7
votes
1answer
157 views

Closed-form of $\int_0^{\pi/2} \arctan(x)\cot(x)\,dx$

I'm looking for a closed-form of the following integral problem. $$I = \int_0^{\pi/2} \arctan(x)\cot(x)\,dx.$$ The numerical approximation of $I$ is $$I \approx 0....
-1
votes
3answers
75 views

Is there a closed-form of $\sum_{n=0}^{\infty }\frac{|B_n|}{n!}=??$ [closed]

Is there a closed-form of $$\sum_{n=0}^{\infty }\frac{|B_n|}{n!}=??$$ where $B_n$ Bernoulli number Thanks
2
votes
0answers
80 views

Tough mathematics question [duplicate]

If $a,b,q=\frac{a^2+b^2}{ab+1}$ are positive integers then $q$ is a perfect square.
0
votes
2answers
69 views

Question about dual of dual of Hilbert space

Let $H$ be a real Hilbert space and let $H'$ be the set of continuous linear functionals on $H$. Then, I know by the Riesz Theorem that for every $L(\cdot) \in H'$, there exists a unique $u\in H$ so ...
2
votes
1answer
161 views

$1+ab$ is a unit if and only if $1+ba$ is a unit. [duplicate]

Let $R$ be a ring with identity and $a,b \in R$ then prove that $1+ab$ is a unit if and only if $1+ba$ is a unit and find the inverse. Then there exist an element say $s \in R$ such that $(1+ab)s =1$ ...
2
votes
2answers
83 views

Square roots of $\Bbb F_p$

Can anyone please help me to show that $\Bbb F_{p^2}$ contains all the square roots of $\Bbb F_p$ where $p$ is a prime? Thanks for any help.
0
votes
1answer
25 views

Simple question , just observe a thing

$x = a + bw + cw^2 , y = aw + bw^2 + c , z = aw^2 + b + cw$ Find the value of $ \dfrac{x^2}{yz} + \dfrac{y^2}{xz} + \dfrac{z^2}{xy}$ Note - Here w is one of the roots of unity. I know the answer ...
1
vote
2answers
2k views

4 sided dice rolls - a probability question

If we had a 4 sided dice (numbers 1,2,3,4 on the faces) and rolled it 5 times and recorded the results. What would be the probability that we rolled the same number exactly 3 times? What about 3 or ...
1
vote
6answers
117 views

is $\cos x < \cos (\sin x)$ in $(0,\pi/2)$?

i need to quickly verify if $\cos x < \cos(\sin x)$ in $(0,\pi/2)$ i dont have any solid ground to prove it except by some trivil way?can anyone show a concrete way to check it?
1
vote
0answers
38 views

Continuous second derivative over the support of a Daubechies4 wavelet

I can not entirely follow the proof from section 3.1.1 from the book "A primer on Wavelets" by Walker. After the first part (listed below), I can grasp the rest so if you could help I would greatly ...
0
votes
2answers
329 views

Show that if the product (the composite) of two linear operators exist, it is linear

I am given this problem to consider and I am unsure how to prove it. It would be nice to see a definition for what it means to be linear or how to check if something is linear because I don't know ...
0
votes
1answer
34 views

Convex Optimization: minimize over unknown convex set starting in center

Essentially I am trying to develop an algorithm to minimize a function over a convex set that I don't know explicitly. However, I have a starting point "deepest in the set" (i.e. with largest norm ...
0
votes
0answers
41 views

Finding the vertices of a square giving the mid point and radius

I'm a programmer with terrible mathematics skills, but I'm getting by with studies. I'm trying to find out the vertices (x,y) of a square, giving the ...
5
votes
2answers
286 views

Find the next term in the sequence. $\frac{7}{3},\frac{35}{6},\frac{121}{12},\frac{335}{36},\ldots $

$\dfrac{7}{3},\dfrac{35}{6},\dfrac{121}{12},\dfrac{335}{36},\ldots $ $\bf\text{Answer}$ given is $\dfrac{865}{48}$ I found that $4^{th}$ differencess of the numbers $7,35,121,335\cdots$ are not ...
0
votes
1answer
23 views

Squaring Polynomial over $\Bbb F_2[X]$ Is Equivalent to Squaring Argument

Thanks to some assistance below, I can now show that if $g(X) \in \Bbb F_2[X]$ then $g(X)^2 = g(X^2)$. Is there some more direct way to prove this special case (not that the original proof is ...
1
vote
2answers
88 views

What exactly is this function doing?

For a set S define the natural isomorphism Char: P(S) → F(S, {0, 1}). Let A be a subset of S. if x is an element of A, then it goes to 1, if it isn't, it goes to 0. I drew a diagram on a paper to ...
0
votes
1answer
56 views

what is the trust region algorithm in optimization?

I see some books that say the trust region work with contour's line .but i can't understand how choose the point with contour's line and sort them? thank you if answer me.
1
vote
2answers
148 views

Proving that the metric space $((0,\infty),d)$ is complete, with $d(x,y)=|\ln x-\ln y|$ [duplicate]

Let $X$ denote $(0,\infty)\subseteq \mathbb{R}$, and let $d:X\times X\to \mathbb{R}$ be defined as $d(x,y)=|\ln x- \ln y|$. Show that $(X,d)$ is a complete metric space. I am taking for granted here ...
2
votes
0answers
72 views

proving inequality using the mean value theorem

Let $f$ and $g$ be two functions from $[0,1]$ to $\mathbb R$. They are continuous on $[0,1]$ and differentiable on $(0,1)$. If $\|f'(t)\| \le g'(t)$ at all points of the interval, prove that $|f(1) - ...
0
votes
1answer
86 views

Showing that if the curvature $\kappa(s) = 0$ for all $s$, then the curve is a straight line

Problem: Show that if $\kappa(s) = 0$ for all $s$, then the curve $\mathbf{r} = \mathbf{r}(s)$ is a straight line. (Here, $\kappa$ represents curvature.) Attempt at solution: If $\kappa(s) = 0$ for ...
3
votes
1answer
171 views

prove inequality by induction — Discrete math

Prove by induction that $∀n ≥ 3$ : $n^{2} + 1 ≥ 3n$ So I know I need to find my base case, would it be: $n=3$ Then calculate the RHS and LSH RHS:$3(3)=9$ LHs: $3^{2} + 1= 10$ we see that the LHS is ...
3
votes
1answer
54 views

is this manifold diffeomorphic to the klein bottle?

Consider the submanifold of $\mathbb{R}^4$ given for the equations $$x_1+x_2x_3x_4 = 0$$ $$x_2 + \sin(x_3x_4)^2 = 0$$ is this $2$-dimensional manifold diffeomorphic to the klein bottle?. I first ...
1
vote
0answers
81 views

Prove that this condition is true on an Zariski open set

Why is there an Zariski open set of $P\in GL(n,\mathbb{R})$ such that $P\,\text{diag}(1,\dots,1,-1)P^{-1}$ can be conjugated by a diagonal matrix $D$ to get an orthogonal matrix? Note that $M=P\,\...
1
vote
2answers
70 views

Definition of differential on manifolds

I'm studying some differential Geometry at the moment and I'm getting a bit stuck with the definition of the differential. It's defined as follows \begin{array}{cl} \phi_{\star,m} : T_{m}M \...
0
votes
0answers
49 views

Probability Question on Blackjack - Looking for verification

In the game of Blackjack, a basic strategy player will end up with a hand consisting of 3 or more cards about 51% of the time. That being said, what is the formula for finding the probability of the ...
3
votes
0answers
87 views

How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
30
votes
11answers
3k views

Is the number $-1$ prime?

From my understanding it's not prime because it's not greater than $0$. So my followup question is why did mathematicians exclude $-1$? The definition of prime is having only two factors. $-1 \cdot ...
0
votes
2answers
35 views

suppose that $G$ is a finite group and $H$ is a subgroup of $G$ that is not cyclic,I want to prove that all conjugates of $H$ are not cyclic.

suppose that $G$ is a finite group and $H$ is a subgroup of $G$ that is not cyclic,I want to prove that all conjugates of $H$ are not cyclic. I supposed that $S$ is a conjugate of $H$ and it is ...
1
vote
1answer
149 views

Test to know if a vector is inside a spherical triangle

Given a spherical triangle defined by $3$ unit vectors on a sphere, how can we test if a vector is contained inside the spherical triangle?
4
votes
6answers
80 views

Prove by mathematical induction that $\forall n \in \mathbb{N}: 20~|~4^{2n} + 4$

Prove by mathematical induction that $\forall n \in \mathbb{N}: 20~|~4^{2n} + 4$ Step 1: Show that the statement is true for n = 1: $4^{2 \cdot 1} + 4 = 20$ Since $20~|~20$, the base case is ...
3
votes
0answers
139 views

Bounded second derivative implies square root of f is Lipschitz.

Can you help me with this exercise? Let $f \in C^2(\mathbb{R}) $ a function $ f(x) > 0, \forall x \in \mathbb{R} $ and $\|f''\|_\infty < \infty $ , prove that $\sqrt f$ is Lipschitz continuous....
-2
votes
1answer
79 views

How many solutions are there to $x^3-y^3=271$ [closed]

How many integer solutions are there to $x^3-y^3=271$.
1
vote
1answer
196 views

Explicit examples of alternating multilinear forms

I think explicit examples of multilinear forms would help me understand them. (I'm studying determinants.) I know they are helpful (only?) for calculating volumes, which is why any alternating ...
3
votes
1answer
145 views

Stochastic matrices with orthogonal eigenvectors

I stumbled across an odd fact while thinking about Markov chains, and I have an odd proof for it. Claim: Let $P$ be the transition matrix of a finite irreducible aperiodic Markov chain, and assume ...
0
votes
2answers
346 views

Convergence criteria of sequences of real numbers

Which of the following conditions implies (imply) the convergence of a sequence $\{x_n\}$ of real numbers? (A) Given $\varepsilon>0$ there exists an $n_0\in\mathbb N$ such that for all $n\ge ...
4
votes
1answer
46 views

What does the binomial theorem applied to integers count?

I have a homework problem that I'm stuck on, but I feel like if I could get help on this one component (not the whole problem) I could make progress. Also, I'll note preemptively that I tried ...
2
votes
1answer
80 views

2014A AMC solution question

From: AMC 10 Q25 Solution I get everything besides the last part. How in the world does he get: $$3k + 2(867 - k) = 2013$$ I don't understand how he got this? What does this mean? Literally ...
0
votes
3answers
50 views

Finding an expression for $f^{-1}$ (function)

I need help with part (a) of this problem. Two functions, f and g are defined by $f(x) = \dfrac{x-1}{ x +1 }$ and $g(x) = mx+c$ (a) Find an expression for $f^{-1}$ I don't know how to make $x$ the ...
0
votes
0answers
43 views

Power Series and their sums

How to find the sum of And what are called the sums got by differentiating it and integrating? I calculated these, but i can't find the name of the sums: 1) 2)
0
votes
3answers
37 views

Suppose that $f: X \rightarrow Y$ is a one to one function and $A \subseteq X$ then $Y - f(A) \subseteq f(X-A)$. Am I on the right track?

Suppose that $f: X \rightarrow Y$ is a one to one function and $A \subseteq X.$ $A, X, Y$, are all sets. I am trying to decipher if the following statements are true or false. If true I will need ...
0
votes
1answer
53 views

Divergence of $\sum^\infty_{n=1} \frac 1 {n\ln2+\ln(\ln2)}$

Show divergence of $\displaystyle\sum^\infty_{n=1} \frac 1 {n\ln2+\ln(\ln2)}$. Since $2>\ln 2 > \ln(\ln2)$, we have: $\frac 1 {n\ln2+\ln(\ln2)}>\frac 1 {2(n+1)}$. Using the comparison ...
1
vote
1answer
62 views

Finding matrix $A$ knowing that $A^2 = B$

Let $B$ be the $3\times3$ matrix $$ \begin{pmatrix} 1&8&5\\ 0&9&5\\ 0&0&4 \end{pmatrix}. $$ How can I find a triangular matrix $A$ with positive diagonal entries such that $A^...
0
votes
0answers
65 views

Is the Taylor expansion of $e^{x}$ an eigenvector of the differential operator on $\mathcal{P}_{n}$?

Wouldn't this be an eigenvector of the differential operator on $\mathcal{P}_{n}$? $$e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} \ldots $$ (corresponding with the eigenvalue $\lambda = 1$) I ...
0
votes
1answer
55 views

Proof of Pentagonal Numbers

Prove that $P$ is a pentagonal number if and only if $\sqrt{24P + 1} ≡ 5\mod 6$ Proof: The pentagonal numbers can be found from the formula: $P = \frac{3}{2}n^2 - \frac{1}{2}n$ That is: $0 = \...
2
votes
0answers
92 views

Dominant Weight

I am reading a paper which begin by a reminder about root system associated to a simple lie algebra $\mathfrak g$. let $\mathfrak h\subset \mathfrak g$ a cartan subalgebra. Question 1: It says that ...
1
vote
1answer
38 views

Can a partially ordered set always be partitioned by its chains?

I would like any references that discuss how to partition a partially ordered set into subsets such that each subset is a chain in the partial order. For example, I am thinking that any partial order (...
0
votes
0answers
61 views

If K $\subset$ L, K Galois over F, L is a splitting field of f(x) over F, can we say L Galois over F

Let $F$ be a field, $f(X) \in F[X]$ an irreducible polynomial of degree $n$ over $F$, $L$ a splitting field of $f(X)$ over $F$, and $\alpha \in L$ a root of $f(X)$. If $K$ is any Galois extension of $...
0
votes
1answer
32 views

is that function must be constant under the following conditions

I'm talking about complex function $f$ is analytic function on a region $D$ that include the point $z=0$. for every $n\in N$ such $\frac{1}{n}$ is in $D$ , the function follows this condition $|f(\...
1
vote
2answers
220 views

How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
3
votes
1answer
77 views

Finding two smallest composite numbers

Find two smallest composite numbers $n$, so that $$2^n\equiv 2 \ \ \ (\text{mod }n)$$ $$3^n\equiv 3 \ \ \ (\text{mod }n)$$ I dont really know how to approach this problem. I could use some hints.

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