text stringlengths 83 79.5k |
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H: parametrization and orientation of the Möbius band
I know that the Möbius band is a nonorientable surface. However, the following exercise seems to contradict this.
A Möbius band can be constructed as a ruled surface by
$x(u,v)=\beta(u)+v\gamma(u)$, where $-1/3<v<1/3$
$\beta(u)=(\cos u, \sin u, 0)$ and
$\gamma(u)=(... |
H: Construct a topological embedding
For an arbitrary discrete space X, construct a compact topological space Y and a topological embedding Y.
I am think about construct $X={0,1}$ equipped with the discrete topology. Topology on Y={∅,{0,1},{1}} which is the Sierpinski space. Thus Y is immediately compact. I try to def... |
H: $4$ and $a_{2n + 1}$ are coprime?
Suppose $a_i$ is a sequence of positive integers. Define $a_1 = 1$, $a_2 = 2$ and $a_{n+1} = 2a_n + a_{n-1}$. Does it follow that
$$ \gcd(a_{2n+1} , 4 ) = 1 $$ ???
Im trying to see this by induction assuming above holds, we need to see that $\gcd(a_{2n+3} , 4 ) = 1$.
But, $\gcd(a_... |
H: Evaluating $\int _{-1}^{e} \frac{1}{x}dx$
Here very easily by the Fundamental Theorem of Calculus
$$\int _{-1}^{e} \frac{1}{x}dx=\ln(e)-\ln(-1)$$
From Euler's identity $e^{i \pi}$=-1 we can easily deduce that $\ln(-1)=i \pi$. Thus the integral becomes
$$\int _{-1}^{e} \frac{1}{x}dx=1-i \pi$$ But we know that calc... |
H: Is this an identification map or not?
The definition of identification map is the following:
A surjective map $f: X \to Y$ is an identification map iff $U$ in $Y$ is open if and only if $f^{-1}(U)$ is open in $X$.
I have an idea of what identification spaces should look like. Like for example you have a space and y... |
H: If arrivals are not occuring at random, why do arrival times still have a probability distribution?
If the arrival times are, say, uniformly distributed U(0,10), I don't quite understand why the arrivals in this case are said to be not occurring at random (this claim was implied in my textbook chapter on queues), s... |
H: Question in analysis from an entrance exam paper
This propped up while I was going through some old question papers.
If $f \in C^1[0,1]$ so that $ \lim \limits_{x\to \infty} \dfrac{x.f(x)}{f'(x)}=2$.Then :
1) Show that for $s<2$, $\lim\limits_{x \to \infty} x^{-s}f(x) \longrightarrow \infty$
2) Find $\dfrac{\int \l... |
H: problem on similar triangles
In the adjacent figure, $\frac{AG}{GD} = \frac{3}{4}$ and $\frac{BD}{DC}=\frac{4}{7}$ and $AE=12 $ cm. Find the length of $EC$.
1) 33
2) 36
3) 44
4) 48
Figure:
I think similarity of triangles is to be used but can't really find the target triangles.
AI: You can use Menelaus... |
H: Find the domain and range of $f(x) = y$
$ y = \sec^{-1}(2x - x^2)$.
If you graph the equation the domain and range can be inferred
I don't how to solve it algebraically.
AI: This is a partial answer
$$
\sec:\mathbb{R}-\{\frac{k\pi}{2}\}\rightarrow(-\infty,-1]\cup[1,+\infty)
$$
$$
\sec^{-1}:(-\infty,-1]\cup[1,+\inft... |
H: Prove that the $\sigma$ - algebras are equal
I want to show that $\sigma$-algebras on $\mathbb{R}$ generated by $(a,b), \ (a,b], [a, b), [a,b], (-\infty, a), (-\infty, a], (b, +\infty), [b, +\infty)$ for $a,b \in \mathbb{R}$ and $a,b \in \mathbb{Q}$ are all equal.
Here is what I've come up with so far:
($\mathcal{M... |
H: Visualize the effect of adding another constraint
I have 2 eqns
$$ x_1+4x_3\leq4$$
$$ x_2+4x_3\leq4 $$
$$x_1\geq0$$
$$x_2\geq x_3\geq0$$
By drawing geometrical figure I have vertices whose co-ordinates is $(0,0,0) , (4,0,0) (0,0,1) ,(0,4,0) ,(4,4,0)$
I want to include an in-equality named $$x_1+x_2+x_4+x_5\geq \ep... |
H: some integral of curvature
I have a problem :
Let $C$ be a curve defined by $C := \{\ (x,y)\in \mathbb{R}^2 |\ x^4 + y^4 = 1\}$,
and let $k$ be its curvature.
Compute the integral $\int_C k$
This is an problem in a exam
What is the definition of the integral?
And how can I evaluate?
AI: Use Global Gauss-Bonnet The... |
H: How find the $\frac{1}{e}\le R\le 1$
Qustion:
let $a_{n}> 0$,and such $\displaystyle\sum_{n=1}^{\infty}a_{n}$ converge,let
$$b_{m}=\sum_{n=1}^{\infty}\left(1+\dfrac{1}{n^m}\right)^na_{n}$$.
show that
$$\dfrac{1}{e}\le R\le 1$$
where the $R$ is the radius of convergence of $\displaystyle\sum_{m=1}^{\infty}b_{... |
H: compare complex number with 0?
im reading the book "what is mathematics" and find the questions.
Calculate $\sqrt{5+12i}$
i followed the hint and wrote the equation $\sqrt{5+12i} = x+yi$ and solved it with these results :
x=3,y=2
x=-3,y=-2
so here my intuition told me maybe I should pick the 3+2i.
but I r... |
H: Finding branch points and branch cuts of arctan
I am studying complex analysis and I do not yet fully understand branch points and branch cuts. I am trying to figure out how it works by looking at the following:
$z \rightarrow \frac{1}{2i} \log(\frac{1+iz}{1-iz})$
(arctan(z)= $\log(\frac{1+iz}{1-iz})$ )
Now how do... |
H: Are complex eigenvalues special?
I've noticed that complex eigenthings are treated as a whole separate topic to real eigenthings (I say things to mean values and vectors).
I see no reason for this distinction, yes it makes the geometric interpretations different.... but it's not like one one has to do a completely ... |
H: How do I get $\int^\infty_{-\infty} ue^{-u^2/2} du = [-e^{-u^2/2}]$
How do I get:
$$\int^\infty_{-\infty} ue^{-u^2/2} du = [-e^{-u^2/2}]^\infty_{-\infty}$$
AI: Oh I should use integration by substitution. Let $y = -u^2/2$ |
H: When can you replace some random variable $X$ with another random variable $Y$?
Is there some condition that one random variable $X$ can be replaced by some other random variable $Y$ provided that $Y$ has the same distribution as $X$?
AI: If you are consistent then renaming will do no harm. Replacing is another thi... |
H: Solve $(z+ \bar{z}=|z^2+1|)$
Solve this equation: $$z+ \bar{z}=|z^2+1|$$
I tried the following.
$$x+iy+x-iy=|z^2+1|$$
$$2x=|z^2+1|$$
$$x=(|z^2+1|)/2$$
and I came to a dead end.
How can I proceed?
AI: Hints: $$z^2+1=(x+iy)^2+1=(x^2-y^2+1)+i(2xy)$$ $$|a+ib|^2=a^2+b^2$$ |
H: Solving inequality. Did I do it right?
Solve the following inequality: $$5(y-2)-3(y+4)\ge2y-20$$
I made calclations and I found:
$$
0\ge2
$$
What does it mean? Are my calculations right?
Here's how I did it:
$$
5(y-2)-3(y+4)\ge2y-20\\
5y-10-3y-12\ge2y-20\\
2y-22\ge2y-20\\
2y\ge2y+2\\
2y-2y\ge2\\
0\ge2
$$
AI: Your... |
H: The limit as $x \to \infty$ of $ \frac {\sqrt{x+ \sqrt{ x+\sqrt x}} }{\sqrt{x+1}}$
$$\lim_{x\to\infty} \frac {\sqrt{x+ \sqrt{ x+\sqrt x}} }{\sqrt{x+1}}$$
I managed to find the limit of the above function, which according to my calculations is 1, but I don't know how to prove my answer is correct.
AI: If you want a ... |
H: How find this limit: $\displaystyle \lim_{n\to\infty} \int_0^1 (1+x)^{-n-1}e^{x^2}\ dx$
I wish to find: $$\lim_{n\to\infty} n\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx\ \ (
> n=1,2,\cdots)$$
I maybe have to evaluate:
$$\int_0^1 (1+x)^{-n-1}e^{x^2}\ dx$$
But I can't, can someone give me some help?
AI: We can partially evalu... |
H: Spot my error in solving a linear system
I almost always get the unit matrix if I try to get to an row reduced echelon form. I probably always make a mistake. Can you spot the error? What illegal operations could a beginner do while trying to solve a linear system?
\begin{array}{}
1 & 1 & 3 \\
-4 & -3 & -8 \\
-2 & ... |
H: What are the values of a and b such that $a^4 + 4 b^4$ is prime
What values of a and b will ensure $a^4 + 4 b^4$ is prime
AI: HINT:
$$a^4+4b^4=(a^2)^2+(2b^2)^2=(a^2+2b^2)^2-2\cdot a^2\cdot2b^2=(a^2+2b^2)^2-(2ab)^2$$
$$=(a^2+2b^2-2ab)(a^2+2b^2+2ab)=\{(a-b)^2+b^2\}\{(a+b)^2+b^2\}$$
For primality one of the two factor... |
H: Volume Integrals
I encountered this problem in Griffiths' Introduction to Electrodynamics. The first problem under volume integrals (Example 1.8). It reads:
Calculate the volume integral of $T = xyz^2$ over the prism in Fig. 1.24.
I understand how to perform integrals, but in this particular one I get lost whi... |
H: Largest palindrome made from the product of two 3-digit numbers
I am doing a python problem and I need to find the largest palindrome made from the product of two 3-digit numbers. So, how would I get all possible combinations?
I imagine it is multiplying all combinations of 100 to 999 but am a bit stuck on how to d... |
H: Solving $\frac{5}{t-3}-2=\frac{30}{t^2-9}$
I need help with solving this equation:
$$
\frac{5}{t-3}-2=\frac{30}{t^2-9}
$$
I tried to solve, but I always get false result. The result should be $-\frac{1}{2}$ but I always get $-\frac{1}{2}$ and $3$.
This is how I did it:
$$\begin{align}
\frac{5}{t-3}-2&=\frac{30}{t^2... |
H: What direction does a vector with more than two entries point at?
Say you are given theses two vectors:
u = (1, -2, 4)
v = (-2, 4, 8)
Since there are three entries, how do you know if they point in the opposite/same/different direction?
AI: Regardless of the dimension, two vectors are in the same direction if the ... |
H: Positive real number has a finite number of binary when is in form $ m/2^n $
Prove that positive real number $ ( x \in \mathbb{R} \ x > 0) $ has a finite number of binary if and only if when is in form $ \frac{m}{2^n} $, where $ m, n \in \mathbb{N} $
I found this solution: Floating point arithmetic
But I don't und... |
H: showing $\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$
I'm having problem with showing that:
$$\arctan(\frac{2}{3}) = \frac{1}{2} \arctan(\frac{12}{5})$$
I would need some help in the right direction
AI: $(3+2i)^2=5+12i$, so $\arctan\frac23+\arctan\frac23=\arctan\frac{12}{5}$.
Alternatively, study this... |
H: How can I show that sample mean has the smallest variance?
Let the population distribution is $N(\mu,1)$.
Sample mean: $\bar{X_n}=\frac{\sum_{i=1}^{n} X_i}{n}$
Then $E(\bar{X_n})=\mu$ and $V(\bar{X_n})=\frac{1}{n}$
It is an unbiased estimator, and as $n \rightarrow \infty$ it converges to $\mu$.
But I want to show ... |
H: Writing a parametrization of the cissoid by using $\theta$
The cissoid of Diocles is the curve whose equation in terms of polar coordinates $(r,\theta)$ is
$$r = \sin\theta \tan\theta, −\frac{\pi}{2}<\theta <\frac{\pi}{2}$$
Write down a parametrization of the cissoid using $\theta$ as a parameter and show that
$$\g... |
H: The definition of Borel sigma algebra
In the text of Probability Essentials by J.Jacod & P.Protter, a theorem:
The Borel $ \sigma $- algebra of $R $ is generated by intervals of the form $(-\infty,a ]$, where $a \in Q$.
As far as I've known the Borel sigma algebra is generated by all open subsets of $R$, which sure... |
H: composition of a strict monic with a section
In Abstract and Concrete Category of Adamek, Herrlich and Stretcker, I'm dealing with exercise 7Dd.
It says that a strict monomorphism $f:A\to B$ followed by a section $g:B\to C$ give rise to a a strict monomorphism $fg:A\to C$.
I try to prove it:
Let $f'$ such that, for... |
H: Limit as n goes to infinity of $(1+x^{n})^{\frac{1}{n}}$
Question: I can't seem to show that the limit of $(1+x^{n})^{\frac{1}{n}}$ as $n\rightarrow\infty$ is $x$, where $x$ is in the interval $[1,2]$.
Attempt: Let $y_{n}=(1+x^{n})^{\frac{1}{n}}$.
Then $\lim_{n\rightarrow\infty}{y_{n}}=\lim_{n\rightarrow\infty}{(1... |
H: Does $\bigcup _{j\in\mathbb{N}}A_j=A$
I'm reading R. Schilling's Measure, Integrals and Martingales and in a proof he makes the following statement.
"Since $A=\bigcup _{j\in\mathbb{N}}A_j$...", is this allways true?
The context:
The object is the verify the third property for a $\sigma$-algebra, that is
$(A_j)_{j\... |
H: Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$
Prove:
$$
{\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\,
\frac{\arccos\left(\vphantom{\huge A}
{\exp\left(\vphantom{\large A}\sqrt{x\,}\right)
\over
\sqrt{\vphantom{\large A}2\,}}\right)}
{1-... |
H: Confused by Example in Herstein's "Topics in Algebra"
The following comes from I.N. Herstein's "Topics in Algebra", just after defining subgroups. He gives the following example
Let $S$ be any set and $A(S)$ be the set of one-to-one mappings of $S$
onto itself, made into a group under the composition of mappings... |
H: Simplify the boolean expression to two literals
Expression:
$$[AB'(C+BD) + A'B']C$$
I start off using the distributive law, and then nowhere to go.
I need help.
AI: Indeed, we do need the distributive law multiple times. It's used in the first line, again in the second line, and once more in the fourth line.
$$\beg... |
H: Integration of $sin^2(x)$ using trig substitution
Hi I'm just learning integration, so I'm sorry if this is really basic.
I know this is true: $\displaystyle \int sin^2(x)d x = \frac{1}{2} (x - sin(x)cos(x)) + C$
Because you can use Power reducing substitution http://www.youtube.com/watch?v=VRNuPqA_Vo8
But my quest... |
H: $\{A_\alpha\}$ closed, locally finite, $f\restriction_{A_\alpha}$ continuous; show $f$ continuous
Let $\{ A_\alpha \}$ be a collection of subsets of $X$; let $X=\bigcup_\alpha A_\alpha$. Let $f:X\rightarrow Y$; suppose that $f\restriction_{A_\alpha}$, is continuous for each $\alpha$. Show that if $\{ A_\alpha \}$ ... |
H: If $X$ invertible is $XX^*$ PD?
I am wondering if $X$ invertible is $XX^*$ positive definite (PD)?
I know $XX^*$ is PSD but looking for PD.
Thanks.
AI: First for any matrix $X$, we have $XX^*$ to be positive semi-definite, since for any vector $a$, we have $$a^*XX^*a = \Vert X^*a\Vert_2^2 \geq 0$$
All we need to s... |
H: Finding an equation for a growth formula
Given a tree that has three nodes each level I want to find the formula that predicts the number of all nodes with a given tree height.
I fitted the data into Numbers with an exponential function and it gave me a numeric formula:
But I'd like to know how to derive a non-nu... |
H: what is the significance of the annulus in laurent series?
I am looking at ways which you can write a function as a series. I am aware that one can use the Taylor series. I am currently trying to understand the Laurent series. I understand there are cases where the Taylor series will not work as all the terms in it... |
H: Triple chain rule problem
$$ f(x) = \log _4 \log _2 \tan x $$
$$ f'(x) = \frac{1}{log _2 (\tan x) \ln (4)} \cdot \frac{1}{\cos ^2x \tan x \ln (2)}= $$
Is there anything else I can do with this besides changing $\tan x$ to $\frac{\sin x}{\cos x}$ and getting rid of cos's square?
AI: As $\displaystyle\frac{d(\ln y)}... |
H: Cross product intersection sets
Here's what I'm, trying to prove.
Let $A, B, C$ be non-empty sets. Prove that $A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)$
First I would need to prove $A \times (B \cap C) = (A \times B) \cap (A \times C)$
If I used distributivity rule then I get: $(A \times B) \... |
H: Does the 13th sphere fit in?
I've been searching for a formula, but couldn't find any. Here is the question:
What is the highest number of equal nonoverlapping spheres that touch a unit sphere? The distance between the points that the outer spheres touch to the inner sphere wont be any smaller than 1 unit, and thi... |
H: The Banach-Steinhaus theorem for seminormed spaces
Assume that we have a vector space $X$ over reals with a countable sequence of seminorms $p_n$ on $X$ such that:
$$
p_n(x)\leq p_{n+1}(x) \textrm{ for } n\in \mathbb N, x\in X,
$$
$$
\textrm{ for } x\in X\setminus \{0\} \textrm{ there is } n\in \mathbb N \textrm... |
H: Expectation of concave function
I want to proof that the function $\phi:\mathbb R\rightarrow\mathbb R: \phi(\lambda)=\mathbb{E}U(w+\lambda X)$, which is everywhere finite-valued, is concave.
$U:\mathbb R\rightarrow\mathbb R$ is concave and the random variable $X$ has zero mean.
My first idea was to use Taylor expan... |
H: 90,8 by 30,1 egyptian division
I am having trouble using the Egyptian method of division.
So far I have
1 30,1
2 60,2
90,8 - 90,3 -> need 5 additional units.
I know at this point I need to take parts of 30,1 but I am not sure which direction I should go from here.
I tried the 2/3 part.
1 ... |
H: Prove by induction that $n < 2 ^n $ where $n \in \mathbb{N}$
Example question in a textbook that I don't understand.
Proof works for n = 1
Setting for k makes $k < 2^k $
Setting for k + 1 makes $k+1 < 2^{k+1} $.
Here, I would be stuck, the book takes the equation to:
$k+1<2^k +1\leq 2^k+2^k = 2 \cdot 2^k=2^{k+1}$... |
H: Convergence of a series
We are considering the series of general term: $(1+\frac{1}{n})^n$
I need to find if this series converges or diverges.
1) The Alembert rule can't be applied since we find the limit equal to 1.
2) I therefore tried rewriting:
$(1+\frac{1}{n})^n= \exp(n\ln(1+\frac{1}{n}))\\= \exp(n(\frac{1}{... |
H: Find functions such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x, y) = h(r)$.
Find all non-constant function $F(x, y)\in C^2(\mathbb{R}^2)$ such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x... |
H: Let $\mathcal{U}(4)$ be a subspace of $\mathcal{P}(4)$ consisting of all polynomials that are even functions
Let $\mathcal{U}(4)$ be a subspace of $\mathcal{P}(4)$ consisting of polynomials that are even functions. Show that there exists a subspace $W \subset \mathcal{P}(4)$ such that $$\mathcal{P}(4) = \mathcal{U}... |
H: Connectedness of a union of sets.
Assume that $E$ and $F$ are conneceted subsets of the metrix space X, such that
$\bar{E} \cap F \neq \emptyset$. Prove that $E \cup F$ is conneceted as well.
When I draw A picture the statement appears pretty logic to me, but I don't know how I should prove it. Here are my attem... |
H: If a subring of a ring R has identity, does R also have the identity?
I know it does not make sense that if a subring of a ring $R$ is commutative, then $R$ is also commutative. (For example, the set consisting of the matrices whose all entries except (1,1)-entry are zero, is a subring of ring of 2x2 real matrices.... |
H: Additive form of a spectral decomposition?
I am in a class on the mathematical foundations of quantum mechanics. My professor has been talking about spectral decompositions, but they are of a different form than the ones I am used to. I have mostly used the form $A=S\Lambda S^{-1}$ where $S$ is a matrix of eigenvec... |
H: Show that if $S\subset \mathbb{R}$, $S\neq \emptyset$, and $S$ is bounded, then $\inf S \leqslant \sup S$
I proved this for a finite and/or countably infinite subset of $\mathbb{R}$, but then it quickly dawned upon me that $S$ doesn't have to be countably infinite or finite. Any hints? thanks!
AI: Let $s\in S$. Th... |
H: How to evaluate indeterminate form of a limit
I can't evaluate indeterminate form of a limit like this:
$$\lim \limits_{x\to \infty} \left (\frac {x-1}{x+4}\right)^{3x+2}$$
I tried to solve this problem by multiplying fractions' top and bottom by the conjugate of the denominator. I did it many times but I don't hav... |
H: Linear Algebra Problem about Direct Sums, Kernel and Image.
Let $V$ be a vector space, $W$ a subspace of $V$ and $T : V \to V$ a linear operator. Suppose $V = \text{Im } T \oplus W$ and $W$ is $T$-invariant. I have to prove that
(a) $W \subseteq \text{Ker }T$, where $\text{Ker }T$ is the kernel of $T$.
(b) If $V$ i... |
H: Explanation of Lagrange Interpolating Polynomial
Can anybody explain to me what Lagrange Interpolating Polynomial is with examples? I know the formula but it doesn't seem intuitive to me.
AI: The Lagrange interpolating polynomial is a tool which helps us construct a polynomial which goes through any desired set of ... |
H: Show that Function Compositions Are Associative
My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse. But my approach requires the associativity of function composition.
Let $f: X \rightarrow Y, g: Y \rightarrow Z, h: Z \rightarrow W$ be functions.
$(... |
H: Defining an inductive set
I'm having some difficulties solving an induction task.
Here is the task i'm working on:
Give an inductive definition of the given language below:
$\{(ab)^n\mid n\in\{0,1,2,\dots\}\} = \{\Lambda,ab,abab,ababab,...\}$
I'm very new to induction. What I dont understand is the statement abov... |
H: Prove Divisibility test for 11
Prove Divisibility test for 11
"If you repeatedly subtract the ones digit and get 0, the number is divisible by 11"
Example:
11825 -> 1182 - 5 = 1177
1177 -> 117 - 7 = 110
110 -> 11 - 0 = 11
11 -> 1-1 = 0
Therefore 11825 is divisible by 11. Note 11825 = 1075*11
I was thinking that we... |
H: Theorem 3.55 Rudin (rearrangement and convergence)
If $\sum a_n$ is a series of complex numbers which converges absolutely, then every rearrangement of $\sum a_n$ converges, and they all converge to the same sum.
Proof: Let $\sum a_n'$ be a rearrangement , with partial sums $s_n'$. Given $\epsilon > 0$ there exis... |
H: Another probability word problem
Again, forgive me if this is basic:
Assume you are taking two courses this semester (A and B). The probability that you will pass course A is 0.835 and the probability that you will pass both courses is 0.276. The probability that you will pass at least one of the courses is 0.981.... |
H: Given the coordinates of two of three collinear points such that $PM=MQ$, find the third point
P,M and Q are three collinear points and PM=MQ.
If P is the point (-1,4) and M is the point (5,8), find the coordinates of the point Q.
What I've Done: I've found the distance MQ which is sqrt 52. I've made the equation... |
H: On functions whose derivative equals zero almost everywhere
Suppose $f: [0,1] \rightarrow \mathbb{R}$ is continuous everywhere and differentiable almost everywhere in $[0,1]$, and $f'(x)=0$ whenever the derivative exists. Is it true that
$f(x)$ equals a constant?
It seems like the answer should be yes. However, I'... |
H: About the multiplication map $I\otimes M\rightarrow M$
so I am learning about flat modules and I found this criterion for flatness. An $R$-module $M$ is flat iff for every finitely generated ideal $I$, we get that the multiplication map $I\otimes_R M\rightarrow M$ is an injection (Eisenbud Proposition 6.1)
I am con... |
H: Is the definition of linearity redundant?
For some two functions f(x) and g(y) and for the transformation T, T is linear if:
1. T(f(x) + g(y)) = T(f(x)) + T(g(y))
2. T(cf(x)) = cT(f(x)) for c in reals.
This definition seems redundant because the first property gives:
T(cf(x)) = T(f(x) + f(x) ... [c times... |
H: How to have this definite integral?
Suppose that a function $f$ of $x$ and $y$ be defined as follows:$$f(x,y) =
\begin{cases}
\frac{21}{4}x^2y & \text{for $x^2 \leq y\leq 1$,} \\
0 & \text{otherwise. } \\
\end{cases}$$
I have to determine the value of integral for which $y\leq x$ also holds.
The answer is $\f... |
H: Finding local normal form of a holomorphic function
So I'm trying to find local coordinates to compute the local normal form of a holomorphic function.
I have $f : \mathbb{P}^1 \to \mathbb{P}^1$ given by $f(z) = \frac{z}{(z-1)^2}$.
Now we have a nice coordinate system $\varphi(z)= \frac{1}{z-1}$ and $ \psi(z) = \fr... |
H: When does the triangle have the smallest area?
The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and the point $D$ is put on the intersection between the line segment $OB$... |
H: If $(4,-1)$ is on the graph of $f$, what point must be on the graph of $g$, where $g(x)= 2f(-2(x+1))-2$?
Given that $g(x)= 2f(-2(x+1))-2$ and the point $(4, -1)$ is on the graph of $f(x)$.
What point must exist for $g(x)$? I don't know how to start this problem.
AI: Hint: If the only thing you know is $f(4)$, then... |
H: Computing the integral $\int(x-ab)x^{a-1}e^{-\frac{x}{b}}dx$
Can someone justify to me why
$\int(x-ab)x^{a-1}e^{-\frac{x}{b}}dx = -bx^ae^{-\frac{x}{b}}$?
WolframAlpha gives the answer but does not explain why. I'm absolutely new to this kind of integration. Thanks!
AI: The right hand side is the antiderivative of ... |
H: Proving Equality of 2 Functions
I have a general question, illustrated by a specific example.
The general question is how to methodically prove that two functions are equal. Much like trying to prove an "if-and-only-if" statement (by first proving the positive, then the converse), or writing a proof by induction (b... |
H: List of (pre-graduate level) exercises
I am about to get my undergraduate degree in (pure) mathematics, but I feel like I'm ill prepared to go through a graduate program. This is why I'm looking for texts like this one http://www.math.kent.edu/~white/qual/list/all.pdf in other fields (algebra, analysis and topology... |
H: How can I prove $288\mid 7^{2n+1}-48n-7$?
How can I prove $$288\mid 7^{2n+1}-48n-7$$ for all nonnegative integers $n$?
My only thought was to write $$7^{2n+1}-7-48n=7(7^n+1)(7^n-1)-48n.$$ This didn't seem beneficial at all. Please help me understand the problem.
AI: Work with induction on $n$. The base case is easy... |
H: Deducing $\sin x + \cos x \ge 1$ from $(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$
So in trig, say I have an acute angle $X$. And one can intuitively conclude that $\sin x + \cos x \ge 1$, but how does the fact that
$$(\sin x + \cos x)^2 = 1 + 2 \sin x \cos x$$
tell me that it is true that $\sin x + \cos x \ge 1$? I... |
H: $\mathbb{R}^n\times\{0\}$ has measure zero in $\mathbb{R}^{n+1}$
I want to show that $\mathbb{R}^n\times\{0\}$ has measure zero in $\mathbb{R}^{n+1}$.
For example, take $n=1$. I want to show that the $x$-axis has measure zero in the plane. I cover it with the sets $[-1,1]\times[-\epsilon/8,\epsilon/8]$, $[-2,2]\tim... |
H: Finding a directional derivative
Find the directional derivative of $f(x,y,z)=3xy+z^2$ at the point $(5,1,−4)$ in the direction of a vector making an angle of $π/3$ with $∇f(5,1,−4)$.
$f_\vec u(5,1,−4)=D_\vec uf(5,1,−4)=?$
I know how to do directional derivative questions but I have no idea about this one. I'm gu... |
H: Proving $mn=nm$ for all $n,m\in \mathbb N$
How can I prove that $mn=nm ,\forall m,n \in \mathbb N$, by using just the following axioms?
$(\forall n \in \mathbb N)(n\cdot 0=0)$
$(\forall m,n \in \mathbb N)(m(n+1)=mn+m)$,
The axioms and properties of addition.
AI: Claim 1: $0 \cdot n = 0$ for every natural number.
... |
H: List all cosets of H and K
Let $G = \mathbb Z_3 \times \mathbb Z_6$, $H = \langle (1,2)\rangle$ and let $K = \langle (1,3)\rangle$. List all cosets of $H$ and $K$.
Can somebody please explain me how to do this problem. First of all I'm having trouble how to obtain the set of $H$ and $K$.
Definition: Let H be a su... |
H: Closure Criterion for convergence of sequences
I know that $\{z\}=\bigcap\{\operatorname{cl}\,\{x_n\mid n\in S\} \mid S\subseteq \mathbb{N}\ \text{and}\ S\ \text{is infinite}\}$ is one of the criteria's of convergence of sequences in a metric space.
Here is my question :
Can the intersection of the closures of the... |
H: Modulo operation of large powers
I came through this property in a cryptography book.
$(ab)\bmod n=\bigl((a \bmod n)(b \bmod n)\bigr)\bmod n$.
There is an example in the book,
$10^n\bmod 3= (10\bmod n)^n$.
Now if I have to calculate $8^15 \bmod 17$ can I calculate $8 \bmod 17$ and multiply the answer (which is... |
H: Two part question about $\sup_y\inf_xf(x,y)\leqslant\inf_x\sup_yf(x,y)$
Background Info:
Let $X,Y\neq\emptyset$
and let $f:X\times Y\to\mathbb{R}$
have a bounded range in $\mathbb{R}$
. Also, let
$f_{1}(x)=\sup\{f(x,y):\: y\in Y\}$ and $f_{2}(y)=\sup\{f(x,y):\: x\in X\}$
Establish the Principle of Iterated S... |
H: $\int^{\infty}_{-\infty} \exp \{-\frac{1}{2} y^2\} \; dy$
$$\int^{\infty}_{-\infty} \exp \{-\frac{1}{2} y^2\} \; dy$$
I tried letting $u = -\frac{1}{2} y^2$ then $dy = - \frac{1}{\color{red}y} du$... but theres still a $y$ term in $dy$?
AI: Hint: If $I$ is the desired integral, then
$$I^2 = \int_{-\infty}^{\infty} ... |
H: Showing that the theory DTO is consistent
Toward the end of Kunen's Models of Set Theory section in his most recent Set Theory text, after talking about relativization, he begins to mention the idea of relative consistency proofs. I've been looking at the following exercise in this section.
Give a purely finitisti... |
H: Alternatinve proof for the principle of the Iterated Suprema
The back of the book gave a proof similar to the proof here Proving principle of the Iterated Suprema, but I proved it following way before I checked the back of the book. Could some one verify this proof?
Let $X,Y\neq\emptyset$
and let $f:X\times Y\to\... |
H: Is it necessarily true that if a polynomial is irreducible in $\mathbb Z_n$ ($n$ is prime) then it is irreducible in $\mathbb{Q}$?
I have played around with a couple examples and I've consistently seen a pattern where the polynomials that are irreducible in $\mathbb{Z}_n$ are irreducible in $\mathbb{Q}$. Can anyone... |
H: Does the limit of function of two variables exist
If $(a,b) \in \Bbb R^2$ and $a+b=1$,
calculate the following limit or demonstrate that there is none:
$$ \lim_{(x,y) \to (a,b)} \frac {y\sin(\pi x)}{x+y-1}$$
I'm doing that: $\lim_{(x,y) \to (a,b)} \frac {y\sin(\pi x)}{x+y-1}=\lim_{(x,y) \to (0,0)} \frac {(y+b... |
H: calculate velocity using parametric functions
if i have the following parametric functions where time is m/s :
x = 8 t
y = -5 t2 + 6 t
and i want to find the initial velocity can i do the following:
v^2 = 8^2 + 6^2
v^2 = 100
v = 10 m/s
is this correct?
also if i were to add in air friction into the horizontal ... |
H: Finding root between two points
The function $f:[0,1]\to \mathbb{R}$ is continuous, $f(0)<0$, $f(1)>0$ and there is one root in between. Using $f(0)$ and $f(1)$, the expression $\frac{1\cdot f(0)-0\cdot f(1)}{f(0)-f(1)}$ would approximate the root.
Question is, if $f'(0)$ and $f'(1)$ are available too, what would a... |
H: Finding the remainder when a polynomial is divided by a product of numbers whose remainders are known
We have a polynomial $f(x)$ with rational roots that leaves remainders $15, 2x + 1$ when divided by the polynomials $x - 3, (x-1)^2$ respectively. What is the remainder when $f(x)$ is divided by $(x - 3)(x-1)^2$?
... |
H: Testing the convergence of the given integral and finding its value
@. Does the integral $$\int_{-1}^1\sqrt{1+x\over 1-x}dx$$ exist ? If so, find it value.
I compared it with $1\over \sqrt {1-x}$ and showed that its convergent. To find its values used the substitution $x=\sin\theta$ after conjugate multiplication... |
H: Injection from $\Bbb N$ to the set of functions from the naturals to $\{0,1\}$.
Let $\Bbb N$ be the set of natural numbers and let $F$ be the set of total functions from $\Bbb N$ to $\{0,1\}$.
Construct a total injective function $g_1\colon \Bbb N\to F$.
Sounds easy, but sorta lost. Thank you.
AI: For each $n\in\B... |
H: Find the fundamental group of torus with two points removed
I'm trying to find a fundamental group of Torus \ {two points}. Any help would be really appreciated.
AI: Can you see that if you remove two points from the torus, that the resulting topological space $X$ deformation retracts to the wedge of three circles... |
H: Find the value of : $\lim_{x\to\infty} \sqrt{x+\sqrt{x}}-\sqrt{x}$
I tried to multiply by the conjugate:
$\displaystyle\lim_{x\to\infty} \frac{\left(\sqrt{x+\sqrt{x}}-\sqrt{x}\right)\left(\sqrt{x+\sqrt{x}}+\sqrt{x}\right)}{\sqrt{x+\sqrt{x}}+\sqrt{x}}=\displaystyle\lim_{x\to\infty} \frac{x-x+\sqrt{x}}{\sqrt{x+\sqrt{... |
H: How is the formula for partition of a set derived.
I don't understand how we get from the first step to the second step. I understand that it would be the product of all of the subsets, but I don't understand the simplification that is made thereafter
AI: You’re starting with
$$\binom{n}{n_1}\binom{n-n_1}{n_2}\bi... |
H: prove Cauchy sequence
I have a problem in this exercise
Suppose that ${(a_n)}$ is a sequence such that ${a_{2n}}$ ${}\le{}{}$ ${a_{2n+2}}$ ${}\le{}{}$ ${a_{2n+3}}$ ${}\le{}{}$ ${a_{2n+1}}$ for all n ${}\geq{}{}$ 0.
Show that this sequence is Cauchy iff $\lim_{n \to \infty} |a_n-a_{n+1}|=0.$
Please suggest me for ... |
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