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

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3answers
152 views

Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere, but I want help with my proof in particular. Let $X$ be a metric ...
2
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4answers
42 views

Showing uniqueness of inverse element of an element of a monoid

Question- If $\langle A,*\rangle$ is a semigroup with identity, prove that every element a belonging to $A$ has at most one inverse. Proof- Let the identity be $e$. Let us assume that $b_1$, $b_2$ ...
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1answer
36 views

Prove ${\sum \limits_{cyc}\frac{a^4+a^2+1}{a^6+a^3+1}\leq\sum \limits_{cyc}\frac{3}{a^2+a+1}}$

If $a,b,c$ are positive real numbers,Prove:$${\sum \limits_{cyc}\frac{a^4+a^2+1}{a^6+a^3+1}\leq\sum \limits_{cyc}\frac{3}{a^2+a+1}}$$ Additional info: We should only use Cauchy and AM-GM. ...
0
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1answer
23 views

For each integer $s$, how many N-tuples with possible elements $\{0, 1, -1\}$ satisfy the condition that the sum of its elements is $s$?

So, we can find the answer using the generating function: $$f(x)=(1+x+x^{-1})^N=x^{-N}\sum_{k=0}^{N}\sum_{m=0}^{k}{N \choose k}{k \choose m}x^kx^m$$ and the number of N-tuples for each integer $s$ is ...
0
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1answer
41 views

If Span(A)=Span(B) then $A \cap B \neq \emptyset$

Let the be V a vector space and $A,B \subset V$ sub-sets of V. If Span(A)=Span(B) then $A \cap B \neq \emptyset$ What I thought is that Span(A)=Span(B) mean that for all $a \in A$,$b \in B$ ...
2
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1answer
15 views

Solution sets/ existence and uniqueness of solutions to $Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x)$

Given $$ Ku-\lambda u=\int^1_0 \frac{x^2}{1+y^3}u(y)dy-\lambda u(x)=f(x) $$ A) For what values of $\lambda$ does there exist a unique solution for all $f\in L^2(0,1)$? B) Find the solution set ...
1
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3answers
58 views

Sequential compactness in $\mathbb{R}$

Well known result: Suppose $f:\mathbb{R}\to \mathbb{R}$ is continuous and let $K$ be a compact set. Then, $f(K)$ is compact. I can prove this using the definition of compactness (finding a ...
0
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0answers
87 views

Proof of Clairaut's Theorem (Weaker Statement)

I would like to ask if such a proof of Clairaut's Theorem (Weaker Statement) is correct Statement: Suppose $f$ is a real-valued function of two variables $x,y$ and $f(x,y)$ is defined on an open ...
2
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1answer
56 views

Show $T: C([0,1]) \rightarrow C([0,1])$ is compact

Consider $T: C([0,1]) \rightarrow C([0,1])$ defined by $$(Tf)(t) := \int_0^1 \kappa_t(s)f(s)ds,$$ where $\kappa:[0,1]^2 \rightarrow \mathbb{R}$ satisfies the following properties: for all $t\in ...
0
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1answer
46 views

Deeper Understanding Mean Value Theorem. Show that $(1+x)^r>1+rx$

Let $r>1$. If $$x>0 \text{ or } -1 \leq x<0,$$ show that $(x+1)^r>rx+1$, This is what ive done: $$f(x)=(x+1)^r-r x-1$$ $$\frac{d }{dx}-r+r (x+1)^{r-1}$$ From this I can see that ...
0
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1answer
32 views

Essential support vs. classical support for a continuous function

The essential support of a function $f:\Bbb R^n\rightarrow \Bbb R$ is defined in the following way: Let's denote $\mathcal A_f=\{\omega \subset \Bbb R^n: \omega \quad \text{open}, \quad f(x)=0\quad ...
0
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1answer
20 views

A Group That Span A Space

Let $A=\{x^2,2x+x^2,x+x^3\}$ Does $A$ span $\mathbb{R}_3[X]$? The basis is defined to be that smallest group that span that space, due to the trivial basis which dimension is 4, there is no smaller ...
1
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3answers
58 views

Prove that if $T=T^*$ and $\sigma(T)=\{\lambda\}$, then $T=\lambda I$

Show that if $T$ is a self adjoint linear operator on a Hilbert space such that the spectrum contains a single point $\lambda$, then $T=\lambda I$. Then, show this is false if $T$ is not self ...
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2answers
316 views

Equality of two iterated square roots

Solve for $x$: $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\dots}}}}=\sqrt{x\sqrt{x\sqrt{x\sqrt{x\dots}}}}$ My attempt: The L.H.S is equal to $\dfrac{1+\sqrt{4x+1}}{2}$ and R.H.S equals $x^2$ Equating both ...
1
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1answer
40 views

Proof of $b^{1/n} \rightarrow 1$ with Bernoulli's Inequality

this is my rough proof of the said statement. I hope you can help me verify it. We want to proof that if $b > 0$ then $b^{1/n} \rightarrow 1$ as $n \rightarrow \infty$. Let $b = (1+a)$, where ...
3
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0answers
31 views

Homology of non-singular projective algebraic variety

I am unsure whether or not the following claim is true or false and whether or not my proof works or not: Claim: Let $V \subset \mathbb{C}P^n$ be a complex $k$-dimensional, non-singular, projective ...
0
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1answer
22 views

Distribution: $f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$. What is its derivative with respect to the parameter $a$ and the limit as $a\to 0$.

Consider the distribution $$ f_a(x)=\frac{H(x+a)-H(x-a)}{2a}$$ Determine the $a$-derivative of this distribution $$ \left < \frac{\partial f_a}{\partial a},\phi \right> = \lim_{h\to0} ...
2
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1answer
42 views

can a group with non-trivial center be isomorphic to a normal subgroup of its group of automorphisms?

i think (tho would be grateful for error-check) that the line of reasoning below suggests any group with trivial center is isomorphic to a normal subgroup of its automorphism group. question does the ...
2
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4answers
69 views

Is this a valid proof for the existence of a rational number between any two real numbers?

Given $a, b \in \mathbb R$ with $a<b$, prove that there exists some $r \in \mathbb Q$ such that $a<r<b$. Before I prove the main statement, there's a lemma I'd like to prove: Lemma ...
0
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2answers
34 views

Trying to disprove a statement - some partial working included

I am trying to find a counter example to show that the statement below is false, but I am having difficulty in trying to find a reasonable argument. Here is the statement: $n^2-12n + 35 \geq 0$ for ...
2
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2answers
40 views

$\operatorname{Span}(A \cup B)= \operatorname{Span}(A)+\operatorname{Span}(B)$

Let there be $A,B\subseteq V$ subspaces of $V$ $\operatorname{Span}(A \cup B) = \alpha_1a_1 + \cdots + \alpha_n a_n + a_{n+1} b_1 + \cdots + \alpha_k b_k=\sum\limits_{i=1}^n ...
2
votes
2answers
227 views

Area of Triangle when 2 Sides and No Angle Known

It is quite possible this question has no answer -- that is, the area cannot be determined from the information given. It's a question I've created myself as I study for the GRE. No trigonometry is ...
2
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2answers
40 views

Spectrum of $Tu=\int^1_0 (x+y)u(y)dy$

Given the operator $$Tu(x)=\int^1_0 (x+y)u(y)dy$$ on $L^2(0,1)$, find the spectrum of $T$. For all eigenvalues, find their multiplicities and the eigenfunctions. The kernel is Hilbert Schmidt ...
2
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1answer
54 views

Verify combinatoric argumentation.

I tried to find all the numbers between 100 and 999, that consist of (pairwise) different ciphers. So the first would be 102 and the last would be 987. I think there are 9*9*8 such numbers, here's ...
1
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2answers
71 views

Limit tending to negative infinity proof

I am new to proofs and would appreciate advice on this proof. Prove that $\lim_{x\to-\infty} \frac1x = 0$. Given $\epsilon>0$ find N such that: if $x < N$ then $\left\lvert\frac1x - ...
1
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5answers
203 views

n is even if and only if n leaves remainder 0, 2, 4, 6 or 8 when divided by 10

"Let n be any given positive integer. Prove that n is even if and only if n leaves remainder 0, 2, 4, 6 or 8 when divided by 10". Am I correct in thinking that with regards to the "if and only if" ...
0
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0answers
50 views

Number theory/proof help appreciated! Includes working.

my question concerns number theory and proofs. I have shown my working for the following two questions but I'm not too sure if I am correct - I feel as if this might be a trick question. Any feedback ...
1
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1answer
31 views

Show there exists a Cauchy subsequence

Let $X$ be a separable reflexive real Banach space and $\{\psi_n\}$ be a dense sequence in $$\{\psi\in X' : ||\psi||_{X'} \leq 1\}.$$ Consider in $X$ the scalar product defined by $$(x | y)_0 = ...
3
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0answers
67 views

Show that $\overline{A} \cup \overline{B} = \overline{A \cup B}$.

Can someone please verify my proof? Show that $\overline{A} \cup \overline{B} = \overline{A \cup B}$. Clearly, $$A \cup B \subseteq \overline{A \cup B}$$ So, $$A \subseteq \overline{A \cup B}$$ ...
3
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1answer
47 views

Show that if $A$ is closed in $X$ and $B$ is closed in $Y$, then $A \times B$ is closed in $X \times Y$.

Can someone please verify my proof? Show that if $A$ is closed in $X$ and $B$ is closed in $Y$, then $A \times B$ is closed in $X \times Y$. Let $x \times y \in X \times Y - A \times B$. Then, ...
3
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1answer
59 views

Weak convergence $f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $f_n^2 \rightharpoonup g$ in $L^1(\mathbb{R})$ implies $f^2\leq g$ a.e.

$f_n \rightharpoonup f$ in $L^2(\mathbb{R})$ and $f_n^2 \rightharpoonup g$ in $L^1(\mathbb{R})$, then $f^2\leq g$ a.e. Could you guys help me check the proof please, thanks! Proof: to show $f^2 ...
2
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3answers
76 views

Show that $h(x) = h(x + \frac{1}{2013})$ for some x in $[0,1]$

Edit: I very much appreciate alternate solutions to the problem, but I would also like to know if there are any problems or suggestions regarding the way I solved it. This is a problem I'm ...
2
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3answers
59 views

Prove that if $n\in\mathbb{N}$ and $n \geq 3$ then $n! + 3$ is composite.

Prove that if $n\in\mathbb{N}$ and $n \geq 3$ then $n! + 3$ is composite. I tried factoring it to show that there are two factors, thus composites but I can't figure out how to get rid of the ...
0
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0answers
29 views

Proof for a relation

Define a relationship $R$ on $\mathbb{Z}$ by declaring that $xRy$ if and only if $x^2 \equiv y^2 (mod 4)$. Prove that $R$ is reflexive, symmetric and transitive. I'm unsure of where to start with ...
0
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1answer
28 views

Check my proof : gcd(a,b)=1=gcd(x,y) => (xa,yb)=gcd(x,b) gcd(y,a)

Note: (x,y) means gcd(x,y) I managed to prove the next Proposition: Let $(a,b)=1=(x,y)$. Then $(x a,y b)=(x,b)(y,a)$. It can be easily be generalized for the case that $(a,b)\neq1$ and or ...
0
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1answer
43 views

Using Wilson's Theorem to prove Fermat's Little Theorem

My book says that the statement "$\forall a \not \equiv 0 (\mod p)$ and $p$ prime, $a^{p-1} \equiv 1 (\mod p)$" follows from Wilson's Theorem. I'd like to know how. This is what I've looked at so far: ...
1
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1answer
17 views

Dimension of $\cal V^-$ if $\cal V$ is a $\Bbb C$ vector space

In Halmo's book 'Finite dimensional vector spaces' there's a question I'm kind of stuck on in chapter 1. $1 (b)$ Every complex vector space $\cal V$ is intimately associated with a real vector space ...
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0answers
41 views

Proof that $S^\perp$ is a subspace of a vector space $V$

Just doing some review for a final exam and would like some feed back on the following proof if anyone would like to help me out. First the premise. Let $V$ be a finite dimensional inner product ...
1
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4answers
42 views

Wilson's Theorem textbook proof question

I'm trying to understand this proof from Stein's Elementary Number Theory, and I understand the pairing of inverses but not the other direction. I have two questions: $1).$ When the proof says, $l$ ...
0
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0answers
21 views

Continuity of $\zeta(a)$: is my proof correct?

Let $A$ be a unital Banach algebra and define $\displaystyle \zeta (a) = \inf_{c \in A: \|c\| =1}\|ac\|$. I tried to prove $|\zeta (a) - \zeta (b)| \le \|a-b\|$ for all $a,b \in A$, could someone ...
0
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2answers
38 views

Showing that two maps are homotopic

Let $X$ be a topological space and let $S^2 \subset \mathbb{R^3}$ be the unit sphere with the metric $d$ inherited from $\mathbb{R^3}$. Show that if $f,g:X\to S^2$ are continuous maps such that ...
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2answers
311 views

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions?

Is this a solution for the problem: $\ a^3 + b^3 = c^3\ $ has no nonzero integer solutions? Suppose $\ a^3 + b^3 = c^3,\ a,b,c \in \mathbb Z^*,\ $then: $c^3 - b ^ 3 = (c - b)((c - b) ^ 2 + 3cb) = ...
1
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1answer
31 views

Show that $\gcd(a,b) = \gcd(a+b,b)$

Show that $\gcd(a,b) = \gcd(a+b,b)$ I understand the proof up till this point: By definition of $\gcd(a,b)$ it implies that $d|a+b$ and $d\mid b$ thus, $d\mid a+b$ By definition of $\gcd(a+b,b)$ it ...
0
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1answer
44 views

Exercise 2, chapter 4, Hatcher.

Show that if $\varphi: X \rightarrow Y$ is a homotopy equivalence, then the induced homomorphisms $\varphi_{*}:\pi_n(X,x_0) \rightarrow\pi_n(Y,\varphi(x_0))$ are isomorphisms, for all n$\in ...
-1
votes
0answers
16 views

Proof that the limit of sequence with upper bound is lower than the upper bound

Can you guys verify this simple proof for me? Here's the hypotheses: If $s_n \leq B$ for all $n \in \mathbb{N}$ and the limit of the sequence is $s$, then $s \leq B$. From the basic definition of ...
1
vote
1answer
48 views

a square in a finite field of odd order

GF(q) is a finite field of order q, where q is odd. Prove that $a\in GF(q), a\neq0$ has a root in $GF(q)$ iff $a^{(q-1)/2}=1$. I tried to prove it this way: Suppose a has a root in ...
4
votes
1answer
30 views

A simple proof varification

A certain pipe can fill a swimming pool in $2$ hours; another pipe can fill it in $5$ hours; a third pipe can empty the pool in $6$ hours. With all three pipes turned on exactly at the same time, and ...
0
votes
0answers
36 views

What is this called specifically?

Imagine you take a radius from the center of the shape, you add up all of the lines as it rotates 360 degrees. The radius is measured from its point of rotation, like (0,0) in Cartesian coordinates,to ...
1
vote
2answers
49 views

3D Cauchy problem for the PDE $ yu_x-xu_y+u_z=0 $

I will answer the question myself but let me know what you think of my correctness. We have the Cauchy Problem $$ yu_x-xu_y+u_z=0 $$ with data $u(x,y,0) = x+y$.
1
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1answer
27 views

Verify Result of a Calculation

In the journal: "A Closed Form Solution for the Similarity Transformation Parameters of Two Planar Point Sets", I cannot get same value for scaling factor for the same problem in the journal. Here is ...