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H: slot machine, probability of stops
could you please tell me , am I correct with my answers in this task ?
slot machine has 3 wheels. Each wheel has 11 stops: a bar and the digits 0,1,2,...,9.
When the handle is pulled, the 3 wheels spin independently before coming to rest.
Find the probablity that the wheels stop o... |
H: Find side length of square with vertices on line $y=x+8$ and parabola $y=x^2$
Let ABCD be a square with the side AB lying on the line $y=x+8$. Suppose C, D lie on the parabola $x^2=y$. Find the possible values of the length of the side of the square.
I'm not sure how to start, I thought of taking the four vertice... |
H: What is the final intuition of Galois solvable groups and radical solutions?
At the end of the fundamental theorem of Galois theory, and after some intermediate moments of clarity realizing, for example, that the subfield lattice is built on fixed elements by different automorphisms, and that the corresponding subg... |
H: Correct way of proving if and only if statements?
When proving that
\begin{equation}
A\iff B
\end{equation}
We generally split the prof into two parts:
\begin{equation}
A\implies B \tag{1}
\end{equation}
\begin{equation}
\tag{2}
B\implies A
\end{equation}
In the cases I have seen, these proofs are completely inde... |
H: Finding the probability that each child gets at least 1 ball when we are distributing 5 DISTINCT balls among 4 children(who are distinct of course).
My approach:-
First, I found the total number of cases(the sample space). We can find that out from, $$4^5=1024$$
Now for favourable cases, I did this:-
I selected any... |
H: Problems with Universal Generalization
I’ve been studying universal generalization recently and, according to the textbooks, $\forall x Q(x)$ can be derived from $Q(a)$, if the variable $a$ is arbitrary. A variable is arbitrary, when it does not appear in any of the undischarged assumptions throughout the derivatio... |
H: Convergence of infinite series of log function
Check the convergence of the infinite series $\sum\limits\frac1{(\log n)^{3/2}}$.
I have tried to use comparison test but got no success.
AI: More generally, For $\sum\limits_{n=2}^\infty \frac{1}{(\operatorname{log}n)^p}$
We use Cauchy Condensation test,
\begin{eqnarr... |
H: Finding first term of arithmetic sequence given first three terms and no common difference
Assuming that the first three terms of an arithmetic sequence are $x, \frac{1}{x}, 1$ and $x<0$.
I seem to be unable to figure out what the first term is.
I know that $a_n = a_1+d(n-1)$ but how do we work out the common diffe... |
H: On the existence of a pullback
I’m not sure about my answer to the following problem:
Problem: Let $A,B$ and $C$ be sets, and let $f:A \rightarrow C$ and $g:B \rightarrow C$ be maps. Show that there exists a set $P$ and maps $h:P \rightarrow A$ and $k:P \rightarrow B$ such that $f \circ h = g \circ h$, and that fo... |
H: Units in $A = \mathbb{Z}_3[x]/(x^2+1)$
Let $A = \Bbb{Z}_3[x]/(x^2+1)$, the quotient ring by the ideal $(x^2+1)$. Which ones are units?
I did this question in a very boring way, merely listing all the possibilities and check. I cannot find an efficient way to find the unit. Is there any efficient way to do this?
AI:... |
H: Exponentiating similar Laplacians
Let $L_c$ be the $n\times n$ Laplacian matrix of the complete graph, and $L$ be the $n\times n$ Laplacian of any simple, connected graph possessing a vertex $k$ of maximum degree $d_k=n-1$. Clearly, there is a column of $L$ that looks like the corresponding column of $L_c$.
Now, co... |
H: UFD is also an ideal of a ring
Is it true that when a UFD is another ring $R$'s ideal, then ring $R$ is also a UFD?
I find an example but I'm not sure: the holomorphic ring $\mathcal O_x$, it's a UFD and the meromorphic ring $\mathcal M_x$, it's also a UFD (I guess but I'm not sure).
AI: If your ideal shared identi... |
H: Proof of the weak Mordell-Weil theorem and showing that torsion part of $A(k)$ is finite
Reading a proof of the weak Mordell-Weil theorem, I'm stuck somewhere. We have the following theorem :
Let $A$ be an abelian variety defined over a number field $k$ and $v$ a finite place of $k$ at which $A$ has good reduction... |
H: Under what circumstances does $\lvert \lambda\rvert \lvert \lvert x \rvert \rvert = \lvert \lvert y \rvert \rvert $ imply $\lambda x = y$
In a proof I saw, we made use of the fact that
for some $y = \lambda_{1}x_{1} + \lambda_{2}x_{2}$ , if we have
$\lvert \lambda_{1}\rvert \lvert \lvert x_{1} \rvert \rvert = \lver... |
H: 2 questions in Statement of Primary Decomposition Theorem and it's Corollary in Linear Algebra
I am self studying Linear Algebra from Hoffman Kunze and I have 2 questions in section 6.8 whose image I am adding->
Questions : (1) In the last paragraph how does one can deduce that $W_{i}'s $ are invariant under T?
(2... |
H: Prove or disprove: $A$ is a subgroup of $G$ if and only if $AA=A$.
I have a question about groups.
I need to prove or disprove:
Let $G$ be a group, and $A$ non-empty subset of $G$. $A$ is a subgroup of $G$ if and only if $AA=A$, where $AA=$ $\{a*a'|a,a' \in A\}$.
If $A$ is a subgroup then of course $AA=A$.
Ho... |
H: A matrix polynomial converging to $A^T$
Does there exist a sequence of matrices $A_i$ such that $$\sum^\infty_{i=0}A_iA^i=A^T$$
i tried inputing $A= 0, I$ but these don't give any substantial information except $$\sum^\infty_{i=0}A_i=I$$
if we take the classical example of a nilpotent matrix $\left[\begin{matrix}... |
H: If $\alpha,\beta,\gamma$ are the roots of $x^3+x+1=0$, then find the equation whose roots are: $(\alpha-\beta)^2,(\beta-\gamma)^2,(\gamma-\alpha)^2$
Question:
If $\alpha,\beta,\gamma$ are the roots of the equation, $x^3+x+1=0$, then find the equation whose roots are: $({\alpha}-{\beta})^2,({\beta}-{\gamma})^2,({\g... |
H: $\lim_{(x,y)\to(0,0)} \frac{x^2y^3}{x^4+2y^6}$ limit calculation
I have tried to write the limit using polar coordination. but I remain with a $cos(\theta)$ in the denominator. thanks for the help
AI: Along the path $y=0$
$$\lim_{(x,0)\to(0,0)} \frac{0}{x^4+0} = 0$$
but along the path $ y=x^{\frac{2}{3}}$
$$\lim_{(... |
H: Simple cardinal arithmetic
How can I see that $$2^{2^\lambda}>2^\lambda$$ ? Is it used here that $\lambda \geq 2^{\aleph_0}$ ?
The reference is here, pages 4 and 8.
AI: Let $X$ be a set of cardinality $2^\lambda$, then there is no bijection between $X$ and its powerset; that is, $2^{2^\lambda}=2^{|X|}>|X|=2^\lambda... |
H: Newton’s method to estimate a root of $f(x)=x^5-3x^2+1$
Taking $x_1=3$ as my initial estimate.
My work so far
The derivative of $f$, which is
$$f'(x)=\frac{d}{dx}(x^5-3x^2+1)=5x^4-6x$$
And, applying Newton's method to the table below.
\begin{array}{|c|c|c|c|}
\hline
x_n& f(x_n) & f'(x_n) & \frac{f(x_n)}{f'(x_n)} & ... |
H: Prove that $\lim(x_n)=0$ using definition of limit of sequences.
Let $x_n:=\dfrac{1}{\ln(n+1)}\space\forall \space n\in N$
Use the definition of limit of sequences to prove that $\lim(x_n)=0$
I tried to use $e^n>n+1\Rightarrow n>\ln(n+1)$ but that gives me $\dfrac{1}{n}<\dfrac{1}{ln(n+1)}$ which doesn't seem of muc... |
H: If $X$ is not full rank, are $X^TX$ or $X^TX + \lambda I_p$ invertible?
Suppose $X$ is a $n \times p$ matrix, where $rank(X) < p$. Since $X$ is not full rank, then it is not invertible.
I'm trying to understand whether functions of $X$ are invertible:
$X^TX$
$X^TX + \lambda I_p$ ($\lambda > 0$ is some scalar, and ... |
H: Question about Rudin (Principles of Math Analysis) theorem 7.26. Why does $Q_n \to 0$ uniformly?
In the proof of the Stone-Weierstrass theorem (7.26), Rudin claims $Q_n \to 0$ uniformly. Can someone explain why this is the case? I don't see how that immediately follows from the bound.
AI: Asserting that a sequence... |
H: Find function $f(x)$ whose expansion is $\sum_{k=0}^{+\infty}k^2x^k$.
I know the expansion of $\frac{1}{1-x}$ is $$1+x+x^2+\cdots+x^k+\cdots$$
So taking the derivative of $$\frac{\partial}{\partial{x}} \frac{1}{1-x}=\frac{1}{(1-x)^2}$$
And subsequently the expansion is (after taking derivative): $$1+2x+3x^2+4x^3+..... |
H: Jensen's inequality tells us variation of $x$ will increase the average value of $f(x)$?
This is from Boyd's convex optimization 6.4.1 stochastic robust approximation (p. 319):
"When the matrix $A$ is subject to variation, the vector $Ax$ will have more variation the larger $x$ is, and Jensen's inequality tells us ... |
H: $C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$
Let $S$ and $T$ be two arbitrary sets and consider the vector spaces $C(S)$ and $C(T)$
generated respectively by S and T. Show that
$C(S \times T)$ is isomorphic to $C(S) \otimes C(T).$
I am starting to read Werner Greub's Multilinear Algebra and I come across t... |
H: Measure 0 of set of points where $f$ is discontinuous
Let $f: [a,b] \rightarrow \mathbb{R}$ be an increasing function. Show that $\{x: f \text{ is discontinuous at $x$}\}$ has measure $0$. Hint: Show that $\{x: o(f,x) > \frac{1}{n}\}$ is finite for each integer $n$. Use the fact that given $f: [a,b] \rightarrow \ma... |
H: $K_0(C(\mathbb{T}^{n})) \cong \mathbb{Z}^{2^{n-1}}$
I am new to this website and I have a question.
I want to show that $K_0(C(\mathbb{T}^{n})) \cong \mathbb{Z}^{2^{n-1}}$ but first I want to show that $C(\mathbb{T}^{n}) \cong C(\mathbb{T} \rightarrow C(\mathbb{T}^{n-1}))$, where $C(\mathbb{T}^{n})$ is the continuo... |
H: How do you find the interval in which a parametric equation will be traced exactly once
I have been all over the internet and I can't find an answer to what seems like a simple question. I want to be able to find the interval for a parametric equation so that it is only traced once. My equations are:
\begin{align}
... |
H: Complex numbers limits
$\lim_\limits{z\to\infty} \sqrt{z-2i} - \sqrt{z-i} ,$ where z is complex no.
How to evaluate this?
I tried by assuming $z = x+iy$ and evaluated $z-2i = x+ i(y-2)$ and $z-1 = x + i(y-1)$ and after putting the value in the given question , I couldn't think of the next step at all
AI: Typically... |
H: Prove that $f_x(1,1)=-f_y(1,1)$
I have a function $f(x,y) \in C^1$. I need to prove that if $f(x,x)=1$ for any real number $x$, then $f_x(1,1)=-f_y(1,1)$.
I tried to prove it using the definition of the partial derivative, but couldn't figure it out.
$$ f_x(1,1)=\lim\limits_{h \to 0} \frac{f(1+h,1)-f(1,1)}{h} = \li... |
H: Trunk in a tree
Just to make things clear:
the author defines a trunk of a tree by formula in [...] in $4.3.1(a)$ in the snippet below.
Is it true that since $\eta_0\in T$ is maximal then for all $\nu\in T$ it holds vacuously that
$lg(\nu)\leq lg(\eta_0)$ ?
AI: No, the definition allows $\nu\in T$ with $\operatorna... |
H: Prove that the series $\sum_{n=1}^\infty {|a_n b_n|}$ and $\sum_{n=1}^\infty {(a_n + b_n)^2}$ converges
Prove that the series $\sum_{n=1}^\infty {|a_n b_n|}$ and $\sum_{n=1}^\infty {(a_n + b_n)^2}$ converges
If the series $\sum_{n=1}^\infty {a^2_n}$ and $\sum_{n=1}^\infty {b^2_n}$ converges.
The part $\sum_{n=1... |
H: Derivative of $y = \log_{\sqrt[3]{x}}(7)$.
Never dealt with a derivative of these type. My approach was $$y = \log_{\sqrt[3]{x}}(7) \iff 7 = (\sqrt[3]{x})^y.$$ Then,
$$\frac{d}{dx}(7) = \frac{d}{dx}\left(\sqrt[3]{x}\right)^y \Rightarrow (\sqrt[3]{x})^y = e^{\frac{y\ln(x)}{3}} $$
From here,
$0 = e^u\dfrac{du}{dx}$ a... |
H: What does w ∈ a, b* mean?
What does $w ∈ a, b^*$ mean?
The context is the language of an automation, which is $L=\{w∈ a, b^*|:|w|$ is even and the central symbols of $w$ are $aa\}$
I really don't understand what $w ∈ a, b^*$ mean, I think it means that $w$ can be $a$ or $b^*$ but that makes no sense because $w$ has... |
H: Show that $\lim_{x\to 0^+} xf'(x)=0$.
Suppose $f$ is a continuous function on $[0,1]$. Suppose $f$ is differentiable on $(0,1)$ and its derivative is continuous on $(0,1)$. Then is it true that
$$\lim_{x\to 0^{+}} xf'(x)=0 \ ?$$
I only thought about functions like $x^{a}$ for $a>0$. It seems to be true. Indeed, in ... |
H: Schur's lemma for finite-dimensional unitary representations
I am reading the book 'Representations of Linear Groups' by Rolf Berndt, and on page 19 they state the following theorem:
'Let $(\pi,\mathbb{C}^n)$ be a unitary matrix representation of a group $G$, i.e. $\pi(g) = A(g)$. Let $M\in GL(n,\mathbb{C}^n)$ be a... |
H: Simplify $\frac{d}{dt}\int_x^t f(t,y)dy$
I am trying to simplify $\frac{d}{dt}\int_x^t f(t,y)dy$ as a part of a proof.
I am somewhat confused on how I can proceed with this. Do I define a function $g(t,y)$ such that $\frac{\partial g}{\partial y} = f(x,t)$ and then say $\frac{d}{dt}\int_x^t f(t,y)dy = \frac{d}{dt}(... |
H: What is the error solving this problem about instantaneous rate of change?
I have the following problem:
A hot air balloon rising straight up from a level field is tracked by
a range finder $150$ meters from the liftoff point. At the moment that
the range finder’s elevation angle is $\frac{\pi}{4}$, the angle is
i... |
H: Show that $X= \ker((u-\lambda)^p) \oplus (u-\lambda)^p(X)$ if $u$ is compact.
Consider the following theorem in Murphy's book "C*-algebras and operator theory"
Can someone explain why the marked lines are true? I think this must be a matter of pure linear algebra but I can't find out the specifics. I.e. my questio... |
H: Find the function $f(x)$ knowing the volumen of the solid of revolution for $a \gt 1$
Find the function $f(x)$ knowing that the volumen of the solid of revolution of $y=f(x)$ around the x axis, for $a \gt 1$ is $V=a^2 -a$ , $f(x)$ continuos and $ f(x) \gt 0$
Basically what I tried to do was use the mean value theor... |
H: Assume that $f$ is holomorphic in $B(0,R)$ and that $|f(z)|\leq e^{-\frac{1}{|z|}} $ for all $0<|z|
Assume that $f$ is holomorphic in $B(0,R)$ and that $|f(z)|\leq e^{-\frac{1}{|z|}} $ for all $0<|z|<R$ then $f=0$
Not sure how to proceed, usually we try Liouville but $ e^{-\frac{1}{|z|}}$ has an essential singulari... |
H: Converging / Diverging sum with a constant power:
I need to prove this sum is diverges/convergent/conditional convergent , but I am pretty sure it is converging to a value:
$$\sum_{k=1}^{\infty} \frac{(1+\frac{1}{k})^{k^a}}{k!}$$
For some constant:
$a >0 , a \in \mathbb{R}$
I tried to prove it using:
$\frac{(1+\fra... |
H: Why a subspace is assumed instead of a vector space?
Why does the following theorem start with "Let $\{u_1, ..., u_p\}$ be an orthogonal basis for a subspace $W$ of $\mathbb{R}^n$" instead of "Let $\{u_1, ..., u_p\}$ be an orthogonal basis for a vector space $W$". In other words, what is the point for starting with... |
H: Suppose $A$ is Artinian and commutative with 1. If $J(A)M=M$, then $M=\{0\}$. $J(A)$ is the Jacobson radical of A.
I just want to know if the following proof is valid for the above theorem. (Note: $M$ is a $A$-module)
Sketch: Since $A$ is Artinian we know that $J(A)$ is nilpotent i.e. there exists $k\geq 1, k\in \m... |
H: Understanding Lang's proof that every closed and bounded set is compact
In his book Serge Lang provides the following proof that every bounded set that is closed is also compact.
" Let S be a closed and bounded set. This implies that there exists $B$ s.t. $|z|\leq B$ where $z\in s$. For all $z$ write $z=x+iy$, then... |
H: Prove that the serie $\sum_{n=1}^\infty \frac{1}{n(n + a)}$ converges
I was trying to solve this problem but i got stucked. I use the Radio Test to compute the convergence interval, but it doesnt works in this case, i need help...
Prove that the serie $\sum_{n=1}^\infty \frac{1}{n(n + a)}$ converges
Calculate the ... |
H: Index of intersection of two subgroups
Let $H$ and $K$ be subgroups of $G$ with indices $3$ and $5$ in $G$.
I need to show that the index of $H\cap K$ is a multiple of $15$.
ATQ
$\frac{|G|}{|H|}$ = 3
So $|G|$ is a multiple of $3$.
Similarly, $|G|$ is a multiple of $5$.
So we conclude that $|G|$ is a multiple of $1... |
H: Proof that a closed set contains all the limit points
In my general topology textbook there is the following proposition:
Let $A$ be a subset of a topological space $(X, τ)$. Then $A$ is closed in $(X, τ )$ if and only if $A$ contains all of its limit points.
And then they give the following proof:
Assume that $... |
H: Limit of a monotone sequence of measurable functions is measurable.
I'm currently studying measure theory as a non mathematician out of self interest but I've come across a typical proof style that I find rather hard. Generally it's showing that a typical set can be written as a countable union of measurable sets. ... |
H: A player rolls four 20-sided dice, takes the lowest value, ignores the rest. What is the probability of this value at least 7?
I'm designing a tabletop game, and I need to figure out how to calculate a few probabilities:
Roll 3 20-sided dice, take the highest value. What is the probability of it being 7 or higher?... |
H: Cubic Spline Interpolation - Constructing the Matrix
Interpolate a cubic spline between the three points $(0, 1), (2, 2) \text{ and } (4, 0).$
I'm trying to understand how to interpolate a given set of points using cubic splines with the help of this solved example. I don't quite get how they arrived at the matri... |
H: Help to solve $y'=y$, building exp function
I come to ask for help building the exponential function as the solution to $y'=y$.
This question is different from :
Prove that $C\exp(x)$ is the only set of functions for which $f(x) = f'(x)$
Since I would like help to prove it using the following arguments :
show tha... |
H: Stuck on a particular step of finding the integral closure of $\mathbb Z$ in $\mathbb Q[\sqrt{n}]$.
Here, $n\in \mathbb Z$ is square-free, $R$ is the integral closure of $\mathbb Z$ in $\mathbb Q[\sqrt{n}]$, and $\alpha = a+b\sqrt{n}$.
I have shown that $a \in \frac{1}{2}\mathbb Z$ statement.
I have shown the ... |
H: Prove that $\int_{x}^{1} \frac{1}{1+ t^2} = \int_{1}^{\frac{1}{x}} \frac{1}{1+ t^2}$ for $x \gt 0$
Prove that $$\int_{x}^{1} \frac{1}{1+ t^2} = \int_{1}^{\frac{1}{x}} \frac{1}{1+ t^2} $$ for $x \gt 0$
Basically what i did is to form the function $$f(x)=\int_{x}^{1} \frac{1}{1+ t^2} - \int_{1}^{\frac{1}{x}} \frac{1}... |
H: Exercise 4.16 in Brezis' Functional Analysis (Counterexample)
The exercise 4.16 in the Brézis book - Functional Analysis, Sobolev Spaces and PDE's, is as follows:
Let $1<p<\infty$. Let $(f_n)$ be a sequence in $L^p(\Omega)$ such that
(i) $f_n$ is bounded in $L^p(\Omega)$.
(ii) $f_n \rightarrow f$ a.e. on $\Omega$.
... |
H: Borel probability measure vs Probability measure
Is there a difference between these two terminologies?
space of all Borel probability measure on $\mathbb R^n$ or some complete, separable metric space.
space of all Probability measure on $\mathbb R^n$ or some complete, separable metric space.
In other words, wh... |
H: A Question on "Stars and Bars" and why it doesn't apply to a problem asked earlier today.
The problem below was presented earlier today and I am wondering why a "stars and bars" approach wouldn't be an appropriate method to solving this problem. The problem reads as follows:
Find the probability that each child get... |
H: Transforming constraints into linear inequality
I want to model the following two constraints in terms of LP, but after trying various ways without success, I wonder if it is possible at all?
Given $x$ and $y_{ij}$ are binary variables. We need the following two constraints:
If $x = 1$, then $\sum_{i = 1}^{n} y_{i... |
H: If $(m,n)\neq 1$, prove $\mathbb{Z}_{mn} \not \cong \mathbb{Z}_{m} \times \mathbb{Z}_{n}$.
I am really struggling with this problem. We just learned isomorphism in rings. We have not learned groups. From reading multiple sources, so far I have:
Suppose $(m,n) = d > 1 \Longrightarrow \frac{m}{d}, \frac{n}{d}$ integ... |
H: A property of the function $\frac{\sin x}{x}$
How can one prove, that $0$ is the only value of $\frac{\sin x}{x}$ taken infinitely often?
What I tried:
To see how the graph looks like https://www.wolframalpha.com/input/?i=%28sin+x%29%2Fx
The function is continuous and has infinitely many positive and negative value... |
H: Sum of dot products of linearly independent vectors
Suppose I have $n$ column vectors of equal length $\{\vec{a}_i\}_{i \in \{1,...,n\}}$ which are linearly independent of one another. Suppose I have a further $n$ column vectors $\{\vec{b}_i\}_{i \in \{1,...,n\}}$, each of the same length as $\{\vec{a}_i\}_{i \in \... |
H: Prove that $0<\lim\limits_{x\to \infty}f(x)<1\implies \lim\limits_{x\to \infty}\big(f(x)\big)^x=0$
Prove that
$$0<\lim_{x\to \infty}f(x)=l<1\implies \lim_{x\to \infty}\big(f(x)\big)^x=0.$$
I'm looking for a delta - epsilon proof for this.
Starting from the LHS, I got to
$$\forall\varepsilon_1 >0,\; \exists m>0,\;\... |
H: two local homeomorphisms
I am being silly here.
Suppose we have two local homeomorphisms $f: E \to X$ and $g: E' \to X$. If $S$ is a sheet of $E$.
Would $g^{-1}f(E)$ be homeomorphic to $f(E)$? My guess is yes as $g$ is a local homeomorphism. Any help would be appreciated!
AI: Take $X$ to be any nonempty space, then... |
H: Why are homogeneous and non-homogeneous first order differential equations called homogeneous an vice versa?
So I have recently been studying differential equations and I am extremely confused as to why the properties of homogeneous and non-homogeneous equations were given those names. It seems to have very little... |
H: Does uniform convergence of a function sequence imply the convergence of the corresponding level set?
Suppose $f_n(x),f(x)$ are both continuous functions with domain $M\subset R^2$ and codomain $R$, and define $L=\{x:f(x)\geq c\}$ and $L_n=\{x:f_n(x)\geq c\}$. Suppose $\underset{x\in M}{sup}|f_n(x)-f(x)|\rightarrow... |
H: Express difference between two positive numbers as a number between $0$ and $1$
It looks like a relatively simple problem, but I found some difficulties finding a way to express the difference between two numbers as a number in between $0$ and $1$ (you can say as a percentage).
Rules are simple: both numbers are al... |
H: Confused About Irreducibility - Fraleigh
Fraleigh says:
"It is worthwhile to remember that the units in $F[x]$ are precisely the nonzero elements of F. Thus we could have defined an irreducible polynomial $f(x)$ as a nonconstant polynomial such that in any factorization $f(x) = g(x)h(x)$ in $F[x]$, either $g(x)$ o... |
H: Prove that $g$ is injective or surjective
I have two questions which are related to mappings as follow:
Given 3 sets E, F, and G such that $f: E \rightarrow F, g: F \rightarrow G$ are two mappings. Prove that:
a. If $g \circ f$ is injective and $f$ is surjective then $g$ is injective.
b. If $g \circ f$ is surjectiv... |
H: What is the set of functions such that any quotients of two of them at infinity is real or infinity?
In this question How to quantify asymptotic growth? I was told that to assume $\lim_{x\to\infty}\frac{f(x)}{g(x)}$ is either real or $\pm\infty$, I have to assume a certain subset of non-oscillating functions for $f... |
H: How is $\mathbb Z[\frac{1}{2}+\frac{1}{2}\sqrt{n}]=\{a+b\sqrt{n} \mid a,b \in \mathbb Z \text{ or } a,b \in \mathbb Z +\frac{1}{2}\}$?
Let $n \in \mathbb Z$ be square-free.
If $n \equiv 1 \pmod 4$, how is $\mathbb Z[\frac{1}{2}+\frac{1}{2}\sqrt{n}]=\{a+b\sqrt{n} \mid a,b \in \mathbb Z \text{ or } a,b \in \mathbb Z... |
H: Conditional probability on dice rolling
Let's roll four dice. What is the probability that there is no "4" on any of the dice conditional on each dice having different values.
My answer is the following:
$$\frac{5 \times 4 \times 3 \times 2}{6 \times 5 \times 4 \times 3}$$
The denominator being all events with diff... |
H: Prove that if $G$ is a finite group in which every proper subgroup is nilpotent, then $G$ is solvable.
Prove that if $G$ is a finite group in which every proper subgroup is nilpotent, then $G$ is solvable. (Hint: Show that a minimal counterexample is simple. Let $M$ and $N$ be distinct maximal subgroups chose with ... |
H: If $y=f(x)=\frac{3x-5}{2x-m}$ find $m$ so that $f(y)=x$.
Question: If $y=f(x)=\dfrac{3x-5}{2x-m}$ find $m$ so that $f(y)=x$.
We have $y=\dfrac{3\left(\frac{3x-5}{2x-m}\right)-5}{2\left(\frac{3x-5}{2x-m}\right)-m} $
How can I find $m$? It is given than $m=3$.
AI: Now, $$\dfrac{3\left(\frac{3x-5}{2x-m}\right)-5}{2\le... |
H: Trignometric Substitution Problem. Can't find right answer
I am trying to solve an integral that involves using Trig Sub (I know it can be also done with partial fraction).
However, no matter how many times I try, I still cannot find the right answer. I hope someone can point out where I did wrong :)
$\displaystyle... |
H: Evaluating limit of the function at $\frac{\pi}{2}$
I'm trying to solve this
$$\lim_{x \to \frac{\pi}{2}} \frac{\cos{x}}{(x-\frac{\pi}{2})^3}$$
I have tried using the L'Hôpital's rule
But I'm stuck at
$$\lim_{x \to \frac{\pi}{2}} \frac{-\sin{x}}{3(x-\frac{\pi}{2})^2}$$
Since the above equation is not in the $\frac{... |
H: What concept of order is introduced in the twentyfold way?
Four of the folds not present in the twelvefold way but introduced in the twentyfold way, rows $5$ and $6$ of the linked table, are defined by the statement that order matters.
However, my understanding is that labeling/de-labeling the elements of the domai... |
H: Let $f(x)$ satisfy Rolle's theorem conditions and have three successive solutions $x_1, x_2, x_3$. How to prove that $f'(x)$ is differentiable?
Rolle's Theorem
Let $f(x):[a,b]\to\mathbb{R}$ where $f$ is differentiable at $(a,b)$ and continuous at $[a,b]$, with $f(a) = f(b)$.
We know from Rolle's theorem that $\exis... |
H: Any finite connected graph with every vertex has degree $\ge 2$ has a circuit
Is my proof for the following statement correct?
Any finite connected graph with every vertex has degree $\ge 2$ has a
circuit.
My attempt: Let $G$ be a finite connected graph. Let $|G|=n$. Suppose that degree of any vertex $\ge 2$. Now... |
H: Let $f$ continuous at $[a,b]$ and differentiable at $(a,b)$ where $f(b)=0$. How to prove that $f'(x_0) = \frac{f(x_0)}{a-x_0}$?
The Problem
Let $f$ continuous at $[a,b]$ and differentiable at $(a,b)$ where $f(b)=0$.
How to prove that:
$$\exists x_0 \in (a,b): f'(x_0) = \frac{f(x_0)}{a-x_0} \quad (1)$$
My solution ... |
H: (Weak convergence $\implies$ strong convergence) $\implies \mathcal{H}$ finite-dimensional
Let $\mathcal{H}$ be a Hilbert space. Show:
$(\forall \psi \in \mathcal{H}: \lim \langle \psi, \phi_{n}\rangle = \langle \psi, \phi_{n}\rangle \implies \lim \phi_{n}=\phi )\implies \mathcal{H}$ finite-dimensional. $(*)$
The ... |
H: Proof Check: $x \leq y+ \epsilon$ for all $\epsilon >0$ iff $x \leq y$.
Synopsis
I want to be sure I'm utilizing proof by contradiction correctly, so please check my proof of the exercise below. It's relatively simple, so it shouldn't take you too much time.
Exercise
Let $x$ and $y$ be real numbers. Show that $x \... |
H: Can $\int_0^\infty f (x) \, dx$ exist if $\lim_{x \to \infty} f(x)$ does not exist?
Is is possible to have a function for which $\lim_{x \to \infty} f(x)$ does not exist, but $\int_0^\infty f(x) \, dx$ exists and is finite?
I think I've found an example actually, but I'm not sure it works. Let $H_n$ be the $n$th ha... |
H: Give an example of a sequence of functions having the following property.
I am finding an example of a sequence $(f_n)$ of differentiable functions on an interval $I$ such that $f_n\to 0$ uniformly on $I$ but $f'_n(x)\to \infty$ for all $x\in I$ as $n\to \infty$.Can someone provide me an example where this can occu... |
H: Defining a family of self-adjoint operators via a bilinear form
I started reading an article and I'm having some trouble understanding a certain family operators they defined. Here the relevant parts:
I'm trying to understand how exactly $L_\sigma$ are defined. I noticed that if I take $\sigma=0$ then I get $B_0(... |
H: Finding the Locus if $z$ Purely Imaginary/Real
If $\omega=\frac{z-1}{z+i}$, find the locus if $\omega$ is:
Purely Imaginary
Purely real
This is for the applying complex numbers topic of an advanced HSC maths course. I was asked to describe the loci.
One of my friends suggested that I rationalise $\omega$ and spli... |
H: Finding the value of a variable in a quadratic
Question:
The number of negative integral values of $m$ for which the expression $x^2+2(m-1)x+m+5$ is positive $\forall$ $x>1$ is?
For me, solving this question if the parameter "$\forall$ $x>1$" was not given would be quite easy. But how do I solve it under the give... |
H: Convergence in probability of a root of an equation
Let $X_1, \dots, X_n$ be iid Uniform $(0, 1)$ random variables, and set $\theta_n$ to be the root of the equation
$$
\sum_{k=1}^n \theta^{X_k} = \sum_{k=1}^n X_k^2.
$$
Apparently, this $\theta_n$ converges in probability to a constant. Is that easy to see?
AI: Fi... |
H: The Explanation of these steps
I was following a lecture on Tangent Spaces, where I find expressions as:
$$(f\circ\gamma\circ\mu)'(0) = (f\circ\gamma)'(\mu(0)).\mu'$$
And in some other place, I find:
$$((f\circ x^{-1})\circ(x\circ\sigma))'(0) = (x\circ\sigma)^{i'}(0).(\partial_{i}(f\circ x^{-1}))(x(\sigma(0))) $$
N... |
H: Interpretation of the convergence in the mean square sense
I'm new to the notion of convergence in the mean square (or convergence in $L^2$). So, I want to ask about the intuition/interpretation of an algorithm whose iterates converge in the mean square to a certain value, say I have the following:
$$ \underset{k \... |
H: Prove: if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely
Prove:
if $\sum^\infty_{n=0}a_nx^n$ converges for every $x$, then $\sum^\infty_{n=0}a_n$ converges absolutely.
I get why the statement is correct (because it means that the convergence of the series doesn't... |
H: equivalence class and partially ordered sets question
the question is -
1.we have 4 sets $A,B,C,D$ , if $AΔB⊆D$ and $BΔC⊆D$ then $AΔC⊆D$
2.given the power set $P$({1,2,3}) and given two relations $A,B∈$P$({1,2,3})$ $ARB$ if and only if $AΔB⊆$ {1,2} and $ASB$ if and only if AΔ{1,2}⊂BΔ{1,2}. which one of the sets is ... |
H: Given $\cos(a) +\cos(b) = 1$, prove that $1 - s^2 - t^2 - 3s^2t^2 = 0$, where $s = \tan(a/2)$ and $t = \tan(b/2)$
Given $\cos(a) + \cos(b) = 1$, prove that $1 - s^2 - t^2 - 3s^2t^2 = 0$, where $s = \tan(a/2)$ and $t = \tan(b/2)$.
I have tried using the identity $\cos(a) = \frac{1-t^2}{1+t^2}$. but manipulating th... |
H: Prove inequality $\tan(x) \arctan(x) \geqslant x^2$
Prove that for $x\in \left( - \frac{\pi} {2},\,\frac{\pi}{2}\right)$ the following inequality holds
$$\tan(x) \arctan(x) \geqslant x^2.$$
I have tried proving that function $f(x) := \tan(x) \arctan(x) - x^2 \geqslant 0$ by using derivatives but it gets really mess... |
H: Evaluate: $\int_0^{\infty}\frac{\ln x}{x^2+bx+c^2}\,dx.$
Prove that:$$\int_0^{\infty}\frac{\ln x}{x^2+bx+c^2}\,dx=\frac{2\ln c}{\sqrt{4c^2-b^2}}\cot^{-1}\left(\frac{b}{\sqrt{4c^2-b^2}}\right),$$ where $4c^2-b^2>0, c>0.$
We have:
\begin{align}
4(x^2+bx+c^2)&=(2x+b)^2-\left(\underbrace{\sqrt{4c^2-b^2}}_{=k\text{ (s... |
H: Compute the $\int \sqrt[3]{1+\sin x}\ dx$ by help of Taylor series.
I want to compute the integral $\int \sqrt[3]{1+\sin x}\ dx$ via Taylor series.
My idea is : find Taylor expansion around zero of the function $f(x)= \sqrt[3]{1+\sin x}=\displaystyle\sum_{n=0}^{\infty} c_nx^n$, and after to integrate the Taylor ex... |
H: Matrix calculation / operation
Assume there is a vector,
$$
A = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_m \end{bmatrix} \in \mathbb R^{mn},
$$
and each subvector $a_i, i\in \{1,2,\cdots,m\}$ are $n$-by-$1$ vector.
The output is
$$
B = \begin{bmatrix} a_1^T & 0 & \cdots & 0 & 0 \\ 0 & a_2^T & \cdots & 0 & 0 \\ \vd... |
H: Proving a vector space has an uncountable basis
I am required to prove that a vector space $V=C[0,1]$ (the space of continuous functions over complex field) has an uncountable basis. The approach I took was to show that there is a subspace $W$ of $V$ with an uncountable basis and concluded that the vector space $V$... |
H: Positive semi-definiteness of the adjoint matrix
I am studying the conditions of positive semi-definiteness of a $(n+1)\times(n+1)$ symmetric matrix $\mathbf{M}$ built in the following way:
$$
\mathbf{M}=\begin{pmatrix}
\mathbf{A} & \mathbf{b} \\
\mathbf{b}^T & c
\end{pmatrix}
$$
where $\mathbf{A}$ is a symmetrix... |
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