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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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Can fundamental theorem of algebra for real polynomials be proven without using complex numbers?

Update 24 Nov 2015: It is solved. Please refer to this arXiv paper. For polynomials with real coefficients, I am trying to prove the following version of fundamental theorem of algebra, which avoids ...
sobasu's user avatar
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25 votes
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Theorem 6.17 in Baby Rudin, 3rd ed: $\int_a^b f \,d\alpha = \int_a^b f(x) \alpha^\prime(x) \,dx$

Here is Theorem 6.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Assume $\alpha$ increases monotonically and $\alpha^\prime \in \mathscr{R}$ on $[a, b]$. Let $f$ be ...
Saaqib Mahmood's user avatar
19 votes
0 answers
644 views

Erratum for Billingsley’s $\textit{Probability and Measure}$, Problem 32.13

This is a verification request for a counterexample that I think I have found for Problem 32.13 on page 427 in Patrick Billingsley’s Probability and Measure textbook (third edition, but the problem ...
triple_sec's user avatar
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18 votes
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723 views

A very general method for solving inequalities repaired

Yesterday, I asked a question about a very general method for solving equations I had found here. As it turned out, there were quite some problems with my method and I got a lot of good feedback. ...
Mastrem's user avatar
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17 votes
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795 views

An Extended Frullani Integral

In the development of Methodology $2$ of This Answer, I found a possible new extension of Frullani's Integral (See Here). Theorem: Let $f$ be Riemann integrable on $[0,x]$ for all $x>0$ and let $a&...
Mark Viola's user avatar
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16 votes
0 answers
711 views

Any diffeomorphism can be locally factorised with several primitive diffeomorphisms.

I ask in this question how to solve an exercise of the text Analysis on Manifolds by James Munkres: the exercise consist to prove a theorem about diffeomorphism with more restrictive conditions. Since ...
Antonio Maria Di Mauro's user avatar
16 votes
1 answer
7k views

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable.

Prove that every compact metric space $K$ has a countable base, and that $K$ is therefore separable. How does the following look? Proof: For each $n \in \mathbb{N}$, make an open cover of $K$ by ...
PandaMan's user avatar
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15 votes
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530 views

Trying to prove a proposition about the nth order derivative of a polynomial by induction - is this correct?

Recently, I decided to try and create a formula for the $n$th order derivative of a polynomial, and I believe I succeeded! I tried to do a proof by induction to confirm this for myself, but since I ...
cdog's user avatar
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14 votes
0 answers
1k views

A detailed and self-contained proof of Tonelli's theorem

Motivation: I have seen the interchange of limit/derivative and integral many times, but don't know how such operation makes sense. I've always desired to remove this uncertainty by giving a ...
Akira's user avatar
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14 votes
0 answers
330 views

Putnam 2018 - Exercise A.5 - proof check

The problem statement is as follows. Let $f:\Bbb R \to \Bbb R$ be an infinitely differentiable function satisfying $f(0) = 0$ and $f(1) = 1$, and $f(x) \geq 0$ for all $x \in \Bbb R$. Show that there ...
dfnu's user avatar
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14 votes
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Theorem 6.12 (a) in Baby Rudin: $\int_a^b \left( f_1 + f_2 \right) d \alpha=\int_a^b f_1 d \alpha + \int_a^b f_2 d \alpha$

Here is part (a) of Theorem 6.12 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition: If $f_1 \in \mathscr{R}(\alpha)$ and $f_2 \in \mathscr{R}(\alpha)$, then $$f_1 + ...
Saaqib Mahmood's user avatar
14 votes
0 answers
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If $f_n \to f$ and $g_n \to g$ in measure and $\mu$ is finite, then $f_n g_n \to fg$ in measure

This is Problem 3.1.5 in Cohn's Measure Theory, 2nd edition. Let $\mu$ be a measure on $(X, \mathcal A)$, and let $f, f_1,f_2, \ldots$ and $g,g_1,g_2,\ldots$ be real-valued $\mathcal A$-measureable ...
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13 votes
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928 views

Fermat's Last Theorem ($n=3$) using the Eisenstein integers

I'm doing the first part of the following exercise in Miles Reid's Undergraduate Commutative Algebra: Exercise 0.18: Prove the cases $n=3$ and $n=4$ of Fermat's last theorem. I'm assuming I should ...
cansomeonehelpmeout's user avatar
13 votes
0 answers
927 views

Topologist's sine curve is a simply-connected space

I am trying to solve the following problem from Hatcher's Algebraic Topology and have written a solution. Could you help me checking my solution, whether I am right? Thanks in advance. $Y$ is simply-...
Sumanta's user avatar
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13 votes
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305 views

Why is counting the number of least prime factors of a sequence of consecutive integers insufficient to resolve Legendre's Conjecture?

I've been thinking a long time about Legendre's Conjecture. A few nights ago, I came across the following argument which is of course too simple to be true. I would greatly appreciate if someone ...
Larry Freeman's user avatar
12 votes
0 answers
3k views

Finding the Green Function of the upper half ball

Find the Green function of $\Omega:=\left\{x\in\mathbb{R}^n:\lVert x\rVert<R, x_n>0\right\}$ and show that the function you've found is indeed a Green function! You are allowed to use the Green ...
user avatar
11 votes
0 answers
179 views

If $a_n \rightarrow a$ Then $\frac{1}{n} \sum_{k=1}^{n} a_k \rightarrow a$

If $a_n \rightarrow a$ as $n \rightarrow \infty$, then can we say that $\frac{1}{n} \sum_{k=1}^{n} a_k \rightarrow a$ as $n \rightarrow \infty$? If we consider $\frac{1}{n} \sum_{k=1}^{n} a_k$ as ...
nyeon_22's user avatar
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11 votes
0 answers
346 views

Is there an elementary argument for $\prod\limits_{p \le n}p < 3^n$ where $p$ is prime.

I was reading Hanson's proof that $\prod\limits_{p^a \le n}p^a < 3^n$ where $p$ is a prime and it occurred to me that there might be a simpler argument for $\prod\limits_{p \le n} p < 3^n$. Am ...
Larry Freeman's user avatar
11 votes
0 answers
550 views

Finite algebraic structures where all hyperoperations (addition, multiplication, exponentiation, tetration, etc.) are well-defined

Let $\langle R, +, \times, \uparrow, \uparrow\uparrow, \uparrow\uparrow\uparrow, \ldots; 0, 1\rangle$ be an algebraic structure with two constants $0, 1$ and where an infinite sequence of binary ...
pregunton's user avatar
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11 votes
0 answers
834 views

Non empty set with zero diameter

Let $A \subset X$ where $X$ is a metric space. by definition diam$(A) = \sup\{ d(x,y), x,y \in A\}$. if $A$ is non empty and has zero diameter, can I conclude that $A$ is a singleton? i reason as ...
user avatar
11 votes
0 answers
342 views

Finding irreducible components of Spec$(R/I^n)$

Let $R= k[x,y,z]/(xy,yz,zx)$. Let $I=(x)$. What are the irreducible components of $\mathrm{Spec}(R/I^n)$ where $n \geq 2$ and $k$ is a field? For solving this problem I'm trying to use following ...
Arpit Kansal's user avatar
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10 votes
0 answers
2k views

Linear independence of tensor product basis $\{ v_i \otimes w_j\}$ for $\{v_i\}$ and $\{w_j\}$ linearly independent.

Show that the set $\{v_i \otimes w_j\}$ is a linear independent subset of $V\otimes W$ when $\{v_i\}$ and $\{w_j\}$ are independent subsets of V and W respectively. I want to find an error in a proof ...
mz71's user avatar
  • 908
10 votes
0 answers
512 views

Find the number of zeros of $f(z)=e^{z-1}-az$ inside unit disk, assuming $\mid a \mid >1$

This is an application of Rouche's theorem, I want to make sure I am doing it correctly: Let $f(z)=e^{z-1}-az$, where $\mid a \mid>1$ and $g(z)=-az$ Now, on the unit circle we have: $$\mid g(z) \...
Mike's user avatar
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10 votes
0 answers
560 views

Is this proof that e is irrational correct?

Yesterday evening, I wrote this proof that e is irrational. I showed it to my maths teacher at school and he thought it was correct. I wondered if anyone else could tell me if it was valid and if ...
Matthew Barber's user avatar
10 votes
0 answers
935 views

A question on the definition of tangent vectors as equivalence classes and directional derivatives

I understand that a tangent vector, tangent to some point $p$ on some $n$-dimensional manifold $\mathcal{M}$ can defined in terms of an equivalence class of curves $[\gamma]$ (where the curves are ...
Will's user avatar
  • 3,305
10 votes
0 answers
5k views

Monotonicity of measures

Let $\mu$ be a measure defined on $\Omega$. Then $\mu(A)\le \mu(B)$ for all $A\subset B\subset \Omega$. pf. Let $A\subset B$, let $C=A^c\cap B$. Then $A\cap C=\emptyset$ and $A\cup C = B$. By ...
mrk's user avatar
  • 3,105
10 votes
0 answers
609 views

Sequence of convex functions converges uniformly

I am working on the following problem. Let $f_{n}: [a, b] \rightarrow \mathbb{R}$ be a sequence of convex functions. Furthermore, for each fixed $x \in [a, b]$, suppose $f(x) = \lim_{n \...
JKL629's user avatar
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9 votes
0 answers
211 views

Prove that $f(x) = x^2 + x$ is uniformly continuous.

Using the definition of uniform continuity, prove that $f(x) = x^2 + x$ is uniformly continuous on $(0,1)$. Proof: We have a function $f:(0,1)\to\mathbb R$. Let $\epsilon > 0$. Note $\forall x,y\in\...
user10101's user avatar
  • 473
9 votes
0 answers
91 views

Understanding Brouwer Separation Theorem with an easier proof

I'd like to undersand better Brouwer separation theorem given in Massey (Proposition 6.5 p.215) since has some smoky parts to me. To lighten the notation we set $D^n := \mathbb{D}^n, S^n := \mathbb{S}^...
jacopoburelli's user avatar
9 votes
0 answers
634 views

About Theorem 3.4 Hartshorne: detailed proof.

I propose a detailed version of part of the proof of Theorem 3.14 from Hartshorne's book Algebraic Geometry. The questions are inserted from time to time within the proof. Thanks for your patience. ...
user avatar
9 votes
0 answers
442 views

Fermat's Last Theorem ($n=4$) using the Gaussian integers

I'm doing the second part of the following exercise in Miles Reid's Undergraduate Commutative Algebra: Exercise 0.18: Prove the cases $n=3$ and $n=4$ of Fermat's last theorem. I would like to know: ...
cansomeonehelpmeout's user avatar
9 votes
0 answers
595 views

Proof of one-side version of Bennett-Bernstein inequality

I'm going to prove the following: For independent random variables $X_i$, $i \in [m]$ satisfying $X_i-E[X_i] \le b$ for some constant $b > 0$. Let $\bar{X} = \dfrac{1}{m}\sum_{i=1}^m X_i$, we have ...
Andrews's user avatar
  • 4,011
9 votes
0 answers
359 views

The volume of $k$-parametrised manifolds when $k=1$ or $k=2$ or $k=3$ agrees respectively with the usual intuitive idea of length or area or volume.

James Munkres, at the chapter $22$-th of the text Analysis on Manifolds, gives the following definition. Furthermore, to point out I say that the function $V\left(D\alpha\right)$ corresponds with ...
Antonio Maria Di Mauro's user avatar
9 votes
0 answers
293 views

Cotangent complex of dual numbers

Let $k$ be a ring, put $k[\epsilon] := k[t]/(t^2)$. What is the cotangent complex of $k[\epsilon] \to k$? I know $\Omega^1_{k/k[\epsilon]]}$ is going to be zero. But I don't see any way around ...
Maanroof's user avatar
  • 524
9 votes
1 answer
349 views

Methods to solve $\int_{0}^{\infty} \frac{\cos\left(kx^n\right)}{x^n + a}\:dx$

Spurred on by this question, I decided to investigate for different functions on the numerator. Here, I went from $\exp(..)$ to $\sin(..) / \cos(..)$. I initially thought I could modify the result ...
user avatar
9 votes
0 answers
309 views

Is Hilbert's space-filling curve measure preserving?

Say $f_n:[0,1]\to [0,1]^d$ is the $n$-th iteration of a $d$-dimensional Hilbert curve touring its range. Is it true that for any open $S\subset [0,1]^d$, then amount of time $f_n$ spends in $S$ is ...
Christian Chapman's user avatar
9 votes
0 answers
1k views

Symmetric linear operator is bounded

Proposition Let $(H, \langle \cdot, \cdot \rangle)$ be a Hilbert space and let $A : H \to H$ be a linear operator defined everywhere. If $A$ is symmetric i.e. if $$\langle Ax,y \rangle = \langle x, ...
Bremen000's user avatar
  • 1,456
9 votes
0 answers
3k views

Show that Minkowski functional is a sublinear functional

A set $C\subseteq X$ is convex if for any $x,y\in C$ and any $0\leq t \leq 1,$ we have $tx+(1-t)y\in C.$ A set $C\subseteq X$ is absorbing if for any $x\in X,$ there exists $t>0$ such that $tx\in C....
Idonknow's user avatar
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9 votes
0 answers
1k views

Isomorphism of rings induces isomorphism of affine schemes

I am new to schemes and I would be very grateful if someone would check my argument below. Important Note. I am following Ravi Vakil's notes: https://math.stanford.edu/~vakil/216blog/...
user350031's user avatar
  • 1,860
9 votes
0 answers
199 views

If the difference of two independent random variables has a mean, so does each variable

This is a proof-verification request; I’m also recording this proof for my own later reference. Any feedback is appreciated. Claim: Let $X$ and $Y$ be independent, real-valued random variables on ...
triple_sec's user avatar
  • 23.6k
9 votes
1 answer
1k views

Uniform limit of one-to-one analytic functions is either constant or one-to-one

Let $U$ be a complex domain, and $(f_n)_{n\in \mathbb{N}}$ be a sequence on one-to-one analytic functions defined on $U$. Suppose that $f_n$ converges to $f$ uniformly on every compact subset of $U$. ...
GaussTheBauss's user avatar
9 votes
0 answers
504 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t e^{-\kappa(t-u)...
Robert W.'s user avatar
  • 724
9 votes
0 answers
2k views

$W^{1,p}$ is separable for $1\leq p<\infty$

I've been asked to prove that the Sobolev spaces $W^{1,p}(\Omega)$, $\Omega$ open in $\mathbb R^n$, are separable for $1\leq p <\infty$ using the map $$i\colon W^{1,p}(\Omega)\to L^p(\Omega)\times ...
batman's user avatar
  • 2,075
9 votes
0 answers
525 views

Differentiation under the integral sign when derivative exists only almost everywhere

Regarding the Theorem 3 from here (or pdf ver.). Let $X$ be an open subset of $\mathbb{R}$, and $\Omega$ be a measure space. Suppose that a function $f\colon X\times\Omega\to \mathbb{R}$ satisfies ...
shall.i.am's user avatar
9 votes
0 answers
294 views

Olympic number theory problem: is this solution fine and sufficiently well written?

Determine all positive integers $m$ such that the ratios $$ \frac{2(5^m+5)}{3^m+1}\quad\text{and}\quad \frac{9^m+1}{5^m+5}$$ are both integers. Attempt at a solution: If the ratios are both ...
Adrian's user avatar
  • 1,687
9 votes
1 answer
286 views

Proof correctness problem

I was watching this talk by Vladimir Voevodsky which was given at the Institute of Advanced Study in 2006. In his talk the first slide he shows has the following written on it: ...
The very fluffy Panda's user avatar
9 votes
0 answers
2k views

Any non-trivial finitely-generated group admits maximal subgroups

I want to solve the following problem from Dummit & Foote's Abstract Algebra: This is exercise involving Zorn's Lemma (see Appendix I) to prove that every nontrivial finitely generated group ...
user1337's user avatar
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9 votes
0 answers
2k views

On the centralizers of $n$-cycles and conjugacy in $A_n$

I'd appreciate comments on the validity of these attempted proofs. Thanks. Let $a$ be an $n$-cycle in $S_n$. a) Show that the centralizer of $a$ in $S_n$ is $\langle a \rangle$. b) Assume that $n$ ...
Alex Petzke's user avatar
  • 8,793
8 votes
0 answers
452 views

Probability of winning a game guessing $k$ random numbers in sequence with optimal strategy

We are playing a game as follows, suppose we have $k$ spaces. One at a time, we will pick $k$ random integers from the range $[1,n]$, without replacement. After selecting a number, we must choose to ...
wjmccann's user avatar
  • 3,105
8 votes
0 answers
251 views

Axler "Linear Algebra Done Right" Exercise 6.B.13

This exercise appears in Section 6.B "Orthonormal Bases" in Linear Algebra Done Right by Sheldon Axler. Inner product spaces, norms, orthogonality, and orthonormal bases have been ...
L. F.'s user avatar
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