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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. ...
Joseph's user avatar
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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 ...
Morten's user avatar
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0 answers
11 views

Torsion subgroup of elliptic curve via birational transformation

Compute the torsion subgroup of $Y^2=X(X-1)(X-2)$. The solution given by my instructor is to observe that the birational transformation $(X,Y)\mapsto (X-1,Y)$ takes the curve to $Y^2=X(X+1)(X-1)=X^3-...
alidixon222's user avatar
-1 votes
0 answers
13 views

The subgroup $B_n \leq \mathrm{GL}_n(F)$ of upper triangular matrices is solvable (Is my proof correct?)

I do my group theory homework again and after some thought I came up with the following proof. It would be nice, if someone could tell me if this is correct. I took some inspiration from this previous ...
Joachim's user avatar
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0 answers
11 views

Do we need products in proof of modular cancellation law?

I'm self-studying proofs using Jay Cummings' "Proofs", in which he provides a proof for the modular cancellation law: Let $a, b, k$ and $m$ be integers with $k > 0$. If $ak \equiv bk \, ($...
schemer's user avatar
3 votes
0 answers
91 views

Trying to Use Difference Equations to Prove Fermat's Last Theorem

I thought Fermat's Last Theorem could be analyzed as a closed-form solution to a difference equation (or at least vice versa) and wrote the following incomplete draft. I apologize for any trivial ...
Daniil Kardava's user avatar
2 votes
0 answers
33 views

Visualize $\mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2 \# \mathbb {RP} ^ 2$ as an immersed surface in $\mathbb R ^ 3$

I have a short question about Munkres chapter 74 question 4 part (b). (b) Show how to picture the $4$-fold projective plane as an immersed surface in $\mathbb R ^ 3$. In the previous part of the ...
Talmsmen's user avatar
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0 answers
17 views

Maximum Line Segments in an n × n Grid Without Loop formation

Exploring Proof for Maximum Line Segments in an (n * n) Grid Without Loop Formation Hello Math SE community, I am investigating how to maximize the number of line segments in an $(n * n)$ grid ...
omkar tripathi's user avatar
0 votes
2 answers
61 views

Surface area of a sphere in the first octant

I am trying to solve an optional geometry problem in MIT's edx calculus course, but I can't figure out why my attempt at this problem is incorrect. The problem is: The four-sided solid shown is the ...
JohnT's user avatar
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2 votes
0 answers
44 views

Approximates $\sin{x}$

I have to find a neighborhood of $x=0$ such that $\sin{x}$ is approximated with an error less than $10^{-n}$. I have thought that from lagrange error I have: $$|\sin{x}-x|\leq \frac{|x|^3}{6}<10^{-...
axi's user avatar
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1 vote
1 answer
19 views

Every planar graph with no cycles of length $3,4,5$ is $3$-colorable.

I'm trying to prove that every planar graph with no cycles of length $3,4,5$ is $3$-colorable. However, I have no opportunity to receive any validation or correction on it, but it would be very ...
ninaPh99's user avatar
0 votes
0 answers
17 views

Item selection probablity after sampling m items out of k and then sampling two item from the m items uniformly.

$m$ items are drawn from $[k] = {1,2,3,...., k}$ according to probability $q(i)$ iid with replacement. That is the probability of item $i$ being drawn is $q(i)$. Then two items $x$ and $y$ are drawn ...
Mario420's user avatar
0 votes
1 answer
22 views

What is the minimum number of pennies that needs to be chosen from a collection so that the restriction is met?

"Discrete mathematics with applications" by Susanna S. Epp contains the following problem: A penny collection contains twelve 1967 pennies, seven 1968 pennies, and eleven 1971 pennies. If ...
Vlad Mikheenko's user avatar
1 vote
0 answers
23 views

Clarification on model satisfaction definition and satisfaction over classes

By the completeness theorem of logic, we have "A theorem $T$ is consistent if and only if it has a model $M$" - whereby $M$ is a model of $T$ if and only if all the non-logical axioms of $T$ ...
Link L's user avatar
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1 vote
3 answers
147 views

Finding the height of a skyscraper

I am trying to solve an optional self-assessment geometry problem in MIT's 18.01x course, but am struggling somewhat with the geometry. I don't know how to fully convey the problem without a picture, ...
JohnT's user avatar
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0 votes
0 answers
16 views

Showing that $σ(X_1, X_2, . . .) = σ({{X_1 ≤ x_1, . . . , X_r ≤ x_r} | x_1, . . . , x_r ∈ \mathbb{R}, r ∈ \mathbb{N}}).$

Let $\left(X_{n}\right)_{n} $ be a sequence of real-valued random variables. Show that $ \sigma\left(X_{1}, X_{2}, \ldots\right)=\sigma\left(\left\{\left\{X_{1} \leq x_{1}, \ldots, X_{r} \leq x_{r}\...
asdfgh jkl's user avatar
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0 answers
56 views

Triangle inequality with condition $a+b+c=2$ [closed]

Problem: Let $ABC$ be a triangle whose side lengths satisfy the condition $a + b + c = 2$. Prove that $$ \sqrt{\frac{a}{a^2+bc}}+\sqrt{\frac{b}{b^2+ca}}+\sqrt{\frac{c}{c^2+ab}} \ge 2$$
Moahmed Amine's user avatar
0 votes
1 answer
16 views

Dominated convergence theorem if $|f_n|^p\le g$ where $p\in [1,\infty)$

Lebesgue's dominated convergence theorem. Let $(f_n)$ be a sequence of complex valued measurable functions on a measure space $(X,\mathcal{A},\mu)$. Suppose that the sequence converges pointwise to a ...
NatMath's user avatar
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1 vote
1 answer
55 views

If $\phi \in C_c^\infty(\mathbb R^n) \cap L^1(\mathbb R^n)$ such that $\| \phi \|_1 = 1$, then the dillations $\phi_t$ preserve these properties?

Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and denote by $L^1(\mathbb R^n)$ the usual space of measurable functions that are integrable. Moreover, let $C^k(\mathbb R^n)$ denote the ...
xyz's user avatar
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2 votes
0 answers
38 views

Rank of an elliptic curve via 2-descent

Let $p$ be a prime with $p\equiv 3\pmod 4$. Prove that the elliptic curve $y^2=x^3+px$ has rank at most $1$. The solution we were given is pretty much a 2-descent argument. Define the curves as ...
alidixon222's user avatar
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0 answers
89 views

Is my proof that $y100-(100-x)y=xy$ correct? [closed]

My name is Aghilas, I am 16 years. Can someone answer my observation? Let's assume we have: $x, y$ : two numbers $a=xy$ $z=100−x$ $s=zy$ $c=y100$ $q=c−s$. We want to verify if the equation $q=a$ is ...
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1 vote
0 answers
47 views

Proof verification on Functional analysis excercise

Let $E$ be a Banach space and $T \in \mathcal{L}(E)$ with $\lVert T \rVert < 1$ . We denote by $T^0 = I$ where $I$ is the identity map of $E$ and $T^k = T \circ \overset{k}{\dots} \circ T$ for $k \...
Daniel García's user avatar
0 votes
0 answers
18 views

If X is a basis for Hilbert space H then exists Y, biorthogonal, with Y a basis for H.

I would like to prove that: given $X = \{x_n\}$, basis of the Hilbert space $H$, then it does exist $Y = \{y_n\}$ with is biorthogonal to $X$. Also, $Y$ is a basis for $H$. I found the first half in ...
Alkianskasf's user avatar
0 votes
1 answer
81 views

Let $A \subseteq [0, 1] \cap \mathbb{Q}$ an infinite subset. Then $A$ has accumulation points.

Let $A \subseteq [0, 1] \cap \mathbb{Q}$ an infinite subset. It is required to show that $A$ has accumulation points, but without using neither the completeness of $\mathbb{R}$ nor concepts like ...
Blue Tomato's user avatar
1 vote
0 answers
38 views

Show that if $f$ is a measurable complex-valued function on $(X,\mathscr{A})$, then $|f|$ is also measurable.

I need to show If $f$ is a measurable complex-valued function on $(X,\mathscr{A})$, then $|f|$ is also measurable. I tried it myself, but don't know if my work is correct or not? Could someone ...
Beerus's user avatar
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1 vote
0 answers
38 views

AP Calculus BC 2024 - Taylor's Inequality/Series Estimation

The free-response questions from this year's AP Calculus BC exam were just released, and I am particularly interested in question 6b. A question of this type appears each year, and I think when I took ...
Sean Roberson's user avatar
0 votes
1 answer
17 views

How to more formally prove this inequality

This is a simple problem I came up with while doing another problem: Given: $n < (n + \frac{1}{2}) < y < (n + 1)$ Prove: $|y - n| > |y - (n + 1)|$ So how I proved it was simply using the ...
Bob Marley's user avatar
0 votes
0 answers
27 views

Let $F$ be a field and $f(x)\in F[x]$ such that $f(x)|g(x),\forall$ non constant polynomial $g(x)\in F[x].$ Show that $f(x)$ is a constant polynomial.

Let $F$ be a field and $f(x)\in F[x]$ such that $f(x)$ divides $g(x)$ for every non constant polynomial $g(x)\in F[x].$ Show that $f(x)$ is a constant polynomial. My solution goes like this: Let $a\...
Thomas Finley's user avatar
0 votes
0 answers
46 views

Is this graph theory proof regarding chip firing correct?

For my olympiad preparation i was reading a handout regarding chip firing games , A single player game which in the context is defined as "Consider an infinite graph such that all vertices have ...
Lucid's user avatar
  • 303
1 vote
0 answers
51 views

$A \subset [0,1]$. $P_a$ is a parabola tangent to OX in $(a,0)$. $B= \bigcup_a P_a \cap [0,a] \times R$. Show: $\lambda_2(B)=0 \iff \lambda_1(A)=0$

$A \subset [0,1]$. $\forall_{a \in A}$ we name as $P_a$ a parabola that is tangent to $OX$ in point $(a,0)$. $B = \bigcup_a P_a \cap [0,a] \times \mathbb{R}$ Show that: $$\lambda_2(B) = 0 \iff \...
thefool's user avatar
  • 1,068
1 vote
0 answers
70 views

Show that $f$ is not differentiable but partially differentiable at $(0,0)$

Let be $f:\mathbb{R}^2\to\mathbb{R}$, where $$ f(x,y):=\begin{cases} x,&x\neq y\\1,&x=y, x\neq 0,y\neq 0\\0,&x=y=0 \end{cases} $$ Show that $f$ is not differentiable but partially ...
Philipp's user avatar
  • 4,523
2 votes
2 answers
217 views

Solution to IMO 2020 P2?

I'm asking to see if my proof of the IMO 2020 P2 is valid. It doesn't follow the typical solution provided so I just wanted to ask if it's correct. The question asks to prove that for all $a, b, c, d$ ...
user1070280's user avatar
-1 votes
0 answers
30 views

Upper and lower integrals, piecewise function [duplicate]

Question: Let $f: [a,b] \to \mathbb{R}$ be a function such that $$f(x) = \begin{cases} x & x\in\Bbb Q \\ x+1 & x\not\in\Bbb Q\end{cases}.$$ Compute the upper and lower integral of f. So I'm ...
jorgecore's user avatar
3 votes
1 answer
51 views

Computing torsion subgroup of elliptic curve

Compute the torsion subgroup of the elliptic curve $y^2=x^3+5x^2+3x+7$. I am only used to computing torsion groups when our equation is in 'short Weirstrass form'; i.e. $y^2=x^3+Ax+B$ for integer $A,...
alidixon222's user avatar
2 votes
0 answers
32 views

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 ...
theta_phi's user avatar
  • 115
1 vote
0 answers
18 views

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 ...
Gonzalo de Ulloa's user avatar
0 votes
1 answer
66 views
+50

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 ...
mathophile's user avatar
  • 3,813
4 votes
1 answer
75 views

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 ...
Alejandro's user avatar
  • 161
0 votes
0 answers
54 views

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 $BA'$ ...
IONELA BUCIU's user avatar
-1 votes
0 answers
30 views

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 ...
AgnostMystic's user avatar
  • 1,694
0 votes
0 answers
48 views

$M_{2\times 3}(\Bbb R)$ is isomorphic to $\Bbb R^5$ [closed]

Is this right way to solve this problem? Can you review it please? Lets consider the next transformation $T:M_{2\times 3}(\Bbb R)\to \Bbb R^5$ given by $T(A)=\langle{a_{11},a_{12},a_{13},a_{21},a_{22}}...
Roma_Rayado's user avatar
0 votes
1 answer
27 views

Kernel of a derivation over an arbitrary ring

Let $ D: R \to R$ be a mapping from a ring $R$ into itself which is a derivation (i.e. linear with respect to addition operation so that $D(x + y) = D(x) + D(y)$ and satisfies Leibnitz's law: $D(xy) =...
giorgio's user avatar
  • 429
0 votes
1 answer
62 views

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 ...
Talmsmen's user avatar
  • 1,190
0 votes
1 answer
67 views
+100

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 ...
IONELA BUCIU's user avatar
2 votes
1 answer
40 views

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 ...
theta_phi's user avatar
  • 115
3 votes
1 answer
44 views

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 ...
xyz's user avatar
  • 1,109
-1 votes
0 answers
36 views

Gram matrix, plane and line

The angle between a plane and a line is unique and can be found through their equations. But I'm given in addition Gram matrix. what it can be used for? $V = \{v \in \mathbb R^{3} $| ${v}_{\ 1}$- ${v}...
Dina_B's user avatar
  • 25
3 votes
2 answers
138 views

Confusion on reading multiple ways of p implies q

So these are the different ways of expressing the conditional statement, and I got these from Rosen's Discrete Math Textbook: So I want to be clear on a couple of things: When we read all these ...
Bob Marley's user avatar
14 votes
3 answers
2k views

43 cookies are randomly given to 10 children. What's the probability each child receives at least 2 cookies?

I wanted to ask 1) if I've solved this puzzle problem correctly, and 2) if there is a shorter or more elegant approach. There are 43 cookies to be given out at random to 10 children. What is the ...
ctesta01's user avatar
  • 504
4 votes
2 answers
93 views

solution-verification Determine the position of M

The problem In the regular quadrilateral prism, $ABCDA'B'C'D'$ the edge of the base is equal to $4 \sqrt{6}$, and the volume is $1152$. Determine the position of the point M on the edge CC', so that ...
Adelina Popescu's user avatar

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