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H: Why can't I use the chain rule to solve this trigonometric integration?
The question is: what is the indefinite integral:
$\int \sin^2(kx) \, \mathrm dx$?
I get the correct answer using trig identities to change the $(\sin(kx))^2$ into $\dfrac{1}{2} - \dfrac{(\cos(2kx))}{2}$ and integrating that. But why can't I ju... |
H: $\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$
Prove: $\alpha > \omega$ is an $\epsilon$-number iff $\beta^\gamma < \alpha, \forall \beta, \gamma < \alpha$.
From left to right:
if $\omega >\beta: \alpha=\omega^\alpha> \beta^\gamma$
if $\omega =\beta: \alpha... |
H: Are limits allowed in a function?
I wasn't sure how to specify my question correctly in my title, so I hope that my language is not too offensive.
What I would like to do is specify a function such as f(x) = .999... * x, such that f(1) represents the largest value less than 1.
I'm not sure if A: this somehow viol... |
H: Max length possible
I have a cabinet that has 15" door. I can use $3$ baskets of $15 \times 15$ in the cabinet. I will like to know if there is any possibility that $15 \times 20$ or bigger basket can fit in this cabinet.
The problem is, it will not turn inside the cabinet if it is too large. Is there any mathema... |
H: Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)
Give an example of $f\in L^1$, $g\in L^{\infty}$, such that $f*g\notin C_0$ (meaning $\lim_{|x|\to\infty}(f*g)(x)\neq0$)
Here's a theorem from my real analysis book:
Assume $1\le p\le \infty$ an... |
H: Computing Fourier sum for infinitely differentiable functions
Let $f\in C^{\infty}(\mathbb{R})$ be a periodic function of period $2L$. I want to show that $$f(x)=\sum_{n=-\infty}^\infty \left(\dfrac{1}{2L}\int_{-L}^Lf(y)e^{-in\pi y/L}dy\right)e^{i\pi nx/L}$$
The sum on the right is equal to
$$\dfrac{1}{2L}\sum_{n=... |
H: Definition of degree
By Hatcher P134, degree is defined from a map $f: S^n \to S^n$ - but degree must be able to applied to all maps. Can I arbitrarily generalize the definition of degree to a map between any two spaces?
For a map $f: S^n \to S^n$ with $n > 0$, the induced map $f_*: H_n(S^n) \to H_n(S^n)$ is an ho... |
H: A question on the restriction of the Euler’s formula
We all know the famous Euler's Formula.
It says that if a polyhedron has F(Faces), V(vertices) and E(edges) then F + V – E = 2.
My question is “is there any restriction on these variables?”
By restriction, I mean something like aF + bV + cE > 0 (for some a, b and... |
H: Orientation-preserving isometry of $R^n$
I am preparing for an exam, and would like to have a rigorous definition of the following:
Orientation-preserving isometry of $R^n$
I know that it is something like the following (feel free to correct my wording):
When the homomorphism $\pi:M_n \rightarrow O_n$ is applied t... |
H: Prove This Bool Expression
Prove $x'z+xyz+xy'z=z$ can you show how you solve this using Boolean Algebra.
My main problem is when I do this $xz (y + y') = 1 $
So $1$ times $x =$ ?
AI: $$ x'z + xyz + xy'z = z $$
$$ x'z + xz (y + y') \ // AssociationRule$$
$$ x'z + xz * 1 \ // \ a + a' = 1; a * 1 = a $$
$$ z (x'+x) ... |
H: Solve for huge linear congruence
How to solve a linear congruence with a very huge number.
For example, 47^27 congruent to x (mod 55)
My idea is to first break this into
47^27 congruent to x (mod 5)
and 47^27 congruent to x (mod 11)
then, by FlT, I can reduce this to:
47^3 congruent to x(mod 5)
and 47^5 congruent t... |
H: Permutation in $S_n$
By book states the following:
"Given the permutation $( 1 , 2)$ in $S_n$, what elements commute with
it ? Certainly any permutation leaving both $1$ and $2$ fixed does.
There are $(n - 2) !$ such. Also $( 1 , 2)$ commutes with itself. This
way we get $2 (n - 2) !$ elements in the group g... |
H: How to solve this mathematically
This is a question given in my computer science class. We are given a global variable $5$. Then we are to use keyboard event handlers to do the following:
On event keydown double the variable and on event keyup subtract $3$ from this variable.
The question is after $12$ presses of a... |
H: Real Analysis Proof Verificaition
Suppose $f$ is defined on all of ${\Bbb R}$, and satisfies, $|f(x)-f(y)|\leq(x-y)^2$ for all $x,y\in {\Bbb R}$. Prove that $f$ is constant.
Basically I have,
$|f(x)-f(y)|\leq(x-y)^2 \Rightarrow \frac{|f(x)-f(y)|}{(x-y)}\leq(x-y)$.
$\Rightarrow \lim_{\ x\to\ y}\frac{|f(x)-f(y)|}{(x-... |
H: Proof for this relation regarding complex numbers
This is what I have to prove :
Re($z_1z_2$) = Re($z_1$)Re($z_2$) - Im($z_1$)Im($z_2$).
I have two complex numbers : $z_1$ and $z_2$. Can anyone give at least some hints?
AI: Why don't you try...?
$$
(a+bi)\cdot(c+di) = \dots
$$ |
H: $f$ is Differentiable on all of ${\Bbb R}$ and $\lim_{\ x\to\infty} f^{'}(x) = 0.$ Show $\lim_{\ x\to\infty} (f(x+1)-f(x))=0.$
Suppose $f$ is Differentiable on all of ${\Bbb R}$ and that $\lim_{\ x\to\infty} f^{'}(x) = 0.$ Show that $\lim_{\ x\to\infty} (f(x+1)-f(x))=0.$
I tried to prove by this logic,
$\lim_{\ x... |
H: Expected number of balls remaining
A box contains $w$ white balls and $b$ black balls. Each time we pick a ball without replacement until there are no white balls left. What is the expected number of black balls remaining in the box ?
AI: Let $X$ be the number of black balls remaining when the last white ball is dr... |
H: Prove that G is not simple
If in a finite group G an element $a$ has exactly two conjugates, prove that G has a normal subgroup $N \ne e$.
I know to find the number of conjugates of $a$ I can use the formula $\frac{|G|}{|N(a)|} = c_a$ where $N(a)$ is the normalizer thus $\frac{|G|}{|N(a)|} = c_a = 2$ but how can I ... |
H: Order of some matrices in $GL(2,p)$ is coprime with $p$
Let $M$ belongs to $GL(2,p)$ where $p$ is a prime number, and $\det M$ generate $GL(1,p)$, so I want to prove that the order of $M$ is coprime to $p$.
I think if $M^{np}=I_2$ that means $M^n=I_2$ but how to do next?
AI: If you look at the answer to this que... |
H: Index of maximal proper subgroup of a solvable group
This is problem 2.7.15 from Hungerford's Algebra:
If $H$ is a maximal proper subgroup of a finite solvable group $G$, then $[G:H]$ is a prime power.
If $G$ is abelian, then it's easy to show that $[G:H]$ is a prime power. I'm stuck on the non-abelian case. Any ... |
H: Differentiation in several variables using projection
Could you tell me how to differentiate a function with several variables? Our teacher gave us an example:
$\pi_1 : (x,y) \rightarrow x, \ \ \ \pi_2: (x,y) \rightarrow y$ - these are differentiable, because they are linear,
Consider $f(x,y) = e^x \cos y$
Let $f_1... |
H: Adjoint of a Matrix Definition
Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree with the adjoint defined here, i.e. transpose of the cofactor matrix?
AI: These are tw... |
H: How find this integral $I=\int\int_{D}|\cos{(x+y)}|dxdy$
Find this
$$
I
=
\int\int_{D}\left\vert\,\cos\left(x + y\right)\right\vert\,{\rm d}x\,{\rm d}y
$$
where
$D
=
\left\{\vphantom{\Large A}\left(x, y\right)\quad
\left.\right\vert\quad
\ \left\vert\,x\,\right\vert + \left\vert\,y\,\right\vert\ \leq\ 2\pi\right\}... |
H: I have confusion while translating propostions to logical expressions
I have following propositions:
p:Grizzly bears have been seen in the area.
q:Hiking is safe on the trail.
r:Berries are ripe along the trail.
I need to convert following compound statement to logical expressions by using logical connectives.
If ... |
H: What is Topology of compact-convergence?
Munkres - Topology p. 283
Definition
Let $(Y,d)$ be a metric space and $X$ be a topological space. Define $B_C(f,\epsilon)$ as the set $\{g\in Y^X : \sup\limits_{x\in C} \operatorname{d}(f(x),g(x)) < \epsilon \}$ for a given compact subspace $C$ and $\epsilon >0$ and $f\in ... |
H: Proving infinite wedge sum of circles isn't first countable
Let $\{S_i\}_{i=1}^\infty$ be a countable family of circles and $\{p_i\}_{i=1}^\infty$ be a family of points such that $p_i\in S_i$. let $X = \bigcup _{i=1}^\infty S_i/\{p_i\}_{i=1}^\infty$ be the topological space obtained from the quotient map $q: \bigc... |
H: $13$ is the largest prime that can divide two successive integers of the form $n^2+3$
How can I prove following problem in number theory?
Show that $13$ is the largest prime that can divide two successive integers of the form $n^2+3$.
AI: Let $p=4k+1$ is a prime number and $q$ is another prime number such that
$$q... |
H: Solving the heat equation - derivative of an integral
I was reading a solution for the heat equation and, at some point, I had to calculate $Q_{x}$, where $$Q(x,t)=\frac{1}{2}+\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\frac{x}{\sqrt{4kt}}}e^{-s^2}ds$$
How can I do this?
AI: First, perform a simple change of variables in ... |
H: Linear algebra statment - rank of a matrix
How can I show that given an $m\times n $ matrix $A$ such that $\operatorname{rank} A = k \leq \min(m,n)$ , then there must exist a $k\times k $ minor of $A$ having $\det \neq 0 $ .
I know that $rankA=k$ implies that there exist $k$ linearly independent rows and $k$ li... |
H: Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$
Prove that $\operatorname{trace}(ABC) = \operatorname{trace}(BCA) = \operatorname{trace}(CAB)$ if $A,B,C$ matrices have the same size.
AI: Is it already known that $\operatorname{Tr}(XY) = \operatorname{Tr}(YX)$ when $X$ ... |
H: Order of an element as a product
I apologise if this has been asked before, but I wasn't able to find it..
I am trying to prove that if $g\in G$, where $G$ is an arbitrary group,$|g| = n$, where $n = ab$, with $\gcd(a, b) = 1$, then there exist $h,k\in G$ such that $g = hk$, and $|h| = a$, $|k| = b$.
So far I have... |
H: Evaluate $\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$
I have been working on this sum for a while. The question asks to evaluate the double integral.
$$\int_0^1\int_p^1 \frac {x^3}{\sqrt{1-y^6}} dydx$$
where $p$ is equal to $x^2$. I know that I have to solve the $y$ integral first and then the $x$. But I don't... |
H: Find the inverse of a matrix with a variable
$$X=
\begin{pmatrix}
2-n & 1 & 1 & 1 & \ldots & 1 & 1 \\
1 & 2-n & 1 & 1 & \ldots & 1 & 1 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
1 & 1 & 1 & 1 & \ldots & 2-n & 1 \\
1 & 1 & 1 & 1 & \ldots & 1 & 2-n\end{pmatrix}_{n\times n}
$$
Which means that ... |
H: Possible number of names from a certain alphabet
I am trying to solve the following problem, but I am a bit stuck. The question is as follows.
The language of a certain island has only the letters A, B, C, D, E. Every place name must start and end with a consonant, consist of exactly 6 letters, contain exactly 2 v... |
H: Problem with integrating by parts
\begin{eqnarray*}
\int x^3\cos 4x \ dx &=& \int x^3(8\cos^4 x - 8\cos^2x+1) \ dx\\
&=&\int 8x^3\cos^4x \ dx - \int 8x^3\cos^2x + \int x^3 \ dx\\
&=&8\int x^3\cos^4x \ dx - 8\int x^3\cos^2x + \frac{x^4}{4}+c \\
&=& ?
\end{eqnarray*}
I'm kind of stuck there. I'm new to integration, m... |
H: prove a limit about convergence of norma
Given $f_n:\Bbb R \to \Bbb R $ converge to 0 on norma 2. Show that:
$$\lim_{n\to\infty}{1\over n}\int_{-n}^n|f_n|dx=0$$
I think it has something to do with C-S inequality but i'm having troubles with it.
AI: Using Daniel Fischer's hint it is enouh to use the C-S inequality:... |
H: Almost uniform convergence of $f_n(x) = x^n$ on the interval $[0, 1]$?
Consider $f_n(x) = x^n$ on the interval $[0, 1]$.
This converges pointwise to
$$f = \begin{cases} 0, & \mbox{if } 0 \le x < 1\ \\ 1, & \mbox{if } x = 1 \end{cases}$$
Now I know $f_n(x)$ doesn't converge uniformly to $f$ but does it converge almo... |
H: What is the Krull dimension of this local ring
I want to know what is the dimension of this ring $\mathbb C[x,y]_{(0,0)}/(y^2-x^7,y^5-x^3)$.
I don't know how to do that. If I suppose $y^2=x^7$ I will get a higher degree of $x$.
AI: We compute in the local ring:
$$y^{6}=(y^2)^3=(x^7)^3=(x^3)^7=(y^5)^7=y^{35}$$
He... |
H: How to find the normal plane from a tangent plane?
$$f(x,y,z)=\frac{x^2}{4} +\frac{y^2}{9} +\frac{z^2}{25}=3 $$
I found the tangent plane from this surface at $P(2,3,5)$ by using the gradient vector, $\nabla F=\langle f_x, f_y, f_z\rangle$. I was wondering if I could find the normal plane at that same point using t... |
H: Is the submanifold compact?
Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$
we know $M$ is a submanifold of $\mathbb R^4$, is $M$ compact?
AI: Yes, because $M$ is closed and bounded.
Closedness:
$M$ is given implicitly on the form $F=0$ with $F:\mathbb{R}^... |
H: Closures of Relations
How to prove that the transitive closure of a symmetric closure of a relation is greater than the symmetric closure of a transitive closure of a relation?
AI: Essential is here that the symmetric closure of a transitive relation is not necessarily transitive. For instance on positive integers ... |
H: Using Euler's method and Taylor polynomial to solve differential equation
Consider the initial value problem
$dy/dx=x+y^2$
with $y(0)=1$
a) Use Euler's Method with step-length $h=0.1$ to find an approximation to $y(0.3)$.
HINT 1: :Numerical methods.
HINT 2: Differential equations videos.
b) Let $P2(x)$ denote the s... |
H: $L^2$ function on finite interval implies $L^1$?
Let $a,b\in\mathbb{R}$. Suppse $f:\mathbb{R}\rightarrow\mathbb{C}$ is an $L^2$ function on the finite interval $(a,b)$. That is, $$\int_{a}^b|f(x)|^2dx<\infty$$ Is it always true that $f$ is an $L^1$ function on the same interval, that is, $$\int_{a}^b|f(x)|dx<\infty... |
H: Does $x \in [0, 1]$ mean $x = 0 \lor x = 1$?
Consider the following notation:
$x \in [0, 1]$
Does this mean that $x$ can be any rational number between 0 and 1 inclusive, or does it mean that either $x = 0$ or $x = 1$?
AI: By definition $[0,1]$ is the set of real numbers $\{x\in \mathbb R \colon 0\le x \le 1\}$. |
H: What does a symplectic vector field means in terms of the physics of a system?
The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain mathematical ideas are meant to capture from ... |
H: Inequation solving. $\frac{(x+2)²}{x+1}<4$
I am trying to solve this inequality, but I always get the wrong score. This is how I did it.
$$
\frac{(x+2)^2}{x+1}<4\\
(x+2)^2< 4(x+1)\\
(x+2)^2 < 4x+4\\
x^2+4x+4 < 4x+4\\
x^2+4x+4-4x-4 < 0\\
x^2<0\\
x<0
$$
I know that I should get $x<-1$ but I always get $0$. What is my... |
H: Simple Algebra Equation
I have a simple part of a question to solve. The problem is my answer is different to the solution in my textbook.
The equation is: $$\frac{5v}{6} = \frac{(\frac{1}{2}a+b+\frac{1}{2} c)v}{a+b+c}$$
I am supposed to get $$\frac{2}{3}(a+b+c) = b$$
But I simply get: $$b=2a +2c$$
I get my answer... |
H: If There are four 2's three 1's and two 0's how many was can you arange them in a 9 Digit number!
If There are four 2's, three 1's and two 0's, in how many was can you arrange them in a 9 Digit number! Using Permutations only.
Show your answer is corrrect by counting it in three different ways and getting the same ... |
H: Determine the core-nilpotent decomposition for P.
Let P be a projector different from the identity. Determine the core-nilpotent decomposition for P.
I know that I can use the formula $Q^{-1} P Q$ and got the result $N =0$, but I don't know how to find $C$
AI: If $P$ is a projector, $P$ is annihilated by $X^2-X=X(X... |
H: Finding $a_n$ such that $x^n+a_1x^{n-1}+\cdots+a_{n-1}+a_n$ cannot be factored when $a_1,\cdots,a_{n-1}$ given
Let $n\ge 4\in\mathbb N$. Suppose that $a_1,a_2,\cdots,a_{n-1}$ are given integers.
Then, here is my question.
Question : Is the following true for any $(a_1,a_2,\cdots,a_{n-1})$ ?
There exists a composit... |
H: Bijectivity of set sequences
I've got this homework problem to prove in my introductory analysis course ... and right now, I really have no idea how to even go about that (and as such, don't really know the right questions to ask). Could you guys maybe give me few hints in the right direction?
Problem 4:
Define $\m... |
H: Homework - algebra, find constants
The question is as follows, I think I solved it partially:
Show that there are $a,b$ real positive numbers such that
$an^7 \leq \frac{n!}{7!(n-7)!} \leq bn^7$
$7\leq n$
my solution for b:
$\frac{n!}{7!(n-7)!} \leq \frac{n!}{(n-7)!} = (n-6)(n-5)(n-4)(n-3)(n-2)(n-1)(n) \leq n*n*n*n*... |
H: Congruence Equation x/20 = 7 ( mod 5)
please tell me how to solve this equation :
x/20 = 7 ( mod 5 )
I tried too many methods but it does not work.
AI: It depends on where you are looking for $x$. Most likely you search for $x\in\Bbb Z$ when you are doing congruence relations; in that case $x$ must be divisible ... |
H: Are integrable, essentially bounded functions in L^p?
Given an arbitrary measure space (of possibly infinite measure), if $f \in L^1 \cap L^\infty$, then by Hölder's inequality, $f^2 \in L^1$, so $f \in L^2$.
Intuition suggests that $f \in L^p$ even for any $1 \le p \le \infty$ (since we have eliminated the only tw... |
H: Trignometry prove question
I am new to this website so Please forgive me for my mistakes.
I have a question of trigonometry to prove and it is as (I dont know how to write theta symbol sorry for it)
$(1-\sin \theta)/(1-\sec \theta) = 2\cot \theta(\cos \theta- \csc\theta)$
Thanks in advance!!!
AI: Sometimes it helps... |
H: Are these statements correct? $A \subseteq f^{-1} \circ f(A)$ and $ f \circ f^{-1}(B) \subseteq B$
We wrote these two statements in class:
$A \subseteq f^{-1} \circ f(A)$
$ f \circ f^{-1}(B) \subseteq B$
where $A$ and $B$ are sets and $f(A)= \lbrace f(x):x \in A \rbrace $ and $f^{-1}(B)= \lbrace x:f(x) \in B \rbrac... |
H: Myhill-Nerode Theorem with constraint
I am trying to to understand the Myhill-Nerode Theorem with the example.
$L = \{{0^i1^j}|\ j > i\}$
I have read some article but still cannot fully understand,what I know about is that I have to choose the subset$\sum^*$ and find two string x and y and z so that if the language... |
H: Find the volume using the triple integral method
Find the volume of a solid bounded by:
$z=0$,
$x^2+2y^2=2$, and $x+y+2z=2$
I got this triple integral: $$\int_{-1}^1\int_{-\sqrt{2-2y^2}}^\sqrt{2-2y^2}\int_0^{1-x/2-y/2}dzdxdy$$ I think it's wrong because I keep getting a negative value. I'd appreciate any help, tha... |
H: Discrete Math need some help!
I'm taking discrete math course now and need some help on this question. THX!!
T/F or unknown?
There is a function that is both $O(n^2)$ and $\Omega(n^3)$.
Given two functions $f(n)$ and $g(n)$, if $g(n) = O(f(n))$ and $f(n) = O(n^2)$, then $g(n) = O(n^5)$.
Given two functions $f(... |
H: Evaluate the integral of a function defined by an infinite series
I need to evaluate $\,\,\,\displaystyle \int \limits_0^{2\pi} \! \sum\limits_{n=1}^\infty \dfrac{\sin nx}{n^3} \, \mathrm{d}x$ and $\displaystyle\int \limits_0^{\pi} \! \sum\limits_{n=1}^\infty \dfrac{\cos nx}{n^2} \, \mathrm{d}x$
I have already pr... |
H: Harmonic and Continuous everywhere but on a curve is harmonic throughout?
Suppose u is a harmonic function everywhere in a domain $\Omega$, but on a curve inside $\Omega$ , say a segment, and is continuous throughout, i.e $u\in C(\Omega)$. Can we conclude that u is harmonic throughout $\Omega$ ?
AI: I am not totall... |
H: Linear dependence on union of intervals
Suppose I have two functions $f,g$ which I want to exam for linear dependence. If I can conclude they are dependent on $(a,b] $ and $ [b,c)$ is it possible to conclude that they are dependent on $(a,b] \cup [b,c) = (a,c)$ ?
For example $f= t^3$, $g=|t^3|$ which are clearly de... |
H: About described DFA
I need to find DFA (or NFA, $\epsilon$-NFA, it's not improtant (I know how to convert between them)) that accept all strings of $0$'s and $1$'s such that every block of five consecutive symbols contains at least two $0$'s.
This is exercise 2.5 c) from Hopcroft's Introduction to Automata Theory.... |
H: Bounded, divergent series with terms approaching zero
Is there an example, or proof that one cannot exist, of a sequence of real numbers $a_n$ such that
(1) $a_n\rightarrow 0$ (perhaps non-monotonically),
and
(2) the sequence of partial sums $\sum_1^N a_n$ are uniformly bounded,
but the sum $\sum_1^\infty a_n$ dive... |
H: Prove that if $\int f^2$ and $\int( f'')^2$ converge, so does $\int (f')^2$
Question:
Let $f: [a,\infty) \to \Bbb R \in C^2$ and the two following integrals converge:
$$\int _a^\infty (f''(x))^2\,dx ,~~~~~~~~~ \int _a^\infty (f(x))^2\,dx$$
Prove that $\int _a^\infty (f'(x)^2)\,dx$ converges as well.
What we tr... |
H: Find $ \lim\limits_{x \to 0^{+}} x^x $
My guess we will have to reformulate the problem in order to be able to use L'Hopital's Rule. Could you give me a hint? Thank you.
AI: Hint: Try $x^x=e^{\ln x^x}= e^{x \ln x}$. |
H: Mean vs Expected Value
I have a probability distribution like so,
$$f(x)=\begin{cases}
1/(2x) & x\geq 2\\
11/40 & x=1
\end{cases}
$$
The question asks to "find the mean for X", which I calculate like so:
$\bar{x}= 1/n \cdot \displaystyle\sum\limits_{i=1}^6 x$
Which gives me, 21/6=3.5.
However, the answer to questio... |
H: Krull dimension of this local ring
I want to know what the Krull dimension of this ring $\mathbb C[x,y]_p/(y^2-x^7,y^5-x^3)$ is, where $p\neq (0,0)$. I know the dimension of it in the origin point, but I don't know other cases.
AI: Since $y^2-x^7,y^5-x^3$ are irreducible polynomials (why?) they form a regular seque... |
H: First Order linear differential equation problem
I need help solving this DE:
$(\cos\theta )v'+v=3$
I really was only able to get it here:
$v'+(\sec\theta\ )v=3\sec\theta $
$I=e^\int\sec\theta\ d\theta$
Any help would be appreciated. Thanks
AI: Hint: It is separable.
Write it as:
$$\dfrac{dv}{d\theta} = (3-v) \sec ... |
H: Example on relative homology
I am trying to prove that
$$H_p(B_{n+1},S_n;\mathbb{A}) \cong \left\{\begin{array}{ll} H_{p-1}(S_n,\mathbb{A}) & \text{if } p\geq2\\\ 0&\text{if } p=1, n\geq 1\\ \mathbb{A} &\text{if } p=1, n=0\\0 & \text{if } p=0 \end{array}\right.$$
For $p=0$ that's ok but I don't understand how to pr... |
H: MLE of fourth moment of normal distribution
Take $X\sim N(0,\theta)$, and let $\phi = E(X^4)$, the fourth moment. What is its MLE, $\hat{\phi}$, and what is the asymptotic distribution of $\sqrt{n}(\hat{\phi} - \phi) $ as $n\to \infty$? Any help with this question would be appreciated, as I really can't think of wh... |
H: Entire function with positive real part is constant (no Picard)
A problem asks to show that an entire function on $\mathbb{C}$ with positive real part must be a constant. I spoke to a professor, and asked why not just use the Picard theorem. He said that we should try to aim the solution at the level of the problem... |
H: $f(x)=\sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n $ is continuous
Let \begin{align} f: \begin{cases} \mathbb{R} &\longrightarrow \mathbb{R} \\ x & \longmapsto \displaystyle \sum_{n=0}^{+ \infty} \frac{(-1)^n}{(n!)^2}\left( \frac{x}{2}\right)^n\end{cases} \end{align}
Show that:
(i) Th... |
H: Probability in Rolling dice 6 times
If I have one die and I'm rolling the dice $6$ times.
What is the probability that in the all $6$ times the result will be the same?
I know that the probability for each number in $6$ sides dice is $\frac{1}{6}$
If I want the result to be 2 in all times, the probability is $(\fr... |
H: What's the difference between these two transformations of functions?
I'm about to graph the transformation of a function, but in this problem I encountered something new. The function transformation looks like this:
y=12(f(x)+2)
Thing is, I've never seen the f encapsulated in parentheses, so I'm unsure what effec... |
H: Variance for random variable X
I have a probability distribution like so,
$$f(x)=\begin{cases}
1/(2x) & x\geq 2\\
11/40 & x=1
\end{cases}$$
I know the mean is $2.775$ so in order to find the variance for $X$, I use
$$E(X) = \sum_{x=1}^6 (x - 2.775)^2 f(x)$$
I have tried subbing in all the values and cannot get any ... |
H: Intuition behind inner product of gradient
Let $f: \mathbb{R}^{n} \to \mathbb{B}$ be continuously differentiable and take the points $x, p \in \mathbb{R}^{n}$. I am familiar with the idea of a directional derivative being given as an inner product of the gradient of a point with another vector in $\mathbb{R}^{n}$. ... |
H: limit of a sequence $(1+1/\sqrt 2+\dots+1/\sqrt n)/\sqrt n$: Cesaro maybe?
What is the limit of:
$$\mathop {\lim }\limits_{n \to \infty } {1 \over {\sqrt n }}(1 + {1 \over {\sqrt 2 }} + ...{1 \over {\sqrt n }})$$
It looks like I need to use Cesaro theorem but I'm not sure how exactly..
AI: Hint
One approach would b... |
H: What does this infinite sum represent?
Is there a better way to write the following?
$$\sum_{i=0}^\infty \left(\frac{2}{3}\right)^i \frac{\left(\dot{f}\right)^i}{f^{2i-1}},$$
where $\dot{f} = df/dt$.
AI: It's $$ \dfrac{f^3}{f^2 - (2/3) \dot{f}}$$
whenever the infinite series converges. |
H: Expected Value Word Problem -Roulette Wheel
The probability a roulette wheel stops on a red number is 18/37.
If it lands on red and you bet red, then you receive double you bet (including your bet).
If you bet $1 on 10 consecutive plays, what is the probability that you make a profit?
My answer:
To make a profit, y... |
H: Normalizing Eigenvectors (Length equal to 1)
I am a bit confused as I have seen texts that normalize the vectors to get unique solutions and others that do not. Is there an empirical rule about when we should set the length of the vector equal to 1 and when not? Thank you.
AI: Sometimes it just makes the vectors ea... |
H: Given two subspaces $N,W$ of $V$ find a linear transformation $T:V\to V$ such that its kernel is $N$ and is range is $W$.
If $N$ and $W$ are subspaces of $V$ such that $\dim(V/N) = \dim W$, then there exists at least one element $A$ of $L(V,V)$ such that $\mbox{ker}(T) = N$ and $\mbox{range}(T) = W$.
Two related pr... |
H: Visualization of rotation in $\mathbb R^3$
I am trying to visualize the following rotation of $\mathbb R^3$, but it is very difficult. I want to get the answer by intuition, and not by using the Rodrigues rotation formula or conjugation of matrices, etc.
Help please.
Problem statement: Determine the matrix that rep... |
H: Simple limit calculation $\lim_{n\to\infty} \sqrt{n(n + 1)}/(n+1)$
Why is this limit equals 1?
$$\eqalign{
& \mathop {\lim }\limits_{n \to \infty } {{\sqrt n \cdot\sqrt {n + 1} } \over {n + 1}} = 1 \cr
& \cr} $$
I tried dividing by n, but it gives 0/0, which isn't so great..
AI: $$\frac{\sqrt n\sqrt{n+1}}{n... |
H: Determination of a constant based on continuity
The following defines function with a constant $b$ to be determined by using the continuity of a function:
$$f(x)=\begin{cases}
\dfrac{x-b}{b+1}, \quad x<0\\[1.75ex]
x^2+b, \quad x>0
\end{cases}$$
In short, for what value of $b$ is $f(x)$ continuous for every... |
H: Tensor product of a ring with itself
If $R$ is a commutative ring then $R \otimes_{R} R \cong R$. Is this still true if $R$ is non-commutative?
AI: For some left $R$-module $M$, we have $R \otimes_R M \cong M$ as abelian groups (or in fact left $R$-modules):
We have the linear map $M \to R \otimes_R M,\, m \mapsto ... |
H: How to solve $\text{ constant} = \sin(2*\theta)\;?$
What would you do to solve $0.587 = \sin(2\theta)$?
I know that this question is rather basic, but I've had no luck trying to find answers online. I was wondering if $\sin$ could be replaced by $\text{opposite} \over \text{hypotenuse}$, but I'm not exactly sure... |
H: Rendezvous problem - Dynamical Systems
I'm learning about graph-based distributed control and there's a problem called "the rendezvous problem" that uses the Laplacian matrix as the state matrix of the system. I have a graph with 4 nodes and already computed the Laplacian matrix, but I cannot figure out how to anal... |
H: Mean Value Property for Continuous Complex Functions
Suppose I have an open set $U$ in the complex plane and a function $g$ that is continuous on $U$.
Let $C(z_0$$,r)$ be a circle fully contained in $U$ of radius $r$ whose center is $z_0$.
I know that if $g$ is harmonic, then the mean value of $g$ over $C(z_0$$,r... |
H: Proof of the nonexistence of an identity $\phi$ involving convolution
The Banach space $L^1(\mathbb{R}^n)$ is an algebra with a product (convolution) which is both commutative and associative. But this algebra does not have a multiplicative identity.
An attempt to show the nonexistence:
If $\phi$ exists, consider $... |
H: Show that $\int_{0}^{1}{\frac{\sin{x}}{x}\mathrm dx}$ converges
As title says, I need to show that the following integral converges, and I can honestly say I don't really have an idea of where to start. I tried evaluating it using integration by parts, but that only left me with an $I = I$ situation.
$$\int \limit... |
H: Prove that $X$ is a finite set
Base case: $7 \in X$
Recursive case: If $x \in X$, either $\dfrac{x}{2} \in X$ (if $x$ is even) or $3 \times x + 1 \in X$ (if $x$ is odd)
Prove that $X$ is a finite set by explicitly listing all of its elements. Show how each element has been derived.
I notice that using my calculator... |
H: Finding a relationship between x and y of a DE
I have the differential equation:
$$ \frac{dy}{dx}=\frac{-5y-xy}{-4x-xy}$$
How do I go about finding a relationship between $x$ and $y$?
AI: $$\frac{dy}{dx}=\frac{-5y-xy}{-4x-xy} = \frac{y(x+ 5)}{x(y + 4)} = \dfrac{y}{y+4} \cdot \frac{x + 5}{x}$$
Now, the strong hint -... |
H: What is $\lim\limits_{i\to 0} \frac{2^n}{\frac{(n+1)\sin((n+1)\theta)}{\sin\theta } - \frac{(n-1)\sin((n-1)\theta)}{\sin\theta }} $?
What is $$\lim\limits_{i\to 0} \dfrac{2^n}{\frac{(n+1)\sin((n+1)\theta)}{\sin\theta } - \frac{(n-1)\sin((n-1)\theta)}{\sin\theta }} $$ where $$\theta=\frac{i\pi}{n} $$ The second page... |
H: lim n tends to infinity, $\frac{\sqrt{n}\sqrt{n+1} }{(n+1)}$
I am interested in the limit $\frac{\sqrt{n}\sqrt{n+1} }{(n+1)}$ as n grows without bound. There is already one question on this site asking about this limit. However this is not duplicate, since my question is rather specific.
I understand the standard w... |
H: Norms Induce by Inner Products (Complex Case)
I have just proved that if $||\cdot||$ satisfy $||u+v||^{2}+||u-v||^{2}=2||u||^{2}+2||v||^{2}$, then there exists an inner product such that $||u||^{2}= \langle u,u \rangle$ is given by $\langle u,v \rangle =\frac{1}{4}(||u+v||^{2}-||u-v||^{2})$. Now, I want to extend ... |
H: Show that a random variable is not dominated
Let $((0,1], \mathcal{B}_{(0,1]}, \lambda)$ be a probability space, and define
$$
X_n = n 1_{(0,1/n]}
$$
This is an example where $\lim E(X_n) \neq E( \lim X_n)$, and the dominated convergence theorem doesn't apply because $X_n$ can't be dominated by an integral random v... |
H: Series convergence test with geometric series
This is my first question on the math stackexchange-website. This is an assignment question, but I've tried to detail my thought process as granularly as possible to show I'm not just being lazy. My goal in asking this question is to fill a gap in understanding.
I'm be... |
H: Show that a positive operator on a complex Hilbert space is self-adjoint
Let $(\mathcal{H}, (\cdot, \cdot))$ be a complex Hilbert space, and $A : \mathcal{H} \to \mathcal{H}$ a positive, bounded operator ($A$ being positive means $(Ax,x) \ge 0$ for all $x \in \mathcal{H}$).
Prove that $A$ is self-adjoint. That is,... |
H: How many ways to seat 9 couple around a round table
You are a host/hostess at your local Applebee’s. You are seating a
group consisting of 9 couples at a round table.
A)In how many different ways can you do this, provided that each couple will sit
together, and all that you care about is their position relative t... |
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