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H: How many maxima and minima an n-degree polynomial can have at most?
There's a statement: "Given a polynomial of degree 6, it may have up to 6 real roots, corresponding to 3 minima and 3 maxima."
Is this true in general?
How to get the number of maxima and minima separately for an arbitary n-degree polynomial?
AI: F... |
H: Given that $x = 5\sin(3t), t\ge 0$: Find the maximum value of $x$ and the smallest value of $t$ for which it occurs.
Given that: $$x = 5\sin(3t), t \ge 0$$
Find the maximum value of $x$ and the smallest value of $t$ for which it occurs.
I have figured out the smallest value by:
$$\frac{dx}{dt}=15\cos(3t)$$
when $... |
H: Why does $\rho'=\sum_i p_i U_i \rho U_i^\dagger$ with $U_i$ unitary imply $\rho'\preceq \rho$?
Let $\rho$ be an Hermitian matrix with unit trace (this is the context I've found this result stated in, but I don't know if these restrictions are necessary for the result).
Suppose $p_i\ge0$ with $\sum_i p_i=1$, and let... |
H: Mathematical writing guidelines
In section 5.3 from the book Book of Proof by Hammack (3rd edition, this link is to the author's website), the author outlines 12 mathematical writing guidelines to help the young mathematician with writing better proofs.
Those guidelines, with their examples are as follows:
Begin ... |
H: Prove that each partial sum of a convergent series of non-negative terms cannot exceed the sum of the series by elementary calculus
Let $\sum_{n=1}^\infty a_n$ be a convergent series of non-negative terms. And its sum is denoted by $S$. Let $S_k$ be the $k$-th partial sum of the series. I would like to prove that $... |
H: For $x≠y$ and $2005(x+y) = 1$; Show that $\frac{1}{xy} = 2005\left(\frac{1}{x} + \frac{1}{y}\right)$
Problem:
Let $x$ and $y$ two real numbers such that $x≠0$ ; $y≠0$ ; $x≠y$ and $2005(x+y) = 1$
Show that $$\frac{1}{xy} = 2005\left(\frac{1}{x} + \frac{1}{y}\right)$$
Calculate $l$:
$$l = \frac{y}{y-x} - \frac{y-... |
H: Quotient algebra is involutive: do we need the ideal do be self-adjoint?
Let $A$ be a $*$-algebra, i.e. an algebra $A$ together with a map $*: A \to A$ such that
$$(a+ \lambda b)^* = a^* + \overline{\lambda} b^*$$
$$a^{**}= a$$
$$(ab)^* = b^* a^*$$
My book then claims that if $I$ is an ideal of $A$ satisfying $I=I^... |
H: Show $( v^\intercal x ) ^{2} = x^\intercal x$ given $\| v \| =1$
Show $( v^\intercal x ) ^{2} = x^\intercal x$ given that $v$ is a unit vector.
I believe that $v$ is ought to cancel out by $v^\intercal v = 1$ but don't see how:
$( v^\intercal x ) ^{2} = ( v^\intercal x )( v^\intercal x ) = \dots ?$
My attempt was t... |
H: Find the integer part of $\frac{1}{\frac{1}{2016}+\frac{1}{2017}+\ldots+\frac{1}{2023}}$
The question is
Find the integer part of $\frac{1}{\frac{1}{2016}+\frac{1}{2017}+\ldots+\frac{1}{2023}}$.
I tried to solve it by using harmonic progression but it is not working.
Kindly solve this question.
AI: Let's observe th... |
H: General matrix decomposition
Let $A \in {\mathbb C}^{m \times n}$. Does there exist a decomposition $A=BC$ for every $k \in \mathbb N$ such that matrix $B$ is $m \times k$ and matrix $C$ is $k \times n$?
AI: We can write $A = BC$ with $B,C$ of sizes $m \times k$ and $k \times n$ if and only if $\operatorname{rank}(... |
H: Where this solution of $f ' (x) = g(x)$ comes from?
I would have liked to solve the following first-order linear ODE for $f(x)$:
$$ f'(x) = g(x) $$
I attempted to solve it like this:
$$ \int f'(x)\,dx = \int g(x)\,dx $$
$$ f(x) = \int g(x)\,dx+C $$
Then I gave $ f'(x) = g(x) $ to WolframAlpha and its solution was q... |
H: Finding the centre of a circle under a specific condition
Question: Consider a circle, say $\mathscr{C}_1$ with the equation $x^2 + (y-L)^2=r^2$. A second circle, say $\mathscr{C}_2,$ with equal radii that has a centre $(x_0,y_0)$ which lies on the line $y=mx$. Find an expression for $x_0$ and $y_0$, in terms of $L... |
H: Question about Sigma Algebra generated by a Random Variable
Given the following exercise:
I need to determine the sigma algebras generated by the random variables X and $S_1$ and show that they are independent under the first measure. $\sigma(S_1)$ should just consist of $\{\emptyset,\{H,T\} , H, T\}$, but I'm not... |
H: Decomposing a matrix into a product of 2 special matrices
Any positive number (i.e., a matrix of size $1\times 1$) $A>0$ can be written as $A=\frac{1}{A}A^2,$ where in the last product one of the factors is necessarily $\geq 1$ and the other one is $\leq 1$.
Let now $A$ be any matrix with positive entries. Is it a... |
H: Proof the theorem from linear algebra
x,y are nx1 vectors of real numbers. Matrix A is a product of x and y'.
$$A=xy^T$$
Prove that for any x and y there exists $\lambda \in \Bbb R$ that meets equation $ A^{k}=\lambda^{k-1}A$, $k \in \Bbb N$.
My idea was to solve it through math induction.
For k=1 we have that $A=\... |
H: Bijective map from a set to a subset of reals?
There is a concept that I have been thinking about quite a lot lately as I am currently self-studying point-set topology:
Say we have a bijective map from one interval, $[a,b]$, to another interval, $[c,d]$, both of which are in $\mathbb{R}$. Also set $c$ and $d$ so th... |
H: How can I find the roots of the polynomial $12x^{4}+2x^3+10x^2+2x-2$?
It's clear that I can divide by $2$, but I don't know what can I do with $$6x^{4}+x^3+5x^2+x-1$$
Is there any algorithm for it or a trick? I have found the roots by an online calculator but I don't know how can I calculate them. Thank you for you... |
H: Projective graded modules
Let $A = \bigoplus_{i = 0}^{\infty} A_i$ be a graded ring (for simplicity, we can take the grading over $\mathbb{N}$), and $M$ a graded $A$-module. In various occasions I have met the statement that $M$ is projective in the category of graded $A$-modules iff it is projective in the categor... |
H: Why is $f_1^2=f_1f_2=1$?
In a proof by induction why $\sum_{i=0}^n f^2_i = f_nf_{n+1}$ for $n\in \mathbb N$ the base case from the solution is $f_1^2=f_1f_2=1$. I assume $f$ denotes a function, but from the exercise there is no information given what $f$ denotes. So why is $f_1^2=f_1f_2=1$?
The exercise is the seve... |
H: Is a collection of open sets is a basis for a topology on $X$ if it gives a basis for a dense subset of $X$?
Let $Y$ be a dense subset of a topological space $X$. Let $\mathcal B := \{U_\alpha : \alpha \in \Lambda\}$ be a collection of open subsets of $X$ such that $\{U_\alpha\cap Y : \alpha \in \Lambda\}$ is a bas... |
H: Universal morphism in the first isomorphism theorem for groups.
I almost don't know anything about categories but I was reading about universal morphisms and I wanted to see this in the context of the first isomorphism theorem for groups. What would the functor $F$ be in this case?
AI: Let $K$ be a normal subgroup ... |
H: When does the subgroup generated by a generator of a group admit a complement?
Let $G = \langle x,y \rangle$ be a finite bicyclic group generated by the two elements $x,y \in G$ and assume that $x \not\in \langle y \rangle,y \not\in \langle x \rangle.$
Is it true that $G = \langle x \rangle \times H$ for some subg... |
H: Find x intercepts of a higher degree polynomial $2x^4+6x^2-8$
I am to factor and then find the x intercepts (roots?) of $2x^4+6x^2-8$
The solutions are provided as 1 and -1 and I am struggling to get to this.
My working:
$2x^4+6x^2-8$ =
$2(x^4+3x^2-4)$
Focus on just the right term $(x^4+3x^2-4)$:
Let $u$ = $x^2$, t... |
H: A matrix that lies in the U(3) group but neither in O(3) nor in SU(3)?
How to find a matrix that lies in the U(3) group but neither in O(3) nor in SU(3)?
I would be grateful for any help!
AI: For example, consider the matrix
$$
A = \pmatrix{i&0&0\\0&1&0\\0&0&1}.
$$ |
H: True or false properties of the eigenvector
$\varphi: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$ an endomorphism with the property $\varphi^2=\varphi$ and $x \in \mathbb{C}^3$ arbitrary where $\varphi(x) \neq 0$. Then $\varphi(x)$ is an eigenvector of $\varphi$
$\varphi: \mathbb{C}^{3} \rightarrow \mathbb{C}^{3}$... |
H: confused about quotienting $\Bbb R^2$ by $\Bbb Z^2$ vs. compactifying $ \Bbb R^2$ first and then gluing sides
Learning a little about quotient spaces and I don't understand something.
(1) Compactify $\Bbb R^2$ to $[0,1]^2$ then glue sides to make torus. (linked post gives example of compactification)
(2) $\Bbb R^2... |
H: How to explain this question and answer on parametric equations from MIT 18.02?
This is a question from MIT's multi-variable course on parametric equations:
And this is the answer:
However, I'm not entirely sure how the answer was arrived. I started off tackling the question by letting P be the point $(1,1,1)$, a... |
H: For every element $g$ in a group $G$ of order $23$, there is $h\in G$ such that $g=h^2$.
For every element $g$ in a group $G$ of order $23$, prove that there is $h\in G$ such that $g=h^2$.
I am not sure how to prove this. I think that, since $G$ has prime order is isomorphic to a cyclic group and therefore abelia... |
H: Problem about the generalized pigeonhole principle
This problem from Discrete Mathematics and its application's for Rosen
What is the least number of area codes needed to guarantee that the 25
million phones in a state can be assigned distinct 10-digit telephone
numbers? (Assume that telephone numbers are of the f... |
H: Proving: $x = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$
How can one expand the function $f_1(x) = x$ on $(−π, π)$in terms of the functions $\cos nx, n = 0, 1, 2, ...$ and $\sin nx, n = 1, 2, ...,$ in a way so that
$$f_2(x) = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$$
is the expansion of the ... |
H: An $A$-algebra $B$ carries the same data as a ring map $A \rightarrow B$
I'm trying to show that an $A$-algebra $B$ has the same data as a ring map $\phi: A\rightarrow B$.
An $A$-algebra $X$ is an $A$-module $B$ that comes equipped with a bilinear operator $\times_B: B \times B \rightarrow B$. So to spell out fully... |
H: Suppose $A$ and $B$ are sets. Prove that $A\setminus(A\setminus B)=A\cap B$.
Not a duplicate of
For sets $A,B$, prove $A \setminus (A \setminus B) = A \cap B$
Showing that $A\cap B = A\setminus (A\setminus B)$
set theory proof of $A\cap B = A \setminus(A\setminus B)$
Using disjunction to prove that $A \setminus (A ... |
H: Would a right triangle with bases $a=i$ and $b=1$ have hypotenuse $c=0$?
Suppose we have a right angle triangle with $a$ and $b$ as bases and $c$ as the hypotenuse, letting
$$a=i$$$$b=1$$
Wouldn't the hypotenuse then be$$i^2+1=0$$
I am finding it hard to understand how this may be possible since the hypotenuse by d... |
H: Integrability of sequence of uniform convergence function defined on unbound domain
Consider $(f_{n})_{n=1}^{\infty}$ being a sequence of function defined on interval $(-\infty,0]$. Assume it is uniform convergence to a function $f$. If each $f_{n}$ is Lebesgue integrable, is $f$ integrable and $\int f=\lim_{n\righ... |
H: On Auslander transpose and stable Hom module
For finitely generated modules $M,N$ over a Noetherian local ring $(R, \mathfrak m)$, define $$P_R(M,N):=\{f\in \text{Hom}_R(M,N): \exists n\ge 0 \text{ and } M\xrightarrow{g} R^n \xrightarrow{h} N \text{ such that } f=h\circ g\}.$$ Then I can prove easily that $P_R(M,N... |
H: Function $f$ with $f(x_1\cdot x_2)=f(x_1)+f(x_2)$ that is not $\log$?
Is the log-function the only function that enables the transformation of a product to a sum:
$$f(x_1\cdot x_2)=f(x_1)+f(x_2)\,?$$
Yes, I can approximate the log function by a Taylor Series, but are there different functions that fulfill this pr... |
H: Proving $\sqrt{x^2-xz+z^2} + \sqrt{y^2-yz+z^2} \geq \sqrt{x^2+xy+y^2}$ algebraically
The question is to prove that for any positive real numbers $x$, $y$ and $z$,
$$\sqrt{x^2-xz+z^2} + \sqrt{y^2-yz+z^2} \geq \sqrt{x^2+xy+y^2}$$
So I decided to do some squaring on both sides and expanding:
$$\sqrt{x^2-xz+z^2} + \s... |
H: Prove that if $A\Delta B\subseteq A$ then $B\subseteq A$.
Not a duplicate of
Prove that if $A \bigtriangleup B\subseteq A$ then $B \subseteq A.$
Prove that if $A \mathop \triangle B \subseteq A$ then $B\subseteq A$
This is exercise $3.5.5$ from the book How to Prove it by Velleman $($$2^{nd}$ edition$)$:
Prove that... |
H: Normed Linear Spaces are Complete iff there unit spheres are complete.
Let N be a nonzero normed linear space, then N is Banach iff $S=\{x:||x||=1\}$ is Complete.
I am able to give a proof for a weaker statement than this, that being
N is Banach iff $\{x:||x|| \leqslant 1\}$
For any Cauchy sequence in N. I asscocia... |
H: Solution to a Nonlinear System of ODEs
In applying the method of characteristics to a problem, I came the across the following set of ODEs:
$$\frac{dz}{dt}=\frac{v_m}{2}z^2\cos(x),\quad\frac{dx}{dt}=v_mz\sin(x).$$
Here $v_m>0$ is a constant. With little hope of a solution, I dejectedly plugged this into Mathematica... |
H: Proof for subgroup of SL(2,q)
In Suzuki's Group Theory I, Theorem 6.21 says
Let $p$ be an odd prime number, and let $\lambda$ be an element of $F$ which is algebraic over the prime field $F_0=GF(p)$. Set $E=F_0(\lambda)$. Let $G$ be defined as follows: $$G=\left\langle\left(\begin{matrix}1&0\\1&1\end{matrix}\right... |
H: Are there sets where it cannot possibly have a metric on it?
To avoid any ambiguity, a metric space, by definition, is a set $X$ with a distance function $d$ such that $d$ satisfies positivity, symmetry property and triangle inequality.
I was wondering does there exist a set where there cannot possibly be equipped ... |
H: Does this $\int_{0}^{\infty}(\frac{\log x}{e^x})^n dx$ always have a closed form for $n$ being positive integer ? what about its irrationality?
It is known that $\int_{0}^{\infty}\left(\frac{\log x}{e^x}\right)^n dx=-\gamma$ for $n=1$ and for $n=2$ we have :$\frac{1}{12}(\pi^2+6(\gamma+\log 2)^2)$ and for $n=3$ we ... |
H: Why is the integral of the unit tangent vector equal to the position vector?
I am working through Apostol's Calculus Vol 1, and was just introduced to curvature. The book just proved that the curvature of a plane curve is given by $\kappa(t)=\left|\frac{d\alpha}{ds}\right|$ where $\alpha$ is the angle the tangent v... |
H: Dihedral group actions on Spheres
As an outsider of algebraic topology, I would like to consult your guidance to understand finite group actions over spheres. I do not imagine how a group acts on a sphere, in particular why a group acts freely on some spheres. I've read that $D_{2n}$ acts non-freely on any sphere. ... |
H: Confusion in proving $\phi: Z_n \to Z_k$ defined by $\phi (x)=x \mod k$ to be a homomorphism.
It is to be proven that if $k|n$ and $\phi: Z_n \to Z_k$ is defined by
$\phi (x)=x\mod k $, then $\phi$ is a homomorphism.
$\phi$ is well defined as: $x=y\implies x\mod k=y \mod k$
To prove that $\phi$ is operation pre... |
H: Conditional Probabilities Paradox
I know which step is wrong in the following argument, but would like to have contributors' explanations of why it is wrong.
We assume below that weather forecasts always predict whether or not it is going to rain, so not forecast to rain means the same as forecast not to rain. We s... |
H: Doubt in Dominated Convergence theorem
From Browder's Mathematical Analysis
On applying Fatou's Lemma to sequence ${(g \pm f_n)}$ we get $\int \liminf (g \pm f_n) \leq \liminf \int(g \pm f_n)$.
My question is how they got as $\int (g \pm f) \leq \liminf \int (g \pm f_n)$.
Is $\int \liminf (g \pm f_n)=\int g \pm f$... |
H: when does $1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-...$converge
when does $1-x-\frac{x^2}{2!}+\frac{x^3}{3!}-\frac{x^4}{4!}+\frac{x^5}{5!}+\frac{x^6}{6!}-\frac{x^7}{7!}...$ converge and diverge
where the pattern is the Thue morse sequence with adding and subtracting. $+--+-++--++-+--+...$.
I plotted it with 31 terms and ... |
H: Find an angle created by lateral edge and the base of the Pyramid
Pyramid $SABC$ has right triangular base $ABC$, with $\angle{ABC}=90^\circ$. Sides $AB = \sqrt3, BC = 3$. Lateral lengths are equal and are equal to $2$. Find the angle created by lateral length and the base.
Here's my attempt, but I didn't get ver... |
H: Expected value of hat game
Question: At each round, draw a number 1-100 out of a hat (and replace the number after you draw). You can play as many rounds as you want, and the last number you draw is the number of dollars you win, but each round costs an extra $1. What is a fair value to charge for entering this gam... |
H: How do I show $\lim_{n \to \infty} \int_0^\infty \frac{n}{n^2+x}\sin(\frac{1}{x})\, dx = 0\,$?
How do I show $$\lim_{n \to \infty} \int_0^\infty \frac{n}{n^2+x}\sin\left(\frac{1}{x}\right)\, dx = 0\,\,?$$ I've tried splitting into the cases where $x \leq 1$ and $x \geq 1$ but I am having trouble finding bounds so t... |
H: Is there more than one way to diagonlize a matrix without using eigenvalue decomposition?
I was trying to answer the question of whether the diagonalization of a matrix is unique and found that it is "unique up to a permutation of the entries." (see: Is a diagonalization of a matrix unique?)
However, I found a slig... |
H: Topologically complete and $G_\delta$ theorem proof
I don't understand the underlined part of the proof. It is very important, but it is not obvious.
Since ($S$, $e$) is complete, $y\in S$.
AI: For the function $g$ (which only serves the purpose of bounded the term to $1$, keeping continuity), we know that $g(t)... |
H: Given a metric, find continuous map that maps to $\mathbb{R}^2$
I wanted to know how to approach the question of:
Given a metric on $\mathbb{R}^2$ does a continuous map $f:\mathbb{R}^2 \to \mathbb{R}^2 $ with the following property:
$$f([0,1]\times[0,1]) = \mathbb{R}^2 $$
exist ?
I currently am a little clueless on... |
H: Equivalency of the maximum and minimum conditions of idempotents of a ring
Let $R$ be a ring with unit, and let $I$ be the set of all idempotents of $R$, that is, all $e\in R$ such that $e^2 = e$. We put a partial ordering $\leq$ on $I$ by saying $e\leq f$ if $ef=e=fe$ or equivalently if $eRe\subset fRf$. We say th... |
H: When is the integral of a function of two variables a continuous function?
Suppose $(\Omega, \mu)$ is a finite measure space. Consider the non-negative functions $f(t,x)$ for $t\in[0,1]$ and $x \in \Omega$. Assume that for every $x\in\Omega$, $t \mapsto f(t,x)$ is continuous and for every $t \in [0,1]$, $x \mapsto ... |
H: Summing pairs from a sequence
Suppose we have an arbitrary sequence $$\{a_k\} = \{a_1, a_2, ..., a_k\} $$
and use it to a create a set as follows
$$A = (a_i+ a_j : a_i, a_j \in \{a_k\})$$
and we wanted to sum over all of the members of this set, would we denote it as $\sum A$ ?
Furthermore, if we wanted to create a... |
H: Suppose $Z\to Y$ is a morphism of schemes, and $f(z)$ is contained in an open affine of $V\subset Y$. Do the specializations of $z$ also lie in $V$?
I am trying to prove Corollary 4.6(f) of Hartshorne Chapter 2.4 using the valuative criterion for separated morphisms:
Suppose $f:X\to Y$ is a morphism between Noethe... |
H: Finding upper bound using Cauchy-Schwarz inequality.
I am learning about the Cauchy-Schwarz inequality and I cam across this question:
Consider the function $f(x) = \frac{(x+k)^2}{x^2 +1}$ where $k>0$ and $x$ is a real number. Show that $f(x)\leq k^2 +1$ for all $x$ and $k>0$ using the Cauchy-Schwarz inequality.
I ... |
H: How to solve this system of ODE: $ u'= - \frac{2v}{t^2}$ and $v'=-u $?
I have this system of differential equations:
$$ \left\{\begin{array}{ccc}u'&=& - \dfrac{2v}{t^2}\,, \\ v'&=&-u
\,.\end{array}\right. $$
I want to find the general solution by deriving an Euler differential equation for $v$ and giving a fundamen... |
H: Tensor contraction and notational problems
I am going through a chapter in my book on tensors, and it gives a basic understanding of tensors.
The question posed is simple: "Show that the contracted tensor $T_{ijk}V_k$ is a rank-2 tensor."
I followed the basic steps outlined earlier in the chapter, and cross checked... |
H: Why we need to mention the scalar product of $\cos (nx), \sin (nx)$?
I found the following text -
The functions $\cos (nx), n = 0, 1, 2, \cdots$ and $\sin (nx), n = 1, 2, \cdots $ which are known to be orthogonal with respect to the
standard scalar product on $(-\pi, \pi)$.
The source of the problem is -
Questi... |
H: Are all finite-dimensional algebras of a fixed dimension over a field isomorphic to one another?
Suppose I have a finite-dimensional algebra $V$ of dimension $n$ over a field $\mathbb{F}$. Then $V$ is an $n$-dimensional vector space and comes equipped with a bilinear product $\phi : V \times V \to V$.
Suppose now t... |
H: Prove $[\mathfrak g,\mathfrak g]$ is an ideal.
I have to show :
Given a Lie algebra $\mathfrak g$, then $[\mathfrak g,\mathfrak g]$ is an ideal.
I was told to use Jacobi's identity, but I am not sure why.
It seems I just have to show that for $x,y,z \in \mathfrak g$, I have $[x,[y,z]]\in [\mathfrak g,\mathfrak g... |
H: How to Calculate an Infinitely Repeating Percent
How do you calculate the percentage of a number in an infinitely repeating function?
Suppose I'm doing an experiment where the inputs cost 100, and there's a 95% chance of success. To figure the average cost of success, I take 100 plus I allow for the 5% chance of fa... |
H: Cauchy-type criterion for uniform convergence of improper integral
Suppose $f=f(x,t)$ is defined on the region $D:=A \times [c,\infty)\subseteq \mathbb R^2,$ and suppose
$$\int_c^{\infty} f(x,t)dt$$
exists for all $x \in A.$ Call this improper integral uniformly Cauchy if, for all $\epsilon >0,$ there exists $M>c$ ... |
H: Why the space of Lipschitz functions from $[0,1]$ to $\mathbb{R}$ with the uniform norm
is not a Banach space ?
We have to find a Cauchy sequence of Lipschitz functions $(f_n)_{n>0}$ such that this sequence does not converge to a Lipschitz function $f$ (for the norm $\vert \vert g \vert \vert_{\infty} = \sup_{t\in[... |
H: Find three primes numbers
Is there any integer $p$ such that $p > 1$ and all three numbers $p$, $p+2$ and $p+4$ are prime numbers? If there are such triples, prove that you have all of them; if there are no such triples, prove why not.
Apart from this, I am given the following information: if $a$ is an integer, the... |
H: Projecting the rectangle onto the plane
Find the area of the parallelogram obtained by projecting the rectangle $(0,0,0),(0,1,0),(2,0,0),(2,1,0)$ onto the plane $x + 4y - 5z = 4$ along $\vec{i} + \vec{j} + \vec{k}$
Now, first I don't understand the question. How does one project a rectangle onto the plane along som... |
H: Prove integral inequality: $\int_{0}^{\frac{\pi}{2}}e^{\sin x}\,dx\geq\frac{\pi}{2}(e-1)$
I am trying to prove $$\int_{0}^{\frac{\pi}{2}}e^{\sin x}\,dx\geq\frac{\pi}{2}(e-1)$$
I found the Taylor series of $e^{\sin x}$ then approximated $\sin x$ as $\frac{2}{\pi}x$. I have no idea what to do next; any suggestions or... |
H: Discounting and Interest
The amount of interest earned on A for one year is 336, while the
equivalent amount of discount is 300. Find A.
Correct answer: $A=2800$
My work:
From the interest, I get that $A(1+i)-A = 336 \iff Ai = 336$
However, I'm not sure how to interpret the discount part. It's clearly not what I ... |
H: Baby Rudin Theorem 3.10(b)
Baby Rudin Theorem 3.10(b):
If $K_n$ is a sequence of compact sets in $\boldsymbol X$ such that $K_n \supset K_{n+1}$ ($n=1,2,3,\dots$) and if
$$
\lim_{n\rightarrow\infty} \operatorname {diam} K_n=0
$$
then $\bigcap_{n=1}^{\infty}K_n$ consists of exactly one point.
Do we require the sets ... |
H: Rotational Volume of $y^2 = \frac{x^3}{2a-x}$ around asymptote $x = 2a$
My approach is to use a shell method over the range $[0, 2a]$. One cylinder will be $C = 2\pi xy \ dx$. Let us only work with the positive quadrant and multiply by two for ease:
https://www.desmos.com/calculator/pvuquz8orz
then the volume is: $... |
H: Question about a continuous function such that $f(x^2)=f(x)$.
Let $f:[0,1]\rightarrow\mathbb{R}$ be a continuous function such that $f(x^{2})=f(x)$, $\forall x\in [0,1]$.
Prove that $f(x^{2n})=f(x)$, $\forall x\in [0,1], \ \ n\in\mathbb{N}$.
Honestly, I don't know how to proceed. I though about induction but I don'... |
H: The Lipschitz condition in the Deformation Lemma
Is it true that if $\varphi$ is a $C^2$ functional on a Hilbert space $X$ and $||\nabla \varphi(u)||, ||\nabla \varphi(v)|| \geq 2\varepsilon$ then
$$
\left| \frac{||\nabla \varphi(v)||}{||\nabla \varphi(u)||} - \frac{||\nabla \varphi(u)||}{||\nabla \varphi(v)||}\rig... |
H: given a density function of $f_x = x^{-n}$ how to compute $\operatorname {var}(X)$?
Say that you are given a density function of $f_x = \frac{1}{5}x^{-n}$ for some $n$ with $x \in [5,\infty]$, how to compute $\operatorname {var}(X)$? Is the gamma function of factorial used in this question?
AI: Remember that $Var(x... |
H: A nilpotent linear operator could be represented by a upper triangular matrix with diagonal entires zero
I am trying to prove the proposition that:
A linear operator $T$ is nilpotent if there exists some positive natural number $k$ such that $T^k=0$. If $T$ is nilpotent, then there is a basis of $V$ such that the m... |
H: Linear Combination to equations by Matrices
Let there be two equations such that a+b=0
a-b=1. Now I have just starting learning linear algebra, so from my understanding 0 is a constant vector with components a and b, 1 is also a constant vector with components a and -b. Now when we convert this into matrix form we ... |
H: How can i evaluate $\int _0^1\frac{\ln ^2\left(x\right)\ln \left(1-x\right)\ln \left(1+x\right)}{x}\:dx$
I want to evaluate $$\int _0^1\frac{\ln ^2\left(x\right)\ln \left(1-x\right)\ln \left(1+x\right)}{x}\:dx$$
I tried integration by parts and shape it in a way that i could expand either $\ln$ terms.
$$-\int _0^1\... |
H: Can a vortex vector field be conservative?
For the following vortex vector field
$$F(x,y)=\left(\frac{2xy}{(x^2+y^2)^2},\frac{y^2-x^2}{(x^2+y^2)^2}\right)$$
If we apply the extended Green's Theorem for an arbitrary simple closed curve $C$ that doesn't pass through the origin and with a circular "hole" $C'$ with rad... |
H: Is it true that every curve defined by a graph of polynomial is regular?
A Regular point on a curve defined as the following
Def.1 (Singular and Regular points of planar curves)
Suppose that $S$ is a curve in $\mathbb{R}^2$ and $a\in S$. If, for every $r>0,S\cap B(a;r)$ is not a $C^1$ graph $~~~($of a function $(a... |
H: Baby Rudin Chapter $4$ Exercise $1$
Suppose $f$ is a real function which satisfies $\lim\limits_{ h\to0}[f(x+h)-f(x-h)]=0$ for every $x$.
Does this imply that $f$ is continuous? Through trial and error I realized this is false but I am not sure that my reasoning is correct. I thought that one could simply consider... |
H: Properties needed to define Derivatives on Topological space
I just started learning topology and was curious about defining derivatives on general topological spaces.
Since we can define continuous functions on Topological spaces, my question is what additional properties one would need to define derivatives on To... |
H: Textbooks to read up on ordinal numbers, or anything about counting beyond $|\mathbb{N}|$
Preferably aimed at undergraduates or something an undergraduate can at least understand.
Thank you
AI: One of the best books in my opinion on Set Theory for an undergraduate is Classic Set Theory by Derek Goldrei. The author ... |
H: The number of n digit numbers which consists of the digits 1 &2 only if each digit is to be used atleast once is equal to 510 then n is equal to:
I tried to solve this questions many times but couldn't even get close to the answer, please help. The answer to this question is 9.
AI: Just solve for $n$ in $2^n-2=510$... |
H: If $n \in \mathbb N$, find $\sum(-1)^r \binom{n}{r}\left(\frac {1} {2^r}+\frac {3^r} {2^{2r}}+\frac {7^r} {2^{3r}} + \cdots \text{m terms}\right)$
If $n \in \mathbb N$, find$$\sum(-1)^r \binom{n}{r}\left(\frac {1} {2^r}+\frac {3^r} {2^{2r}}+\frac {7^r} {2^{3r}} + \frac {15^r}{2^{4r}} + \cdots \text{upto m terms}\r... |
H: How to express block diagonal matrices in mathematical notation?
Suppose I have a block diagonal matrix that looks like this
$$C = \begin{bmatrix} A & 0\\ 0 & B\end{bmatrix}$$
then I would write $C = \mbox{diag} (A, B)$, however, what if I have a block diagonal matrix that is mirrored, i.e.
$$D = \begin{bmatrix} 0 ... |
H: Formula for calculating weighting percentages given individual grades and final grade
I was wondering what the general formula is for calculating the weight (%) of each category in a gradebook given the individual grade for each category and the final grade. For example, lets say category 1 was a 96%, category 2 wa... |
H: Derivative of vector
I have troble to understand derivative of vector.
In scalar case $y=f(x)$, the follow is truth
$$\frac{dy}{dx}=\left(\frac{dx}{dy}\right)^{-1}$$
In vector case, $\mathbf{y}=(y_1,y_2)$, $\mathbf{x}=(x_1,x_2)$
$\mathbf{y}=f(\mathbf{x})$
$$\frac{d\mathbf{y}}{d\mathbf{x}}=\left(\frac{d\mathbf{x}}{d... |
H: Finding a PDF of a sum of random variables
The following is given:
$$X,Y\ \textit{are independent}$$
$$X∼exp(1),Y∼exp(2)$$
$$Z=e^{−X}+e^{−2Y}$$
And I want to find:
$$f_Z(1)=?$$
As a part of my solution I do the following:
$$Z=U+V$$ where $$U=e^{−X},V=e^{−2Y}$$
Using one dimensional transformation we get:
$$U∼Uni(0,... |
H: $\mathbb{E}[|Z|]$ with $Z=X-Y$
If $X\perp Y\sim \operatorname{Exp}(1)$ and $Z=2X-Y$, how can I compute $\mathbb{E}[|2X-Y|]$?
Generally speaking, what is $\mathbb{E}[|X+Y|]$ and $\mathbb{E}[|X-Y|]$?
In our case, knowing that $X\in [0,+\infty)$, is it correct saying that $\mathbb{E}[|2X-Y|]\leq |2|\mathbb{E}[|X|]+|-1... |
H: are all points of outside of Mandelbrot set connected
Mandelbrot set is connected. That is to say within a mandelbrot set for any pair of points there is a path within the set, connecting these points.
What abouthe set of all other points? Is there a pair of points that cannot be connected without going through the... |
H: How to quickly compute matrix derivatives
I have studied the mathematics behind autoencoders. In a proof, a minimization problem is rewritten several times by taking the derivative regarding matrices/ vectors.
Notation: $W_1$, $W_2$ are matrices. $b_1$, $b_2$ and $x$ are vectors.
The first problem in example is :
$... |
H: Why can't one cancel partial differentials?
I have a question about the below formula:
$$\frac{dz}{ds} = \frac{dz}{dx} \cdot \frac{dx}{ds} + \frac{dz}{dy} \cdot \frac{dy}{ds}$$
Ok. I understand what this means. Small change of s makes small change of x and y and thus the sum makes small change of z.
However, I si... |
H: Variant of Quotient Metric is an Ultrametric
Let $(X,d)$ be a metric space and define an equivalence relation $\sim$ on $X$. Then
$$
d'([x],[y]):= \inf\{d(x',y'): x' \in [x],\, y' \in [y]\},
$$
may fail the triangle inequality, where $[x]$ is the equivalence class of $x\in X$ under $\sim$ (and similarly for $[y]$)... |
H: Maximum volume of a cuboid box
A rectangular sheet of a fixed perimeter with sides having their lengths in the ratio 8: 15 is converted into an open rectangular box by folding after removing squares of equal areas from all four corners. If the total area of removed squares is 100, the resulting box has maximum volu... |
H: Asymptotics for $\sum_{p
I would like what is the asymptotic behaviour of
$$f(x) := \sum_{p<x} \log\left(\log(p)\right),$$
as $x \to +\infty$,
where the sum runs over prime numbers. It is well-known, by the prime number theorem, that $\sum_{p<x} \log(p) \sim x$. But here the extra log would require to study $\prod_... |
H: Number of ways to distribute ice cream to children.
A certain school of 10 children is visiting the local ice-cream factory to see how ice-cream is made. After the demonstration, the factory has 15 scoops of vanilla ice-cream and 2 scoops of chocolate ice-cream to distribute to the kids. How many ways can the ice-... |
H: Why does the plot of $f(x)=|\cos x|-|\sin x|$ look almost piecewise linear?
I recently stumbled upon an interesting plot that I - even until today - could not quite explain:
It's the plot of $f(x) = \lvert \cos(x) \rvert - \lvert \sin(x) \rvert$. I mean this is almost piecewise linear...
I tried to derive this sha... |
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