# All Questions

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### How to differentiate $a(t-1)+bt+(1-t)\int_{0}^{t}\frac{dB_s}{1-s}$

someone can help me to differentiate $$a(t-1)+bt+(1-t)\int_{0}^{t}\dfrac{dB_s}{1-s}?$$ I've tried but I really don't know how to do with the last part.. Thank you somuch for your help
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### Prove that the kernel of a group homomorphism $\phi$ is a subgroup and that $\phi$ is injective

I am solving the following exercise: Let $\phi : G_1 \rightarrow G_2$ be a homomorphism (where $G_1$ and $G_2$ are groups) and $\ker \phi := \{ g \in G_1 \mid \phi(g) = e \}$ now I have to ...
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### Integers and funtional equation [duplicate]

Let $\Bbb{Z}^+$be the set of all non-negative integers where $n$ and $k$ are given natural numbers. We consider the following non-decreasing function, $$f:\Bbb{Z}^+ \to \Bbb{Z}^+$$ such that ...
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### Integral Inequality bounded by $\sup \{|f''(x)|-f''(y)| , |x-y|\leq r\}$

Let $f \in C^2 [a,b]$. Define $$\omega_2(r)= \sup \{||f''(x)|-|f''(y)|| \, : \, |x-y|\leq r\}$$ we can prove that $\omega_2(r)$ is continuous. The following lemma is given in Convergence rates of ...
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### Why is this class recurrent?

In our reading we had the following example for a Markov chain. I cite from the reading: Here we have three communicating classes: $\left\{0\right\}, \left\{1,2,3\right\}$ and ...
187 views

### Designing a deterministic finite automata

How would I go about designing a deterministic finite automata to recognize the language L = {λ, ab, abab, ababab, . . . } consisting of strings that start with ‘a’, end with ‘b’, and alternate in ...
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### Dominated convergence for sequences with two parameters, i.e. of the form $f_{m,n}$

Let $f_{m,n}(x)$ be a sequence (dependent on $m$, $n$) of Lebesgue integrable functions on $\mathbb{R}$. Suppose that $f_{m,n}(x)\to 0$ as $m,n\to+\infty$, for almost $x\in\mathbb{R}$; in addition, ...
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### If $n = m^2 + 1$ and $x$ is a square modulo $n$, then how to show that $n - x$ is also a square modulo $n$?

I see that if $x \equiv y^2 (\text{mod } n)$, then $n - x \equiv m^2 - y^2 + 1 \equiv (m+y)(m-y) + 1 (\text{mod } n)$. However, I'm not sure how to proceed from there. I'm a complete beginner at ...
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### Term for functions with infinite derivatives [closed]

Functions that include a negative indice such as x-1 or similar have an unlimited number of derivatives, so f'(x), f''(x), and fn(x) exist. Is there a technical term for functions like these? I've ...
56 views

### Conditional Posterior Distribution Based on Two Simultaneous Signals

I am trapped by such a problem. Assume the state variable $\theta$ is (prior) normally distributed $N(\eta, \sigma^{2}_{0})$. Now we have two independent signals about $\theta$. Signal 1 is ...
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### Why is $\int_{\mathbb{R}^3} |p\rangle \langle p| d\lambda(p)=id$?

As I have written in the headline, I am curious how the relation $\int_{\mathbb{R}^3} |p \rangle \langle p| d\lambda(p)=id$ that physicists use, where $|p\rangle$ is the eigenfunction to the ...
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### What is the values of $a$ and $b$ without using the L'Hôpital's Rule

Suppose that $$\frac{(2x)^x-2} {a(x-1)+b(x-1)^2}\to 1$$ as $x \to 1$. ThenWhat is the values of $a$ and $b$ without using the L'Hôpital's Rule? Thanks for your help!
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### measurability in backwards martingales

$X$ is a backwards martingale with $X_0\in L^1$ According to the convergence theorem:$X_{-n}\to X_{-\infty}$ a.s. But how to get the conclusion that $X_{-\infty}$ is $\mathcal F_{-\infty}$ ...
I have code $C$ over $F_p$ with generator matrix which looks like \$G = \begin{pmatrix} 0 &0& 0& 1& 0& 1& 1 &1\\ 1& 0 &0& 0 &1 &0 &1& 1\\ ...