For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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Possible proof of infinite twin prime conjecture

I have an idea for proving the infinite twin prime conjecture that would set up a correspondence between primes. Since they've been proven infinite, twin primes would be shown infinite. Here it is: ...
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0answers
6 views

Elementary proof that a particular language is not regular

I want to show that given an alphabet $A = \{ L, R \}$, the language $$ \mathcal{L} = \{ x_{1} \ldots x_{n} \in A^{*} : \# \{ j \leq n : x_{j} = L \} = \# \{ j \leq n : x_{j} = R \} \}$$ cannot be ...
0
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2answers
49 views

Discrete math 4 proofs

I have a few questions that i have answered but i am not sure if its proof enough, if you could help me out and tell me if i am correct or not thta would be great. Question 1: For all sets A,B,C we ...
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0answers
20 views

Connected sets prove that definitions are equivalent

I found the following two definitions of connected set. I couldn't really see how they were equivalent so I tried to prove it. Definition: Two subsets $A$ and $B$ of a metric space $X$ are said to ...
1
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1answer
27 views

Linear transformation representation proof

I am wanting for someone to go over what I have and possibly correct my mistakes. Or any comments on the techniques, etc. I want to prove that if $V$ and $W$ are vector spaces over some field $F$, ...
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0answers
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Eigenvalues of $MA$ versus eigenvalues of $A$ for orthogonal projection $M$

Suppose that $M$ is symmetric idempotent $n\times n$ and has rank $n-k$. Suppose that $A$ is $n\times n$ and positive definite. Let $0<\nu_1\leq\nu_2\leq\ldots\nu_{n-k}$ be the nonzero ...
5
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1answer
33 views

Where is the error in this proof of the Hodge theorem?

Let $(M,g)$ be a closed smooth Riemannian manifold. The following is the decomposition part of the Hodge theorem: Theorem The canonical map $\mathscr{H}^k(M)\to H^k(M)$ from harmonic $k$ ...
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0answers
11 views

A graph $G=(V,E)$ is connected and a vertex $s \in V$ is not a vertex separator iff $G-s$ is connected - requires $deg(s) > 3$?

I was asked to prove that $G$ is connected and $s$ is not a vertex separator iff $G-s$ is connected, given that $deg(s) > 3$. I'm struggling to understand why I need the $deg(s) > 3$ part. One ...
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3answers
42 views

What is the significance of using prime numbers in proving: $x$ is a multiply of $y$?

I came to a problem where it asks me to prove, for example, $n^4-n^2$ is a multiple of $12$. Now, factorize the multiple: $n\times n\times (n-1)\times (n+1)$. Here we have $3$ consecutive integers. ...
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2answers
33 views

prove limit of exponential function without concept of logarithm

The question is, prove that if a real number $x>1$, then $\lim_{n\to\infty}x^n = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure ...
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1answer
40 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
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2answers
27 views

Vacuous statements and explosion

So my understanding of vacuous statements is as follows: For any statement $P$, the statement $(\forall x \in \emptyset)(P(x))$. This can be argued as follows: Assume for contradiction $\neg [(\forall ...
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4answers
73 views

If $B\subset A$ and $f:A\to B$ is injective prove it's a bijection between $A$ and $B$

I want to show that if $B\subset A$ and $f:A\to B$ is an injective function then there's a bijection between $A$ and $B$. I believe my "proof" is wrong, I probably use too much "intuition" when I ...
1
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1answer
21 views

Projection maps are open

I want to show $p_x: X\times\ Y \to X$ is an open map. Here's my proof: Let $W \subset\ X\times\ Y$ be open subset, then $W = \bigcup U_\alpha \times\ V_\beta$, for $U_\alpha, V_\beta$ are open ...
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1answer
17 views

How to convert sentence into logic formula

Hi I wanted to know if I have converted this sentence into propositional logic correctly. This is the sentence At least two of the propositions $p$, $q$ and $r$ are true. and this is my answer ...
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0answers
33 views

Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
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1answer
30 views

Geometrically interpreting complex numbers.

Prove that $|e^{i \alpha} - e^{i \beta}| |e^{i \gamma} - e^{i \delta}| + |e^{i \beta} - e^{i \gamma}| |e^{i \alpha} - e^{i \delta}| = |e^{i \alpha} - e^{i \gamma}| |e^{i \beta} - e^{i \delta}|$ ...
1
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1answer
23 views

Proof for $x\le -1 \implies x^3-x\le 0$?

Here is my proof: Let $x\in \mathbb{R}$, assume $x\le -1$ Then $x^2\ge 1$ Then $x^3\le -1$ Since $x\le -1$ $x^3\le x$ Then $x^3-x\le 0$ Therefore $x\le -1 \implies x^3-x\le 0$ Therefore ...
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3answers
54 views

For $\int f < \infty$, the measure of the set of points where $f=\infty$ is zero.

I fear this question was already discussed here, but I was not able to find it. Please remove if it is a duplicate. Prove: For a function $f\geq 0$, if $\int f < \infty$, then the measure of ...
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0answers
33 views

Variety of the ideal.

Hey I am trying to understand the inclusion $Z(I) \setminus Z(J) \subset Z(I:J)$through the standard definition of a variety (not the closure). I will be borrowing results from this answer. $$Z(I) = ...
6
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1answer
64 views

Basic question $|x^2| < 9$

I have a rather basic question. Let's assume that $|x^2| < 9$, where $x\in \mathbb{R}$. Then everyone knows that $x \in$ (-3,3). However, I have trouble arriving at the answer based on basic ...
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1answer
44 views

Proving the volume of sphere by using tiny volumes

How can I prove the volume of sphere, by using many cones starting at the center of the sphere? It doesn't have to be cones, pyramids also work.
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2answers
64 views

Construct $f: X\to Y$ such that $f(p)=p$

Let , $X=[-1,1]\times [-1,1]$ and $Y=\{0\}\times \left[-\frac{1}{2},\frac{1}{2}\right]$. Construct an example of a continuous map $f:X\to Y$ such that $f(p)=p$ for each $p\in Y$. I construct a ...
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0answers
19 views

Give an example of limits that misbehave under conjugation of function

My quest: Find real valued functions $f(x)$ and $g(x)$ such that $f \rightarrow b$ as $x\rightarrow a$ and $g\rightarrow c$ as $x\rightarrow b$ but $g(f(x)) \nrightarrow c$ as $x\rightarrow a$ I ...
0
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1answer
27 views

Strictly monotone functions and continuity

Let $f : X \to I$ be a strictly monotone surjective function mapping $X \subseteq \mathbb{R}$ to an interval $I \subseteq \mathbb{R}$. Then is $f$ necessarily continuous? Without loss of ...
0
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1answer
30 views

Real Analysis, Folland 3.4.26, Differentiation on Euclidean Space

Background Information - A Borel measure $\nu$ on $\mathbb{R}^n$ will be called regular if i.) $\nu(K) < \infty$ for every compact $K$ ii.) $\nu (E) = \inf\{\nu(U): E\subset U, U \ ...
2
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4answers
41 views

Showing that Harmonic numbers are $\Theta(\log n)$, intuitively

I wish to verify that Harmonic numbers $H_n = \sum_{k=1}^{n} \frac{1}{k}$ are $\Theta(\log n)$. One idea I have is to approximate the sum with an integral: $$\int_{1}^{n} \frac{1}{k} ~dk = \log(n) - ...
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0answers
35 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
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2answers
27 views

Congruence problem $12x\equiv3\pmod{45}$ [on hold]

$$12x\equiv3\pmod{45}$$ Find all possible solutions to above congruence and show procedure in detail.
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0answers
38 views

Probability that a given function is prime…

If we have a set of primes $p_1$, $p_2$, ... , $p_n$, we can easily construct a function of their product: $$f(\alpha) = \alpha \left( \prod_{k=1}^n{p_k} \right) + 1, \alpha \in \mathbb{N}$$ I'm ...
0
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1answer
27 views

Limit of a sequence of measurable functions is measurable: alternative proof

If $f_n$ is an infinite sequence of measurable functions, let $\lim\limits_{n \rightarrow \infty} f_n(x) = f(x)$. Prove that $f$ is measurable. The proof (as I was taught it) is as follows. If ...
0
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1answer
59 views

Is there anything wrong in the following proof?

Problem. Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces and let $U\subseteq X$ and $V\subseteq Y$ such that $U$ and $V$ are respectively open in $X$ and $Y$. Show that $U\times V$ is open in ...
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0answers
34 views

Prove that $\sqrt 5$ is irrational given that Let $n$ be an integer. [on hold]

Prove that $\sqrt{5}$ is irrational given that Let $n$ be an integer. Prove that if $5|n^2$ then $5|n$. Hint: consider the contrapositve and consider cases. Also I considered the fact that $n$ cannot ...
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2answers
37 views

What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
1
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1answer
43 views

Linear Alg. Short proof on determinant

Hi can I get a quick check on my proof to see if it is correct. proof
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2answers
42 views

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $\prod_{n=1}^\infty (1-a_n)$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$.

Suppose $1>a_n>0$ for $n\in \mathbb{N}$. Prove that $$\prod_{n=1}^\infty (1-a_n)$$ converges if and only if $\sum_{n=1}^\infty a_n<\infty$. I know this question is similar to one I just ...
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4answers
78 views

Is this argument valid for a proof?

Please kindly forgive me if my question is too naive, i'm just a prospective undergraduate who is simply and deeply fascinated by the world of numbers. My question is: Suppose we want to prove that ...
2
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1answer
51 views

Deductive logic counter-intuitive result

I am working on a small proof in deductive logic. Here is what must be proved: $(\exists x \in T \mid A \implies P(x)) \implies A \implies (\forall x \in T \mid P(x))$ To me that looks unprovable ...
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1answer
16 views

Uniformly convergent in a set implies uniformly convergent in the set closure, too.

Let $f_n$:$X\rightarrow \mathbb{R}$ be a sequence of functions uniformly convergent in $X\subseteq \mathbb{R}$ . Suppose that each $f_n$ is continuous in the closure of $X$. Then $f_n$ is also ...
3
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1answer
64 views
+50

Differential equation where Picard-Lindelöf can not be applied

My question is the following : Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and let $u:[a,b]\to\mathbb{R}$ be a $C^1$ function such that $$\forall t\in[a,b],u'(t)=f(u(t))\text{ and ...
1
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1answer
52 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
0
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1answer
40 views

Finding a closed formula for: $1\cdot2\cdot3+2\cdot3\cdot4+…+(n-2)\cdot(n-1)\cdot(n)$ [duplicate]

As I calculated the sum of the serie above doesn't exist(sum doesn't converge). How can I prove it using the double computing(combinatorical method)?
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3answers
64 views

Alternative Proof that $\sqrt{p}$ is Irrational when $p$ is Prime

I have found various proofs that $\sqrt{p}$ is irrational on this site, but I didn't find one similar to the one that I am about to post, so I am wondering if it is free of logical problems. Here is ...
0
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0answers
29 views

How to calculate the length of this plane curve (loxodrome/rhumb line)?

I am trying to calculate the length of a (what I believe is) a loxodrome, using differential geometry. I am given a curve $\gamma(t)=\big(\theta(t),\varphi(t)\big)\subset \mathbb S^2$ that ...
3
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0answers
30 views

Properties of the Discrete Logarithm Problem

I am self-studying Hoffstein's An Introduction to Mathematical Cryptography, and this is problem 2.3 (p. 107-08). Let $p$ be a prime and $g$ an element in $\mathbb{F}_p^*$ with order $r$. ...
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1answer
21 views

Prove regular expression with induction

I need help proving the following regular expression via induction. I have the base case (easy of course) but I'm having a difficult time determining the inductive case. A regular expression over ...
3
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3answers
87 views

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+\left\lfloor 2t\right\rfloor =52 $. My effort Using the fact that for any number $x$ we have that $x=\left\lfloor x\right\rfloor+\{x\}$ (where $\{x\}$ is the fractional ...
0
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1answer
30 views

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups.

Prove $(H \times 1)(1\times K)= H\times K$ where $H,K$ are groups. Suppose $x=ab,a\in H\times 1,b\in 1\times K$ Then $x=(h,1)(1,k)$ where $h\in H,k\in K$ Hence $x=(h,k)\in H\times K$ Let ...
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2answers
54 views

Proof Verification: If $x$ is a nonnegative real number, then $\big[\sqrt{[x]}\big] = \big[\sqrt{x}\big]$

Let $x$ be a nonnegative real number and denote $[x]$ as the greatest integer less than or equal to $x$. We will attempt to prove that $\big[\sqrt{x}\big] = \big[\sqrt{[x]}\big]$. First suppose that ...
0
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0answers
54 views

Hypothetical proof of Goldbach's conjecture? [on hold]

Goldbach's conjecture: Every even number greater than 4 is the sum of two prime numbers. An equivalent statement is; For all $n\geq 2$ there exists a number $e(n)$ such that $n-e(n)$ and ...