text stringlengths 83 79.5k |
|---|
H: Does string substitution have a definition, similar to the one for string homomorphism in terms of monoid morphism of the free monoid?
string homomorphism is defined in formal language theory as:
A string homomorphism (often referred to simply as a homomorphism in formal language theory) is a string substitution s... |
H: A comparision between two subadditivity
I have a simple question about the subadditivity of inf,
it's easy to show that
$\inf_x (f+g)(x) \ge \inf_x f(x) + \inf_y g(y)$.
And for norm we have
$\inf_{x\in X,y\in Y}(\|x\|+\|y\|)\le \inf_{x\in X}\|x\| +\inf_{y\in Y}\|y\|$.
My qustion is why they look quite similar but w... |
H: Plane wave solutions to the Dirac Lagrangian
I was studying Tong's lecture notes and there's a specific mathematical step I do not see how to derive (page 108); specifically, I do not see how to derive $(5.9)$ and $(5.10)$
Let's assume the following equation holds (which is a sum of plane wave solutions for $(-i\ga... |
H: Prove $\left[\ln(n + 1)\right]^p - \left[\ln(n)\right]^p \to 0$ as $n \to \infty$, $p \geq 1$.
Problem: I am trying to prove that for each real number $p \geq 1$, that the sequence $\left(\left[\ln(n + 1)\right]^p - \left[\ln(n)\right]^p \right)_{n \geq 1}$ converges to $0$.
I can do it for $p = 1$ using the contin... |
H: Open and connected in normed space implies path-connected
Suppose that $V$ is an open subset of a normed space $X$. Then $V$ is connected iff $V$ is path-connected.
My attempt:
The implication $\Leftarrow$ is a well-known result. My question is about the other implication:
It suffices to show that every $x\in V$ ... |
H: Determine a basis in given subspace
Let $V$ be the subspace of $R^4$ consisting of the vectors $ x = (x_1, x_2, x_3, x_4)^T$
whose coordinates satisfy the equation $ x_4 = x_1 + 2x_2 − 3x_3$ . Determine a basis in $ V$
containing the vector $ v = (1, 1, 1, 0)^T$
I figured it out by myself that we can find 1 other... |
H: Closed operator bijective
Definition. Let $X$ and $Y$ normed space. A linear operator $T\colon X\to Y$ is said closed if the graph of operator $T$ is closed in the product topology. That is $T$ is closed if and only if for each sequence $\{x_n\}\subseteq X$ such that $\{x_n\}$ converges in $X$ and $\{Tx_n\}$ conve... |
H: Can we generate any random variable by transforming a uniform?
Let $X$ be a real-valued random variable with cdf $F$.
Question: Can we always find a transformation of a uniform random variable that has the same distribution as $X$? That is, given $U \sim \mathrm{Unif([0,1])}$ can we find a (measurable) function $\v... |
H: How do I find the ODE in Sturm-Liouville form when given two functions
I have not had any success in finding the Sturm-Liouville ODE corresponding to the ODE solutions
$\{x^m \cos(nx), x^m \sin(nx)\}$ where $n,m \in \mathbb {R}$.
Thanks in advance!
AI: Consider
$$(\frac{y}{x^m})’’+n^2 \frac{y}{x^m} = 0.$$
Now, can... |
H: Let $G\subset\Bbb R$ be a Borel set, show that the Borel sets of $G$ (as a subspace) are the same as the Borel subsets of $\Bbb R$ included in $G$
Let $B$ be the Borel sigma-algebra over $\Bbb R$ (real numbers).
Let $G\subset R$ be a Borel-set. And $A_0$ the family of all subsets of $G$ which have the form $G\cap O... |
H: Suppose that $x$ and $y$ are real numbers. Prove that if $x\neq0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$.
Not a duplicate of
Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$
Prove that for any real numbers $x$ and $y$ if $x \neq 0$, then if $y=\frac{3x^2+2y}{x^2+2}$ then $y=3$.
This ... |
H: Marginal Distributions of given CDF
I have the CDF given by :
$$F(x_1, x_2) = e^{-(-x_1-x_2)^{1/\beta}}$$
with $x_1,x_2 \leq 0$ and $\beta \geq 1 .$
I need to find the marginal distribution functions. However when I try to apply the limit to infinity for any of these two random variables I get something not deter... |
H: precompact compact differece
the definition of precompact on a metric space (X,d) shall be: $\forall r >0 \exists F\subset X \ \mathrm{ finite, s.t. } \ X= \cup_{x\in F} B_r(x).$
As all open sets in (X,d) are balls, why is that condition not sufficient, and we also need completeness?
Further, can one define precomp... |
H: Derivative of $\frac{d}{d\theta }\left(\left(\theta \:-1\right)\sum\limits_{i=0}^n\:\ln\left(x_i\right)\right)$
I'm doing some maximum likelihood estimation exercise and I got stuck on the derivative of the following part: $\frac{d}{d\theta }\left(\left(\theta \:-1\right)\sum\limits_{i=0}^n\:\ln\left(x_i\right)\rig... |
H: Flux of $(x,y,z^2)$ through $\{(x,y,z): 2(x^2+y^2)
Let $$D=\{(x,y,z): 2(x^2+y^2)<z<3\}$$
and the vector field $$F(x,y,z)=(x,y,z^2)$$
I wanto to compute the flux of $F$ through $\partial D$
Applying the divergence theorem, I should compute
$$\int_D2+2z \ dxdydz$$
To evaluate the integral I pass to cylindrical coor... |
H: Proving the isomorphism $A \otimes B \cong B\otimes A$ of the tensor products of abelian groups $A,B$ given the definition by the quotient groups.
For two Abelian groups $A$ and $B$ we define their tensor product
$A\otimes B$ as the quotient of the free Abelian group on the set of
formal generators $\{a \otimes b ... |
H: Showing a prime is inert
This is a part from the book "Generic Polynomials
Constructive Aspects of the Inverse Galois Problem" which I don't understand. This is also pretty much the counterexample given in: https://www.jstor.org/stable/1969410?seq=1.
Let $L_2$ be the unramified extension of $\mathbb{Q}_2$ of degree... |
H: The "symmetric" property of Day convolution.
This question has to be divided into the following parts:
The definition of Day convolution in nlab
To define Day convolution, it assumes that $V$ be a closed symmetric monoidal category with all small limits and colimits, and $C$ be a monoidal category.
see https://nc... |
H: Folland Real Analysis Exercise 5.15 - A unique linear tranformation $S$ such that $T = S \circ \pi$ where $\pi$ is the projection onto $X/N(T)$
Consider the following problem, Exercise 15 of chapter 5 of Folland's Real Analysis, 2nd edition:
Suppose that $X$ and $Y$ are normed vector spaces and $T \in L(X, Y)$. Le... |
H: Prove that a function is distance preserving.
Let $V$ be a real inner product space, $\{x_{n}\mid n\in\mathbb{N}\}$ is an orthonormal basis.
I want to show that $\phi : V \rightarrow l_2\;$ which takes each $v\in V$ to $\left\langle v,x_{n}\right\rangle _{n=1}^{\infty}\in l_{2}$, is distance preserving.
$l_{2}=\{(... |
H: Kernel of complex tori morphism, is this elementary assertion true?
Let $\Lambda_1, \Lambda_2$ be lattices of $\mathbb{C}$ and $T:\mathbb{C}\rightarrow\mathbb{C}$ be a $\mathbb{C}$-linear map such that $T(\Lambda_1) \subset \Lambda_2$. This induces complex tori morphism $\varphi:\mathbb{C}/\Lambda_1 \rightarrow \ma... |
H: How to Prove $\int_{0}^{\infty}\frac {1}{x^8+x^4+1}dx=\frac{π}{2\sqrt{3}}$
Question:- Prove that
$\int_{0}^{\infty}\frac {1}{x^8+x^4+1}dx=\frac{π}{2\sqrt{3}}$
On factoring the denominator we get,
$\int_{0}^{\infty}\frac {1}{(x^4+x^2+1)(x^4-x^2+1)}dx$
Partial fraction of the integrand contains big terms with their ... |
H: Examples of Continuous Probability Distributions with Slowly Varying Upper Tail and Infinite Expectation
$L(x) = 1 - F(x)$ is the slowly varying upper tail, where F(x) is a continuous probability distribution function with infinite expectation. That is $\int_{0}^{\infty} u dF(u) = \infty$. I am unable to construct ... |
H: Combinatorics problem about weight of balls. (POSN Camp $2$)
If Jack has $25$ white balls and $63$ black balls and all black ball weight are less than $26$ grams , all white ball weight are less than $64$ grams. (All weight are in integer and some balls may have the same weight.) Prove that Jack can select some whi... |
H: Confused about a step in a proof; For a closed set, $F$, and a point $x\in F$, there's an open interval around $x$ and contained in $F$?
I've underlined the step in the proof that isn't clear to me.
I know what I said in the title is wrong; but I'm not really sure what motivates the step that I underlined.
This is... |
H: Solving $Ax=0$ for non-negative $x$
This seems like a fairly elementary problem but I have not been able to find an a suitable way to this problem. Any advice would be greatly appreciated.
The problem is simple. I have a large matrix and would like to find a vector in its nullspace such that the components are gre... |
H: Proof that pullback is smooth by considering charts only
I'm reading up about pullbacks and pushforwards. If $\psi:M\to M'$ is smooth and $f:M'\to\mathbb{R}$ is a smooth function, then we define the pullback as $\psi*f=f\circ\psi$. By definition of a smooth map between manifolds then, the pullback of $f$ is also a ... |
H: Determine the dimension of the subspace
The vectors $a, b, c, d $ form a basis of $ R^4$. Determine the dimension of the
subspace generated by the vectors $a + b, c + d, a + c, b + d$.
I think combined vectors($a+b,c+d,a+c,b+d$) somehow must be dependent, however, I could not prove.
AI: Generally a good way to tack... |
H: Computation of the commutator of permutations of $J_n$
I've been slowly going through some of the material in Serge Lang's Alegbra, and I've just stumbled upon some computations that's puzzling me at the moment. It's a specific step in the proof of a theorem, namely:
Theorem: If $n\geq 5$ then $S_n$ is not solvabl... |
H: A linear map $T:V\rightarrow V$ can be written as $T=T_2T_1$ for some linear map $T_1$ and $T_2$.
Question: Let $V$ be a finite dimensional vector space over $\mathbb{R}$ and $T:V\rightarrow V$ be a linear map. Can you always write $T=T_2T_1$ for some linear maps $T_1:V\rightarrow W$, $T_2:W\rightarrow V$, where $W... |
H: ${\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}+\frac{\mathrm{d} y}{\mathrm{d} x}-2\cdot y}={4\cdot x\cdot e^ {- 3\cdot x }-17\cdot e^ {- 3\cdot x }}$
I have the equation $${\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}+\frac{\mathrm{d} y}{\mathrm{d} x}-2\cdot y}={4\cdot x\cdot e^ {- 3\cdot x }-17\cdot e^ {- 3\cdot x }}$$ I think ... |
H: How to convert degrees of a circle into positive and negative X and Y values
I am hoping this is a very simple equation that can solve this (I am not a mathematician) but considering I have values in degrees (0 - 360), I am trying to convert that back into plus and minus X and Y values.
So between 0 and 180 degrees... |
H: For about real Eigen vector.
Is there any complex hermitian matrix, which have real Eigen vector. If there any, please give such example. Thanks.
AI: Well, yes. See for instance $\begin{pmatrix}1&0&0\\ 0&1&i\\ 0&-i&1\end{pmatrix}$ which has an eigenvector in $(1,0,0)^t$. It is true that if a Hermitian matrix has a ... |
H: Integrate $\int_0^1 \ln{\left(\ln{\sqrt{1-x}}\right)} \mathop{dx}$
Problem says to integrate $$\int_0^1 \ln{\left(\ln{\sqrt{1-x}}\right)} \mathop{dx}$$
I try $u=1-x$ and got
$$\int_0^1 -\ln{2}+\ln{(\ln{u})} \mathop{du}$$
Then $t=\ln{u}$
$$-\ln{2}+\int_{-\infty}^0 e^t \ln{t}$$
Now what?
AI: You are on the right trac... |
H: Derivative of Input in nonlinear State Space representation
I am dealing with obtaining an space state representation of a nonlinear differential equation that arises from an inverted pendulum. It includes some terms that reflect the fact pendulum is controlled in a PID similarly fashion.
The original EOM would be ... |
H: Prove the following equality with determinants
Show that
$$
\begin{vmatrix}
b_1+c_1 & c_1+a_1& a_1+b_1 \\
b_2+c_2 & c_2+a_2 & a_2+b_2 \\
b_3+c_3 & c_3+a_3 & a_3+b_3 \\
\end{vmatrix}
=
2\begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & b_2 & c_2 \\
a_3 & b_3 & c_3
\end{vmatrix}
$$
I can only see the brute-force approach.... |
H: Finding an irreducible polynomial over the rationals
I'm very confused by a homework question.
"Find the irreducible polynomial for $ \sin{2\pi/5}$ over Q.
I found that $16t^{4}-20t^{2}+5=0$ but this is not monic? This is also irreducible by Eisenstein, but minimal polynomials are always monic?
AI: If you want a mo... |
H: bound for the ratio of Gamma functions
Let $x \in R$, $N$ is a natural number.
How to bound from above
$$
\frac{\Gamma(1-1/x)}{\Gamma(N+1-1/x)}
$$
AI: Repeated application of the fact that $\Gamma(x+1)=x\cdot\Gamma(x) \text{ for } x\notin-\mathbb{N}\cup\{0\}$ yields
$$\frac{\Gamma(1-1/x)}{\Gamma(N+1-1/x)}=\prod_{i=... |
H: Finite-Type Algebra Property Under Change of Base
I was wondering how to prove the following assertion. Any help would be appreciated!
Suppose we have to ring homomorphisms: $f: A \rightarrow B$ and $g: B \rightarrow C$.
Then $C$ being finite type over $A$ implies that $C$ is finite type over $B$.
AI: $C$ being fin... |
H: Solve: $3x(x+y-2)=2y$ with $y(x+y-1)=9x$
This is a question from my Regional Mathematics Olympiad. It is from the topic- polynomials.
AI: Multiply both equations,
$$3x(x+y-2)y(x+y-1) = 18yx$$
Cancelling $x$ and $y$, we obtain a solutions $x=0$ and $y=0$
Next assume $x+y=t$ and solve quadratic.
$$(t-2)(t-1)=6$$
$$(t... |
H: Should I use the "variation of parameters" here?
I am trying to understand how I should approach this kind of equation. $${x^2\cdot \left(\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}\right)+6\cdot x\cdot \left(\frac{\mathrm{d} y}{\mathrm{d} x}\right)+6\cdot y}={\frac{x+4}{x^3-4\cdot x^2}}$$ $x>4$
I think I should use the ... |
H: if 3 random variables are independent, then any pair is independent given the remaining random variable
How do I prove the following:
$X,Y,Z$ are independent, then any pair is independent given the remaining
random variable.
I know that $E(X|Z)=E(X) $ and $E(Y|Z)=E(Y)$
so $E(XY)=E(X)E(Y)=E(X|Z)E(Y|Z)$
does it imply... |
H: There exists a nonnull subset $A\in \mathcal{A}$ such that : $ \{f_n\}, \{g_n\}\text{ and }\{h_n\} \text{ are uniformly integrable on }A $?
Let $(E,\mathcal{A},\mu)$ be probability space.
Lemma. Suppose $\{f_n\}$ is a sequence in $\mathcal{L}^1_\mathbb{R}$ such that
$$
\sup_n\int_{E}|f_n|d\mu<+\infty.
$$
Then the... |
H: Restriction of a quotient map to its inverse image is a quotient map
If $q:X\to Y$ is a quotient map. Is it true that $q|_{q^{-1}(B)}:q^{-1}(B)\to B$ is also a quotient map for any subset $B$ of $Y$.
It seems that it is not true but i'm not getting any counter example. can anyone help. Thanks in advance!
AI: Counte... |
H: Suppose that $f:[0,1]\to M$ is locally injective. Show that $f$ is piecewise injective.
Consider a metric space $M$ and $f: [0,1]\to M$ locally injective, i.e. for each $x\in M$ there is a neighborhood on which $f$ is injective. Show that $f$ is piecewise injective, i.e. there exists $0= t_0 < t_1 < \dots < t_N = ... |
H: Taylor series for $\ln(\frac{1-z^2}{1+z^3})$
Taylor series for $\ln(\frac{1-z^2}{1+z^3})$
I've tried to $$\ln(1-z^2)-\ln(1+z^3)=\sum (-1)^{3n-1}z^{2n}-\sum(-1)^{4n-1}z^{3n}$$
I didn't manage to make it one series any help is good
AI: I think you mean $-\sum_{n\ge1}\frac1nz^{2n}+\sum_{n\ge1}\frac{(-1)^n}{n}z^{3n}$. ... |
H: true or false- continuous functions
I'm having some hard time deciding if those sentences are true or false:
$1$. If $f$ is continuous on $\mathbb{R}$ then if $\left|f(x)-x\right|<1$ for every $x$ on $\mathbb{R}$ then $f$ is getting every real value on $\mathbb{R}$.
$2$. If $f$ is continuous on R then if $\left|f(... |
H: Find the PDF of $Y = X^2 + 3$ where $X \sim Poisson(\lambda)$.
I am following the CDF method to calculate the PDF oF Y. Up to this, I have done:
$$F_Y = P(Y \leq x)$$
$$F_Y = P(X^2+3 \leq x)$$
$$F_Y = P(X \leq \sqrt{x-3)}$$
$$F_Y = e^{-\lambda}.\sum_{k=0}^{\sqrt{x-3}} \frac{\lambda^k}{k!}$$
Now I have to differenti... |
H: How do I integrate $f(z)=3\cdot\operatorname{Im}z$ on a curve?
Let $f(z)=3y$, where $z=x+iy\in\mathbb C$, and let $\gamma(t)=t+it^2$, for $t\in[0,1]$. Find $\int f(z)\ dz$.
If the problem read "Let $f(z) = 3z$" for example, I would first find $\gamma'(t)= 1+2it$.
Then, I would find $\int f(\gamma(t))(\gamma'(t))\... |
H: Determine sequence with generating function $(2x-3)^3$
Determine the sequence associated to the following generating function: $(2x-3)^3$
I know that the sequence $\left(\begin{pmatrix} 3 \\ 0 \end{pmatrix}, \begin{pmatrix} 3 \\ 1 \end{pmatrix}, \begin{pmatrix} 3 \\ 2 \end{pmatrix}, \ldots\right)$ has the generatin... |
H: Does the pullback of the metric tensor have the form $\phi^* g(X,Y)=g(\phi X,\phi Y)$?
I have seen in many textbooks that the pullback of an arbitrary tensor field of type (r,s) under the diffeomorphism $\phi:M \rightarrow N$ is defined as
$\phi^* T(\eta_1,\dots, \eta_r, X_1, \dots, X_s) = T( (\phi^{-1})^*(\eta_1),... |
H: Differential equation with initial conditions problem: how do we solve $yy'' - 2(y')^2 - y^2 = 0$?
I have problem with solving following equation with initial conditions:
$$y*y''-2(y')^2-y^2=0 $$
$$y(0)=1; y'(0)=0 $$
The problem is that i've tried substitution $ u(y)=y' $
and I end up with $$u'*u-2u^2/y =y $$ which... |
H: natural logarithm of a complex number $\ln(a+bi)$
I am trying to find a formula for $\ln(a+bi)$, is my working correct?
$$a+bi=re^{i\theta},\,\,r=\sqrt{a^2+b^2},\,\,\theta=\frac{|b|}{b}\arctan\left(\frac ba\right)$$
and so:
$$\ln(a+bi)=\ln(re^{i\theta})=\frac{\ln(r^2)}{2}+i\theta$$
$$\therefore \ln(a+bi)=\frac{\ln(... |
H: What are the critera for a family of languages to qualify as a type of language in Chomsky hierarchy?
Chomsky hierarchy consists of several types of languages: regular, CFL, CSL, r.e..
What are the critera for a family of languages to qualify as a type of language in Chomsky hierarchy?
All the four types of langua... |
H: Lie algebra of Lorentz Group $O(1,3)$
Let
$$
O(1,3)=\{A\in GL_4(\mathbb R):A^TgA=g\}
$$
where $g$ is the diagonal matrix with $1$ on the first diagonal entry, and $-1$ on the other diagonal entries. I want to show that the Lie algebra consists of matrices $X$ such that $gXg=-X^T$. As I've already shown the spin hom... |
H: If the integral of a continuous function vanishes for every compact set, the function is identically zero.
Let's consider the continuous function
$$
f:\mathbb{R}^n\to \mathbb{R}
$$
Is it true that if $$\int_C fdx=0$$ for every compact set $C$ then the function is $f\equiv 0$?
I would say it is true, but I wouldn't ... |
H: There exists no $M_n(\mathbb{C})$ module of dimension coprime to $n$
Let $m,n\in\mathbb{N}:\gcd(m,n)=1$ prove there is no $M_n(\mathbb{C})$-module $V$ s.t. $\dim_{\mathbb{C}}V=m$
I think that since $V$ is an $M_n(\mathbb{C})$-module we should have that $\dim_{\mathbb{C}}V\mid n$ and thus we are done but is that r... |
H: Let $f(u)=\tan(u)$ and $g(x)=x^8$
I have solved the following:
$$f'(u)=\sec^2(u)$$
$$g'(x)=8x^7$$
However, these two have been giving me issues. This is my progress so far:
$$f(g(x))=\tan(u)^8$$
$$f'(g(x))=\sec^{10}(u)$$
Where did I go wrong here? Also, how would I go about in solving the following (I would just wa... |
H: If $f:\mathbb{R}^d\to\mathbb{R}^d$ is a homeomorphism, then $\lim_{\| x \| \to \infty} \| f(x) \| = +\infty$
If $f:\mathbb{R}^d\to\mathbb{R}^d$ is a homeomorphism, then $\lim_{\| x \| \to \infty} \| f(x) \| = +\infty$.
My attempt:
I need to show that $\forall M\in\mathbb{R}: \exists R>0: \forall x\in\mathbb{R}^d:... |
H: Proving that $zJ''(z)+J'(z)+zJ(z)=0$
In my complex analysis book there is the following problem that I'm having some trouble solving:
Consider the function:
$$J(z)=\sum_{n\geq0} \frac{(-1)^n}{(n!)^2} \left(\frac{z}{2}\right)^{2n}$$
1 - Determine the radius of convergence of this power series.
2 - Show that:
$$zJ''... |
H: Is $x*y=x$ associative and/or commutative on $\Bbb Z$?
Let $*:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{Z}$ on the integers by the formula $x*y=x$ for any $x,y\in\mathbb{Z}$. Decide whether $*$ is associative and/or commutative.
To the best of my understanding this means I'm taking a Cartesian product $(x,y)$ an... |
H: If a series converges and its sequence is monotonic, why it can not be a monotonic increasing function?
I have been wondering this for a while
If $$\sum a_{n}$$ converges and {$a_{n}$} is a monotonic function
Why must {$a_{n}$} be decreasing. Why can't it be increasing
AI: Let me answer this question in a more intu... |
H: Find the range of an irrational function
Find the range of given function: $y=-\sqrt{3x^2+4x+3}$. I don't know how to find the range of an irrational function. Can someone explain to me? Also it would be awesome if there are some excersises about this topic! (You can't use calculus here)
AI: \begin{align}
y
&=-\sqr... |
H: Prove that $ \kappa\times\lambda=\lambda $
let $ \kappa<\lambda $ and assume $ \aleph_{0}\leq\lambda $
prove that: $ \kappa\times\lambda=\lambda $
So, my attempt, based on the fact that i already proved for infinite cardinals $ \lambda $ that it follows : $ \lambda\times\lambda=\lambda $
if $ \kappa\neq0 $
choos... |
H: Dimension of product of affine varieties is the sum of dimensions of each variety
How do I prove that the dimension of the product of affine varieties is the sum of dimensions of each affine variety?
I am aware that similar questions had been asked in Dimension of product of affine varieties and Dimension of a tens... |
H: Proof that the union of connected sets where the intersection of the closure of one with the other is non-empty.
The problem says:
Prove that if $(X,d)$ is a metric space and $A, B$ are connected subsets of $ X$, then if $cl(A)\cap B\neq\emptyset$, $A\cup B$ is connected.
To show this, I supposed the contrary, that... |
H: Example of a group of order $15$ satisfying some conditions
Suppose that a group $G$ of order $15$ has only one subgroup of order $3$ and only one subgroup of order $5$, then I need to prove that $G$ is cyclic.
If I can show that $\exists a\in G$ such that $|a|=15$, then the result is proved.
By Lagrange's theor... |
H: Let $A\in M_{n\times n}(\textbf{F})$. Then a scalar $\lambda$ is an eigenvalue of $A$ if and only if $\det(A - \lambda I_{n}) = 0$.
Proposition
Let $A\in M_{n\times n}(\textbf{F})$. Then a scalar $\lambda$ is an eigenvalue of $A$ if and only if $\det(A - \lambda I_{n}) = 0$.
MY ATTEMPT
We say that $\lambda$ is an e... |
H: probability with two variables - Taking balls out of the basket
There's 3 balls in the basket - White, red and black. Three people chose one ball after the other with return.
$X$ is the various colors that got chosen.
$Y$ is the number of people that chose white.
I need to calculate $P(X+Y≤3|X-Y≥1)$.
So I got that ... |
H: Why does the characterization of Gaussian primes really work?
Citing https://en.wikipedia.org/wiki/Gaussian_integer :
A Gaussian integer $a + bi$ is a Gaussian prime if and only if either:
one of $a, b$ is zero and absolute value of the other is a prime number of the form $4n + 3$ (with $n$ a nonnegative integer)... |
H: Is $f(x)=\sin x$ integrable?
I'd like to know if the following reasoning to prove that $f(x)=\sin x$ is not integrable is correct, or what's the mistake I'm making.
Consider the following definition and Corollary taken from Folland's Real Analaysis:
Definition: Consider a measure space $(X,M, \mu)$. If $f:X \to \Bb... |
H: Vectors and Cross Product in 3D
I set the vector V as (a,b,c). I know that you have to multiply out the two vectors, so that is what I did. I then got multiple equations which I used to find the values of a, b, and c. The answer that I got was (2,2,-5), however, that solution only works for the equation on the bot... |
H: How to calculate $\int \frac{dx}{\cos(x) + \sin(x)} $?
I did it by the method of integration by parts, with
$$
u=\frac{dx}{\cos(x) + \sin(x)},\quad dv=dx$$
so
$$
\int \frac{dx}{\cos(x) + \sin(x)} =
\frac{x}{\cos(x)+\sin(x)}
- \int \frac{x(\sin(x)-\cos(x))}{(\cos(x)
+ \sin(x))^{2}}
$$
Where,
$$\int \frac{x(\sin(x... |
H: Equivalence relation on the complement of a subspace.
I'm trying to solve this problem: Let $V$ be a real vector space with dimension $n$ and $S$ a subspace of $V$ with dimension $n-1$. Define an equivalence relation $\equiv$ on the set $V\setminus S$ by $u\equiv v$ if the line segment
$$L(u,v)=\{(1-t)u+tv: t\in [0... |
H: Computing $\lim_{x\rightarrow 0}{\frac{xe^x- e^x + 1}{x(e^x-1)}}$ without L'Hôpital's rule or Taylor series
This limit really stamped me because i'm not allowed to use L'Hôpital's rule or Taylor's series, please help!
I think the limit is $\frac{1}{2}$, but i don't know how to prove it without the L'Hôpital's rule ... |
H: $\lim_{n\rightarrow \infty} E[\min(|X_n-X|,1)]=0\quad\Rightarrow\quad\lim_{n\rightarrow \infty} E[|X_n-X|]=0$
Consider a sequence of measurable functions $X_n$ and $X$ measurable on some probability space $(\Omega,\mathcal{F},P)$.
I want to show, that following holds
$$\lim_{n\rightarrow \infty} E[\min(|X_n-X|,1)]=... |
H: proof that $\mathbb{R}=\mathbb{Q}\cup\mathbb{Q}^{'}$
I'm not sure if this is correct or not, I'm trying to find a proof or disproof that $\mathbb{R}=\mathbb{Q}\cup\mathbb{Q}^{'}$, that means, the set of the real numbers is the union of the rational and irrational numbers, if possible I'd like a more explicative ans... |
H: Continuing previous discussion?
Regarding the following question's answer : Seeking a combinatorial proof of $2n^{n-3} = \sum_{m=1}^{n-1}\binom{n-2}{m-1}m^{m-2}(n-m)^{n-m-2}$
I have some additional questions:
1. why m can't be 0 or n?
2. Plus what (n-2 / m-1) refers to? why are we choosing m-1 from n-2?
AI: Conside... |
H: Find the recurrence relation for the probability that the number of successes is divisible by $3$
(Feller Vol.1, P.285, Q.21) Let $u_n$ be the probability that the number of successes in $n$ Bernoulli trials is divisible by $3$. Find a recursive relation for $u_n$.
Answer: $u_n = q^n + \sum_{k=3}^n {k-1 \choose 2}... |
H: Some question about proving $\displaystyle\limsup_{n\to\infty}|\cos{n}|=1$ by using density of $\{a+b\pi|a,b\in\mathbb{Z}\}$
I have seen Proving $\displaystyle\limsup_{n\to\infty}\cos{n}=1$ using $\{a+b\pi|a,b\in\mathbb{Z}\}$ is dense and got this question.
Hagen von Eitzen gave the solution as following:
Pick an i... |
H: Intersecting families
Suppose A, B, C are 3 subsets of the set {1,...,n} Where each pair has nonempty intersect. Is there any intersecting family F from {1,...,n} subsets where F cardinality is equal to 2^(n-1) and F contains A, B and C?
AI: HINT: If $A\cap B\cap C\ne\varnothing$, it’s pretty easy to find such an $... |
H: Computing Binomial Distribution Expectation
How do I compute $E(2^{X}3^{(1-X)})$ given $X \sim Bin(1,p)$. Note that $X = 1$ with probability $p$ and $0$ otherwise.
AI: You mean that $X$ is an indicator random variable with probability of success equal $p$? In that case, if $X=1$ the expectation equals 2 and if $X=0... |
H: Is the nullspace of a matrix's transpose equal to the nullspace of its RREF's transpose?
Consider a matrix $ A $ and its RREF $ B $. Are $ Null(A^\intercal) $ and $ Null(B^\intercal) $ equal?
How should I go about this problem?
AI: No. Conceptually, the row reduced echelon form of a matrix $A$ must be in the form o... |
H: Probability of a Uniform Random Variable
Let $X_1, X_2, X_3$ be iid Uniform (0,1) random variables.
How do I find the probability that $X_{\min} = \min[X_1,X_2,X_3]$, is between 0 and 1/2?
AI: For the min not to be $\le 1/2)$, all have to be $\gt 1/2$.
$P(X_{min}\le 1/2)=1-P(X_{min}\gt 1/2)=1-\prod_{k=1}^3 P(X_k\gt... |
H: Is it possible to make curvature of sine wave equal to that of a parabola?
Suppose there is a symmetric parabola pointing downwards now we only consider the part above the x axis so is it possible to make curvature of sine wave equal to that part above x axis
So that it coincides with that part of parabola above x ... |
H: Another proof of $\displaystyle\limsup_{n\to\infty}|\cos{n}|=1$
I have seen a proof of $\displaystyle\limsup_{n\to\infty}|\cos{n}|=1$ by using density of $\{a+b\alpha: a,b\in \mathbb{Z}\}$ in $R$, where $\alpha$ is irrational.
Here I give another proof of as following:
See this article, a special case is that the... |
H: How to prove statement regarding directional derivatives and gradients
The question asked me to prove that
$$\|\nabla f\|^2 = (D_{u}f)^2 + (D_{v}f)^2$$
whenever vectors $u$ and $v$ are perpendicular.
How can I prove this?
AI: Hint:
$$\Vert \nabla f \Vert^2=(\nabla f\boldsymbol{\cdot}\underline{u})^2+(\nabla f \bold... |
H: Property of closure relation.
I am reading $\textit{On the Foundations of Combinatorial Theory II. Combinatorial Geometries}$. They give a definition.
A closure relation on a set $S$ is a function $A\mapsto \bar{A}$ defined for all subsets > $A\subseteq S$, satisfying
$$
\overline{A}\subseteq S,\quad A\subsete... |
H: Continuous Distribution and Probability density function Question
Suppose that $X_1, X_2, X_3$ denote a random sample of size three from a continuous distribution with probability density function (pdf)
$f_X(x) = \frac{1}{\theta}e^{\frac{−x}{\theta}}, x>\theta.$
Consider the following three estimators for $\theta:\... |
H: Nested Interval Property and uniqueness
Is the common element contained within $\bigcap\limits_{i = 1}^{\infty} I_j$ where $I_j$ is a closed interval unique?
AI: Hint: Consider $I_j = [-1/j, \ 1\!+ \!1/j]$
With this in mind, what must be true of nested closed intervals $I_j$ if they are such that $\displaystyle \bi... |
H: Showing $\sum_{k=0}^{n+1} \binom n k \frac{(-1)^k}{(n+k)(n+k+1)} = \sum_{k=0}^{n+1} \binom {n+1} k \frac{ (-1)^k}{n+k}$
The background context is this old MSE question here. Essentially I was trying to write up a proper response for the question, but I'm stuck on one part myself. One goal for the question is to pro... |
H: Definition of a rational number.
In high school, in definition of rational number we used to say,
"A number, which can be written in the form $\frac{p}{q}$, where $p,q(\neq0)\in\Bbb{Z}$ is called rational number."
But now I am realising that this is not a definition but it is a characterisation of rational numbers.... |
H: Why is $f(x+h) = 3x+3h-1$ when $f(x)=3x-1$?
$f(x+h) = 3x+3h-1$ when $f(x)=3x-1$
Is there some kind of factoring that gets you $3x+3h-1$? It looks the 3h comes out of nowhere.
AI: $$
f(x) = 3x - 1 \implies f(x+h) = 3(x+h) - 1 = 3x + 3h - 1
$$ |
H: Find all $(x,y)$ pairs : $x,y$ $\in \mathbb{Z}$ such that :- $x^4 - 4x^3 - 19x^2 + 46x = y^2 - 120.$
So here is the Question :-
Find all $(x,y)$ pairs : $x,y$ $\in \mathbb{Z}$ such that :-
$$x^4 - 4x^3 - 19x^2 + 46x = y^2 - 120.$$
What I tried :- I factored the LHS and got as :- $$x(x - 2)(x^2 - 2x - 23) = y^2 - 12... |
H: Proof Hints: Teichmüller-Tukey Lemma
Synopsis
Why this is NOT a duplicate (as far as I'm aware).
After having read the other posts on StackExchange about this lemma, many of them rely on concepts like well-ordering and other things I haven't learned yet. All I have learned so far is the axiom of choice and basic ca... |
H: Expected value for random variables based on Poisson distribution
I have to calculate the limit of the sequence $E((\frac{X_n-n}{\sqrt{n}})^+)$, where $X_n$ is Poi(n) and $\xi^+$ means "Maximum of $\xi$ and $0$". I get $e^2/\sqrt{2\pi}$ and I am not sure that this is correct. My calculation is straightforward, usin... |
H: IMO 1992 Problem 6
For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares.
a.) Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$.
Now solution says for part a)
Represe... |
H: Name of Math Symbol $"\mapsto"$ in the expression $\mathbf{x} \mapsto A \mathbf{x}$
I do not know the name of the math symbol with the arrow bracket pointing to the right. Any help identifying it or resource to find it would be helpful. I already checked the LA symbols on Wikipedia and no look. Thanks!
AI: The sym... |
H: how to derive the cross enthropy equation from the information theory?
This is the q(xi) distribution of information theory
(1/2) is bit information and $l_{i}$ is the length of the number of bit
then the cross enthropy derived to
I'm lack of experiecne to handling the log so can you show me how $-E_{p}[ln(q... |
H: $A$ could've done a job in 6 hours longer, $B$ in 15 hours longer and $C$ in twice the time needed if they work together. How long together?
I am trying to solve this question:
A, B and C do a piece of work together. A could've done it in 6 hours longer, B in 15 hours longer and C in twice the time. How long did it... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.