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

learn more… | top users | synonyms

1
vote
1answer
26 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 ...
0
votes
0answers
27 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 ...
2
votes
0answers
20 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 ...
0
votes
1answer
31 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 ...
3
votes
2answers
74 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
vote
1answer
28 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 ...
1
vote
2answers
58 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
votes
1answer
51 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
votes
1answer
34 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 ...
0
votes
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 $a\equiv b\pmod m$ ...
0
votes
0answers
10 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 ...
1
vote
0answers
49 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, ...
0
votes
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 : ...
1
vote
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 ...
0
votes
2answers
37 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 ...
0
votes
2answers
52 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 . ...
0
votes
2answers
29 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
votes
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 $f=g$ ...
1
vote
1answer
61 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 ...
0
votes
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, ...
3
votes
2answers
52 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 ...
1
vote
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
votes
2answers
34 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 ...
1
vote
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 ...
0
votes
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
vote
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 ...
0
votes
1answer
13 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
votes
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 ...
6
votes
5answers
194 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) ...
1
vote
0answers
47 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 ...
2
votes
0answers
33 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 ...
0
votes
0answers
12 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
vote
1answer
60 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
votes
1answer
27 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. ...
0
votes
0answers
24 views

Show that $\cup A_n$ is connected.

Can someone please verify my proof or offer suggestions for improvement? Let $\{A_n\}$ be a sequence of connected subspaces of $X$, such that $A_n \cap A_{n+1} \neq \varnothing$ for all $n$. Show ...
4
votes
2answers
57 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 ...
0
votes
1answer
87 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
votes
1answer
96 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 ...
2
votes
0answers
37 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
votes
3answers
82 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
votes
0answers
20 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 ...
1
vote
0answers
30 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 ...
2
votes
1answer
42 views

True or False. Non parallel lines in 3-space.

Two non parallel lines in 3 space must intersect in at least one point. True or False? I say false because you can have two perpendicular lines on x and y, but on a different "level" of the z-axis. ...
1
vote
1answer
20 views

Problem involving pseudomonotone mappings on Banach space

I have the following question regarding mappings on a Banach space $X$. If anyone has an idea or hint as to how to resolve this question it would appreciated. Let $X$ be a Banach space, $X^{*}$ its ...
3
votes
0answers
78 views

Is this proof correct? Divergence of $\int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \, \mathrm{d}x $

Problem: Show that $$ \int_{1}^{\infty} \left| \frac{\sin x}{x} \right| \,\mathrm{d}x $$ diverges. I know that there are many questions in which this problem is solved, but I want to know if my ...
1
vote
3answers
42 views

Finding the minimum distance from the origin to the surface $xyz^2=2$

This was an old exam question I was looking at for a friend, although it's been a while since I've done this stuff: Q. Find the shortest distance from the origin to the surface $xyz^2=2$. I ...
5
votes
2answers
57 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
0
votes
1answer
26 views

If B span $V$ and |B|=n so B is linear independent

Let there be V a vector space, $dim(V)=n$. B spans V, |B|=n, prove that B is linear independent Let there be $B \subseteq S $ where S span $V$ too, Because |B|=n then |S|>n and B is a basis, ...
0
votes
0answers
29 views

In which of the three topologies does the following sequence converge?

Can someone please verify my proof or offer suggestions for improvement? Notations: $d(x, y) = |x-y|$: Standard metric on $\mathbb{R}$ $\bar d(x, y) = \operatorname{min}\{1, d(x, y)\}$: Standard ...