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H: Prove that $\exists !c \in \mathbb{R} \exists ! x \in \mathbb{R} (x^2 + 3x + c = 0)$
This is an exercise from Velleman's "How To Prove It". I am struggling with how to finish the final part of the uniqueness proof, so any hints would be appreciated!
a. Prove that there is a unique real number $c$ such that there ... |
H: All integer solutions of $x^3-y^3=2020$.
Find all integer pairs $(x,y)$ satisfying $$x^3-y^3=2020\,.$$
First,
$x^3-y^3=(x-y)(x^2+xy+y^2)=2020$
and $2020=2^2\cdot 5 \cdot 101$.
But what next? Can it be worked out by using modulo? Or how?
Any idea? Thanks in advance.
AI: There are no solutions, because $x^3,y^3\eq... |
H: If $\lambda \not= 0$ and $\lambda x = 0$, then $x = 0$
I am trying to use the definition of vector spaces to prove that, if $\lambda \not= 0$ and $\lambda x = 0$, then $x = 0$.
One proof I have seen begins as follows:
$\lambda 0 = 0$ for each $\lambda$ since $\lambda 0 = \lambda(0, + 0) = \lambda 0 + \lambda 0$.
... |
H: power series of $f(z)=\frac{1}{z^2+1}$ at $1$
In an exercise I am asked to find the power series of the function $f(z)=\frac{1}{z^2+1}$ centered at the point $1$.
My approach:
$$ \begin{align}
\frac{1}{z^2+1} & = \frac{1}{z^2+ 2 - 1}
\\
\\& = \frac{1}{2}\cdot\frac{1}{1-\frac{1}{2}(1-z^2)}
\\
\\& = \frac{1}{2} \su... |
H: Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$
Prove that $0$ is the only $2\pi$-periodic solution of $\ddot{x}+3x+x^3=0$.
I don't know how to deal with this non-linear differential equation. I tried to consider $\ddot{x}(t+2\pi)+3x(t+2\pi)+x^3(t+2\pi)=0$ but with no success...
I need ... |
H: Evaluate limit of $\lim_{x\to 1} \frac{e^{(n!)^x}-e^{n!}}{x-1}$
$$\lim_{x\to 1} \frac{e^{(n!)^x}-e^{n!}}{x-1}$$
Since it is $e^{(n!)^x}$ and not $e^{x×(n!)}$ so I can't using the general form for $\frac{e^x-1}{x}$ as $x\to 0$ could somebody help? It'd better if without using l'hopital rules
AI: Let
$$f(x)=e^{(n!)... |
H: Solving system of linear equations using orthogonal matrix
I'm given the following matrix:
$$
A = \begin{pmatrix}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}} \\
0 & 1 & 0 \\
\frac{1}{\sqrt{2}} & 0 & \frac{1}{\sqrt{2}}
\end{pmatrix}
$$
We're asked to determine if this matrix is orthogonal. I did this successfull... |
H: Show that each $\hat x$ is a member of $C_0(\Delta)$
Suppose $A$ is a commutative Banach algebra without a unit, let $\Delta$ be the set of all complex homomorphisms of $A$ which are not identically $0$. Each $x\in A$ defines a function $\hat x$ on $\Delta$, given by
$$\hat x(h)=h(x)\quad (h\in\Delta).$$
If we give... |
H: Getting value of x by differentiating a given equation
If we consider an equation $x=2x^2,$ we find that the values of $x$ that solve this equation are $0$ and $1/2$. Now, if we differentiate this equation on both sides with respect to $x,$ we get $1=4x.$ Now, I know that it is wrong to say that the value of $x=1/4... |
H: Isomorphism between two partially ordered sets.
I want to define an Isomorphism $\phi:\left\langle [n]\times[m],\leq_{Lex}\right\rangle \rightarrow\left\langle [n\cdot m],\leq\right\rangle $
I understand how to write down this isomorphism by hand: lets say $n,m = 2$, then we define:
$(0,0) \rightarrow 0$
$(0,1) \ri... |
H: Find parametric equations for the midpoint $P$ of the ladder
The following problem appears at MIT OCW Course 18.02 multivariable calculus.
The top extremity of a ladder of length $L$ rests against a vertical
wall, while the bottom is being pulled away.
Find parametric equations for the midpoint $P$ of the ladde... |
H: Is there a matrix $A \neq 0$ such that $A\in F^{2\times 2}$ and $A^2=0$?
Any hints? i don't know how to disprove the statement I looked at the multiplication with parameters and looked at the different cases but there were not enough information
$F$ is a field
AI: Take the field $F_2$,
then consider the matrix,
$... |
H: Convergence of a specific series
The assignment that I am having trouble with is as follows:
a) Use the fact that $\lim_{n\rightarrow\infty}n^3\left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)=\frac{1}{6}$ to show that
$$\sum_{n=0}^\infty\left(\frac 1n-\sin\left(\frac 1n\right)\right)$$
converges.
b) Determine ... |
H: Is $\int_{-a}^{a}fg^{\prime}+\int_{-a}^{a}fg^{\prime\prime}=\int_{-a}^{a}f^{\prime}g+\int_{-a}^{a}f^{\prime\prime}g$ true?
Using integration by parts, I have been trying to figure out if $$\int_{-a}^{a}f(x)g^{\prime}(x)dx+\int_{-a}^{a}f(x)g^{\prime\prime}(x)dx=\int_{-a}^{a}f^{\prime}(x)g(x)dx+\int_{-a}^{a}f^{\prime... |
H: Showing that the intersection of a cylinder and a plane is an ellipse
One of the questions in my homework was:
"Show that the curve $\vec{r}(t)=\cos t \vec{i}+\sin t \vec{j}+(1-\cos t)\vec{k}$ is an ellipse by showing that it is the intersection of a cylinder and a plane. Find equations for the cylinder and the pla... |
H: Count the number of dimes?
You are given a bag with 100 coins. The bag only has pennies, dimes and half-dollars. The bag has at least one of each coin. The total value of the coins in the bag is worth $5, How many dimes are in the bag?
AI: Let $P$ be the number of pennies, $D$ the number of dimes, and $H$ the n... |
H: Expected hitting times for simple random walk on a hypercube
Setup
In an $n$-dimensional hypercube $C_n = \{0,1\}^n$, we define the Hamming distance of two vertices $d(A,B)$ to be the number of coordinates in which they differ. (e.g. $d((0,0,1), (1,0,1)) = 2$.)
A simple random walk on the vertices of $C_n$ has $1/n... |
H: Connectedness and complexity in Polish spaces
I was wondering: How complex can connected subsets of Polish spaces be?
Are there connected non-Borel subsets of a Polish space? Given $X$ Polish space (not totally disconnected), does it have proper analytic ( $\boldsymbol{\Sigma}_1^1(X)\setminus \boldsymbol{\Delta}_1^... |
H: Can two planes in $\mathbb{R}^4$ not intersect and also not be parallel?
Intuitively, I was thinking we can approach this the same way as two lines in 3D space, where they may not intersect nor be parallel.
I think I've come up with the example of ${(0,x,y,0):x,y \in R},{(1,x,0,y):x,y \in R}$, where the planes are ... |
H: How to prove this series Cauchy product?
Let the first series be $\sum_{k=0}^{\infty} u_{k}(x-a)^{k}$.
Let the second series be $\sum_{k=0}^{\infty} v_{k}(x-a)^{k}$
Their convergence areas are $R_{1}$ and $R_{2}$.
I don't know how to prove that $f(x)*g(x)=\sum_{k=0}^{\infty} w_{k}(x-a)^{k}$, where $w_{k}=\sum_{k} u... |
H: how to construct radical function around a cusp?
I read this construction from the paper "Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one": Let M be a rank 1 locally symmetric space. On each cusp $\hat{M}$ of $M$ there exists a proper function $r: \hat{M} \to R_+$ such... |
H: Sum of product of NB combinations
While trying to solve the PMF of two independent NB random variables, I end up with a summation of the product of two combinations:
$$\sum_{j=0}^{k} {j+r-1 \choose j} {k-j+s-1 \choose k-j} $$
According to the textbook, it should be equal to
$$ {k+r+s-1 \choose k} $$
But how can the... |
H: Question about $x \sin(1/x)$ at $x = 0$
I know that $f(x) = \sin(1/x)x$ takes the value $0$ at $x=0$. What else can we say about the function $f(x)$ at $x=0$? More specifically, is $f$ continuous at $x=0$? Is it even differentiable at $x=0$? Thank you!
AI: The function $f(x)= x \sin(1/x)$ is not $0$ at $x=0$ as it... |
H: Exponential map on Lie groups being a diffeomorphism
Consider a Lie group $G$ and let it's Lie algebra be $\mathfrak{g}$. Let the exponential map be denoted by $\exp: \mathfrak{g} \to G$.
Given any $g \in G$, does there exist an open set $O \subset \mathfrak{g}$ with $g \in \exp(O)$ such that the exponential map r... |
H: connectedness of A={$(x,y):x \in \mathbb{Q}\,$ or$ \, y \in \mathbb{Q}$}
i have the following problem about connectedness, prove that in the Euclidean plane
A={$(x,y):x \in \mathbb{Q}\,$ or$ \, y \in \mathbb{Q}$} is connected
I have tried in several ways, but it causes me a problem, that some of the coordinates hav... |
H: Finding the subgroups of cyclic groups
Currently trying to work out all of the subgroups of $X \times Y$, $X = C_3 = \langle x \rangle$ and $Y = C_3 = \langle y \rangle$.
I know that $X = \{1, x, x^2\}$ and $Y = \{1, y, y^2\}$. I also know that
$$X \times Y = \{(1,1), (1,y), (1,y^2), (x,1), (x,y), (x,y^2), (x^2, ... |
H: find the dim S.
Problem taken from Apostol calculas Volume $2$ page No: $13$ books
Let $P$ denote the linear space of all real polynomials of degree $\le n$, where $n$ is fixed. let $ S$ denote the set of all polynomials $f$ in $P$, satisfying the condition given below . find the dim S.
$1.$$f$ is even.
$2.$ $f$ i... |
H: Saddle point approximation gives a null result
So I want to compute the following integral $$I=\int_0^1 x\sqrt{1-x}\exp \left(a^2x^2\right) dx$$
where $a>>1$. If we try to do a Saddle point approximation
\begin{align}
I&=\int_0^1 f(x)\exp \left(a^2g(x)\right)\\
&\approx f(x_0)\exp\left(a^2g(x_0)\right)\sqrt{\fr... |
H: What am I doing (simple uniform converge problem) $f_n(x)=n^2x^2e^{-xn}$
consider the function $f_n(x)=n^2x^2e^{-xn}$ I am asked if it uniform converge on $A=(a,\infty) \quad a>0$
So it easy to see that in converge to $0$, but when I wanted to check uniform converge I noticed that the function getting maximum on $x... |
H: Not understanding unexplained notation
I am reading a set of lecture notes on Functional Analysis and I have come across a not introduced notation when reading a corollary of Hahn-Banach Extension Theorem. Could anybody please explain what does $S_{X^*}$ represent?
The corollary is the following:
Let $X$ be a norme... |
H: Why is probability sometimes calculated using ordered pairs of outcomes rather than unordered pairs?
For example, if we are tossing two coins, where each coin falls on either heads ($H$) or tails ($T$), we have the following possible outcomes: $\{H, H \}$, $\{H, T \}$, $\{T, T \}$.
However, when solving some exerci... |
H: Proof for pumping lemma for new kind of CFL
I have a context-free grammar $(V,\Sigma,R,S)$ that is defined by the condition that every production in $R$ has to be on one of the following two forms:
$A\to uBv$ where $A,B\in V$ and $u,v\in\Sigma^*$
and
$A\to u$ where $A\in V$ and $u\in\Sigma^*$
The pumping lemma for ... |
H: Homeomorphism between two circles
I'm trying to understand how homeomorphism works and have found a result that I find somehow distant from my intuition.
Consider the functions:
$$f(x)=\left\{\begin{matrix}
x-5 & x \in 2\mathbb{S}^1+5\\
x& x \in \mathbb{S}^1
\end{matrix}\right.$$
$$g(x)=\left\{\begin{matrix}
x+5... |
H: How is this function injective?
I'm currently studying Galois Theory and I came across this theorem.
Theorem Let $E$ be a field, $p(x)\in E[x]$ an irreducible polynomial of degree $d$ and $I = \langle p(x) \rangle$ the ideal generated by $p(x)$. Then $E[x]/I$ is an extension field of $E$.
The proof first uses the f... |
H: Confusion about the Definition of Smooth Functions on a Manifold
I am slightly confused about the definition of smooth functions on a smooth manifold given in An Introduction to Manifolds by Loring Tu (Second Edition, page no. 59). The definition is given below.
I am confused because I don't see how $f\circ \phi^{... |
H: transform a 2 dimensional ode 1 system to 2nd order one dimension system
Given a matrix $M$ of $2 \times 2$, and an ode: $$y'=My$$
let $y=(v_1,v_2)$.
find a second order ode such $v_1,v_2$ are solutions.
AI: We have $y'=My$. I consider $M = \begin{bmatrix}m_{11} &m_{12} \\ m_{21} &m_{22}\end{bmatrix}$, and the solu... |
H: Matrix equation solution (what condition a matrix needs to fulfill to make the equation possible)
Let $A$ be an $m \times n$ matrix such that $\mathrm{rank}(A) < n$. If one were to set the equation
\begin{equation*}
A Z = B A,
\end{equation*}
what is the condition that matrix $Z$ needs to fulfill such that the abo... |
H: Continuous functions from compact Hausdorff spaces to the interval
Let $X$ be a compact Hausdorff space and let $C(X,I)$ be the set of all continuous functions from $X$ into the closed interval $[0,1]$. If we equip $C(X,I)$ with the the topology of uniform convergence is $C(X,I)$ compact? I'm inclined to think not,... |
H: Fibonacci Rabbit's variation
Okay so I am trying to understand modifications to the famous fibonacci rabbit problems so I can make a generalized website for it as a pet project, where people just need to input paramaters and it will generate the tree like structure and the recurrence relation if possible.
One pair... |
H: Prove that the collection is a basis that generates standard topology of the real line
Attempt:
We can use a Lemma in munkres to show this. That is, if for any open set (in usual topology) we can always find some member of collection $\mathscr{C}$ inside the open set. In other words,
Let $(c,d)$ be any open set in... |
H: Finding the mean sales price
everyone.
I have the following problem: a factory produce valves, with 20% chance of a given valve be broken. The valves are sold in boxes, containing ten valves in each box. If no broken valve is found, then they sell the box for 10 dollars. With one broken valve, the box costs 8 dolla... |
H: Eigenvectors for eigenvalue with multiplicity $\mu = 2$
I'm looking for a way to determine linearly independent eigenvectors if an eigenvalue has a multiplicity of e.g. $2$. I've looked for this online but cannot really seem to find a satisfying answer to the question. Given is a matrix A:
$$ A = \begin{pmatrix} 1 ... |
H: Find the saddle point of $F(x_1,x_2,x_3,y_1,y_2,y_3)=(x_1-2x_2+x_3)y_1+(2x_1-2x_3)y_2+(-x_1+x_2)y_3$
Finding the saddle points of $F(x_1,x_2,x_3,y_1,y_2,y_3)=(x_1-2x_2+x_3)y_1+(2x_1-2x_3)y_2$+$(-x_1+x_2)y_3$ subject to the constraints $x_1+x_2+x_3=1, y_1+y_2+y_3=1$. Show that the saddle point is $x=(\frac{1/3}{1/3}... |
H: Basic question about vector subspaces
As part of some question in a more advanced course in linear algebra I'm using this claim which I'm pretty sure is true but a little confused about how to justify it.
Given a vector space $V$, $\dim V=n \in \mathbb{N} $, and a subspace $W \subset V$ (meaning $\dim V > \dim W$)... |
H: How to evaluate $\int_{|z|=2} \frac{|z| e^z}{z^2} dz$?
The only thing that I know is that the result should be $\displaystyle 4\pi i$. Could you give me a hint/suggestion?
I thought about using the Residue's theorem, where if $\displaystyle f(z) = \frac{|z|e^z}{z^2}$, it has a 2nd order pole in $0$, but after that,... |
H: Elements of quotient ring $\mathbb{Z}_3[x]/I$ being represented as $ax^2 + bx + c + I$ by Euclidean Algorithm?
I came upon this problem in http://sites.millersville.edu/bikenaga/abstract-algebra-1/quotient-rings-of-polynomial-rings/quotient-rings-of-polynomial-rings.pdf, but I don't understand how he applied the Eu... |
H: Inequality with measure and weight
Let $(M, \mu)$ be a measure space and $g\in L_{loc}^{1}(M)$ be a positive non-zero function. How can one show that for $f$ measurable and $a>0$ $$\mu\left\{|f|> \frac{1}{a}\right\}\leq a^{2}\frac{\int_{M}{f^{2}gd\mu}}{\int_{M}{gd\mu}}?$$
Thanks in advance!
AI: This is not true. Ta... |
H: Use of the $\subset$ and $\subseteq$ symbols in the definition of a power set and re-defining the power set with these symbols.
In my Mathematics textbook, the definition of the power set of a given set is given as follows :
$$P(A) = \{X : X \subseteq A \}$$
Now, this is used to say that the power set of a given se... |
H: An increasing arithmetic sequence of positive integers has $ {a_{19}}=20$ and ${a_{a_{20}}}=22$ . Find ${a_{2019}}$
Question:- An increasing arithmetic sequence of positive integers has $ {a_{19}}=20$ and ${a_{a_{20}}}=22$ . Find ${a_{2019}}$
I Found this question on a blogspot.On Seeing the statement of question i... |
H: To which values of $a$ is the following an inner product space?
Given the following inner space over $\mathbb{R}^{2}$:
$$
\left< \left(\begin{pmatrix}
x_{1}\\
x_{2}
\end{pmatrix} ,\begin{pmatrix}
y_{1}\\
y_{2}
\end{pmatrix}\right) \right> =x_{1} y_{1} -3x_{1} y_{2} \ -3x_{2} y_{1} +ax_{2} y_{2} \
$$
To whic... |
H: Relation between an increasing function and its Riemann integral
I have recently been trying some questions on Riemann integral. Got stuck in one of the problems which says:
Suppose $f$ is an increasing real-valued function on [$0$,$\infty$] with $f$($x$) $\gt$ $0$ for all $x$ and let $g$($x$)=$\frac{1}{x}$$\int_{0... |
H: Limit of integrals where both bounds and integrand depend on $n$
I want to find the limit
$$\lim_{n\rightarrow\infty}\int_{[0,n]}\left(1+\frac{x}{n}\right)^n e^{-2x} \, d\lambda(x)$$
I have noted the following: It is well known that $(1+\frac{x}{n})^n$ converges to $\exp(x)$ in the limit. So I'm certain the limit ... |
H: Does null average against every smooth function implies independence?
Are these assertions equivalent?
$f:\mathbb{S}^1\times \mathbb{S}^1\to\mathbb{C}$ is such that
$$ \int_0^{2\pi}\int_0^{2\pi}f(x,y)\psi(y)dydx=0$$
for all $\psi\in C^{\infty}(\mathbb{S}^1).$
$f:\mathbb{S}^1\times \mathbb{S}^1\to\mathbb{C}$ is su... |
H: Is it correct to state that $\langle x(t),x(t)\rangle' = 2\langle x'(t),x(t)\rangle$ for an arbitrary inner product?
Let $x:\textbf{R}\to\textbf{R}^{3}$ be a differentiable function, and let $r:\textbf{R}\to\textbf{R}$ be the function $r(t) = \|x(t)\|$, where $\|x\|$ denotes the length of $x$ as measured in the usu... |
H: An exponential distribution that represents time between events
You're responsible for maintaining four ATMs (E,W,N, and S). The time between failures for each ATM is exponentially distributed with mean time between failures 6 hours, 5 hours, 8 hours, and 8 hours, respectively. The ATMs can be serviced between 8 A.... |
H: Every integer $α>2$ can be expressed as $2a+3b$
I am confused about a homework problem I have, and don't really know where to begin. I need to prove this. Any idea of where I can start. I am not necessarily looking for a solution, but a place to begin. The statement is that
Show that every integer $α > 2$ can be wr... |
H: Why $f(x)=\Sigma_{n=0}^{\infty}\frac{x^n}{n\ln(n)}$ $\in c^n$
consider $f(x)=\sum_{n=0}^{\infty}\frac{x^n}{n\ln(n)}$ and $h(x)=\sum_{n=0}^{\infty}\frac{\sin(nx)}{1+n^4}$.
why $f(x)\in c^n$ in $[\frac{-1}{8},\frac{1}{9}]$ and $h(x)\in c^2$ in $\mathbb{R}$
I think h(x) is a periodic function and I think it can get de... |
H: Accuracy of low rank approximation
I am currently studying about randomized low-Rank Approximation of a matrix. In the problem's statement, given $m$ x $n$ $A$,it is referred that we want to minimize
$\|A-Q_{k}Q^{T}_{k}A\|$
and for this reason, we seek a $m$ x $l$ $Q$ with orthonormal columns that approximates well... |
H: Find $\lim_{n \to \infty} n^2 \int_{n}^{5n}\frac{x^3}{1+x^6}dx$
Question:Find the limit $\lim_{n \to \infty} n^2 \int_{n}^{5n}\frac{x^3}{1+x^6}dx$
I tried to convert it into $\frac{0}{0}$ indeterminate form,then applying
L'Hospital's rule but the expressions in numerator are not nice to integrate.I do not know oth... |
H: False Assumption in Rank Calculation - Flawed Argument?
I've gotten an exam back, and I think I've found somewhere that I can snag some marks, but I'm not sure about the quality of the argument presented in the question.
Let $B$ be a matrix that is obtained from changing the value of exactly one entry of a matrix $... |
H: Evaluate $\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$
Evaluate: $$\sum_{n=0}^{\infty} \frac{{\left(\left(n+1\right)\ln{2}\right)}^n}{2^n n!}$$
I am not sure where to start. The ${\left(n+1\right)}^n$ term is obnoxious as I can't split the fraction. Perhaps this can be turned into a... |
H: Negation of "Either X is true, or Y is true, but not both"
Negation of "Either X is true, or Y is true, but not both"
My attempt:
If seems that let X be true and Y be true, not X for X is false and not Y for Y is false. In order for the above statement to be True, we need:
The negation of both X and Y to be true: ... |
H: Let $s_k(n)$ denote number of digits in $(k+2)^n$ in base $k$ , evaluate $\lim_{n→∞}\frac{s_6(n)s_4(n)}{n^2}$.
Let $s_k(n)$ denote number of digits in $(k+2)^n$ in base $k$ , evaluate $\lim_{n→∞}\frac{s_6(n)s_4(n)}{n^2}$.
How to find out the number of digits in a particular base? Any hint for the problem is apprec... |
H: Evaluate $\lim_{n \to \infty} \sum_{j=0}^{n} \sum_{i=0}^j \frac{i^2+j^2}{n^4+ijn^2}$
I am asked to evaluate: $$\lim_{n \to \infty} \sum_{j=0}^{n} \sum_{i=0}^j \frac{i^2+j^2}{n^4+ijn^2}$$
I am not experienced with double summations, but I tried simplifying the expression above into:
$$\lim_{n \to \infty}\frac{1}{n^2... |
H: Helix around Helix around Circle
I'm trying to find the parametric equations for a helix around a helix around a circle (helix on helix on circle)
That is: I would like to start with a circle, add a helix around it and a helix around the helix.(See video)
I'm ok even if the second helix is not perfectly orthogonal ... |
H: Property of solution to a Cauchy problem
Let $I\subseteq \Bbb R$ an interval, let $w$ differentiable on $I$ such that $$w'(t)\le L|w(t)|\qquad \forall t\in I$$ for some $L>0.$ Let $t_0\in I$.
Prove that $$w(t_0)\le 0\implies w(t)\le0\quad \forall t>t_0$$ and $$w(t_0)\ge0\implies w(t)\ge0\quad \forall t<t_0$$
AI: $w... |
H: Odd-dimensional $\mathbb{R}$-vector space has a one-dimensional $\varphi$-invariant subspace
Let $V$ be an $\mathbb{R}$-vector space with odd dimension, and let $\varphi$ be an endomorphism on $V$. Show $V$ has a one-dimensional $\varphi$-invariant subspace.
I already know that $\ker f(\varphi)$ is a $\varphi$-in... |
H: Do the functions f(x), g(x), and h(x) exist so that f'(x)=g(x), g'(x)=h(x), and h'(x)=f(x), but none of the functions are multiples of each other?
I know that there are functions where, if you take the derivative of that function a multiple of n times, the $n^{th}$ derivative of that function is equal to the origin... |
H: Lebesgue measure of specific subset of $[0,1)$
For any $x\in [0,1)$ we assign its binary representation $(x_1,x_2,\dots,x_n,\dots)$ without $1$ in period. Let $\{n_k\}_{k=1}^{\infty}$ be some increasing sequence of natural numbers, $\{a_k\}_{k=1}^{\infty}$ be some sequence of $0$ and $1$. Let $A=\{x\in [0,1): x_{n_... |
H: Evaluating $\lim\limits_{n \to \infty} \sqrt[n]{\frac{n!}{\sum\limits_{m=1}^n m^m}}$
Evaluate: $$\lim_{n \to \infty} \sqrt[n]{\frac{n!}{\sum_{m=1}^n m^m}}$$
In case it's hard to read, that is the n-th root. I don't know how to evaluate this limit or know what the first step is... I believe that: $$\sum_{m=1}^n m^... |
H: Volume of tetrahedron.
Let $D$ be a tetrahedron with corners $(0,0,0), (2,0,0), (0,6,0), (0,0,4)$. Find the volume of $D$ by setting up a triple integral.
The equation of the plane containing these points is $$6x+2y+3z-12=0.$$ My question is how do I set up the bounds on the three integrals to get a volume? Thanks... |
H: constant polynomial
Can you give me a counter example of this ;
Let $ P: = P (x, y) $ is a polynomial function with positive coefficients : $$P (x, y)=\sum\limits_{i+j\leq n}^{n} a_{ij}x^{i}y^{j}), \quad a_{ij}\geq 0 ,\forall i,j$$ $ a, b \ge 0 $ (edit $a\neq b$) such that $ Q_1 (x) := P (x, b) $ is constant fu... |
H: Sine Parametric function exercise
Find the biggest negative value of $a$ , for which the maximum of
$f(x) =sin(24x+\frac{πa}{100})$ is at $x_0=π$
The answer is $a=-150$, but I don't understand the solving way. I would appreciate if you'd help me please
AI: Recall that the general solution of $\sin\theta = 1$ is:
$... |
H: Manipulating the product of the dot product of multiple vectors is producing a paradox
Unless otherwise indicated all defined vectors lie on the surface of the unit sphere (i.e their norm is 1).
Let's take the following expression:
$(V\cdot Z_1)(V\cdot Z_2) = V^TZ_1V^TZ_2$
Given that the dot product is commutative ... |
H: Evaluate the integral $\int_{|z-1|=2} \frac{1}{z^2 - 2i} dz$
It should be solved with the Cauchy's Integral Formula. The solution given is $\displaystyle \frac{\pi}{2} + i\frac{\pi}{2}$ but I obtained $0$.
I did this:
Let $C = \{|z-1|=2\}$
$$\displaystyle\int_C \frac{dz}{z^2-2i} = \int_C \frac{dz}{(z-(1+i))(z+(1+i... |
H: can we conclude $f(m)=\sqrt2(\log m)^{1/2}$?
If $$f(m)=\sup\{s: s^2/2\leq \log m\}$$
then can we conclude that $f(m)=\sqrt{2}(\log m)^{1/2}?$
AI: $$ \frac{s^2}{2}\leqslant \log m\iff |s|\leqslant \sqrt{2\log m} $$
Thus $f(m)=\sqrt{2\log m}$. |
H: find the solution of $2x\sin{\left(\frac{y}{x}\right)}dx+3y\cos{\left(\frac{y}{x}\right)}dx-3x\cos{\left(\frac{y}{x}\right)}dy=0$
A solution of the equation
$$2x\sin{\left(\frac{y}{x}\right)}dx+3y\cos{\left(\frac{y}{x}\right)}dx-3x\cos{\left(\frac{y}{x}\right)}dy=0$$
I know the answer $c\sin(3y/x)$
but I don't know... |
H: Concept of 2-variable function for operators on an $n$-dimensional inner product space
I'm reading the book "Finite-Dimensional Vector Spaces (2nd Ed)" by PR Halmos. The concept of a 2-variable function (or polynomial) for operators is introduced in Theorem 1 of Section 84 on page 171 in the following setting:
Two ... |
H: How do the Averages Work?
I am trying to figure out the average items sold per customer for the year 2019
I have multiple customers per day, who each have a random number of items. Sometimes a customer makes more than one purchase a day - about 6% of the time. In that case - all sales for that customer are consider... |
H: If two random variables follow the same distribution, does it mean X=Y?
The question here is:
Find the covariance of $X$ and $Y^{2}$ when
$X $~ $N(0,1)$
$Y $~ $N(0,1)$
Do $Cov(X, Y^{2})$ and $Cov(X, X^{2})$ have the same value?
Or, is $Cov(X, Y^{2})$ equal to $Cov(X, X^{2})$ ?
AI: This distinction is the entire po... |
H: Is the following statement true: $((A \Rightarrow B) \wedge (B \in C)) \Rightarrow A \in C$?
Just came up with problem that seems to be very basic, but I can't figure it out. I'm pretty certain it's not true. A, B are logical sentences and C is a set. Can you prove that below statement is true, or give a counterexa... |
H: $A$ is positive definite iff $\det(A_k) > 0$
Let $A$ be a symmetric $n \times n$ matrix, and $V = \mathbb{R}^n$. Define $\langle v,w \rangle = v^t A w$ with $v,w \in V$. Show that $A$ is positive definite iff $\det(A_k) > 0$ for $1 \leq k \leq n$ where $A_k$ is the $k \times k$-matrix which is the left upper $k \ti... |
H: How many paths are possible such that the area of the square cut off is exactly half the area of the entire square?
A vertical polygonal path will be formed by picking one point from each row of the four by four grid of points below (Fig. 1), and then connecting these points sequentially from top to bottom. The are... |
H: Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$
For $a,b,c>0; abc=1.$ Prove$:$ $$(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$$
My proof by SOS is ugly and hard if without computer$:$
$$\left( {a}^{2}+{b}^{2}+{c}^{2} \right) ^{3}-9\,abc \left( {a}^{3}+{b} ^{3}+{c}^{3} \right)$$
$$=\frac{1}{8}\, \left( b-c \right) ^{6}+{\frac ... |
H: How to solve $\frac{2s^s}{(1-s)^{1-s}}\leq 3$?
How to compute that
$$\sup_s\{s\geq 1: \log \frac{2s^s(1-s)}{(1-s)^s}\leq \log 3\}?$$
AI: If you consider the function $$f(s)=\frac{2s^s}{(1-s)^{1-s}}$$ you have
$$f'(s)=2 (1-s)^{s-1} s^s (\log (1-s)+\log (s)+2)$$ which is zero when
$$\log (1-s)+\log (s)+2=0 \implies s... |
H: $\alpha\in L$ algebraic over $K$ implies $\beta\in L$ algebraic over $K$ iff $\beta$ algebraic over $K(\alpha)$.
The fact that $\beta\in L$ algebraic over $K$ implies $\beta$ is algebraic over $K(\alpha)$ is obvious. Since $\alpha\in L$ is algebraic over $K$, we have $[K(\alpha):K]=n<\infty$. Assume $\beta$ is alge... |
H: Every nonzero homomorphism of a field to a ring is injective?
I've read the following theorem:
And am trying to understand what was done there. I think it is the following: We want to prove that
$$\text{Non zero hom of field to ring} \implies \text{hom is injective}$$
I think they used the contrapositive:
$$\ov... |
H: How to find the probability of countable infinite sets?
consider the set A = {1,2,3,...n}. If all subsets of A are equally likely to be chosen, what is the probability that a randomly selected subset of A contains 1?
AI: The total number of subsets of A is given as
$$n(\phi(A)) = 2^{n} $$
The way I get this is th... |
H: Closure of family of hypersurfaces over a punctured disk
Let $\Delta\subset \mathbb C$ be an open disk, $\Delta^*=\Delta\setminus\{0\}$ the punctured disk. Let $p:\mathcal{X}\to \Delta^*$ be a family of irreducible projective hypersurfaces of degree $d$. In other words, we have a diagram
$\require{AMScd}$
\begin{CD... |
H: Understanding the fulfillment of a condition required for applying the pasting lemma.
Here is the solution of the question (which asks us to prove that the relation of homotopy among maps $X \rightarrow Y$ is an equivalence relation):
My question is about the last part of proving transitivity. Why while applying... |
H: Is $(X,T)$ is door space?
A topological space $(X, T)$ is said to be a door space if every subset of $X$ is either an open set or a closed set (or both).
Is the following given statement is true/false ?
If$ X$ is an infinite set and $T$ is the finite-closed topology, then $(X, T)$ is a door space.
My attempt :
I t... |
H: Pigeonhole Problem with Subsets
Let $S$ be an arbitrary subset of $\{1, 2, ..., 99\}$ with $|S|=10$. Prove that there are two different subsets $A$ and $B$ (don't have to be disjoint) of $S$ so that
$$\text{the sum of all the elements in $A$} = \text{the sum of all the elements in $B$}$$
Ex. $S=\{1, 2, 3, 4, 5, 6, ... |
H: What is the sum of n terms 1.5.9+5.9.13+9.13.17…?
How to solve this?
My attempt is to write the $r^{th}$ term which is $$(4r-3)(4r+1)(4r+5)$$
Then let $p=4r+1$
The $r^{th}$ term becomes $$(p-4)(p)(p+4)=p^3-16p$$
Now we know summation $\sum p^3 =[\frac{n(n+1)}{2}]^2$ and $\sum p = \frac{n(n+1)}{2}$.
Therefore, $$\su... |
H: How do we denote the $0$ vector in a Quotient Space $V /W$?
Just $0$ or $0 + W$? Unfortunately my textbook has no mention of the notation for this, nor can I find clarification for this online.
AI: I think the most standard way to write it is just $0$, but you can also call it $0 + W$ or $[0]$. There is no fixed no... |
H: Closed form of $\int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx$
Can a closed form solution for the following integral be found:
$$\int_0^\infty \arctan^2 \left (\frac{2x}{1 + x^2} \right ) \, dx\,?$$
I have tried all the standard tricks such as integration by parts, various substitutions, and pa... |
H: Another proof by contradiction $\sqrt{2}$ is irrational.
There's a famous proof that $\sqrt{2}$ is irrational by assuming $\sqrt{2}=p/q$ for relatively prime $p$ and $q$ and then proving that this leads to $p$ and $q$ being both even which contradicts with them being coprime.
Now there's something I noticed that m... |
H: Simplify $4^3\sin^4(20^\circ)\sin^2(70^\circ)-4\sqrt3\sin^3(20^\circ)\sin(70^\circ)+3$
I was trying to solve a question where the two sides of a triangle were $$\frac{a\sin(20^\circ)}{\sin(70^\circ)}$$and $$\frac{a\sin(60^\circ)\sin(30^\circ)}{\sin(70^\circ)\sin(40^\circ)}$$ and the ange between them was $70^\circ$... |
H: $A \subset B$ be a faithfully flat extension of domains and $B$ is integrally closed then $A$ is also integrally closed.
Let $A \subset B$ be a faithfully flat extension of integral domains. If $B$ is integrally closed then I have to show that $A$ is also integrally closed.
Assuming $L,K$ be the field of fractions ... |
H: Determine if the statement is true or false. NBHM 2014 PhD question.
If $f$ and $g$ are continuous functions on $\mathbb{R}$ such that $\forall x \in \mathbb{R}$, $f(g(x))= g(f(x))$. If there exist $x_0 \in \mathbb{R}$ such that $f(f(x_0)) = g(g(x_0))$ then there exist $x_1 \in \mathbb{R}$ such that $f(x_1)= g(x_1)... |
H: How do concepts such as limits work in probability theory, as opposed to calculus?
When I am flipping a fair coin and say that as the number of trials approaches $\infty$ the number of heads approaches $50\%$, what do I really mean?
Intuitively, I would associate it with the concept of a limit, as used in calculus:... |
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