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H: prove that $\int(f(x)+g(x))dx= \int f(x)dx+\int g(x)dx$
Let $f,g$ be two functions defined on $A$. Supposed that $F$ and $G$ are anti-derivative of $f$ and $ g$. Prove that
$\int(f(x)+g(x))dx= \int f(x)dx + \int g(x)dx$
Here is what I got.
Let $H(x)$ be a function such that $H'(x)=f(x)+g(x)$
Since $F$ and $G$ are... |
H: Find the sum of $\sum_{n=1}^\infty (-1)^{n+1} x^{2n-1}$
What is the sum of the following series? $$s(x) = \sum_{n=1}^\infty (-1)^{n+1} x^{2n-1}$$ $$x \in (-1, 1]$$
I would use Taylor series for $\sum_{n=1}^\infty x^n = \frac{1}{1-x}$ but I don't know how to treat the $(-1)^{n+1}$.
So again, what is the sum of $s(x)... |
H: Unsure about notation with matrix
I'm not sure about a notation:
Let $X= A\mathbb R^3$ where $A$ is a $3$x$3$ matrix. What is $X$? I think it should be the image of $\mathbb R^3$ under the transformation with matrix $A$ but I am not sure.
AI: You've got it right. It's the image of the transformation given by $A$. ... |
H: One point set in $[0,1]^{A}$ is not $G_\delta$ when A is not countable
I need to prove that one point set in $[0,1]^{A}$ is not $G_\delta$ when A is not countable
I tried something like this: assume that $\{x\}$ is $G_\delta$ for some x $\in$ $[0,1]^{A}$
then $\{x\} = \bigcap_{i=1}^{\infty } U_i$
I know that eac... |
H: How can I show the equality of integration for shifting simple functions over $\mathbb{R}$
Let $\phi(x) = \sum_{k=1}^n a_i\chi_{E_i}(x)$ be a simple function on $\mathbb{R}$ with finite support. I want to show that
\begin{equation}
\int_\mathbb{R} \phi(x) = \int_\mathbb{R} \phi(x+t).
\end{equation}
It seems rather... |
H: Fundamental Theorem and Integral Problem
$F'(1)$ given that
$$F(x) = \int_{5}^{x^9}\frac{1}{5+t^2}dt .$$
So far I have simplify the problem to
$$F'(x)=\frac{9x^8}{5+x^9} .$$
So what I'm wondering is do I replace x with 1 $F'(1)$ or do I replace x with the $\int_{5}^{x^9}$ and subtract them?
Can anybody please h... |
H: For which values of $x$ does the sequence not converge?
I have a wee problem.
$$x_{i+1} = \frac{2}{18} (2x_i+1)^2$$
So - for what values of $x_1$ does the sequence not converge?
Now, some pesky classmate has borrowed my notes, so I ran out of ideas pretty quickly. I'm sure the solution is quite obvious, but for t... |
H: inverse trig and an algebra nitpick?
Let's say you have:
$-cot(y)=FOO$.
Normally, you would multiply through by (-1)
$cot(y)=-FOO$
and finally,
$y=cot^{-1}(-FOO)$
But, is it valid to go from
$-cot(y)=FOO$
to
$y=-cot^{-1}(FOO)$ ?
Does this only work in certain situations?
Even/Odd function?
(By the way, how can ... |
H: Naive Bayes to Predict a class label
How do I use Naïve Bayes to predict a class label for a test sample
$(A=1, B=1, C=1)$
I know Bayes Theorem is:
$$P(C|A) = [P(A|C) P(C)]/P(A)$$
I have no idea how to do this, please help.
AI: Informally, what Bayes' rule here calculates is: "What is the probability that $C$ occ... |
H: How to naturally understand Inclusion-Exclusion Principle.
Workbook question:
Professor is giving an exam. (A) How many ways are there to assign five proctors to the three exam rooms, so that each exam is proctored by at least one person?
Answer:
$n = 5, a = 3$
$3^5 - \binom{3}{1}(3-1)^5 + \binom{3}{2}(3-2)^5 = 1... |
H: $|a_n|$ is diverging. Prove $a_n$ has a subsequence converging to a finite limit
$|a_n|$ is diverging.
How do you prove $a_n$ (without absolute value) has a subsequence converging to a finite limit?
I know that if a sequence has two subsequences converging to different numbers, then the sequence is diverging. May... |
H: How is the action of scalar ring $S$ on $M\otimes_R N$ well-defined?
Let $\sum^k m_i \otimes n_i = \sum^l m_j' \otimes n_j'$ be two representations of the same tensor in the abelian group $M \otimes_R N$, where $M$ is an $(S,R)$-bimodule and and a left $R$-module. How is the action of $S$ on $M\otimes_R N$, defi... |
H: One to one correspondence between sets of base 2 and base 4 sequences
We need to find a one to correspondence (injection) between the set of all binary sequences and the set of all quaternary $\{0,1,2,3\}$ sequences.
This is what I came up with:
$B=$ binary sequence
$Q=$ quaternary sequence
$\phi : Q \to B = [f(1\s... |
H: Prove positive, semi-definite
Let $A \in M_2(\mathbb C)$ and let $A^*$ denote the conjugate transpose of $A$. I need to show that $A^{*}A$ is positive semi-definite.
So I need to show $$\quad\quad \langle A^{*}\!A\,h,\;h\rangle \geq 0,\mbox{ for all }h \in M_2(\mathbb C)$$.
I don't know how to approach this, any... |
H: $( x \cdot y ) \mod 37 = 1$
I am doing a paper for my security class. I have this equation which I'm trying to understand
$$( x \cdot y ) \mod 37 = 1 $$
e.g. if $x = 8$ and $y = ?$ ; which has to be the inversion of x . then y in this case is 14.
My question is how do I solve a... |
H: General Topology Question
$Y = [-1, 1]$ induced by the subspace topology from $\mathbb{R}$.
$A = (-1, -1/2)\cup(1/2, 1)$ and $B = (-1, -1/2]\cup[1/2, 1)$.
a) Are $A$ and $B$ open or closed in Y with the subspace topology?
b) Are $A$ and $B$ open or closed in $\mathbb{R}$ with the standard topology?
My attempt at ... |
H: An ordered list of tuples whose elements sum to $n$
I am trying to find an method for creating an ordered list of tuples whose elements sum to $n$.
For instance if $n=15$, then the list should include the tuple
$(5,5,5)$ because $5+5+5=15$, $(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)$ because $1+1+1+1+1+1+1+1+1+1+1+1+1+1+1=15... |
H: Absolutely continuous, strictly positive measures
Let $\mu$ be a strictly positive $\sigma$-finite measure on $\mathbb{R}^{n}$ that is absolutely continuous with respect to the Lebesgue measure, $\lambda$. One way to think about the absolute continuity requirement is as a lower bound: the $\sigma$-ideal $N$ of $\m... |
H: How to show that $\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)$?
Let $\mathbb{Z}[i]$ be the ring $\{a+bi:a,b\in\mathbb{Z}\}$ and $\mathbb{Z}[x]$ the ring of polynomials over $\mathbb{Z}$. If $(x^2+1)$ denotes the ideal generated by $x^2+1$, how to show that
$$\mathbb{Z}[i]\cong \mathbb{Z}[x]/(x^2+1)?$$
Attempt: If we d... |
H: Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
Assume that $x+\frac{1}{x} \in \mathbb{N}$. Prove by induction that $$x^2+\frac1{x^2}, x^3+\frac1{x^3}, \dots , x^n+\frac1{x^n}$$ is also a member of $\mathbb{N}$.
I have my base, it is indeed true for $n=1$..
I can assume it is true for $x^k+x^{-... |
H: Are the eigenvectors of a power matrix A^k the eigenvectors of the matrix A?
If $x\in\mathbb C^n$ is an eigenvector of $B\in\Bbb C^{n\times n}$ and $B=A^k$ for a certain $k\in\Bbb N$, is $x$ an eigenvector of $A\in\Bbb C^{n\times n}$?
AI: Not necessarily. Take $A=\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$. Then ... |
H: $\exists x Px \land \exists x Qx$ does not imply $\exists x (P x \land Q x)$
I am pretty confused by this.
We know that $\phi : = \exists x Px \land \exists x Qx $
does not imply $\psi : = \exists x (P x \land Q x)$,
as for the model $M$ with domain $\{0,1\}$ with $P := \{0\}$ and $Q := \{1\}$,
we ha... |
H: Do p-norms of two discrete probability distributions 'rank' them equivalently?
Please, forgive me if this is an elementary question, as well as my the sloppy phrasing and notation.
Suppose we have two discrete probability distributions $p = {\lbrace p_i \rbrace}$ and $q={\lbrace q_i \rbrace}$, $i=1,\dots,n$, where ... |
H: Solution to the limit of a series
I'm strugling with the following problem:
$$\lim_{n\to \infty}(n(\sqrt{n^2+3}-\sqrt{n^2-1})), n \in \mathbb{N}$$
Wolfram Alpha says the answer is 2, but I don't know to calculate the answer.
Any help is appreciated.
AI: For the limit: We take advantage of obtaining a difference of ... |
H: Equivalence class of $T$ on $\mathbb{R} \times \mathbb{R}$ given by $(x,y) T (a,b)$ iff $x^{2}+y^{2}=a^{2}+b^{2}$
What is the equivalence class of $T$ on $\mathbb{R} \times \mathbb{R}$ given by $(x,y) T (a,b)$ iff $x^{2}+y^{2}=a^{2}+b^{2}$
I can see that the equivalence class cannot be negative, as the square of an... |
H: Generated equivalence relations in logics
Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. We introduce the following notation
$$ \varphi[A,x_1, \dots x_n] := \{ (x_... |
H: Find the eigenvectors and eigenvalues of A
scale by 2 in the x direction, then scale by 2 in the y direction, then projection onto the line y = x
AI: OK. To find fhe eigenvalues compute the solutions of $\det(xI-A)=0$ i.e. $$\det\left(\begin{array}{ccc} x-1 & -1 \\
-1 & x-1 \end{array}\right)=(x-1)^2-1$$ from this... |
H: Let A = {1,{1},{2},{1,2}} and complete with ⊆ or ∈:
Can someone check over my answers? I'm having trouble getting intuition for things like this so any tips you guys may have is helpful too. Also is {1,2} the same thing as saying {1}∪{2}?
Let A = {1,{1},{2},{1,2}}
a) Fill in the blank with ⊆ or ∈:
1) Ø __ A My Ans... |
H: Symbol of equivalence for comparing two equations?
If I need to compare two distinct equations, such as $y=\log_a x$ and $x=a^y$, is there any symbol that I can use in order to compare them instead of using words: "[equation 1] is equivalent to [equation 2]"? Note that I can't use the equals sign ($y=\log_a x = x=a... |
H: Primes taking the form $a+nb$
If $a,b$ are two coprime positive integers, can we find infinitely many positive integer $n$, such that $a+nb$ is a prime?
AI: Yes. This is Dirichlet's theorem on primes in arithmetic progressions. The proof is difficult, and is a landmark in the history of number theory, combining ide... |
H: Lower Bound on Log function
In this paper "Papandriopoulos, J.; Evans, J.S., "SCALE: A Low-Complexity Distributed Protocol for Spectrum Balancing in Multiuser DSL Networks," Information Theory, IEEE Transactions on , vol.55, no.8, pp.3711,3724, Aug. 2009"
The authors used the following lower bound on the log functi... |
H: Reflexive, symmetric, and transitive relations
On $A = \left \{1, 2, 3, 4 \right \}$
$\left \{(1,1), (2,1), (1,2)\right \}$ is NOT reflexive because there's no $(2,2)$ in the set. It is symmetric. However, it is NOT transitive.
I'm confused as to why it is not transitive. I thought since $1 R 2$ and $2 R 1$ and $1... |
H: How to develop intuition in topology?
Is there any efficient trick (besides doing exercises) to develop intuition in topology?
The question is general but i would like to add my view of things.
I started to teach myself topology through several books a couple of months ago. I already passed the point of being overw... |
H: Quick question about covering maps
Let $p:E\rightarrow B$ be a covering map and $b \in B$ so there exists a neighborhood $U$ of $b$ such that
$$p^{-1}(U)=\bigcup V_\alpha \text{ (disjoint union)}$$
and each $p\restriction_{V_\alpha}:V_\alpha\rightarrow U$ is a homeomorphism. Does it follow that each $V_\alpha ... |
H: Real Analysis proof for boundedness for function
a) Let $f : [a, b] → \mathbb{R}$ be a (not necessarily continuous) function with the property that, for every $x ∈ [a, b]$, there is a number $δ_x > 0$ for which $f$ is bounded on the neighborhood $Vδ_x (x)$ of $x$. Prove that the function f is bounded on the interva... |
H: Graph Theory - Leaves vs. # of vertices degree 3+
I am studying Problem 35, Chapter 10 from A Walk Through Combinatorics by Miklos Bona, which reads...
Prove that a tree always has more leaves than vertices of degree at least 3.
I feel like there should be an inductive argument with respect to n, the amount of ve... |
H: Find the eigenvectors and eigenvalues of A geometrically
$$A=\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}=\begin{bmatrix} 1/2 & 1/2 \\ 1/2 & 1/2\end{bmatrix} \begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix} \begin{bmatrix}2 & 0 \\ 0 & 1\end{bmatrix}.$$
Scale by 2 in the $x$- direction, then scale by 2 in the $y$- direction... |
H: Let $f: A \rightarrow B$, $D \subseteq A$, and $E \subseteq B$. Prove that $f^{-1}(B - E) \subseteq A - f^{-1}(E)$
Let $f: A \rightarrow B$, $D \subseteq A$, and $E \subseteq B$. Prove that $f^{-1}(B - E) \subseteq A - f^{-1}(E)$
Proof: Let $x \in f^{-1}(B-E)$, then $x \in f^{-1}(B)$ and $x \notin f^{-1}(E)$...
...... |
H: Help understanding a counting and probability exercise
I need help in trying to understand the answer to this exercise.
[Question]
A club is considering changing its bylaws. In an initial straw vote on the issue, 24 of the 40 members of the club favored the change and 16 did not. A committee of 6 is to be chosen f... |
H: Probability that the majority of 3 classifiers are wrong
3 classifiers. Each classifier has $0.7$ accuracy and makes its error independently.
How do I calculate the probability that the majority of three classifiers are wrong?
how I'm trying to solve it:
(3 choose 2) * 0.3 * 0.3 * 0.7 * 0.3 * 0.3 * 0.3
AI: Hint: Ma... |
H: If $f$ is entire and $|f|\geq 1$, then show $f$ is constant.
I know I'm going to use Liouville's Theorem, but my main question is why is $1/|f(z)|$ entire as well if $f$ is entire? Is this just a basic property: if $f$ is entire, then $1/f$ is entire? Thanks for the help.
AI: If $f$ is holomorphic at $z_0$ and $f(z... |
H: For any nonempty set X construct a surjection
For any nonempty set X construct a surjection: $S:P(X) -> x$ (Hint: Do not forget to identify for all S ∈ P(X) )
So I know a possible solution is P({X}) -> x. But I do not understand why. If someone can explain this more in depth or provide a more intuitive example, tha... |
H: Prove that the box dimension of $\{0,1,\frac{1}{2},\frac{1}{3},...\} $is$ \frac{1}{2}$
I'm supposed to consider the difference $\frac{1}{n+1}-\frac{1}{n}$ and let it equal to $\epsilon$. Hence $\epsilon=\frac{1}{n(n+1)}$. But how do I show that the number of boxes of size $\epsilon$ to cover the set is $N(\epsilon)... |
H: Is indicator function integrable?
f is discontinuous on [0,1] so it's not integrable? but I think the answer is yes, it is integrable. But i dont know how to prove that.
any hint would be great. thanks
AI: I assume you are talking about Riemann Integrability. The following result is useful to check the Riemann int... |
H: The limit of $\frac{1}{x}$, as $x \to 0$ doesnt exist, or does it?
Obviously, if you approach $0$ from left you get $-\infty$, if you approach from right you get $+\infty$. Ergo, the limit doesnt exist.
But what if we work in the number system of where, to real numbers we adjoin a single unsigned infinity? In that ... |
H: A standard proof of $\pi_1(\mathbb{S}^1) = \mathbb{Z}$ using universal covering spaces
I am looking for "a standard proof of $\pi_1(\mathbb{S}^1) = \mathbb{Z}$ using universal covering spaces", as suggested by the book Homotopy Type Theory (p. 255). What is this proof? Where can I find it?
AI: The result here is th... |
H: Proving $\lim_{x\to 3} (x^2-5x+1)=-5$ by the $\epsilon -\delta$ definition of a limit.
Prove that $\displaystyle\lim_{x\to 3} (x^2-5x+1)=-5$ by the $\epsilon -\delta$ definition of a limit.
What I've done so far:
$\forall \epsilon >0 \exists \delta\ni 0<|x-3|<\delta\Rightarrow 0<|x^2-5x+1-5|<\epsilon\\\Rightarrow... |
H: finite sum of riemann function
find a closed form for the sum of the zeta function $\zeta(k)$ for $k$ runs from $1$ to $n$.
I need this to find the sum of an infinite series involving the zeta function at the natural numbers. Any help is nice.
AI: Since you aren't very specific with what you're asking, I only have ... |
H: Energy functions and Lyapunov
So I found that $(\pm 1,0)$ and $(0,0)$ are steady states and its trace of the linear system is always $-1$. This implies all three points are sinks (fixed points).
Is the question for (a) implying I need a Lyapunov function for EACH point?
I know for $(0,0)$, I would do $L_0 = y^2/... |
H: If |f| is Riemann integrable, then f is Riemann integrable???
So i am stuck here.. how do i prove the first & second inequalities?
Also if |f| is Riemann integrable, then f is Riemann integrable.
I think it's ture but i dont know how to prove it.
any hints would be appreciated!
Thank You
AI: Answer is No.
Counter ... |
H: True OR False. Provide a proof or a counterexample
True OR False. Prove a proof or a counter-example:
If A∩B = Ø then ℘(A)∩℘(B) = Ø
AI: Suppose $A$={$c:c=2M$ for any natural $M$} and $B$={$y:y=2M+1$ for any natural $M$}. Okay so $A\cap B$ is the empty set. Take $f:x\mapsto 2x$. Now $f(x)\cap f(y)$ is not the empty... |
H: Showing set is a vector space
Say I have a set of vectors with multiplication and addition both defined. To prove that it is a vector space I have to confirm the eight axioms. When I check the distributive property for scalar multiplication:
$ r(u+v) = ru + rv $
(where r is a scalar and u,v are vectors) is the addi... |
H: P-adic expansion construction
Can anyone teach me about p-adic expansion? especially the case where we have to expand a square root.
I need to know how to construct them. for example: the 7-adic expansion of $\sqrt{305}$.
This is a general question and not an assignment or anything, so i cannot post the progress.
... |
H: What kinds of sets are added by Cohen forcing?
I am trying to get a feel for what kinds of sets are added by forcing. I apologize in advance, this question might be hard for me to put precisely into words. Let's give a very simple example:
Let $\mathbb{P}$ be the set of all finite partial functions $p:\omega\to 2... |
H: Linear Differential Equations word problem
I had this quiz on LDE and I wasn't sure how to do this problem...I know how to do LDE but I couldn't come up with the equation to get me started. Any ideas on how to do this?
AI: Here is a start. Recalling the Newton's second law
$$ F = m a = m\frac{dv}{dt}. $$
where $m... |
H: proving identity for statistical distance
How do I show the following identity?
Let $\vec{\rho}_X$,$\vec{\rho}_Y$ denote the probability distributions over a finite set $R$ respectively.
Prove that
$\Delta(\vec{\rho}_X,\vec{\rho}_Y)=\max_{S\subseteq R}P_X(S)-P_Y(S)$ where the maximization is taken over all subsets... |
H: Prove that if $F: A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is one-to-one
Prove that if $F: A \rightarrow B$ and $F^{-1}$ is a function, then $F$ is one-to-one
Proof: Suppose $F$ is not one-to-one. Then there exist $x_{1}, x_{2} \in A$ such that $F(x_{1}) = F(x_{2})$ where $x_{1} \neq x_{2}$. Now, at th... |
H: Can we write $||A - B|| \leq ||A||$?
I am confused with the very basic question related with the matrix norm.
Can we write $||A - B|| \leq ||A||$ ?
Thanks for the help and time.
AI: No you cannot; let $A$ be the zero matrix. Then we get a contradiction. |
H: Expected number of times before winning the lottery $n$ times
Let $p,n$ be positive integers. Suppose that every time you buy the lottery, you have a $\dfrac1p$ chance of winning it (independently of other times). What is the expected number of times you have to buy the lottery before you win $n$ times?
Intuitively... |
H: Topologies of equivalent metrics
On $X= {\bf R}^2 - \{ (x_1,0)|\ x_1>0 \}$ define two metrics :
$d(x,y) = |x-y|$ and $d_2$ is a path metric.
Then $$ d(x,y) \leq d_2(x,y),\ d_2(x,y)\leq C(x,y)d(x,y) $$
Here if $x_n=(n,1/n),\ y_n = (n,-1/n),\ n>0$ then $$ \lim_{n\rightarrow \infty} C(x_n,y_n) = \infty \ (\ast)$$
... |
H: Finding Calculus or Probability error in basic continuous probability problem
I am having a lot of difficulty spotting my error in the following probability problem.
The joint probability density function of $X$ and $Y$ is given by $f(x,y) = c\left (y^2-x^2\right)e^{-y}, \ \ \ \ -y \le x \le y,\ \ \ 0 \lt y \lt... |
H: $[[x,y],z]=[x,[y,z]] \Rightarrow [x,y]=0$?
I got the next problem:
Let $A$ be a Lie algebra, prove that if the bracket associates $([[x,y],z]=[x,[y,z]]$) then the bracket is zero $([x,y]=0)$.
Can't get the result using the properties (alternating, Jacobi identity, anticommutativity), i think that the result is fals... |
H: Calculus: Find an upper bound for an estimate of the area
Using n=6 rectangles, find an upper bound for an estimate of the area under the parabola y=x^2 from x=0 to x=1. Hint: use the right side of each rectangle as its height.
This is a calculus problem part of integral. I dont understand how to solve it.
AI: Assu... |
H: Maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$
Is there an expression for the maximum of a sequence $\left({n\choose k} \lambda^k\right)_k$ (i.e. $\max_{k\in\{0,\ldots,n\}}{n\choose k}\lambda^k)$ in terms of elementary functions of $n$ and $\lambda$?
This seems like a simple calculus problem but my u... |
H: Standard norm of $\mathbb{R}^3$
I am going through the paper, Energy of a Knot by Jun O'Hara. Let me quote from the Definition 1.1 of Section 1 on the first page:
Let $f:S^1 = \mathbb{R}/\mathbb{Z} \to \mathbb{R}^3$ be an embedding of class $C^2$ such that $|f'(t)| = 1$ for all $t \in S^1$, where $|.|$ denotes the ... |
H: integrate $3x^3/(1-x^2)^{1/2}$ by trigonometric substitution
Please can someone integrate
$$
\int \frac{3x^3}{\sqrt{1-x^2}}dx
$$
by trigonometric substitution?
Thanks, all.
AI: Let $x = \sin t$, so that $dx = \cos{t}dt$. Now we have
$$\cos^2 t = 1 - \sin^2 t = 1 - x^2$$
Thus our integral can be re-written as
$$\int... |
H: How find this limit $I=\lim_{x\to\infty}\left(\sin{\frac{2}{x}}+\cos{\frac{1}{x}}\right)^x$
Find this limit :
$$I=\displaystyle\lim_{x\to\infty}\left(\sin{\frac{2}{x}}+\cos{\frac{1}{x}}\right)^x$$
note $x=e^{\ln{x}}$
$$I=\exp\left(\lim_{x\to\infty}x\ln{\left(\sin{\frac{2}{x}}+\cos{\frac{1}{x}}\right)}\right)$$
and ... |
H: Real Analysis: ε-δ definition to prove that f is a continuous function.
Let $f:R\backslash \{1 \} \to R$ be defined by $f(x)= \frac{1}{(1-x)}$. Use the $\epsilon$-$\delta$ definition to prove that $f$ is a continuous function.
My attempt is
let $c \in R \backslash \{1\}$. $$|f(x) - f(c)| = \Big|\frac{1}{1-x} - \f... |
H: Fourier Transform of $\frac{1}{(1+x^2)^2}$
I need to find the Fourier Transform of
$f(x) = \frac{1}{(1+x^2)^2}$
Where the Fourier Tranform is of $f$ is denoted as $\hat{f}$, where $\hat{f}$ is defined as $$\hat{f}(y)=\int_\mathbb{R}f(x)e^{-ixy}dx$$
I think I need to use the Fourier Inversion Theorem. From this th... |
H: Proving that $\cos(n^a t)$ doesn't converge to $1$
Looking at the graph of the functions $\cos(n^a t)$ ($a>0$) it looks obvious that they don't converge to $1$ as $n \rightarrow \infty$. For integer odd $a$ it's easy to prove this directly, as $\cos(n^a \pi/2)=\cos(m \pi/2)=0$ for some odd $m$ in this case. I've tr... |
H: Evaluate the series $\lim\limits_{n \to \infty} \sum\limits_{i=1}^n \frac{n+2}{2(n-1)!}$
Evaluate:
$$\displaystyle \lim_{n \to \infty} \sum_{i=1}^n \frac{n+2}{2(n-1)!}$$
Is there a theorem to be applied here? Or is there way to use telescoping series?
Please help.
AI: Assuming the problem to be $\displaystyle \li... |
H: Number theory: 2 numbers within a set with same difference
You have the numbers 1,2,3...,99,100. From that set you have to choose 55 different numbers.
Show that:
There are 2 numbers with a difference 9,10,12,13
Show that there aren't neccessarily 2 with a difference 11.
I have no idea how to do this, it seems ... |
H: Elementary set theory, Cantor-Bernstein-Schröder usage, check my proof
I have a question, I was asked to show that $[0,1]$ and $\mathbb R$ are of equal cardinality using the Cantor-Bernstein-Schröder theorem.
I would just like some feedback, if I solved it correctly:
Let $f:[0,1]\to \mathbb R$, $f(x)=x$. it is clea... |
H: Showing: family is linearly independent
The Following text is not a mathematical proof, yet, it's just a collection of ideas I have.
I need your help to make it a math. proof :)
$ (f_n)_{n \in N} $ is given by $ f_n : R \to R, x \to sin(2^{-n}x).$
Show that the family $ (f_n)_{n \in N} $ is linearly independent.
We... |
H: understanding Exponent of $y= a^{mx}$
Why does $y=a^{mx}$ imply $y=e^{log(a)\cdot mx}?$
AI: By definition $a=e^{\log(a)}$, with $\log$ the natural logarithm, which is the inverse of an $e$-power.
Thus, $y=a^{mx} = (e^{\log(a)})^{mx} = e^{ \log(a) mx }$, where the last equality hold because in general $(b^c)^d = b... |
H: What kind if distribution is the problem $p(x-y = k| x>y)$?
We choose randomly two numbers $x,y$ out of $\{1,2,...,n\}$. Calculate $P(x-y=k | x>y), 1 \leq k \leq n-1$.
What I said:
We obviously need a random variable here $X$ that, in the case that $x>y$, represents the equation $x-y$. The outcome of $X$ can be ... |
H: Find the cardinality of these sets
Question from my homework im struggling with
Find the cardinality of these sets:
1) the set of all sequences of natural numbers
2) the set of all arithmetic series (difference between 2 numbers is the same,example 11,9,7 ...)
3) the set of all rising arithmetic series (difference... |
H: Question on Convergence of Improper integrals
Question is to check which of the following improper integrals are convergent?
$$\int _1^{\infty} \frac{dx}{\sqrt{x^2+2x+2}}$$
$$\int _0^5 \frac{dx}{x^2-5x+6}$$
$$\int _0^5\frac{dx}{\sqrt[3]{7x+2x^4}}$$
I was having a stupid mindset that :
"$\textbf{Improper integrals s... |
H: Prove $x^n y^m \leq \frac{n^n m^m}{(n+m)^{n+m}}$ for $x + y = 1$
If $x > 0$, $y > 0$, x and y are real, and $x + y = 1$, prove that
$x^n y^m \leq \frac{n^n m^m}{(n+m)^{n+m}}$
for all positive integers n, m.
My proof attempts have been to apply two dimensional induction on n and m, but I haven't had much success. As... |
H: ABC Conjecture: Simple example showing $\epsilon$ is necessary
I was looking over Lang's discussion of the abc conjecture in his famous Algebra tome. He says
We have to give examples such that for all $C>0$ there exist natural numbers $a$,$b$, $c$ relatively prime such that $a+b=c$ and $|a|>C N_0(abc)$. But trivia... |
H: On notation of $n$-dimensional Gaussian distribution
could anyone please help me with the following question?
Let $\Sigma$ be the covariance matrix of an $n$-dimesnional Gaussian distribution. The generic requirement for $\Sigma$ is to be symmetric and positive definite. However, what's happening when $\Sigma$ is ... |
H: in every triangle we can inscribe a circle
I am trying to show that in every triangle we can inscribe a circle. I reduced it to following: in every triangle there must be a point in the interior, such that there are three points on the triangle (each one on one side) such that the distance from any one of them to t... |
H: Effective Enumeration of Set when Membership is Semi-Decidable
I have a set $S$, where $x \in S$ is semidecidable, i.e. there is a function $f(\cdot)$ that returns 1 in finite time if $x \in S$ but will not halt if $x \notin S$*. I also have an effective enumeration $A_0, A_1, \dots$ of $A$, where $S \subset A$. Is... |
H: Commutative rings as images of domains
It is well known that the quotient ring $R/I$ of a commutative ring $R$ is an integral domain if and only if $I$ is a prime ideal.
If $R$ is a commutative ring with identity, is it always possible to find an integral domain $D$ and ideal $I \subset D$ such that $R \cong D/I$ ?... |
H: n-th derivative of exponential function $\;e^{-f(x)}$
Is there a closed-form expression for $n$-th derivative of exponential function below ($n>0$)?
$$
\frac{d^n}{dx^n}\large e^{-f(x)}
$$
AI: See Faa di Bruno's identity generalizing the chain rule to higher derivatives:
$${d^n \over dx^n} f(g(x))=\sum \frac{n!}{m_1... |
H: How prove this matrix inequality $\det(B)>0$
Let $A=(a_{ij})_{n\times n}$ such
$a_{ij}>0$ and $\det(A)>0$.
Defining the matrix $B:=(a_{ij}^{\frac{1}{n}})$, show that $\det(B)>0?$.
This problem is from my friend, and I have considered sometimes, but I can't. Thank you
AI: The statement is false. Counterexample:
$$
B... |
H: Meaning of the statement, almost sure convergence
what is the meaining of $$ \lim_{n \to \infty}P(\sup_{m \ge n} |X_m -X|>\epsilon) \to 0 $$ forall $\epsilon > 0.$ I found this statement , while trying to understand almost sure convergence , i.e $X_n \to X$ iff the above statement holds.
AI: Let ${X_n}$ be a seque... |
H: Nullspace and different solutions
$\begin{pmatrix}
R_{11} & \cdots & R_{1A} \\
\vdots & \ddots & \vdots \\
R_{S1} & \cdots & R_{SA} \\
1 & \cdots & 1
\end{pmatrix}
\begin{pmatrix}
x_1 \\
\vdots \\
x_A
\end{pmatrix}
=
\begin{pmatrix}
R_{11} & \cdots & R_{1A} \\
\vdots & \ddots... |
H: On idempotent elements that are contained in center of a ring
Let $e$ and $f$ be idempotent elements of a ring $R$. Assume that $e,f$ are contained in center of $R$. Show that $Re=Rf$ if and only if $e=f$
Help me a hint to prove it.
Thank in advanced.
AI: For $r\in R$, we have $r\in Re$ if and only if $re=r$.
Ind... |
H: Integration without using parametrization .
I would like to integrate the following line integral without using parametrization .
I wanted to integrate the following $$\int_C \frac{1}{z-a} dz$$ , where $C$ is a a curve along $|z-a| =r$ .
Using parametrization i can easily get the answer , but i wanted to integrat... |
H: Show that $3^{2^n}-1$ is divisible by $2^{n+2}\,\, \forall n \in \Bbb N$
I am stuck on the following problem:
Use the principal of induction to prove
that $3^{2^n}-1$ is divisible by $2^{n+2}\,\, \forall n \in \Bbb N$
My Attempt: Let us denote the statement by $ \,P(n) \colon 3^{2^n}-1$ is divisible by $2... |
H: Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4, p$ is a prime.
I know th... |
H: Showing the linear independence between two row equivalent matrices
I can prove that if A and B are row equivalent matrices, then the column vectors of A are linearly independent iff the column vectors of B are linearly independent.
However, does this result also hold for row vectors? That is, is it true that if A... |
H: What are the names in English for Alterando, Invertendo, Componendo and Dividendo?
I am writing an article in English but don't want to use the Latin names. What are their English equivalent?
AI: I think those rules are called by their latin names, if one ever would give it names! Something like 'Corollaries of cr... |
H: Finding $\lim (3^x-e^x)/(7x^{15}+5x^{25})$ as $x\to+\infty$
I have to find the limit of $(3^x-e^x)/(7x^{15}+5x^{25})$ as $x\to+\infty$ using only notable limits (I can't use Taylor series or de l'Hopital's method). I'm stuck in finding this limit, I tried to substitute x with ln(y) so that I can use logarithm's pro... |
H: Is the difference of two percentages also a percentage?
If I have some share of a product in some sector as 2012-50% 2013-60%
If I find share change for 2013-12 then should it be 10% or 0.1?
AI: Yes, the shares from $2012$ to $2013$ show an increase of $10\%$, which corresponds to a net change of $$\dfrac{60\% - 5... |
H: Evaluate $\iint$ of $f(x,y)=xy$ in polar coordinates, where $R$ is $(x-2)^2+y^2=4$ in quadrant 1
I'm not sure if I converted correctly from cartesian to polar, but I know that I made a mistake along the way.
$f(x,y)=xy$, where $R$ is $(x-2)^2+y^2=4$ in quadrant 1
$$\int_{0}^{{\pi/2}}\int_{0}^{2}(r\cos\theta r\sin\t... |
H: Why the geometric series diverges for x<=-1
The geometric series is defined as:
$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + ...\tag{for $|x|<1$}$$
I know that for $x>=1$ it must diverge, of course. But I want some proof that it diverges when $x=1$ and $x<1$.
AI: If $x=1$ then
$$1+x+x^2+...+x^n=1+1+..1+=n+1$$
which goes t... |
H: Integral in hyperbolic coordinates
all.
My homework problem is the following:
Define $D=\{(x,y)\mid x,y>0, 1\leq x^2-y^2\leq 9, 2\leq xy\leq4 \}$.. For a continuous function $f:D\rightarrow\mathbb{R}$, use the hyperbolic coordinates from Exercise 7 to show that
$$
\int_D[x^2+y^2]dxdy=8
$$
In exercise 7, the functio... |
H: $\lim_{n\to\infty}a_n$ and Cauchy condensation
Let $(a_n)$ be a decreasing sequence which converges to 0.
If $2^n a_{2^n}\rightarrow 0$, does it follow that $na_n\rightarrow 0$?
AI: Yes.
Let $k:=k_n$ be so that $2^k \leq n <2^{k+1}$. Then
$$a_{2^k} \geq a_n \geq a_{2^{k+1}}$$
and hence
$$ na_{2^k} \geq na_n \geq na... |
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