text stringlengths 83 79.5k |
|---|
H: A bishop on a grid
Suppose that we have an $n\times m$ chessboard and bishop on the square $(1,1)$. It starts to move diagonally with the following rules:
If bishop is in any corner square except $(1,1)$, it stops moving.
If bishop meets a boundary square which is not a corner, it changes the direction of motion b... |
H: Trigonometry - How do I simplify this expression?
We have the expression $$ 13 \sin [ \tan ^{-1} (\dfrac{12}{5}) ] $$
Apparently the answer is 12, and I have to simplify it, and I'm assuming it means I have to show it's 12, without using a calculator.
Normally I show my own work in the questions, but in this case I... |
H: On subspace verification
I am struggling with the following Problem:
\begin{align}Y= \lbrace (x^4-y^4,0,0,0) \mid x,y \in \mathbb{R} \rbrace \subset \mathbb{R}^4 \end{align}
Question, is the given Set a subspace of $\mathbb{R}^4 ?$
(Answer given by my tutor: Yes)
I thought about it as follows \begin{align}(x^4-y^4,... |
H: Why can't you count real numbers this way?
Sorry but this is probably a naive question.
Why can't you generate real numbers by a*10^b, the same way as rational numbers by a/b? a and b could be integers so that you would start counting real numbers like:
a\b 0 1 -1 2 -2
0 0 0 0 0 ... |
H: Finding the limit of $\frac{\sqrt{x}}{\sqrt{x}+\sin\sqrt{x}}$
How would one find the limit of
$\displaystyle\lim_{x\to 0}\frac{\sqrt{x}}{\sqrt{x}+\sin\sqrt{x}}$
I know I have to use the L'Hospital rule.
$\displaystyle\lim_{x\to 0}\frac{\frac{1}{2}x^{-1/2}}{\frac{1}{2}\frac{1}{\sqrt{x}}+\frac{1}{2}\frac{1}{\sqrt{x}... |
H: Question about SOT and compact operators
I need some help with functional analysis / Hilbert space theory. If you have a favorite text to recommend, please let me know~
Here is my question:
Given $v_t$ be the "squeeze operator" on $H=L^2[0, 1]$, where $v_t: L^2[0,1] \to L^2[0, \frac{2-t}{2}]$ acts on $f \in L^2[0,1... |
H: Is the proof of the claim correct? Is the claim true?
We say that an integer a is divisible by the nonzero integer b, if a = bc for some integer c: When a is divisible by b, we write b | a and say b divides a.
Claim: Let a and b be nonzero integers. If a | b and b | a, then a = b.
AI: The claim is false: $2\mid (-2... |
H: A question about convex set
I need to prove the closed set $C\subseteq \mathbb{R}_{+}$ is a convex. And let $x$, $y$ be arbitrary given in $C$, I have proved that $1/2(x+y)\in C$. Then does this means $C$ is convex ?
AI: If $C$ is a closed subset of $\Bbb R_+,$ then it does indeed suffice to prove that $C$ is midpo... |
H: Probability of a random graph being bipartite
We start from an "empty" graph with $n$ vertices standing alone. Each vertex has $s$ chances to choose one vertex each chance as its neighbor, uniformly and independently from the $n$ vertices, including itself, with replacement. A vertex chooses its neighbors one by on... |
H: What is a difference between these two definitions of $T_3$ space?
Definition(1)
A space $X$ is $T_3$ iff 'For any closed set $F$ and a point not in $F$, there exist non overlapping open neighborhoods.
Definition(2)
A space $X$ is $T_3$ iff 'For any nonempty closed set $F$ and a point not in $F$, there exist non ov... |
H: How can I prove by induction that $9^k - 5^k$ is divisible by 4?
Recently had this on a discrete math test, which sadly I think I failed. But the question asked:
Prove that $9^k - 5^k$ is divisible by $4$.
Using the only approach I learned in the class, I substituted $n = k$, and tried to prove for $k+1$ like th... |
H: Two polynomials $r_1, r_2 \in R[X]$ are equal if and only if the cofficients $a_i, b_i$ are equal for all $i, 0 \leq i \leq n$ - Purely a definition?
I've read that two polynomials $r_1, r_2 \in R[X]$ on the form $r = a_nX^n + ... + a_1X + a_0$ are equal if and only if the cofficients of $r_1, r_2$: $a_i, b_i$ are ... |
H: For $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$
Prove that for $x>0$, $x + \frac1x \ge 2$ and equality holds if and only if $x=1$.
I have proven that $x+ \frac1x \ge 2$ by re-writing it as $x^2 -2x +1 \ge0$ and factoring to $(x-1)^2\ge0$ which is true because you cannot square something and... |
H: Partitioning techniques for finding large matrix determinents
I'm in a linear algebra class and we're doing determinants right now. I got this matrix to do:
$\begin{matrix} 2 & 1 & 0 & 0 & 0 \\ 3 & -1 & 2 & 0 & 0 \\ 0 & 4 & 1 & -1 & 2 \\ 0 & 0 & -3 & 2 & 4 \\ 0 & 0 & 0 & -1 & 3 \end{matrix}$
It wouldn't be hard to ... |
H: Limit of sequence :$ x_n = \frac{2n^2 + 3}{n^3 + 2n}$
Consider the sequence $ x_n = \frac{2n^2 + 3}{n^3 + 2n}, n \in \mathbb{N}$. Show that $ \lim_{n\to \infty} x_n = 0$
I have no idea how to find my $n_{\epsilon} $ such as $ n > n_{\epsilon} \Rightarrow \left| \frac{2n^2 + 3}{n^3 + 2n} \right | < \epsilon $ . I'... |
H: A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six.
I have this question as an example in my maths school book. The solution given there is:-
E = the man reports six
P(S1)= Probability that six actually occurs = $\frac{1}{6}$
... |
H: Extension of a linear map to a commutative graded algebra
Let's fix the notation, $V=\bigoplus_{i\geq 0}{V^i}$ is a graded vector space and $\Lambda V$ is the free commutative graded algebra on $V$. I have been struggling to understand this example:
Consider a graded vector space $V$ with basis $\{a, b\}$ such th... |
H: Can you help solve this cubic in root x?
Here's the original equation:
$$\frac{1}{\beta}arctan\left(\frac{\sqrt{2rx+x^{2}}}{r}\right)+\left(r\sqrt{2rx+x^{2}}-r^{2}arctan\left(\frac{\sqrt{2rx+x^{2}}}{r}\right)\right)=\frac{\pi}{2\beta}$$
which I've expanded:
$$\frac{\sqrt{2}}{\beta\sqrt{r}}x^{1/2}+\left(\frac{8\b... |
H: How to calculate this limit
I need find the follow limit: $\lim_{n\to\infty}\sqrt[n]{1^{\pi}+2^{\pi}+\cdots+n^{\pi}}$. Please help me. Thanks for your attention.
AI: Will use the property: If $\lim_{n\to\infty}a_n=a, \lim_{n\to\infty}b_n=a$, then a sequence $(c_n)$ at property $a_n\leq c_n \leq b_n$ for all $n\in\m... |
H: If 4 people have 5 different cars to choose from and two people cannot pick the same. How many different ways could people pick the cars?
If 4 people have 5 different cars to choose from and two of those people cannot pick the same(the remaining two people could have the same car). How many different ways could peo... |
H: Finding the limit of $x \sin\frac{\pi}{x}$
How can I find the limit of the following
$x\rightarrow\infty$
$x \sin\frac{\pi}{x}$
I did
$\dfrac{\sin\frac{\pi}{x}}{\frac{1}{x}}$
I took the derivative using l hospital and got.
$\dfrac{-1x^{-2} \cos \dfrac{\pi}{x}}{-1x^{-2}}$
Simplying I get
$\cos \frac{\pi}{x}$ but I ... |
H: A question about compact Hausdorff space
Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. $f=0$), then $F=X$???
AI: Yes, by Urysohn's Lemma : If $F \neq X$, then there i... |
H: Odds of 2 players meetings in a 8 person single elimination tournament
I have a 8 person tournament.
For the sake of this problem let's say odds of winning are 50% for each player.
What is the formula to figure out the odds of any 2 players meetings at any point in the tournament.
AI: with random seeding each pair ... |
H: Prove the following with equivalence statements.
I need to prove the following statement with equivalence statements.
$\exists x \in D,(P(x) \Rightarrow Q(x)) \ \text{is equivalent to} \ (\forall x \in D, P(x)) \Rightarrow (\exists x \in D, Q(x)))$
At the moment, I don't see how they can be possible equivalent as ... |
H: Specify limit of x and y in math equation
I have the following linear equation
$y = -0.13x + 3$
This calculates a ratio (y) depending on a computer size in gigabytes. The domain in negative is irrelevant ( x > 0) and the minimum ratio(y) is 1.7. How can I modify this function to better integrate those limits?
AI: T... |
H: Find the area of the largest rectangle
A rectangle is formed by bending a length of wire of length $L$ around four pegs. Calculate the area of the largest rectangle which can be formed this way (as a function of $L$).
How am I supposed to do this? If I'm interpreting the question correctly, a square would have an... |
H: At how many points is this function continuous?
Question:
Let $f$ be a function with domain $[-1, 1]$ such that the coordinates of each point $(x,y)$ satisfy $x^2 + y^2 = 1$. What is the total number of points at which f is necessarily continuous?
My Answer:
I think the answer should be zero. Here's why: the graph... |
H: Negating $(\forall a \in A)(\exists b \in B)(a \in C \leftrightarrow b\in C)$?
I'm not quite sure how to go about doing this. When negating I know the quantifiers themselves will be negated meaning that $\forall$ would become $\exists$ and vice-versa. Also I know that $\leftrightarrow$ can be written for example as... |
H: Abby, Bob, Chris, and Dan have 5 vehicles to choose from how many ways can this be done?
Abby, Bob, Chris, and Dan have 5 types vehicles to choose from a red car, red truck, green jeep, brown suv, and grey convertible. More than one person can have the same type of vehicle. If Abby and Bob cannot have the same co... |
H: How to prove this set is a closed set?
List item
Ex: how to prove the sphere
$$\left\{ (x,y,z)∈ℝ^3∣x^2+y^2+z^2=1 \right\}$$ is a closed set??
I tried to use the definition of the closed set,but it did not work out for me.
AI: My hint would be: show that the "inside" is open and the "outside" is open.
To add a ... |
H: Convergence almost everywhere and convergence in measure
Let $(\mathbb{R},\mathcal{L},m)$, let $f_{n}(x)=n\chi_{[0,\frac{1}{n}]}$ then the sequence converges to $0$ everywhere except at $x=0$ thus $f_{n}$ converges a.e.
Then in my book (Folland) we have that if $f_{n}\to f$ a.e and $|f_{n}|\le g\in L^{1}$ then $f_{... |
H: Finding the limit of $\frac{\sqrt{1+x^2}}{x^2}$
I am kind of confused when it comes to finding this limit:
$\displaystyle\lim_{x\rightarrow\infty}\frac{\sqrt{1+x^2}}{x^2}$
I did
$\dfrac{\dfrac{1}{2}\dfrac{1}{\sqrt{\arctan(x)}}}{2x}$
then I am kind of stuck I know I can multiply the complex fraction and get
$\dfrac{... |
H: How can prove that $-(-x)=x$?
I need to prove the following property, but I don't know how: $$-(-x)=x.$$
Please help me. Thanks for your attention.
AI: $-(-x)=-(-x)+0=-(-x)+x+(-x)=[-(-x)+(-x)]+x=0+x=x$ |
H: Simple upper bound for $\binom{n}{k}$
I remember seeing an upper bound for the binomial $\binom{n}{k}$ with an exponential function, something like $\binom{n}{k}\leq \left(ne/k\right)^k$. What exactly is it, and are there other similar good upper bounds for $\binom{n}{k}$?
Edit: As the link in Macavity's comment sh... |
H: Is every normed vector space, an inner product space
Let $V$ be a vector space over $\mathbb{C}$. If $V$ is an inner product space, then $V$ is normed (where the norm is defined as $\|x\|=\sqrt{(x,x)}$ ). Now if $V$ is normed, does it follow that $V$ is an inner product space ? I suspect no. I would like to see a... |
H: Area of a circular segment.
See the picture below:
How can I calculate the area in black, using no handy formulas which will give me the answer if I plug in the right values? I had the idea to take $\displaystyle \int_{0.5r}^{r}$, but the problem is I don't know which function to take the integral of.
AI: The area... |
H: Is the set $\{0,1\}$ opened or closed?
I think it is neither open nor closed because it contains none of boundary points and interior point.
I'm not sure if its correct, help please!
AI: $\{0,1\}$ is closed in $\Bbb R,$ since its complement is $(-\infty,0)\cup(0,1)\cup(1,\infty),$ which is a union of open sets, and... |
H: Boolean formula over 64 Boolean variables X
This question comes from this homework assignment from ECS20 at UC Davis.
Chess is played on an 8 x 8 board. A knight placed on one square can move to any unoccupied
square that is at a distance of two squares horizontally and one square vertically, or else two squares... |
H: How to write the proof for this?
Let $a,b,c \in \mathbb{Z}$, and $a \neq 0$. Use a proof by contradiction to show that if $(a \nmid (bc))$ then $(a \nmid b)$. The symbol $\nmid$ stands for "does not divide".
I got the layout, but I don't know how to go about this.
Assume x in D:
Assume P(x)
Assume ¬Q(x)
... |
H: Derived algebra of a lie algebra contained in an ideal
Let $\mathfrak{g}$ be a Lie algebra over $\mathbb{R}$ or $\mathbb{C}$. Assume $\mathfrak{i}$ is an ideal with $\mathfrak{g/i}$ abelian. Then the derived algebra $[\mathfrak{g},\mathfrak{g}]\subseteq \mathfrak{i}$. I don't see why this is true. I am new to Li... |
H: Evaluating integral of $\int e^{-ax} \,dP$ where $P$ is the normal distribution $N(\mu,\sigma^2)$.
I realize questions regarding integrating the normal distribution are numerous, but I wasn't able to find an already answered question that helped me with this. The integral is:
\begin{align*}
\int_{\mathbb{R}} e^{-ax... |
H: Theorem for calculating the coordinate of standard basis vectors with respect to given basis
Is there a theorem that tells how to calculate the coordinates of each of the standard vectors in $\mathbb{R}^n$ with respect to a given basis for $\mathbb{R}^n$?
AI: Let $M$ be the matrix whose columns are the vectors of t... |
H: $\frac{dx}{dt} = |x|^{1/2}$
Im looking to find 4 solutions to the ODE :
$\frac{dx}{dt} = |x|^{1/2} , x(0)=0$.
Clearly, $x=0$ is one solution.
Using seperation of variables for $x>0$ yields $x= t^2/4$ as another solution, and if we consider $x<0$, I find that $x = -t^2/4$. Could someone give a hint as to where I am... |
H: Fourier Series: Shifting in time domain
I am reading "Fourier Transformation for Pedestrians" from T. Butz. He speaks about what happens to the Fourier coefficients when the function is shift in time. I have copied the equation I have a problem with:
I don't understand the logic behind going from $f(t-a)$ in the f... |
H: Integrals Regulated functions
stuck on an example for this question,
Give an example of a regulated function $f \colon [a,b] \to \mathbb{R}$ with the properties that $\forall x \in [a,b] f(x) \ge 0 , \int_a^b f = 0$ and there is $c \in [a,b]$ with $f(c) > 0$
I think a function that fits though would be one where $f... |
H: Probability I lost when my friend told me I lost
I make a bet with a friend. There's a 1/999 chance that I will lose. I don't directly know the results of this bet so there's a 99/100 chance that he will tell the truth about the results regardless of whether he wins or loses. Suppose he tells me that I lost. What i... |
H: Difficult Integral Question
I'm trying to evaluate the following integral;
$$\int e^{(x^2 - z^2)} (2x \cos(2xz) - 2z \sin(2xz)) dz$$
I've tried splitting it up, and using integration by parts, but it just isn't coming out in a simple way. I've been stuck on this for hours. I'm sure there's some rule or trick I can ... |
H: A question from GRE math sub 9367, problem 59
Two subgroups H and K of a group G have orders 12 and 30, respectively. Which of the following could NOT be the order of the subgroup of G generated by H and K?
A. 30
B. 60
C. 120
D. 360
E. Countable infinity
A is the answer because H, with order 12 that doesn't divide ... |
H: Integrals of continuous functions (as an approximation of step functions)
I need to..
Show that a continuous function $f \colon [a,b] \to \mathbb{R}$ with the properties $\forall x \in [a,b] f(x) \ge 0$ and $\int_a^b f = 0$, must be identically 0.
Now i can see why this is true. Its continuous hence 'smooth' betwee... |
H: What is meant by 'the completion of Z'?
In the first chapter of Algebraic Number Theory (lecture notes collected by Cassels-Fröhlich), page 28 has the following paragraph:
"We suppose now that $k$ is a finite field of characteristic $p$ with $q=p^m$ elements. Denote by $\bar{\mathbb{Z}}$ the completion of $\mathbb{... |
H: what's the difference between a rational number and an irrational number?
I tried to understand the difference between rational numbers and irrational numbers. I understand what is a rational number (a number that can be expressed as the ratio of two numbers p/q).
what makes an irrational number, irrational?
how do... |
H: Why would $f_n(x) = (\lfloor 2^nf(x)\rfloor/2^n)\wedge n$ converge to $f(x)$?
Why would $$f_n(x)=\frac{\lfloor 2^nf(x)\rfloor}{2^n}\land n$$
converge to $f(x)$?
I saw this step in the proof of change of variable formula in Rick Durrett's Probability Theory and Examples.
AI: Write $f(x)$ in base two. Multiplying by ... |
H: Show that $x$ is square free iff for any $y,z$ positive integers $x=yz \Rightarrow \mathrm{hcf}(y,z) = 1$
Show that x is square free if and only if
$$x = yz\Rightarrow\mathrm{hcf}(y,z) = 1$$
where x and y are positive integers. I have tried using coprime factorisation leading to
$$1 = jy + kz$$
But cant get any fur... |
H: What is the proof that SVM can be used to solve the least squares problem with norm equality constraint?
I've seen it claimed that the solution to the minimization problem:
$$\begin{align*}
\arg \min_{b} \quad & {\left\| A b \right\|}_{2}^{2} \\
\text{subject to} \quad & {\left\| b \right\|}_{2} = 1
\end{align*}$$
... |
H: Can someone explain the solution to this statement?
Say C: set of courses
P(x,y): 'x is a prerequisite for course y'
statement: 'some courses have several prerequisites'
symbolically:
∃ x ∈ C, ∃ y ∈ C, ∃ z ∈ C, P(y, x) ∧ P(z, x) ∧ y ≠ z
I don't really understand how you get the symbolic expression from the verba... |
H: To what extent can I square both sides of an absolute equation?
I am working on some absolute equation problems like the following:
$$\begin{align}
& {|x-4|} \lt 1 \\
& 1 \le |x| \le 4 \\
& |x+3| = |2x+1|
\end{align}$$
Now, for both of these equations, I simply squared both sides to get rid of the absolute and the... |
H: Why is $|z-a|=\rho$ equivalent to $|z|^2-a(z+\overline{z})=\rho^2-a^2$?
I have some problems to understand the following statement from a book about reflections in poincare half-plane modell:
For $z,\overline{z} \in \mathbb{C}$ and $ a,\rho \in \mathbb{R}$ we have:
$$|z-a|=\rho \quad \text{ equivalent to } \quad |... |
H: Is this symbolic expression correct?
Say C: set of courses
P(x,y): 'x is a prerequisite for course y'
statement: 'some courses have the same prerequisites'
Is this symbolic expression correct?
If not, how would I write this with implication? Also how would I write this without implication?
∃ x ∈ C, ∃ y ∈ C, ∃ z ∈ C... |
H: Placing different color balls into distinguishable boxes
In how many ways can you place 4 red balls, 5 blue balls, and 6 yellow balls in 4 distinguishable boxes? (Balls with same color are indistinguishable)
AI: HINT: If you had only the $4$ red balls, this would be a standard stars-and-bars problem; the same is tr... |
H: Is $f =g$ when $g=\limsup f_n$?
f and g are two functions . Is $f =g$ $\mu$ a.e when $g=\limsup f_n$ when n -> infinity ?
we have $f_n$ --> f $\mu$ a.e , $f_n$ is measurable for all n in N.
I think they will equal because when f_n - > f , g = limsup f = f at each x that makes f_n --> f.
AI: Yes these are the same... |
H: Finding $\int \frac{x^2}{(a^2-x^2)^{\frac{3}{2}}}dx$ using trigonometric substitution. Where did I go wrong?
Evaluate the following integral using trigonometric substitution
$$\int \frac{x^2}{(a^2-x^2)^{\frac{3}{2}}}dx$$
I used the substitution $x=a \sin(u)$, then $dx = a \cos(u) du$. The integral then becomes:
$$\... |
H: Differentiating both sides of a non-differential equation
I'm working on solving for $t$ in the expression $$\ln t=3\left(1-\frac{1}{t}\right)$$ and although I can easily tell by inspection and by graphing that $t=1$, I'd like to prove it more rigorously.
I got stuck trying to solve this algebraically, so I tried t... |
H: Generators of permutation group
I want to proof that $S_n$ is generated by the set of transpositions ${(1,2),(1,3), \ldots , (1,n)}$ using that $(k,j) = (1,k)(1,j)(1,j)$ but I don't know how to continue. I know this is a easy problem but I dont know what to do.
AI: Any permutation $\sigma \in S_n$ can be written as... |
H: Why if $n \mid m$, then $(a^n-1) \mid (a^m-1)$?
My Number Theory book says that for $n, m$ be positive integers and $a>1$, then
$(a^n -1)\mid(a^m -1)$ if and only if $n\mid m$.
I understand the proof for only if part, but in if part the autor says "it is clear". However a tried to prove that but a get stuck. Can ... |
H: Is the Koch Snowflake a Compact Space?
I am taking an introductory topology class, and we recently defined the notion of compactness. Earlier in the chapter, the Koch snowflake is described, and I am wondering: is the Koch snowflake a compact set? Intuitively I think the answer is yes: it is an infinite union of cl... |
H: Using the partition theorem in probability questions
So there is one box and it contains 1 white ball and 1 red ball. When a ball is drawn, it is replaced and another ball of that colour is added to the box.
$A_i$ : the event that the ball is red on round $i$
So for example, $P(A_1)$ is $\frac 12$ and $P(A_2|A_1)=\... |
H: $\sigma\mathcal C$ is the $\sigma$-algebra generated by $\mathcal C$. Show $\sigma\mathcal C\subset\sigma\mathcal D$ if $\mathcal C\subset\mathcal D$.
If $\mathcal{C}$ and $\mathcal{D}$ are two collections of subsets of $E$. How do I prove the following:
$$\mathcal{C}\subset\mathcal{D}\implies\sigma\mathcal{C}\subs... |
H: Finding the CDF of a random variable that has uniform distribution of outcomes.
I want to find the CDF of a a random variable $X(\omega) = e^\omega$, with the sample space $\Omega = [-1,1]$. The outcomes of $\Omega$ are uniformly distributed.
What I've managed so far is to get to this point:
$F_X(x) = P(X<= x) = P(... |
H: Partial Fractions and power of a factor with $x^2$
I just started working with partial fractions and hit a wall with splitting this one:
$$ \frac{3x^2 + 2x + 1}{(x + 2)(x^2 + x + 1)^2} $$
I get here:
$$ \frac{Ax + B}{(x^2 + x + 1)^2} + \frac{Cx + D}{x^2 + x + 1} + \frac{E}{x + 2}$$
Then on to:
$$ (Ax + B)(x + 2) + ... |
H: Find the number of possible triangles
An interview question. We are given three positive integers p, q, r such that:
p + q + r = 27
and p<q<r.
Find the number of triangles that are possible using p, q, and r.
AI: HINT: Since $\frac{27}3=9$, it’s clear that $r$ must be at least $10$. In order for $p,q$, and $r$ to ... |
H: How to prove $x-y = x+(-y)$ in ring theory.
Okay, I have talked with a lot of people about this silly question. And I have thought about this way longer than is good for me. Everybody seem to disagree with me, and that is the reason I think I can't get this out of my head, because I feel like pure mathematically I'... |
H: derivative of a summation with variable upper limit
Is the following statement correct and if yes does it need to satisfy specific requirement to be correct:
$${ d \over dt} \sum_{j=1}^{N(t)} f(t,j) = \sum_{j=1}^{N(t)} {df(t,j) \over dt} + f(t,N(t)) {dN(t) \over dt}$$
AI: There are two approaches for this problem: ... |
H: Is the abelianization of a subgroup $H$ a subgroup of the abelianization of a group $G$?
Let $G$ be a finite group and $H<G$. Then, is it true that $H^{ab} < G^{ab}$, that is, the abelianization of $H$ is a subgroup of the abelianization of $G$?
To me, it would make sense if is was indeed true. However, I do not kn... |
H: I have a recursively defined function, and another function involving powers of a matrix. How can I show that they are equal?
The problem is
Let $A$ be the $n \times n$ adjacency matrix of a graph $G=(V,E)$ on $n$ vertices, i.e. $A=(a_{ij})$ and
$$a_{ij}=\begin{cases}
1 & ij\in E \\ 0 & ij\notin E
\end{cases}$... |
H: Are power series in a normal matrix themselves normal?
Are (convergent) power series in a normal matrix themselves normal? I have looked around for this result, and not found it. How might we prove it?
AI: Yes, because a matrix is normal if and only if it is unitarily diagonalizable, we can simultaneously diagonali... |
H: Simple Newton's method problem
Estimate the number of iterations of Newton's method needed to find a root of $f(x)=\cos(x)-x$ to within $10^{-100}$.
The answer is $7$ iterations, but I have no idea how it was solved by my instructor.
AI: The idea behind the reasoning is the quadratic convergence of Newton's algorit... |
H: $\frac{dy}{d \theta} = {e^y\sin^2(\theta)\over {y\sec(\theta)}}$
Please help me solve the above differential equation. I'm confused as to the steps required to obtain the answer
AI: Hint: The equation is separable, so rewrite and integrate both sides as:
$$\displaystyle \int e^{-y}~ y ~dy = \int \cos \theta~ \sin^2... |
H: Prove that $x^3 + x^2 = 1$ has no rational solutions?
Is this enough for a proof?:
$$x^3+x^2 = 1$$
I would factor and get: $x^2(x+1) = 1$
I would show that $x = \sqrt1$, which is rational but then what else would I have to show? $x+1=1$ which gives me $x=0$ and since $x$ cannot equal to $0$ as this would make the s... |
H: Suppose $g$ is even and let $h=f \circ g$. Is $h$ always an even function?
I came across one of the following problems in my homework set:
$$ \text{Suppose} \, g \, \text{is even and let} \, h=f \circ g. \text{Is} \, h \, \text{always an even function?}$$
I came to the conclusion through examples that "yes" the an... |
H: Does my proof make sense?
Theorem:
For groups $(\Bbb R,+)$ and $(\Bbb R,*)$ (both only dealing with positive integers) there is a function $\phi$ that turns $(\Bbb R,+)\to(\Bbb R,*)$ and vice versa.
Proof:
Assume $(\Bbb R,+)\to(\Bbb R,*)$. So there is a function where elements $x_1,x_2$ going from additive oper... |
H: Is this language regular ? [automata]
Is this a regular language :
$$L = \{w : w \in \{a,b\}^*\text{ and }abw = wba\}$$
Does my automata only need to start with $a$ and $b$, then loop on $a,b$ and finish with $b\to a$, or do I don't understand the language?
AI: HINT: Prove that $L$ is generated by the regular expr... |
H: $\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]] = \mathbb{E}[yp] = p\mathbb{E}[Y]$
Its given $\mathbb{E}[X \mid Y] = yp$
$$\mathbb{E}[X] = \mathbb{E}[\mathbb{E}[X \mid Y]] = \mathbb{E}[yp] = p\mathbb{E}[Y]$$
What I do not understand is the 3rd to last step. Can I just change $\mathbb{E}[y]$ to $\mathbb{E}[Y]$ li... |
H: If $\alpha$ is even show that $\beta \alpha \beta^{-1}$ is even.
Right then I have $\alpha,\beta \in S_n$ for some n and that $\alpha$ is even. I want to show that $\beta \alpha \beta^{-1}$ is even.
What I came up with:
I know that I can write $\alpha$ and $\beta$ as the product of 2-cycles (if n > 1) and I've show... |
H: Finding the associated matrix of a linear transformation to calculate the characteristic polynomial
Let $T : M_{n \times n}(\Bbb R) \to M_{n \times n}(\Bbb R)$ be the function given by $T(A)=A^t$ (the transpose of $A$).
I need to find the minimal polynomial and the characteristic polynomial of $T$. So, to find the ... |
H: Symmetry groups in algebra
Recently I was going over online notes regarding symmetry groups and I came across the following notation:
$S_3=\{1,x,x^2,y,xy,x^2y\}$ is generated by $\{x,y\}$. What does this mean? Aren't the elements in $S_3$ of the form $\{(12),(123),(23),(132),e, (13)\}$. Can someone please explain?
... |
H: If I weigh 250 lbs on earth, how much do I weigh on the moon?
One of my homework questions is to determine how much a 250 lb person weighs on the moon. I first googled a calculator for this and found that the weight is 41.5 lbs. So I tried to derive it myself and I cannot seem to get the correct answer.
Here is w... |
H: Notation for Permuting Sets
If I have some arbitrary sets $A_i : i \in I$ and I want to permute their intersections pairwise, how would I write such a permutation? Would I use some permutation tensor? Essentially I want to permute $A_i \cap A_k \, \, \forall i,k \in I$. How would I notate this formally?
AI: You wan... |
H: Deriving the Laurent series
How do I derive the Laurent series
$$\frac{1}{z^2}=\sum_{n=2}^{\infty} \frac{(-1)^n(n-1)}{(z-1)^n}$$
from $$\frac{1}{(1-z)^2}=\sum_{n=0}^{\infty}(n+1)z^n$$
It looks like I can do some sort of substitution $$z'=\frac{1}{z-1}$$
however I cannot simply to the result.
Any hint would be appre... |
H: how to show that $A(x)\nabla u\in L_\mathrm{loc}^{2}(\Omega) $ for $u\in H_\mathrm{loc}^{1}(\Omega)$
Let $\Omega\subset \mathbb{R}^n$ be a connected open set containing $0$, $u\in H_\mathrm{loc}^{1}(\Omega)$, $A(x)\leq C|x|^{-1+\epsilon}$, where $\epsilon$ is small, and we also have
$$
\|\nabla u\|_{L^2(|x|\leq R)}... |
H: probability of 4 of a kind from a deck of 52
5 cards from a deck of 52 , how many ways of four of a kind can be dealt ?
I have (13c1) to determine the rank of card
so (13c1)(4c4) is all i can think of until now
the answer is 2(13c2)(4c4)(4c1)
can someone explain how does this work ?
AI: There are indeed $\binom{... |
H: Setting up a triple integral in cylindrical coordinates
I'm confused on how to get my $\theta$ limits for my triple integral. The question reads as follows:
Let D be the region inside a cylinder whose base in the $xy$-plane is the circle $r=3\cos\theta$ and whose top is in the plane $z = 5 - x$. Set up an interate... |
H: Group isomorphism from subgroup of $U(n)\times \mathbb{Z}_n$ to $D_n$ the dihedral group of order 2n.
I have a group $G_n = U(n)\times \mathbb{Z}_n$ with the operation $(a,x)(b,y) = (ab,ay+x)$ and I have a subgroup $H_n = \{(a,b) \in G_n | a = \pm 1\}$ which I want to show is isomorphic to $D_n$ the dihedral group ... |
H: How find the minimum of the value of $n$ such $n^2\equiv 1\pmod{1007}$
let $n>1$ is positive integers,How find the minimum value of $n$,such
$$n^2-1\equiv 0\pmod {1007}$$
My try:
$$n^2-1=(n+1)(n-1)$$
and $1007=19\cdot 53$
so I guess $n_\min=1006$, But How prove it?
AI: You want $x^2\equiv1\pmod{19}$ and $x^2\... |
H: Need help in determining the volume of styrofoam used with dimensions $2.50ft + 1.50ft + 1.00ft$
So to elaborate on the title, the question is this:
The average density of Styrofoam is $1.00 \frac{kg}{m^3}$. If a Styrofoam cooler is made with outside dimensions of $3.00ft$ $x$ $2.00ft$ $x$ $1.50ft$ and inside dime... |
H: Differential equation application question
The air in a room with volume $200m^3$ contains 0.15% carbon dioxide initially. Fresher air with only 0.05% carbon dioxide flows into the room at a rate of $2m^3/min$ and the mixed air flows out at the same rate.
a) Find the amount of carbon dioxide in the room as a functi... |
H: How to prove a number is not a prime number (without a computer)
Show that
$$5994937829$$ is not prime number
How can I use math methods to prove it, and I know that this be proven using computer. But I can use only math methods to solve it.
AI: If $p = 5994937829$ were prime, then Fermat's little theorem impl... |
H: Linear transformation / Polynomial Question
$T:P_{3}\rightarrow P_{3}$ defined by $T(p(t))=tp'(t)+p(0)$ is a linear transformation.
Determine whether $T$ is invertible.
If yes, find $T^{-1}(q(t))$, where $q(t)$ is a polynomial of degree at most three.
Thank you.
AI: Hint: Just take a polynomial $p \in P_3$, say $... |
H: Evaluating $\lim_{x\to 0} \dfrac{\sqrt{1-x}-\sqrt{1+x}}{x^2-3x}$
$$\lim_{x\to 0} \dfrac{\sqrt{1-x}-\sqrt{1+x}}{x^2-3x}$$
I am stuck at radicals. Division by 1/x doesn't help.
AI: HINT:
Rationalize the numerator $$\sqrt{1-x}-\sqrt{1+x}=\frac{(1-x)-(1+x)}{\sqrt{1-x}+\sqrt{1+x}}=\frac{-2x}{\sqrt{1-x}+\sqrt{1+x}}$$
T... |
H: What is an example of a non-convex region?
In complex analysis, the proof of Morera's Theorem
"for $f\in C(D) $ such that D is a region, if for any triangle $\triangle$ in D, $\int_{\triangle}f = 0$ is True, then f is analytic in D."
splits the proof into two cases: for convex and non-convex regions D.
I'd like so... |
H: Find a particular solution for second order ODEs using undetermined coefficients method
Match the appropriate form of the particular solution labelled A through J with the differential equations below. Enter K if all of the particular solutions are incorrect.
$$y''-5y'-24y = 3xe^{2x}, (1)$$
$$y''-4y'+4y = -3xe^{2x... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.