0
votes
2answers
47 views

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
4
votes
1answer
845 views

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 ...
2
votes
0answers
25 views

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 ...
1
vote
0answers
91 views

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 ...
0
votes
0answers
90 views

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 ...
0
votes
2answers
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 ...
2
votes
2answers
55 views

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, ...
1
vote
1answer
28 views

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 ...
2
votes
2answers
60 views

Roots of simultaneous power sum equations (numerically or otherwise)

I'm a physicist, and I've come across a problem in my research where I need to solve a set of equations looking like (e.g. in 3D) $$r_1 + r_2 + r_3 = k_1$$ $$r_1^2 + r_2^2 + r_3^2 = k_2$$ $$r_1^3 + ...
1
vote
1answer
34 views

Using the comparison test to determine series' convergence

I'm having trouble figuring out how to use the comparison test to check if an infinite series converges or diverges. I put two problems that I have to solve, does anyone have any input on this? The ...
1
vote
1answer
401 views

Finding a basis such that the $\mathcal{B}$-matrix is diagonal for orthogonal projection and reflection

Find a basis of the transformations such that the $\mathcal{B}$-matrix is diagonal. Orthogonal projection $T$ onto the plane $3x_1+x_2+2x_3=0$ in $\mathbb{R^3}$. Reflection $T$ about ...
2
votes
1answer
471 views

Give an example of a maximal ideal in a noncommutative ring which is not prime

While trying to find an example, I came up with this: Since if $J$ is an ideal of a the ring $M_n(R)$, where $R$ is a commutative ring, then $J=M_n(I)$ for some ideal $I$ of $R$. IF I could show that ...
1
vote
1answer
49 views

Calculating future data based previous data

The sales volume of the next month is predicted by the data in the past. The sales volume is changed greatly from month to month, but the annual fluctuation pattern is almost the same every ...
0
votes
1answer
113 views

How do I solve two equations in two unknowns?

In my free time, I've been challenging my mind with IQ problems. I found this question: Jim has as many sisters as he has brothers, but his sister has twice as many brothers as she has ...
0
votes
1answer
240 views

Find smallest number bigger than y that is multiple of x

I can't seem to find an answer for this, as all the topics regarding multiples deal with integers... I need to find the smallest number after x that is multiple of 0.36. For example if x = 3000, ...
0
votes
2answers
40 views

The vertices of a triangle are A(-1, 1) B(4,0) and C(1,6) Find the equation of the altitude of the triangle ABC drawn from A.

I need some help understanding the process of how you go about answering this question: The vertices of a triangle are A(-1, 1) B(4,0) and C(1,6) Find the equation of the altitude of the triangle ABC ...
4
votes
2answers
92 views

Does there exist an $a_0$ such that the sequence $a_{n+1} = 2a_n + 1$ is prime for all $n \ge 0$?

I believe I see that $a_n = 2^n(a_0+1) - 1$ but am somewhat uncertain where to proceed afterwards. I am a complete beginner at number theory and would appreciate it if someone could point me in the ...
0
votes
1answer
78 views

Combinatorial proof of an identity of Striling number of first kind

I can prove this identity using induction but i was looking for a combinatorial proof for this identity regarding stirling numbers of first kind. How should i proceed? Where, Thanks in advance.
0
votes
2answers
29 views

Limit $\lim_{n\to \infty }\frac{13^n+(-5)^n}{4\dot\ 13^n+(-11)^n }$

How can I calculate this limit ? $$\lim_{n\to \infty }\frac{13^n+(-5)^n}{4\dot\ 13^n+(-11)^n }$$
2
votes
2answers
189 views

Log base 10 not equal to log while differentiating?

I was looking at sample questions from my textbook and I came across something interesting that I need a little help understanding The question was to find the derivative of: $\log_{10} ...
0
votes
2answers
65 views

Showing there is no invertible function $f: \mathbb{R} \to \mathbb{R}$

I'm wondering whether there is an invertible function $f: \mathbb{R} \to \mathbb{R}$ such that $f(-1)=0$, $f(0)=1$ and $f(1)=-1$. I think it's not but I'm missing a real proof. The easiest would be ...
0
votes
1answer
40 views

Prove that a polynomial an irreducible g has no multiple root in C

I was looking at a question from Artin from Algebra which says that an irreducible polynomial g in F[x] where F is subfield of $\mathbb{C}$. So as per the proofs I have seen so far says as - ...
8
votes
1answer
175 views

Integer functions

For $x>0$ consider the following three functions: $$\begin{align} f(x)&=x+1;\\g(x)&=2x;\\t(x)&=3x \end{align}$$ Let $A(x)$ be the minimum number of operations using only functions ...
1
vote
3answers
93 views

Using Residue Theorem to calculate the integral

for $$I=\int_{|z|=1}{z^m \cos\left(\frac{1}{z}\right)}\,dz$$ where $m=0,1,2...$ Is the singularity $z=0$ or there are some other singularities? if it is $z=0$, what's order of pole?
1
vote
2answers
165 views

Permutaion and Combination Problems

Hey folks I'm having some issues with permutation and combination problems. 1) Make a 3 digit even number without repeated digits, using 0, 4, 5 , 6, 7. Also the first digit cannot be 0. I ...
0
votes
0answers
103 views

Prove this relation between truncated SVD and eigen decomposition?

For a real matrix $M$, we have a full SVD $M=USV^T$ and a truncated SVD $M_{k}=U_kS_kV_k^T$. The truncated SVD means (matlab grammar): $U_k=U(:,1:k), S_k=S(1:k,1:k), V_k=V(:,1:k)$. Based on the ...
2
votes
3answers
91 views

Density of sets

I have got a problem on whether a set is dense or not but not quite sure on how to approach it. Consider the space $M_2(R)$ with its usual topology.Consider the set $ S$ of all matrices with both ...
1
vote
1answer
42 views

Bounded and absolute convergent real sequences

I have problems with this question: Denote $E$ the set of all real sequences $\{a_n\}$ such that $|a_n|\le 1$ for every positive integers $n$, $l^1$ be the set of all real sequences $\{a_n\}$ such ...
1
vote
3answers
108 views

Fixed points of: $\dot{x}=\sin(y) \qquad \dot{y}=\cos(x)$

How can you find the fixed points of this system: $\dot{x}=\sin(y)\\ \dot{y}=\cos(x)$ Normally I would suggest that you find the points when both functions are equal to 0.
0
votes
1answer
129 views

Sherman Morrison Formula for hermitian updates

I have a problem in which, in principle I can apply twice Sherman-Morrison formula but it seems to me that for this case, there should be a simpler solution so my question is "May the process ...
0
votes
2answers
13 views

Find the solutions of an equation with arctan?

I have to show that $1$ and $\frac{-1}{\sqrt{3}}$ are (maybe not) solutions of the following equation: $arctan(x)+arctan(x\sqrt{3})= \frac{7\pi}{12}$. How can I do that ? Thank you in advance
0
votes
4answers
190 views

If matrix $AB=A$, does it mean B must be an identity matrix?

If matrix multiplication $AB=A$, does it mean $B$ must be an identity matrix? If not, why? What conditions? $A$ is not a zero matrix.
2
votes
0answers
67 views

Are stable manifold for gradient flows embedded submanifold?

Generally, the stable manifolds $W^s(p)$ of a diffeomorphism $\phi:M\to M$ is no embedded submanifold. The injective immersion $$ E^s:T_p^sM\to M $$ does not need to be a homeomorphism onto its image ...
2
votes
1answer
75 views

Finite Group and normal Subgroup

Let $d,m\in \mathbb{Z}$ with $d,m\geq 1$ and $\gcd(d,m)=1$. Let $G$ be a group of order $dm$ and define the set $X:= \{g\in G | g^d=1\}$. Show: if $H$ is a normal subgroup of $G$ with order d then ...
5
votes
0answers
100 views

Number of restricted ways to two-color a necklace [duplicate]

There are $n$ beads placed on a circle, $n\ge 3$. They are numbered in random order as viewed clockwise. Beads for which the number of the previous bead is less than the number of a next bead are ...
0
votes
2answers
48 views

$\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$

I am trying to show the following: $\dfrac{1}{||x||}$ not Lebesgue integrable in $\mathbb {R^d}$ on set $E=\{x\in \mathbb {R^d}: ||x||≥1\}$ I tried to use Fubini's theorem and the fact that ...
6
votes
1answer
156 views

Differential in Lie groups

I am trying to make sense of the Lie group machinery and relate it to the calculus. Suppose that $\psi(t)=\phi(s)\phi(t), s, t \in I$. Where $\phi(t)$ is a one-parameter subgroup of the Lie group ...
2
votes
3answers
158 views

Prove that $S= \{ (x,y) : x^2 - y^2 <1 \}$ is open in $\mathbb{R}^2$

Prove that $S= \{ (x,y) : x^2 - y^2 < 1 \}$ is open in $\mathbb{R}^2.$ The question itself is rather easy and trivial by observing the $S$ in $\mathbb{R}^2$ geometrically. But if we are ...
1
vote
1answer
51 views

Simplify this fraction with square roots; application to arctangent equation.

I need your help. I don't know how to simplify: $\frac{-1+\sqrt{3}+\sqrt{4+2\sqrt{3}}}{2\sqrt{3}} $ and $\frac{-1+\sqrt{3}-\sqrt{4+2\sqrt{3}}}{2\sqrt{3}}$ Thank you in advance. I found $1$ and ...
2
votes
1answer
259 views

Modify the closest-pair algorithm to use the $L_\infty$ distance.

I'm trying to understand the closest pair of points problem. I am beginning to understand the two-dimensional case from a question a user posted some years ago. I'll link it in case someone wants to ...
2
votes
1answer
61 views

Lower bound for expectation of squared log?

Is there a (tight) lower bound for $\mathbb{E}[(\log x)^2]$ where $x$ is a non-negative random variable? Jensen's inequality doesn't seem to apply here since the squared of a log isn't convex. Thanks! ...
1
vote
1answer
12 views

Obtaining an expression between $s'(n,r)$ and $s(n,r)$

I've a doubt in this: We're given $[x]_n=(x)(x-1)\ldots (x-(n-1))$ and $[x]^n=(x)(x+1)\ldots (x+n-1)$ . Now as we can write : $[x]_n=(x)(x-1)\ldots (x-(n-1))=a_0+a_1x+a_2x^2\ldots ...
2
votes
1answer
85 views

On the equality $\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}= \sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_m}$

Let $k,m\in \mathbb{N}$. Let $a_1,a_2,\cdots,a_k\ >0$ and $b_1,b_2,\cdots,b_m \ >0$ such that $$\sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}= ...
0
votes
1answer
50 views

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 ...
0
votes
1answer
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 ...
1
vote
1answer
52 views

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 ...
1
vote
2answers
74 views

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!
0
votes
1answer
32 views

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} $ ...
2
votes
3answers
63 views

Code is not cyclic for any q

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\\ ...
0
votes
1answer
124 views

On a decomposition of a conditional distribution

I am trying to make some sense out of equation (7) in the recent paper of Peter van Leeuwen: "Representation errors and retrievals in linear and nonlinear data assimilation" ...

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