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

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2answers
43 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
142 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 ...
2
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2answers
106 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
67 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 ...
2
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0answers
42 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
174 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
207 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
298 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
65 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 ...
4
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1answer
96 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
45 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
169 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 ...
1
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1answer
59 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
131 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 ...
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1answer
63 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 ...
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2answers
96 views

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

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
36 views

Congruence 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 $a\equiv b\pmod m$ ...
3
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1answer
55 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 ...
6
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1answer
331 views

Perfect Map $p:\ X\to Y$, $Y$ compact implies $X$ compact

I was assigned the following homework problem for an introductory course in topology: Let $p:\ X\to Y$ be a closed continuous surjective map such that $p^{-1}(\{y\})$ is compact for each $y\in Y$. ...
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0answers
83 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
42 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
70 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
92 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
86 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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2answers
40 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 $V(C)$ ...
2
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1answer
39 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 $f=g$ ...
2
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1answer
74 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
58 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, ...
3
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2answers
56 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
47 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 $k$ ...
4
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2answers
62 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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1answer
99 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
104 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
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1answer
122 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 ...
2
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0answers
251 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 ...
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1answer
22 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
57 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 ...
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5answers
225 views

Alternate Proof for $e^x \ge x+1$

This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it: Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) ...
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0answers
130 views

Deduction of usual Cayley-Hamilton Theorem from “Determinant Trick”

Here is a statement of a standard theorem in commutative algebra (see page 60 of this book): Theorem. ("Determinant Trick") Suppose that $R$ is a commutative ring with $1$. Let $M$ be a finitely ...
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0answers
37 views

Projection measures and integrability

Let $(M, \mathcal{A}, \mu)$ a probability space, $Y$ compact metric space. Consider $\mathcal{M}(\mu)$ be the space of probability measures $\eta$ on $M\times Y$ such that $\pi_{*}\eta=\mu $ where ...
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0answers
16 views

$w \in \operatorname{Span}(T) \leftrightarrow [w]_b\in \operatorname{Span}[T]_b$?

$$w \in \operatorname{Span}(T) $$ lets apply coordinate function on both sides $$[w]_b=\alpha_1t_1+\cdots+\alpha_nt_n=[t_1,\ldots,t_n]_b$$ $$\operatorname{Span}(T)=\sum \beta_it_i=\left[\sum ...
1
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1answer
131 views

Using resultants to check if multivariate polynomials have a common factor - is my proof correct?

Proposition: Let $f, g \in \mathbb R[x,y,z]$. Then the condition that $f, g$ have a common polynomial factor is an algebraic condition on their coefficients. By algebraic condition, I mean there is a ...
3
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1answer
44 views

Unsure about alternative angles proof using contradiction

Prove that alternate interior and exterior angles are equal if the lines who are crossed by the third line is parallel. Could someone review my proof? I used a proof by contradiction. ...
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2answers
233 views

Is this proof that there are no perfect, odd, integer square numbers legitimate?

Assumptions: Any even number times any other number is always an even number. An odd number times an odd number is always an odd number. An even number plus an even number is even, and an odd number ...
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1answer
102 views

If $\lim_{x \to \infty}f(x)=a$ then $\lim_{x \to \infty}f'(x)=0$ - whats wrong with the proof?

Here's how i would prove this. Since we have that $\lim_{x \to \infty}f(x)=a$ this implies that $\lim_{x \to \infty}f(x + 1) - f(x)=0$ By mean value theorem we have that $\lim_{x \to ...
0
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1answer
127 views

Few Questions about analysis in Rudins book

I have been looking at intro to real analysis. I am using the text book "Principals of Mathematical Analysis, third edition" by Walter Rudin. I have some questions about things I found confusing and ...
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0answers
47 views

Kernels of induced maps in cohomology

Let $k$ be a field, and let $A$ and $B$ be two commutative $k$-algebras. Suppose $\varphi,\psi:A\to B$ are maps of $k$-algebras such that there are algebra automorphisms $G:A\to A$ and $F:B\to B$ ...
2
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3answers
91 views

Calculate $\int\frac{dx}{x\sqrt{x^2-2}}$.

The exercise is: Calculate:$$\int\frac{dx}{x\sqrt{x^2-2}}$$ My first approach was: Let $z:=\sqrt{x^2-2}$ then $dx = dz \frac{\sqrt{x^2-2}}{x}$ and $x^2=z^2+2$ $$\int\frac{dx}{x\sqrt{x^2-2}} ...
0
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0answers
39 views

Proof that the $sqrt[k]{z}\, z \in \mathbb N$ counts the amount of numbers less than or equal to z with a $k-$exact power

Empirically, it can shown that $$\mathrm{Floor}[\sqrt[k]{z} ] \,, z \wedge k\in \mathbb N $$ is equal to the amount of numbers which have a $k-$exact root. For example, $\sqrt 36 = 6$ means that there ...
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0answers
51 views

Proving isomorphism between a quotient space of continuous functions.

This is really a follow-up question to this question, in the sense that it arose from that question. You don't need to read that question for this to make sense. To be proven: Let $V=\mathcal ...