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Questions tagged [upper-lower-bounds]

For questions about finding upper or lower bounds for functions (discrete or continuous).

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1answer
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Bob selects four points on a $10\times 10$ square.

Bob selects four points on a $10\times 10$ square. Is it true that two of them are less than $\sqrt{101}$ units apart? I know how to prove things like this for five points. These seems ...
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0answers
18 views

Bound of the Subtraction of Two Inverse Matrix

Let $\bf{A, A'_k}\in \mathbb{R}^{NxN}$ be symmetric positive definite. For some $1\le k \le N$, $\bf{A'_k}$ is defined as $$\bf{A'_k} = \begin{pmatrix} \bf{A}_{(1:k-1), (1:k-1)} & 0 \\ 0 & \...
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0answers
27 views

Bounds on L1 norm of product of matrices [on hold]

I have a matrix $W \in R^{n \times k}$ whose $m^{th}$ column is $w_m$, and $|w_m|_{\infty} < 1$ and $|w_m|_{1} < \lambda$ where $\lambda$ is known. I am multiplying it with a matrix $X \in R^{k \...
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0answers
53 views

Upper bound for the probability of violating a set of conditions

Let $X_1,\ldots,X_n$ be independent and normally distributed $\mathcal{N}(\bar{x},\sigma^2)$ random variables. Let $g:{\rm R} \to [0,\bar{g}]$ be a decreasing bounded function. Let $a$, $\lambda$ and $...
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2answers
44 views

Find the upper bound of $|\frac{d^2}{dx^2}(e^{-x^2})|\leq6$ in $x\in[0,1]$

Show by finding the second derivative of $e^{-x^2}$ that for all $x\in[0,1]$ $$|\frac{d^2}{dx^2}(e^{-x^2})|\leq6$$ (if you obtain a better bound, that is fine ) My Try let, $f(x)=e^{-x^2}$...
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0answers
75 views

An Application of the Phragmen-Lindelof Theorem

This is Chapter 6, section 4, exercise 7 (page 141) from Functions of One Complex Variable by John B. Conway, Second Edition. Exercise 6.4.7: Let $G=\{z:\Re (z)>0\}$ and let $f:G\rightarrow\mathbb{...
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0answers
39 views

How to solve this optimisation problem when I have varied profits associated with different bags?

Recently I came across an optimization problem. Suppose we have $n$ bags each with varied capacity say $c_j$ i.e. capacity of $j$th bag and $m$ items. There can be multiple instances of these $m$ ...
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0answers
8 views

Bounding the Cokurtosis

Summary I have four random variables $A_1$, $A_2$, $B_1$, $B_2$ and want to bound their Cokurtosis $K(A_1, A_2, B_1, B_2) := E\left((A_1-E(A_1))(A_2-E(A_2))(B_1-E(B_1))(B_2-E(B_2)) \right)$. I am ...
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43 views

Upper bounds for certain finite sums involving restriction $\sum\limits_{n=1}^{N}{{{\left| {{a}_{n}} \right|}^{2}}}=1$

Let us suppose that ${{a}_{n}}\in \mathbb{C},\text{ }n=1,2,\ldots N,$ are $N$ complex numbers such that $\sum\limits_{n=1}^{N}{{{\left|a_n\right|}^{2}}}=1$, and define finite sums ${{S}_{1}}={{\left| \...
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0answers
28 views

Using Least common multiple to establish a lower bound for a ratio of primorials

Let $x\#$ be the primorial for $x$. Let $\text{lcm}(x)$ be the least common multiple of $\{1, 2, 3, \dots, x\}$. It occurs to me that for $x \ge 4$, it is straight forward to find a lower bound of $\...
1
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1answer
34 views

Upper bound for probability of truncated normal random variable

Suppose that $X$ and $Z$ are iid with distribution ${\mathcal N}(\mu,\sigma^2)$. $X$ conditional on $d<X<+\infty$ has a truncated normal distribution with support $(d,+\infty)$. Letting $Y$ ...
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1answer
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What is the rate of growth of $M_n := \max_{1 \le i \le n} U_i^{(n)}$, where $U_i^{(n)} \sim \operatorname{Uniform}[0,n]$?

On pp. 370-374 (see this previous question) of Cramer's 1946 Mathematical Methods of Statistics, the author shows that for any continuous distribution $P$, if $X_1, \dots, X_n \sim P$ are i.i.d., then ...
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0answers
14 views

A question related to induction and recursive sequences

I've been struggling with this problem for a long time now. It's Q2 part (b). Anyone have any ideas?Here it is
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1answer
47 views

Show that $\sum_{k=0}^{N} |P(k)| \leq C(N) \int_{0} ^{1} |P(t)| dt $.

I’m attending a functional analysis course and I am given to solve this problem as an exercise but I’m a little bit disoriented and I don’t know what tools I can use to get it. Show that, for each $...
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0answers
15 views

Show that for a function $f$ bounded by $M$ on a disk $D_{r}$ show that $|f^{n}(z)|\leq n!M/\delta^{n}$ for $D_{r-\delta}$

Suppose that a function $f$ is analytic in the open disk $$D_{r}=\{z\in \mathbb{C}:|z|<r\}$$ where $r>0$, and there is a number $M\in\mathbb{R} $ such that $|f(z)| \leq M$ for all $z \...
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1answer
19 views

What is the least upper bound of the sequence defined by the following piecewise function:

What is the least upper bound of the sequence defined, for $n\in\mathbb{N}$, by $$a_{n}=\begin{cases} 3/n, & \text{if }n\text{ is odd;}\\ 1/n, & \text{if }n\text{is even.} \end{cases}$$ I ...
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0answers
10 views

Finding an Upper bound of a matrix

How do I find the upper bound of $\left( v^{T}U_{n}D_{nn}U_{n}v\right)$ where $v$ is the covariance between each point $x_i$ in a domain $D$ and a new location $x*$. $U_{n}\;\text{and}\; D_{nn}$ are ...
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0answers
27 views

Solve a linear problem using bounded variables method

Consider the following $$\min 3x_1+4x_2\\ s.t. 4x_1+3x_2\ge12 \\ 3x_1+4x_2\le12\\ x_1,x_2\ge0$$ Substitute the first restriction by $x_1\le3$ and solve the LP by bounded variable method. Attempt ...
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1answer
36 views

Sum of terms with recurrence relation

I have the following sequence, where $s$ is some positive multiple of 4: \begin{equation} L_n = \begin{cases} \frac{(s-2)!}{2^{s/4-1}(s/2)!(s/4-1)!}, & \text{for $n=1$} \\ \\ L_{n-1}\cdot\frac{2(...
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1answer
47 views

Cardinality bounds on set algebra

Assume two sets $A$ and $B$. Consider the following set of inequalities: $\max \left(\lvert A \rvert, \lvert B\rvert \right) \le \lvert A \cup B\rvert \le \lvert A \rvert + \lvert B \rvert $ $\lvert ...
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2answers
36 views

Find the best upper bound of $\varepsilon$ such that $g(x) = 1 + \varepsilon f(x) > 0$, $f(x)$ bounded on a closed set?

Consider a smooth function $f : [0, 1] \to \mathbb{R}$. Moreover, consider: $$g(x) = 1 + \varepsilon f(x),$$ for $x \in [0, 1]$ and some $\varepsilon > 0$. Which are the conditions on $\...
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1answer
17 views

Give upper and lower bounds for $\left|\frac{\sin^3{na}-2}{\left(1+\frac{\cos b}{2}\right)^n}\right|$

Give upper and lower bounds for $\left|\frac{\sin^3{na}-2}{\left(1+\frac{\cos b}{2}\right)^n}\right|$ The only way that book explained up to this point in other examples uses this kind of technique: ...
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0answers
29 views

Limits and continuous functions with Graph

Graph is given: $\lim_{x→2^-} f(x)$ $\lim_{x→2^+} f(x)$ At $x=2$, is the function continuous from the left or continuous from the right? $\varliminf_{x→2} f(x)$ (lim inferior) $\varlimsup_{x→2} f(x)$ ...
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2answers
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Can we say a function is “unbounded” when we mean it''s tending to infinity?

I'm watching the Limits series on Khan Academy. In many videos Sal repeatedly says that although some people say that functions that tend to infinity have a limit infinity. (For example, in this video,...
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0answers
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Constructions of big $k$-uniform hypergraphs - is there a better result?

We have the following result (and I know its proof): There exists a $k$-uniform hypergraph on the set $\{1,2,\ldots,d\}$ with $|F_1 \cap F_2| <\varepsilon k$ for any two edges $F_1, F_2$ and with ...
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31 views

Calculating upper and lower bound

Suppose there is a function $\ f(x)= 19n^2/5n +1-n $ I want to calculate upper and lower bound. But I had this confusion that whether I have to calculate in terms of n^2? Because dominating term is n^...
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1answer
47 views

Can't seem to find the lower bound for this function

Been reading about algorithms. I am trying to find the lower and upper bounds for the function f(n). not very familiar with mathjax so i used mathtype. how do i proceed with the lower bound. ...
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1answer
21 views

Prove that if a sequence converges then $\lim{x_n} = \lim \sup {x_n}$ or $\lim{x_n} = \lim \inf {x_n}$

Given a convergent sequence $\{x_n\}$ prove that either: $$ \lim_{n \to\infty}\{x_n\} = \lim_{n \to \infty} \sup \{x_n\} $$ or $$ \lim_{n \to\infty}\{x_n\} = \lim_{n \to \infty} \inf \{x_n\} $$ ...
0
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1answer
22 views

Greatest lower bound

All I have to do is show from definition that $A$ must have at most one greatest lower bound if $A$ is a subset of $\mathbb{R}$ and is not empty. My thoughts are if $A$ is not bounded below, then it ...
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1answer
28 views

Prove that the sequence is bounded : [closed]

How to mathematically prove that the sequence $ \{a_n=\dfrac{3}{3^n} \}$ is bounded ?
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2answers
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Show that the sequence $a_n = (1 + \frac{1}{n})^n$ is bounded. [duplicate]

Show that the sequence $$a_n = \bigg(1 + \frac{1}{n}\bigg)^n$$ is bounded.
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1answer
47 views

Bound for $(1-\frac{1}{n})^t$

I'm having trouble proving that: For any constant $\epsilon > 0$ and $n > 1$: $$ \left(1-\frac{1}{n}\right)^{n lg\left(n^{\epsilon}\right)} \leq \frac{1}{n^{\epsilon}}$$ I'm using $lg(n)$ as $...
0
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1answer
22 views

Minimum value of a continuous function.

Let $f:[a, b]\to (0, \infty)$ be a continuous function. Let $$F:[a, b]\times [a, b]\to (0, \infty): F(x, y) =\frac{f(x)}{f(y)}$$ Then I am interested in the lower bound on $F$. If it is $1$ then how ...
0
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1answer
37 views

Proving an expression is non-negative on the interval $(-1,1)$

I'm having trouble proving that the following expression is non-negative for $\alpha^2<1$ and $\theta > 0$: $$\frac{1}{(1-\alpha)^2}+\frac{1}{(1+\alpha)^2}+\frac{1}{(1-\alpha+\theta)^2}+\frac{1}{...
1
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1answer
35 views

Lower bound of the entropy over the probability distribution

I am trying to show that: $$\inf_{z: 1^Tz=1} \sum_{i=1}^m {z_i \log z_i} = -\log m$$ I thought about Jensen inequality or induction, but none of them provided me something.
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0answers
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Proof about ordering-preserving function

Let $f: C_1 \to C_2$ be an order preserving function. Assume that for $A \subset C_1$ there exist $Sup(A) \in C_1$ and $Sup(f(A)) \in C_2$. Prove that $Sup(f(A)) \leq f(Sup(A))$. The statement seems ...
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0answers
33 views

Prove $\{|C_a^n|\}$ is decreasing starting from some $k$ given $C_a^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}$

Let $C_a^n$ be defined as: $$ \begin{cases} C_a^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}\\ n\in \mathbb N \\ a \in \mathbb R \\ C_a^0 = 1 \end{cases} $$ Prove that $\{|C_a^n|\}$ is decreasing ...
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3answers
28 views

Finding bounds of the set: $ A = \{ \frac{m\cdot n}{m+n}: m,n \in \mathbb N \} $

I'm having some problems with this. I know that lower bound will be $\dfrac{1}{2}$. Should I just find $m,n$ for which $\dfrac{m\cdot n}{m+n} \lt \dfrac{1}{2} + \epsilon $? Also I'm not sure how to ...
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0answers
9 views

Find upper bound for $\frac{b(abc)^2 (1+bx)x}{a((1+s)(1+bx)+bs(T-x))^2}$

I want to find an expression for an upper bound for $$F(x) = \frac{b(abs)^2 (1+bx)x}{a((1+s)(1+bx)+bs(T-x))^2},$$ where $a,b,s, T$ are constants, $a,b,s$ are positive, and $x \leq T$, and subject to ...
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1answer
30 views

fundamental theorem of algebra in the complex plane

My question is about the following lemma (where the Extreme Value Theorem is assumed); $(1)$Let $f : \mathbb{C} → \mathbb{C}$ be any polynomial function. Then there exists a point $z_0$ $∈ \mathbb{...
2
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2answers
62 views

One lower bound for $(1+x)\log(1+x)-x$

Problem When studying Chernoff bound, one result is used without proof and reference, which is $$ (1+x)\log(1+x)-x\geq \frac{x^2}{\frac{2}{3}x+2} $$ I am wondering how this is proved. What I Have ...
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vote
0answers
26 views

Best error term in $\sum_{(n,q)=1}\frac{1}{n}$ (harmonic series with coprimality condition)

It is very well known and not difficult to prove that $\displaystyle\sum_{\substack{0<n\leq X\\ \\(n,q)=1}}\frac{1}{n}=\left(\log(X)+\gamma+\sum_{p|q}\frac{\log(p)}{p-1}\right)\frac{\phi(q)}{q}+O\...
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1answer
25 views

Showing that a second order differential equation has unique bounded solution

I'm trying to show that $x''+bx'+x=\cos(t)$ has a unique bounded solution, for $b<0$, $b\in\mathbb{R}$. I believe I understand that it has a unique solution as all coefficients and the non-...
0
votes
1answer
24 views

Using the chebychev inequality in the absence of st.dev but known max value

A random variable X takes the maximum value of 80, and has a mean equal to 50. Give the best upper bound on P(X<=20). So is it possible to use the Chebychev inequality here. Note that both values ...
0
votes
0answers
28 views

Lower bound for $\|x-y\|$

For $x$,$y$ in Hilbert space $\mathcal{H}$ I want a lower bound for \begin{equation} \|x-y\|_{\mathcal{H}}^2 \end{equation} I know \begin{equation} |\ \|x\|_{\mathcal{H}}-\|y\|_{\mathcal{H}}\ |\leq\|...
1
vote
2answers
53 views

Prove that $C_{3 \over 2}^n$ is bounded given $C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}$

Let: $$ \begin{cases} C_{a}^n = \frac{a(a-1)(a-2)\dots(a-n+1)}{n!}\\ C_{a}^0 = 1 \end{cases} $$ Prove $C_{3 \over 2}^n$ is bounded. I've started with finding a reduced formula: $$ C_{3\over 2}^n ...
0
votes
1answer
9 views

Coercive bilinear form for maximum norm

Let $f$ be a differentiable function. Denote a bilinear form by $$b(f,f) = \int_{0}^{1} \bigg( \frac{d f(x)}{dx} \bigg)^{2} dx.$$ Given $f(0) = 0,$ we want to show that $$a \cdot b(f,f) \geq ||f||_{\...
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votes
5answers
53 views

Show $x_{n+1} = {1\over 2}x_n^2 - 1$ is bounded below and unbounded above and $x_n$ is increasing.

Let: $$ \begin{cases} x_{n+1} = {1\over 2}x_n^2 - 1\\ x_1 = 3\\ n\in \mathbb N \end{cases} $$ Show that the sequence $x_n$ is bounded only below and is increasing. I've started with the ...
0
votes
2answers
29 views

Let $A$ be a set and $s$ its supremum. Given an $\epsilon>0$, can I assure that I can find and $a\in A$ that meets: $s-\epsilon<a<s$?

Let $A$ be a set and $s$ it's supreme. Given an $\epsilon>0$, can I assure that I can find and $a\in A$ that meets: $s-\epsilon<a<s$ ? I tried saying that $s>a\,\, \forall a\in A$, and ...
2
votes
1answer
72 views

Lower bound for sum of Hecke eigenvalues

Let $\lambda$ be weakly multiplicative, $\lambda(n)\geq0$, $p$ prime and $S(x)=\sum_{n\leq x}\lambda(n)\log(\frac{x}{n})$ for real $x$. How can I show $S(x)\gg \left(\sum_{p\leq \sqrt{x/3}}\lambda(p)\...