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6 votes
Accepted

Is this a valid proof that a subset of $\mathcal{L}(V, W)$ is not closed under addition

Okay, at this point there's slightly too much to say to fit into the comments so I'll convert them into an answer. This is heading in the right direction (you want to show that $S$ is not closed ...
Qiaochu Yuan's user avatar
5 votes

Basic Solution to the Heat Equation

As of your final answer, it is indeed a correct one. In more general terms, you would need to find a so-called fundamental solution $\Phi(x,t)$, which is the solution to the problem: $$ \left\{ \begin{...
Egor Larionov's user avatar
5 votes

Let R be a ring with three elements. Show that $x^2=y^2$ if $x,y$ are non-zero elements of R.

If $x$, $y$ are non-zero, and distinct, then $x+y$ cannot be $x$, nor $y$, so it's $0$ (there are only $3$ elements). Hence $x^2=y^2$.
orangeskid's user avatar
3 votes
Accepted

solution-verification | Find the minimum of the expression $E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}$

$$ E(x)= \frac{-2x^4+8x^3-8x^2-9}{x^4-4x^3+4x^2+3}=-\frac{2x^2(x-2)^2+9}{x^2(x-2)^2+3}=-\left(2+\frac3{x^2(x-2)^2+3}\right) $$ $$ x^2(x-2)^2+3\ge3 \implies \frac1{x^2(x-2)^2+3}\le \frac13\implies-\...
RandomGuy's user avatar
  • 1,807
3 votes

Solution-verification: Solve $3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr)$

I think it's accepted that the algebraic calculations once you substitute $x=a+b$ are valid so let's talk about why you can make this substitution. Most steps in solving equations are "if and ...
A.M.'s user avatar
  • 4,030
3 votes
Accepted

Solution-verification: Solve $3x^2-6x+4 = 6\{x\}\bigl(\lfloor x\rfloor - \{x\}\bigr)$

Lets try the common way: $n\leq x<n+1.$ Then $\lfloor x\rfloor=n$ and $\{x\}=x-n.$ Putting in the equation we get $$3x^2-6x+4=6(x-n)(2n-x)$$ $$3x^2-6x+4=-6x^2+18nx-12n^2$$ $$(3x-4)^2=12-18x+18nx-...
Bob Dobbs's user avatar
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2 votes
Accepted

Ex. 8.21 in Einseidler & Ward FA book: where do we need that $G$ is abelian?

In Step 4, if you $\ell\in\bigcap_{j=1}^n X_{g_j}$ and apply Step 3 with $g=g_{n+1}$, you get a functional that is invariant under $g_{n+1}$. What is not clear (and not true in general) is that the ...
MaoWao's user avatar
  • 15.5k
2 votes
Accepted

Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

To be more general, for $a,b\ge 0$ we have $$ \sqrt{ab}+\sqrt{a(a+b)}+\sqrt{b(a+b)}\le \left(\frac12+\sqrt{2}\right)(a+b). $$ Since $$ \sqrt{ab}\le\frac{a+b}{2},\ \sqrt{a}+\sqrt{b}\le \sqrt{2}\cdot\...
cybcat's user avatar
  • 776
2 votes

Show that : $\sqrt{[x]\cdot \{x\}} +\sqrt{x \cdot \{x\}} + \sqrt{[x]\cdot x} \leq 2x$

Alternatively, you can recognize the scalar product and use the Cauchy-Schwarz inequality $$\left|\langle u,v\rangle\right|\le\|u\|\cdot\|v\|$$for the vectors $$u=\left(\sqrt{\lfloor x\rfloor},\sqrt{\{...
PinkyWay's user avatar
  • 4,670
2 votes

Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

I don't think you're proving what you were asked. You said that you were asked to show that $f'(x)$ is increasing but instead showed that $\frac{f(x)-f(a)}{x-a}$ is, and $f'(x)$ and the other ...
David G's user avatar
  • 142
2 votes
Accepted

Basic Solution to the Heat Equation

There is a conceptual bridge between stochastic models of motion of particles by random steps in time and the smooth Gaussian as the probability density of the final distribution the the position ...
Roland F's user avatar
  • 3,224
2 votes

Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$

This is not ok. $0\ge 0$, but $\frac{x}{0}\le x$ is false (meaningless, even). You have to work hard to make that kind of bound. The integrand is not dominated by $ue^{-u}$. It suffices to show: $$\...
FShrike's user avatar
  • 42.7k
1 vote
Accepted

Precisely sketch the mapped region and write the equations of its boundaries.

This region is a triangular region with vertices at $0$, $1$, and $-1$. This is not correct. The vertices are at $i$, $1$ and $-1$. jjagmath has already provided a good geometric answer. So, in the ...
mathlove's user avatar
  • 145k
1 vote
Accepted

Find the Cosine of the Angle Between the Plane $(MND)$ and the Plane $(ABC)$

a) $\angle(AD',MN)= \angle(BC',AO)= \angle OAB$ I think this should be $$\angle(AD',MN)= \angle(BC',AO)= \color{red}{\angle{AOB}}$$ Then, the tangent of the wanted angle is equal to $\frac{AB}{BO}=\...
mathlove's user avatar
  • 145k
1 vote

Prove that $\sup⁡(A\cup B)=\max\{\sup⁡(A),\sup⁡(B)\}$

Your proof looks fine after following the suggestions in the comments. But you may consider writing it in plain English to make it easier to understand. Let $A, B \subseteq \mathbb{R}$ be bounded from ...
K. Jiang's user avatar
  • 8,679
1 vote
Accepted

What is the fault in this method of finding second moment of area of a circle

You did not account for the second moment of the triangle itself. The parallel axis theorem applies, as it always does. For this purpose you can treat each segment as a triangle with base parallel to ...
David K's user avatar
  • 100k
1 vote

Is implication true if two statements are always the case?

I think some people misunderstand that if a statement $C$ implies both $B$ and $A$, then in general $A \Leftrightarrow B$ does NOT hold. Therefore, in my opinion, you need to specify your question a ...
Noctis's user avatar
  • 286
1 vote
Accepted

Minimum number of edges to be removed from a square grid so that no rectangles remain.

Your solution is not fully logically justified. Here are two problems with your proof. Your method does not necessarily cover all rectangles. Just look at the case $n=2$. ...
Mike Earnest's user avatar
  • 78.3k
1 vote
Accepted

Prove $\frac{d}{dx} \frac{f(x)-f(a)}{x-a} \geq 0$ if $f(x)$ is convex without twice differentiability.

By the mean value theorem, there is some $c \in (a, x)$ with $f'(c) = \dfrac{f(x) - f(a)}{x - a}$. Then $$\frac{d}{dx} \frac{f(x)-f(a)}{x-a} = \frac{f'(x)(x-a) - (f(x)-f(a))}{(x-a)^2} = \frac{f'(x) - ...
arkeet's user avatar
  • 7,799
1 vote
Accepted

solution-verification | compare to angles in a rectangular parallelipiped

Because $YD=2a\sqrt{2}, DN=3a\sqrt{2}, DN=a\sqrt{38}$ by the thorem of the cosinus the $cos \angle YDN = \frac{11}{24}$ I think $DN=a\sqrt{38}$ is not correct. It should be $\color{red}{YN}=a\sqrt{38}...
mathlove's user avatar
  • 145k
1 vote

Redundancy in the definition of uniform spaces

No. If you take $\Phi = \{\Delta\}$, it satisfies Definition B. However, it is not a uniformity.
J.-E. Pin's user avatar
  • 40.7k
1 vote

Showing that $k[x_1,\ldots,x_n]/\mathfrak{a}$ is a finite dimensional vector space over $k$ assuming basic linear algebra and min amount of abs alg.

You have the right idea, but you are overcomplicating things. You don't need Thm 2, for example. Assume that for each $i$, there is some $i$ for which $x^d_i\in\mathfrak{a}$. Then, for all $m_i\geq ...
GreginGre's user avatar
  • 15.3k
1 vote
Accepted

Let R be a ring with three elements. Show that $x^2=y^2$ if $x,y$ are non-zero elements of R.

To spell out CJ Dowd's comment: if $R$ is any ring, then let $\lambda\colon R \to \text{Hom}_{ab}(R,R)$ (where target of $\lambda$ is the ring of group homomorphisms from the additive group of $R$ to ...
krm2233's user avatar
  • 5,123
1 vote

Find $\displaystyle\lim_{n \to \infty} \int_0^\infty \frac{1+\frac{x}{\sqrt{n}}e^{-x/n}}{(x+1)^2} \, dx$

Since there are different scales at play ($\sqrt{n}$ and $n$) in this kinds of situations is often useful to split the integral in intervals that grow as $n\rightarrow\infty$ and/or use the following ...
Mittens's user avatar
  • 40.7k

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