Questions tagged [upper-lower-bounds]
For questions about finding upper or lower bounds for functions (discrete or continuous).
2,356
questions
0
votes
0
answers
30
views
Help me to get an upper bound for $f(x)=(1+x+\frac{x^2}{3})e^{-x}$
My problem is the following. Let $f$ be the function defined on $\mathbb{R^+}$ such that $f(x)=(1+x+\frac{x^2}{3})e^{-x}$. I have the inequality : $$f\left(\frac{b}{x}\right)\leq c\leq f\left(\frac{a}{...
- 1
0
votes
1
answer
33
views
Finding the bound to a probability with Markov's inequality [closed]
I have the following question that I am struggling to solve. And I would appreciate it if someone could give me some hints on solving it:
Let x be a random variable with probability density 1/4 for 0&...
- 11
-1
votes
2
answers
62
views
Upper bound on the finite sum of $\sum_j x^j/j!$ [closed]
How can I derive an upper bound on the following finite summation,
\begin{equation}
S = \sum_{j=1}^k \frac{x^j}{j!},
\end{equation} where $0 < x$, in terms of $x$ and $k$ (it's perfectly fine to ...
- 23
0
votes
0
answers
27
views
Average of a sampled set
I have a set of natural numbers, denoted by A. The average of A is X. Then, I sample every item of A independently and add it to another set, denoted by A' with probability of p. My question is - what ...
- 3
2
votes
2
answers
68
views
Upper and lower bounds for $\displaystyle \sum\limits_{\text{cyc}}a^2b$
When dealing with 3-variable inequalities, often we meet expressions like $\displaystyle \sum\limits_{\text{cyc}}a^2b$ ($a,b,c>0$), which becomes a problem if we attempt to use the pqr method. The ...
- 468
0
votes
0
answers
48
views
Divisors of $x^2-1$ in Brocard's Problem
In this post, I was curious if the divisor bound could be improved for the product of two consecutive even numbers. It seems most likely it cannot by much. How could the upper bound of $\sigma_0(x^2 - ...
- 33
1
vote
1
answer
33
views
Worst Case Solution for directed Chinese Postman Problem
The Problem
Let $G$ be a directed graph with $n$ vertices. How long is a shortest circuit that visits every edge in the worst case? That is how long is the solution to the directed Chinese Postman ...
- 11
3
votes
1
answer
79
views
Upper bound on number of divisors of $x^2 -1$
I am trying to find an upper bound on $\sigma_0(x^2 - 1)$, where $\sigma_0$ is the number of divisors and $x$ is some odd integer. From this post, we get an upper bound of
$$\sigma_0(x^2 - 1) \leq (x^...
- 33
1
vote
0
answers
60
views
Lower bound of Hypergeometric function ${}_2F_1(d,1,d+1, z)$ [closed]
I am looking for a (non-trivial) lower bound of the Gauss hypergeometric function ${}_2F_1(d,1,d+1, z)$ where $d\in\mathbb{N}$ and $0\leq z <1$. Ideally, the bound would be valid $\forall d\in\...
- 11
1
vote
2
answers
42
views
Proving the sequence sastifying a specific property is bounded
Let $a_1, a_2,...$ be a sequence of positive numbers. Assume that for
any positive integers $k,l,m,n$ satisfying $k + n = m + l,$ we have the following equality $$ \dfrac{a_{k}
+ a_{n}}{1 + a_{k}a_{n}...
- 160
2
votes
0
answers
49
views
How much can the operator norm of a matrix increase upon deleting an entry?
By "deleting an entry," I just mean replacing it with $0$. Here's an example. The matrix $\begin{bmatrix}
1&1\\1&-1
\end{bmatrix}$ has norm $\sqrt 2$, but the matrix $\begin{bmatrix}...
- 86
0
votes
0
answers
8
views
Generalization bound with covering number
I would like to bound generalization error using covering number.
For the problem I’m working on, the hypothesis class is bounded and covering number is finite. Also, I have already know the covering ...
- 1
0
votes
2
answers
70
views
Proof Verification: Supremum and Infimum of $A=\{n+\frac{(-1)^n}n|n\in\mathbb N\}$
Problem
If $A=\{ n + \frac{(-1)^n}{n} | n \in \mathbb{N}\}$, prove that $A$ is not bounded above but bounded below and $\inf(A)=0$.
My answer
Let $A=B\cup C$ where $B=\{ n - \frac{1}{n} | n \in \...
- 129
-2
votes
0
answers
27
views
Is possible to find an upper bound for this expression?
For $n=1,2,3,...$ the numbers $a_n,b_n,c_n$ are positive integers and $d_n$ is rational number, such that,
$$0<a_n-b_ne<\frac{1}{4^nn!}\\
0<e-\frac{c_n}{d_n}<\frac{1}{n\:n!}$$
Moreover, $n!...
- 2,687
1
vote
1
answer
275
views
Proving a lower bound for this double integral
Let $f:\mathbb R^n\to\mathbb R$ be a smooth function. Let $k>0$ and consider the cutoff function $T_k:\mathbb R\to\mathbb R$ defined for any $t\in\mathbb R$ as
$$T_k(t)= \int_0^t 1_{\mathbb R\...
- 3,277