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Jacod and Protters Proof of characteristic uniqueness of functions, extending...
I want to fill in the details in Jacod and Protter's proof of the uniqueness of characteristic functions. The problem is that we have something that works for a compact set, and I want it to work for...
View ArticleFrom $\int_{B(x_j,r_j)} F_1 dV\leq \int_{B(x_j,r_j)} F_2 dV,\forall j$ to...
Let $\{x_{j}\}$ be a sequence of points in $\mathbb{R}^n$, let $\{r_j\}$ be a countable set of positive real numbers, and let $F_1,F_2$ be non-negative Lebesgue integrable functions. Suppose thatthe...
View ArticleProblem with the definition of maximal solution of a differential equation
During my classes we defined an extension of a solution $\psi$ of a differential equation:$y'=f(x,y)$ on an interval $J$ as a solution $\phi$ on an interval $I$ s.t.: $J \subseteq I$ and $...
View ArticleUnderstanding the Proof of Proposition 5.1.3 from Measure Theory by Donald Cohn
QuestionI am self-studying Measure Theory by Donald Cohn. I got stuck on a step of his proof of Proposition 5.1.3. Here is the proposition, its proof, and where I got confused:Proposition 5.1.3$\quad$...
View ArticleOn minimal eigenvalue [closed]
Is it true that $\min\left(\lambda_{\min}(M_{12}),\lambda_{\min}(M_{13}),\lambda_{\min}(M_{23})\right) \le \frac{2}{5}$ where $M = \mathbf{x}_{1}\mathbf{x}_{1}^{\top} +...
View ArticleDifference between finite partial sums from two divergent series
Fix a sequence $(r_i)_{i\in\mathbb{N}} \subseteq (0, 1)$ such that $\lim_i r_i=0$ and $\sum_{i\in \mathbb{N}} r_i=\infty$. According to the answer in this post, for any $c>0$ there exists...
View ArticlePolynomial $f(x)$ has positive coefficients and only real roots. How many...
Let$$f(x)=a_n \ x^n+a_{n-1} \ x^{n-1}....+a_1 \ x+a_0$$be a $n$-th degree polynomial with positive coefficients such that all of its roots are real. Choose any number terms from this expression...
View ArticleArgue for changing the sum of a double sequence into the sum of a sequence
Let's say I have a nonnegative, double-indexed sequence (like a table extending to infinity in two directions) $(a_{ij})_{i,j\in \mathbb{N}}$, and assume $\sum_{j}a_{ij}=1/2^{i}$.By theorem 8.3,...
View ArticleAnalyzing the asymptotic behavior of a given function $f(\mu, t)$ as $t \to...
Consider the following improper integral$$f(\mu, t) = \frac{2}{\pi} \int_0^\infty \frac{\sin u}{u} \frac{\left(U+u\right) \exp\left(-\frac{\mu}{t} \, u\right) - 2U \exp\left(-\frac{\mu}{t} \, U...
View ArticleUniform Convergence using Abel's test for a series based on convergence of a...
A problem from uniform convergence of series:$$\sum_{i=1}^\infty a_n$$ is convergent then show that $$\sum_{i=1}^\infty \frac {nx^n(1-x)}{1+x^n} a_n$$ and $$\sum_{i=1}^\infty \frac...
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How to show the arguments in the proposition below?
In the proof of the given , a function $\Phi_{3}$ is defined as:$$ \Phi_{3}(x)=\sup \{ \dfrac{\Phi_{2}(xy)}{\Phi_{1}(y)}:y\geq 0\}$$ where $\Phi_{1}$ and $\Phi_{2}$ are $N$- functions with...
View ArticleFind $d_k$ that minimize $\sum_{k=1}^\infty d_k\cdot\exp(-(d_1+\cdots+d_{k-1}))$
N.B.: The question has been edited after Sangchul Lee's answer to include an extra condition that is required for the originalproblem to make sense.Consider a strictly increasing sequence of strictly...
View ArticleIf $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x)...
If $f$ is continuous, nonnegative on $[a, b]$, show that $\int_{a}^{b} f(x) dx = 0$ iff $f(x) = 0$"$\Rightarrow$" Assume by contradiction that $f(x) \neq 0$ for some $x_0 \in [a, b]$. Without loss of...
View ArticleIf $\lim\limits_{n \to \infty}f(\sum\limits_{k=0}^N x_k n^k)=0$, Does this...
This problem is in my problem book:Let $f :[0,\infty) \to R$ be a function such that, for each real $ x_0, x_1 > 0$ , the sequence ${f(x_0+x_1 n)}$ converges to zero. Does the $\lim\limits_{x\to...
View ArticleInfimum/Supremum: How could you have quickly completely this MCQ?
Consider sequence $$B=\left\{\frac{(-1)^n}{\sqrt{2n+1}}:n\in\mathbb{N} \right\}$$ What is the $\sup B$ and $\inf B$?(A) $\sup B=\frac{1}{\sqrt[3]{2}},\inf B=-1$(B) $\sup B=\frac{1}{\sqrt[3]{2}},\inf...
View Articlemotivation of additive inverse of a Dedekind cut set
My understanding behind motivation of additive inverse of a cut set is as follows :For example, for the rational number $2$ the inverse is $-2$. Now $2$ is represented by the set of rational numbers...
View ArticleDedekind Cut additive inverse
Let $\alpha$ be a Dedekind cut and define $\alpha^* :=\{x\in\mathbb{Q}|\exists r>0\space \text{such that} -x-r\notin\alpha\}$. I need to show that $\alpha^*$ is a Deddekind Cut and the additive...
View ArticleLimit value depending on a function's asymptotycs
I want to compute the limit of the following fucntion when $n\to\infty$:$$f(n)=\frac{2 n \left(1-\frac{1}{n}\right)^{g(n)}}{2-3 n \ln \left(1-\left(1-\frac{1}{n}\right)^{g(n)}\right)}$$It depends on a...
View ArticleAdditive inverse of a Dedekind cut -- is my definition alright?
I am struggling to understand why the Additive Inverse is typically defined as such:$$\alpha^∗ := \{x \in \mathbb Q | \exists r > 0\text{ such that }−x−r\notin \alpha\}$$or in another form$$\alpha^*...
View ArticleCardinality of a set of continuous functions on $[0,1]$ satisfying some...
Let $S$ be the set of all non-negative continuous functions $f$ on $[0 ,1]$ satisfying$\int _0 ^1 \sin ^2 (x)f(x)dx=\int _0 ^1 \cos ^2 (x)f(x)dx = \int _0 ^1 \sin (x) \cos (x)f(x)dx =1 $Then $S$ is(a)...
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