prove that there are no fixed points in a non-autonomous system
Consider the non-autonomous dynamical system
$$
\dot x = f(x,t)
$$
with $x \in \mathbb{R}^n$. This may be converted to an autonomous system of dimension $n+1$ with $t = x_{n+1}$ and $\dot x_{n+1} = 1$.
Question: how can one prove that the new system ha... |
H: Set theory: proving of set identities
Suppose we have four sets A, B, C and X.
How can one prove the following identity:
$(A \cap B \cap C \cap \neg X) \cup ( \neg A \cap C) \cup ( \neg B \cap C) \cup (C \cap X)=C$
I tried to apply here any of basic set identities like Distributive Law or Associative Law but it ... |
H: Amount of points on an elliptic curve over $F_q$
Assume I have these two elliptic curves:
\begin{align*}
E:Y^2&=X^3+b_2X^2+b_4X+b_6\\
E':Y^2&=X^3+gb_2X^2+g^2b_4X+g^3b_6,
\end{align*}
over $\mathbb{F}_q$, where $g$ is not a square in $\mathbb{F}_q$, and $\mathbb{F}_q$ does not have characteristic $2$. I know that $... |
H: Ordinary generating function and 1/(1 - x)
I do believe it's dummy question, but I would be grateful if one explains me why following generating function is valid. I'm novice in the topic and intuitively I can't understand why it's true.
It's well known OGF with 1, 1, 1, .... is generating function for
$\frac{1}{(1... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.