# Questions tagged [solution-verification]

For posts looking for feedback or verification of a proposed solution. "Is this proof correct?" is too broad or missing context. Instead, the question must identify precisely which step in the proof is in doubt, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.

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### Proof of the Single Value Decomposition

I am working through the a proof of the single-value decomposition, from Strang's 'Introduction to Linear Algebra, 4th edition'. I have included the proof as shown in the book at the end of this post. ...
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### Proof of irreducible elements are prime elements in PID

$\textbf{Theorem}$: If $R$ is a principal ideal domain, then $p$ is prime $\iff$ $p$ is irreducible. $Proof:$ $(\Longleftarrow)$ If $p$ is irreducible, then $(p)$ is maximal in the set of all proper ...
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### Different Formulations of Differentials as Generating Transformations: $e^{t A}f(x) = h(x,t); A=\frac{\partial_x}{g'(x)} \& h(x,t)=f(g^{-1}(g(x)+t))$

For context and introduction please see: Perspective on Differentials Generating Transformations of Functions - with Fourier Transformations $e^{a\partial_x} f(x) = f(x+a)$ Here I used Fourier ...
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### Strictly convex function for minimization problem

I have the following function $$F\left(x,\lambda,\zeta\right)=\sqrt{\frac{1+\zeta^2}{\lambda-\mu x}},$$ due to a minimization problem of the kind $\operatorname{min}\int_0^1 F(x,y(x),y'(x)) dx$, i ...
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### If $f:\mathbb{R}-\{-1,K\}\rightarrow\mathbb{R}-\{\alpha,\beta\}$ is a function given by $f(x)=\dfrac{(2x-1)(2x^2-4 px+p^3)}{(x+1)(x^2-p^2x+p^2)}$

If $f:\mathbb{R}-\{-1,K\}\rightarrow\mathbb{R}-\{\alpha,\beta\}$ is a bijective function defined by $f(x)=\dfrac{(2x-1)(2x^2-4 px+p^3)}{(x+1)(x^2-p^2x+p^2)}$, where $p\geq 0$, then which if the ...
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### Not sure if this is a valid way of proving local maximum

I am unsure if my argument makes sense: Let $f(x) = 0$ if $x$ is irrational and $f(x) = \frac{1}{q}$ if $x = \frac{p}{q}$ with $p,q$ in lowest terms. The exercise asks to find all local maximum and ...
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### solution-verification | compare to angles in a rectangular parallelipiped

the problem Let $ABCDA'B'C'D'$ be a rectangular parallelepiped with $AB=3a, BC=2a,AA'=6a, a>0$. We denote by $M$ and $N$ the means of $AA'$ and $CC'$ and by $P$ the intersection of the lines $BA'$ ...
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### Number of onto functions from a set of 4 elements to a set of 3 elements [duplicate]

I want to compute the number of onto functions from $\{1,2,3,4\} \to \{1,2,3\}$. Here is my attempt: To count the onto functions from a set with $4$ elements to a set with $3$ elements, we apply the ...
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### Find $\pi_1(K\#\dots\#K\#\mathbb S ^ 2)$

Find $\pi_1(K\#\dots\#K\#\mathbb S ^ 2)$ where $\#$ is the connected sum of two topological surfaces and $K$ is the Klein bottle. We induct on the number of Klein bottles considered in the connected ...
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### solution-verification| calculate a trigonometric function of the angle of the plane (ABC) with the plane $\alpha$

the problem The triangle ABC has vertex A in a plane $\alpha$ and is projected onto this plane according to the isosceles right triangle AMN, with $AM=MN=a\sqrt{2}$ M being the projection of B and N ...
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### Perspective on Differentials Generating Transformations of Functions - with Fourier Transformations $e^{a\partial_x} f(x) = f(x+a)$

An analytic function $f(x)$ can be transformed by the exponential of a differential operator. The most known and easiest example is $e^{a \partial_x} f(x) = f(x+a)$ Generally this is shown by Taylor ...
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### Equivalent characterisation of the space $L^p_{\operatorname{loc}}(\Omega)$, where $\Omega$ is a non-empty open subset of $\mathbb R^n$?
Consider the euclidian space $\mathbb R^n$, where $n \in \mathbb N$ is an arbitrary integer, equipped with the usual Lebesgue measure. Moreover, let $\Omega \subset \mathbb R^n$ denote an arbitrary ...