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H: Galois group of $\mathbb{Q}(\sqrt 2+\sqrt 3):\mathbb{Q}$ compared to $\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}$
I have the following rather trivial question, but I can't seem to figure out.
If look at the galois group of $\mathbb{Q}(\sqrt2+\sqrt3):\mathbb{Q}$ then the minimal poly is $(x^2-5)^2-24$ and the roots are $... |
H: Compute $\int_0^\infty \frac{dx}{1+x^3}$
Problem
Compute $$\displaystyle \int_0^\infty \frac{dx}{1+x^3}.$$
Solution
I do partial fractions
$$\frac{1}{x^3+1}= \frac{2-x}{3 \left( x^{2}-x+1 \right)}+\frac{1}{3 \left( x+1 \right)}.$$
But we could simplify the left one $$\frac{2-x}{3\left( x^{2}-x+1 \right)} = \frac{2}... |
H: Finding order properties in the relation $aSb \iff \exists k \in \mathbb{N} : b = ak$
An order relations exercise I just did. I think it's fine, but the second proof felt a bit too wordy or discursive, instead of going straight to the point with brief and accurate statements. How could I improve that?
Over $\mathb... |
H: Is Aluffi's "Algebra: Chapter 0" a good introduction to algebra?
I'm teaching myself following Algebra: Chapter 0 and up until now (I'm at chapter 3) I'm enjoying myself. In spite of that I have some doubts.
It's not a standard text. All of the category theory, although I really like it, makes me feel like I'm not... |
H: Help with Input and Output relationships?
Here's the question: Give three examples of input-output relationships in real life that cannot have negative values in the practical range? Explain why their range cannot have negative values?
It's a confusing question because how can a relationship have both and input an... |
H: Graphing of Ceiling Functions
How do I graph the function $\lceil x^2\rceil$ (this is ceiling not just brackets). Any explanation is appreciated so I can understand how to!
AI: Remember that the ceiling function rounds a real number up to the next integer. So, the range of your function will only consist of intege... |
H: Determining equivalence classes of certain pairs for the relation $(a,b)R(c,d) \iff a^2 + 7b^2 = c^2 +7d^2$
This is an equivalence relations exercise. It has two parts. The first is about proving that the relation is of equivalence, which seems to be fine to me, but I'll put it there anyway. With the second part, h... |
H: How to approach/solve this integral?
Could somebody suggest how to approach or solve this integral:
$$
\int_{0}^\infty e^{-a t}{2+t-2\sqrt{1+t}\over t^2}{\rm d\,}t,
$$
where $a>0$ ? It is not a homework. I tried to use residuum calculation but did not find a path encircling the pole of order 2 at zero that would be... |
H: Example of invertible maximal ideal that is not generated by one element
Could anyone give me an example of an invertible maximal ideal of some integral domain which is not generated by one element?
AI: $$(2, 1+\sqrt{-5}) \subseteq \mathbf Z[\sqrt{-5}]$$ |
H: Induction proof strategy - backward induction
Normally, when using induction, I assume a statement is true for n, then I will try to show the same statement is also true for n+1.
In the problem I have now, is is correct if I assume a statement is true for n+1, then show that the statement is true for n, the the who... |
H: AMC Problem Help 12B 2010
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$ , and $a_3=\tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$?
The AMC website has a solution to this, and I wish I could post that here. I would like to know how else this could be solved, not using ... |
H: Product of non-disjoint k-cycles
I have two $k$-cycles $\alpha=(a \dots c \dots b \dots)$ and $\beta=(a \dots b \dots c \dots)$ and $\alpha \neq \beta^{-1}$. How to show that the product $\alpha \beta$ does not result in a cyclic permutation (just one cycle permutation).
Or, perhaps it has a counter example.
Thanks... |
H: How to prove Riemann integrable with partitions
I'm quite stuck on how to prove that this function:
$$
f(x) = \begin{cases} 1 & x \in [0,{1\over 2}) \\ x - {1\over 2} & x \in [{1 \over 2}, 1] \end{cases}
$$
is Riemann integrable. I've tried setting the partition $P_1 = \{0,{1\over 2} - \delta, {1\over 2} + \delta, ... |
H: Show that two matrices with the same eigenvalues are similar
First assume that $A$ and $B$ are $p \times p$ matrices and that $\lambda_1,\ldots , \lambda_p$ are distinct eigenvalues of $A$ and $B$. I want to show that $A$ and $B$ are similar.
Here is my approach:
The goal is to show that there is a nonsingular $p ... |
H: Proving at boolean algebra
Must prove that
$$(X+Y
)=X+(X.Y')$$
i tried a lot of ways, using logic things and expanding this things, but cant reach the Y.
$$(X+Y
)=(X+X).(X+Y')$$
Whats the possible prove to this?
AI: It should go like this:
\begin{align*}
X + Y &= (X + Y) \cdot 1 = (X + Y) \cdot (X + X') = XX + XX'... |
H: Regular module endomorphisms into itself
Let $k$ be a field and let $A$ be an algebra over $k$. Denote by $End_A (A)$ the set of all endomorphisms of the regular $A$-module $A$ into itself. Fix $a \in A$, and define the A-module homomorphism $r_a : A \rightarrow A$ by $r_a(x) = x \cdot a$.
Clearly, $\{r_a: a \in A\... |
H: Solve for $\sin^2(x) = 3\cos^2(x)$
I am trying to solve the following equation for x. The textbook's answer and my answer differ and I keep getting the same answer. Could someone please tell me what I'm going wrong? Notice that this is "sin squared x" and 3 * "cos squared x"
$\sin^2x = 3\cos^2x$ //Just rewriting th... |
H: Help me to prove $\operatorname{Span}(X)=F$
Let $X \subset F$ be a subset with the following property: every linear transformation $A:E \rightarrow F$ whose image contains $X$ is surjective. Prove that $\operatorname{Span}(X)=F$.
my doubt: since $\operatorname{Im}(A)=F \supset X$, then $X\subset \operatorname{Span}... |
H: Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$
Another homework problem from my Group Theory class.
Let $a,b$ be elements of a group, $G$. Let $\operatorname{ord}(a)=m$ and $\operatorname{ord}(b)=n$.
Let $a$ and $b$ commute. Prove: If $m$ and $n$ are relatively pr... |
H: Optimisation: Minimise series
Let $a_i\geqslant 0$ for $i=1,\ldots,n$.
Show how to minimize $$\sum_{i=1}^n\frac 1 {a_i+x_i}$$ subject to $$\sum_{i=1}^n x_i = b$$ where $x_i\geqslant 0$ for $i=1,\ldots,n$ and $b>0$.
I'm stuck on how to do this problem.
AI: If we ignore the constraint that $x_i\ge0$ we can use stan... |
H: If $f^{-1}(I)$ is connected for connected $I$ then $f$ is monotone.
Let $f:[0,1] \to \mathbb R$.
Claim: If $f^{-1}(I)$ is connected for all connected $I \subseteq \mathbb R$ then $f$ is monotone.
Assume $f(0) \leq f(1)$. I want to show that $f$ is monotone increasing.
I would appreciate some hints. Arguing by co... |
H: Notation for the Set of All Finite $n$-Tuples from a Set $A$
Let $A \ne \emptyset$. Let $S = \{(a_1, a_2, \ldots , a_n) : a_i \in A$ and $ n \in \mathbb{N}\}$.
Now I'm curious if there is a more concise (and standard) way of writing this set down?
AI: You could write $A^{<\omega}$ or $\bigcup_{n \in \mathbb{N}} A^... |
H: Parametrization of $S^3$ embedded in $\mathbb R^4$?
I would like to know of any parametrization of the standard 3-sphere:
{$(x_1,x_2,x_3,x_4): x_1^2+x_2^2+x_3^2+x_4^2=1$} embedded in $\mathbb R^4$.
I know of parametrizations for $S^1$, for $S^2$ , but I cannot think of how to parametrize $S^3$ as above. The closes... |
H: Convolution composed with an invertible matrix
Let $T$ be an invertible $n \times n$ matrix and let $(h \circ T)(x)$ mean $h(Tx)$.
Take functions $f,g$.
Does it hold that $(f*g) \circ T = |det(T)| (f \circ T) * (g\circ T)?$
I have had some thoughts about using the fact that $f * g = g*f,$ but I cannot see wholly ho... |
H: proof of differentiatiable function
prove that x^(1/3) is differentiable at a with f(a)'=((a^(1/3))^-2)/3 for all a not equal to 0.
I tried a epsilon-delta proof with limes theorem, and or that does not work or I am making somewhere mistake, if any one can help I would really appreciated it!Thank you!
AI: Hint:
Yo... |
H: RGB to HSV Color Conversion Algorithm
I'm a programmer looking to build an RGB to HSV color converter. I found an algorithm, but I have very little mathematical background and I'm not quite sure what's going on with it. A step-by-step breakdown of exactly what is happening would be tremendously helpful so that I ... |
H: What is $\mathbb{R}/\mathbb{Z}$ isomorphic to?
Intuitively, I see how is related to $\{e^{i\theta} : 0 \le \theta \le 2\pi \}$. I tired to use the first Isomorphism theory where it states that the image of φ is isomorphic to the quotient group G / ker(φ). Now, I am so confused now since the kernel doesn't work ou... |
H: A question about strongly continuous.
I am reading a book about C*-algebra. In the book,
Let $\phi$ be a linear functional on $B(H)$ ($H$ denotes a Hilbert space), if $\phi$ is strongly continuous, therefore, there exist vectors $\xi_{1}, \xi_{2},...,\xi_{n}$ in $H$ and $\delta>0$ such that $|\phi(a)|\leq1$, whene... |
H: Jacobian for a Cartesian to Polar-Coordinate Transformation
I have a simple doubt about the Jacobian and substitutions of the variables in the integral.
suppose I have substituted $x=r \cos\theta$ and $y=r \sin\theta$ in an integral to go from cartesian to polar-coordinate. If I use simple area rule or the standard... |
H: Find the first three terms of the Maclaurin Series
Determine using multiplication/division of power series (and not via WolframAlpha!) the first three terms in the Maclaurin series for $y=\sec x$.
I tried to do it for $\tan(x)$ but then got kind of stuck. For our homework we have to do it for the $\sec(x)$. It is... |
H: In the context of algorithms what does "bookkeeping scheme" mean?
In this paper, begining of page 5 is written:
The partial costs are then equivalent to
[...]
with the bookkeeping entities
[...]
The bookkeping scheme enable fast evaluation of the cost function.
Briefly, what is a "bookkeeping scheme"?
A... |
H: Prove that every subset of $X$ is connected in the particular point topology on X, and in the excluded point topology on X.
Let $X$ be a set and assume $p\in X$. Prove that every subset of $X$ is connected in the particular point topology on $X$ and in the excluded point topology on $X$.
http://en.wikipedia.org/wik... |
H: How do I solve this simple inequality algebraically?
How do I solve this inequality:
$\frac{1}{x} < 0 $
Its deceptively tricky. I've spent some time thinking about it, but came up with nothing. The answer is obviously $x < 0$, but how do I derive that algebraically?
Can the result be derived by performing algebraic... |
H: Why does this have a complex component?
Why does:
$$(-2)^{\frac{2}{3}}$$
have a complex component?
I thought it would be equal to:
$$((-2)^2)^{\frac{1}{3}}$$
$$= 4^{\frac{1}{3}}$$
which doesn't have a complex component.
But Wolfram Alpha says it has a complex component:
http://www.wolframalpha.com/input/?i=%28-2%29... |
H: A chain of compact subsets whose union is an open subset
Show that there is a sequence $\{K_n\}$ of compact subsets such that $ K_1 \subset K_2 \subset K_2^\circ \subset K_3 \subset K_3 ^\circ \subset \cdots$ such that the nonempty open subset in $\mathbb{C}$, $O$, $O = \bigcup_{n=1}^{\infty} K_n$
My Attempt:
Let... |
H: Why does this series diverge? $\sum_{n=1}^\infty \frac{n-1}{4n-1}$
So taking my original problem: $\sum_{n=1}^\infty \frac{n-1}{4n-1}$
I treated it like a limit problem as took the sum to be $\frac{1}{4}$ and since that is $<1$ for this geometric series, I assumed it converges. But it diverges and I don't really u... |
H: Are $A$ and $B'$ are independent events when $A$ and $B$ are independent events?
It is quite basic I think. But I am not really sure with my opinion below. Are $A$ and $B'$ are independent events when $A$ and $B$ are independent events? In my opinion the answer is yes. But what do you think?
AI: Yes, for sure. B=1... |
H: $SO(3)$ with minimal and maximal trace.
Let $O(3)$ be the set of $3 \times 3$ orthogonal matrices. Let $SO(3)$ be a subset of $O(3)$ such that det($A$)=1 for all $A \in SO(3)$. Show that there is a matrix with minimal trace in $SO(3)$ and show that there is a matrix with maximal trace in $SO(3)$.
I know the identi... |
H: Find the absolute min and max value on given interval
$f(t) = t\sqrt{16 − t^2}$ on interval $[−1, 4]$. I was able to find the critical points which are 2√2 and -2√2. All was left for me to do was to evaluate both my critical and end points into the given function. I ended up with the points: $(-1,-3.87) ; (4,0) ; (... |
H: real integrals using residues
How to evaluating this integral using residues where $a>0$:
$$\int _0^{\infty }\frac{x^3dx}{x^5-a^5}$$
Any help is appreciated
AI: As I said, the integral posted diverges. That said, let's evaluate the following real integral:
$$\int_0^{\infty} dx \frac{x^3}{x^5+a^5}$$
To do this via ... |
H: Integral of Logistic Distribution
$$\int_{-\infty}^\infty \frac{xe^x}{(1+e^x)^2} dx=0.$$
The integral represents the mean of the Logistic Distribution which is supposed to be zero. I've tried the following substitution: $u=\frac{1}{1+e^x}, du=\frac{e^x}{(1+e^x)^2} dx$ which gives:
$$\int_{1}^0 \ln(\frac{... |
H: How prove this $\binom{n}{m}\equiv 0\pmod p$
let $p$ is prime number,and such $p\mid n,p\nmid m,n\ge m$
show that
$$p\>\Big|\>\binom{n}{m}$$
I know that: if $p$ is prime number,then
$$\binom{n}{p}\equiv \left[\dfrac{n}{p}\right] \pmod p$$
But I can't prove my problem,Thank you
AI: Let us write $n$ and $m$ in base $... |
H: connectivity and graph construction
this might be very stupid question for regular maths students, but I had the following thought after reading about $2$ connected graphs, and thought about asking it. Now $G$ is $2$ connected is equivalent to (for a cycle $C$) $C = G_1\subset G_2\ldots G_n = G$, where $G_{i+1}$ is... |
H: Conjugate class in the dihedral group
List all the conjugate classes in the dihedral group of order $2n$ and verify the class equation.
The dihedral group is generated by two elements $r$ and $s$.
The order of $r$ is two since $r^2=e$ and $s$ is $n$ since $s^n = e$. And I know all elements can be produced as eithe... |
H: Finding the Integer solution to $a^3+b^2+c^2=2013$
this is my first time posting and I hope someone can help because this question has been driving me crazy... For a bit of background I am a sophomore math major in college, I have taken math up through vector calculus and Differential equations - I came across this... |
H: What are the steps to take the derivative of this function?
This calculus derivation is giving me an extremely difficult time. I am having trouble understanding how to manipulate the $e^t$. The function is
$$P*{e}^t/[(1-q)*{e}^t]$$
I want to take the derivative with respect to $t$ so $d/dt$. The answer to this prob... |
H: Any predetermined sequence in the decimal expansion of an irrational number
I came up with this question in a random math discussion with my friend. I am wondering if one can always find a predetermined sequence of numbers, such as 123456, 33333, in the decimal expansion of a given irrational number, say, pi. Since... |
H: Almost sure convergence of sample mean
I have a sequence of events $(A_n : n \in \mathbb{N})$ with $\mathbb{P} (A_n) = 1/n^2$ for all $n$. We have $X_n = n^2 1_{A_n} - 1$, and $m_n = (X_1 + ... + X_n )/n$ is the sample mean. I would like to show that $m_n \to -1$ almost surely as $n \to \infty$. Any ideas on how to... |
H: $L^p$ Spaces, Young's Theorem, Convolutions, and Minkowski's Inequality
I need to show
\begin{align}
\|f*g\|_p \le \|f\|_p\|g\|_1
\end{align}
By using the generalized Minkowski inequality instead of just Young's Theorem. I have spent a lot of time, but I keep hitting a dead end. Thanks a million in advance!
AI: $$\... |
H: How to find this Linear Transformation
Q. Find the Linear Transformation $T:V_3\rightarrow V_3$ , such that
$T(0,1,2)=(3,1,2)$
$T(1,1,1)=(2,2,2)$
I tried considering $(0,1,2),(1,1,1)$ as basis, it doesnt seem to work that way. Just need some pointers in the right direction !
AI: Consider,
$$ T(a(0,1,2)+b(1,1,1)) = ... |
H: Metric space question
Let (M,p) be a metric space and suppose that ${x_n}$ is a sequence in (M,p) so that $x_n -> x$ and $x_n->y$. prove x=y
Let $E>0$. then,
$p(x_n,x)->0$
$lim$ $n->inf$
$p(x_n,y)->0$
$lim$ $n->inf$
Suppose $p(x,y)=abs(x-y)$
$abs(x_n-x)<E$ and $abs(x_n-y)<E$
Thus, x=y. is this ... |
H: Definition of an active hyperplane
We are learning about the Geometry of Duality in Linear Programming, and my prof uses the terminology active hyperplane. I'm wondering what the formal definition of this is. I can't seem to find any other references to this online.
From my understanding if we have a linear program... |
H: Equality question
Hi I'm a bit confused with this?
$\frac{1}{x} < 0 \iff x\frac{1}{x} < x\times 0 =0 \iff 1 < 0$
This was another question that I saw which was $\frac{1}{x} < 0$ but when I multiplied by $x$ I got $1<0$?
Can anyone explain this to me?
AI: Multiplication by a negative number reverses inequalities. |
H: How to determine the equivalence classes of a relation?
I don't fully understand how to find the equivalence classes of a relation.
Over $\mathcal P(E)$, where $E = \{1,2,3,4,5,6\}$, $ARB \iff |A\cap\{1,2\}| = |B\cap\{1,2\}|$
From what I've seen, people try to make up a formula of some sort that calculates a set ... |
H: A physics related question about an infinitely long pipe.
This is a really nice question I found some days ago, so I translated it into English to share.
Suppose we have a water pipe which is infinitely long, with water
flowing in it. We know that if a molecule of water in the pipe is at a
point with the coord... |
H: Zero, the Additive Identity, as the Multiplicative Annihilator
In the structures I have encountered so far, I have always seen a zero, which is usually defined as the additive identity. For example:
$\exists 0 \in \mathbb{Z}$ s.t. $\forall a \in \mathbb{Z}, a + 0 = 0+a = a$
It just so happens to be that whenever ... |
H: Initial value problem $x'=e^{-|x|}$, $x(0)=0$
Consider $x=x(t)$ and $\frac{dx}{dt}=e^{-|x|}$, $x(0)=0$. Now I want to find the solution of this initial value problem. I want to solve it for $t\geq0$ and $t\leq0$. If $t\geq0$:
$\int_0^xe^{-s}ds=\int_0^t1dv$. Thus $e^{-x}=t+1$ and $x=-\ln(1+t)$.
For t<0:
$\int_x^0e... |
H: The limit of sequence $(\frac{n \sin(2n)}{n^2 + \cos(n) + 4})$
I have trouble evaluating the limit of the sequence $(\frac{n \sin(2n)}{n^2 + \cos(n) + 4})$. Could anyone help me? Thank you!
AI: The comment pretty much says everything you need, anyway, here is an alternative approach:
$$0 \le \Big|\frac{n\sin(2n)}{n... |
H: Subgroup of a nilpotent group
Let $G$ be nilpotent and $H \le G$. Let $P_1,P_2,\ldots,P_k$ be the Sylow subgroups of $H$. Is it true that $H = P_1 P_2 \cdots P_k$?
I know that when $G$ is nilpotent, it is the direct product of its Sylow subgroups, but is that true for a subgroup of $G$ as well?
AI: Yes,because a su... |
H: Simple question about complex $e^{i}$ and angles
I'm working with angles.
I have a hard time figuring something.
In electric physics, I have an equation describing an AC voltage function, this way
$V_{x} = 0.0469 \cdot e^{-j \cdot 1.083}\cdot e^{j(200\pi \cdot t)}$
Well, I can't get why it does in the solutions. In... |
H: Solving ODE by contraction mapping
Let a and c be real numbers. Solve the initial value problem y'(x) = ay(x), y(0) = c on the interval [0, 1/2a] with the help of the contraction mapping theorem.
I understand that solving this ODE is equivalent to finding the fixed point of a contraction map, but im not sure about... |
H: The determinant of a linear transformation on a finite vector space
Given a finite vector space $V$ and a linear transformation $f : V \rightarrow V,$ is it true that for any two ordered bases of $V$, call them $a$ and $b$, the determinant of the matrix of $f$ with respect to $a$ will always equal the determinant o... |
H: exponential generating function of $\frac{1}{(1-x)^2}$
Hey just had a quick question related to coming up with an explicit formula for $a_n$ with relation to $F(x)=\frac{1}{(1-x)^2}$. I know to compute such a function I have to use the derivative of $\frac{1}{1-x}$ to obtain the summation of $(n+1)x^n$, but I am un... |
H: Prove by induction $n^3 < 3^n$. What is the value of $n_0$?
Prove by induction for $n \geq n_0$, $n^3 < 3^n$. What is the value of $n_0$?
AI: $n^3 < 3^n$ when $n \ge 4$, so $n_0 = 4$. This is our base case since $64 < 81$.
Assume the result to be true for $n=k$, then $k^3 < 3^k\implies 3k^3 < 3^{k+1}$.
We want to s... |
H: Finding pattern (2,7,8,3,5)
What would replace $Z$ in the following sequencing? $$\begin{matrix}
2 & 3 & 4 \\
7 & 6 & 5 \\
8 & 7 & 1 \\
3 & 0 & Z \\
5 & 4 & 3 \\
\end{matrix}$$
$a)\ 1$
$b)\ 2$
$c)\ 4$
$d)\ 7$
Note. I've added my answer below. If you have another way t... |
H: Probability, mathematical symbol
Good day,
Would like to ask about the meaning of ^ in P(S^B) as shown in the image below. Thanks for your help!!
Regards,
Math noob
AI: The $\wedge$ symbol is the and operator, so on the first line, $P(S \wedge B)=P(S)P(B)$ says "The probability of both $S$ and $B$ equals the probab... |
H: Most unusual form of mathematical induction
After reading the algebraic proof of Fundamental Theorem of Algebra, where induction was carried out on "The highest power of $2$ dividing $n$", which I regard to be unusual and brilliant at the same time, I wondered if there were other problems where similar unusual indu... |
H: Verify two martingale properties
I have a process $S_n=X_1+...+X_n$ where all the $X_i$ are iid and $E[X_1]=\mu, Var(X_1)=\sigma^2$ and $\phi(\theta)=Ee^{\theta X_1}$
Now I want to prove two things.
(1) $S_k^2-\sigma^2 k$ for $k\ge0$ is martingale iff $\mu=0$
(2) $e^{\theta S_k}\phi(\theta)^{-k}$ is a martingale fo... |
H: A detail in the proof of Stone representation Theorem
Let $(\mathcal{B},\sqcap,\sqcup,\leq)$ be a Boolean algebra. Let $x,y\in\mathcal{B}$. I want to prove the following implication:
$$x\sqcap y'\leq 0\Rightarrow x\leq y$$
where $y'$ is the complement of $y$.
I have checked that this works in the case $\mathcal{B}... |
H: Why is $\sin(t)\cos(t)$ equal to $\frac{1}{2}\sin(2t)$?
I know that $\sin(t)\cos(t)$ is equal to $\frac{1}{2}\sin(2t)$ but I do not understand why, please explain it to me!
AI: $\sin 2t =\sin (t+t)= \sin t \cos t+ \cos t\sin t =\ldots$. |
H: How to find $x^4+y^4+z^4$ from equation?
Please help me.
There are equations: $x+y+z=3, x^2+y^2+z^2=5$ and $x^3+y^3+z^3=7$. The question:
what is the result of $x^4+y^4+z^4$?
Ive tried to merge the equation and result in desperado. :(
Please explain with simple math as I'm only a junior high school student. Thx a l... |
H: Power series question
$$\sum_{k=1}^{\infty}\left({-2 \over D } \right)^k \left[\frac{\Pi_{i=1}^{k}(2i-1+ \lambda) }{(2k)!}x^{2k}\right]$$
Provided that for $k=0$ the series is $1$.
What function hold this kind of series? Is this a cosine series? if YES then what is the final answer in term of cosine function?
AI: I... |
H: What sets are Lebesgue measurable?
I cannot detect the fallacy in the set of the following statements in my inconsistent notes:
A sigma algebra is a set of the sets in the generating set closed under the set operations countable union, countable intersection, set difference, relative complement.
A Borel set is a s... |
H: Is the product of two numbers both less than one less than one
I'm bad at mathematics, and I wanted to know something.
Say there are two numbers $a$ and $b$ where $a, b \in \Bbb R$ $-1 < a < 1$ and $-1 < b < 1$
Is it necessary that $a \times b < 1$?
Edit: I was in hurry and didn't notice the big mistake I did
AI: Y... |
H: Convergence or divergence of integral $\int_1^\infty \frac{x \sin x}{\sqrt{1+x^5}}dx$
I'm struggling with how to show that
$$
\int_1^\infty \frac{x \sin x}{\sqrt{1+x^5}}dx
$$
either diverges or converges.
If we call the integrand $f(x)$ then
$$
f(x)\leq g(x)=\frac{x}{\sqrt{1+x^5}}\forall x\in[1, \infty)
$$
so I tr... |
H: Some basic questions about matrix rings and reversibility.
Neither commutative rings nor division rings are viable approaches to studying rings of matrices. However, there is a very cool notion of a reversible ring, which looks like it can fill this void. I have a few basic questions, but first, here's a little inf... |
H: limit of a sequence $(1+1/\sqrt 2+\dots+1/\sqrt n)/\sqrt n$ - need a review of my solution
$$\eqalign{
& {a_n} = {1 \over {\sqrt n }}(1 + {1 \over {\sqrt 2 }} + ... + {1 \over {\sqrt n }}) \cr
& = {1 \over {\sqrt n }} + {1 \over {\sqrt n \sqrt 2 }} + ...{1 \over n} \cr} $$
Now, it's easy to see the sequence ... |
H: Question on proof of weak compactness of $L^p$
Suppose $L^q(X,\mu)$ is separable (i.e. admits a countable dense subset). I wish to prove that every sequence $\{f_n\}$ in $L^p$ that satisfies $\sup_n \|f_n\|_p < \infty$ has a weakly convergent subsequence, i.e. a subsequence $f_{n_k}$ and $f \in L^p(X,\mu)$ such tha... |
H: Numerical range of a matrix contains the convex hull of the eigenvalues.
I am stuck with the following question.
Question: Let $A \in \mathbb{C}^{m \times m}$ be arbitrary. Let $W(A)$ be the numerical range i.e. the set of all Rayleigh quotients of $A$ corresponding to a all nonzero vectors $x \in \mathbb{C}^m$. Sh... |
H: Functions exercise for $f(x) = \begin{cases} x \textrm{ if } x \le 3 \\ 11 - 2x \textrm{ if } 3 < x\end{cases}$
Could you check on my answers? Any other observation is appreciated. I'm a bit new to the injective and surjective topics.
Given $f : \{1,2,3,4,5\} \rightarrow \{1,2,3,4,5\}$ defined by
$$f(x) = \begin{c... |
H: show that the even numbers 2k+2,2k+4,...,4k,4k+2 are congruent mod m to...
(first post, hello!)
I'm having a bit of trouble with the following problem:
let k be a positive integer and let $m = 4k + 3$
show that the even numbers $2k+2, 2k+4,..., 4k, 4k+2 $ are congruent mod
$m$ to the negatives of the odd numbers... |
H: Tensor Multiplication - Why should we use permutations?
I'm reading a book on multilinear algebra, and the author first establishes this easy isomorphism: if $V_1,\dots,V_k$ are vector spaces over the field $K$ and if $\sigma\in S_k$, then there is an isomorphism $f_\sigma : V_1\otimes\cdots\otimes V_k\to V_{\sigma... |
H: A limit problem involving repeated cosines
I was playing around on my calculator and I found something interesting:-
Let's say I take some value $x$ in degrees and apply the following operation: $cos(cos(cos(cos....(x)))))...)$. This always seems to converge to the value $0.999847741531...$, regardless of $x$. It d... |
H: Proof using properties of an isosceles or right-angle triangle
Given a $\triangle ABC$ with sides $AB=BC$ and $\angle B=100^\circ $,
prove that $$a^3 + b^3 = 3a^2b$$
where $a=AB=BC$ and $b=AC$,
I have tried to use simultaneously the sine and cosine rules as well as the Pythagorean Theorem with all my attempts fail... |
H: Problem in convex analysis
I found this problem in one of the old exams for convex analysis:
Let $A \subseteq \mathbb{R}^n$ be a convex set and $f:A \rightarrow \mathbb{R}$ a convex function.
a) Show that $f^{-1}(-\infty,a)$ is a convex set for every $a \in \mathbb{R}$.
b) Find an example when $f^{-1}(0,\infty)$ is... |
H: Premeasure on an algebra: $\sigma$-additive $\Rightarrow$ finite additive?
I have a very short question concerning a proof. If I have an algebra $\mathfrak{A}$ and a set function $\mu\colon\mathfrak{A}\to [0,\infty]$ for which I have to show that it is a premeasure on $\mathfrak{A}$, then I have (besides other th... |
H: Can closed sets in real line be written as a union of disjoint closed intervals?
It is known that open sets in real line can be written as a countable union of disjoint open intervals. (link) I'm curious that if there is similar statements for closed sets in real line.
AI: Closed sets are the inverse of open sets. ... |
H: What's the name of this equation? Please give me some document about it.
I have a equation. Can someone help me?
AI: It is an inhomogeneous linear elliptic equation with homogeneous Dirichlet boundary conditions. If $\mu$, $\beta_1$ and $\beta_2$ are constant, then then it is said to have constant coefficients. |
H: Decide the limit of the sequence $\{n^3/2^n\}_{n=0}^\infty$
Decide the limit of the sequence $\{n^3/2^n\}_{n=0}^\infty$
I've verified that the sequence is indeed monotonic decreasing for $n \ge 11$ using induction. Also, the sequence is bounded. This implies the sequence is convergent, and I know the limit is $0$.
... |
H: Why are differential forms more important than symmetric tensors?
In differential geometry, differential forms are totally anti-symmetric tensors and play an important role. I am led to wonder why do we not study totally symmetric tensors as much as forms. What properties of differential forms makes them so useful ... |
H: Probability Problem on Divisibility of Sum by 3
From the 3-element subsets of $\{1, 2, 3, \ldots , 100\}$ (the set of the first 100 positive integers), a subset $(x, y, z)$ is picked randomly. What is the probability that $x + y + z$ is divisible by 3?
This is a math Olympiad problem. I would welcome a good soluti... |
H: Set theory proof with empty set
Prove that if $A \times B = A \times (C \setminus B) $
$then: A \times (B \bigcup C) = \emptyset$
I get that $B=(C\setminus B)$ so that means either C is an empty set or C and B have no element in common. (Striked text is probably wrong)
What can I do from here on ?
Thanks.
AI: We ... |
H: Limit of a function when x approaches infinity
I need to prove the following limit using definition only:
$$\lim_{x\to -\infty} \frac{(7x+3)}{(x-1)} =7.$$
The definition is: for any $\epsilon >0$. there is a $\delta$ so $x<\delta \implies |f(x)-L|<\epsilon$
In order to show that $|f(x) - L| < \epsilon$ I assume... |
H: Richardson Iteration
Given the Richardson Iteration, $x_{n+1} = x_n + \alpha(b-Ax_n)$ (with
$\alpha$ a scalar constant). To which polynomial $p(A)$ at step $n$
does this iteration correspond to?
My first idea would be to write out this recursion, for example if $n=2$, then
$x_3 = x_2 + \alpha (b-Ax_2) $
$= (x... |
H: Pre-multiplying and post-multiplying matrices give the same diagonal elements?
If
$$X = \left[ \begin{array}{ccc}
3 & 4 & 1\\
4 & 1 & 3\\
1 & 3 & 4\end{array} \right]$$
find the possible matrix $Y$ such that:
$$XY - YX = I$$
The method my professor gave us was that if we observe the diagonal elements of $XY$... |
H: Need help with this question. A graph has K10 as a subgraph. What does this tell us about the size of a maximum matching?
Need help with this question. A graph has K10 as a subgraph. What does this tell us about the size of a maximum matching? (Does it give us an upper bound? A lower bound? Or does it tell us noth... |
H: Is the sum of this series convergent? and how to find the sum?
I don't know if this is a duplicate, but I can't seem to find how to prove that the sum of this series is convergent, this series is actually the area hyperbolic tangent function.
An additional question is: if this series is convergent, then how to find... |
H: Prove that d(.;A) is continuous
I need help with the following proof, which my professor added for practice (but not as homework). I am completely lost here.
Let $A$ be a nonempty subset of a metric space $X$. Define $d(\cdot ,A) : X \to [0,\infty)$ by
$$d(x,A) = \inf\{d(x,a) : a \in A\}.$$
Prove that $d(\cdot,A)... |
H: Drawing two numbers from a set
Two numbers $X_1$ and $X_2$ are drawn randomly from the set
$\{1,2,...,n\}$ without replacement. Find $P(X_2 > X_1)$.
Now I know that once we choose $X_2$ we have $X_2 - 1$ options for $X_1$ to hold the condition that $X_2 > X_1$ but the fact that we choose $X_1$ first kind of mak... |
H: Help using substitution to evaluate $\int_3^6(-x^2+2x+3)^2\,dx$
Can you point out where I go wrong in integrating $$\int_3^6(-x^2+2x+3)^2\,dx$$
I run into trouble trying to substitute in $u=-x^2+2x+3$ and then get
$$\frac{du}{dx}=-2x+2$$
$$\frac{du}{-2x+2}=dx$$
$$\int_3^6u^2\,\frac{du}{-2x+2}$$
$$\left[\frac13 (-x^... |
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