1
vote
1answer
73 views

Is the set of sentences in peano arithmetic countably infinite?

Im wondering if the set of all sentences, which would be wffs in which there are no free variables, is countable infinite or not. I imagine that since the number of axioms in PA is countably ...
1
vote
1answer
45 views

Every continuous function $f:X\times Y\rightarrow \mathbb{R}$ is the uniform limit of functions

Let $X$, $Y$ metric compact spaces. Prove that every continuous function $f:X\times Y\rightarrow \mathbb{R}$ is the uniform limit of functions of the form $\varphi ...
2
votes
1answer
66 views

What is the technical term for the $n$-dimensional generalization of the unit interval?

What is the technical term for an $n$-dimensional generalization of the unit interval $[0, 1]$? Would we call an $n = 1,2,3,...$ dimensional generalization of the unit interval an $n$-cube?
3
votes
3answers
392 views

Determinant of matrix $A^3 + 2A^2 - A - 5I$ Given the eigenvalues of A

So A is a 3 by 3 matrix with eigenvalues -1, 1, 2. And I have to find the determinant of $$A^3 + 2A^2 - A - 5I$$ Let $u$ be the eigenvector for the eigenvalue -1. Let $S = A^3 + 2A^2 - A - 5I$ then ...
1
vote
1answer
74 views

$\overline{A\cdot B}=\overline{A}\cdot\overline{B}$ for $A,B\subseteq\Bbb R$? [duplicate]

If $A$ is a closed set in $\Bbb R$ and $B$ a compact set in $\Bbb R$, let $A\cdot B=\{a\cdot b:a \in A,b\in B\}$. Question: Is $\overline{A\cdot B}=\overline{A}\cdot\overline{B}$? Note: ...
1
vote
1answer
131 views

How To Simulate Mirrors/Reflection?

If light is hitting a Parabolic Trough defined by $y=x^2$ at a 60 degree angle from vertical so that the effective cross-section of the modified parabola is paramaterized by: x=t, y=t^2, z=tcot(60). ...
0
votes
1answer
56 views

Extraneous solutions to simple equations

I had an interesting thought during my procrastination: is it legal to take an equation, say $3 = a * b * c$ and do the following: $3 = abc$ $0 = abc - 3$ $0 / a = bc - 3/a$ $0 / b = ...
2
votes
1answer
304 views

On why the Vitali Covering Lemma does not apply when the covering collection contains degenerate closed intervals

I believe I have a fundamental misunderstanding of the concept of the Vitali Covering Lemma. Definition - A closed bounded interval $[c, d]$ is said to be nondegenerate provided $c < d$. ...
5
votes
5answers
140 views

evaluation of $\lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n}$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n}$ $\bf{My\; Try}::$ $\displaystyle \lim_{n\rightarrow ...
1
vote
1answer
324 views

Tensors = matrices + covariance/contravariance?

I have read several topics on tensors but it is still not clear to me. Tensors are different from matrices because they contain additional information about how do they transform. To fully specify a ...
2
votes
2answers
324 views

If $f(z)$ is entire, and it is constant in $\{z:|z|\le 1\}$, then $f$ is constant in $\mathbb{C}$?

If $f(z)$ is entire, and it is constant in $\{z:|z|\le 1\}$, then $f$ is constant in $\mathbb{C}$? (I'm asking this because I need to prove that a function is constant, so I wonder if I'm already ...
0
votes
1answer
69 views

A basic question on boundedness of Riemann integrable function

I want to prove that Riemann integrable implies function is bounded by contradiction. Suppose function is not bounded, then I want to find an $\epsilon$ such that for all partition $P$ there exist a ...
0
votes
3answers
88 views

Cauchy convergent sequences

Suppose that $(a_n)$ and $(b_n)$ are convergent sequences and that $b_n > 0$ for all $n$. Is it true that $(a_n / b_n)$ is Cauchy? If it is true, prove it. If it is not true, give a counterexample ...
0
votes
1answer
71 views

Find the transformation that converts a square with diagonal vertices (0 , 3) and (-3 , 6) into a unit square at the origin.

Find the transformation that converts a square with diagonal vertices (0 , 3) and (-3 , 6) into a unit square at the origin.
0
votes
1answer
103 views

Connected subsets of a metric space

I have to prove the following result: Suppose $X$ is a metric space, $Z$ is a metric subspace of $X$ and $S \subset Z$ Then $S$ is a connected subset of $X$ if, and only if,$S$ is a connected subset ...
0
votes
2answers
149 views

prove that this operator is not compact

Let $g\in C[0,1]$ be a continuous function and $g\ne 0$. Let $G:C[0,1]\to C[0,1]$ the operator defined by: $G(f)(x)=f(x)g(x)$. I proved that the operator is linear and continuous. I want to prove that ...
3
votes
1answer
86 views

Stone's theorem for 1-parameter groups of unitary multipliers?

Let $A$ be a nonunital C*-algebra and let $M(A)$ denote its multiplier algebra. Let $(u_t)_{t \in \mathbb{R}}$ be a strictly continuous 1-parameter group of unitary multipliers. That is, $u_t x \to x$ ...
4
votes
1answer
116 views

Computing Coefficients for Generalized Combinatorial Sets

I'm new to Combinatorics and am wondering if there is a well-known generalization to the binomial coefficients in the following sense: The binomial coefficients, $n \choose d$, can be considered as ...
0
votes
1answer
54 views

Complement of A or B

I have a small general question.. Let's say we have two events $A$ and $B$. Is the probability that $A$ or $B$ will happen, the complement of the event that the complement of $A$ and the complement ...
2
votes
1answer
59 views

how to show f is bijective?

Suppose that the set-mapping $f:X\rightarrow Y$ of one-dimensional domains of $\mathbb{C}$ induces an isomorphism $f^0:\mathcal{O}(Y)\rightarrow \mathcal{O}(X)$ defined by $g\mapsto g\circ f$ of ...
0
votes
2answers
74 views

Sequence and convergence of subsequences

Suppose that $(a_n)$ is a sequence. Assume that both $(a_{2n})$ and $(a_{2n+1})$ converge to the same $L$. Prove carefully that $(a_n)$ also converges to $L$ I was thinking that $(a_{2n})$ and ...
1
vote
4answers
54 views

Find the radius of convergence and interval of convergence

Seems like you are suppose to do the root test to come up with the answer. but the 2x-5 in the numerator concerns me. the (-5) part. The root test says that the series has to have positive terms. ...
13
votes
1answer
172 views

Order of an element in a finite cyclic group

Let $G$ be a cyclic group of order $m$ generated by an element $a$. I want to show that the order of $a^k$ is $m/d$, where $d:=\gcd(k,m)$. I have a simple proof, but want to make sure I haven't ...
0
votes
0answers
39 views

Limit of the expectation of the sum

Show that for $g(t)= E \left\{\sum_{n=3}^{\infty}\frac{(iut)^{n}}{n!}\right\}$ that $\lim_{t \to 0} \frac{|g(t)|}{t} =0$. I think I should bound it and then use LDCT, but I'm having trouble doing ...
1
vote
1answer
78 views

Find all positive integers $n$ such that..

Find all positive integers $n$ such that $1!+\ldots+n!$ divides $(n + 1)!$ I think I know that the only two positive integers are $1$ and $2$. Proving it inductively has been a problem for me ...
4
votes
1answer
822 views

Expected number of steps in a random walk with a boundary

Let's say I am trying to climb a flight of $N$ stairs. Each time I want to take a step, I flip a fair coin. Heads means I take a step up; tails means I take a step down. If I'm at the bottom of the ...
4
votes
4answers
854 views

How to conclusively determine the interior of a set

I'm fairly confident I understand the meaning of the 'interior' of a set, but I can't figure out how to prove conclusively what said interior is for a given set. Consider this definition: Let $S$ ...
-1
votes
4answers
189 views

Finding a finite set from two uncountable sets

I am trying to find a finite set such that an infinite set A and an infinite set B in $A - B$ will result in a finite set. I saw an answer from someone else as all reals $>=0$ - all reals $> 0$ ...
1
vote
1answer
678 views

Is there integrable function sequence which is uniformly converges to not integrable function?

Is there any example (Riemann) integrable function sequence which is uniformly converges to not integrable function?
-1
votes
2answers
42 views

How to I prove the equivalence of $f(S_1 \cup S_2)$ and $f(S_1) \cup f(S_2)$ (Discrete Mathematics) [duplicate]

If $S_1$ and $S_2$ are both subsets of some arbitrary set $A$, then how do I prove that $f(S_1\cup S_2) = f(S_1) \cup f(S_2)$ for ALL cases I understand that it is true, but I don't know how to prove ...
0
votes
1answer
133 views

Understanding Poisson Distribution Question?

I have the following question: I have the following formula: However, it is unclear to me how to extract what I need. Would U be equal to 17 for an average of 17 per minute and X be equal to 25? ...
0
votes
1answer
53 views

using Taylor's formula in a proof

Prove that $1+\frac{1}{n} < e$ for all $n$ in the natural numbers. How does this connect to Taylor's formula? I know that $e^x > 1+x$ for $x>0$, but then where does Taylor's formula come in ...
2
votes
0answers
62 views

Analytical solution nonlinear partial differential equation

Do you know how I can solve nonlinear PDEs analytically i.e does the perturbation method work? e.g $$a^2u_{tt} - u_{xx}+ f(u)=0$$ where $f$ is nonlinear in $u$, with boundary condition. what the ...
1
vote
3answers
273 views

Evaluate the limit without L'Hospital

$$\lim_{x \to 0} \frac{1-\cos(\sin(4x))}{\sin^2(\sin(3x))}$$ How can I evaluate this limit without using the L'Hospital Rule? I've expanded $\sin(4x)$ as $\sin(2x+2x)$, $\sin(3x) = \sin(2x + x)$, but ...
1
vote
2answers
41 views

Let $Tx = 1+\log(1+e^x)$. Show that $T$ has no fixed points.

Let $Tx = 1+\log(1+e^x)$. Show that $T$ has no fixed points. This is what I have: We say that $T$ has a fixed point if $Tx=x$. $$Tx = 1+\log(1+e^x) = x$$ $$\log(1+e^x) = x-1$$ $$1+e^x = e^{x-1}$$ ...
0
votes
1answer
70 views

Definition for $\lim(s_n)$ and $\limsup(s_n)$

Can someone provide me the definition of a (finite ) $\lim (s_n)$ and how it correlates to the definition of $\limsup(s_n)$? $\lim(s_n)=+\infty$ if $\forall M>0, \exists N=N(M)\in \Re$ ...
3
votes
4answers
266 views

What's a good reference to study multilinear algebra?

This semester I'm taking a course in linear algebra and now at the end of the course we came to study the tensor product and multilinear algebra in general. I've already studied this theme in the past ...
0
votes
1answer
29 views

Inverse theorem on product of two convergent sequences

Suppose I have two sequences, $a_n$ and $b_n$. I know that: $\lim_{n\to\infty} a_n=1$ and that $\lim_{n\to\infty} a_nb_n=c$. Does this mean that $\lim_{n\to\infty} b_n$ converges? If so, by ...
0
votes
2answers
146 views

Partial Fractions Decomposition, Differentiating between linear and nonlinear

My confusion is between $$\dfrac{(x^2+1)}{(x^3)(x+1)}= A/(x) + (Ax+b)/(x^2) + (Cx+d)/(x^3)$$ or $$\dfrac{(x^2+1)}{(x^3)(x+1)}= A/(x) + B/(x^2) + C/(x^3)$$ Also in the following case, I think the ...
1
vote
3answers
51 views

Prove that $\nabla\langle Ax,Ax\rangle = 2A^TAx$

Prove that $\nabla\langle Ax,Ax\rangle = 2A^TAx$. My book uses this property to prove the $2-norm$ of a matrix $A$ is the square root of the spectral radius of $A^TA$. That is $$||A||_2 = ...
0
votes
2answers
56 views

Help with a trigonometric limit

Find the limit and determine if the function is continuous at the point that is being approached: $$\lim_{y \to 1}\;\; \mathrm{sec}\;(y\;\mathrm{sec}^2y \;-\;\mathrm{tan}^2y\;-\;1)$$ My try: I ...
1
vote
2answers
407 views

Proving that a set is an orthonormal basis

Any ideas on how to quickly show that $$ \left( \frac{1}{\sqrt{2\pi}}, \frac{\sin(x)}{\sqrt{\pi}}, \frac{\sin(2x)}{\sqrt{\pi}}, ..., \frac{\sin(nx)}{\sqrt{\pi}}, \frac{\cos(x)}{\sqrt{\pi}}, ...
1
vote
1answer
110 views

Combinatorics Graph Theory Proof problem

I am struggling with 9.31 from A Walk Through Combinatorics by Miklos Bona. The problem statement reads: There are several people in a classroom; some of them know each other. It is true that if ...
0
votes
1answer
83 views

Least upper bound and greatest lower bound

Find the $\sup E$ and $\inf E$. i) $E$=$(0,1]$ ii) $E$=$\{x \in Q : x^2 < 2\}$ i) $\sup E$ = $1$, $\inf E$ = $1$ ii)$\sup E$ = $\sqrt 2$, $\inf E$ = $DNE$ I got these answers using my ...
0
votes
3answers
101 views

The irrationality of the square root of 2 [duplicate]

Is there a proof to the irrationality of the square root of 2 besides using the argument that a rational number is expressed to be p/q?
0
votes
0answers
102 views

Monte Carlo Integration

I was reading this document (I will reproduce the equation): ...
1
vote
3answers
75 views

How do you factor $(10x+24)^2-x^4$?

I tried expanding then decomposition but couldn't find a common factor between two terms
1
vote
1answer
121 views

A very interesting problem of vector calculus

Consider the scalar fields $f(\theta, \phi)$ and $g(\theta, \phi)$ defined on the unit surface $S^2$, where $\theta$ is the co-latitude ($0$ at the north pole and $\pi$ at the south) and $\phi$ is ...
1
vote
2answers
61 views

How do i convert $\frac{1}{2+x}$ to a summation?

I am given the summation for $\frac{1}{1-x}$. I get that I need to sub in $-x$ for $x$. I don't get how I am supposed to know where I put the $2$. I am not sure if there is a systematic procedure ...
2
votes
1answer
48 views

Find $\mathbb{C}[x^2,x^3]\cap \mathbb{C}[(x-1)^2, (x-1)^3]$.

Find $\mathbb{C}[x^2,x^3]\cap \mathbb{C}[(x-1)^2, (x-1)^3]$. I am trying to find the above subring. I would prefer hints more than complete solutions.

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