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1 vote

Proving that the function $f(x) := |x|^{\frac{\lambda - n}{p}} (1- \psi(x))$ satisfies two specific properties related with limits and supremums.

Your approach for property 1 seems sound. Let's proceed to address property 2. For property 2, we need to show that the expression tends to zero as $\xi$ approaches zero. To do this, let's expand the ...
Aueriga's user avatar
  • 355
2 votes
Accepted

How does this proof that for every infinite set $A$ there exists an injection $f : \mathbb{N} \rightarrow A$ rely on the axiom of choice?

Your proof, as written, uses a slightly stronger principle, Dependent Choice. The reason is that the choice of $f(k)$ depends on the previous choices. If, for example, you chose that $f(0)=a$, then $f(...
Asaf Karagila's user avatar
  • 394k
1 vote

Behavior of quadratic form on basis

I'm not going to delve as much into details as Nicholas did, I'm only going to point out that you might have misused the bilinearity. Indeed, let me use the bilinearity of the map $\psi(\cdot, \cdot)$ ...
Egor Larionov's user avatar
1 vote

Behavior of quadratic form on basis

You're right if $\mathbb F$ has characteristic different from 2. When $\mathbb F$ has characteristic 2, then the statement is just false as written: there is a (non-unique) matrix $a_{ij}$ ...
Nicholas Todoroff's user avatar
2 votes
Accepted

Am I using Euclid's Lemma in this informal argument for the fundamental theorem of arithmetic?

You need to be a little more careful with the "meaning that we can cancel out..." part. How would you express this formally? Since $p_1$ divides $\prod_{j=1}^{s}q_j$ then we can show almost ...
Cameron Buie's user avatar
1 vote
Accepted

$f(x)=\sqrt{\frac{x-1}{x-3}}$ and $g(1)=e$ which of the following options is/are correct?

I've confirmed that your solution of $g(x)=e^{\frac{1}{2-x}}$ satisfies the differential equation. $g$ is undefined (vertical asymptote) at $x = 2$, and continuous everywhere else. (i) Since $f'(x)$ ...
Dan's user avatar
  • 15.2k
0 votes

Prove or Disprove: Is there a connected planar graph with an odd number of faces where every vertex has a degree of 6?

As noted by MJD, it is well-known that there exists no planar graph that has a minimum vertex degree of 6 or more. Therefore, such a graph cannot exist. But because of this, both statements In a ...
return true's user avatar
1 vote

$f(x)=\sqrt{\frac{x-1}{x-3}}$ and $g(1)=e$ which of the following options is/are correct?

Posting this to let you know my thought on the matter. I am not sure whether this is a valid question or not. Domain & range $\mathbb{R} \to \mathbb{R}$ for $f(x)$ will involve Imaginary Numbers ...
Prem's user avatar
  • 9,978
0 votes

$f(x)=\sqrt{\frac{x-1}{x-3}}$ and $g(1)=e$ which of the following options is/are correct?

I think if $f'(x)$ is discontinuous at $x=1,x=3$.You can't just say $g(x)$ should be discontinuous at $x=1,x=3$,because they didn't have direct contact.We can consider a counterexample,$f(x)=\frac{1}{...
Rick C's user avatar
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1 vote
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Constrained Optimizer as Projection of Unconstrained Optimizer

The claim is true if $H$ is one-dimensional. The claim does not hold in general. Here is a counterexample. Take $H=\mathbb R^2$, $C=\{ x\in \mathbb R^2: \ x_2=0\}$. Define $$ f(x) = (x_1-x_2)^2 + (...
daw's user avatar
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4 votes

Prove or Disprove: Is there a connected planar graph with an odd number of faces where every vertex has a degree of 6?

You are definitely on the right track it seems. The total degree of $G$ is $2E$ and so we have $6V=2E$ which, when dividing both sides by 2 gives $3V=E$. Since the graph is planar, we use Euler's ...
Red Five's user avatar
  • 2,064
1 vote

Apply Fubini's theorem to $\int^1_0\int^2_{y^2}x^2y-y^2x$

The region of integration is not a rectangle: Reversing the order of integration will give you the sum of two double integrals.
John Wayland Bales's user avatar
1 vote
Accepted

Open and dense subsets of continous functions with respect to L1-norm

The solution of the density can be made simpler. The functions $f_k(x)=kx^{k^2-1}$ satisfy $\|f_k\|_1=1/k$ and $\|f_k\|_\infty=k.$ For any function $f\in X$ we have $\|(f+f_k)- f\|_1\to 0$ and $$\|f +...
Ryszard Szwarc's user avatar
2 votes
Accepted

Continuity implies surjectivity if the the limits in both infinities are infinite

A pretty straightforward proof by contradiction can show what you're after. Suppose $\exists a$ such that $\forall x, f(x)\neq a$ Note $\exists x_1$ such that $f(x_1)<a$. This is because if $\...
cambridgecircus's user avatar
1 vote
Accepted

Directly computing dimension of $M_P/M_P^2$

The dimension is certainly no more than $2$. Any element of $\langle\overline{X},\overline{Y}\rangle$ is some residue class of $f(X,Y)X+g(X,Y)Y$ and the relations $X^2=XY=Y^2=0$ mean all higher order ...
FShrike's user avatar
  • 40.6k
0 votes

Prove generalized associative law

Here is a more general proof for semigroups that gives a formula for enumerating all such expressions and as a trivial consequence the recursion identity for Catalan numbers - namely for any magma $(S,...
Ethan Splaver's user avatar
8 votes

Folium of Descartes - what is this point P?

Yes, that is the correct idea. At the point $P$, the derivative $dy/dx$ is undefined; or equivalently, the derivative $dx/dy = 0$. The function $f(x,y)$ satisfies reflection symmetry about $y = x$; ...
heropup's user avatar
  • 137k
0 votes

Is this Proof for the Integral of Binomial Coefficients Correct?

$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} ...
Felix Marin's user avatar
  • 89.7k
-1 votes

Prove that there does not exist a value $z_{0}$ on a line $\overline{ab}$ for which $f(b)-f(a)=f'(z_0)(b-a)$

So the equation can be also written as, $f(b) - f(a) = f'(z)(b-a) \\ \Rightarrow \int_{a}^{b}f'(t)dt = f'(z)(b-a)$ Now obviously the above statement is only true when f'(t) doesn't change in interval (...
Harsh Chaudhari's user avatar
1 vote

Understanding inductive proof of $\sum_{{m=k}}^{N} {m\choose k} = {N+1\choose K+1}$

Seems to me that your confusion comes from the ideia of the induction itself, and not from this particular proof. Hence, I wil try to give you a insight on what is going on in the induction process. ...
peE_'s user avatar
  • 39
3 votes

Understanding inductive proof of $\sum_{{m=k}}^{N} {m\choose k} = {N+1\choose K+1}$

Can someone please help me understand how the argument of mathematical induction serves to prove this identity? There is no such thing as a step 2 separate from step 3. In ordinary induction, the ...
Julio Di Egidio's user avatar
0 votes

Given $d(\vec{x},A):=\text{inf}\{\lVert\vec{x}-\vec{y}\rVert:\vec{y}\in A\}$, show that $f^{-1}(\{0\})=\overline{A}$

As someone else said, your proof that $\bar{A}\subset f^{-1}(\{0\})$ is fine. Your proof that $f^{-1}(\{0\})\subset \bar{A}$ is lacking though. First, notice that if $A=\varnothing$, the result is ...
Ayoub's user avatar
  • 1,656
0 votes

Given $d(\vec{x},A):=\text{inf}\{\lVert\vec{x}-\vec{y}\rVert:\vec{y}\in A\}$, show that $f^{-1}(\{0\})=\overline{A}$

For any metric space $(X,\rho)$ we have $z\in f^{-1}(\{0\})\iff f(z)=0\iff d(z,A)=0\iff \inf \{\rho (z,y) |y\in A\}=0\iff \forall \epsilon >0 \exists y\in A:\rho(z,y)<\epsilon\iff \forall \...
SK_'s user avatar
  • 631
3 votes

In which points is $f: \mathbb{R}P^2 \rightarrow \mathbb{R}^3$ an inmersion.

Your chart $\phi_1$ is the one you would use if working with the projective space as the quotient $\Bbb{R}^3\setminus\{0\}\to\Bbb{RP}^2$, not $S^2\to\Bbb{RP}^2$. Thus your whole calculation is wrong. ...
peek-a-boo's user avatar
2 votes

Finding a Fixed Point for nested Squareroots

With square roots you can use the binomial theorem $$ |\sqrt{3+2y}-\sqrt{3+2x}|=\frac{2|y-x|}{\sqrt{3+2y}+\sqrt{3+2x}} \le \frac{|y-x|}{\sqrt3}. $$ This is sufficient for the fixed-point theorem. You ...
Lutz Lehmann's user avatar
0 votes

Given $d(\vec{x},A):=\text{inf}\{\lVert\vec{x}-\vec{y}\rVert:\vec{y}\in A\}$, show that $f^{-1}(\{0\})=\overline{A}$

For $\overline{A} \subseteq f^{-1}(\{0\})$ suppose $d(x,A)>0$. Take $2\varepsilon = d(x,A)$. Suffices to note that $B(x,\varepsilon)\cap A = \emptyset$. If not, then there exists $y\in A$ such that ...
AlvinL's user avatar
  • 8,699
1 vote

Finding a Fixed Point for nested Squareroots

This works with a little bit more caution: $\phi'(x)<1$ is not sufficient, but we need more strongly $\phi'(x)\le k$ where $k$ is a constant that is strictly less than $1$. Surely this is easy in ...
Just a user's user avatar
  • 15.4k
1 vote

Given $d(\vec{x},A):=\text{inf}\{\lVert\vec{x}-\vec{y}\rVert:\vec{y}\in A\}$, show that $f^{-1}(\{0\})=\overline{A}$

Your proof for $\text{Cl} A \subseteq f^{-1} (\{ 0 \})$ seems fine. The proof of the reverse inclusion is not correct: how does the existence of $y \in A$ such that $d (x_0, y) \ge 0$ imply that $x_0 ...
K. Jiang's user avatar
  • 7,275
1 vote

Convergence of series $\sum_{n=1}^{\infty}x_n^{\alpha}$ subjected to $x_1\in(0,1),x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$.

Thanks for @ Sangchul Lee's helpful comments, in the follwing I give the proof of $$0<x_n<1,\quad \lim_{n\to\infty }x_n=0.$$ Proof: Note the following first: $$x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}=...
Riemann's user avatar
  • 7,355
0 votes
Accepted

Calculating the expected cost of defective products

The hint should actually say $$\operatorname{E}[Y^{\color{red}{2}}] = \sigma^2 + \mu^2.$$ To use the hint, you would apply the linearity of expectation: $$\begin{align} \operatorname{E}[C] &= \...
heropup's user avatar
  • 137k
0 votes

statistical proof with Moment generating functions

Let $X, Y, Z$ denote random variables with a $\Gamma(2,1), L(1)$ and exp(1) distribution respectively, and let subscripts denote independent copies. We know that $Y \overset{d}{=} Z_{1} - Z_{2}$, and ...
minginator's user avatar
1 vote

Proof $E(|X|) = \sqrt{\frac{2}{\pi}}$

I realised that in my comment above I was thinking of $\mathbf{E}(X^k)$ for $k \in \mathbf{N},$ for which you do need some form of analytic continuation. Your integral $\mathbf{E}(|X|) = \int\limits_{-...
William M.'s user avatar
  • 7,577
1 vote

How do I check if this number is transcendental?

It looks as if the zeroes in the decimal expansion of your number may occur in long enough chunks to make it a Liouville number and thus transcendental. If not, you can probably modify your ...
Ethan Bolker's user avatar
  • 95.7k
0 votes

Generalization of $\max$ and $\min$ of $Ax^2+Bxy+Cy^2$ with given $x^2+y^2=k$.

Eliminating $x$ in $$ \cases{ z = A x^2+B x y +C y^2\\ x^2+y^2=k } $$ we obtain $$ p(y) = A^2 k^2-2 A^2 k y^2+A^2 y^4+2 A C k y^2-2 A C y^4-2 A k z+2 A y^2 z-B^2 k y^2+B^2 y^4+C^2 y^4-2 C y^2 z+z^2 $$ ...
Cesareo's user avatar
  • 33.5k
2 votes

Generalization of $\max$ and $\min$ of $Ax^2+Bxy+Cy^2$ with given $x^2+y^2=k$.

The linear algebra way to do this is to evaluate the objective as. $x^T\pmatrix {A & \frac {B}{2}\\\frac {B}{2} & C} x^T$ And the maximum will be associated with the largest eigenvalue. The ...
user317176's user avatar
  • 11.3k
0 votes

Generalization of $\max$ and $\min$ of $Ax^2+Bxy+Cy^2$ with given $x^2+y^2=k$.

You can rotate the picture to make $B = 0$. When you do so the determinant $B^2 - 4AC$ and the trace $A + C$ are unchanged. In other words, if the function $Ax^2 + Bxy + C^2y^2$ transforms into $A' x^...
Zarrax's user avatar
  • 45.1k
1 vote

Prove the interior of a finite intersection of sets is equal to the finite intersection of interiors

If you work in metric spaces (which I assume) then your proof is correct. For countable intersections this is indeed false. For example, in $\mathbb{R}$ consider $A_n==(-\frac{1}{n}, 1+\frac{1}{n})$. ...
Mark's user avatar
  • 40.4k
0 votes
Accepted

If $X = g(Y)$, then Is the sigma-algebra generated by $X$ a subset of the sigma-algebra generated by $Y$?

Let $\mathcal B$ be the Borel $\sigma$-algebra of $\mathbb R$. The definition is that $\sigma(X)=\{X^{-1}(B):B\in\mathcal B\}$ and $\sigma(Y)=\{Y^{-1}(B):B\in\mathcal B\}$. Notice that a continuous $g:...
Liding Yao's user avatar
  • 1,835
0 votes
Accepted

Is my proof that $X \setminus (Y\cup Z)=(X\setminus Y)\cap (X \setminus Z)$ valid?

Theorem. Let $R,S,T$ be sets. Then $R\setminus (S\cap T)=(R\setminus S)\cup (R\setminus T).$ Proof. $x\in R\setminus (S\cap T)$ $\iff (x\in R) \land (x\notin S\cap T)$ $\iff (x\in R)\land (x\notin S \...
Will Fitchet's user avatar
0 votes

Given $E=\{ p_n : n \in \mathbb{N}\}$ and $\lim_{n\rightarrow \infty} p_n =p$, prove $Cl(E)=E \cup\{p\}$ and $Cl(E)$ is compact

You seem to implicitely assume that the ambient space is metrizable, but let us generalize your two results in an arbitrary topological space. Proof that $F:=E\cup\{p\}$ is compact (even in a non-...
Anne Bauval's user avatar
  • 35.8k
0 votes

A little problem in combinatorics (to understand)

In the three colour case, if the three colours chosen are $A,B,C$ chosen in $\frac{n(n-1)(n-2)}{6}$ ways with $n=8$, then one of the three will appear twice, so you have the possible patterns ...
Henry's user avatar
  • 157k
0 votes
Accepted

A little problem in combinatorics (to understand)

Board is like this : 2 Colour Case : We have to choose 1 Colour for 1 & 4 We have to choose 1 Colour for 2 & 3 Rotation will make it $[8 \times 7] / 2$ 4 Colour Case : We have to choose 1 ...
Prem's user avatar
  • 9,978
0 votes

Equality of transcendence degree and local dimension for non-algebraically closed fields

$\def\trdeg{\operatorname{trdeg}} \def\p{\mathfrak{p}} \def\m{\mathfrak{m}} \def\n{\mathfrak{n}}$Even the OP didn't ask for it, for completeness I will explain how one deduces the result over a not ...
Elías Guisado Villalgordo's user avatar
0 votes

Prove that $\binom{2n}{n} < 2^{2n-2}$

Just as Gary said, by Induction, we first assume $\binom{2n}{n}<4^{n-1}$ is true. We need to prove that $$\binom{2(n+1)}{n+1}<4^{(n+1)-1}$$ $$LHS=\frac{(2n+2)!}{(n+1)!(n+1)!}=\frac{(2n+2)(2n+1)}{...
Gwen's user avatar
  • 1,565
0 votes

Finding the circle that is at the minimum distance from all randomly generated points.

Instead of finding the best fit circle using the minimax approach, I used the following error function and minimized it. The error function is $ E = \displaystyle \sum_{i=1}^N \bigg( \sqrt{(P_i - C)^...
i don't know what i am doing's user avatar
0 votes

How to prove that: $|e^i|^2=1$

I assume that your definition of $e^z=\exp(z)$ is $$\exp(z):=\sum_{k=0}^\infty \frac{z^k}{k!}$$ for each $z\in\mathbb{C}$, and that the property $$\exp(z+w)=\exp(z)\cdot\exp(w)$$ has been shown in ...
Dilemian's user avatar
  • 1,035
-1 votes

How to prove that: $|e^i|^2=1$

Any complex number in Euler's form is defined as $|z|e^{iarg(z)}$ for z=a+ib. So if you have any complex number in Euler form as $e^{ix}$ its |z| will always be 1 as we can see by comparing. Therefore,...
Toshiv's user avatar
  • 13
0 votes

How to prove that: $|e^i|^2=1$

$e^{i\theta}=\cos{\theta}+i\sin{\theta}$ and therefore has modulus 1 for any angle $\theta$ since $|\cos{\theta}+i\sin{\theta}|=\sqrt{\sin^{2}{\theta}+cos^{2}{\theta}}=1$. Raising 1 to any power does ...
Red Five's user avatar
  • 2,064
0 votes

How to prove that: $|e^i|^2=1$

You should distinguish the absolute function in $\mathbb{R}$ and the modulus in complex analysis. They are different in nature. Regard a complex number $z$ on an Argand diagram, you can somehow regard ...
Angae MT's user avatar
  • 1,180
0 votes

Where is the error in evaluating this integration? :$\int_0^{\frac{\pi}{2}} \frac{dx}{\cos(x)+\sin(x)}$

Another approach for integrals involving quotients of trigonometric functions is to employ the Weierstrass Substitution: $$ t = \tan\left(\frac{x}{2}\right) \longrightarrow dx = \frac{2}{1+t^2}\:dt, \...
David Galea's user avatar

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