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H: What does "analytic" mean in the context of control system
Consider LTI causal and finite-dimensional system $G$ and let $\hat{G}$ be its Laplace transform.
We say that $\hat{G}$ is stable if it is analytic in the closed right half-plane (Re $s>0$).
What "analytic" means here?
I know that $\hat{G}$ is stable if d... |
H: Fast way to determine $P^{-1}$ for diagonalising a $3 \times 3$ matrix?
Diagonalise the following matrix:
$$A = \begin{pmatrix}-1&-2&2\\ \:4&3&-4\\ \:0&-2&1\end{pmatrix}$$
I know I have to represent the matrix $A$ as $A = P^{-1}SP$, where $P$ is a matrix with only the eigenvalues of $A$ in the diagonal, and $P$ i... |
H: What is the technique I should use here $\phi_{1}(x)=e^{x^{2}}, y^{\prime \prime}-4 x y^{\prime}+\left(4 x^{2}-2\right) y=0$
I am asking to solve this differential equation while I am given one of its solutions, I am not familliar with this kind of equation, it reminds me ODE but the coeficciant are constants so I ... |
H: Create plot for random walk
Is there any tool, that can create such a plot?
AI: The position of a randomly walking object is given by:
$$
S_n=\sum_{i=1}^{n}X_i
$$where in its simplest form, it is assumed that $X_i$s are iid and $\Pr\{X_i=1\}=\Pr\{X_i=-1\}={1\over 2}$. To sketch $S_n$ in terms of $n$, you can consi... |
H: Conjugate of quaternion doesn't give expected result
I'm using this site to play with quaternions.
All of my quaternions are unit quaternions.
I find quaternion of some Euler Angles(x, y, z) by using the website -inputs are degree and ZYX order Euler- and then by inputting the conjugate of the founded quaternion, ... |
H: Norm of convolution of $f$ and $g$ where $f \in L^1(R)$ and $g \in L^p(R)$
Here is the question:
For $f \in L^1(R)$ and $g \in L^p(R)$, define $f*g(x)=\int_{- \infty}^\infty f(x-y)g(y)dy$ .
prove that $f*g\in L^p(R)$ and $||f*g||_p\le||f||_1||g||_p$
This question is from Real and Complex analysis by Walter Rudin... |
H: Monopoly: pricing.
The total costs of a monopolistic firm are CT = 10q + 2 * (q ^ 2).
Assuming that the firm decides to produce q* = 10 and that for that level of production the price elasticity of demand is equal to:
| epsilon q, p | = 3,
determine the price charged by the company.
My partial solution:
Marginal co... |
H: If $AX=BX$ where $A$, $B$, $X$ are all square matrices, when can I calim $A$ = $B$?
My matrix knowledge is on the rusty end and this question has bothered me for a while now recently. I have always assumed that if $A$, $B$, $X$ are all square matrices, and when $AX=BX$ where $X$ is invertible, I can claim $A$ = $B... |
H: Let $a\in\mathbb{C}, |a|=1$ and $c$ an irrational real number. Prove: $a^c$ is dense in the unit circle.
Let $a\in\mathbb{C}$ a complex number such that $|a|=1$ and $c$ an irrational real number.
Prove: The set $a^c$ is dense in the unit circle.
The problem is taken from Notes on Complex Function Theory by Donald... |
H: Taylor series for 1/(1-x).
I am trying to write a Taylor series for $$ f(x)=\frac{1}{1-x}, \ x<1 \ .$$
In most sources, it is said, that this function can be written as a Taylor series, if $$ \left| x \right|<1. $$
However, I don't get the same condition for x.
Because $$ f^{\left(n\right)}\left(x\right)=\frac{n!}... |
H: positive semidefiniteness of the Hessian of $f\circ g$
Let $\Omega\subset\mathbb{R}^n$ be open, $g\in C^2(\Omega,\mathbb{R}^n)$ and $D^2g(x)$ be positive semidefinite for all $x\in\Omega$. Let furthermore $f\in C^2(\mathbb{R},\mathbb{R})$ with $f',f''\geq 0$. Show that $D^2(f\circ g)(x)$ is positive semidefinite, f... |
H: Show that $g_n$ converges to $g$ uniformly.
Problem
Let $f:\Bbb{R}\times[0,1]\rightarrow\Bbb{R}$ be a continuous function and $\{x_n\}$ a sequence of reals converging to $x$. Define
$g_n(y)=f(x_n,y),\hspace{0.5cm}0\le y\le1$
$g(y)=f(x,y),\hspace{0.9cm}0\le y\le1$.
Show that $g_n$ converges to $g$ uniformly on $[0,... |
H: Solution to partial differential equation by separating variables
Could someone please show me how to calculate this math problem?
By separating the variables, find the solution to the partial differential equation
$$\frac{\partial^{2} u}{\partial x^{2}}-\frac{1}{4} \frac{\partial^{2} u}{\partial t^{2}}-u=0$$
in th... |
H: Continuity of a two variables function - $\epsilon-\delta$ definition
Let's suppose we have a function $$u=u(x,t):[-\pi,\pi]\times(0,+\infty)\to \mathbb{R}$$ such that $u$ is continuous on $[-\pi,\pi]\times[t_0,+\infty)$ for every $t_0>0$.
Can we use the $\epsilon-\delta$ definition (ie, definition of continuity be... |
H: Vector space over field span the same space .
Let V be vector space over field $R$, and $v_1, v_2, v_3 \in V$.
Prove that span(B) = span(A), when $B= \{v_1 + 2 v_2, v_1 + v_2 - v_3,
5v_3\}$ and $A = \{v_1, 4v_2, 6v_3\}$
It's clearly to prove that when $v_1, v_2, v_3$ are independent.
But I don't have idea how t... |
H: Simply Connected Complex Domains
I am studying Complex Analysis using Sarason's text. In it he says that a domain $G \subset \mathbb{C}$ is simply connected if the extended complement $\bar{\mathbb{C}} \setminus G$ is a connected set. Throughout the text a domain is a nonempty, open connected set.
Is there a way to... |
H: A problem concerning a parallelogram and a circle
Sorry for the ambiguous title. If you can phrase it better, feel free to edit.
"A parallelogram $ABCD$ has sides $AB = 16$ and $AD = 20$. A circle, which passes through the point $C$, touches the sides $AB$ and $AD$, and passes through sides $BC$ and $CD$ at points ... |
H: Euclidean algorithm worst case: why never be more than five times the number of its digits (base 10)?
I read the Euclidean algorithm of Wikipedia page(https://en.wikipedia.org/wiki/Euclidean_algorithm), but I was stuck at worst-case proof. At the second paragraph, it says:
For if the algorithm requires $N$ steps, ... |
H: Does the operator $(\hat{f}\cdot m )^\vee$ maps Schwartz in it self?
Given $m \in L^\infty$ and $\phi \in \mathcal{S}$ a Schwartz function, is it true that $(\hat{f}\cdot m)^\vee$ is a Schwartz function??
I trying to prove this so I could conclude that operator of the form $(\hat{f}\cdot m)^\vee$ maps $\mathcal{S}... |
H: Split $C[0,1]$ into direct sum of two infinite-dimensional subspaces
How to split $C[0,1]$ in the form of a direct sum of two infinite-dimensional subspaces?
AI: A more concrete decomposition. For every continuous function $f$ of $[0,1]$ let $f^*$ be the function $f(x)+f(1-x)$ and $f^{**}$ be the function $f(x)-f(1... |
H: Measure theory problem - Integrable functions- Showing f=0 a.e
Suppose $f \in L^1(R) $ and satisfies $\limsup_{h\to0}\int_R |\frac{f(x+h)-f(x)}{h}| = 0$ then show that $f=0$ a.e.
I'm not really sure how to approach this problem ( I have a feeling that we need to use the dominated convergence theorem but no idea how... |
H: Calculate $\lim_{n \rightarrow \infty} \phi_{\frac{S_n}{n}}(t)$
They give me $X_1, X_2, X_3, ...$ random variables that are independent and with the same distribution and they ask me to calculate $\lim_{n \rightarrow \infty} \phi_{\frac{S_n}{n}}(t)$.
Obs: $S_n = X_1 + ... X_n$ and $E(X_i) = \mu$.
Well my first idea... |
H: Self-adjoint operator $H$ is not in the $C^*$-algebra generated by unitary operator $U$?
Let $U, H$ be operators on Hilbert space $L^2\left(\mathbb{T}, \frac{d \theta}{2\pi}\right)$($\mathbb{T}$ is unit circle $S^1$), for any $f\in L^2\left(\mathbb{T}, \frac{d \theta}{2\pi}\right)$,
$$
(Uf)(e^{i\theta}) = e^{i\the... |
H: The natural domain of $f(x)=\frac{\sqrt{5−x^2}}{(x−1)(2x−1)}$
Background
As there is a radical in the numerator, this restriction would need to be applied first.
$$f(x)=\frac{\sqrt{5−x^2}}{(x−1)(2x−1)}$$
Thus, $x=-\sqrt{5}$, and for the denominator, $x=1, x=\frac{1}{2}$
The natural domain would then be: $[-\sqrt{5}... |
H: Taylor series for function of two variables
In my textbook taylor series for function of two variable has been written like this:
$$ f(a+h,b+k)=f(a,b)+f_x(a,b)h+f_y(a,b)k+\frac 1 2 (f_{xx}(a,b)h^2+ 2hkf_{xy}(a,b)+f_{yy}(a,b)k^2)+h^2+k^2)^{\frac{3}{2}}B(h,k) $$
in which $B(h,k)$ is a bounded function around the cent... |
H: Is $f(p)=\beta p^{\alpha}$ the unique nonnegative function on $[0,1]$ satisfying $\frac{f(p)}{f(1-p)}=\left(\frac{p}{1-p}\right)^{\alpha}$?
Let $f(p)\geq 0$ be a function on $[0,1]$. Suppose
$$
\frac{f(p)}{f(1-p)}=\left(\frac{p}{1-p}\right)^{\alpha}
$$
for some constant $\alpha>0$ and all $p\in [0,1]$. A specif... |
H: Prove there isn't an increasing $\omega_1$ sequence on real set
Although I have read that it's quite easy to prove there isn't an $\omega_1$ increasing sequence on real set I spent a lot of time figuring out why it happens and finally I think I made it, but I'm not sure about it. Here is my approach.
First of all, ... |
H: Calculate $\int_{1}^{\phi}\frac{x^{2}+1}{x^{4}-x^{2}+1}\ln\left(x+1-\frac{1}{x}\right) \mathrm{dx}$
$$\int_{1}^{\phi}\frac{x^{2}+1}{x^{4}-x^{2}+1}\ln\left(x+1-\frac{1}{x}\right) \mathrm{dx}$$
Insane integral! So far I have tried to complete the square for the denominator then substitute and use taylor series for t... |
H: Calculating the arc length of a difficult radical function
I am fairly new to calculus and self learning integration from home has been challenging so I'm sorry if I make any mistakes. I want to work out the arc length of:
$y = \sqrt{7.2 (x-\frac {1}{7}}) - 2.023, [0.213, 0.1.27]$.
I have used the definition of a d... |
H: In Wikipedia's Statement of The Universal Approximation Theorem, is it taking identity activation on the output layer with no bias?
See the Universal Approximation Theorem (arbitrary width) on wikipedia or below.
The universal approximation theorem (arbitrary width) is talking about a neural network with 1 hidden l... |
H: Convergence of $e^x$
I am working with the Maclaurin series for $f(x)= e^x$. I am in the point of proving that the series converges to $f(x)$ for all $x$, using Taylor's theorem with remainder I have to show the following:
$$
\mathop {\lim }\limits_{n \to \infty } \frac{{\left| x \right|^{n + 1} }}{{(n + 1)!}} = 0.... |
H: Calculating the value of a determinant
$\begin{vmatrix}
1 & 2 & 1 & -2 & 1 & 4\\
-3 & 5 & 8 & 4 & -3 & 7 \\
2 & 2 & 2 & -1 & -1 & -1 \\
1 & 2 & 3 & 4 & 5 & 6 \\
2 & 4 & 2 & -4 & 2 & 8 \\
3 & 5 & 7 & 11 & 13 & 17 \\
\end{vmatrix} $
I tried to make an upper or under triangle matrix, where the value of the determin... |
H: When a Radon measure finite in bounded sets?
Let $(X,d)$ be a locally compact separable complete metric space (locally compact Polish metric space) and $\sigma$ a Radon measure i.e. a Borel measure finite in compact sets $\sigma(K)<\infty$ and regular: For every $A \in \mathcal{B}(X)$
$$\sigma(A)=\inf\{ \sigma(O): ... |
H: Submodule generated by $N\cup N'$ is subset of $N+N'$
I want to show that for submodules $M,M'$, the submodule generated by $M\cup M'$ is equal to the submodule $M+M'$. The inclusion $M+M'\subseteq$ is easy from the closedness of the submodules under addition. However, I don't have a clue how to do the reverse incl... |
H: Using expected value of number of attendees to an event to calculate expected revenue
In the problem you pre-sell 21 non-refundable tickets values at 50 dollars to an event and can only accommodate 20 people. But in the event the 21st person shows up you must pay the person who's out $100. Each person has a 2% of... |
H: Clarification for maximizing the Rayleigh Quotient
Suppose I have a Hermitian matrix $M$ with orthonormal eigenbasis $\{x_1, \ldots x_n \}$, then for a unit vector $v$, I am reading about maximizing the Rayleigh quotient $R(M,t) =\frac{t* M t}{t^*t}$ = $t^*Mt = \lambda_i t_i^2$, where $t_i$ is the $i'th$ co-ordinat... |
H: Ratio of area covered by four equilateral triangles in a rectangle
The following puzzle is taken from social media (NuBay Science communication group).
It asks to calculate the fraction (ratio) of colored area in the schematic figure below where the four colored triangles are supposed to be equilateral. The sides... |
H: ODE $ y''-3y'+2y=4(2-a)e^{-ax}$
I have to solve the system
$y'=-y-2z+2e^{-ax}$,
$z'=3y+4z+e^{-ax}$.
I expressed z from the first equation and plugged it into the second equation. The result is equation $y''-3y'+2y=4(2-a)e^{-ax}$. How to solve this equation?
Any help is welcome. Thanks in advance.
AI: $$y''-3y'+2y=4... |
H: An ultranet $x_\lambda$ is frequentely in $Y$ if and only if it is residually too.
Definition
If $x_\lambda$ is a net from a directed set $\Lambda$ into $X$ and if $Y$ is a subset of $X$ then we say that $x_\lambda$ is redisually in $Y$ if there exsit $\lambda_0\in\Lambda$ such that $X_\lambda\in Y$ for any $\lambd... |
H: Help in solving $\lim_{x\to 0}\frac{x-e^x+\cos 2x}{x^2}$
The question is
$$\lim_{x\to 0}\frac{x-e^x+\cos 2x}{x^2}$$
I tried solving it
as follows
My method
My answer is $-2$ but actual answer is $-5/2$ so where is my method wrong?
AI: The answers given above are all giving methods to solve the problem. But this ... |
H: Show that in [0,1] with its usual topology there exists a net having no convergent strict subnet
I only have difficulties in the final step: Show that {$x_y:y\in I$} has no convergent strict subnet.
My efforts:
With the construction, $I$ is a minimal uncountable well-ordered set. Thus it has the following properti... |
H: Subset of $M_2(\mathbb{R})$ isomorphic to a field?
Consider matrices of the form $\begin{bmatrix} a & b \\ -b & a-b \end{bmatrix}$ with entries from $\mathbb{R}$, closed under addition and matrix multiplication (see Unital rings within matrices ). This forms a unital and commutative ring.
Moreover, the determinant ... |
H: How to find $\lim_{n \to \infty} \left( \frac{2^n - 1}{2^n} \right)^{\log_2 n}$
I am trying to show that something occurs with high probability, and my final expression is
$$\lim_{n \to \infty} \left( \frac{2^n - 1}{2^n} \right)^{\log_2 n}$$
Based on trying some very large numbers it seems that this does indeed c... |
H: Proving that $\tau=\bigcap \limits_{i\in I} \tau_i$ is a topology on $X$
In my general topology textbook one exercise ask me to prove the following:
If $\tau_1,\tau_2,...,\tau_n$ are topologies on a set $X$, then $\tau = \bigcap \limits_{i=1}^n \tau_i$ is a topology on $X$.
If for each $i \in I$, for some index s... |
H: The operator norm is defined based on the supremum or equivalently the maximum.
The definition is
$$\|A\| = \sup_{x \neq 0} \frac{\|Ax\|}{\|x\|} = \max \left\{ \|Ax\|: \|x\|=1 \right\}$$
but how is maximum coming in place?
AI: I assume here you mean that you are working in a finite dimensional normed space. In that... |
H: Evaluate $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}$
Evaluate $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}$
My attempt:
Let $\sqrt{a+b+\sqrt{\left(2ab+b^2\right)}}=\sqrt{x}+\sqrt{y}$
Square both sides: $a+b+\sqrt{\left(2ab+b^2\right)}=x+2\sqrt{xy}+y$
Rearrange: $\sqrt{\left(2ab+b^2\right)}-2\sqrt{xy}=x+y-a-b$
That's where... |
H: schauder basis vs isomorphisms
My doubt is about a space with schauder basis. If a space has schauder basis so can i say that it has an isometric isomorphism with some space? For an exemple: $c$ has a schauder basis, can i considere another set a dual space who it is isomorphic?
I am thinking this because i know th... |
H: Almost everywhere differentiable (composition)!
Let $f:\mathbb{R}\to\mathbb{R}$ be a function differentiable almost everywhere and $g$ a function defined by
$$g(t)=\arctan(f(t))$$
I read in a paper that $g$ is differentiable almost everywhere. Can someone tell me why $g$ is differentiable almost everywhere?
AI: If ... |
H: Gradient in linear regression with weights
From 3.3 in Pattern Recognition in Machine Learning, I am asked to obtain weights for a regression with a weighted square loss function.
That is, $E(w,x) = \sum_{j=1}^n r_j(y_j - x_j^Tw)^2$
where $r_j$ is the weight for example $j$. I'm trying to formulate this as a vector... |
H: If a random variable $X$ is integrable, how do we show that $X^2$ is also integrable?
If we know that a random variable $X$ is integrable, i.e. $\mathbb{E}(|X|) < \infty$, how do we show that $X^2$ is also integrable?
AI: As pointed out in the comments, the claim isn't true. For example, we can consider a random va... |
H: Is there a geometric analog of absolute value?
I'm wondering whether there exists a geometric analog concept of absolute value. In other words, if absolute value can be defined as
$$
\text{abs}(x) =\max(x,-x)
$$
intuitively the additive distance from $0$ to $x$, is there a geometric version
$$
\text{Geoabs}(x) = \... |
H: Support Vector Machines (SVMs), unclear math steps
I am studing the maths behind the Support Vector Machines (SVMs), but there are two not clear steps.
Following the video of 16. Learning: Support Vector Machines (MIT OpenCourseWare, minutes 14:24), we have the following steps
$Width = (\bar{x}_{+}-\bar{x}_{+}) \cd... |
H: Union of intervals for $f(x)=\frac{\sqrt{7−x^2}}{(x−3)(4x−2)}$
Background
From the radical, we know $7−x^2≥0$ so $x^2≤7$, so $−\sqrt{7}≤x≤7$
$$x≠3, x≠\frac{1}{2}$$
The natural domain would then be: $[−\sqrt{7},\frac{1}{2})∪(\frac{1}{2},3)∪(3,\sqrt{7}]$
However, I checked a calculator and the solution was: $[−\sqrt{... |
H: Change of variables in basic differential equation from Hamming's Art of Doing Science and Engineering
I'm reading Richard Hamming's "The Art of Doing Science and Engineering" and I'm trying to figure out whether there's an error in the book or I'm misapplying some change of variable rules.
In Chapter 2 he derives ... |
H: easiest proof of the Prime Number Theorem to study and teach?
I know there are several variants of proofs for the Prime Number Theorem.
Which one is the easiest one to study and then re-teach?
By easiest, I mean those that assume minimal knowledge beyond secondary school mathematics.
For example, most school leaver... |
H: Point of indeterminacy of the projection map (exercise I.4.3 Hartshorne)
I am trying to prove that the morphism $\varphi : W = {\mathbb{P}}^2 \setminus \big\{ [0:0:1] \big\} \rightarrow {\mathbb{P}}^1$ given by $\varphi([a_0 : a_1 : a_2]) = [a_0 : a_1]$ cannot be extended to the point at infinity.
My approach is as... |
H: Showing that $(\mathbb{Q},+)$ has no maximal normal subgroup
In A Course in Abstract Algebra Volume: Author(s): Vijay K. Khanna, S.K. Bhamri it is proven that:
$\mathbb{Q}$ with $+$ has no maximal normal subgroup.
Proof: Let $H$ maximal normal subgroup of $\mathbb{Q}$. Then $\mathbb{Q}/H$ is simple and so $\mathbb... |
H: Direct proof that integral of a function does not depend on the $\sigma$-algebra used to define it?
If $\mathcal{G}\subset\mathcal{F}$ are two $\sigma$ algebras on a set $X$, $\mu$ is a nonnegative measure on $(X,\mathcal{F})$ and $f:X\to[0,+\infty]$ is $\mathcal{G}$-measurable, then there are two possible definiti... |
H: What is the "most obtuse" triangle that can fit on a sphere?
Often, when someone introduces the idea of non-euclidean geometry they give the examples of spherical and hyperbolic geometry. To help visualize these concepts, they'll usually compare the sum of the angles of ordinary triangles in each of these geometrie... |
H: Separable first order differential equivalence: how can I get expected outcome?
I am told a colony at any time has a population at time $t$ according to the rule:
$$\frac{dB}{dt} = kB + I$$
Where $B$ is the colony size, $k$ is a positive constant, and $I$ is some constant. I want to show that the colony size at any... |
H: A probability counterexample for the measure $Q(A) = \int_{\Omega} X \mathbb{1}_A \mathbb{1}_B \ \text{d} \mathbb{P}$
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space, $B \in \mathcal{A}$ and $X : \Omega \to (0,\infty)$ an integrable random variable. Define $$Q : \mathcal{A} \to [0,\infty) \quad \tex... |
H: The angle between a line and a normal vector
The problem I am trying to solve is below:
What is the angle formed by the line
$(1,2,0) + t(-1,2,1)$ and a normal vector of the plane $x+y-z = 4?$ Give your answer in degrees.
I am having a little bit of trouble solving this problem. I have watched videos and looked at ... |
H: Calculating derivative of linear distance
This is an exercise from Morris Kline's "Calculus: An Intuitive and Physical Approach".
If an object moves along a circle of radius $R$, its position can be described by specifying the angle $\theta$ through which it has rotated. The derivative of $\theta$ with respect to ... |
H: Show $\det(F_n)=1$ for all $n$
Consider the $n\times n$ matrix $F_n= (f_{i,j})$ of binomial coefficients
$$f_{i,j}=\begin{pmatrix}i-1+j-1\\i-1\end{pmatrix}$$
Prove that $\det(F_n)=1$ for all $n$.
My current idea is to apply Leibniz formula for determinants and induction, but it seems too complicated. Any better id... |
H: Difference b/w $p$ and $P(X)$ i.e. output of Binomial Distribution ($BD$)? Is it possible to have $p=1$ but $BD=0$?
I have a confusion with Binomial Distribution. For Binomial Distribution we use the formula:
$P(x) = {n \choose k} \cdot p^x \cdot (1-p)^{n-x}$
Now let's suppose $p=1$ but if we put this $1$ in $P(x)$... |
H: Showing $f(h)=g^{-1}h^{-1}gh$ for any $g \in G, h\in H$ where $H$ is normal subgroup of $G$.
Let $H$ be a finite normal subgroup of $G$. Show that for any $g\in G$, the map $f:H \to H$ defined by $f(h)=g^{-1}h^{-1}gh$ is bijective.
Edit to problem:
Let $H$ be a finite normal subgroup of $G$. Let $g\in G$ have order... |
H: Evaluate $\lim\limits_{n \to \infty}\frac{|\sin 1|+2|\sin 2|+\cdots+n|\sin n|}{n^2}.$
It's well-known that
$$\lim\limits_{n \to \infty}\frac{|\sin 1|+|\sin 2|+\cdots+|\sin n|}{n}=\frac{2}{\pi},$$which can be obtained by the uniform distribution.
Can it be used directly to solve the present problem?
AI: Yes, you may... |
H: Transversality of two mappings and diagonal
I've never taken differential topology and am confused by the definition of transversality, and while trying to solve the following I got stuck.
Given smooth manifolds and maps $f:M\to N$ and $g:P\to N$, show that $f$ and $g$ are transversal to each other if and only if ... |
H: Convergence of real integral
I'm trying to analyze the convergence of following integral:
$$
\int_{0}^{1}\frac{dx}{\sqrt[3]{x(e^{x}-e^{-x})}}
$$
Currently not being able to get a hint on how to proceed to do it, any help is really appreciated.
So far I've tried:
1.
$$
\frac{dx}{\sqrt[3]{x(e^{x}-e^{-x})}} >= \frac{d... |
H: Suppose $\lim_{n\to\infty}f(i,n)=\infty$ for all $i$, does that mean that $\lim_{n\to\infty}f(n,n)=\infty$ as well?
Suppose that $f:\mathbb{N}\times\mathbb{N}\rightarrow \mathbb{R}$ is a function such that for each $i\in\mathbb{N}$, as $n\rightarrow \infty$,
$$ f(i,n) \rightarrow 0.$$
Does this imply that $f(n,n) \... |
H: Proving convexity of a set
The problem
Given a convex set $S \subset \mathbb{R}^n$. Prove that $\overline{S}$ is also convex.
What i've tried:
To solve this problem i've tried using the definition that $\overline{S}$ is composed by the interior set and the frontier set of $S$. Hence, $S$ is convex, so, the interior... |
H: Polynomials of degree $n$ with $\Bbb{F}_p$ are always reducible in $\Bbb{F}_{p^n}$
This is a rather basic question, but I can't seem to find any reference here on StackExchange. Is it true that, given a polynomial $p(x) \in \Bbb{F}_p[x]$ of degree $n$, we have that $p(x)$ is always reducible in $\Bbb{F}_{p^n}[x]$?
... |
H: Can we say that , for a parabola , no such point exists inside the parabola which is midpoint of more than one chord?
I was reading about the properties of parabola, amongst which one of the property was that parabola has no centre.
I tried to prove it by considering four parametric points on the parabola i.e. $P_1... |
H: How to prove $\tanh^{-1}(\sin x)=\sin^{-1}(\tan x)$
Here's what I attempted:
$$ y =\tanh^{-1}(\sin x)$$
$$\tanh y=\sin x$$
But I don't know what to do after this. Please help me.
AI: These two are not equal. Let $x = \pi/3$ for instance. Then
$$\text{arctanh}(\sin(x)) \approx 1.32$$
(per Wolfram), while you can't e... |
H: Is this norm equivalent to the $\ell_1$ norm?
I am studying for my qualifying exams and was asked to prove or disprove that the following norm is equivalent to the $\ell_1$ norm:
$$\lVert x \rVert' = 2\left\lvert \sum_{n=1}^{\infty}x_n \right\rvert + \sum_{n=2}^{\infty}\left(1+\frac{1}{n}\right) \lvert x_n\rvert$$
... |
H: Convergence of double series involving minimum
Determine the convergence of the series $$\sum_{n,m\in\mathbb{Z}:|n-m|>10,|m-10|>0}\min\{|n|^{-10},|n-m|^{-10}\}.$$
I tried solving this using "an integral test", saying
$$
\sum_{n,m\in\mathbb{Z}:|n-m|>10,|m-10|>0}\min\{|n|^{-10},|n-m|^{-10}\} \le \int_{|x-y|>10,|y-1... |
H: If a circle and parabola touch each other and also have common root then what's the relationship between their coefficient?
Actually this question is from physics (projectile motion) but i believe its related more to maths here equation of parabola
$$y=ax-5x^2-5(ax)^2$$
And that of circle
$$x^2+y^2=(a/5(1+a^2))^2$$... |
H: Proof Verification for Baby Rudin's Chapter 4 Exercise 4
I'm trying to prove:
$f: X \to Y$ is continuous, $g: X \to Y$ is continuous, $E \subset X$ is dense in $X \implies$ $f(E)$ is dense in $f(X)$ and $\forall p \in E, g(p) = f(p) \implies g(p) = f(p) \forall p \in X$.
My attempt:
First, we show that $f(E)$ is ... |
H: Confusion over solution to Linear Transformation from P2 to P3
I'm trying to understand the solution to the question below. I warrant I'm probably confused over the notation.
In the question (attached below) it says that transformation T(p)[x] = xp(x-3), with standard basis for P2 and P3.
I assume you need to break... |
H: would n dimensional extreme point of Polytope hiding form all n-1 dimensional coordinate projection?
Given a Polytope $X \subset R^n$ and assume we have nonempty extreme point set $P(X)$.
We can do have $n$ different $n-1$ coordinate projection (namely hiding one coordinate). We use $h(X, i)$ indicates $i$th coordi... |
H: Expected value of coin game
You have four coins and I have four coins. We both throw the four and if your four sides equal to mine, I will give you 2 dollar and otherwise you give me 1. Will you do it?
I want to calculate the expected value of the game, and I'm not sure why this is wrong. Let X be the probability t... |
H: Sum of geometric series when exponent is $2n$, not $n$?
I have a probability below which denotes the chance of catching a fish.
$$P = \left(\frac{1}{4}\right) + \left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^2 + \left(\frac{1}{4}\right)\left(\frac{3}{4}\right)^4 + \dots $$
I can find a generalized form of $P$ by ... |
H: Probabilty with Combinations concept check
When we calculate the probability of a random selection of $3$ students being all boys, from a group of $6$ boys and $4$ girls, then we can just multiply $\mathbb P(\text{1st being boy}) \times \mathbb P(\text{2nd being boy}) \times \mathbb P(\text{3rd being boy})$ i.e. $\... |
H: Proof of the change of variables formula without using the Monotone Convergence Theorem
I recently encountered the problem Exercise 36 in Tao's An Introduction to Measure Theory. The link of an online version of this problem is here. Now I quote this problem as follows:
Exercise 36 (Change of variables formula)
Le... |
H: Find out the limit of $f(z)=(z-2)\log|z-2|, z\neq 2$ at the point $z_0=2$, or explain why it does not exist.
Question: Find out the limit of $f(z)=(z-2)\log|z-2|, z\neq 2$ at the point $z_0=2$, or explain why it does not exist.
My approach: Let $z=x+iy$, where $x,y\in\mathbb{R}$. This implies that $f(z)=f(x+iy)=(... |
H: Can the range of a linear transformation contains the null space?
Let $V$ be a finite vector space, and let $T$ be a linear transformation $T:V\rightarrow V$. If $\operatorname{null}(T)=\operatorname{span}\{\phi\}$, can $\operatorname{ran}(T)$ contains $\phi$, where $\phi$ is not the trivial vector?
I know that
$\... |
H: Bessel processes and Brownian motion
In class, we talked about Bessel process as a process which solves the SDE:
$$
dB=\frac{n-1}{2B}dt+\sum_{i=1}^n\frac{W_i}{B}dW_i
$$
Where $W_1,W_2,...,W_n$ are independent, standard Brownian motions.
We then showed that $B=||W||=(\sum_{i=1}^n(W_i)^2)^{1/2}$ solves this equation.... |
H: $(W_1\cap W_2)^{0}=W_1^0+W_2^0$
If $W_1$ and $W_2$ are subspaces of a finite dimensional vector space $V$, then $$(W_1\cap W_2)^{0}=W_1^0+W_2^0$$.
Attempt Suppose $f\in W_1^0+W_2^0$. Then $f=f_1+f_2\in W_1^0+W_2^0$ ,where $f_1\in W_1$ and $f_2\in W_2^0.$
Now for $z\in (W_1\cap W_2)$, $f(z)=(f_1+f_2)(z)=f_1(z)+f_... |
H: Can this inductive proof that $\sum_{i=0}^n2^{2i+1}=\frac23(4^n-1)$ be simplified?
The general structure of equations I've used for the inductive step for proofs with a summation is something like:
We'll prove that $\sum_{i = 0}^{n + 1} (\text{something}) = (\text{closed form expression})$
\begin{align}
\sum_{i = 0... |
H: What Is Bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$
Hello every what is bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$?
I tried to square up and I got $100^{200}$ or $99^{99} \cdot 101^{101}$ and I don't have an idea how to continue.
AI: Taking logarithms, we see that we want to compare $f(100)$ an... |
H: I want to show some function as the norm of two vectors.
I want to see $[\sin(x-y)]^2$ is rewritten by the square of the norm of the difference of two vectors like following (It is a sample and it's not right).
$$
\| (\sin(x) , \cos(x) \sin(x) ) - (\sin(y) , \cos(y) \sin(y) )\|
$$
Please tell me how to consider thi... |
H: How can I integrate $\int\frac{e^{2x}-1}{\sqrt{e^{3x}+e^x} } \mathop{dx}$?
How can I evaluate this integral $$\int\dfrac{e^{2x}-1}{\sqrt{e^{3x}+e^x} } \mathop{dx}=\;\;?$$
My attempt:
I tried using substitution $e^x=\tan\theta$, $e^x\ dx=\sec^2\theta\ d\theta$, $dx=\sec\theta \csc\theta \ d\theta.$
$$\int\dfrac{\tan... |
H: Can I get $\frac{f(x)}{g(x)}$ integrable when $g(x)\neq0$ for $f(x),g(x)$ is integrable
Can I get $\frac{f(x)}{g(x)}$ (meaning $\int_{a}^{b}\frac{f(x)}{g(x)}dx$ is
convergent)integrable when $g(x)\neq0$ for $f(x),g(x)$ is integrable?
When $g(x)\geq C>0$, $\frac{f(x)}{g(x)}$ is integrable. But if $g(x)$ is just inf... |
H: On characterisation of smooth $G$ equivariant morphisms between Product manifolds with $G$ action
In particular I am interested in the following!
Let $M$ be a smooth manifold and $G$ be a Lie Group. Let $\rho: (M \times G)\times G \rightarrow M \times G$ be the smooth action of $G$ on $M \times G$ given by $(m,g).g... |
H: On $(0,\infty)$, the metrics $d(x,y)=|x-y|+|\frac1x-\frac1y|$ and $d_e(x,y)=|x-y|$ are equivalent.
Problem
Let $Y=(0,\infty)$ and define the metric $d(x,y)=|x-y|+|\displaystyle\frac1x-\displaystyle\frac1y|$ on $Y$. Let $d_e(x,y)=|x-y|$ be the usual Euclidean metric on $Y$, then show that both the metrics $d$ and $... |
H: a problem regarding projection maps on finite dimensional vector space
Let $V$ be a finite dimensional vector space over a field $F$ of characteristic zero. Let $E_1 , E_2, ...,E_k$ be projections of $V$ such that $E_1+E_2+...+E_k=I$. Show that $E_iE_j = 0$ for all $i\neq j$.
Hint: Use the trace function.
Using t... |
H: Conjunctive Normal Form evaluates true when atleast half of the clauses are true.
This is an Exam question.
Which of the Following is TRUE about formulae in Conjunctive Normal form?
-For any formula, there is a truth assignment for which at least half the clauses evaluate true.
-For any formula, there is a truth as... |
H: if $(X,\tau)$ is an $T_1$- space, then every subset of $X$ is a saturated set
In an general topology exercise I have to prove that if $(X,\tau)$ is an $T_1$- space, then every subset of $X$ is a saturated set (i.e. it is an intersection of open sets).
My approach:
Because $(X,\tau)$ is a $T_1$- space, then every s... |
H: Prove that $\frac{d^4\phi}{dx^4}=f(x)$ using integration by parts and fundamental theorem
A function f is continuous on $[0,∞)$ and $\phi(x) = \frac{1}{3!}\int_0^x(x−t)^3f(t)dt, x ≥ 0$. Show that
$\frac{d^4\phi}{dx^4}=f(x),\forall x\geq0$.
I can prove this using newton Leibnitz theorem but can't see how to do it by... |
H: Show that an operator is continuous
Let $X := C^1 [0,1]$ and define $||f|| := |f(0)| + \sup_{0\le t \le 1} |f'(t)|$. Now, consider the operator $T: X \to \mathbb{R}$ defined as
$$Tf = \int_0^1 f(t)dt$$
Show that T is continuous.
I know this should be proved with the closed graph theorem, but I want to ask if this... |
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