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

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1answer
17 views

Adjoint of a compact operator between Hilbert space is compact

Let's $H$ be a Hilbert space and $T:H \rightarrow H$ a compact map. Show that the adjoint operator $T^*$ is also compact. I've been thinking a lot about this problem and this is what I've done: It's ...
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3answers
53 views

What is wrong with the proof of this incorrect formula for the probability of $A\cap B$?

The probability that events $A$ or $B$ (or possibly both at once) will occur is $P(A\cup B)$. Since we can think of $A\cup B$ as the set-theoretic analog of inclusive logical disjunction $\vee$, I ...
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1answer
40 views

What is wrong with the following induction argument?

I found a problem on a note on induction. The problem went like this: "Let $n$ be a non-negative integer. Suppose we are given a triangle and n points inside it, with no three of the given $n + 3$ ...
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1answer
31 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
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1answer
47 views

Proof on $F(x)=\sum_{n=1}^{\infty}f\left(\frac{x}{n}\right)$ uniform convergence and differentiability

Let $f$ be a function of $C^{\infty}$ class, such that $f(0)=0=f'(0)$. Prove that if $x\in\mathbb{R}$ and $$F(x)=\sum_{n=1}^{\infty}f\left(\frac{x}{n}\right)\ ,$$ then $F(x)\in\mathbb{R}$ and $F$ is ...
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1answer
17 views

One-point sets are G$_\delta$ in first-countable $T_1$ spaces

Can someone please verify my proof or offer suggestions for improvement? I am aware that there is a similar question elsewhere. I only need help with my proof in particular. Show that in a ...
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0answers
13 views

Mother Wavelets are symmetrical or not

Are mother wavelets symmetrical i.e can we write $\psi(-x)= \psi(x) ?$ As we know that $\psi^{a,b}(x)=|a|^{-1/2}\psi(\frac{x-b}{a})$. Then $\psi^{a,b}(0)=|a|^{-1/2}\psi(\frac{-b}{a})$. So can we write ...
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1answer
32 views

Infinite topological space with cofinite topology is not Hausdorff

I found a proof to the question, but mine is completely different (sort of). Is this correct? If $X$ were Hausdorff, then consider $u,v \in X$ with disjoint neighbourhoods $U, V$ that separates ...
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1answer
46 views

Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
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2answers
28 views

$Ker(T) \subseteq V$ Is A Subspace

Let $V,W$ be a vector space over a field $\mathbb F$, and $T$ a linear transformation $T:V \rightarrow W$ $Ker(T) \subseteq V $ to prove that $Ker(T)$ is a subspace can we say that: by definition ...
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0answers
156 views

Proof on why $0-1+2-3+4-\ldots\neq-1/4$

When reviewing my notes on series' convergence, I thought of applying a workaround on why $\sum_{n=0}^{\infty}(-1)^nn$ should or shouldn't be $-1/4$ (I recalled this page). I started by considering ...
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2answers
23 views

Linear Transformation $T_{A}$ Is invertible $\iff$ A Is invertible

Let $T_{A}$ be the linear mapping corresponding to the matrix A, and $A \in F^{n*n}$ $T_{A}$ Is invertible $\iff$ there is $T_{A}^{-1}$ so $T_{A} \circ T_{A}^{-1}=I $ $T_{A} \circ T_{A}^{-1}(v)=v$ ...
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2answers
30 views

Proving that only the linear codes pass parity check

An exercise in my book goes as follows: Let $C$ be a binary $(n,k)$ linear code with parity-check matrix $H$. We know $Hc=0$ for all $c\in C$. Show that $Hw=0$ implies $w\in C$. My idea: Let ...
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1answer
43 views

General formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Is such proof correct?

I have a very simple case: Find general formula for $\frac{a^{2n+1}-a^{2n-1}}{a^n-a^{n-1}}$. Of course dividing one by another was quite simple with outcome: $a^n(a+1)$. However I would like to prove ...
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2answers
28 views

Convergence of integral, that is absolutely convergent, proof

Can you think of any proof on convergence of improper integral, that is absolutely convergent? It is so obvious, that I really don't know where to start. Triangle inequality gives us ...
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2answers
36 views

The maximum no. of edges in a DISCONNECTED simple graph…

... on n vertices when it is not connected being equal to (1/2)(n - 1)(n - 2)... I can see that for n = 1 & n = 2 that the graphs have no edges... however I don't understand how to derive this ...
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2answers
44 views

Linear Code (9,5): Is my Parity Check correct?

I have an exercise about designing parity checks for the Hamming (9,5) group code with minimum distance $3$. I use the following notation for the generator matrix: $$ ...
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1answer
52 views

Check my proof by contradiction…

The question is to prove the following by contradiction. There does not exist a smallest positive non-zero rational number. What I tried... There does exist a smallest positive non-zero rational ...
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0answers
29 views

Characterisation of absolutely continuous measure on the real line

Let $\lambda, \nu$ be two Radon measures on $\Bbb R$ such that $\lambda(\Bbb R)< \infty$. Show that the following are equivalent: $\lambda \ll \nu$; $\forall \epsilon>0$ there exists ...
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3answers
85 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
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3answers
50 views

Show that $f(x)=x/\sqrt{x^2+1}$ is a bijection of $\mathbb R$ onto $\{ y: -1<y<1\}$

I am looking for help in regard to a practice question about functions. The question is Show that a function $f$, defined by $f(x)=x/\sqrt{x^2+1}$ , $x \in \Bbb R$ is a bijection of $\Bbb R$ onto ...
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1answer
34 views

Vector spaces - Multiplying by zero vector yields zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
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0answers
36 views

Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
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0answers
33 views

Finding an analytic function such that real part is the given function.

I am reading the book Complex Analysis by Lars V Ahlfors. In the book he uses a nice method without involving integration to evaluate $f$ given that the real part of the function is $U$. The method ...
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1answer
41 views

How to prove a function from a set of triples to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection?

Let $Y=\{y_1, y_2, y_3, y_4,y_5\}$ The function from the set of triples $(y_{i_1},y_{i_2},y_{i_3})$ where $i_1 \le i_2 \le i_3$ to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection given by ...
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2answers
82 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
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1answer
31 views

Proving IMVT using delta-epsilon

Let's assume $f(a)<0$ and $f(b)>0$. IMVT claims that there's $c\in(a,b)$ such that $f(c)=0$. The Proof: Consider $$A = \{ a\le x\le b : f(x) < 0 \}$$ That's a non-empty set and therefore, by ...
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2answers
63 views

Exercise about truth functions in J.R.Shoenfield's “mathematical logic”

The first exercise in Joseph R. Shoenfield's "mathematical logic" is: An n-ary truth function $H$ is definable in terms of the truth functions $H_1,\dots,H_k$ if $H$ has a definition ...
0
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1answer
54 views

If $f$ is strictly convex prove that $f(x + f'(x)) \geq f(x)$ for every $x$.

Let $f$(maps from $R$ to $R$) be twice differentiable function and strictly convex. Prove that for every $x$ from $R$ it is true that $f(x + f'(x))\geq f(x)$. Let's suppose otherwise i.e. let $c$ be ...
2
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1answer
35 views

Show that $X$ is Hausdorff if and only if the diagonal $\Delta = \{(x, x):x \in X\}$ is closed in $X \times X$

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify it or offer suggestions for improvement? Show that $X$ is Hausdorff if ...
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0answers
19 views

Congurence proof of modulus equivalence

I would like some advice if I have approached this problem correctly please: let $a,b,m,n \in \mathbb{Z}$ and $m,n > 0$. Prove that if $a\equiv b \pmod n$ and $m|n$, then $a\equiv b\pmod m$ ...
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0answers
15 views

What are the continuous automorphisms of $\Bbb T$?

I wanted to check my reasoning on this problem. From standard Pontrjagin duality arguments, it's not hard to see that the continuous homomorphisms of the torus (to itself) are nothing more than the ...
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0answers
50 views

Quotient of local ring is of finite length

My objective is to show that $\mathcal{O}_{P}/(f,g)$ is of finite length as a $\mathcal{O}_{P}$-module. $\mathcal{O}_{P}$ is the local ring of $P = (0, 0)$. In other words it's $k[x, ...
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1answer
21 views

Use the universal cover to prove $\gamma * \gamma$ is nullhomotopic if $\gamma$ is a loop in the projective plane

Use the universal cover to prove $\gamma * \gamma$ is nullhomotopic if $\gamma$ is a loop in the projective plane Well, If I was not asked to prove it this way, I could have argued like : ...
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2answers
45 views

Proving that $\sqrt{pq} \ne (p + q)/2$ implies $p \ne q$ using the contrapositive

Prove by the contrapositive method, that if $p$ and $q$ are positive real numbers with the property that $\sqrt{pq}$ is not equal to $(p+q)/2$, then $p$ is not equal to $q$. The proof is easy enough ...
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2answers
42 views

Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$

Let $f(x)$ and $g(x)$ be irreducible polynomials over a field $F$ and let $a,b \in E$ where $E$ is some extension of $F$. If $a$ is a zero of $f(x)$ and $b$ is a zero of $g(x)$, show that $f(x)$ is ...
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2answers
53 views

$Rank(A+B)\leq Rank(A)+Rank(B)$ [duplicate]

Let there be $A,B$ matrices. Let $C=A+B$ $Span(Col(C))\subseteq Span(Col(A))$ because C is a linear combination of A . $Span(Col(C))\subseteq Span(Col(B))$ because C is a linear combination of B . ...
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30 views

Prove that $G$ is Hamiltonian.

Let $G=(V,E)$ be a connected graph which is not a tree. Prove that if for every cycle $C$ of the graph G and for any $v \in V(G)- V(C)$ there are more than $\frac{|C|}{2}$ edges from $v$ to $V(C)$ ...
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1answer
24 views

Convergence in $L^p(0,T;L^q(\Omega))$

If $\Omega\subset\mathbb{R}^3$ is bounded, $$f_n\to f\mbox{ in }L^q(0,T;L^p(\Omega)),\,1\leq q<\infty,\,1\leq p<2 $$ and $$f_n\to g\mbox{ weak-star in } L^\infty(0,T;L^2(\Omega)),$$ then $f=g$ ...
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1answer
64 views

Can $xy=0$ be the image of an algebraic morphism $\mathbb A^2 \rightarrow \mathbb A^2$?

Suppose we have an algebraic morphism $f:\mathbb{A}^2\rightarrow \mathbb{A}^2$. Can the image of $f$ be the zero locus of the polynomial $xy$? I think not, at least not if we're working over ...
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1answer
35 views

Prove that for a bounded self adjoint operator, $\langle Tx,x\rangle \geq 0$ is equivalent to $\sigma(T)\subset [0,\infty)$

Prove that for a bounded self adjoint operator, the following are equivalent: A: $\langle Tx,x\rangle \geq 0$ B: $\sigma(T)\subset [0,\infty)$ What I have said so far: Since $T$ is self adjoint, ...
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2answers
53 views

Proving an convexity-looking inequality

If $0 \le \alpha \le 1$ and $0 \le \lambda \le 1$, then $$\lambda^\alpha x^\alpha +(1-\lambda^\alpha) y^\alpha \ge (\lambda x + (1-\lambda)y)^\alpha$$ whenever $0 \le y \le x$. This looks ...
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2answers
32 views

Can I perform the quadratic formula on polynomial with complex coefficient?

2 weeks ago, we had a Math test on complex number. One of the question was: Let $z=x+iy$ be a non-zero complex number, where $x,y \in \mathbb{R}$. Given that $z+\frac{1}{z} = k$, where $k$ ...
3
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2answers
39 views

Prove that there is no bipartite graph on $14$ vertices with this degree sequence.

Prove that there is no bipartite graph on $14$ vertices with degree sequence: $$6, 6, 6, 6, 6, 6, 6, 6, 5, 3, 3, 3, 3, 3.$$ I assume a vertex with degree $5$ breaks this graph from being ...
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0answers
17 views

Gallai & Milgram path covers theorem from Diestel

I have a question about the theorem of Gallai and Milgram stating that every directed graph has a path cover $P$ such that one can make an independent set of $G$ by picking vertices from each of the ...
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1answer
23 views

Proof with parallelogram inside a parallelogram

Prove that $PBRS$ is a parallelogram. (Note: $P$ and $Q$ are respectively the middles of the sides $AB$ and $CD$) Now the corrections give the following method: $PBQD$ is a parallelogram $BR ...
2
votes
1answer
30 views

Show that a finite union of compact subspaces of a topological space $X$ is compact.

I am aware that there is a similar question elsewhere, but I need help with my proof in particular. Can someone please verify my proof or offer suggestions for improvement? Show that a finite ...
1
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0answers
30 views

Prove that every subset of $\mathbb{R}$ is compact in the finite complement topology.

I need help with my proof in particular. I am aware that there is a similar question elsewhere. Can someone please verify my proof or offer suggestions for improvement? Prove that every subset of ...
1
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1answer
18 views

Regularity of Wavelets

In theorem 2.9.2, where we are discussing the Regularity of wavelets. The proof begins by showing the uniform boundedness of the function before the proof of holder inequality in two parts one for ...
2
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3answers
36 views

What is the set with characteristic function $\chi_A(x) + \chi_B(x)-\chi_A(x)\chi_B(x)$?

Suppose that $A$ and $B$ are subsets of $X$ Find the subset $C$ whose characteristic function is given by: $\chi_C(x)=\chi_A(x) + \chi_B(x)-\chi_A(x)\chi_B(x)$ The answer given is ...