0
votes
0answers
2 views

Bound this expression

Here is a problem I have encountered in the course of trying to apply the Berry-Esseen theorem to a sum of random variables, and I would like a hint as to how to approach it, or a reference. I believe ...
1
vote
0answers
3 views

Recursion and Time Complexity Concept

I have been solving this question which appeared in one of the entrance exam. The question is as follows: ...
0
votes
0answers
3 views

extension of log concave functions

So I need to prove something or see if its true and I don't know how to write in nice math text because this is my first question, so please bear with me We have a CDF Fh[s] for 0<=s<=1. The ...
2
votes
2answers
14 views

Matrices of rank 1

Let the n x n complex matrix A have rank 1. Prove: $A^2$ = c(A) for some scalar c. What I know is that all matrices having rank 1 have rows based on a scalar multiple of the other. This should ...
3
votes
0answers
29 views

How do we find u(x)?

I want to know how to find u(x) in the below question: $$u''(x)+{e^u}^{(x)} = 0\\ x \in[0,1]\\u(0) = u(1) = 0$$ Please explain briefly how this was done?? Thanks!!
0
votes
0answers
20 views

quadratic equation

A garden is in the shape of a rectangle, $20$m by $8$m. Around the outside is a border of uniform width and in the middle is a square pond. The area which is not occupied by either border or pond is ...
0
votes
0answers
4 views

the partial derivative of poisson kernel w.r.t. theta tends to 0 as r tend to 1?

the partial derivative of poisson kernel w.r.t. theta tends to 0 as r tend to 1? how to show it ,thanks!!! seem it diverge,because we have a 'n' term after diff.?
0
votes
0answers
7 views

Analytic Solution of Second Order Nonlinear odes

Any idea how to find analytic solution of the following ode. $y''+0.1 y'+y^{5} = sin (t)$ I will really appreciate your response! Shah
0
votes
2answers
24 views

Prove by induction that $r_0 + r_1a + r_2a^2 + \cdots + r_{n−1}a^{n−1} < a^n$.

Let $a$ be a natural number greater than $1$. Prove that for all integers $r_0 , r_1 , \cdots , r_{n−1}$ with $0 ≤ r_j < a$, we have: $$ r_0 + r_1a + r_2a^2 + \cdots + r_{n−1}a^{n−1} < a^n ...
0
votes
0answers
5 views

What's the significance to the $m$ in the notation $L(n,m)$ for the Lens space?

I'm reading a quick example (Example 12.13 of Topological Manifolds by John Lee) of the construction of the lens space $L(n,m)$. Basically, let $$S^3=\{(z_1,z_2)\in\mathbb{C}^2:|z_1|^2+|z_2|^2=1\}$$ ...
0
votes
0answers
17 views

How prove this konw$\sin{A}:\sin{B}:\sin{C}$ then Find $\sin{(2A)}:\sin{(2B)}:\sin{(2C)}$

let $x,y,z>0$ is give numbers, and the postive number $k$ such $$\dfrac{x^2}{x^2+k}+\dfrac{y^2}{y^2+k}+\dfrac{z^2}{z^2+k}=1$$ in $\Delta ABC$, ...
1
vote
1answer
11 views

In every polygon circumscribed about a circle, there exist three sides that can form a triangle.

How can one show that in every polygon circumscribed about a circle, there exist three sides that can form a triangle? (This was posted by another user and then deleted while I was typing my answer.) ...
0
votes
0answers
11 views

Beautiful property of every single circunscribed polygon ever

Show that in any circumscribed polygon, there exist three sides which could form a triangle. Been on it for a while starting with quadrilaterals and trying to connect some properties and proved for ...
0
votes
2answers
14 views

Essential supremum of a measurable function

Suppose $f\colon(X,\mu)\to [0,\infty]$ is measurable. Let $S$ be the set of all real $\alpha$ such that $\mu (f^{-1}((\alpha,\infty]))=0$ If $S=\emptyset$, put $\beta=\infty$. If $S\neq \emptyset$, ...
0
votes
0answers
10 views

Is equicontinuity a necessary condition for the result of Arzela- Ascoli Theorem

The result shows the existence of a uniformly convergent sub-sequence of the original sequence i.e. some kind of limiting behavior. So, can equi-continuity be relaxed to some kind of limiting ...
0
votes
3answers
26 views

Prove $\lim_{n \rightarrow \infty} \frac{1}{n}\cdot \frac{3 + \frac{1}{n}}{4 - \frac{1}{n}} = 0 $

Prove $\lim_{n \rightarrow \infty} \frac{1}{n}\cdot \frac{3 + \frac{1}{n}}{4 - \frac{1}{n}} = 0 $ Let $\epsilon > 0$ be arbitrary. I want to find $N$ such that $n \in \mathbb{N}$ guarantees $ ...
0
votes
0answers
5 views

Nonhomogeneous Poisson process

Let $\lambda:[0,\infty)\rightarrow [0,\infty)$ be a continuous function and $N$ be a Poisson process with rate $1$. Define $\Lambda(t)=\int_0^t{\lambda(x)dx}$ then how do we prove that $$ ...
0
votes
0answers
15 views

How to compute the triple integral over this region?

The region K is defined by: $ 0 \leq y+x \leq 4-(x-y)^2 $ $ y^2+x^2 \leq z \leq 2(1+x+y) $ So far I could reform the terms and it looks something like this: $ 0 \leq y+x \leq ...
1
vote
2answers
18 views

Question about the boundedness of the following sequence

I would like to know whether the following sequence grows without bounds. For any small real $b < 1$, define a sequence $a_n= \frac {(2n)!}{n!}b^n$ It seems to me that the factorial is so large ...
1
vote
1answer
9 views

Lattice of a POSET Realtion

Given a set $S=\{1,2,3,4,5,6,7,8\}$, defined by a partial order relation Divisibility. Now consider all 4 elements containing sub-graphs, out of which $\{1,2,4,8\}$ is a Lattice obviously . Is ...
0
votes
1answer
27 views

Beauiful wordly combinatorics.

Show that the number of m-letter words using letters T O W N(maybe not using some of them) in which the number of Ts and Os are equal, is equal to the number of 2m-letter words only using Ts and Os ...
0
votes
2answers
15 views

Unusual Probability Question

Wheel of fortune: Assume that the probability for an angle φ is P(φ) = λ$φ^2$. The game pays $1000 times the angle. a) What is λ? b) Find the expectation and the variance of the game. Comments: ...
0
votes
0answers
7 views

Design new data structure

Design a data structure and a search algorithm to solve the following problem. It should be able to answer the query in O(log2 n) time. A high-level description of your data structure, the search ...
0
votes
3answers
37 views

Proving the inequality $x^2\sin{x}+x\cos{x}+x^2+\frac{1}{2}>{0}$ [on hold]

Prove the inequality for real number $x$ $x^2\sin{x}+x\cos{x}+x^2+\frac{1}{2}>{0}$
0
votes
1answer
19 views

Modulus proof with gcd [on hold]

Let $a,b,m,k\in\Bbb Z$ such that $m\ge2$ and $k\ne0$. Let $d=\gcd(k,m)$. Prove the following: If $a\equiv b\bmod m$ and $k$ is a common divisor of $a$ and $b$, then $\left(\dfrac ...
3
votes
1answer
24 views

Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v

I wrote a proof for the above question, but I am not sure whether it is right or not since I assumed linear independence. Here's the proof: Let $u$,$v$ be linearly independent vectors in $V$. ...
0
votes
0answers
20 views

Cyclic subgroup of U19 [on hold]

Consider the cyclic subgroup of U19. a. List all it’s generators. b. List all distinct subgroups of U19 and their elements.
4
votes
1answer
33 views

Beautiful invariant algebra problem

Eugene wrote a 100 numbers on a board. He added 1 to each number and the product didn't change. He did the same thing k times, each time the product didn't change. What is the maximum k? I guess the ...
2
votes
2answers
20 views

Find the Power Series

How would one write $f(z) = \frac{1}{1-wz}$ as a power series? ( Where $z,w$ are in $C$.) Would it just be $\sum_{n=0}^{\infty} (zw)^n$?
-1
votes
0answers
10 views

prove or disprove these statements [on hold]

how can we prove or disprove these two claims? ([x] is floor of x) 1) for all x in R, for all e in R+, there exists some d in R+, for all w in R, if |x-w| < d --> |[x] - [w]| < e 2)the negation ...
1
vote
0answers
35 views

Beautiful combinatorics and logic problem

In some country they use golden and platinum sand. Gold can be exchanged to platinum and vice versa, using an exchange rate defined by postive integers g and p as follows: $x$ grams of gold is ...
2
votes
1answer
32 views

Find Max Of $M=|z^{3}-z+2|$

Give a complex number $z$: $|z|=1$ Find Max $M=|z^{3}-z+2|$ Could someone help me solve this ?
2
votes
0answers
14 views

The ideal $(x_1,x_2,x_3,…)$ (with infinitely many variables) in the ring $K[x_1,x_2,x_3,…]$ is not finitely generated.

How can I show that the ideal $(x_1,x_2,x_3,.........)$ (with infinitely many variables) in the ring $K[x_1,x_2,x_3,....]$ is not finitely generated. I cannot complete the arguments as I thought if ...
1
vote
0answers
10 views

proof for equality of measures for case the case that $F_{\mu}$ be the completion of $F$

Let $(\Omega,F, \mu)$ be a measure space, and let $F_{\mu}$ be the completion of $F$ relative to $\mu$. If $A \subset \Omega$, define: $\mu_{0}(A)=\sup \{\mu(B: B \in F, B\subset A \}$, ...
1
vote
0answers
13 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
42 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 ...
2
votes
2answers
33 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to\infty.$$ Obviously, this sequence of functions ...
1
vote
0answers
15 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
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 ...
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 ...
2
votes
1answer
34 views

A density question

Let $\theta \in \mathbb{R} \setminus \mathbb{Q}$. Is the set $\{ (2n+1) \theta \bmod 1: n \in \mathbb{N} \}$ dense in $[0,1]$?
2
votes
1answer
32 views

Proving that if $f$ is Riemann integrable and $1/f$ is bounded then $1/f$ is Riemann integrable

I have to prove the following Suppose $f$ is Riemann integrable on $[a,b]$ and $1/f$ is bounded on $[a,b]$. Prove that $1/f$ is Riemann integrable on $[a,b]$. My attempt: Since $1/f$ is bounded ...
-1
votes
0answers
18 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
20 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 ...
-2
votes
0answers
33 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 ...
2
votes
0answers
33 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]
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
2answers
47 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 ...
1
vote
1answer
32 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 ...
1
vote
1answer
27 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?

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