1
vote
1answer
73 views

Why zeta(2) in these inifinite sums?

The infinite sum of the reciprocals of these two sequences have zeta(2) in the result. The value is not in OEIS. A000326 A002411 Edit---rolled back the changes. Both $\frac{1}{2}$ and $2$ are ...
1
vote
0answers
14 views

Multiway tree to Binary Tree

A multiway tree T can be represented as a binary tree T~ by using the firstChild and nextSibling pointers. If we think of the firstChild link as being the left link and the nextSibling link as being ...
3
votes
2answers
45 views

Unbounded sequence that does not diverge to $+ \infty$ or to $- \infty$

I'm trying to find an example of sequences such that $$a_n \to + \infty \quad \text{and} \quad b_n \to 0$$ $$a_n b_n \text{ is unbounded but does not diverge to }+ \infty \text{ or} - \infty$$ Is ...
14
votes
0answers
76 views

Time-optimal control to the origin for two first order ODES - But wait, the node is unstable? Hard-mode active!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...
6
votes
1answer
85 views

Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
0
votes
2answers
49 views

Discriminant of $x^2+x-1$

I'm working on a homework problem, and am worried I'm going crazy. I believe $x^2+x-1$ is irreducible, but it's discriminant is $1^2-4(1)(-1)=5$ is positive, which would make it reducible (since ...
2
votes
1answer
17 views

This equation define a regular surface?

Consider the function: $f(x,y,z)=xyz^2$ Its gradient is $\nabla f=(yz^2, xz^2, 2xyz)$ then the critical points are all in the sets $\{(x,y,0): x,y\in \mathbb{R}\}, \{(0,0,z): z\in \mathbb{R}\}$. My ...
0
votes
0answers
26 views

Questions concerning upper densities

Let $D(X) = \limsup_{N \rightarrow \infty} \big( \frac{|A \cap \{1,...,N\}|}{N} \big)$ represent the upper density of set $X$; $FS(X)$ be the set of all finite sums of terms/elements in $X$. Note: $0 ...
0
votes
1answer
11 views

monotone class theorem failure for a class of subsets that is not a field

show the monotone class theorem fails if $F_{0}$ is not assumed to be a field. Monotone class theorem: Let $F_{0}$ be a field of subsets of $\Omega$, and $C$ a class of subsets of $\Omega$ that is ...
0
votes
1answer
31 views

A simple question in calculus (equivalence of limits).

So I want to prove the next equivalence: where D-lim, is $\lim_{n\rightarrow \infty , \ n \notin M \subset \mathbb{N}}$. The easy part, mainly $\Rightarrow$ I did I think good. I am having ...
0
votes
1answer
33 views

binary addition

Can any direct me to any resources online that teach how to approach binary addition such as this/ working with more complex binary arithmetic? I know the basics of binary addition and carrying the ...
2
votes
0answers
34 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
4
votes
2answers
45 views

Proofs that rely on an infinite matrix

If I have an operator $A\in B(\mathcal{H})$ that can be "identified" with an infinite matrix with countably many entries, is it in any way unrigorous to do actual calculations with the picture we have ...
1
vote
0answers
17 views

Finding the explicit form of the recursive function $P_{1(n)}=\left\lceil\frac12P_{1(n-1)}\right\rceil+\left\lfloor\frac12P_{2(n-1)}\right\rfloor$

I'm trying to find the explicit form of the recursive function $$P_{1(n)}=\left\lceil\frac12P_{1(n-1)}\right\rceil+\left\lfloor\frac12P_{2(n-1)}\right\rfloor\;.$$ First, let me explain what this ...
1
vote
0answers
37 views

How prove this $ (x+z\cos{(2B)}+y\cos{(2C)})^2+(y\sin{(2C)}-z\sin{(2B)})^2=0$

Nice quetion: let $x,y,z$ is postive numbers,and the postive $k$ such ...
1
vote
1answer
24 views

Prove that $S$ is a subring of $\mathbb{Z}_{28}$

Question: $S=\{0,4,8,12,16,20,24\}.$ Prove that $S$ is a subring of $\mathbb{Z}_{28}$ Confusion 1: This might be a dumb question, but when we refer to $[4]$ in $S$, for example, is that the congruent ...
0
votes
1answer
16 views

Describe smallest algebra, monotone class, $\sigma$-algebra

I'm trying to understand better the concepts of monotone classes, algebras and $\sigma$-algebras so I came into the following problem. For the family $E := \{∅, \mathbb{N}, \{2\}, \{2, 4\}, \{2, 4, ...
3
votes
1answer
85 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
0
votes
0answers
31 views

Solution of this nonlinear equation

How do I solve this equation for $y$? I can see there is a trivial solution $y=0$ but how do I get $y$ as a function of other variables? Does anyone know how to use MATLAB's fzero function to find ...
1
vote
1answer
34 views

A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding $A^{-1}$

The question is: A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding A^-1. I have looked at other similar questions on this site: 1. Here 2. and Here But they use ...
0
votes
1answer
30 views

In what base does the equation $x^2 - 11x + 22 = 0$ have solutions $6$ and $3$?

If we have below equation and know that $6$ and $3$ are answers of this equation, how to obtain the base used in the equation? $$x^2 - 11x + 22 = 0$$ Partial result The base is not $10$. (Because ...
1
vote
0answers
16 views

Law of large numbers

Consider the following question: A coin has the probability of landing of head equal to 1/4 and is flipped 2000 times. Use the law of large numbers, find a lower bound to the probability ...
0
votes
1answer
4 views

Finding the domain of this trigonometric function

how can I find the domain of this function? f(x) = (xsin(x) + cos(x) / 1 - cos(x)) + (|X| - 2 / x^2 -4) I assume we don't want the dominator to be zero so f(x)1 ...
0
votes
2answers
29 views

ergodic system has a.e. dense orbits

one more question: Let X be a metric space with probability measure $\mu$ and T. X $\to$ X ergodic. => f.a.e. x the orbit $O_x=\{T^n(x) : n \in Z\}$ is dense in X so I have to show that the set B ...
9
votes
6answers
655 views

Is there always a prime number between $p_n^2$ and $p_{n+1}^2$?

The following table indicates that there is a prime number p between the square of two consecutive primes. $$ \displaystyle \begin{array}{rrrr} \text{n} & p_n^2 & p_{n+1}^2 & \text{p} \\ ...
0
votes
1answer
8 views

Pentagon Inscribed Circle Radius

Square ZENG has a perimeter of 12, with midpoints Y and I on sides ZE and ZG, respectively. Find the radius of the largest circle that can be inscribed in pentagon GIYEN. I got past all the simple ...
-1
votes
1answer
45 views

If $\lim (S_n)=s$, does it follow that $\lim (S_{n+1}) = \lim (S_{n+2}) = s$?

I have proven $\lim (S_n)=s$, where $S_n$ is a sequence. Am I allowed to say $\lim (S_n) = \lim (S_{n+1}) = \lim (S_{n+2}) = s$?
-1
votes
0answers
34 views

Ziploc Conjecture [on hold]

A flexible plastic bag width $w$ and height h is filled with a liquid of volume V almost fully to roundness, sealed at top and placed on a flat table with its height approximately vertical. Prove ...
1
vote
1answer
28 views

Equation: $2\sqrt{1-x}-\sqrt{1+x}+\sqrt{1-x^2}=3-x$

$2\sqrt{1-x}-\sqrt{1+x}+\sqrt{1-x^2}=3-x$ Could someone help me solve this problem?
0
votes
1answer
26 views

The primes such that removing digits from the right end leaves another prime

The number 73,939,133 is prime. Keep removing a digit from the right end. Each of the remaining numbers is prime. How to find other numbers with this property?
0
votes
0answers
20 views

Help with Java coding [on hold]

I've recently started taking an intro java class(I apologize in advance for my lack of knowledge) and I've run into some trouble. I was required to make a program that contains 3 data fields, 2 ...
0
votes
0answers
21 views

Set builder notation question

I have that G is the set of real numbers of the form $m+n\sqrt2$ for $m,n \in \mathbb{Z}$ So I think that it is either: $G=\{m+n\sqrt2 \in \mathbb{R}, m,n \in \mathbb{Z}\}$ or $G=\{m+n\sqrt2 \in ...
10
votes
3answers
151 views
+50

Value in retracing mathematicians' steps (specifically Galois)?

So I'd like to learn Galois Theory, which I am probably not "qualified" for in an ordinary sense (I've never done abstract algebra, and I'm just now learning linear algebra in my vector calculus ...
0
votes
1answer
14 views

Volume of a cone made from circle

Circle X has a radius of 15. Points D and F are located on circumference of the circle. Given that angle DXF measures 48°, find the volume of the cone that is formed by aligning the two straight ...
1
vote
2answers
38 views

Show that 47 divides 5^{23}+1

Show that $47$ divides $5^{23}+1.$ My attempt: Since 47 is prime and 47 does not divide 5, by Fermat's Little Theorem, $5^{47-1} \equiv 1$ (mod 47) $5^{46} \equiv 1$ (mod 47) Now I noticed that ...
2
votes
1answer
258 views

Incomplete Fermi-Dirac integrals and polylogs

The complete Fermi-Dirac integrals $$ F_s(x) = \frac{1}{\Gamma(s+1)} \int\limits_{0}^{\infty} \frac{t^s}{e^{t-x}+1} \: dt $$ are related to the polylogarithms, see http://dlmf.nist.gov/25.12#iii $$ ...
0
votes
1answer
14 views

Analysis of IQ scores given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...
3
votes
1answer
125 views
+100

Fourier series of $\sqrt{1 - k^2 \sin^2{t}}$

I'm struggling with a Fourier series. I need to find the Fourier series of the following function. That's the function under study: $f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$. The function ...
0
votes
2answers
50 views

If $|G|=p^2$ and $p$ is prime then $G\simeq \mathbb Z_{p^2}$ or $G\simeq \mathbb Z_p\times \mathbb Z_p$?

Let $G$ be a finite group of order $|G|=p^2$ where $p$ is a prime. How can I show $G\simeq \mathbb Z_{p^2}$ or $G\simeq \mathbb Z_p\times \mathbb Z_p$? Notice $|G|=p^2$ implies $G$ is abelian ...
3
votes
1answer
17 views

Minimize distance to a given point subject to a number of linear inequality

I'm trying to find a point that has minimal distance to a known point and satisfies a number of linear inequality. Example in two dimensions and one inequity: $min\{$distance to $(50,70)$ | ...
0
votes
2answers
37 views

Sequences of polynomial functions converging uniformly on $[a,b]$ to a continuous function not a polynomial

What is (are) the necessary and sufficient condition(s), if any, for a sequence of polynomial functions to converge uniformly on a given (finite) closed interval $[a,b]$ to a continuous function not a ...
0
votes
2answers
19 views

Checking if transformation T(p(x)) is diagonalizable?

Say you have a transformation of $P_3$ to $P_3$ defined by, say, $T(p(x)) = p'(x) + p''(x) + p'''(x)$. How would you determine if this is diagonalizable? Do I sub in a standard basis of ...
2
votes
1answer
12 views

Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
0
votes
0answers
8 views

Unconstrained Optimal Control - $J = \frac{1}{2}x^2(2) + \frac{1}{2} \int_{0}^{2}(u^2 - 2xu)dt$

I've been given the following unconstrained optimal control problem, but I feel like I've made a mistake at some point. The system $\dot x = -x + u$, where u = u(t) is not subject to any ...
1
vote
0answers
19 views

Formulating recurrence relation

Alice and Bob worked in a restaurant and received n currency notes in total as tips. Every note has a value (either 1 dollar, 5 dollar or 10 dollar) written on it. The currency notes are arranged from ...
0
votes
0answers
22 views

error function inverse limits

$$ \lim_{x \to 0+} e^{-(\operatorname{erfc}^{-1}(2x) - 2\operatorname{erfc}^{-1}(x))^2}(e^{\operatorname{erfc}^{-1}(x)^2} - e^{\operatorname{erfc}^{-1}(2x)^2}) $$ Where $\operatorname{erfc}(x) = ...
1
vote
2answers
57 views

Integration of $F(\sum_k x_k)$ over positive orthant

Problem Suppose we integrate some function $F\left(\sum\limits_{k=1}^n x_k\right)$ over the positive orthant $[0,\infty)^n$. Show that this this is proportional to the integral ...
3
votes
2answers
199 views

Existence of a symmetric matrix $A$ such that $XA=Y$.

Let $X,Y$ be vectors in $\mathbb{C}^n$, and assume that $X\ne0$. Prove that there is a symmetric matrix $B$ such that $BX=Y$. This is an exercise from a chapter about bilinear forms. So the ...
2
votes
1answer
40 views
+50

Pullback of principal Cartier divisors along a field extension

I tried the following problem in Liu's book, 7.3.1 but I don't see where it was needed that $X$ is integral - maybe someone can help me here. Is the following true without supposing that $X$ is ...
4
votes
3answers
264 views

Does “Big Data” Have a Ramsey Theory Problem?

I'm erring on the side of conservatism asking here rather than MO, as it is possible this is a complex question. "Big Data" is the Silicon Valley term for the issues surrounding the huge amounts of ...

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