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

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“Necessary” condition for Power Diophantine Equation.

Motivation: Brocard’s problem $n!+1$ being a perfect square Observations: Given a power Diophantine equation of $k$ variables with a “general solution” (provides infinite integer solutions) to ...
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1answer
13 views

A convergent sequence of non-expansions converges uniformly on a totally bounded domain

Here's a theorem that I tried to prove: Let $V,d_V$ and $W,d_W$ be metric spaces and $(f_n)_n$ a sequence of non-expansions that converges to a function $f:A \subseteq V \rightarrow W$: $$ f_n:\ A ...
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1answer
21 views

limit of convergent sequence of contractions is a contraction

Let $(f_n)_n$ be a sequence of functions $f_n: [0,1] \rightarrow [0,1]$ that (pointwise) converges to a function $f$. Suppose that all $f_n$ are contractions, prove that $f$ is a contraction as well. ...
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2answers
38 views

Can we differentiate equations without changing the solutions?

Suppose we have $x^2+4x+3=0$ We differentiate both sides to get $2x+4=0$. So $x=-2$ But $(-2)^2+4(-2)+3=0$ $4-8+3=-1$ is not equal to $0$. Which tells that we cannot differentiate equalities. ...
2
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1answer
30 views

Inner Product: prove that $\langle w, v+v' \rangle = \langle w, v \rangle + \langle w, v' \rangle$

We could use linearity in the first argument, homogeneity in first argument, and conjugate symmetry properties of the dot product. So this was my attempt at proving this: We know that $\langle ...
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4answers
2k views

Is this theorem already in existence?

I came up with this theorem (shown below) a few months ago, and I haven't been able to find anything like it on the web. This theorem will give you the quadratic expression that results from ...
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1answer
14 views

Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$

Let $M$ be a metric space, let $K \subset V \subset M$, $K$ compact, $V$ open. Prove that there exist $r>0$ such that $\bigcup_{x \in K} B(x,r) \subset V$ I came up with a proof, but there is ...
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2answers
12 views

Upper Bounds Of Integers Intersection

Prove\Disprove: $A$ is bounded from above $\iff$ $A\cap \mathbb{Z}$ is bounded from above. Let $A=\{a\in \mathbb{Q} \setminus \mathbb{Z}: a<0\}$ is bounded from above, $A\cap ...
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2answers
27 views

Upper Bounds Of Sub Sets

Prove/Disprove: Let $A\subseteq B\subseteq \mathbb{R}$ If $B$ is bounded from below so $A$ is bounded from below. $B$ is bounded from below $\rightarrow$ $x\leq B$. Let there be $z\in A$, ...
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1answer
38 views

The set of algebraic numbers is countable: is this proof correct and well written?

Problem: prove that the set of all algebraic numbers is countable. My proof: Let $f: \bigcup^{\infty}_{n=1} \mathbb{Z}^n \rightarrow \mathcal P(\mathbb{C})$ be a function associating an ordered ...
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0answers
47 views

Prove if A and B are skew symmetric then A+B is skew symmetric

If $A^T$ = $-A$ which means A is skew symmetric then prove that $(A+B)$ is also skew symmetric. I managed to prove it like this: $(A+B)^T$ = $A^T$+$B^T$ =$(-A+-B)$=$-(A+B)$ Therefore ...
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0answers
17 views

Any ambiguity in my proof of weakly stationarity for a time series model?

my lecturer said that there is an ambiguity in my proof but I cannot see any of them. All I wanted to do is for a time series $${X_t} = C + {\phi _1}{X_{t - 1}} + \cdots + {\phi _p}{X_{t - p}} + ...
1
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1answer
54 views

Did I solve exercise 4.5.4 (b) of 'How to Prove it' by velleman correctly and concisely?

4.5.4 Suppose R is a strict partial order on A. Let S be the reflexive closure of R. (b) Show that if R is a strict total order, then S is a total order. Suppose R is a strict total order. ...
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1answer
36 views

what is wrong with this proof? (proving the transitive property of Big O)

So the problem is if $f(n) \in O(g(n))$,and $g(n) \in O(h(n))$ then $f(n) \in O(h(n))$ Assume $f(n) \geq 0, g(n) \geq 0, h(n) \geq 0$ Proof: From assumptions, ...
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1answer
18 views

prove that $a +b_1 +2 \leq 3 * (b_2 + 1)$ using the following inequalities

prove $$a +b_1 +2 \leq 3 * (b_2 + 1)$$ Here are your assumptions: $$(a + 1) * 3 < b_1 + b_2 + 2$$ $$b_1 + 1 < (b_2 + 1 ) * 2$$ $$(a+1) * 3 \geq b_1 + b_2 + 1$$ $$(a + 1) \leq (b_1 + b_2 + 1 ...
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0answers
18 views

On bilinear forms: if $B$ is non-degenerate, then $(U_1\cap U_2)^{\perp_L} = U_1^{\perp_L} + U_2^{\perp_L}$

I think I have a proof, but I'm not completely certain it is correct: Because $B$ is non-degenerate, it follows that if $W \leq V$, then $W = W^{\perp_L \perp_R}$. It is evident, too, that $(U_1 + ...
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1answer
53 views

Prob. 14, Sec. 2.10 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: application to a system of equations?

Let $M$ be a non-empty subset of a normed space $X$, and let $M^a$ denote the subspace of the dual space $X'$ that consists of all those bounded linear functionals that vanish at each point of set ...
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2answers
51 views

Is this a correct proof that these two norms are not equivalent?

Consider $(C[0,1],\|\cdot\|_1)$ and $(C[0,1],\|\cdot\|_2)$ where the norms are $$ \|f\|_1 = \int_0^1 |f(x)| dx$$ and $$ \|f\|_2 = \left (\int_0^1 |f(x)|^2 dx \right )^{1\over 2}$$ I tried to ...
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1answer
54 views

If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple

I'm looking for verification: My claim: If $G$ is a finite group and $H$ is a (proper)subgroup of index $k>1$, where $k! \leq |G|$, then $G$ is not simple. Proof: Consider the set of left cosets ...
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0answers
45 views

Normal Exponential Convolution proof

I am seeing a scientific paper where they explain background correction modelled as two random independent variables one with exponential distribution and the other with normal distribution. ...
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1answer
33 views

Residues and poles, proof with poles

Proof that statements i)If $f_1$ and $f_2$ have residues $r_1$ and $r_2$, show that the residue of $f_1+f_2$ at $z_0$ is $r_1+r_2$. ii)If $f_1$ and $f_2$ have simple poles at $z_0$ show ...
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0answers
93 views

Very tentative proof of Beal's Conjecture?

I'm a high school student, so please point out my mistakes nicely and in layman's terms :) Thanks! Ok. Beal's Conjecture: If $$a^x+b^y=c^z$$ where $a$, $b$, $c$, $x$, $y$, $z$ are whole numbers; $x, ...
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1answer
53 views

Prove that $\mu(x,z)\geq \min \left \{ \mu(x,y),\mu(y,z) \right \}$ for $x,y,z\in \{ 0,1\}^{\mathbb{N}}$.

Let $x,y,z\in M:=\{ 0,1\}^{\mathbb{N}}$ and define $\mu(x,y)=\min\{ n\in \mathbb{N}\mid x_{n}\neq y_{n}\}$. I want to show that $\mu(x,z)\geq \min \left \{ \mu(x,y),\mu(y,z) \right \}$. I have ...
0
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1answer
25 views

Prove: functions with bounded derivatives are Lipschitz continuous

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $M > 0$, $f: [a, b] \rightarrow \mathbb{R}$ be a function which is continuous on ...
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2answers
36 views

Showingthat $\mathbb{R}^2$ with operation $\langle (a,b), (c,d) \rangle = ac - bd$ is not an inner product space

We define the "inner product" as $\langle (a,b),(c,d)\rangle = ac-bd$ on $R^2$ We want to verify if this is an inner product space. I'm saying it's not because it does not satisfy the positivity ...
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1answer
35 views

Geometric proof that (symmetry w/r to $x$ and $y$ axes) $\implies$ (symmetry w/r to origin)

I'm trying to prove that reflecting a point about the x and y axes is equivalent to reflecting it about the origin. Is my proof valid? How could I improve it? Proof: Take a point $a$ in the first ...
2
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1answer
16 views

Where is “homogenuity” used in this proof?

Theorem Let $R$ be a ring and $P\in R[X_1,...,X_n]$ be a symmetric polynomial and $p_1,...,p_n$ be the elementary symmetric polynomials. Then, there exists a polynomial $Q\in R[X_1,...,X_n]$ ...
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0answers
6 views

Is my proof showing that if products of powers of elementary symmetric polynomials are the same, then the power is same, correct?

Below proof is assuming this lemma, which can be proven easily: Let $\leq$ be a monomial ordering on $R[X_1,...,X_n]$. Let $f,g\in R[X_1,...,X_n]$ be nonzero polynomials such that $LC(f)$ is ...
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1answer
21 views

Help with Proving : Period estimation for for concatenated sequences

Assuming I have two 8-bit random number sequences $s[n]$ and $d[n]$ which each have a period of $X$ and $Y$ respectively. Therefore: $$s[n+X] = s[n]\\ d[n+Y] = d[n]$$ If they were concatenated ...
0
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1answer
42 views

Spivak Ch1 Proof Critiques

I've started working through Spivak's Calculus. I'm going into senior year after this summer, took the AP Calculus BC test last year, and wanted to get a firmer foundation in calculus before I take ...
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1answer
23 views

$\lbrace \lim f_n(x) \rbrace$ is a Borel set if each $f_n$ is borel

Suppose for all $n$ that $f_n:\mathbb{R}\to \mathbb{R}$ is Borel measurable. What follows is an attempt of the proof that $\lbrace x: \lim_{n\to \infty} f_n\rbrace$ is Borel measurable, but I am a bit ...
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2answers
73 views

Proof that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$

I tried to prove one of the examples in my Abstract Algebra book that stated: Prove that if $a,b \in G$ and $a^4b = ba$ and $a^3 = e$ then $ab = ba$ I went about just saying that $a^4b = ba ...
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2answers
43 views

Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ then $Z(G)$ is a group

So my challenge is: Prove that if $Z(G) = \lbrace x \in G: gx = xg \text{ for all }g\in G\rbrace$ where $G$ is a group, then $Z(G)$ is a group Unlike this question: To show that the center is a ...
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1answer
26 views

Characterization of subsets of $\mathbb{R}^n$ of the form $X+Y$

The following comes from the mathematical tripos exam at Cambridge: Let $X,Y \subset \mathbb{R}^n$, and define $X+Y = \{x+y : x \in X, y \in Y\}$ Prove or disprove each of the following: (i) If ...
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2answers
55 views

Proof that odd perfect numbers cannot consist of single unique factors?

I'm a high school student, so please point out my mistakes nicely :) So we already know odd perfect numbers cannot be in the form of a square, but how about that they cannot be in this form: ...
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38 views

How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...
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1answer
49 views

Understanding a part of a proof involving Hilbert-Schmidt norm

I came across a proof I do not seem to understand fully, a screenshot is provided below. my concerns are the following: Why does the fact that $||T||_2 = ||UT||_2$ for every unitary U, allow us ...
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2answers
69 views

Proving that $\sum_{i=2}^n(5i-4)=\frac{n(5n-3)-2}{2}$ for all $n\geq 1$ by mathematical induction

I have this question: Show, using mathematical induction, that for all natural numbers $n$, $$6 + 11 + 16 + 21 + \cdots + (5n-4) = \frac{n(5n-3)-2}{2}$$ I am confused in that that question states ...
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1answer
24 views

Prove that if $f$ is an invertible function and $g$ is an inverse, then the codomain of $g$ is equal to the domain of $f$ and vice versa

I am trying to show, without using the bijection properties, what is above. Assume $f$ is an invertible function and $g$ is an inverse of $f$. For $f \circ g $ to be well defined then the image of ...
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2answers
44 views

Proof that $n\Bbb Z \leq \Bbb Z$ and are the only subgroups of $\Bbb Z$

My challenge is Prove that if $n = 0,1,2,\ldots$ and $n\Bbb Z = \lbrace nk: k \in \Bbb Z \rbrace$, show that $n\Bbb Z$ is a subgroup of $\Bbb Z$ and are the only subgroups. I handled the first ...
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1answer
34 views

Multiplying two inequalities

Suppose we have two inequalities $$a\leq x\leq b\tag{1}$$ $$c\leq y\leq d\tag{2},$$ where $a,b,c,d>0$. Then can I conclude that $$ac\leq xy\leq bd\quad ?$$ My attempt: Since $a,b,c,d>0$ and ...
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3answers
47 views

The equation of a tangent to a circle at a given point

18. Show that the equation of the tangent $PT$ at the point $P \left(\frac{1}{5}, \frac{3}{5}\right)$ on the circle $$x^{2} + y^{2} + 8x + 10y - 8 = 0$$ is $3x + 4y - 3 = 0$. Find ...
1
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2answers
29 views

Proof of the right and left cancellation laws for Groups

I was asked to proof the right and left cancellation laws for groups, i.e. If $a,b,c \in G$ where $G$ is a group, show that $ba = ca \implies b=c $ and $ab = ac \implies b = c$ For the first ...
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0answers
52 views

Ito's lemma - mistake in text book?

Ito gives us $$dW = \dfrac{dW}{dX} dX + \left(\frac{dW}{dt} + \frac{1}{2} \frac{d^2W}{dX^2}\right) \, dt$$ We have a function $W(t) = 1 + t + E^{X(t)}$. My text book says that $$dW = e^{X(t)} \, dX + ...
2
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1answer
18 views

Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...
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5answers
66 views

Show $\sum n e^{-na}$ converges for $a>0$

Is there any test or property in particular I can use to show $ \sum n e^{-n a}$ is convergent for $a>0$ ? I think it is obvious that from looking at the function that this is convergent, since ...
2
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1answer
62 views

Showing Uniform convergence of $\frac{n x}{1 + n \sin(x)}$

I want to prove for all $a\in \left(0,\frac{\pi}{2}\right]$, $ \ f_n\to f$ uniformly on $\left[a,\frac{\pi}{2}\right]$. Also, how is this different from $f_n \to f$ uniformly on $\left(0, ...
2
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0answers
34 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
2
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1answer
25 views

Proof that there exists an $x \in G$ such that $xa = b$

So this is my challenge: Let $G$ be a group and $a,b \in G$. Then $xa = b$ has a unique solution I went about saying that $xa = b \iff xaa^{-1} = ba^{-1} \iff x = ba^{-1}$. $ba^{-1} \in G$ ...
3
votes
0answers
34 views

Why is this class closed under difference?

We have two independent random variables $X\perp Y$ involving three spaces: $(\Omega,\mathcal{A},P), (E,\mathcal{E}), (F,\mathcal{F}).$: $$X:\Omega \rightarrow E,\ Y:\Omega\rightarrow F$$ My book says ...