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H: Assess the limit: $ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$
Compute the following limit:
$$ \lim_{n\to\infty} \frac{1}{n}\int_0^n \frac{\arctan(x)}{\arctan{\frac{n}{x^2-nx+1}}}dx$$
I'm looking for an easy approach if possible.
AI: $$\arctan \left(\frac{n}{x^{2}-nx+1}\... |
H: What's the definition of limit of sets(esp. ordinals) in set theory?
By definition of exponent operator on ordinals, we have
$$0^\omega=\lim_{\xi\to\omega}0^\xi$$
However, Note that $0^\xi$ is not increasing, so if we still let $\lim_{\xi\to\omega}0^\xi=\sup\{0^\xi|\xi<\omega\}$ then it is followed by $$0^\omega=\s... |
H: Open set whose boundary is not a null set
I've just seen a theorm about a bounded set $A \subset \mathbb{R}^n$.
$$\chi_A \text{ is Riemann integrable} \Longleftrightarrow \partial A \text{ is a null set}$$
Then, I wonder if there's any open set whose boundary is not a null set. Can you give me some example for that... |
H: Uniqueness theorem for harmonic function
So far in complex analysis books I have studied about Uniqueness theorem: If $f$ is analytic in a domain $D$ and if its set of zeroes has a limit point in $D$ then $f\equiv 0$ on $D$, I want to know is this result holds for harmonic functions?
AI: In general harmonic functio... |
H: Height of this triangle?
Each edge of the following cube is 1 and C is a point on the edge.
What would the height of triangle be in this case , how would you measure it?
AI: Assuming you mean "height" as "altitude passing through point $C$". You might want to clarify if this is not correct:
The height of the t... |
H: Formula to estimate sum to nearly correct : $\sum_{n=1}^\infty\frac{(-1)^n}{n^3}$
Estimate the sum correct to three decimal places :
$$\sum_{n=1}^\infty\frac{(-1)^n}{n^3}$$
This problem is in my homework. I find that n = 22 when use Maple to solve this. (with some programming) But, in my homework, teacher said fin... |
H: Finding the equation of the tangent plane to the surface $z=x^2+ y^2$ at the point $(1,2)$
Just wanted to ask a simple question I've forgotten how to solve (lost my mind completely).
Find the equation of the tangent plane to the surface $z=x^2 + y^2$
at $(1,2)$.
The fact it has $3$ variables is what is putting ... |
H: Ordered numbers
Let $0<a<b<1$, can we find a point $x\in (a,b)$ such that $a<x^{2}<x<b$. I know that we can find $x$ such that $a<x<b$ and this $x$ will satisfies $0<x^{2}<x<b$, but I'm not sure how to choose such $x$ with $a<x^{2}<x<b$?
AI: This is not true. For example, take $a=\frac{1}{4}$ and $b=\frac{1}{3}$.... |
H: Can an ordered field be finite?
I came across this question in a calculus book.
Is it possible to prove that an ordered field must be infinite? Also - does this mean that there is only one such field?
Thanks
AI: Recall that in an ordered field we have:
$0<1$;
$a<b\implies a+c<b+c$.
Suppose that $F$ is an orde... |
H: Evaluating $\int \frac{x^3}{(x^2 + 1)^\frac{1}{3}}dx$
$$\int \frac{x^3}{(x^2 + 1)^\frac{1}{3}}dx$$
I am suppose to make a $u$ substitution and to make this a rational integral and then evaluate it from there but I have no idea how to do that. There aren't any good examples of this in the book and I can not find any... |
H: Inequality $|f(1)-f(0)|\le g(1)-g(0)$ if $|f(t)|\le g'(t)$
I have the question:
Let $I=[0,1]$ be the closed interval, $f:I\to\mathbb{R}^n$ and $g:I\to\mathbb{R}$ differentiable, with $|f(t)|\le g'(t)$, for all $t\in I$. Show that $|f(1)-f(0)|\le g(1)-g(0)$
The book suggests to use the same trick in proving the Me... |
H: Formal proof of De Morgan's laws for quantifiers
Consider the set of inference rules for first order logic (analogous to the ones listed here : http://en.wikipedia.org/wiki/Sequent_calculus#Inference_rules)
I am stuck in proving the following rule
$$\vdash_{\gamma} \neg \forall x.\phi \implies \exists x. \neg \phi ... |
H: Evaluating $\int \frac{dx}{x^2 - 2x}$
$$\int \frac{dx}{x^2 - 2x}$$
I know that I have to complete the square so the problem becomes.
$$\int \frac{dx}{(x - 1)^2 -1}dx$$
Then I set up my A B and C stuff
$$\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{-1}$$
With that I find $A = -1, B = -1$ and $C = 0$ which I know is ... |
H: high school line equation question
I hit stumbling block below.
But i try everything that I can think of but i failed to find any satisfactory answer.
my workout is that I find mid point between point B and C.
and then using A and earlier mid point to find slope which I know is perpendicular.
But line equation y = ... |
H: Spivak Calculus Prologue
I'm completely blown away by the difficulty of Spivak. I've managed to work through the first 3 problems, but I feel I'm missing something important to solve these basic inequalities in his 4th problem:
$$x^2 + x + 1 > 2$$
&
$$x^2 + x + 1 > 0$$
any suggestions?
AI: Complete the square:
$$
x... |
H: Evaluating $\int \frac{1}{(1-x^2)^{3/2}} dx$
I am struggling to evaluate the following integral:
$$\int \frac{1}{(1-x^2)^{3/2}} dx$$
I tried a lot to factorize the expression but I didn't reach the solution.
Please someone help me.
AI: Hint:
Set $x=\sin(t)$, then everything will turn out very well.
This often help... |
H: Evaluate $\lim_{x \to \infty} \frac{1}{x} \int_x^{4x} \cos\left(\frac{1}{t}\right) \mbox {d}t$
Evaluate $$\lim_{x \to \infty} \frac{1}{x} \int_x^{4x}
\cos\left(\frac{1}{t}\right) \mbox {d}t$$
I was given the suggestion to define two functions as $g(x) = x$ and $f(x) = \int_x^{4x}\cos\left(\frac{1}{t}\right)dt$ s... |
H: Tricky radius of convergence: $\sum\limits_{n=0}^\infty\cos\left(\alpha\sqrt{1+n^2}\right)z^n$
I encountered the following power series, and while I know a couple of ways to determine radius of convergence, I wasn't able to figure out how to evaluate the appropriate limit to get said radius. Can anyone help?
What ... |
H: Proof Using Truth Tables
Pleae forgive the very basic question, but I know nothing really of formal logic and so
would appreciate some feedback.
The truth table defining the implication operator
P Q P implies Q
T T T
T F F
F T T
F F T
together with the negatio... |
H: Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form
As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work:
$\displaystyle\left(1 + \exp\frac{2i\p... |
H: Given $x^2 + 2y^2 - 6x + 4y + 7 = 0$, find center, foci, vertex/vertices
So the equation is:
$$ x^2 + 2y^2 - 6x + 4y + 7 = 0 $$
Find the coordinates of the center, the foci, and the vertex or vertices.
What I did was put the equation in the form:
$$
\frac{(x-3)^2}{4}+ \frac{(y+1)^2}{4} = 1
$$
Now based on that, I... |
H: Fourier and integral
Given the below trigonometric series:
$1 + \sum_{n=1}^{\infty} \frac{2}{1+n^{2}}\cos (nt)$
Where $f(t)$ is the value of the series.
Can I then deduce that $\int_{-\pi}^{\pi} f(x) dx$ is $2\pi$? I ask because
the series for $f(t)$ looks like a fourier series and I can then recognize that $1 =... |
H: Explanation of how models can differ on $\omega$?
Assuming set theory (here, ZF) is consistent, there is a model $V$ of ZF, the universe of all sets.
So, there is a $\omega^V\in V$. A set $A\in V$ is countable iff a bijection $f\in V$ exists between $A$ and $\omega^V$.
By the downward Löwenheim–Skolem Theorem, th... |
H: Let $f$ be a holomorphic function on D = $ ( z\in C : |z| <1 ) $ such that $ | f(z)|\leq1$.
Let $f$ be a holomorphic function on D = $ ( z\in C : |z| <1 ) $ such that $ | f(z)|\leq1$. Let $ g : D: \rightarrow C $ be such that
$ g(z) = \frac{ f(z)} {z} $ if $z\in D $, $ z\neq 0$ and $ g(0) = \ f' (0) $ .
I have ... |
H: Fourier integral
how could i evaluate the following integral ?? $$\int_{-\infty}^\infty dt \frac{\exp(-iut)}{|at|^{1/2+ib}}$$
here $a$ and $b$ are positive real numbers.. how can i make this integral ? thanks.
$ |x| $ means the absolute value function
I think this integral is related to the Mellin transform $$\int_... |
H: Hilbert space linear operator question
Let $\mathcal{H}$ be the vector space of all complex-valued, absolutely continuous functions on $[0,1]$ such that $f(0)=0$ and $f^{'}\in L^2[0,1]$. Define an inner product on $\mathcal{H}$ by
$$\langle f,g\rangle=\int_0^1f^{'}(x)\overline{g^{'}(x)}dx
$$ for $f,g\in\mathcal{H... |
H: Composite number
Determine for what numbers $n$ the number $n^4 + 4$ is a composite number.
Sorry about my English. I found $n^4 + 4 = (n^2 + 2n +2)(n^2 -2n + 2)$, but i don't know what to do from here.
AI: You did the non-obvious part, and are now essentially finished! Determine the $n$ for which one of your term... |
H: Why is this covering map doubly periodic?
The universal cover of the torus $T$ is the complex plane $\mathbb{C}$. If $p: \mathbb{C} \to T$ is the covering map, why is $p$ doubly periodic?
AI: Since $\mathbb C$ is simply connected, it is a universal covering space. Any two covering maps $\mathbb C\to\mathbb T$ are r... |
H: there exist analytic function with $f(\frac{i^n}{n})=-1/n^2$
Does there exist analytic function with $f(\frac{i^n}{n})=\frac{-1}{n^2} \forall n\ge 2$, well I guess Yes, beacuse $g(z)=f(z)+z^2$ has zero set $\{-\frac{i}{n}: n \text{ odd}\}$ which has limit point zero, hence $f(z)=-z^2$ Is my answer is correct?
AI: I... |
H: Can the argument of an algebraic number be an irrational number times pi?
This is mainly out of curiosity. Let $\nu$ be an algebraic number. Can Arg($\nu$) be of the form $\pi \times \mu$ for an irrational number $\mu$?
AI: Yes. Try $\alpha + i \beta$ where $\alpha$ and $\beta$ are rational and nonzero and $\arc... |
H: Multiplication in the field $F = \mathbb{Z}_2[x]/f(x)$
Let $f(x) = x^6 + x + 1$ and define the field $F = \mathbb{Z}_2[x]/f(x)$
Compute the following in this field:
1. $(x^5 + x + 1)(x^3 + x^2 +1)$
I start by multiplying (in $\mathbb{Z}_2[x]$):
$(x^5 + x + 1)(x^3 + x^2 +1)$ = $(x^8 + x^7 + x^5 +x^4 + x^2 + x + 1)... |
H: Index notation clarification
Previously, I have seen matrix notation of the form $T_{ij}$ and all the indices have been in the form of subscripts, such that $T_{ij}x_j$ implies contraction over $j$. However, recently I saw something of the form $T_i^j$ which seems to work not entirely differently from what I was us... |
H: In which cases is the inverse of a matrix equal to its transpose?
In which cases is the inverse of a matrix equal to its transpose, that is, when do we have $A^{-1} = A^{T}$? Is it when $A$ is orthogonal?
AI: If $A^{-1}=A^T$, then $A^TA=I$. This means that each column has unit length and is perpendicular to every o... |
H: Is there any orthogonal matrix P that makes a symmetric A, diagonal by $PAP^{-1}$?
Given a symmetric matrix A. Is there any orthogonal matrix P that makes $PAP^{-1}$ diagonal?
I've found at wikipedia this:
The finite-dimensional spectral theorem says that any symmetric matrix whose entries are real can be diagonal... |
H: Distribute pennies for children
We distribute n pennies to k boys and l girls, so that (to be really unfair) we require that each girls gets at least one penny. In how many ways can we do this?
AI: I'm assuming each boy and girl is labeled (that is, giving 3 pennies to boy 1 and 5 to boy 2 is counted as distinct fr... |
H: Maximally entropy preserving irreversible functions. (CS related)
The topic/problem is related to hashing for data structures used in programming, but I seek formal treatment. I hope that by studying the problem I will be enlightened of the fundamental limitations of hashing. So, the purpose is educational.
Let's c... |
H: Given a symmetric matrix $A$, are there any matrices $B$, $C$ that $BAC = I$?
Given a $4 \times 4$ symmetric matrix $A$, are there any matrices $B,C$ that:
$BAC = I_{4}$ ?
I've thought of $B$ being a orthogonal matrix $P$ ($B=P$) and $ C = P^{T}$ so we get
$PAP^{T} = \begin{bmatrix}\lambda_{1}&0&0&0\\0&\lambda_{2}... |
H: What does face-width mean?
What is the meaning of the term face-width? I have seen the term used as a property of an embedding of a graph on a surface. I haven't found a definition.
AI: See here, here, and here for the definitions of face-width. |
H: Impossibility of certain methods of proof?
There are many methods available for proving a given statement: direct proof, proof by induction, proof by contrapositive, proof by contradiction, etc. In some cases there is an obvious method that should be employed, such as using induction to prove that $\displaystyle\s... |
H: Why is the range of arctan $[ -\frac{\pi}{2} , \frac{\pi}{2}]$?
I've been taught in school and it says on Wikipedia that the range of arctan is $[ -\frac{\pi}{2} , \frac{\pi}{2} ]$.
Why isn't it $[0,\pi]$ ?
AI: As the graph of the function $\,\tan x\,$ show, this is a very not $\,1-1\,$ function onto the reals, so ... |
H: Proving a derivative equality
How can I prove the following equality?
$$
\frac{1}
{{n!}}\frac{{d^n }}
{{dx^n }}\left( {\left( {x^2 - 1} \right)^n } \right) = \sum\limits_{k = 0}^n {\left( {\frac{{n!}}
{{k!\left( {n - k} \right)!}}} \right)} ^2 \left( {x + 1} \right)^{n - k} \left( {x - 1} \right)^k
$$
And withou... |
H: Uniform Convergence on a Closed and Bounded Interval
Let $f_n\colon [a,b] \to \mathbb{R}$ be a sequence of continuous functions converging uniformly to a function $f$. Show that if each $f_n$ has a zero then $f$ also has a zero.
Thanks for any help.
AI: Since each $f_n$ has a zero, there is a number $x_n \in [a,b]... |
H: Norm and invertibility of operator $\left(-\Delta+\lambda I\right)$ with $\lambda>0$.
Let $\lambda>0$ and $n\geq 1$. Prove that the operator
$$-\Delta+\lambda I:H^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)$$
is invertible and find the norm
$$\left|\left|\left(-\Delta+\lambda I\right)^{-1}\right|\right|_{L^2(\mathbb{R}^n)... |
H: Determine if a Turing Machine M, on input w, will move its head to the left, at least once
Here is a problem from my formal languages class
Consider the following problem: Determine if a Turing Machine M, on input w, will move its head to the left, at least once.
Is this problem decidable?
Can Rice's... |
H: How to verify this function is continuous?
Recently, I'm reading a paper "Spaces with a regular Gδ-diagonal" of A.V.Arhangel’skii's. I can't understand the function $d$ in the example 9 is continuous. Could someone help me? Thanks ahead:)
AI: The space is $X=X_0\cup X_1\cup U$, where $X_0=\Bbb R\times\{0\}$, $X_1=\... |
H: Set theory puzzles - chess players and mathematicians
I'm looking at "Basic Set Theory" by A. Shen. The very first 2 problems are: 1) can the oldest mathematician among chess players and the oldest chess player among mathematicians be 2 different people? and 2) can the best mathematician among chess players and th... |
H: Finding Polynomial Limit
I am asked to find the Limit for:
$$\lim_{x\rightarrow -∞}(x^4+x^5) $$
The first thing I am tempted to do is divide the numerator and denominator of this fraction by the highest power of x, in this case $x^5$.
$$\lim_{x\rightarrow -∞}\frac{\dfrac {x^4+x^5}{x^5}}{\dfrac1{x^5}}$$
Continuin... |
H: Discontinuous function sending compacts to compacts
I know that the condition that $f(X)$ is compact if $X$ is compact should not be sufficient to say that $f$ is continuous, but I can't come up with an example of such discontinuous $f$. What is it?
Thanks
AI: Let $f:\Bbb R\to\Bbb R$ be such that $f(x)=0$ if $x\l... |
H: equivalence of $E[X_\infty]=1$ and $X$ is a u.i. martingale on $[0,\infty]$
Let $(X_t)$ be a strictly positive supermartingale on $[0,\infty)$. Hence $X_t$ covnerge to $X_\infty$ a.s. Now how can I show the following: $E[X_\infty]=1$ is equivalent to $(X_t)$ is a uniformly integrable martïngale on $[0,\infty]$.
hul... |
H: Evaluating $\lim_{y \to 0^+} (\cosh (3/y))^y$
Evaluating $$\lim_{y \to 0^+} (\cosh (3/y))^y$$
This is what I have tried:
$L = (cosh(3/y))^y$
$\ln L = \frac{\cosh(3/y)}{1/y}$, applying L'Hopital's rule, I get:
$\ln L = \frac{-3y^2(\sinh (3/y))}{(y^2)}$
$\ln L = 3\sinh (3/y)$
Now I seem to be stuck in a loop betwee... |
H: Multiplication inverse for dedekind cut
Let $\alpha \in P_R$ be a cut.
Since there exists a cut that is not $\{q\in Q\mid q<r\}=r^*$ for every $r\in Q$, $\alpha$ doesn't need to be of the form $r^*$.
Let $$\gamma= 0^* \cup \{0\} \cup \{q\in P_Q\mid\text{ there exists }r\in P_Q\text{ such that }r>q\text{ and }1/r \n... |
H: Question about flat modules and exact sequences
I have a basic question about exact sequences. I want to show that if I have that whenever $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is exact then $0 \to A \otimes N \to B \otimes N \to C \otimes N \to 0$ is then $N$ is flat.
So let $\dots \to A \xrightarrow... |
H: Computing: $\lim_{x\rightarrow0} \frac{\log(1+x)}{x^2}-\frac{1}{x}$
Could the following limit be computed without L'Hopital and Taylor? Thanks.
$$\lim_{x\rightarrow0} \frac{\log(1+x)}{x^2}-\frac{1}{x}$$
AI: Here's an approach. Note that you can write the limit as
$$\lim_{x\to 0} \frac{\log(1+x)-x}{x^2}$$
and use th... |
H: Probabilty of one person getting a pair
A dealer is using a standard deck of $52$ cards. One extra ace of spades is put into the deck.
So now he got $53$ cards in the deck with two ace of spades in total.
The dealer deals $4$ hands with $5$ cards each. What is the probability that
one $5-$card hand contains the two... |
H: question about the bracket process of brownian motion
Suppose I have a multidimensional brownian motion $W=\{W_t\}$. Why is the following true:
$$\langle W^k,W^l\rangle_t = \delta_{k,l}t$$
where $W^k$ denotes the k-th coordinate, $\langle \cdot,\cdot\rangle$ denotes the bracket process and as usual $\delta_{k,l}$ t... |
H: Why parametrise a curve in this way (on the unit circle)?
I saw papers saying something like "let $\gamma:S^1 \times [0,T] \to \mathbb{R}^2$ parametrise a curve. The second interval above just makes it time dependent, but why parametrise (for fixed time) the curve on S^1, the unit circle? I think it's to make it cl... |
H: Proving that $A=\{(-2)^n : n \in \mathbb{N} \}$ is unbounded
I am trying to prove that $A=\{(-2)^n : n \in \mathbb{N} \}$ is unbounded.
What I did was first to show that for every $n \in \mathbb{N}$ if $n$ is even then $(-2)^n = 2^n$ and if $n$ is odd then $(-2)^n = -2^n$ (I did it by induction on $n$).
Then I sh... |
H: Showing that the space $C[0,1]$ with the $L_1$ norm is incomplete
Can anyone think of a relatively easy counter example to remember, which demonstrates that the space $C[0,1]$ with the $L_1$ norm is incomplete?
Thanks!
AI: This example works to show $C[0,1]$ is not complete with respect to the $L^p$ norm for all $1... |
H: Linearly dependent vectors over finite fields
My problem is as follows:
Assume you have a vector space of dimension $(d + 1)$, with values over $GF(q)$.
Every vector in this vector space can be regarded as an element of the extension field $GF(q^{d+1})$. It is well known that every element of a finite field can be ... |
H: A problem about conditional expectation.
I'm studying advanced probability theorem by myself and have encountered a exercise:
Let $A>0$ be a constant, $\xi$ be a $r.v.$ such that $E|\xi|<\infty$ and
$P(\xi\leq x) = P(-\xi\leq x),\quad x\in\mathbb{R}$
Compute the conditional expectation $E(\xi\ |\ \xi I_{\{|\xi|\leq... |
H: Number of $4 $ digit numbers with no repeated digit.
Number of $4$ digit numbers with no repeated digit is
$4536$
$3024$
$5040$
$4823$
Well, I am very much weak in combinatorics. Please help.
AI: Ok so lets write down any old $4$ digit number
$abcd$
How many choices do we have for the digit $a$? We have $9$ choi... |
H: Rank and determinant of $D$ , an $n\times n$ real matrix, $n\ge 2$
Let $D$ be a $n\times n$ real matrix, $n\ge 2$. Which of the following is valid?
$\det(D)=0\Rightarrow \mathrm{rank}(D)=0$
$\det(D)=1\Rightarrow \mathrm{rank}(D)\neq 1$
$\det(D)=1\Rightarrow \mathrm{rank}(D)\neq0$
$\det(D)=n\Rightarrow \mathrm{rank... |
H: Needed an example to understand the concept .
I would be glad if anyone could provide me example of a Polar Set .
Polar set is defined as follows: Given a dual pairing $(X,Y)$ the polar set or polar of a subset $A$ of $X$ is the set $A^0$ of $Y$ such that :
$$A^0 = \{y \in X : \sup |\langle x,y\rangle|\le1 \}$$
A... |
H: Limit exercise from Rudin: $\lim\limits_{n \to \infty} \sqrt{n^2+n} -n$
This is Chapter 3, Exercise 2 of Rudin's Principles.
Calculate $\lim\limits_{n \to \infty} \sqrt{n^2+n} -n$.
Hints will be appreciated.
AI: Hint:
$$\frac{\sqrt{n^2+n}-n}{1} = \frac{\sqrt{n^2+n}-\sqrt{n^2}}{1}\times \frac{\sqrt{n^2+n}+\sqrt{n^2}... |
H: how to visualize binomial theorem geometrically?
How does $ \binom{n}{k} $ 'n choose k' get involved with coefficient of $ (a+b)^n $. Is there any intuitive geometrical picture (interpretation) that it seems obvious?
AI: Hint: Imagine writing $(a+b)^n$ as $(a+b)(a+b)\dots(a+b)$, and then multiplying out all the bra... |
H: Solving an equation with precondition
How do I solve the following equation:
$$x + y\ne0\text{ and }\frac{1}{x+y}=x$$
Wolfram Alpha came up with this solution
$$x\ne0,\:y=\frac{1-x^2}{x}$$
but I don't know how to get there.
thx
alex
AI: First af all, $x + y \neq 0$ since it is a denominator. Therefore $1=x(x+y)$. I... |
H: Solving P vs NP with computer
Is it possible to build a computer program that would (eventually) bring a solution to the P vs. NP question?
AI: Nobody knows. I suppose if there is a polynomial-time algorithm for 3-SAT (or some other NP-complete problem) then a computer could find it and prove P = NP. And if there i... |
H: How to check if a matrix is positive definite
I want to know how to check if a matrix M is positive definite
,assume that M is 3x3 real numbers matrix
I think one way is to put the matrix in a quadratic form $X^TMX$
, where X is a vector $X^T=[x_1 x_2 x_3]$
, my question is if I found that $X^TMX = ax_1^2 + ... |
H: The Frobenius-Nakayama Formula
I am currently reading a paper where it refers to the usual Frobenius-Nakayama formula describing quotients of an induced module. It is refering to the following result:
If $k$ is a field, $P$ is a subgroup of a group $G$, $F$ is a $kP$-module and $W$ is a $kG$-module, then we have t... |
H: Example computation of $\operatorname{Tor_i}{(M,N)}$
Let $M = \mathbb Z / 284 \mathbb Z$ and $N = \mathbb Z / 2 \mathbb Z$.
Can you tell me if my computation of $\operatorname{Tor_i}{(M,N)}$ is correct:
(i) First we want a projective resolution of $M$:
$$ 0 \to \mathbb Z \xrightarrow{\cdot 284 } \mathbb Z \xrightar... |
H: If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?
This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success.
Let $M$ be a monoid and $u\in M$. Suppose there exists a unique $z\in M$ su... |
H: How is uncountability characterized in second order logic?
How is uncountability characterized in second order logic?
Also, why is this characterization of uncountability "absolute" in the way that FOL's characterization of uncountability is not?
A very direct answer will be much appreciated.
Much thanks.
AI: The d... |
H: Basic probability - either event occurs but not both
I'm taking a graduate course in probability and statistics using Larsen and Marx, 4th edition and I'm struggling with a seemingly basic question.
If A and B are any two events, not mutually exclusive: $$P((A \cup B) ^\complement) = 0.6, P(A \cap B) = 0.2$$
What i... |
H: multiplication of a trigonometric series
Let $f(x)$ be the value of a trigonometric series, which converges uniformly on $\left[ -\pi, \pi\right]$. If I multiply $f(x)$ with $e^{iax}$ where $a\in\mathbb{N}$ will the result then be a trigonometric series which converges uniformly?
AI: Let $f_n$ denote the n-th parti... |
H: $C^1$ function questions
$f(x,t)$ is a function defined on the set $$S = \cup_{t \in [0,T]} A(t) \times \{t\}$$ where $A(t)$ is an open subset of $\mathbb{R}^n$ that depends on $t$ in some way.
I am told that $f \in C^1(S)$. Since the set $S$ is not closed (I think, even though $\{t\}$ is closed), we cannot say tha... |
H: A problem about Riemann-Lebesgue lemma
Let $f$ be an integrable function over $[a,b]$. Prove that:
$$\lim_{n \rightarrow \infty } \int _a ^b f(x)|\sin(nx)| dx= \frac {2}{\pi} \int _a^b f(x) dx.$$
AI: Hint: Prove the theorem for characteristic functions of intervals, then prove the theorem for step functions. Approx... |
H: Cheat sheet for various Mathematic topics
I was searching through the web to find out any webpage which contains the consolidated list for all mathematical cheat sheets in various topics in one place. I couldnt find one. It would be good if we can share the known cheat sheet links in this post for various mathemati... |
H: Riemann Surfaces Question (Complex Analysis)
Let $X$ be a compact Riemann surface, and denote by $m_X$ the following field:
$ m_X := \{ f:X \to \mathbb{P}_\mathbb{C} : f- \text{meromorphic} \} - \{\infty \} $
What is the natural injection of the field of rational functions $\mathbb{C}(z)$ into $m_X$ ?
p.s- $\math... |
H: Resources for matrices and its applications
I was preparing some presentation slides on basics of matrices and its application. Even though, many of the participants are familiar with basic matrix operation, I planned to explain them by starting from Matrix addition, Matrix subtraction, Matrix Multiplication and Ma... |
H: is this continuous and differentiable
Let $I=\{1\}\cup\{2\}$, for $x\in\mathbb{R}$,$f(x)=\operatorname{dist}(x,I)=\inf\{|x-y|:y\in I\}$
Then
1.$f$ is discontinuous some where on $\mathbb{R}$
2.$f$ is continuous on $\mathbb{R}$ but not differentiable only at $1$
3.$f$ is continuous on $\mathbb{R}$ but not differen... |
H: A limit that involves prime numbers
Let be $p_{n}$ the nth prime number and $(a_{n}), n\geq1$ such that:
$$a_{n}=\frac{1}{p_1}+\frac{1}{p_2}+\cdots+\frac{1}{p_n}$$
By using this result,
$$\lim_{n\rightarrow\infty} \frac{p_{1}}{p_{1}-1} \frac{p_{2}}{p_{2}-1}\cdots\frac{p_{n}}{p_{n}-1}=\infty$$
I have to prove that... |
H: Explanation of Zeta function and why 1+2+3+4+... = -1/12
Possible Duplicate:
Why does $1+2+3+\dots = {-1\over 12}$?
I found this article on Wikipedia which claims that $\sum\limits_{n=0}^\infty n=-1/12$. Can anyone give a simple and short summary on the Zeta function (never heard of it before) and why this odd r... |
H: A list of basic integrals
I am in need of a list of basic integrals for my upcoming ODE test,
I have searched on Math.SE for a post that might help but I didn't find such a post.
When I write 'basic' I don't necessarily mean immediate integrals (such as $e^x,\sin x$) but also 'useful' ones such as $\ln x$. If there... |
H: what is derivative of determinant map
Possible Duplicate:
Derivative of Determinant Map
consider $v=(v_1,v_2)\in \mathbb{R}^2$ ,$w=(w_1,w_2)\in\mathbb{R}^2$ consider the determinant map
det:$\mathbb{R}^2\times \mathbb{R}^2$ define by $\det(v,w)=v_1w_2-w_1v_2$. The derivative of the determinant map at $(v,w)\in\m... |
H: Proving ${p-1 \choose k}\equiv (-1)^{k}\pmod{p}: p \in \mathbb{P}$
Possible Duplicate:
Prove $\binom{p-1}{k} \equiv (-1)^k\pmod p$
The question is as follows:
Let $p$ be prime. Show that ${p \choose k}\bmod{p}=0$, for $0 \lt k \lt p,\space k\in\mathbb{N}$. What does this imply about the binomial co-efficients $... |
H: Why is $\log_{-2}{4}$ complex?
With the logarithm being the inverse of the exponential function, it follows that $ \log_{-2}{4}$ should equal $2$, since $(-2)^2=4$. The change of base law, however, implies that $\log_{-2}{4}=\frac{\log{4}}{\log{-2}}$, which is a complex number. Why does this occur when there is a r... |
H: solving a ODE with periodic boundary conditions
Help me please to solve this problem:
$u_{xx}+(\cos x+\cos^{2} x)u=e^{\cos x - 1}$
Thanks a lot!
AI: The starting point could be changing the dependant variable:
$$\cos x=t$$
$$\frac{du}{dx}=\frac{d(\cos x)}{dx}\frac{du}{d(\cos x)}=-\sin x\frac{du}{d(\cos x)}$$
$$\beg... |
H: Are these proofs correct? (Number Theory)
I'm finishing Chapter 1 of Apostol's book Introduction to Analytic Number Theory. I have made almost half of the 30 problems posed. I have some doubts on the proofs I produce, since sometimes I seem to assume extra information, or seem assume things that are obvious, when ... |
H: Limit involving $(\sin x) /x -\cos x $ and $(e^{2x}-1)/(2x)$, without l'Hôpital
Find:
$$\lim_{x\to 0}\ \frac{\dfrac{\sin x}{x} - \cos x}{2x \left(\dfrac{e^{2x} - 1}{2x} - 1 \right)}$$
I have factorized it in this manner in an attempt to use the formulae.
I have tried to use that for $x$ tending to $0$, $\dfrac{\sin... |
H: Diamonds of ideals, part 3
I'd like to wrap up the line of questioning started first in this question and then continued in this question.
The only variant left to try is:
"How close can you get to the Diamond lattice with two-sided ideals of a ring?"
Naturally, the commutative example in the first post is an ex... |
H: Prove that $R \otimes_R M \cong M$
Let $R$ be a commutative unital ring and $M$ an $R$-module.
I'm trying to prove $R \otimes_R M \cong M$ but I'm stuck. If $(R \otimes M, b)$ is the tensor product then I thought I could construct an isomorphism as follows:
Let $\pi: R \times M \to M$ be the map $rm$. Then there e... |
H: Compute close formula for a sum.
Let $k$ be a integer. How can we compute the close formula for
$$ \sum_{m=0}^{k} (m+1)(m+2)(2m+3)(3m+4)(3m+5)? $$
AI: We can expand the polynomial we are summing over to give a degree 5 polynomial in $k$, as follows:
$$(m+1)(m+2)(2m+3)(3m+4)(3m+5)=18m^{5}+135m^{4}+400m^{3}+585m^{2}... |
H: Arclength of the curve $y= \ln( \sec x)$ $ 0 \le x \le \pi/4$
Arclength of the curve $y= \ln( \sec x)$ $ 0 \le x \le \pi/4$
I know that I have to find its derivative which is easy, it is $\tan x$
Then I put it into the arclength formula
$$\int \sqrt {1 - \tan^2 x}$$
From here I am not sure what to do, I put it in w... |
H: "Negative" versus "Minus"
As a math educator, do you think it is appropriate to insist that students say "negative $0.8$" and
not "minus $0.8$" to denote $-0.8$?
The so called "textbook answer" regarding this question reads:
A number and its opposite are called additive inverses of each other because their sum is z... |
H: Simplification of the Expected Value via CDF: Does it work for ALL Probability Distributions?
If a random variable $X$ has a density $f$, then the expected value can be simplified:
$$\mathbb{E}[X]=a+∫_{a}^{b}(1-F(x))dx,$$
where $F$ is the cumulative distribution function, $F(x)=\Pr(X≤x)$.
My question is: Does this... |
H: Inverses and orders in the group $Z_5$
Ill like some guidance to solve this kind of question ( I have many like this) and I have no clue what I need to do here:
I need to find
$$g^{-1}$$
and
$$o(g)
$$
when $G$ is $Z_{_5}$.
adding - is I have G is $Z^x_{_5}$ what group is this????
AI: So $\mathbb{Z}_5 = \{[0], [1... |
H: boundary of multiple sets
Is it possible for a boundary point of a set to be a boundary point with respect to multiple other sets, in the sense that any neighborhood of a point of A contains points of BOTH B and C? If so can you give me an example? If so is it then the case that an entire boundary set can be a boun... |
H: Area of a surface of revolution of $y = \sqrt{4x+1}$
$y = \sqrt{4x+1}$ for $1 \leq x \leq 5$
I really have no idea what to do with this problem, I attempted something earlier which I will not type up because it took me two pages.
$$y = \sqrt{4x+1}$$
$$\int 2 \pi \sqrt{4x+1} \sqrt{1 + \frac{4}{1+4x}}dx$$
$$2 \pi \i... |
H: Is the support of a random variable those values where the graph of its distribution is not "flat"?
In the literature the support, $S$, of a random variable $X$ is defined as the smallest closed subset of real line $\mathbb{R}$ with probability $1$. Looking to prove that $S$ is where the graph of $X$'s cdf, $F$, is... |
H: How is this an example of a linear system?
Consider the following transfer function:
$$\frac{Y}{X} = \frac{A_0}{\omega_o^2 s^2 + 1} $$
or something similar that is supposed to represent an undamped block on a spring. I encountered the following question about it which has me flummoxed:
Linear systems, when given a ... |
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