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H: How many permutations of presentations is possible in this conference?
I came across the following solved question in a book and am having difficulty understanding the answer given for it.
Q: A conference is to have eight presentations over the course of one day, consisting of three long presentations, and five sho... |
H: Complex linear transformation, show maps center of circle to center of circle
If $L$ is a complex linear transformation with $L(z)=Az+B$, $A \neq 0$, $B \in \mathbb{C}$ and $L$ maps the circle $C_1$ onto the circle $C_2$ where
$$C_1=\{z \in \mathbb{C}:|z-z_0|=R_1>0\} \text{ and } C_2=\{w \in \mathbb{C}:|w-w_0|=R... |
H: Finding eigenvalues and eigenfunctions for a BVP
Find the eigenvalues and eigenfunctions for $$y'' + \lambda y = 0, y(0) = 0, y'(\pi/2) = 0$$
According to my book we must check 3 cases: $\lambda < 0$, $\lambda = 0$, $\lambda > 0$.
I started with $\lambda > 0$, found the general solution, applied the boundary condit... |
H: $\surd (I^e)=(\surd I)^e$
I'm trying to solve this question:
I'm having troubles to prove the $\surd (I^e)=(\surd I)^e$ in the part (ii). I'm trying a lot proving the inclusions $\subset$ and $\supset$ without any success, I really need help.
Thanks in advance
AI: Let $r \in \sqrt{I}$. That means $r^n \in I$ for ... |
H: Is this formula $\Sigma_1^{ZF}$?
This is probably a simple question.
Let $\varphi(x)$ be a formula expressing $dom(x)\not\in Ord$. I want to know whether $\varphi$ is a $\Sigma_1^{ZF}$ formula or not, meaning whether there is a $\Sigma_1$ formula $\psi(x)$ so that $ZF$ proves $\forall x (\varphi(x)\leftrightarrow ... |
H: Determining if the relation is an equivalence relation
I'm needing help in determining if the relation
$$R=\left\{(f,g)\mid \exists k,\forall x\in\Bbb Z, \ f(x) = g(k)\right\}$$
where f, g: $\Bbb Z \rightarrow \Bbb Z$ is an equivalence relation.
More specifically, i don't seem to understand the part where it says ... |
H: Area of an elliptic?
I'm looking for an analytic way to calculate the area of an elliptic described by $${x^2 \over a^2} + {y^2 \over b^2}=c^2$$
I saw it before, but now i've forgotten. I remember we set $x=a \cos x$ and $y=a \sin x$ but I don't remember what we did after that!
AI: Certainly, if you are looking for... |
H: Question: Expansion of algebra in matrices
I have a problem that I would like to check:
Expand $(A+B)^3$ where $A$ and $B$ are matrices.
Is this right?
$$
A^3+A^2B+ABA+AB^2+BA^2+BAB+B^2A+B^3
$$
Thanks.
AI: Yes, you've got it.
It is probably easiest to start off with
$$
(A+B)^2=A(A+B)+B(A+B)=A^2+AB+BA+B^2.
$$
Fro... |
H: Is $\{\sin x,\cos x\}$ independent?
Is $\{\sin x,\cos x\}$ linearly independent in $\mathbb{R}^n$?
I thought they were not because I can write $\cos x=\sin (x+\pi/2)$.
My professor on the other hand said it was independent and his proof is as follows:
If $\{\sin x,\cos\}$ is independent in $[0,2\pi]$, then it will... |
H: Show that the product of upper triangular matrices is upper triangular
I have a question. Prove that the product of an [arbitrary] number of upper triangular matrices of [arbitrary] size with [undetermined] upper triangular entries is upper triangular using induction? Should I use transfinite induction? I don't eve... |
H: row echelon vs reduced row echelon form
I apologize if this is a very basic question.
I understand the difference between the two forms, but i was curious when row echelon from is enough. where is row echelon form used?. Why shouldn't I always go for reduced row echelon form?
AI: In addition to lessening the workl... |
H: 2 Questions about the Opposite category
1. If you have a category $\frak{A}$ how are the morphisms of its opposite category $\frak{A}^{op}$ defined?
2. How would you also show that $(\frak{A}^{op})^{op}=\frak{A}$?
Thanks!
AI: Here is a category.
Here is its opposite category.
Simple as that. Even though you mig... |
H: Linear dependence of multivariable functions
It is well known that the Wronskian is a great tool for checking the linear dependence between a set of functions of one variable.
Is there a similar way of checking linear dependance between two functions of two variables (e.g. $P(x,y),Q(x,y)$)?
Thanks.
AI: See http://e... |
H: What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$
While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question:
What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$?
Thanks!
AI:... |
H: Evaluate the limit $\lim_{n\to\infty} \left(\frac{3n}{3n + 1}\right)^n$
I've been trying to evaluate this limit and can't seem to find a way around.
I know how to show that
$\lim_{n\to\infty}\left(1 + \frac{1}{n}\right)^n = e$ but I can't find a way to apply it.
AI: Let's apply just what you know
$$\lim_{n\to+\inft... |
H: The ubiquitous "helper function" $\frac{f(z) - f(a)}{z - a}$
I've been looking at basic complex analysis recently, and have noticed (am imagining?) something which I've never really paid attention to before: The "helper function"
$$g(z) = \frac{f(z) - f(a)}{z - a}$$
At first, I didn't give it a second thought real... |
H: When would one use factorials in probability?
What should a question say so that you know you must use factorials to solve it?
Would the word distinguishable be a keyword?
AI: Here, the factorials you mention don't have anything to do with probability - rather, they have to do with combinatorics. This is because... |
H: How to show this subset is proper?
Let $f:R\to S$ a commutative ring homomorphism. I'm trying to prove that $Q^c=f^{-1}(Q)$ is a primary ideal if $Q$ is a primary ideal.
Curiously, I'm stuck only in the easiest part, to show that $Q^c$ is proper.
Any help is welcome.
thanks in advance.
AI: I assume that you take yo... |
H: Linear Operators satisfying $S^n=0$ but $S^{n-1}\neq 0$
I need help with part (c). I could do part (a) and part (b).
AI: Hint: If $S^{n-1} \neq 0$, then there exists $x$ such that $S^{n-1}x \neq 0$. Consider the vectors $\{x, Sx, S^2x, \ldots, S^{n-1} x \}$, and try to draw the connection with the "shift" property... |
H: What happens when a probability actually occurs?
Sorry if I am mixing up the model with the reality, but when for instance a low probability occurs, what happens with the rest of the probability? Philosophically I find it hard to argue that any other probability than 1 has occured.
For instance, 80 % probability w... |
H: a question on orbit in ergodic theory
For the map $T: [0,1]\to [0, 1]$ defined by $Tx=10x\pmod{1}$,
how to use the decimal expansion to construct a $x$, such that the orbit of $x$, say $\theta_x=\{T^nx: n\geq 0\}$ is dense in $[0, 1]$ and $T^px\to 0$ as $p\to \infty$, where $p$ runs over all prime number ?
AI: Som... |
H: How to calculate a big combination $\binom nr$
How to calculate a big combination such as $$\binom {10^{80}}{10^{10}}$$, using software or by hand, or at least can we get an acceptable approximation.
AI: You can use Stirling’s approximation; even the simplest form,
$$n!\approx\sqrt{2\pi n}\left(\frac{n}e\right)^n\;... |
H: Simplifying $\sin(4x)\cos(4x)$
Simplify $\sin(4x)\cos(4x)$ using double angle or compound trigonometry.
Can someone please show me how its done, Ive tried several times but no where near the answer.
AI: The double angle formula is $ \ 2 \sin\theta\cos\theta = \sin(2\theta) \iff \sin\theta\cos\theta = \frac{1}{2}... |
H: $n=\dim V$. Then $V=\ker(T^n)\oplus\mathrm{range}(T^n)$
I trying to solve the following problem. The question is from a past exam.
Suppose that $V$ is a finite dimensional vector space over a field $K$. Let $T: V\rightarrow V$ be a linear operator. If $n=\dim V$, then $V=\ker(T^n)\oplus\mathrm{range}(T^n)$
Attemp... |
H: Given $a,b,c>0$ and $a^5+b^5+c^5=3$. Is $a+b+c\leq 3$ always true?
Given $a,b,c>0$ and $a^5+b^5+c^5=3$. Is $a+b+c\leq 3$ always true?
I tried many ways to prove it and to find a counterexample, but I couldn't. Please help. Thanks.
AI: There are many ways to prove this.
One way is to use the fact the $2^{nd}$ deri... |
H: How to prove the Milner-Rado Paradox?
For every ordinal $\alpha<\kappa^+$ there are sets $X_n\subset\alpha$ $(n\in\Bbb{N})$ such that $\alpha=\bigcup_n X_n$ and for each $n$ the order-type of $X_n$ is $\le\kappa^n$.
[By induction on $\alpha$, choosing a sequence cofinal in $\alpha$.]
I tried to prove this problem... |
H: If six straight lines and five circles intersect each other, then what are the maximum possible number of distinct points of intersection?
This is how I went about doing it. I know that if there are 'n' number of straight lines they intersect each other in nC2 ("n choose 2") ways. Therefore, the points of intersect... |
H: two dimensional linear differential equation with $1$ eigenvector
I have the following linear differential equation:
\begin{equation}
x' = \begin{pmatrix}3&-4\\1&-1\end{pmatrix}x
\end{equation}
The corresponding characteristic equation is:
\begin{equation}
\lambda^2-2\lambda+1 = (\lambda-1)(\lambda-1) \implies \lam... |
H: "IFF" (if and only if) vs. "TFAE" (the following are equivalent)
If $P$ and $Q$ are statements,
$P \iff Q$
and
The following are equivalent:
$(\text{i}) \ P$
$(\text{ii}) \ Q$
Is there a difference between the two? I ask because formulations of certain theorems (such as Heine-Borel) use the latter, while others... |
H: In a square grid ($6 \times 6$) that comprises 25 small unit squares each of side 1 cm, how many rectangles (not squares) are there in the grid?
This comes under Combinatorics under intersection of parallel lines. I calculated the number of rectangles to be $\binom62 \times \binom62 = 225$. But how does one subtrac... |
H: using the conditional to abbreviate formulas
i hope you're all doing well. I was reading a paper recently that started out with a language $L$ with a set $PV$ of propositional variables, Boolean connectives $\neg, \vee$, and some modalities called $\Box$ and O.
It then says "$\wedge, \rightarrow,$ and $\leftrighta... |
H: "range of function" vs "target of function"?
Page 14 of Fundamentals of Computer Graphics states that if we have a function like this:
...the set that comes before the arrow is called the domain of the function, and the set on the right-hand side is called the target.
...The point f(a) is called the image of a, an... |
H: Can we avoid talking about proper classes by talking about models?
Intuitively, we can define the ordinal numbers $\mathsf{On}$ as the closure of $\{0\}$ with respect to successorship $x \mapsto x \cup \{x\}$ and (set-sized) unions. Arguably the most natural way to implement this idea is to write $\mathsf{On} = \bi... |
H: linearly independence of $e^{a_1x},... e^{a_nx}$
$a_1,\ldots,a_n$ are real different numbers. Prove that the functions $e^{a_1x},...,e^{a_nx}$ are linearly independent in $Fun(R,R)$.
My way to try to prove it:
I assumed: $b_1e^{a_1x} + \cdots + b_n e^{a_nx} = 0$, and we want to show that $b_1 = \cdots = b_n = 0$... |
H: Series expansion of $\ln(\sec x + \tan x)$?
I'm looking for series expansion of $\ln(\sec x + \tan x)$ ?
I tried to differentiate and then find an expansion then integrating but found nothing.
AI: $$\frac{d}{dx}\ln(\sec x+\tan x)=\frac{\sec x\tan x+\sec^2x}{\sec x+\tan x}=\sec x=\sum_{n\ge 0}\frac{(-1)^nE_{2n}}{(2... |
H: NBA round robin probability
Let’s say there is a team who you expect to win 75% of its games in a given 82-game NBA regular season (and the probability of winning each game = 75%). What is the probability that the team will never lose consecutive games at any point during the 82-game season.
AI: You can make a Mar... |
H: Simplify the trigonometric equation using double angle and compound angles.
Simplify the trigonometric equation $\dfrac{1-t^2}{1+t^2}$ where $t=\tan\dfrac{x}{2}$.
using double angle and compound angles.
I've worked up to the point where I converted the equation into $\sec$ form after substituting the $\tan\dfrac{x... |
H: Addition and Subtraction of Convergent Series
I have a simple question and I can't find the answer of it: If a series $a_{n}$ can be proven to converge and a series $b_{n}$ too, will the series $a_{n}+b_{n}$ converge? Same goes with the subtraction of them.
Thank you
AI: $\{a_n\},\{b_n\}$ converge implies $\exists ... |
H: Prove triangle inequality
I want to prove that
$d(x,y) = 1- \sum_i {\min(x_i, y_i)}$ where $\sum_i {x_i} = \sum_i {y_i} =1$
and $\forall i: x_i, y_i \geq 0$ satisfies the triangle inequality.
The domain of $d$ therefore is $\mathcal{X} \times \mathcal{X} \to \mathcal{X}$ with $\mathcal{X} = \{x | x \in \mathbb{R}^d... |
H: why is $E[E[Y|X]] = E[Y]$
I have a derivation from my book, I have a problem with the very first line:
$$
\begin{align}
E[E(Y|X)] &= \int_{-\infty}^\infty E(Y|x)f_1(x)dx <- \text{why dx}\\
&= \int_{-\infty}^\infty\int_{-\infty}^\infty yf(y|x)f_1(x)dydx\\
&=\int_{-\infty}^\infty y \int_{-\infty}^\infty f(x,y)dxdy\\... |
H: What does $a\equiv b\pmod n$ mean?
What does the $\equiv$ and $b\pmod n$ mean?
for example, what does the following equation mean?
$5x \equiv 7\pmod {24}$? Tomorrow I have a final exam so I really have to know what is it.
AI: It’s a bit late to be learning a basic definition, but here it is: $a\equiv b\pmod n$ mean... |
H: How to show there exists an infinite sequence satisfying $a_0 = x$ and $(a_n,a_{n+1}) \in R$.
Intuitively, we can use the fact that
(i) for all $a \in \mathbb{R}$, there exists $b \in \mathbb{R}$ such that $a < b$
in order to conclude that
(ii) there exists an infinite sequence $a : \mathbb{N} \rightarrow \mathbb{... |
H: Integration of a matrix over a hypersphere
Can anybody please help me on this one please?
$\int_{B({\bf x}_0;R)} \frac{1}{2} ({\bf x} - {\bf x_0})({\bf x} - {\bf x_0})^{T} d{\bf x}$
Here, $B({\bf x}_0;R)$ is a hypersphere(ball?) with radius $R$ with center $\bf x_0$.
AI: Suppose that $\mathbf{x}$ is a column vector... |
H: Inverse formula for sg(specific gravity) to plato
I am making a class (in software) in wich several algorithms are used to convert values.
Now almost all functions need to go two way, so sg_to_plato(sg) and plato_to_sg(plato)
These algorithms I gather from online sources and literature. Is is all working within rea... |
H: Variance of summation of Bernoulli variables
Let $X_1,\ldots,X_n$ be independent Bernoulli variables, with probability of success $p_i$ and let $Y_n =\frac1n\sum\limits^n_{i=1} (X_i - p_i )$
a) find the mean and variance of $Y_n$
b) show that for every $a>0, \lim\limits_{n\to\infty} P(Y_n<a)=1$
Now for the mean, ... |
H: find $0 < l < 35$ such that $l^5 \equiv 3\pmod {35} $
I have to find some $0 < l < 35$, such that $l^5 \equiv 3\pmod {35} $.
I tried to use suggestions from my previous question,
So I tried:
$l^5 \equiv 3\pmod {35} $ => $35 | l^5 - 3$, I find some matching $l$'s
But I didn't find any matching number. I can't u... |
H: Calculating the derivation of $F(t):=\int_0^{\infty}\frac{e^{-x}-e^{-tx}}{x}\, dx$
As the title says, i have to calculate $F'(t)$ for
$$
F(t):=\int_0^{\infty}\frac{e^{-x}-e^{-tx}}{x}\, dx.
$$
What I already have is
$$
F'(t)=\lim\limits_{h\to 0}\frac{F(t+h)-F(t)}{h}=\lim\limits_{h\to 0}\int_0^{\infty}\frac{-... |
H: characteristic polynomial and eigenvalues of $T(A)={ A }^{ t }$
Let $V=M_2(\mathbb R)$ and
$T(A)={ A }^{ t }$.
I was asked to find the characteristic polynomial of $T$ and it's eigenvalues, and finally to say if $T$'s diagonalizable.
Is there a way to solve this without actually finding a matrix representation... |
H: Find Limit of the given function
Find the limit
$$\lim_{x\rightarrow 1}\ln(1-x)\cot\left({{\pi x}\over2}\right)$$
AI: Putting $1-x=y$
$$\lim_{x\rightarrow 1^{-}} \ln(1-x)cot({{\pi x}\over2})$$
$$=\lim_{y\to0^+}\ln y\tan \frac {\pi y}2$$
$$=\lim_{y\to0^+}\frac{\ln y}{\cot \frac {\pi y}2}\text{ which is of the form }... |
H: Constant growth rate?
Say the population of a city is increasing at a constant rate of 11.5% per year. If the population is currently 2000, estimate how long it will take for the population to reach 3000.
Using the formula given, so far I've figured out how many years it will take (see working below) but how can I ... |
H: Every subspace of $\mathbb{R}^n$ is a solution space of a homogeneous system of linear equation.
All solution of $AX = 0$ where $A$ is a $n \times n$ matrix and $X$ is a column vector form a subspace of $\mathbb{R}^n$. All the subspaces of $\mathbb{R}^n$ are of this type. How to prove this result? Linear Algebra: s... |
H: Solving a linear system with complex eigenvalues
I have the system:
\begin{equation}
x' = \begin{pmatrix}5&10\\-1&-1\end{pmatrix}x
\end{equation}
The corresponding characteristic equation is:
\begin{equation}
\lambda^2-4\lambda+5 \\
\implies \lambda_1 = 2+i \land \lambda_2 = 2-i
\end{equation}
I am having trouble s... |
H: Need a source for the following result: From $v \in H^1$ and $f$ Lipschitz it follows that $f(v) \in H^1$.
I am looking for a source of the following (or a similar) result:
If $v \in H^1(\Omega,\mathbb{C})$ on a bounded domain $\Omega$ and $f: R(v) \to \mathbb{C}$ is Lipschitz continuous, then $f(v) \in H^1(\Omega... |
H: Quadratic Polynomial Question - Solving for a coefficient using the discriminant
This question has been troubling me:
A parabola whose equation is of the form $y = Bx^2$ (where B is a constant) has the line $20x - y + 20 = 0$ as a tangent. Find $B$.
The explanation says, basically; "The line is a tangent if only on... |
H: Cannot understand while reading simplicial=singular homology
I was reading http://www.math.toronto.edu/mgualt/MAT1300/Week%2010-12%20Term%202.pdf
, and I can't understand the last paragraph of pg 29, and the first paragraph of pg 30.
It says that "To compute the singular group $ H_n (X ^ k , X^{k-1} ) $, consider ... |
H: Inverse Polynomial in a ring R
I just started working on my Bachelor-Thesis in IT-Security and therefore try to understand the NTRUencryption algorithm. It operates on polynomials in a Ring.
My problem is that I don't understand how someone computes the inverse of a polynomial in such a ring. I just tried to follo... |
H: If $ A^2=0$ , prove that $A$ doesn't neccesarily have a row of zeros
Question
$A^2 \in M_{n \times n} (F), A^2=0, n\ge 3$.
Prove that it's not true that A necessarily has a row of zeros.
Thoughts
We thought that the matrix must be nilpotent, but therefore it's main diagonal is 0s (and must mean that there's a line ... |
H: Does $z^i=i^z$ have any solutions, beside $z=i$?
Does this equation have any solutions:
$$\large{z^i=i^z}$$
Putting polar form of $z$ is better for LHS, But rectangular form is suitable for RHS !
What to do? Thanks!
AI: $$
\frac1z\log\left(\frac1z\right)=\frac1i\log\left(\frac1i\right)=-\frac\pi2
$$
Thus, there... |
H: Combinatorics problem arising from physics
I'm currently studying general mechanics where the following problem came up:
Assume we have the space $\Gamma = \mathbb{R}^6$ which we are dividing in small cells $C_i$ . Let $f(\vec{x}, \vec{v})$ be a probability distribution on our $\Gamma$. The amount of elements in a ... |
H: Extending our language with a new function symbol
Given an arbitray first-order theory (not necessarily a set theory) and definable predicates $P(*)$ and $Q(*,*)$ in the language of that theory, if we adjoin a new function symbol $f$ together with the axiom
$$\forall x(P(x) \Rightarrow \exists y(Q(x,y))) \Rightarro... |
H: Prove $(1-\cos x)/\sin x = \tan x/2$
Using double angle and compound angles formulae prove,
$$
\frac{1-\cos x}{\sin x} = \tan\frac{x}{2}
$$
Can someone please help me figure this question, I have no idea how to approach it?
AI: $$\dfrac{1-\cos x}{\sin x}=\dfrac{1-(1-2\sin^2\frac x2)}{2\sin\frac x 2\cos\frac x2}... |
H: Inequality involving tangent provided $\tan\theta\geq 1$
If $\tan\theta\geq1$, then
$$\sin\theta-\cos\theta\leq\mu(\cos\theta+\sin\theta)\implies\tan\theta\leq\dfrac{1+\mu}{1-\mu}.$$
Why? I get as far as the obvious $$\tan\theta\leq1+\mu(\cos\theta+\sin\theta)$$
AI: Assume for the moment that we are in quadrant I (... |
H: How to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists?
I have 2 groups $U_5$ and $U_{12}$ , ..
$U_5 = \{1,2,3,4\}, U_{12} = \{1,5,7,11\}$.
I have to determine whether an isomorphism $\varphi: {U_{12}} \to U_5$ exists.
I started with the "$yes$" case: there is an isomorphism.
So I searched an ... |
H: $a, b, c, d$ are positive integers， $a-c|a b+c d$, and then $a-c|a d+b c$
$a, b, c, d$ are positive integers， $a-c|a b+c d$, and then $a-c|a d+b c$
proof: really easy when use $a b+c d-(a d+b c)$
however my first thought is, $a-c| a b+c d+k(a-c)$, and set some $k$ to prove, failed.
question1 : is this method cou... |
H: Infimum, supremum of a set problems
I am solving some infimum/supremum problems, and my book has different answers for some of the problems.
Let $A = \{ x \in \Bbb N | x^2 < 5\}$ find sup A and inf A, their answer is sup A = $\sqrt5$, inf A = $-\sqrt5$.
I think this is wrong, since A is a finite set, its clear sup ... |
H: Number of Permutations without the the "diagonal terms"
If I have a set of n numbers (we can say n is 5, to create a concrete example), then there are n! (5!) different ways of arranging these numbers.
How many of these don't use the "diagonal terms" - i.e, the first term isn't a 1, the second term isn't a 2, and ... |
H: calculate angles between O'clock hands
suppose that now it is $1:50$, we need to calculate angle between these hands first because we have $12$ hour system per day and night and they are equal, each hour corresponds $360/12=30$, from $10$ to $1$ we have $30+30+30=90$, but i want to know what should be de... |
H: Mysterious Matrix Norm
Given a matrix $M$,
does anyone know the name and the definition of the following norm?
$$
\|M\|_*
$$
Thanks in advance,
Francesco.
AI: That's the Schatten norm.
It's defined like:
$ \lVert A \rVert_* := \text{tr}(\sqrt{AA^T}) $ with tr is the trace of the matrix and $A^T$ is the transpose.
I... |
H: $u\in W_0^{1,p}(\Omega)$ but it's extension by zero does not belong to $W^{1,p}_0(\mathbb{R}^N)$
My problem is the following: I want to find a bounded domain $\Omega\subset\mathbb{R}^N$ such that if $u\in W_0^{1,p}(\Omega)$, $p\in (1,\infty$), then the extension by zero of $u$ to $\mathbb{R}^N$ is not in $W_0^{1,p}... |
H: Definability of Kolmogorov Complexity?
This paper claims to have a proof of Godel's Second Incompleteness Theorem using Kolmogorov Complexity: http://www.ams.org/notices/201011/rtx101101454p.pdf
As far as I can tell, it seems to assume that Kolmogorov complexity (over some language or Turing Machine) is definable i... |
H: Solve equation using combinations of integers from 0 to 9 in Maple
Display answers for $x$ using all combinations of $0$ to $9$ integers for $a$ and $b$
$\dfrac{1}{x^{2}}=\dfrac{a^{2}}{s^2}+\dfrac{b^{2}}{t^{2}}$
The values for $s$ and $t$ are known values and must be entered by the user
AI: What I could do for you ... |
H: Does every normal space have countable basis?
I know that every regular space with a countable basis is normal. But my question is if the converse is true?
Normal spaces are obviously regular but does every normal space have a countable basis?
Can someone help me please?
AI: No, the discrete topology on an uncount... |
H: For what integers $n$ does $\phi(2n) = \phi(n)$?
For what integers $n$ does $\phi(2n) = \phi(n)$?
Could anyone help me start this problem off? I'm new to elementary number theory and such, and I can't really get a grasp of the totient function.
I know that $$\phi(n) = n\left(1-\frac1{p_1}\right)\left(1-\frac1{p_2}\... |
H: Is $\Bbb{6Z}$ a free group?
I'm trying to understand the concept of free groups , and from what I've learned
so far , a group $G$ is called a free group , if there is a subset $S ⊂ G$ such that
any element of G can be written uniquely as a product of elements of S , and their
inverses .
So , is $\Bbb{6Z}$ a free g... |
H: How to solve a ratio question
Studying for the GRE. In the GRE guide, it says that
If the ratio is $2x:5y$, and this equals the ratio $3:4$, what is the ratio of $x:y$?
I tried cross multiplying but I don't get the answer. It says the answer is $15:8$. I get $8:15$. Which step am I missing?
AI: We are given: $$\... |
H: How to solve percentage of new
I am good with all percentage questions except finding the original price of something. If I had a coat that cost $120 after an 8% increase, how do I formulate the original price before the increase?
AI: let the original price be $x$$
so, after 8% increase, the price would be,
$$x+\df... |
H: invariant sub space
So I preparing myself to a test in linear algebra and I scanned the last years test and I reached a question which I do not understand why is it like that.
True or false:
$ \forall T\colon V \rightarrow V$
$\exists 0_{V} \neq W \neq V $
so that $T(W) \in W$
So the answer is false and as example... |
H: $\displaystyle f: \Bbb C \to \Bbb C$ is analytic function
let $\displaystyle f: \Bbb C \to \Bbb C$ be an analytic function. For $z = x + iy$, let $u, v: \Bbb R^2 \to \Bbb R$ be such that
$u(x,y) = \Re f(z)$ and $v(x,y) = \Im f(z)$. Which of the following are correct?
1.∂2u∂x2+∂2u∂y2 = 0
2.∂2v∂x2+∂2v∂y2 = 0
3.∂2u∂x∂... |
H: Triangle inequality, is this implication correct?
$a, b,$ and $c$ are not necessarily sides of a triangle, but they are positive numbers.
The question was:
Given that $a+b>c$ , does this imply that $\sqrt a +\sqrt b > \sqrt c$ , and that $a^2+b^2>c^2$ ?
The one with the square roots is easily proven by assuming it... |
H: Operations on negative integers
I was trying to teach my younger sister some math, and it drifted on to integers, and operations on negative integers. So questions like:
a) $-3+2 = ?$
b) $2- (-3)= ?$
c)$-3 -2 = ?$
had to be answered. So, I did not want to say that because minus of minus is plus, so the answer to b)... |
H: How to find the corresponding eigenfunction after determining the eigenvalues?
I was reading this page (http://www.jirka.org/diffyqs/htmlver/diffyqsse25.html) example 4.1.4, which says:
Again $A$ cannot be zero if $\lambda$ is to be an eigenvalue, and $sin(\sqrt {\lambda} \pi)$ is only zero if $\sqrt {\lambda}=k$ ... |
H: solving equations by the method of substitution
$\dfrac{a}{x}+\dfrac{b}{y}=\dfrac{a}{2}+\dfrac{b}{3},$
$x+1=y$
We have to solve for $x$ and $y$.I have tried to solve for them by finding value of $x$ or $y$ from the second equation and place them in the second.It is obvious that the answers would be $2$ and $3,$but ... |
H: Algebraic Transformation query...
I'm boning up on Algebra, and I'm looking into Algebraic Transformation.
I understand the basic concept - but I'm confused by two self assessment questions.
The two questions, from what I can see, are almost similar but have quite different ways of arriving at the end result.
Now, ... |
H: Generalizing a theorem about indentations around simple poles
Assume the function $f(z)$ has a simple pole at $z_{0}$.
There is a theorem that states that if $C_{r}$ is an arc of the circle $|z-z_{0}| = r$ of angle $\alpha$, then $$\lim_{r \to 0} \int_{C_{r}} f(z) \, dz = i \alpha \, \text{Res}[f,z_{0}].$$
But wh... |
H: Evaluate the integral: $\lim \limits_{n\to\infty}\int_0^1\frac{nx}{nx^3+1}$
Evaluate the integral:
$$\lim \limits_{n\to\infty}\int_0^1\frac{nx}{nx^3+1}dx$$
I'm pretty much stuck on how to solve this one:
$$\int_0^1\frac{nx}{nx^3+1}dx$$
or even getting the improper integral.
What can i do?
AI: If you think that your... |
H: Proof of $ \lim_{y\to\infty} (\tan\frac{x}{y})\cdot y = x$?
At lunch a coworker was talking about how to calculate, say, the 100th digit of pi using a square around the circle, then a pentagon, etc, basically you end up taking the limit of the circumference as the number of sides n goes to infinity.
So I tried work... |
H: Simple question about full time derivative
Let's have full time derivative equation
$$
\frac{F(\mathbf r (t), \mathbf p (t), t)}{dt} = \frac{\partial \mathbf r }{\partial t}\frac{\partial F}{\partial \mathbf r} + \frac{\partial \mathbf p }{\partial t}\frac{\partial F}{\partial \mathbf p } + \frac{\partial F}{\parti... |
H: Does this function have a minimum?
$$f(x, y, z) = xy - xz$$
My textbook asks to find the minimums of various different functions, and this is one of them. But I don't think this has a minimum. If $X=(x, y, z)$, then $f(tX)=t^2f(X)$, so if $f(X)$ is ever negative, the function will tend to $-\infty$ along the line g... |
H: A statement equivalent to $\exists ! x P(x)$
I'm trying to write a statement in logic symbols that says there is a unique $x$ such that $P(x)$ is true. I've heard of writing this down as $\exists !x P(x) $. But I think I'm not allowed to use the symbol $\exists !$. Now I'm trying to write an equivalent statement wi... |
H: Strange graph theory problem
Let $G$ be a graph where every vertex has degree 1 or 3. Let $X$ be the set of all vertices of degree 1. Suppose there exists a set of edges $Y$ such that by removing these edges from $G$, each component of the remaining graph is a tree which contains exactly one vertex in $X$. Determi... |
H: what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by 47?
Can any one please tell the approach or solve the question
what is the remainder when $1!+2!+3!+4!+\cdots+45!$ is divided by $47$?
I can solve remainder of $45!$ divided by $47$ using Wilson's theorem but I don't know what must be the approach... |
H: Laurent-Development of zero function
The following "idendity" contradicts the uniqueness of the Laurent Development:
$$
0=\frac{1}{z-1}+\frac{1}{1-z}=\frac{1}{z}\frac{1}{1-1/z}+\frac{1}{1-z}=\sum_{n=1}^{\infty}\frac{1}{z^n}+\sum_{n=0}^{\infty}z^n=\sum_{n=-\infty}^{\infty}z^n
$$
1) Where is the mistake? 2)... |
H: Finite ways to write $1 =\sum_{i=1}^{h}\frac{1}{n_i}$
Let $h\geqslant 1$ an integer.
Can we show (simply), without using group actions, that
there exists a finite number of decomposition of the form
$\displaystyle 1 =\sum_{i=1}^{h}\frac{1}{n_i}$, with $n_i$ positive
integers.
P.S.: With group actions, I got... |
H: Find the Wronskian of the Functions
Find the Wronskian of the functions $f(t)=6e^t\sin{t}$ and $g(t)=e^t\cos(t)$. Simplify your answer.
please list out all steps as simple as possible
thank you
AI: Recall, that given $$f(t)=6e^t\sin{t}\quad g(t)=e^t\cos(t)$$
The Wronskian of $f(t), g(t) = W(f,g)(t)$
$$W(f, g)(t) =... |
H: Induction of inequality involving AP
Prove by induction that
$$(a_{1}+a_{2}+\cdots+a_{n})\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right)\geq n^{2}$$
where $n$ is a positive integer and $a_1, a_2,\dots, a_n$ are real positive numbers
Hence, show that
$$\csc^{2}\theta +\sec^{2}\theta +\cot^{2}\t... |
H: Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?
I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges .
I tried it though a small code piece in python so that I can have a lot of data points to an... |
H: Venn diagram counting problem?
Suppose that
Set A has 5 elements
Set B has 6 elements
Set C has 7 elements
$\Omega$ has 10 elements
Determine the maximum and minimum number of elements the following set can have:
$ A^{c} \cup (B \cap C)$
I can see that there can be a maximum of 6 elements in $B \cap C$ but from ... |
H: Number of attempts required to increment a counter based on probability of success
Suppose I have a counting variable C that gets incremented every time a particular condition is met. Let's call this a "successful attempt."
Suppose also that I have a non-linear function F() expressing the probability of an attempt... |
H: Part of a solution to a mathematical induction problem I don't understand
There's a part in the solution that I can't understand, I think it's just something basic that I'm missing. In the solution it says:
$$T(k) \leq 2(c(k/2)^2 \log(k/2)) + k^2$$
Then it became
$$T(k) \leq ( ck^2 \log(k/2) ) / 2 + k^2$$
P.S: I f... |
H: Understanding associators as natural transformations
Reading Baez and Stay's "Rosetta Stone," and trying to understand the definition of monoidal category on page 12, I read that a monoidal category requires a natural isomorphism called the associator, assigning to each triple of objects $X, Y, Z \in C$ an isomorph... |
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