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H: Finding number of roots of the polynomial $\,\,x^3-x\,$ in $\,\,\Bbb Z /6 \Bbb Z$
The number of roots of the polynomial $x^3-x$ in $\Bbb Z /6 \Bbb Z$ is : $1,2,3,6.$
I have to pick the correct choice.
Since $\,\,\Bbb Z /6 \Bbb Z=\{\overline 0,\overline 1,\overline 2,.....,\overline 5\}$ and we see $\overline 0... |
H: What if $(a) \land (b) \iff (c)$ and $(c)$ is false?
My question is about the logical inverse of this Lemma:
Lemma: Let $g$ be a real analytic function. Then we have the equivalence $((a)∧(b))⇔(c)$, where the statements $(a),(b)$ and $(c)$ are given by:
(a) $g$ has infinitely many real zeros and the set of those z... |
H: Calculating a Fraction's Reciprocal
Is there any way or equation that allows me to calculate the reciprocal of any fraction?
I mean if i have 5/6 and i need it's reciprocal by using a formula or an equation to calculate it.
Is there or not?
thanks all
AI: Flip it, ${}{}{}{}{}{}{}$ (provided the numerator is not $0$... |
H: Taylor's theorem for vector valued functions
I'm reading about linear and nonlinear programming and on one page I have the following statment (I have highlighted the areas where I have problems and drawn questions for them in the bottom of it):
Proposition $5$. Let $f\in C^2$. Then $f$ is convex over a convex set ... |
H: Matrix powers and recurrence relations
The nth Fibonacci number can be found by raising the matrix $\begin{pmatrix}1 & 1 \\ 1 & 0 \end{pmatrix}$ to the nth power. Are there other recurrence formulas that can be solved like this? This yields faster algorithms for computing them.
AI: Yes - all constant coefficient ho... |
H: Solution verification: $\lim\limits_{x\rightarrow \infty} \frac{\sqrt{x}+x^2}{2x-x^2} = -1$
I am trying to find the following limit
$$\lim_{x\rightarrow \infty} \frac{\sqrt{x}+x^2}{2x-x^2}$$
and I did the following steps:
\begin{align}
\require{cancel}
&\lim_{x\rightarrow \infty} \frac{\sqrt{x}+x^2}{2x-x^2} \\
&\... |
H: Solve the following Differential Equation $x \ln x\ \mathrm{d}x+(y-\ln x\ \mathrm{d}y)=0$
I want to solve the following equation:
$$x \ln x\ \mathrm{d}x+(y-\ln x\ \mathrm{d}y)=0$$
How can i solve it? Thanks.
AI: It's strange that there is no $dx$, I am assuming $dy$ just means $y' = dy/dx$ really.
So we must solve
... |
H: Proper method for solving quadratic equations with exponents
$(\sqrt {x^2-5x+6}+\sqrt{x^2-5x+4})^{x/2}$ + $(\sqrt {x^2-5x+6}-\sqrt{x^2-5x+4})^{x/2}$ = $2^{(x+4)/4}$
I have found out, by trial and error method, that $x=0$ and $x=4$ satisfy this equation. But is there a proper way to solve this equation and get the s... |
H: Prove that the centralizer subgroup is normal in the normalizer subgroup
To my dear friends with gratitude.
I want to get help proving centralizer of a nonempty subset of a group is a normal subgroup in the normalizer of that set in the mentioned group.symbolically:
$C_G (S)\trianglelefteq N_G (S)$
AI: Let $z \in ... |
H: Sequence of the form $pn^2+qn+r$
A sequence 192, 360, 576 is formed by multiplying the corresponding two different arithmetic progression. What is the eighth term of the sequence?
Solution says that answer will be of the form $pn^2+qn+r$. Why is this, and how can I find it?
AI: Let's let the first arithmetic progre... |
H: Limit of functions
I need to give a counterexample to the following statement:
If $ \lim_{x \to 0} \left( \frac {f(x)}{g(x)} \right) = 1 $, then $ \lim_{x \to 0} \left( f(x) - g(x) \right) = 0 $.
The problem is I think this statement is correct for any functions because:
$$ \lim \left( \frac {f(x)}{g(x)} \right... |
H: Why does this function not have any extrema?
Why does the following function not have any extrema: $z=(xy-1)^2 +x^2$? I calculated that $\dfrac{dz}{dx}=2((xy-1)y+x)$ and $\dfrac{dz}{dy}=2(xy-1)x$ which are both zero when $x$ and $y$ are zero. What's my mistake?
AI: It is possible that the function has a saddle poin... |
H: Haar measure of $SO(3)$ obtained from $SU(2)$
I am reading 'Analysis on Lie groups, an introduction' by Faraud and don't understand the following statement
… the image by the map Ad of the Haar measure $\mu$ of $SU(2)$ is equal to the Haar measure $\nu$ of $SO(3)$.
We know that $Ad:$ $SU(2)\rightarrow SO(3)$ is a... |
H: Find probability that only one event will occur
I have a problem with such simple task:
Probabilities of two independent events $A_1$ and $A_2$ are equal repectively $p_1$ and $p_2$. Find the probability that only one of the events will occur.
The answer given in the book is $P=p_1+p_2-2 \cdot p_1 \cdot p_2$
Howeve... |
H: What exactly is a linear space?
I often stumble upon books using the term "linear space" (outside of Linear Algebra) and I have never been totally comfortable with this. Perhaps I am over complicating this, but my intuition says that a "linear space" is one which has some sort of basis, say $\{a, b\}$, and all the ... |
H: Weierstrass... thing
There is in my maths text-book this property/theorem given under the name of Weierstrass property/theorem:
Let $ (a_n) $ be a sequence of real numbers.
a)If $ (a_n) $ is monotonically increasing and has an upper bound, then $ (a_n) $ is convergent.
b)If $ (a_n) $ is monotonically decreasing and... |
H: Orthogonal Matrix statements/proofs
I am currently learning about Orthogonal matrices and have three statements that I can not figure out.
Here are three Proofs that I have written down but can not figure how to prove them:
If $Q$ is an orthogonal matrix, then $Q^{-1}$ is orthogonal.
If $Q$ is an orthogonal matrix,... |
H: Proving that $\Bbb{R}_\ell$ is finer than $\Bbb{R}$.
Let us take the two topologies $\Bbb{R}_\ell$ and $\Bbb{R}$. The book "General Topology" by Munkres says that $\Bbb{R}_\ell$ is finer than $\Bbb{R}$. This article says that every open set of $\Bbb{R}$ has to be an open set in $\Bbb{R}_\ell$ for the latter to be f... |
H: If $f^n$ is mixing then $f$ is mixing?
Let $(X,\mathcal{A},\mu)$ be a probability space and $f:X\to X$ be a measurable map that preserves $\mu$. Fix $n\in \mathbb{Z}^+$.
It's not hard to see that $f$ ergodic does not necessarily imply $f^n$ ergodic. For example, take $X=\mathbb{Z}/4\mathbb{Z}$ with the uniform prob... |
H: Does this follow from the definition of the LambertW function?
The LambertW function $W(s)$ also called ProductLog seems to satisfy this relation:
$$-W(s) = \underbrace{-s e^{-s e^{\cdot^{\cdot^{-s e^{-W(s)}}}}}}_n$$
Or truncated:
$$-W(s) = -se^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e^{-s e... |
H: Proof about boundedness of $\rm Si$
$\def\Si{{\rm Si}}$
I want to prove the boundedness of
$$\Si(x) := \int_0^x \frac {\sin \xi} \xi d\xi$$
as part of a homework (about the non-surjectivity of $\mathcal F : L^1(\mathbb R) \to C_0^0(\mathbb R)$). As a first step, I found (by means of plotting)
$$\frac{\sin x}{x} \le... |
H: How find the limit $I=\lim_{m\to 0,n\to 0}(m^2-2n)^n$
find the limit
$$I=\lim_{n\to 0}\lim_{m\to 0}(m^2-2n)^n$$
my try:
$$I=\lim_{n\to 0}(0^2-2n)^n=\lim_{n\to 0}(-2n)^n$$ is not exsit,
my try is true? Thank you
AI: The limit is not well-defined, i.e. the value depends on the choice of a sequence $a_j = (m_j, n... |
H: Suppose that $0 \le f(n) \le 1$, why $\lim_{n \to \infty} (1 - f(n))^n = 0 \iff \lim_{n \to \infty} f(n)n = \infty$?
Suppose that $0 \le f(n) \le 1$ and consider two eqauation:
$$ \lim_{n \to \infty} (1 - f(n))^n = 0 \tag{A} $$
$$ \lim_{n \to \infty} f(n)n = \infty \tag{B} $$
It seems that A and B are equivalent. ... |
H: Proof that $\bigcap_{n=1}^\infty J^n=0$ in commutative noetherian ring
If we let $R$ be a commutative noetherian ring. Then $\bigcap_{n=1}^\infty J^n=0$ where $J$ is the jacobson radical of $R$
Proof.
Denote $X=\bigcap_{n=1}^\infty J^n=0$. Then let $XJ=Q_1\cap \ldots \cap Q_n$ be a primary decomposition of $X$. We ... |
H: Compute homotopy classes of maps $[T^{2},T^{2}]$
How to compute $[T^{2},T^{2}]$ the set of homotopy class of continuous maps $f:T^{2}\longrightarrow T^{2}$?
Thanks.
AI: Since $T^2=S^1\times S^1$, you get
$$
[T^2,T^2]=[T^2,S^1]\times [T^2,S^1].
$$
So, the problem reduces to finding $[T^2,S^1]$. Since $S^1$ is the f... |
H: number of automorphisms for group in order 169
Let $G$ be a group with order 169. Prove number of automorphisms is at least 143.
I thought that 169 is 13 squared so maybe G isomorphic to $ Z_{169} $ but I dont have any idea. How can I solve it?
AI: Let $p$ be prime and $G$ a group of order $p^2$.
If $G$ is cycli... |
H: Fields of positive characteristic
Let $\mathbb F$ be an infinite field of characteristic $p>0$. It is true that every element of $\mathbb F$ is algebaric over the prime subfield $\mathbb F_p$ of $p$ elements?
AI: Consider ${\mathbb F}_p (x)$, the field of rational functions over ${\mathbb F}_p$. Is $x$ algebraic o... |
H: $\epsilon$-$\delta$ Verification of a Lipschitz Function
A function $f:D\rightarrow \mathbb{R}$ is said to be a Lipschitz function provided that there is a nonnegative number $C$ such that
$|f(u)-f(v)|\le C|u-v|$ for all $u,v\in D$.
Show that a Lipschitz function satisfies the $\epsilon$-$\delta$ criterion on $D$.... |
H: Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$
I am stuck with the following problem:
Consider the equation $\,\,x^{2007}-1+x^{-2007}=0.\,$Let $\,m$ be the number of distinct complex non-real roots and $\,n$ be the number of distinct real roots of the above equation. Then $\,m-n\,$ is
1.$\,0$
2.$\,2006$
3... |
H: Last removed number in Sieve of Eratosthenes
I want to find the last deleted number in Sieve of Eratosthenes when applied on numbers below 1000. How can I find it?
AI: HINT: If $n<1000$ is composite, it will be removed when its smallest prime factor is processed. $31<\sqrt{1000}<32$, so the smallest prime factor or... |
H: What's wrong with that proof?
What wrong with this proof?
$(-1)=(-1)^{\frac{2}{2}}=(-1)^{2\times \frac{1}{2}}=\sqrt{1}=1$ then $1=-1$
AI: $x^{\frac{1}{2}}$ is a multiple-valued "function", since in general $x$ has two square roots. One could also write:
$$\sqrt1=-1$$ |
H: Proving something with Wilson theorem
I need to prove that $x^2\equiv -1\pmod p$ if $p=4n+1$.
($p$ is prime of course...)
I need to use Wilson theorem.
AI: We have, $p-r\equiv-r\pmod p$
Putting $r=1,2,\cdots,\frac{p-1}2$ and multiplying we get $$\prod_{1\le r\le \frac{p-1}2}(p-r)\equiv(-1)^{\frac{p-1}2}\prod_{1\le... |
H: What is the probability that the number $5$ comes up on exactly two of three loaded dice?
I roll three different loaded dice. For the first die, the probability of getting a $5$ is $0.7$, for the second die the probability of getting a $5$ is $0.48$, and for the third die the probability of getting a $5$ is $0.38$.... |
H: How to construct a contractible space but not locally path connected?
I am looking for a space which is contractible and not locally path connected.
I know the cone $CX$ of every space $X$ is contractible. Besides, it seems that if $X$ is locally path connected, so is $CX$. Therefore, I need to find a space which ... |
H: conditional probability that randomly chosen
In a certain village sports club, 46 % of members play football, 36 % of members play cricket, and 17 % of members play both games. What is the probability (between 0 and 1) that a randomly chosen member does not play football given that he/she plays cricket?
Give your s... |
H: Find the probability of two random real numbers $x$ and $y$ between $0$ and $2$, where $\min(x,y) < 2/3$
Here is a picture of what I did so far.
http://sdrv.ms/HhxIvu
I got a result of $\frac59$, because the total area is $4$, and I'm subtracting the square with side of $\frac43$.
Can anyone confirm that this is th... |
H: Why are singleton sets connected?
Why are singleton sets connected? I know that a subset $S$ of a metric space $X$ is connected if and only if given subsets $U$ and $V$ of $X$:
i. $U$ and $V$ are open
ii. The intersection of $U$, $V$, and $S$ equals the empty set
iii. $S$ is a subset of the union of $U$ and $V$.
A... |
H: if neither f nor g is differentiable at x=a. is $f+g$ differentiable at $x=a$?
Suppose that $f$ and $g$ are defined on R and that neither f nor g is differentiable at x=a. prove or disprove: f+g is not differentiable at x=a.
I know how to show if f and g are differentiable at x=a.then f+g is differentiable at x=a.... |
H: Prove that f is a constant function
Let $f$ be a function defined on R and suppose that there exists $M>0$ such that for any $x,y∈R$, $|f(x)-f(y)|≤M|x-y|^2$. Prove that $f$ is a constant function.
I don't even know how to start, I know that I need to show that $f(x)$ equal to some number , zero for instance. I thin... |
H: Simplification of a trilogarithm of a complex argument
Is it possible to simplify the following expression?
$$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$
where
$$\large\xi=\frac{\sqrt[3]3}6\sqrt[3]{27+\sqrt3\,\sqrt{243-\pi^6... |
H: Application of Main Homomorphism Theorem
This is related to this question. I just didn't want a prolonged discussion in the comments.
Let $\phi: G \to G'$ be a homomorphism. Let $G$ be a finite group. Let $K \leq G$ be the kernel of $\phi$. Let $I \leq G'$ be the image of $\phi$.
Let $H' \leq G'$. Find a formu... |
H: What can we conclude from "f is not little-o of g"?
Given two functions $f$ and $g$, what does "$f$ is not $o(g)$" mean ? What can we conclude from this statement ?
I know "$f$ is $o(g)$" means the limit at infinity of $\frac fg$ is zero.
So does "$f$ is not $o(g)$" mean the limit at infinity of $\frac fg$ is diffe... |
H: Can an uncountable group be generated from a single element?
First question : can an uncountable group be cyclic?
Ok so my though is if $G$ is generated by i then for $x\in G$ we have $x=i^n$ for integer n, so then it must be countable. Is there a way to generate an uncountable group in some way from a single eleme... |
H: Graph Theory Complements
Let G be a simple graph with n vertices. What is the relation between the number of edges of G and the number of edges of the complement G'?
In the example below, I noticed that by adding the vertices and edges of G and the edges of G' you get a total of 10. Then the number of edges in G' ... |
H: are any two vector spaces with the same (infinite) dimension isomorphic?
Is it true that any 2 vector spaces with the same (infinite) dimension are isomorphic? I think that it is true, since we can build a mapping from $V$ to $\mathbb{F}^{N}$ where the cardinality of $N$ is the dimension of the vector space - where... |
H: Trying to prove that the sequence: 3, 3, sin(1), 3, 3, sin(2), 3, 3, sin(3), 3, 3, sin(4), $\ldots$ does not converge to 3
$\textbf{Proof}$:
I need to show that $\exists \epsilon > 0$, such that, $\forall N \in \mathbb{N},\ \exists n \ge N$, such that, $\ \left | a_n - 3 \right | \ge \epsilon$
Let $\epsilon = 2$, t... |
H: Continuous real valued functions and inner product space?
Let $V$ be the space of all continuous real valued functions on the interval $[1,4]$ with the inner product defined by:
$$\langle f,g\rangle = \int_1^{4} f(t)g(t)\,dt.$$
(i) Find an orthonormal basis of the space $W$ of polynomials of degree less than or... |
H: lagrange multiplier slope
I was reading the link given in the thread's last comment. I understood initial part. I understand that in case of the hill, if we take any point on that hill, the gradient of the original function will always point towards the peak of the mountain. But I am a bit confused about the gradie... |
H: I'm a beginner in Abstract algebra and am trying to show the homomorphism
You know that map where G goes to H and then we also know a surjective group homomorphism G to G/N exists. How do I connect the G/N to H? As in, how do I show that there is a surjective map between them? Also, how come showing that a map is w... |
H: Given identity map $U:\ell^n_a\rightarrow \ell^n_b$ for $\mathbb{R}^n$, how to computer operator norm $\forall a,b$?
If $U:\ell^n_a\rightarrow \ell^n_b$ is the identity map of the underlying vector space $\mathbb{R}^n$, then how do you compute the operator norm $U$ for all possible values of $a$ and $b$?
AI: Suppos... |
H: What is a polynomial and how is it different from a function?
I have a problem that asks me to find a polynomial $P(x)$ so that $P(3)$ is 9.
Now I can say with certainty that $P(x)$ can be $x^2$. This is a second degree polynomial.
But what about functions such as $\frac{1}{x^2+3}$, are these not polynomials? If no... |
H: Hamming Code Error Detection
I am learning few things about hamming code and error detection so my question may sound stupid. So i know that lets i ahve (7,4) hamming code and i made transpose of parity check matrix H(t). Now say my code word was Y="1001000" now i need to find the error i know the procedure that yo... |
H: Why it does not produce a Klein bottle?
I cannot understand why the action $\mu : (\mathbb{Z}\oplus \mathbb{Z})\times \mathbb{R}^2 \longrightarrow \mathbb{R}^2 $ given by $\mu((m,n), (x, y)) = (x+ m, (-1)^m(y + n))$ does not produce the Klein bottle $K$. If $X = \mathbb{R}^2 / (\mathbb{Z}\oplus \mathbb{Z})$, then ... |
H: Non-elementwise Matrix Derivatives
Let A,B,C,D,X be matrices.
I'd like to perform a Gradient Descent minimization to the loss functin
$$ tr[(AXBX^TC-D)^T(AXBX^TC-D)] $$
My question is, how to take the gradient efficiently w.r.t. $B$?
I have came up with the following:
$$ \frac {\partial} {B_{ij}} tr[(AXBX^TC-D)^T(... |
H: What function can achieve the following?
So I have some values which are computed linearly. But I want to stress the middle range more so I want the values to be "transformed" into something like this:
So basically, say for $x$ between $0$ and $100$, it should start steeper, $f(x) - x$ should be biggest in the mid... |
H: Largest Circle in a Polygon
My polygon is given by $P=$$\left\{x\geq 0, y\geq 0, 3x-4y\leq 2, 4x+3y\leq 12\right\}$
Now trying to find the largest circle inscribed inside these half-planes. But whenever I formulate it as an LP problem, the answers don't make sense. I'm using the method of Chebyshev Center and these... |
H: Markov-chain properties
I have some questions about a Markov-chain $(X_n)$ on a finite state-space $S$ with transition matrix $P$. A function $f:S\rightarrow\mathbb R$ is a columns vector and $Pf$ therefore a matrix multiplication. Now there are three points I am not aware of:
$\mathbb E^x$ means that $X_0=x\in S$... |
H: differentiability with complex numbers
Let $f:\mathbb{C}\rightarrow \mathbb{C}$ defined by $f(z)=z^{3}$. Prove that does not exist a point $z_0$ for the line segment that joins $z_1=1$ and $z_2=i$ such that
$f(z_2)-f(z_1)=f'(z_0)(z_2-z_1)$
...any idea how to start, please?
AI: Hint: This is really equivalent to as... |
H: Can I use this trick for my proof?
I am given the following problem, with a hint that states that I should use the General Lebesgue Dominated Convergence Theorem.
Let $\{f_n\}$ be a sequence of integrable functions on $E$ for which $f_n \rightarrow f$ a.e. on $E$ and $f$ is integrable over $E$. Show that $\int_E \l... |
H: Confused by a proof in Strichartz' book on Fourier Transforms
Hi I'm confused by a proof on page 53 in Strichartz book on Fourier Transforms. Specifically, in the first equation on page 53, why is it valid to interchange the action of the distribution with the integral? I know that distributions are linear, but int... |
H: Using an Interpretation, Prove Two Equations Are Not Equivalent
I have been working on this problem for hours and cant seem to understand how to go about doing it. The question is to prove that
Is not equivalent to
by giving an interpretation which is a model for one, but not for the other.
I don't see how simpl... |
H: What is a 0-ball?
I'm reading a paper that says $\bigcap V_{T,X}$ is either empty or a closed $l$-ball where $T \subset S$ is a subset of points $S$ and $\operatorname{card}{T} = m + 1 - l$ where $m$ is the dimension of the smooth manifold $\Sigma$ that the points $S$ are sampled from. My question is what happens w... |
H: Rotation of 2D polar graph in a 3D space along some fixed axis?
Does there exist some systematic way of rotating a 2-D polar graph $r=f(\theta)$ around some axis in a 3D space?
For example: $f(\theta)=cos(\theta)$ in 2-D looks like:
If we want to rotate the above plot along the y-axis (in 3D of-course) the plot s... |
H: Is This Interpretation A Model Of This Formula
I have a formula
and an interpretation I:
Where D is the domain, and R1 is the set of relations.
I am trying to prove or disprove that I is a model for A. I believe that it is. However, I am unsure how I would go about proving it. Couldn't any random assortment of tw... |
H: Proof of $gcd(f_{n},f_{n+2})=1$ for natural numbers
I'm going to use the Principle of Mathematical Induction to prove the above statement.
Base cases:
$(n=1)$ $f_{1}=1, f_{3}=2$ so $gcd(1,2)=1$
$(n=2)$ $f_{2}=1, f_{4}=3$ so $gcd(1,3)=1$
Assume that $gcd(f_{n}, f_{n+2})=1$ holds for some natural numbers. I want to s... |
H: Proof of if $A \times B = A \times C$, and $A \neq \varnothing$, then $B=C$
Proof: suppose $A \times B = A \times C$
Then $\frac{A \times B}{A} = \frac{A \times C}{A}$
Therefore $B=C$
Is this proof valid?
AI: Your argument, while suggestive, does not mean anything because there is no division operation defined for ... |
H: How to approach factoring problems?
Generally speaking, how should I approach a problem involving factoring? I usually don't have a problem with the more typical forms, but sometimes I just don't know what to do.
My calc2 question is this:
The given curve is rotated about the y-axis. Find the area of the resultin... |
H: $\sum_{n=0}^{14}\tan(12n+1^\circ)$
I often fail to find trigonometric sums such as the one in the question shown in the following.
When I tried the question, I first led $z=e^{i\pi/180}$.
After simple calculations, I obtained
$\sum_{n=0}^{14}\tan(12n+1)=\sum_{n=0}^{14}\dfrac{z^{24n+2}-1}{z^{24n}+z^2}$
How can I ... |
H: Surfaces of genus g
The problem: give maps $f:\Sigma_{g}\longrightarrow\Sigma_{h}$ not homotopic to a constant map with $0<g<h$.
Any idea would be helpful.
AI: The existence of such a map is guaranteed by $K(G,1)$ theory because $\Sigma_g$ is aspherical for $g\geq 1$. To construct a map explicitly, try to come up w... |
H: What is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$?
Suppose $a_1, a_2,\dots , a_n$ are $n$ positive real numbers with $a_1a_2 \dots a_n = 1$.
Then what is the minimum value of $(1 + a_1)(1 + a_2). . .(1 + a_n)$ ?
I think $(1 + a_1)(1 + a_2). . .(1 + a_n)$ takes its minimum value when $a_1=a_2=\dot... |
H: Clear explanation about uniform continuity.
Can anyone explain the uniform continuity clearly with picture if possible?? I have read the section on this topic in my text book but I am still not clear on this. Thanks.
--edit
In my text book, it gives two definitions of a continuous function. One with sequence in the... |
H: Short Prove or Disprove: $R_1 \cap R_2$ is an equivalence relation
Suppose $R_1, R_2$ are both equivalence relations defined on nonempty set $A$. Prove or disprove: $R_1 \cap R_2$ is an equivalence relation.
What method (if any) would you take to prove this in as few sentences as possible?
AI: Let $R=R_1\cap R_2.$... |
H: Is my proof correct? (Conformal equivalence of two circular annuli)
I want to show that the two annuli $$A=\{r<|z-z_0|<R\} $$
$$A'=\{r'<|z-z_0'|<R'\} $$
are conformally equivalent (i.e. there exists a biholomorphic map between the two) iff $$\frac{R}{r}=\frac{R'}{r'}. $$
Sufficiency:
Suppose the ratios of the radii... |
H: A question on the isomorphism induced by a homotopy equivalence.
Now, I am learning a proof that a homotopy equivalence induces an isomorphism. However, since I am a beginner in algebraic topology, I cannot fully understand the proof.
Suppose $\varphi:X\to Y$ is a homotopy equivalence. To prove $\varphi_\ast:\pi_1(... |
H: Limit evaluation: very tough question, cannot use L'hopitals rule
I found a very tough limits question online. The question asks you to evaluate the limit $$\lim_{x \to 0}\frac{(x+4)^\frac{3}{2}+e^{x}-9}{x}$$ without using L'Hôpitals rule.
I tried to treat the top as a radical expression with the $e^x-9$ grouped an... |
H: Question on Proof of the Contraction Mapping Theorem
Contraction Mapping Theorem
If $T\colon X\to X$ is a contraction mapping on a complete metric space $(X,d)$ then there is exactly one solution $x\in X$.
Proof:
Let $x_0$ be any point in $X$. We define a sequence by
$$x_{n+1}=Tx_n, \qquad \text{for } n\geq 0.$$
De... |
H: Representing a double for-loop as a series.
What would be the best way to represent the sum formula (series) of this for-loop:
for (int i = 1; i <= n; i++)
for (int j = 1; j <= n; j+=i)
count++
What process is best to determine this?
EDIT:
Sum in this sense: $\sum \limits_{k=1}^N k^2$
AI: Notice that t... |
H: Solving trigonometry equation
Please help me understand how to solve this for $0\leq x\leq360 $
I seem to have a problem with equations with powers.
$$3\sin^2 x-3\cos^2x+\cos x-1=0 $$
thinking that I would start by simplifying:
$$3 (\sin^2 x- \cos^2x)+\cos x - 1=0 $$
How I wish the equation in the bracket was i... |
H: Calculate the complex integral $\oint_{|z|=1}\sin{\frac{1}{z}} dz$
How do I calculate this complex integral?
$$\displaystyle\oint_{|z|=1}\sin\left ({\displaystyle\frac{1}{z}}\right ) dz$$
I made the Taylor series for this:
$$\displaystyle\sum_{n=0}^\infty \displaystyle\frac{(-1)^n*(1/z)^{2n+1}}{(2n+1)!}$$
But now I... |
H: Define $f$ on $\Bbb R$ by $f(x)=x^3$ for $x\ge0$ and $f(x)=0$ for $x\lt 0$. Find all $n\in\mathbb N$ such that $f^{(n)}$ exists on all of $\Bbb R$.
Define $f$ on $\Bbb R$ by $f(x)=x^3$ for $x\ge0$ and $f(x)=0$ for $x\lt 0$. Find all $n\in\mathbb N$ such that $f^{(n)}$ exists on all of $\Bbb R$.
I am studying for ... |
H: The choice of the eigenfunction of Laplacian
$M$ is a closed Riemannian manifold and $\lambda_1>0$ is the first nontrivial eigenvalue of $\Delta$. Can we find a eigenfunction $f$ of $\lambda_1$ such that $\mathop {\sup }\limits_M f - \mathop {\inf }\limits_M f = 2$ and $\mathop {\inf }\limits_M f \geq-1$?
AI: Yes. ... |
H: If $f(x)=\int_0^x x^2 \sin {t^2}~dt $, find $f'(x)$.
Stumbled with this problem
If $f(x)=\int_0^x x^2 \sin {t^2}~dt $, find $f'(x)$.
How do you solve problems like this?
AI: HINT: $f(x) = x^2 \cdot g(x)$, where $g$ is the integral after the $x^2$ has been taken out.
So $f'(x) = (x^2)'\cdot g(x) + g'(x) \cdot(x^... |
H: Trig equations solution
Solving the following for x:
$$
\frac{3\cos(2x)+5\cos(x)-1}{\sqrt{-\cot(x)}}=0
$$
The solution says that the answer is $x=-\frac{\pi}{3}+2\pi k$ where $k$ is an integer.
I am not sure why there is the minus sign in front. Can someone please help me out?
AI: First off, there is a domain issue... |
H: Determinant from matrix entirely composed of variables
I don't want the answer, but I'd love to kick in the right direction. I'm really not sure how to approach this question.
$$\begin{align}
& -6 = det\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix} \\
& x = det\begin{bmatrix}
a & b & c \\
2d... |
H: Cyclotomic Polynomial Evaluated at 1
I noticed that when trying to evaluate the $n$th cyclotomic polynomial at $1$, $\Phi_n(1)$, we run into an issue while using the explicit formula $\Phi_n(x)=\prod_{d|n}(x^d-1)^{\mu(n/d)}$. In particular, $$\Phi_n(1)=\prod_{d|n}(1^d-1)^{\mu(n/d)}=\prod_{d|n}0=0?$$ This is certain... |
H: Does the principle of complete induction imply the well-ordering principle
Proof: Assume the PCI. Let $T$ be a nonempty subset of $\mathbb{N}$. Then $T$ has some element $x$. Then $\{1,2,...,x-1\}$ is a subset of $\mathbb{N} - T$. By the PCI, $x$ is an element of $\mathbb{N}-T$. This is a contradiction, because $x$... |
H: Proof that changing a finite number of terms in a series does not change where or not it converges
I want to prove the following theorem:
Changing a finite number of terms in a series does not change whether or not it converges, although it may change the value of its sum if it does converge
I don't know how to b... |
H: Drawing a lattice for a set partially ordered by divisibility
I am a little bit confused as to how I would draw a lattice for the set $S= \{1,2,3,4,6,9,12,18\} $when it is partially ordered by divisibility.
I was able to prove that it has a greatest lower bound, and a least upper bound for all elements $x,y \in S$... |
H: The limit of a sequence when at $n-1$
Suppose $\sum\limits_{n=1}^{\infty} a_n$ is a series that converges.
Therefore, $\lim\limits_{n \to \infty} S_n$ exists, where $S_n$ is the sum of the first $n$ terms of the series.
So, let $\lim\limits_{n \to \infty} S_n = L$.
How do I formally justify that $\lim\limits_{n \to... |
H: Does this hold: $(1+\sqrt[n]{M})^n=2^n\cdot\sqrt M$
$(1+\sqrt[n]{M})^n=2^n\cdot\sqrt M $ (TRUE/FALSE)
My try: Using binomial theorem, I got $$(1+\sqrt[n]{M})^n=\sum_{k=0}^n\binom{n}{k}\big(\sqrt[n]{M}\big)^{n-k}=\sum_{k=0}^n\frac{n!}{(n-k)!\cdot k!}\big(M^{\frac{1}{n}}\big)^{n-k}.$$
I don't know what to do next. ... |
H: All Partial sums of two given sequences are bounded by a positive constant
Let $\theta \in \mathbb R$ be a non-integer multiple of $2\pi$. Prove that the sequences $(\sin(n\theta))_{n \in \mathbb N}$ and $(\cos(n\theta))_{n \in \mathbb N}$ verify $|S_N|\leq K$ where $K>0$ and $S_N=a_1+...+a_N$ for a given sequence ... |
H: Maclaurin series of: $ f(x) = {x + 5\over1-x^2}$.
I'm trying to get the Maclauren series of:
$ f(x) = {x + 5\over1-x^2}$.
I am sure there is some trick here, the result according to Mathematica is:
$5 + x + 5x^2 + x^3 + 5x^4 + x^5 + 5x^6 + \ ...$
I defined:
$f^n_m = {x^n\over(1-x^2)^m}$
$f^n_m = 0$ for $n < 0 $ (ju... |
H: Complex Numbers and Transformations
If a transformation t acts by rotating every point of the plane around the origin by $\pi/5$ clockwise and then proceeds to translate it by vector $v$ = $(1,2)$.
How do I describe this transformation by complex numbers? (Define the function t(z) so that the image of any point z ... |
H: Logial Entailment vs. Material Conditional: binding free variables?
I think that now I DO understand basic logic manipulations.
I DO understand why (camels have feathers) -> (Michigan has a lot of great lakes) is true.
Nevertheless, we can still write $\mathscr{A}$ for (camels have feathers) and $\mathscr{B}$ for (... |
H: In a cylic group of order $12$, we can find an element $g \in G$ such that $x^2 = g $ has no solutions.
$$ \textbf{PROBLEM} $$
If $G = \{ g^n : 0 \leq n \leq 11 \} $. Then we can find an element $a
\in G$ such that the equation $x^2 = a$ has no solutions.
$$ \textbf{ATTEMPT} $$
My claim is that the multiples of ... |
H: Extended Pigeonhole Principle: How to prove it?
A version of the pigeonhole principle is:
(1) If m objects are put in n boxes and n < m, then at least one box
contains at least ceil(m/n) objects
An alternate (more generalized) version is:
(2) For a nonempty finite collection of integers (not necessarily
dist... |
H: Proving a Sequence Does Not Converge
I have a sequence as such:
$$\left( \frac{1+(-1)^k}{2}\right)_{k \in \mathbb{N}}$$
Obviously it doesn't converge, because it alternates between $0,1$ for all $k$. But how do I prove this fact?
More generally, how do I prove that a sequence does not converge? Are there any neat... |
H: Help with solving $\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$
Given
$$\int_0^n \exp(-rt)\exp\left(\frac sre^{-rt}\right).dt$$
Can you please show the step(s) involved to reach this next line in the textbook:
$$\left[-\frac 1s\exp\left(\frac sr e^{-rt} \right) \right]_{t=0}^{t=n}$$
Is it done (or can it... |
H: Induction: $2^n = \sum_{v=0}^{n} \binom{n}{v}$
I have to prove the following identity for $n \in \mathbb{N}$:
$\displaystyle 2^n = \sum_{v=0}^{n} \binom{n}{v}$
Is there a way to show it through induction? Or is there a easier way? My steps so far:
$\displaystyle n=2: 2^2=\sum_{v=0}^{2} \binom{n}{v} \Rightarrow 2 \... |
H: Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$
How do I prove:
Let
$\lim\limits_{n\to \infty} a_n = \infty$
and
$\lim\limits_{n\to \infty} b_n = \infty$
Prove: $\lim\limits_{n\to \infty} a_n ⋅ b_n = \infty$
Thank you
AI: Notice can find and $\alpha $ such that $0 < \alpha < \lim b_n$. For large $n$, in part... |
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