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H: What is $\mathfrak{gl}(\infty)$
As title says, I know what is $\mathfrak{gl}(n,\mathbb{C})$, but what is $\mathfrak{gl}(\infty)$? Where can I find good reference for this?
AI: This is the finitary Lie algebra
$$
\mathfrak{gl}(\infty)=\bigcup_{n\in \mathbb{N}}\mathfrak{gl}(n).
$$
The simple ones have been classified... |
H: Reducing ordinals in the representation of a limit ordinal as $\bigcup_{\beta < \alpha} \beta$
Let $\alpha$ be a limit ordinal such that $\kappa$ < $\alpha$ < $\kappa^{+}$
where $\kappa$ is initial ordinal (so $|\alpha| = \kappa$).
I want to know whether or not I can find a sequence {$\beta_{i}$} where $ i < \kapp... |
H: Showing $\sum_{r=1}^\infty \frac{1}{(r-z)^2}$ is holomorphic on $\mathbb{C}\setminus\mathbb{N}$
Ok, so as per the question title I'm wanting to show that
$\displaystyle\sum_{r=1}^\infty \dfrac{1}{(r-z)^2}$ is holomorphic on $\mathbb{C}\setminus\mathbb{N}$.
I'm thinking that if I show that
$$\int_{\gamma} \sum_{r=1}... |
H: The Mean of a multivariate function
Jane and Jack each toss a fair coin twice.
Let X be the number of heads Jane obtains.
Let Y be ther number of heads Jack obtains.
Define U = X + Y.
Find the mean and variance of U.
I have tried to first find f(U) and am unable to.
Would it be:
$$ {4 \choose u}(0.5)^{u}(0.5)^{4-u}... |
H: Are there p-adic manifolds?
Is there anything resembling a manifold on the field of p-adic or complex p-adic fields?
If so is there a connection to algebraic geometry as rich as in the reals?
AI: Yes, there are. One source to learn about them is the second half of Serre's book Lie Algebras and Lie Groups, and anot... |
H: vector problem in parallelopiped
I want to find out absolute volume the parallelopiped
I have not got that how they did with this vector notation, $h= \vec{A}\cdot \vec{n} $
The volume will be $\vec{A}\cdot (\vec{B} \times \vec{C}) $
AI: Note that $|\vec{n}|=1$ and $\vec{A}\cdot \vec{n}=|\vec{A}||\vec{n}|cos... |
H: $(X,d)$ m.e., with $Y \subset X$: $Y$ is open, $Y$ is connected it's equivalent to another property.
Let $(X,d)$ be a metric space and let $Y \subset X$: $Y$ is open. Prove that $Y$ is connected if and only if there aren't $A,B \subset X$ non-empty such that $Y=A \cup B$ and $A \cap \overline B=\emptyset$ and $\ove... |
H: how the monotonicity of $\frac{1}{(logn)^n}$ ,with induction or an other way??
Could you tell me how to show the monotonicity of $\frac{1}{(logn)^n}$ ? With induction or with an other way?
AI: Let us define $a_{n}=\frac{1}{(log(n))^{n}}$
$log(n+1) > log(n)$ for all natural numbers $n$ as the function $f(x)=log(x)$... |
H: Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$
Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and $S_n=\sum\limits_{j=1}^n X_j$... |
H: Summation of a floored square root
I am working on a little something and have hit a roadblock of sorts. I have arrived at this equation:$$\sum_{n=1}^{r}\left\lfloor\sqrt{2nr-{n}^{2}}\right\rfloor$$
I am attempting to find some way of solving this summation as a function of r. If this is possible, could I get some ... |
H: Square root in Banach algebra
Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin 10.30) we know that there exists a "square root" of $a$, i.e. an element $b\in A$ such that $... |
H: Simple logarithm properties proof
I was reading a school algebra book about logarithm function (on $\mathbb{R}^+$). There were several properties without proof. So I decided to prove 2 of them myself.
The first property:
$log_a(x\cdot y) = log_ax + log_ay$
Proof
By definition of logarithm we know, that: $a^{log_ax}... |
H: Limit at infinity involving $e$
I am to find the limit of $$\lim_{x \to \infty} \left(1+\frac{x}{5x^3+x^2+8}\right)^ {\dfrac{x^3+8}{x}}$$
I could not find the proper substitution here. I would be happy if someone could shed some light. Thanks.
AI: $$
\begin{align}
\lim_{x\to\infty}\left(1+\frac{x}{5x^3+x^2+8}\righ... |
H: Is identity map one to one and onto?
Im reading a chapter of compactness in Real Analysis, Carothers, 1ed. Actually, identity map has been involved in and I've captured its definition:
Equivalent Metrics
As a last topic related to both compactness and uniform continuity, we discuss several notions of equivalence f... |
H: Prove that subsequence converges to limsup
Given a sequence of real numbers, $\{ x_n \}_{n=1}^{\infty}$, let $\alpha =$ limsup$x_n$ and $\beta = $ liminf$x_n$.
Prove that there exists a subsequence $\{ x_{n_k}\}$ that converges to $\alpha$ as $k \rightarrow \infty$.
Not sure how to start this without since I'm no... |
H: For a sequence of non negative numbers, if the series converges, then the series of the sequence raised to p also converges if p>=1
Let $p \geqslant 1$ and let $(a_n)$ be a sequence of non-negative numbers. Then if $\sum\limits_{n=1}^\infty a_n$ converges, so does $\sum\limits_{n=1}^\infty a_n^p$. Prove this statem... |
H: Euler numbers grow $2\left(\frac{2}{ \pi }\right)^{2 n+1}$-times slower than the factorial?
Stirling's approximation of the factorial for even numbers is given by
$$ (2n)! \sim \left(\frac{2n}{e}\right)^{2n}\sqrt{4 \pi n}. \tag{1} $$
Further, the Euler numbers grow quite rapidly for large indices as they have the f... |
H: Show that there is a number between 1 and 1000 such that there is a perfect square
Show that there exists an integer $n \in S = \{1,2, \ldots, 1000\}$ such that
$$\prod_{i\in S-\{n\}}i!$$
is a perfect square.
I was thinking in trying to prove it by contradiction using the Pigeonhole Principle.
AI: First observe tha... |
H: Evaluating the integral of a delta function multiplied by an exponential
How does one evaluate the integral below? I don't think we can use integration by parts here. I am basically stuck.
$$ \int_{-1}^{6}{(2+e^{-t})\delta(t-2)} dt $$
$$ = \int_{-1}^{6}{2\delta(t-2)dt} + \int_{-1}^{6}{e^{-t}\delta(t-2)dt} $$
AI: As... |
H: Concrete Mathematics - Towers of Hanoi Recurrence Relation
I've decided to dive in Concrete Mathematics despite only doing a couple of years of undergraduate maths many years ago. I'm looking to work through all the material whilst plugging gaps in my knowledge no matter how large they are.
However, solving recurre... |
H: Using euclidean algo to find d (RSA encryption)
The questions says "let p = 5, q =11, n = 55 tocient(n) = 40. e=7.
Use the Euclidean algo to find the value of d.
This is driving me crazy. Here's what I did:
40=5(7) +5
7= 1(5) + 2
5= 2(2) +1
2 = 2(1) + 0
So the GCD is 1.
Now to get 1 = de +f*tocient(n) to find d.
1... |
H: Finding the moment generating function of the product of two standard normal distributions
The following question is on my homework assignment that I cannot figure out:
Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find the moment generating functio... |
H: How to split the difference in game theory?
There are two parties in talks to settle a law suit.
The expected value of going to court for the plaintiff is $\$3,155$
The expected value of going to court for the defendant is $-\$10,244$
The defendant could give up $3,155 + \$1$ and settle the matter - and at first b... |
H: epsilon delta approach to a problem
The following is what I would like to show.
$$\lim_{x \to5}\frac{1}{x-3}=1/2$$
Given any $\epsilon>0$, $|\frac{1}{x-3}-\frac{1}{2}|\le\frac{|x-5|}{|x-3|}\le2\epsilon$
How do I get rid of $|x-3|$ here to get to
$|x-5|<\delta$ ?
My friends told me something about $\delta=\min\{... |
H: Cardinality of the union of disjoint set, each of which have a cardinality of $\aleph_{0}$?
Consider three sets $A$,
$B$, and
$C$
such that $A\cap B=\emptyset$
, $A\cap C=\emptyset$
, $B\cap C=\emptyset$
($A,B,C$
are pairwise disjoint) and $\overline{\overline{A}}=\overline{\overline{B}}=\overline{\over... |
H: $\forall x,y,z \in \mathbb{R} (x\cdot 0 + y \cdot 0 + z \cdot 0=0) \to (x=y=z=0)$
$\forall x,y,z \in \mathbb{R} (x\cdot 0 + y \cdot 0 + z \cdot 0=0) \to (x=y=z=0)$?? If is true, why? Thanks in advance!
AI: If $+$ and $\cdot$ mean usual addition and multiplication of real numbers, and if $0$ means the particular zer... |
H: When does equality hold in the Minkowski's inequality $\|f+g\|_p\leq\|f\|_p+\|g\|_p$?
I would like to see a proof of when equality holds in Minkowski's inequality.
Minkowski's inequality. If $1\le p<\infty$ and $f,g\in L^p$, then $$\|f+g\|_p \le \|f\|_p + \|g\|_p.$$
The proof is quite different for when $p=1$ and... |
H: Problem concerning Numerical Solutions of Nonlinear Systems of Equations (Burden and Faires)
I need help with this particular question:
The nonlinear system
$3x_1 - \cos (x_2 x_3) - \frac{1}{2} = 0$
$x{_1}^2 - 625x{_2}^2 - \frac{1}{4}=0$
$\exp ^{-x_1x_2} + 20x_3 + \frac{10\Pi -3}{3}=0$
has a singular Jacobian ma... |
H: Question about automorphisms
Let $G$ be a finite abelian group and let n be a positive integer relatively prime to $|G|$.
a. Show that the mapping $ϕ(x)=x^n$ is an automorphism of $G$.
b. Show that every $x ∈ G$ has an $n^{th}$ root, i.e., for every $x$ there exists some $y∈G$ such that $y^n=x$.
a.To show that the ... |
H: How to show integral is related to complementary error function?
I have to somehow show that:
$$
\int_0^\infty{\frac{e^{-a t}}{t^{1/2}(t+x)}} \textrm{d}t=\frac{\pi}{\sqrt{x}} e^{a x} \textrm{erfc}(\sqrt{ax})
$$
I've tried substituting $u=\sqrt{t}$ to get
$$
2 \int_0^\infty{\frac{e^{-a u^2}}{u^2+x}} \textrm{d}u
$$
w... |
H: Proving a function is bounded given that the function is uniform continuous over a bounded, closed domain
Let's say I have a function $f : I \rightarrow \mathbb{R}$ that is uniformly continuous, and $I$ is a colsed and bounded interval. Then $f(I)$ is closed by the following:
Assume $f(I)$ is not closed, and that i... |
H: Why are these relations not posets?
I was hoping you guys could help me clarify why these relations are or arent posets. I gave my thought process that resulted in the wrong answer.
a)(Z, =) poset
b)(Z, !=) not a poset
c) (R,==) poset
d) (R, <) not a poset
A) I thought it wouldn't be a poset, but it is. I had assu... |
H: How to prove that if n and k are integers with 1 ≤ k ≤ n, then k*(n C k)=n(n−1 C k−1) combinatorally?
I am having with combinatorial proofs. My professor says to come up with a scenario so that we can connect both sides by double counting but I am clueless.
AI: There is a group of $n$ people. We want to give $k$ p... |
H: Prove that $\left | \cos x - \cos y \right | \leq \left | x - y \right |, \forall x,y \in \mathbb{R}.$
This is one of the problem in my text book where the section which the problem is stated talks about the mean value theorem and Rolle's theorem. By looking at this, I have no idea where to start. Can I have some h... |
H: Known probability for one month, what is the probability for 100 months?
This is a pretty famous probability problem:
The probability that a driver will have an accident in 1 month equals 0.02. Find the probability that in 100 months, he will have 3 accidents (Papoulis, 1984).
AI: the probabilities of outcomes in... |
H: Correctly representing a $2^n < n!$ statement
$$2^n < n!$$ After an inductive proof I determined that $2^n < n!$ is valid only for values greater than or equal to $4$.
So. How do I represent this conclusion?
Is this correct?
$$\forall n \in \mathbb{Z}^+, n \geq 4 \implies 2^n < n!$$
AI: Yes is correct! But better ... |
H: Application of Sylow's theorem to direct products
Question: Let $p $ be a prime number. Suppose that $|A|=N $ and $|B|=M $, and let $p^n $ be the largest power of $p $ that divides $N $ and $p^m$ is the largest power that divides $M $. Consider the group $A \times B$. Let $P$ be a p-Sylow subgroup of A and ... |
H: smooth lie group action
Let $\theta:G\times M\to M$ be a smooth left action of a Lie group $G$ on the manifold $M$.
Suppose $G$ is compact and $M$ is Hausdorff. Let $K$ be a compact set in $M$. Is it true that $G_K:=\{g\in G:(g\cdot K)\cap K\neq \emptyset \}$ is closed?
AI: If $y_n \in G_K$ and $y_n \to y \in G$. L... |
H: Evaluation of $\int_\Gamma e^{-z^2}\ dz$
My question is simple. Prove the following equality
$$\int_\Gamma e^{-z^2}\ dz = \int_{-\infty}^\infty e^{-x^2}\ dx$$
where $\Gamma = \{ z\in {\bf \mathbb C}| \ {\rm Im}\ (z) = c \}$
AI: Consider
$$\oint_C dz \, e^{-z^2}$$
where $C$ is a rectangle having vertices $-R$, $R$,... |
H: Generalized distributive laws proof feedback
I'm currently learning proofs and elementary set theory. I would like to have feedback on my proof since I'm self-studying. Are some part superfluous or unclear? My proof goes as follows:
I will prove that $(\bigcap_i X_i)-A = \bigcap_i(X_i - A)$, where "$-$" is the set ... |
H: Group of order 35, and normal subgroup of order 7
G a group of order 35, H a normal subgroup of order 7. Prove that if g in G has order 7, then g is in H.
What I'm doing is invoking Sylow's third theorem to show that there exist a single subgroup of order 5 and a single subgroup of order 7. And then the direct pr... |
H: Can we determine which statements are incomplete due to Godel?
Due to Godel's incompleteness theorems we know that there are true statements in a system that cannot be proven with that system. My questions are 1) can we tell which statements in a system are the ones that cannot be proven, and 2) can we choose thro... |
H: Proving a set is countably infinite $\left \{q \in \Bbb Q:q=\dfrac{a}{b}\ \text{where $a$ is even and $b$ is odd} \right \}$
Proving a set is countably infinite
$$\left \{q \in \Bbb Q:q=\dfrac{a}{b}\ \text{where $a$ is even and $b$ is odd} \right \}$$
I am not sure how to go about solving this problem. I know it ... |
H: When proving a statement by induction, how do we know which case is the valid 'base'?
For example proving 2^n < n!, 4 is the 'base' that works for this exercise, then starting from there we prove p + 1 considering p has to be at least 4 and we have our result. However, I believe determining the first valid value wi... |
H: Proving $n^n \ge n!$, is induction necesary?
I am trying to prove that:
$n^n \geq n!$ is valid for a $x$ set of numbers.
So, I am trying an inductive process.
However, case $P(0)$ doesn't seem to work because I have read somewhere that $0^0$ is undefined, but I've also read that $0^0$ is $1$.
What approach do you r... |
H: Does this sequence converge or diverge? $a_n = (-1)^n\frac{n^2+n+2}{2n^2+3n+4}$
We had this question on an exam and I don't know if I got it right or not. We were asked to justify our answer. I said the sequence diverges since the limit bounces between $-\frac{1}{2}$ and $\frac{1}{2}$ but now I think I got it wro... |
H: How do I identify repeated irreducible factors?
I thought my solution was correct - but it seems like that's not the case. Can anyone possibly explain to me why I'm wrong?
AI: It's clear that $x^2 + 1$ is an irreducible quadratic. On the other hand, using the fact that $x^2 - 1 = (x - 1)(x + 1)$, we can rewrite th... |
H: Proving discontinuity at $x=0$
I would like to prove that the function below is discontinuous at $x=0$. $$f(x)=\begin{cases}\sin\left(\frac{1}{x}\right) & \text{for } x\neq 0\\
0 & \text{for } x=0\end{cases}$$
What I have so far is this: (not much)
$$\left|\sin\left(\frac{1}{x}\right)-0\right|=\left|\sin\left(\fra... |
H: Is $L^\infty(X, \mu)$ dual of $L^1(X, \mu)$ if X is discrete
Is $L^\infty(X, \mu)$ dual of $L^1(X, \mu)$ if X is discrete and $\mu$ is not finite?
Like, if X = {x,y} and $\mu({x}) = 1$ and $\mu({y}) = \infty $, then $L^\infty(X, \mu)$ is not dual of $L^1(X, \mu)$ since, Let $f(x) = 1$ and $f(y) = 0$ And I can def... |
H: Dirac Delta identity
Reading through the proof on the Helmholtz decomposition of a vector field, I came across the following identity:
$$\delta(x-x')=-1/4\pi*\nabla^2*(|x-x'|)^{-1}$$
Does anyone have any insight on how to prove/derive this identity? Thanks in advance. Here's the article for reference:
http://facul... |
H: Good Textbook on Topology
I have one year calculus and one year linear algebra background. In addition, I have had one semester study in metric space analysis. Can anyone suggest some good textbooks on topology, please? A reader-friendly text with plenty of examples would be ideal. Thank you!
AI: I strongly recomme... |
H: How many ways are there for $2$ teams to win a best of $7$ series?
Case $1$: $4$ games: Team A wins first $4$ games, team B wins none = $\binom{4}{4}\binom{4}{0}$
Case $2$: $5$ games: Team A wins $4$ games, team B wins one = $\binom{5}{4}\binom{5}{1}-1$...minus $1$ for the possibility of team A winning the first fo... |
H: Optimizing bounds?
Pretty much trying to optimize the bounds of a function (or window when using a graphing calculator).
So the function is $=e^{-x}\sin(x)$, with $x\geq0$.
When taking the derivative I get $-\sin(x)+\cos(x)$. Setting it equal to $0$, I get $x=\pi/4$ and $x=5\pi/4$.
Graphing it on my calculator, it... |
H: Why, while checking consistency in $3\times3$ matrix with unknowns, I check only last row?
I would like to know whether my thinking is right. So, having 3 linear equations,
$$
\begin{align}
x_1 + x_2 + 2x_3 & = b_1 \\
x_1 + x_3 & = b_2 \\
2x_1 + x_2 + 3x_3 & = b_3
\end{align}
$$
I build $3\times 3$ matrix
\begin{bm... |
H: Definition of Left Translation of a function on a topological group
In Folland's A Course in Abstract Harmonic Analysis, he defines for a function $f$ on a topological group $G$, and $y\in G$,
$$L_{y}f:x\to f(y^{-1}x)\text{ and }R_{y}f:x\to f(xy)$$
He then remarks that the $y$ is inverted so that the map $y\mapsto ... |
H: Write a grammar that generates the strings over {a,b} starting with a
The answer is:
S -> aA, A -> aA, A -> bA, A -> a, A -> b, S -> a
Any idea how they got this?
AI: Let's write a few strings from the language, some are:$a,aa,abbbb,aabaabbb,...$.
There is only one condition, every string in this language should ... |
H: good reference for the Laplace method
I was wondering if someone could suggest good reference about the Laplace method.
One source I have now is just a short section (appendix) of Stein's Complex Analysis textbook.
I wish it starts with basic materials and provide rather precise error bounds, and also treats mult... |
H: Sum of 2 Standard Guassian variables
We know that when X and Y are independent and normally distributed, then X+Y is also Normally distributed. But now, let's remove the hypothesis of Independence. Can anyone please prove or provide a counter example for the following statement
When $X$ and $Y$ are Gaussian random... |
H: Drawing Graph on 5 vertices.
Draw a graph on 5 vertices that contains no clique of size three (that is, no triangle) and no anti-clique of size three (that is, three vertices none of which is connected to any other).
Here are a few graphs which I made. The problem I'm having is I don't quite understand what the q... |
H: Lottery Ball problem
There are two sets of numbered balls. One set consits of white balls numbered $1-10$ the other is blue balls numbered $1-20$. To play you select two white balls and two blue balls. What is the probability that your ticket contains exactly one matching white number and two matching blue numbers?... |
H: Area of a right angled triangle is an even integer
Actual Question is :
Let $ABC$ be a triangle in the plane such that $BC$ is perpendicular to $AC$. Let $a,b,c$ be the lengths of $BC$, $AC$, and $AB$ respectively. Suppose that $a,b,c$ are integers and have no common divisor other than $1$. Which of the following s... |
H: Generally true that $\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y)$?
Is it generally true that
$$\frac{\mathrm d}{\mathrm da} \int_{-\infty}^{a-y} f(x)\, \mathrm dx = f(a-y),$$
where $a$ and $y$ in the expression are constants?
To give context to the question, I am reading about the... |
H: Suppose that a cube is inscribed in a sphere of radius one. What is the volume of the cube? my reasoning vs answer
Now my reasoning is that, s^2 + s^2 = 2^2, where s is the side of the cube, giving, s^3 = 2 sqrt 2. But the answer and explanation here is different: http://math.acadiau.ca/aumc/hints4.pdf
how is the d... |
H: Why is this binomial coefficient bounded thus?
Source: Miklos Bona, A Walk Through Combinatorics.
$$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$
The RHS is the upper bound of the Ramsey number $R(k,k)$.
How can I prove the inequality without using mathematical induction? I've merely expanded the LHS to obtain $... |
H: $3+33+\dots+33\ldots3={10^{n+1}-9^n-10\over 27}$
I need help to show by Induction
$3+33+\dots+33\ldots3={10^{n+1}-9^n-10\over 27}$
Thank you.
AI: I think the problem has a typo.
It should be $$f(n):3+33+\cdots+\underbrace{33\cdots 33}_{n \text{ digits}}=\frac{10^{n+1}-9n-10}{27}$$
Let $f(n)$ holds true for $n=m$
$$... |
H: Is statistical dependence transitive?
Take any three random variables $X_1$, $X_2$, and $X_3$.
Is it possible for $X_1$ and $X_2$ to be dependent, $X_2$ and $X_3$ to be dependent, but $X_1$ and $X_3$ to be independent?
Is it possible for $X_1$ and $X_2$ to be independent, $X_2$ and $X_3$ to be independent, but $X_... |
H: Application of pigeonhole principle
Select $11$ different numbers from $f\{1,2,...,20\}$. Prove that two of your numbers, $a$ and $b$, will differ by two.
Clearly this is an application of the pigeonhole principle. However, I'm not sure how to write up a coherent proof.
AI: Consider the following sets: $$\{1, 3\}, ... |
H: Show combination of affine functions and logs has at most one zero
For $x>0$, let
$$
f(x)=(x+2){\sf log}(x)-(x+1){\sf log}(x+1)
$$
Can anybody show that the equation $f(x)=0(x > 0)$ has at most one
solution.
AI: We compute
$$
f'(x) = 1 + 2/x + \log x - 1 - \log (x+1) = 2/x - \log (1 + 1/x).
$$
We want to show that ... |
H: Is $\mathbb{Z}$ isomorphic to a direct subproduct of the family $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?
Is $\mathbb{Z}$ isomorphic to a direct subproduct of the family $\left\{ \mathbb{Z}_{n}\right\} _{n>1}$?
AI: I'm assuming that $\mathbb Z_n$ means $\mathbb Z/n\mathbb Z$. I'm also assuming that "subproduct" mea... |
H: Can conclude that $N = f\left(K\right)\oplus f\left(L\right)$?
Let $K,L$ be two submodules of an $R$-module $M$ with the property that $K+L=M$ and $f:M\longrightarrow N$ is an $R$-epimorphism. Assume that $K \cap L = \ker f$. Can conclude that $N = f\left(K\right)\oplus f\left(L\right)$ ?
AI: It's obvious (from s... |
H: Let $\left\{ f_{i}:M_{i}\longrightarrow N\right\} _{i\in I}$, can $Im\left(\oplus_{i\in I}f_{i}\right)=\sum_{i\in I}Im\left(f_{i}\right)$
Let $\left\{ f_{i}:M_{i}\longrightarrow N\right\} _{i\in I}$ be a family of $R$-homorphism from $R$-module $M_{i}$ to $R$-module $N$. Do we have $Im\left(\oplus_{i\in I}f_{i}\rig... |
H: Evaluate $\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$
Question is to Evaluate :
$$\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots$$
what all i could do is :
$$\frac{1}{3}+\frac{1}{4}\frac{1}{2!}+\frac{1}{5}\frac{1}{3!}+\dots=\sum_{n=1}^{\infty} \frac{1}{(n+2)n!}=\sum_{n=1}^... |
H: Area of triangle $OAB$
Question is :
Consider a circle of unit radius centered at $O$ in the plane. let $AB$ be a chord which makes an angle $\theta$ with the tangent to the circle at $A$ .find the area of triangle $OAB$
What all i could do was is just draw the picture and even in that i am not sure if he mean angl... |
H: Why is $ 2\binom nm^2
$\forall n\geq2 \forall m\geq2,$ $$ 2\binom nm^2<n^{2m}.$$
Why is the above inequality, which is equivalent to $ \binom nm<\frac{n^m}{\sqrt 2}$, true?
AI: $$\binom nm =\frac {n (n-1) ... (n-m+1)}{m!} < \frac{n^m}{m!}$$
Thus, if $m \geq 2$
$$\binom nm^2 < \frac{n^{2m}}{m!^2} < \frac{n^{2m}}{2}$... |
H: Is the number 100k+11 ever a square?
Is the number 100k+11 ever a square?
Obviously all such numbers end in 11. Can a square end with an 11?
AI: Let $a^2=100\cdot k+11$. Then easily $a$ must be odd.
So $(2n+1)^2=100\cdot k+11$. It follows $2(n^2+n)=50\cdot k+5$, which is impossible. |
H: How prove this $f(x)=g(x)=h(x)=0$?
let polynomial $f(x),g(x),h(x)\in C[x]$, and such
$$f^2(x)=xg^2(x)+xh^2(x),$$
prove or disprove
$$f(x)=g(x)=h(x)=0$$
I know solve this follow problem
let polynomial $f(x),g(x),h(x)\in R[x]$, and such
$$f^2(x)=xg^2(x)+xh^2(x),$$
prove or disprove
$$f(x)=g(x)=h(x)=0$$
then I... |
H: Restricting the ordering of a given permutation
Let $p_1p_2p_3...p_n$ be a randomly-selected n-permutation.
Why is $P(p_1>p_2>p_3)=\frac 16$?
(P denotes probability.)
AI: There are $n!$ ways to arrange $n$ distinct numbers. The condition $p_1 > p_2 > \dotsc > p_n$ singles out one of these ways. If all permutations ... |
H: Difference between quaternions and rotation matrices
This is a really simple question, I guess. Do quaternions cover the same set of rotations as rotation matrices? I assume the answer is yes, they both can represent SO(3), but I'm unsure about the sources which prove that.
Extending the question a bit further, whe... |
H: recipe for infinitely many irrational numbers - or is it?
What if we write 0. and then throw a coin and depending on the result continue the number with 1 or 0 and continue this process indefinitely. It seems like a recipe for producing irrational numbers.
Are these numbers really irrational? Are they transcendenta... |
H: How find this linear equation $-x_{1}-x_{2}-x_{3}-\cdots+(2^n-1)x_{n}=2^na$?
let $a\in R$,then solve this follow equation:
$$\begin{cases}
x_{1}-x_{2}-x_{3}-\cdots-x_{n}=2a\\
-x_{1}+3x_{2}-x_{3}-\cdots-x_{n}=4a\\
-x_{1}-x_{2}+7x_{3}-\cdots-x_{n}=8a\\
\cdots\cdots\cdots\cdots\cdots\cdots\\
-x_{1}-x_{2}-x_{3}-\cdots... |
H: Let $\{a_n\}$ be a positive monotonic decreasing sequence of real numbers. Show $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$
Let $\{a_n\}_{n=1}^{\infty}$ be a monotonic decreasing sequence of positive real numbers. Show that $\lim_{k \rightarrow \infty} \frac 1 k \sum_{n=1}^k a_n = \inf a_n$
... |
H: Complex numbers inequality
Let $z$ , $w$ $ \ \in \mathbb{C}$ with $|z|$ , $|w| < 1$. Then prove that $|\frac{z-w}{1-z\bar{w}}| < 1$.
What I have noticed : $|z-w| = |w||\frac{z\bar{w}}{|w|} -1| $ but I don't know how to proceed.
AI: $$|\frac{z-w}{1-z\bar{w}}| < 1<=>|z-w|<|1-z\overline w|<=>(z-w)(\overline z-\overlin... |
H: Differential forms as functionals on curves
Please give me a reference to a book or lecture notes where the following stuff is studied.
Let $M$ be a Riemann surface with boundary $\partial M$ (but not necessarily, any smooth $n$-dimensional manifold suffices if the following statements make sense in this case).
Wh... |
H: Can $f(x)>g(x)$ be implied from $\frac{df(x)}{dx}\gt \frac{dg(x)}{dx}$?
I am new to functions. My question is Can $f(x)>g(x)$ be implied from $\frac{df(x)}{dx}\gt \frac{dg(x)}{dx}$?
AI: Given that $f(a)>g(a)$ at $a$ and $f'(x)\geq g'(x)$ for all $x>a$, you can deduce $f(x)>g(x)$ for all $x>a$ inductively. But the... |
H: What the symbol $\subseteq$ represents generally?
My book says that $\subset$ is used to represente any subset, proper or improper, needing in this case to show the anti symmetric property of sets. ($A = B \iff A \subset B \, \, \wedge \,\, B \subset A)$
And $\subseteq$ is used specifically to represent improper su... |
H: Numerical Integration of $g(x)=4\sqrt{1-x^2}$
The limits of integration are $[0,1]$ and thus the result should be $\pi$. My book suggests an elegant way to evaluate the integral by Monte Carlo Integration but I was wondering, can we reach the same result with an easier way? Thank you.
AI: Hints:
$$\sin u:= x\implie... |
H: On derivatives...
I have a quick question here. I hope someone can help. I haven't done calculus for a long time so I seem to missed out on details.
If $x=g^{-1}(y)$ and $g$ is monotonic and is differentiable for all $x$, how did it happen that $$\frac{d}{dy}g^{-1}(y)=\frac{1}{dg(x)/dx}|_{x=g^{-1}(y)}$$
I tried to... |
H: Show two graphs are not isomorphic
I know this graphs are not isomorphic. However they have the same number of vertex and edges, and the same degree sequence, is not the most easy case.
If im correct, the graphs are isomorphic if evey posible bijection between vertex preserve adjacencies, then if i find just one... |
H: Do Symmetric problems have Symmetric solutions?
There is a general notion of Symmetry in mathematics,
that of an object being constant under some transformation.
If we think of our object as being a "mathematical problem" we can see certain mathemtical problems have symmetries in them, in terms of how the problem i... |
H: quotient of two entire functions
Suppose we have a quotient of two entire functions , i.e. $g(z)=\frac{f(z)}{h(z)}$ where $f,h$ are entire functions in complex plane. If $f$ and $h$ have the same set of zeros, what can we tell from the order of these zeros? For example, if we have $f(z_0)=h(z_0)=0$, what can we say... |
H: Lebesgue measure of algebraic irrational numbers in $\mathbb{R}$
Find all Lebesgue measurable subsets $A \subset\mathbb{R}$ such that all $B\subset A$ is measurable.
I argued that if the measure is positive then $A$ is an interval so we can construct the Vitali set and thus it'd have non-measurable subsets. So $A$ ... |
H: $K$ and $L$ homeomorphic, then $C(K)$ is isomorphic to $C(L)$
Can someone sketch the proof (or give me some reference) of the following fact : If $K$ and $L$, compact and Hausdorff spaces, are homeomorphic then the lattices $C(K)$ and $C(L)$ are isomorphic. (I am aware this is half of Kaplansky theorem, but I'm cu... |
H: Confusion about extension of functionals
Assume that $\Omega\subset \mathbb{R}^N$ is a bounded set and take a function $f:\Omega\to\mathbb{R}$ which belongs to $L^2(\Omega)$ but does not belongs to $L^q(\Omega)$ for $q>2$.
Define a bounded linear map $T:L^2(\Omega)\to\mathbb{R}$ by $$\langle T,g\rangle=\int_\Omega ... |
H: meaning of the symbol $Z_n^*$ in discrete mathematics
I was reading discrete Mathematics, and i found a symbol $$Z_n^*.$$ I don't know what it means.
The text says that the "image" with the multiplication operator is an abelian group.
can any one explain.
AI: maybe the author is talking about the group of unit elem... |
H: Relatively prime polynomials
If $f(x)$ is relatively prime to $p(x)$ in $F[x]$ prove that there is a polynomial $g(x) \in F[x]$ such that $f(x)g(x) ≡ 1_F \pmod{ p(x)}$.
Now it has just occured to me that this is a field we are working in and not a ring because we are using an $F$. This is just an assumption, so I d... |
H: How do i get $b$ from ${b\over2-3ab}=2$
I have this expression:
$${b\over2-3ab}=2$$
So now i try to get $b$
I started with:
$${2-3a} = {1 \over 2}$$
But how you can see $b$ gets lost, what did i wrong?
Thanks!
My teacher said the right answer would be $b = {4\over 6a+1}$
AI: As I see it, you did two steps:
$$
\frac... |
H: How prove this $\frac{a}{(bc-a^2)^2}+\frac{b}{(ca-b^2)^2}+\frac{c}{(ab-c^2)^2}=0$
let $a,b,c\in \mathbb{R}$, if such
$$\dfrac{1}{bc-a^2}+\dfrac{1}{ca-b^2}+\dfrac{1}{ab-c^2}=0$$
show that
$$\dfrac{a}{(bc-a^2)^2}+\dfrac{b}{(ca-b^2)^2}+\dfrac{c}{(ab-c^2)^2}=0$$
Does this problem has nice methods?
My idea:let
$$(ca-b... |
H: Closed subspace of Hilbertspace
Let $X$ be a norm closed subspace of a Hilbert space $\mathcal H$. Is it true that if $x_n \in X$ converges weakly to $x \in \mathcal H$, then also $x \in X$ ?
AI: Yes. Let $Y=X^\perp.$ Then for every $y\in Y$ we have $\langle x,y\rangle=\lim_{n\to\infty} \langle x_n,y\rangle=0.$ Hen... |
H: Work done by gravitational force
In my calculus class we learned about line integrals, and for homework we have exercise to find work done by gravitational force on material dot with mass $m$ which follows path of the elipse $\frac{x^2}{a^2}+\frac{z^2}{c^2}=1$ in second quadrant in positive direction (clockwise).
I... |
H: The meaning of $Z_a$, where $Z$ is a partitioning of $A$ and $a \in A$
I am unsure about the common usage of subscripting a set with something, more precisely something which might be a member of that set or some other set (as opposed to, say, subscripting a set with an integer for reasons of indexing).
The text w... |
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