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H: Domain and range of a multiple non-connected lines from a function?
How do you find the domain and range of a function that has multiple non-connected lines?
Such as, $ f(x)=\sqrt{x^2-1}$. Its graph looks like this:
I'm wanting how you would write this with a set eg: $(-\infty, \infty)$.
P.S. help me out with th... |
H: Let $A$ and $B$ be $n \times n$ complex matrices. Then
Let $A$ and $B$ be $n \times n$ complex matrices.Then
a) $\lim_{k→∞} A^k =0 ⇔$ all eigenvalues of A have absolute value less than $1$
b) $e^A\cdot e^B=e^{(A+B)}$
c) If $A$ and $B$ are nilpotent, then so is $A + B$.
d) $I + A + A^2 + A^3 + \dots$ converges ⇒ $A$... |
H: Sequence $f_n$ of continuous functions on $[0,1]$, such that $0 \le f_n \le 1$ and $\lim\limits_{n\to\infty}\int_{0}^{1} f_n(x)\,\mathrm dx=0$
Let $f_n$ be a sequence of continuous functions on $[0,1]$, such that $0 \le f_n \le 1$ and $$\lim_{n\to\infty}\int_{0}^{1} f_n(x)\,\mathrm dx=0.$$
a) The sequence ${f_n (x)... |
H: Mean-value for three functions
If $f,g,h$ are continuous functions on $[a,b]$ which are differentiable on $(a,b)$, then prove that there exists $c\in(a,b)$ such that $$f(a)[g(b)h'(c)-h(b)g'(c)]+h(a)[f(b)g'(c)-g(b)f'(c)]=g(a)[f(b)h'(c)-h(b)f'(c)]$$
This looks very much like (generalized) mean-value theorem, so I'm... |
H: If $G=(V,E)$ is a planar graph, all vertex degrees are $3$, all faces are of five/six edges, how many five-edged faces are there?
Given a graph $G = (V,E)$, a planar graph where every vertex has degree $3$ and all faces are five-edged or six-edged. How many five-edged faces are there?
It was a question in one of ... |
H: multiple choice question on topology of matrices
Which of the following are true?
a)The set of symmetric positive definite matrices are connected.
b)The closed unit ball centred at $0$ and of radius $1$ of $l_1$ is compact
c)The set of invertible matrices on R forms a set of measure $0$.
d)The set of symmetric matr... |
H: number of solution of polynomial with parameter
Given the polynomial $P(x) = x^5 - 20x^2 + a$ and $a\in\mathbb{R}$, for which values of $a$ this polynomial have only one solution?
from the fact that $\lim_{x\to+\infty}{P(x)} = +\infty$ and $\lim_{x\to-\infty}{P(x)} = -\infty$ there is $a_0$ such that $P(a_0)<0$ and... |
H: Average number of $U[0,1]$ random numbers to reach $x$
Let $P(x)$ be a random process where you keep selecting random numbers, uniformly distributed between 0 and 1, until the sum reaches $x$.
From memory, the expected value of the number of terms needed by $P(x)$ is $e^x Q(x)$, where $Q(x)$ is a piecewise polynomi... |
H: At which points the given function $f$ is not differentiable?
Let $f(x) = \cos (|x-5|) + \sin (|x-3|) + |x + 10 | ^ 3 - (|x| + 4 )^2$;
at which points the function $f$ is not differentiable?
Since $|x-a|$ is not differentiable at any real number $x= a$. so, the function $f$ is not differentiable at $x=5,3,-10$ and ... |
H: Differentiability of multivariable functions, and its relation to the chain rule.
I'm struggling with the conditions for the applicability of the chain rule.
$${df(C(t))\over dt} = \mathrm{grad}f(C(t))\cdot C'(t)$$
Where $C$ is in $\Bbb R^n$ and $f$ is differentiable in the following sense:
$f:\Bbb R^n\rightarrow \... |
H: Pell-like equations and continued fractions
Why does the continued fraction method work? Could be applied in order to solve, for example, $x^{17}-19y^{17}=1$ ?
AI: I tried that for cubes. It was not that successful.
With $x^2 - 19 y^2 = 1,$ you are guaranteed an infinite set of solutions. Also, using Lagrange's met... |
H: Differential equation on $\Bbb R$
We have a differential equation on $\Bbb R$ of the form
$$\frac {d^2}{dx^2}u = \chi_{[0,1]},$$
where $\chi_{[0,1]}$ is the characteristic function of the interval $[0, 1] ⊂ \Bbb R$. I want to find a generalized solution for this differential equation.
I also want to know that wil... |
H: Two-form wedge product
Consider a two-form $\alpha$, then $d\alpha \wedge d\alpha$ is not necessarily zero.
Is this statement true?
Consider $\beta = \alpha \wedge d \alpha$. Then $d\beta = d(\alpha \wedge d \alpha) = d \alpha \wedge d \alpha + \alpha \wedge d^2 \alpha = d\alpha \wedge d \alpha.$
Is this state... |
H: Why aren't the strong LLNs and CLT contradicting each other?
Given $n$ i.i.d. random variables $\{X_1, X_2, \dots , X_n\}$, each with mean $M$ and variance $V$, both strong and week LLNs seem to say that the average of the $n$ random variables, $S_n = \frac{X_1 + X_2 + \dots + X_n }{n}$, approaches $M$, as $n \to \... |
H: How to derive compositions of trigonometric and inverse trigonometric functions?
To prove:
$$\begin{align}
\sin({\arccos{x}})&=\sqrt{1-x^2}\\
\cos{\arcsin{x}}&=\sqrt{1-x^2}\\
\sin{\arctan{x}}&=\frac{x}{\sqrt{1+x^2}}\\
\cos{\arctan{x}}&=\frac{1}{\sqrt{1+x^2}}\\
\tan{\arcsin{x}}&=\frac{x}{\sqrt{1-x^2}}\\
\tan{\arccos... |
H: Is it possible to solve this PDE
It would be pretty sweet if I could solve this for $A$. Is it possible?
$$\frac{dA}{dx}+\frac{dA}{d\tau}=wx\tau$$
where $w$ is a constant and $x$ is a function of $\tau$.
It might help that it is also known that:
$$w\tau=\frac{d^2A}{dx^2}$$
So the equation to be solved could also b... |
H: calculate $ \pi_{1}\left(T_{2}\#T_{2}\right) $
How can I calculate $\pi_{1}\left(T_{2}\#T_{2}\right)$ ?
I know that this is a covering space of a tori which means that this group is a subgroup of $\mathbb{Z}^{2}$.
Thanks!
AI: Write the connected sum as the union of the tori with disk removed; the intersection of th... |
H: Density function as derivative (Self-study)
I'm trying to do the Society of Actuaries' example problems. I am having trouble with no. 62, which says:
A random variable $X$ has CDF
$$ F(x) =
\begin{cases} 0 & \text{for $x < 1$} \\
\frac{x^2 - 2x + 2}{2} & \text{for $1 \leq x < 2$} \\
... |
H: Finding the range of $f(x) = 1/((x-1)(x-2))$
I want to find the range of the following function
$$f(x) = \frac{1}{(x-1)(x-2)} $$
Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please help me in solving this problem .
AI: HINT:
Let $$y=\frac1{(x-1)(x-2)}$$
... |
H: Linearly independent rows and matrices
Let M be an $m \times n$ matrix whose rows are linearly independent. Suppose that $k$ columns $c_{i_1}, ... , c_{i_k}$of M span the column space of M. Let C be the matrix obtained from M by deleting all columns except $c_{i_1}, ... , c_{i_k}$. Show that the rows of C are also... |
H: Prove that $\dfrac{(n^2)!}{(n!)^n}$ is an integer for every $n \in \mathbb{N}$
Prove that $\dfrac{(n^2)!}{(n!)^n}$ is an integer for every $n \in \mathbb{N}$
I know that there are tools in Number theory to proves the required but I want to use the tool that says that if you can prove that an expression solves a co... |
H: For a $k$ form $\eta$ and a $l$ form $\omega$, what is $d(\eta\wedge\omega)?$
For a $k$ form $\eta$ and a $l$ form $\omega$, what is $d(\eta\wedge\omega)?$ Thank you very much for your help and guidance!
AI: Use the Leibniz rule, i.e.
$$d(\eta\wedge\omega)=d\eta\wedge\omega+(-1)^{k}\eta\wedge d\omega.$$
It is one o... |
H: Airy function and modified Bessel function
I have got a question concerning the Airy functions in relation to the Bessel function.
From Wiki, it is possible to see how
$$ Ai(x)=\frac{1}{\pi}\sqrt{\frac{x}{3}}K_{1/3}\left(\frac{2}{3}x^{\frac{3}{2}}\right) $$
The question is: how can the Airy function retrieve a 0.35... |
H: Set Relations Quick Question
Can someone please explain how this answer was reached? I know that relation of A is just A * A but wouldn't that just be $4^{2}? $
Let A = {1, 2, 3, 4}. How many relations are there on a set A?
Solution: $2^{4^2} = 2^{16} = 131,072$
AI: By definition, a relation on $A$ is a subset of ... |
H: Inequality with square root and squaring each side of the inequality
My book says if I take $\sqrt{x^2 + y^2} \lt 1,\;$ and it says if I "square each side of the inequality" the result will give the inequality $\;x^2 + y^2\lt 1,\;$ but I don't understand the concept.
If you find the square root of $5^2$ isn't that ... |
H: Simplifing formulas using tensor notation
Im trying to symplify formulas like:
$$\operatorname{div}(\operatorname{rot}\vec{F}),\qquad \operatorname{rot}(\operatorname{rot}\vec{F}) $$
or something more strange like:
$$\operatorname{rot}(\vec{r}\operatorname{div}(r^4\operatorname{grad}(r^4)))$$
To do this I want to ... |
H: Can saturation drop when passing to an elementary extension.
Suppose I have a model $\mathcal{U}$ with some theory $T$ such that $\mathcal{U}$ is $\kappa$-saturated for some infinite cardinal $\kappa$. If $\mathcal{V} \succeq \mathcal{U}$ is an elementary extension of $\mathcal{U}$, is it true that $\mathcal{V}$ is... |
H: When $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$?
maybe this is a stupid question . Anyway, when $(\mathbb{Q}(\zeta_n) : \mathbb{Q}) = 2$, i.e., $\phi(n) = 2$ (where $\phi$ is the Euler's totient function)?
Thanks in advance.
AI: Hint: if $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, then
$$\varphi(n)=(p_1-1)p_1^{a_1-1}\c... |
H: If the difference of cubes of two consecutive integers is a square, then the square can be written as the sum of squares of two different integers.
How can i prove the statement that if the difference of cubes of two consecutive integers is an integral power of 2, then the integer with power 2 can be written as the... |
H: Proving a minimum vertex cover for a family of 3-uniform hypergraphs
Let $H_n$ be a $3$-uniform hypergraph. For every $\{a, b, c\} \subseteq [n]$ we have an edge $\{\{a, b\}, \{a, c\}, \{b, c\}\} \in H_n$.
What is the minimum vertex cover of $H_n$?
The following construction is a vertex cover of $H_n$ (call it $C... |
H: Why $x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$?
Why $f(x) = x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$ (where $\zeta = \zeta_7$ is a primitive root of the unit) over $\mathbb{Q}$?
Of course it's irreducible by the Eisenstein criterion, however it appa... |
H: How to solve $u'' + k u + \epsilon u^3 = 0$?
I am looking at the project of my ODE class, there is one problem saying we have to solve $u'' + k u + \epsilon u^3 = 0$. The problem gives us some values of $k$, $\epsilon$ and says you should experiment with different initial values with Euler's method. I have solved t... |
H: How to prove that: $|f(x+h)-f(x)| \leq 2|f(x)|$ for continuous $f$ and sufficiently small $h$.
If I have a continuous function $f$, how can I prove that $$|f(x+h)-f(x)| \leq 2|f(x)| $$ for all sufficiently small $h$?
for example if I suppose that $f$ is not continous. I know only that $f\in L^{1}(\mathbb{R})$ how ... |
H: Confused about a limit proof and Big O.
I gave an incorrect proof here : How can evaluate $\lim_{x\to0}\frac{\sin(x^2+\frac{1}{x})-\sin\frac{1}{x}}{x}$
I am confused as when considering the mistakes in my proof it seems the limit cannot be $0$. The method must thus be completely wrong even more wrong than the comme... |
H: How do we derive new inference rules?
I've been toying with a system of inference rules for propositional logic. I can use the system to prove theorems; but my question is, can I use the system to obtain new inference rules?
Here are the details. We begin with 5 inference rules, and 0 axioms.
(1) Proof by contradic... |
H: Need help identifying notation, groups, $(G:1)$
Let $G$ be a finite group and let $p$ be a prime integer.
(a) Show that if $p$ divides $(G:1)$, then $G$ contains an element of order $p$. (You may assume this holds if $G$ is abelian)
My only question is what does the notation $(G:1)$ refer to? I am preparing for... |
H: Maximum of a trigonometric function without derivatives
I know that I can find the maximum of this function by using derivatives but is there an other way of finding the maximum that does not involve derivatives? Maybe use a well-known inequality or identity?
$f(x)=\sin(2x)+2\sin(x)$
AI: The idea is to use $\sin^2 ... |
H: Big Rudin 1.40: Open Set is a countable union of closed disks?
Reading through Big Rudin, I have come across the following statement in the proof of Theorem 1.40:
Let $S \subset \mathbb{C}$ be a closed set [in the topology induced by
the complex modulus]. Let
$\Delta = \{z \in \mathbb{C}: |z-\alpha|\leq \epsi... |
H: combinatorics: Calculating number of different possibilities
This is a HW question
Part a) A normal pizza can have upto 3 toppings out of possible 18 choices, or can be one of four speciality pizzas. Calculate the number of different pizzas possible (incliding the pizza with no toppings).
I came to the conclusion ... |
H: Can an eigenvalue $\lambda_i=0$?
I was doing some work with diagonalization of a matrix $A$ in order to find a matrix $P$ such that $\,P^{-1}AP\,$ was diagonal. In order to that I set $\;\lambda I_{n}=0\;$ and found the characteristic polynomial and its roots.
When I factored my characteristic polynomial I obtained... |
H: failed application of magicry in Taylor expansion of $1/x^2$ near $x=2$
It's straightforward to find the Taylor expansion for $\frac{1}{x^2}$ near $x=2$ using the the Taylor series definition.
This is turns out to be $\frac{1}{4} - \frac{1}{4} (x-2) + \frac{3}{16}(x-2)^2 + \cdots$
I was trying to be cute by finding... |
H: Marginal density, indicator function
I'm trying to figure out how to find the marginal density for $Y$ of the following function:
$$f_{X,Y}(x,y)=\begin{cases} 1 &\text{if } 0\le x \le2, \max(0,x-1)\le y \le \min(1,x) \\
0 & \text{otherwise}\end{cases} $$
$$f_Y(y)=\int_{-\infty}^\infty f_{X,Y}(x,y) \, dx = \int_0^2 ... |
H: A question concerning the Hilbert space trace
I am stuck with an equation regarding the trace on a Hilbert space $H$. The trace is defined in the book by Pedersen ("Analysis now", Sect. 3.4) as follows.
We choose an orthonormal basis $\{ e_j \mid j \in J\}$ for $H$, and for every positive operator $T \in B(H)$ (the... |
H: How do you formally prove that a function in several variable is really a function
Let say for example that we define $f:\mathbb{R}^{3}\longrightarrow \mathbb{R}^{3}$ such that $f(x,y,z)=(y^{2},xz,xy^{2})$. My informal argument would be just that there is only one object that can be defined having three real numbe... |
H: Show $X=\left\{x \in [0,1]: x \neq \frac1n\text{ for any }n \in \Bbb N\right\}$ is neither compact nor connected
I am stuck on the following question:
Let $X=\{x \in [0,1]: x \neq \frac1n: n \in \Bbb N\}$ be given the subspace topology. Then I have to prove that $X$ is neither compact nor connected.
Can someon... |
H: An identity for Lagrangian function
For $a_1,a_2,\cdots,a_n$, let $f_i(x)$ be
$$f_i(x)=\frac{\prod_{j\neq i}(x-a_j)}{\prod_{j\neq i}(a_i-a_j)}.$$
For $b_1,b_2,\cdots,b_n$, let $g_i(x)$ be
$$g_i(x)=\frac{\prod_{j\neq i}(x-b_j)}{\prod_{j\neq i}(b_i-b_j)}.$$
Can we prove such an equality:
$$\sum_{i=1}^n f_j(-b_i)g_i... |
H: Greatest integer $n$ where $n \lt (\sqrt5 +\sqrt7)^6$
I'm really not sure how to do this. I factored out a power of $3$ and squared so that I have $2^3 (6+\sqrt{35})^3 \gt n$ , and I know that if I can prove that $12^3-1 \le (6+\sqrt{35})^3 \lt 12^3$ I am basically done, but I don't know how to do that. Any help is... |
H: Can $\mathbb{R}$ be obtained as union of disjoint translations of a dense subset?
Doing some homework I had to find out maximal subgroups of $\mathbb{R}$ and my first approach was that subgroups of $\mathbb{R}$ are discrete or dense and, of course, a maximal subgroup $G$ can't be discrete and $\mathbb{R}/G \simeq \... |
H: Does $n \log\left(\cos\left(\frac{\pi\,n!}{n^2}\right )\right ) \neq 0 \implies n = p$?
Let $p$ denote a prime $> 3$.
Take any $\text{odd }n\geq3,\;n \in \mathbb{N}$.
How could one (dis)prove that:
$$n \log\left(\cos\left(\frac{\pi\,n!}{n^2}\right )\right ) \neq 0 \implies n = p$$
Although it doesn't seem practical... |
H: What is the difference between a ball and a neighbourhood?
I am presently reading chapter two of Rudin, Principles of Mathematical Analysis (ed. 3). He provides the following definitions:
Definition: If $\boldsymbol{x} \in \mathbb{R} ^ k$ and $r > 0$, the open ball $B$ with center at $\boldsymbol{x}$ and radius $r$... |
H: Why doesn't sign (appear to) change in inequality?
Given the equation $\frac{1}{x}\gt -1$, I would assume one would work it as
$$\frac{1}{x}\gt-1$$
$$x\cdot\frac{1}{x}\gt-1\cdot x$$
$$1 \gt -x$$
$$-1\cdot 1 \gt -x \cdot -1$$
$$-1 \lt x $$
which is incorrect, because the answers should be $x\gt0$ or $x\lt-1$. When I... |
H: Understanding Equivalence and Relations
Can someone please explain these answers? I have reviewed the slides and read about properties of equality but I still don't understand how to apply it to these sets.
For each the following relations on the set of integers list all that apply
(Reflexive, Symmetric, Antisymme... |
H: $k$-cells: Why $a_i < b_i$ instead of $a_i \le b_i$
In Rudin, The Principles of Mathematical Analysis, there is the following definition:
Definition: If $a_i < b_i$ for $i=1,2,...,k$, the set of all points, $ \boldsymbol{x} = ( x_1, x_2, ..., x_k )$ in $ \mathbb{R} ^ k $ whose coordinates satisfy the inequalities ... |
H: Commuting squares in abelian categories
Here $A,B,C$ and $D$ are all objects in an Abelian category.
$\require{AMScd}
\begin{CD}
A @> >> B @> >> C;\\
@VVV @VVV @VVV\\
D @> >>E @> >> F;
\end{CD}
$
The square $ABCDEF$ commutes (the outer square ) and the square $BEFC$ also commutes. Is it true that the square $ADBE$... |
H: Showing that a transformation $T:\mathbb R^3 \to \mathbb R^2$ is linear
OK, I am trying to prove the following transformation is linear, and find the basis for $\ker(T)$ and Im$(T)$ (also denoted in our textbook by $N(T)$ and $R(T)$ ). Then we're suposed to find the nullity and rank of $T$.
$T: \Bbb{R}^3 \rightarr... |
H: Motivation for Jordan Canonical Form
I took linear algebra and understood the proof that linear operators on a vector space over an algebraically closed field have a Jordan Canonical Form. Why should I care about this theorem? I understand that it can be useful in doing some computations, but it seems that these co... |
H: $\lim_{x\rightarrow 0^+}x^x$
How can I calculate $\lim_{x\rightarrow 0^+}x^x$?
I can only write it in the form $e^{x\ln x}$. I would like to use L'Hospital rule somehow, but I can't write it in form of fractions.
AI: HINT:
$$y=x^x\iff \ln y=x\ln x=\frac{\ln x}{\frac1x}$$ which is of the form $\frac\infty\infty$ as ... |
H: what is no. of positive eigen value of symmetric matrix A with some given relationship
Suppose A is a 3*3 symmetric matrix s.t.
$$\begin{pmatrix} x & y & 1 \\
\end{pmatrix} A \begin{pmatrix} x \\
y\\
1\end{pmatrix} = xy -1 $$
let p be the no. of positive eigen value of A and q = rank(A)-p, then
p=? q=?
what ... |
H: related rates (calculus) questions
The minute and hour hands of the GPO clock are 2m and 1.5m long respectively. How fast are their ends approaching at (a) 2 o'clock (b) 6 o'clock?
A fuel storage tank is in the shape of a right circular cone of base diameter 10m and height 20m and is being filled at a constant rat... |
H: Remainders of compactifications are images of the Stone-Čech remainder.
I need to show that if $\gamma X$ is a compactification of $X$, then $\gamma X\setminus X$ is the continuous image of $\beta X\setminus X$. I know that there exists a continuous function from $\beta X$ onto $\gamma X$ that is the identity on $... |
H: Proving principle of the Iterated Suprema
Let $X$ and $Y$ be nonempty sets and let $h : X\times Y \to R$ have bounded range in $\mathbb{R}$. Let $F: X \to\mathbb{R}$ and $G : y \to \mathbb{R}$ be defined by $F(x):=\sup\{h(x,y) : y\in Y\}$, and $G(y) := \sup\{h(x,y) : x\in X\}$. Establish the Principle of the Iterat... |
H: Let $A$ and $B$ be $n \times n$ complex matrices. Pick out the true statements.
Let $A$ and $B$ be $ n \times n$ complex matrices. Pick out the true statements:
a) If $A$ and $B$ are diagonalizable, so is $A + B$
b) If $A$ and $B$ are diagonalizable, so is $AB$
c) If $A^2$ is diagonalizable, then $A$ is diagonaliz... |
H: If a morphism between affine schemes is dominant, is the corresponding ring morphism injective?
Suppose we have $\phi$ a ring morphism from $A$ to $B$, let $X=\operatorname{Spec}A$ , $Y=\operatorname{Spec}B$ and $\psi$ is the induced morphism of affine schemes. Is it true that if $\psi$ dominant, than $\phi$ is in... |
H: Substituting an equation into itself, why such erratic behavior?
Until now, I thought that substituting an equation into itself would $always$ yield $0=0$. What I mean by this is for example if I have $3x+4y=5$, If I substitute $y=\dfrac {5-3x}{4}$, I will eventually end up with $0=0$. However, consider the equatio... |
H: Which of these statements about biholomorphic functions $f \colon D(0, 1) → D(0, 1)$ is true?
$f \colon D(0, 1) → D(0, 1)$ is a biholomorphic function.
a) $f$ must be constant
b) $f$ must have a fixed point
c) $f$ must be a rotation
d) $f$ must fix the origin.
Any such map looks like $e^{i\alpha}{(z-a)\over (1-\b... |
H: Theorem by Whitney
For $0<k<\infty$ and any $n$-dimensional $C^k$ manifold the maximal atlas contains a $C^\infty$ atlas on the same underlying set by a theorem due to Whitney. Could someone please point me to where I can find the theorem and a proof thereof?
AI: Morris Hirsch's Differential Topology text contains ... |
H: Why isn't the Ito integral just the Riemann-Stieltjes integral?
Why isn't the Ito integral just the Riemann-Stieltjes integral?
What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral:
$$\int_0^Tf(t)\;dB(t).$$
So what if we can't apply the change of variables for... |
H: Sentential Logic
Below is a question comes from the book How to Prove It written by Daniel J. Velleman.
Let $P$ stand for the statement, “I will buy the pants” and $S$ for the statement “I will buy the shirt.” What English sentences are represented by the following expressions?
$\lnot(P \land \lnot S)$
The given ... |
H: How does this vector addition work in geometry?
I saw the accepted answer to the question: Finding a point along a line a certain distance away from another point!
I am not getting how to use it actually to find the coordinates of the new point at a given distance. This is because I am confused between how to trans... |
H: Determine the trace and determinant of a block upper triangular matrix
Let $A$ be a $5\times5$ real skew symmetric matrix with entries and $B$ be a $5\times5$ real symmetric matrix whose $(i,j)$-th entery is the binomial coefficient ${i \choose j}$ when $i\ge j$. Now, if
$$C= \begin{pmatrix}A & A+B \\ 0 & B\\ \end{... |
H: how to prove a parametric relation to be a function
For example lets suppose that I have given the functions $f:\mathbb{R}\longrightarrow \mathbb{R}$ and $g:\mathbb{R}\longrightarrow \mathbb{R}$. If my relation is $R=\{(x,(y,z))\in \mathbb{R}\times \mathbb{R}^{2}: y=f(x) \wedge z=g(x)\}$ How to prove formally (from... |
H: How to find solutions to this equality $\; \mathrm{x} = \mathrm{a^2x \, (1-x)\,(1-ax\,(1-x))}$
We have the following equality:
$$ \mathrm{x} = \mathrm{a^2x \, (1-x)\,(1-ax\,(1-x))}$$
Some of the solutions I found:
$\mathrm{x} = 0$
Also for $\mathrm{a}=0$, every $\mathrm{x}$ is a solution I believe
I tried getti... |
H: Are the real numbers a nontrivial simple extension of another field?
Is there a proper subfield $K$ of the real numbers and a real number $\theta$ such that $\mathbb R = K(\theta)$?
I thought of this question earlier idly wondering about what the structure of the poset of all subfields of $\mathbb C$ looks like and... |
H: Defining holomorphic function as an integral in $\mathbb{R}^n$
Let $U$ be a bounded connected subset of $\mathbb{R}^n$. Let $f:\mathbb{C} \times U \to \mathbb{C}$ such that
1) for all $(u_1,\ldots,u_n) \in U$ the function $z \mapsto f(z,u_1,\ldots,u_n)$ is holomorphic
2) for all $z\in \mathbb{C}$ the function
$(u... |
H: Prove that $\sum_{i=1}^{m-1} i^k$ is divisible by $m$
Prove that $\sum\limits_{i=1}^{m-1} i^k$ for odd numbers $m,k \in \mathbb{N}$ is divisible by $m$.
Because $m \mid m^k$, it is equivalent to the following:
Prove that $m \mid \sum\limits_{i=1}^{m} i^k$ for odd numbers $m,k \in \mathbb{N}$.
Say $m = 2t + 1$ w... |
H: local isometry for riemannian manifolds is not transitive
Let $(M_1,g_1)$ and $(M_2,g_2)$ be Riemannian manifolds of the same
dimension, and let $\phi: M_1 \to M_2$ be a smooth map.
We say that $\phi$ is a local
isometry if $g_2 (\phi_* X, \phi_* Y ) = g_1 (X, Y )$
for all $m \in M_1$ and $X, Y \in T_m M_1,$ where ... |
H: Relation in probability
As part of the solution of an exercise I have the following relation:
$$\sum_{k=0}^{\infty}k(1-p)^{k-1}=\frac{1}{(1-(1-p))^2}$$
Where $p$ is a probability.
I don't understand where this is coming from?
AI: Hints:
For $\,x\in\Bbb R\;,\;\;|x|<1\,$ , we have the well known power series developm... |
H: Linearly independent elements over a vector space over $\mathbb R$.
Let $n\geq3$ be an integer, let $u_1,u_2,u_3,\ldots,u_n$ be $n$ linearly independent elements over a vector space over $\mathbb{R}$. Set $u_0=0$ and $u_{n+1}=u_1$ and define $v_i=u_i+u_{i+1}$ and $w_i=u_{i-1}+u_{i}$ for $i=1,2,\ldots,n$, then
$v_1... |
H: What is the semantic of square brackets after the set denoting coefficients of polynomial?
I have the following excerpt:
Unless stated otherwise, we assume all polynomials take integer coefficients, i.e. a polynomial $f \in \mathbb{Z}[{\bf y}, x]$ is of the form
$$f(y, x) = a_m · x^{d_m} + a_{m−1} · x^{d_{m−1}} + ... |
H: Tangent of circumscribed circle
I found a solution online which it said :
"It's easy noted that $AG.AE$ = $AD^2$ = $AF^2$ (Using tangent of circumscribed circle)"
I found this not obvious at all. I know that $AD = AF$ but why it had to equal to the product of two inline line?
AI: Nevermind, I've found it! It's cal... |
H: Product and Sum of Polynomial Roots
The ratio of the sum of the roots of the equation, $8x^3+px^2-2x+1=0 $ to the product of the roots of the equation $5x^3+7x^3-3x+q=0 $ is $3:2$. What is the value $\frac{p-q}{p+q}$?
Well I found out the sum of the roots of 1st equation is $-\frac{p}{8}$ and product of roots of s... |
H: If $E(X^2)$ exists, $E(X)$ exists
How do I prove this? Is it even true?
I proved the obvious cases, but don't know how to prove/disprove the $0\lt x\lt 1$ case.
IF $E(X^2)$ exists, $\sum_{x\in\omega}x^2P(x)\lt \infty$. Therefore:
$$\forall x \le 0:\space \sum_{x\in\omega}xP(x) \le \sum_{x\in\omega}x^2P(x) = E(X^2)\... |
H: What exactly are limits?
So as far as I know, a limit of a function is the $y$ value that we get as $x$ gets closer and closer to the limit but remains distinct from the limit.
Also, limit when approached from right should be the same as limit when approached from left or else the limit does not exist.
The proble... |
H: Which of the following groups is not cyclic?
Which of the following groups is not cyclic?
(a) $G_1 = \{2, 4,6,8 \}$ w.r.t. $\odot$
(b) $G_2 = \{0,1, 2,3 \}$ w.r.t. $\oplus$ (binary XOR)
(c) $G_3 =$ Group of symmetries of a rectangle w.r.t. $\circ$ (composition)
(d) $G_4 =$ $4$th roots of unity w.r.t. $\cdot$ (mult... |
H: not injective/not surjective linear maps
Let $S$ be the vector space of real sequences, and for $x=(x_1,x_2,\dots)$ define $\alpha(x)=(0,x_1,x_2,\dots)$ and $\beta(x)=(x_2,x_3,\dots)$. The problem was asking for few other things to do, but I got stuck at showing that the first is not injective, while the second is ... |
H: How to define Conway's class of all games rigorously?
I'm reading John Conway's On Numbers And Games.
In the course of defining the surreal numbers, Conway defines the term game more or less as follows.
$(\emptyset,\emptyset)$ is a game.
If $L$ and $R$ are sets whose elements are games, then $(L,R)$ is a game.
All... |
H: Orthogonal projection $\alpha$ of $\mathbb{R}^3$ onto a plane
Let $\Pi$ be the plane in $\mathbb{R}^3$ that contains $\textbf0,\textbf j,\textbf k$. Show that the orthogonal projection $\alpha$ of $\mathbb{R}^3$ onto $\Pi$ is a linear map.
I'm more interested in finding this orthogonal projection, as I still don't ... |
H: Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$
We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$
I have tried and it gets confusing.
AI: $$\frac{\sin\theta-\cos... |
H: Probability $P(A < B)$
Given two independent and continuous random variables $A$ and $B$ with cumulative distributions $F_A$ and $F_B$, show that $$P(A<B) = \int_{-\infty}^{\infty} F_A(x)\, F'_B(x)\,dx.$$
Is this something obvious and available in text books ?
AI: Let the density functions be $f_A(x)$ and $f_B(y... |
H: Complex integral, correct?
I am supposed to do the integral $$ \int_{\gamma_2} \frac{\sin(z)}{z+\frac{i}{2}} dz$$ where $\gamma_2:[-\pi, 3\pi] \rightarrow \mathbb{C}$ , $\gamma_2(t)=\exp(it)$ for $ t\in [-\pi,\pi]$, $\gamma_2(t)=(1+t-\pi)\exp(it)$ for $t\in [\pi,2\pi)$ and $\gamma_2(t)=(1+3\pi-t) \exp(it)$ for $t\i... |
H: Analyticity of Laplace transform
Let $f(x)$ be a bilateral Laplace transform of a measure $\mu$:
$$
f(x)=\int_{-\infty}^{+\infty} e^{-xt} d\mu(y).
$$
Suppose that $f(x)$ converges absolutely in $(a,b)$, and $(a,b)$ do not contain the origin.
It is always true that $f(x)$ is analytic in $(a,b)$? Or it is true just f... |
H: Question on a theorem on Riemann surfaces
In the book "Lecture on Riemann surface" of Forster, in the page 23, there is a theorem as follows:
Suppose $X$ and $Y$ are Riemann surfaces, $p: Y\rightarrow X$ is an unbranched holomorphic map and $f:Z\rightarrow X$ is any holomorphic map. Then every lifting $g:Z\rightar... |
H: A Differential operator.
What are the fundamental solutions for the operator
$$\mathcal D=a{\partial^2\over\partial x_1^2}+b{\partial^2\over\partial x_2^2}$$
on $\Bbb R^2 $ with standard cordinates $(x_1,x_2)$.
Here $a,b\in \Bbb R$.
Definition: A distribution $E\in D'(R)$is called a fundamental solution of an ... |
H: Open sets and projections maps in metric spaces
Let $E_1$ and $E_2$ metric spaces and $E=E_1\times E_2$ a metric spaces with some metric $d$. Let $\pi_1$ and $\pi_2$ the projections maps of $E_1\times E_2\rightarrow E_1$ and $E_1\times E_2\rightarrow E_2$ respectly, i.e, $$\pi_1(x, y)=x,\,\,\,\,\,\,\,\,\,\pi_2(x, ... |
H: weak derivative of a vector valued function
Consider $T>0$ and $U$ a opensubset of $R^n$ ,bounded and with smooth boundary. Consider ${\Omega}_T = U \times (0,T]$. Let $u: {\Omega}_T \rightarrow R $ a smooth function.
Define $h : [0,T] \rightarrow H^{1}_0 (U)$ given by $h(t) = u(. ,t)$ .
What is the weak derivati... |
H: A question on boundary of open set
Let $\mathbb Q$ and $\mathbb R$ have the usual topologies. Is there an open set $N\subseteq \mathbb Q\times \mathbb R$ (with product topology) such that for every non-empty open subset $U\subseteq N$, the boundary of $U$ is not compact?
AI: Sure. Let $N = (0,1)^2 \cap (\mathbb{Q}... |
H: Proving that $\iint_S (\nabla \times F) \cdot \hat{n} dS =0$
I have the following question:
Prove that $$\iint_S (\nabla \times \vec{F}) \cdot \hat{n} dS =0$$ for any closed surface $S$ and twice differentiable vector field $\vec F:\mathbb{R^3} \to \mathbb{R^3} $ .
I need to prove this using Stokes' theorem.
T... |
H: The most general linear map of $\mathbb{R}^n$ to $\mathbb{R}$
Find the most general linear map of $\mathbb{R}^n$ to $\mathbb{R}$.
Obviously, dot product is one. However, I'm not sure how do I show that it's the most general of them? Or maybe, something else is even more general? Hint appreciated!
AI: As with any tw... |
H: Operator Norm of a Matrix composed of Standard Basis and Fourier Basis
Let $\mathbf{A}_n$ be an $n\times 2n$ matrix (where $n=2^k$) composed of Fourier basis and standard basis; that is,
$$\mathbf{A}_n = \begin{bmatrix}\mathbf{I}_n & \mathbf{F}_n\end{bmatrix}$$
where $\mathbf{I}_n$ is the identity matrix and $\mat... |
H: What is the dimension of an open subset of an affine euclidean space? And why?
My question comes from an exercise from my differential geometry book:
An open subset $M \subset \mathcal E$ is a smooth manifold of maximal dimension, i.e. $dim(M) = Dim(\mathcal E)$, and - for all p $\in$ M -
$T_pM = E$
I used a restri... |
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