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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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Bounds on a logarithmic function

Let $\alpha_i$, $i=1,\dots,n$, and $k$ be positive real numbers and consider the following function: $$ f(k_1,\dots,k_n) = \sum_{i=1}^n \log(1 + k_i\alpha_i) $$ where $k_i:=\max\{0,\mu-1/\alpha_i\}$ ...
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26 views

Cramer-Rao Lower Bound for a Conditional Likelihood Function

I'm here looking for assurance that my interpretation is correct. Let the likelihood function under consideration be a conditional likelihood given by $$p(r|x;\theta)$$ where $r$ is some random ...
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Real analysis: least upper bound and greatest lower bound for infinite series

What is the least upper bound and greatest lower bound in $y>3$? I think the least upper bound is $3$ because all other elements are greater than $3$ and greatest lower bound does not exist.
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3answers
232 views

Bound on $\sum\limits_{n=0}^{x}{\sin{\sqrt{n}}}$

Using Desmos and Mathematica, I was able to find a function $g(x)$ that seemingly estimated the function $$f(x)=\sum_{n=0}^{x}{\sin{\sqrt{n}}}$$ I found that $${g(x)=2\sqrt{x}*\sin{\left({{\sqrt{...
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9 views

Lower bound on the number of trials

For $0\leq p_1,\ldots,p_k\leq 1$, it is known that that $\frac{1}{k}\sum_{i=1}^k p_i=q$. Taking an upper bound on $$\frac{1}{k}\sum_{i=1}^k (2p_i-1)^{2t}(1-p_i)\leq\delta$$ for $\delta<1$, can we ...
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2answers
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Is there a simpler way to show that $x_n = {1\over n^3} \sum_{j=1}^n \sum_{i=1}^j i$ is bounded?

Let $n \in \mathbb N$ and: $$ x_n = {1\over n^3} \sum_{j=1}^n \sum_{i=1}^j i $$ Show that the sequence $x_n$ is bounded. I've solved this by expanding the sums and then finding closed forms of ...
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1answer
49 views

Upper bound for the sine integral

For all $x\in\mathbb{R}\backslash\{0\}$ the cardinal sine function $\text{sinc}(x) = \sin(x)/x$ is trivially bounded by $$ |\text{sinc}(x)| \le \frac{1}{|x|},$$ since $\sin(x)\le 1$. I am wondering ...
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30 views

Lower bound on the value of a polynomial

I am interested in the following algorithmic question: Given as input an univariate polynomial $p(x) \in \mathbb{Q}(x)$ of degree at most $d$ as a list of coefficients, and $\epsilon \in \mathbb{Q}$, ...
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Reasoning about the lower bound of ${{2n}\choose{n}}$

A standard lower bound is ${{2n}\choose{n}} > \frac{4^n}{2n}$. For example, see this Wikipedia article. It occurs to me that for higher $n$ using elementary arguments, this can be greatly ...
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2answers
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lower bound and upper bound of expression ($1-\frac{1}{n})^{\log n}$

Let's say that $f(n)=\bigl(\frac{n-1}{n}\bigr)^{\log n}$ ( I know bounds of $\bigl(\frac{n-1}{n}\bigr)^{n}$ ). Is there any way I can get good upper bound of $f(n)$ when n is positive? Thanks in ...
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1answer
120 views

$u$ harmonic and $\frac{u(x)}{|x-x_0|^n}$ bounded, then $u$ identically $0$

Let $\Omega\subset\mathbb{R}^N$ be a connected open set and $u$ an harmonic function in $\Omega$. Suppose that there exists $x_0\in\Omega$ such that there is a neighborhood $V\subset\Omega$ of $...
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24 views

Upper bound of a quadratic expression

We know that a quadratic constraint of the form $$ x^TA^TAx+b^Tx+c\leq0. $$ is equivalent to the following SOC constraint $$ \left\| \begin{array}{c}(1+b^Tx+c)/2\\Ax\end{array}\right\|_2 - (1-b^Tx-...
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0answers
37 views

Prove Least Upper Bound of a Set

How do we prove for $$S = \{ y \in \mathbb R | y^3 \leq x\}$$ $x$ is $\alpha^3$ for which $\alpha$ is the least upper bound of S? I try to rule out $\alpha^3 < x$ and $\alpha^3 > x$ but I have ...
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Lower bounds on $\epsilon$-covers of arbitrary manifolds

Let $M \subset \mathbb{R}^d$ be a $k$-dimensional manifold embedded in $\mathbb{R}^d$. Let $\mathcal{N}(\epsilon)$ denote the size of the minimum $\epsilon$-cover $P$ of $M$, that is for every point $...
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Bounds on inverse Frobenius norm

It's easy to show that $\|A^{-1}\|_F = \frac{\|A\|_F}{\det(a)}$ when $A$ is $2\times2$. It is clearly not true for larger matrices, but is there a known upper bound that uses the direct norm and the ...
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1answer
107 views

Upper bound of integral $\int_0^b\frac{1}{x+2^x} dx$

Is it known a closed form of the integral: $$\displaystyle\int_0^b\dfrac{1}{x+2^x} dx?$$ Using the Talenti inequality I found the following bound: $$\displaystyle\int_0^b\dfrac{1}{x+2^x} dx\lt\ln\left(...
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0answers
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Derivation of ELBO upon the Existence of Conditional Latent Variable Model

I am reading the recently published paper from DeepMind, "Neural Scene Representation and Rendering" and especially its "Supplementary Materials". Following is the page 1 and it's pretty hard for ...
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Inequalities / estimates for a lower bound of the $L^2$ inner product of a quantity and its derivative?

For some numerical analysis of a fluid, I am wondering if there is any inequality that provides a lower bound (in the $L^2$ norm), for the $L^2$ inner product of a quantity with its derivative. In ...
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48 views

Find the supremum and infimum of $A=\{x\in \mathbb{Q}: 4<x^2\leq7\}$

I have to find the supremum and infimum of $A=\{x\in \mathbb{Q}: 4<x^2\leq7\}$ My proof: $4<x^2\leq7\implies 2<x\leq\sqrt{7}$ I will prove that $\sup A=\sqrt7$ and $\inf A=2$ Since $\...
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2answers
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Show that the $\sup S=1/2$ where $S= \{(-1)^n/n : n \in \mathbb{N} \}$

Show that the sup(S) where $S = \{\frac{(-1)^{n}}{n} : n \in \mathbb{N} \}$ is sup(S) = $\frac{1}{2}$ Attempt: In order to show $\frac{1}{2}$ is the supremum I must show: $$\forall \epsilon > 0, ...
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Determining the error bound on an estimate with upper and lower limits…

I have determined the upper and lower limits of an approximation i.e. the solution lies within these two limits. Just a side note; this is a transverse vibration question for beams with continuous non-...
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2answers
68 views

Prove $f(x)\sim e^{-x}(\frac{1}{x}-\frac{1}{x^2}+\frac{1\cdot2}{x^3}-\dots).$

Let $\displaystyle f(x)=\int_x^\infty \frac{e^{-t}}{t}dt.$ Use integration by parts to show that $f(x)\sim e^{-x}(\frac{1}{x}-\frac{1}{x^2}+\frac{1\cdot2}{x^3}-\dots).$ We need to show that $f(x)\sim ...
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38 views

To find the maximum load on a N bin problem.

Consider a scenario, where we have $n$ different colored balls ($C_1, C_2 ... C_n$). Each colored ball ($C_{i}$) contains at least one or more instances of it ($1 \leq |C_i| \leq N$). Where $N$ is ...
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2answers
42 views

Is $\frac{\sqrt{x}}{\ln(x)-1}$ strictly increasing for $x \ge 21$

I was thinking the prime counting inequalities from Pierre Dusart. In particular, I was interested in thinking about when $x \ge 5393$ $$\frac{x}{\ln(x)-1} < \pi(x)$$ I was using Excel with $\...
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1answer
95 views

What would be the standard way to show for $n \ge 148, \pi(n) < \dfrac{n}{4}$

As I understand it, using the Prime Number Theorem, it is well known that for any positive integer $x$, there exists an integer $N$ such that for all $n \ge N, \pi(n) < \dfrac{n}{x}$ For many ...
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2answers
68 views

Lower bound for complex polynomial beyond circle or radius R [duplicate]

If we have a polynomial with $c_i$ a complex number $$c_nz^n + c_{n-1}z^{n-1} + \cdots + c_1 z + c_0$$ then $$|P(z)| > \frac{|c_n|R^n}{2}$$ When |z| > R for some R I have tried using the triangle ...
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1answer
37 views

What is the motivation for Gaussian Tail Bounds?

Perhaps I am missing something here, but I'm not seeing the value of having an upper tail bound for a Gaussian random variable. In my statistics class, we motivate tail inequalities for situations in ...
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2answers
46 views

Does the empty set have a supremum? Is it $-\infty$? [duplicate]

I need to give an example of this, or provide an argument for why it's impossible. Two sets $A$ and $B$ with A intersection $B = \emptyset$ $\sup A = \sup B$, $\sup A$ is not an element of $A$ and $\...
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Lower bound on absolute smallest root of a polynomial

Given a polynomial $p(x)=\sum_k a_k x^k$ with $x,a_k\in \mathbb{R}$, I am looking for a lower bound on roots $r_i$ of the following kind: $$ \min_i |r_i| \geq ? $$ So far, I found this paper which ...
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5answers
70 views

how to show that $ \frac{\theta e^t(1-\theta)}{(1-\theta+\theta e^t)^2} \leq\frac{1}{4}$?

let $0 \leq \theta \leq 1$ , then how to show that $\forall t\in R$ $$ \frac{\theta e^t(1-\theta)}{(1-\theta+\theta e^t)^2} \leq\frac{1}{4}?$$ This is a step of a proof of hoffeding's lemma.
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Need explanation of answer for finding $f = O(g)$ or $f = Ω(g)$ for $f(n) = [log(n)]^2$, $g(n) = n^{log(n)}$

I have $f(n) = [log(n)]^2$, $g(n) = n^{log(n)}$ And answer $f !=O(g)$, but $f=Ω(g)$, $\forall n>=17$. With explanation: $f(n) = e^{nlog(n)}$, $g(n) = e^{log(n)^2}$ $lim_{n->\infty}\frac{log(...
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1answer
20 views

Finding if g(x) is lower or upper algorithm bound for f(x)

Looking at time complexity explanations and found some general rules for determining big-O notation, which made sense, but then found some problems listed as being for the same stuff that are like: ...
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1answer
34 views

Proving the supremum

I'm given these two sets $A\subset (0,+\infty ),$ inf$A=0$ and $A$ is not upper bounded $B=\left \{ \frac{x}{x+1}:x\in A \right \}$ and I have to find the supremum. Here's the solution my book ...
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4answers
42 views

Prove sequence (of integrals) is bounded.

I've got the function $f(x)=x^2-6x+8$ and the sequence of integrals $I_n=\int_3^4 (x^2-6x+8)^n dx$ $ n>0 $ I have to prove that $I_n$ is convergent. We are taught that a sequence converges ...
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0answers
34 views

Upper-bound on sum of two vectors

I have two vectors $a$ and $b$. The resultant of the two vectors is denoted as $c = a+b$. I wish to find the upper bound of $\mid c \mid$ in terms of $\mid a \mid$. Suppose $\phi$ is angle between $a$ ...
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2answers
39 views

Is $\frac{y-2}{x^2+(y-2)^2}$ bounded?

Is $$\frac{y-2}{x^2+(y-2)^2}$$ bounded between $[-1,1]$? I do not think so, but I am not able to prove that. Maybe with a counterexample? Thanks!
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4answers
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With $\alpha>0$, showing that $\int_0^1 t^\alpha(1-t^2)^kdt$ goes to $0$ as $k\to+\infty$

My idea was to compute the derivative of the integrand, in order to find its maximum in $[0,1]$. It turns out it is attained at $t=\sqrt{\frac\alpha{\alpha+2k}}$. Then the claim follows from $$0<\...
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2answers
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Bound of derivative of Lipschitz continuous function

I have a function $f(x) = g(x)^Tx$, where $f: \mathbb R^n\to \mathbb R$, $g(x):\mathbb R^n \to \mathbb R^n$, $x\in \mathbb R^n$. Furthermore assume $g(0) = 0$, $\|g(x)\|$ is bounded and also the ...
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1answer
63 views

Sum of alternating ratio of combinations

Let $0 < \ell \leq b \leq a$. Let $a > b+\ell$. I am trying to upper bound the following expression such that the upper bound evaluates to $\leq 1$ (clearly 1 is an upper bound but i want ...
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2answers
172 views

Show that $\int_0^2 \int_0^2 \frac{x}{1+\ln(x^2y^2)} \,\mathrm{d}x \,\mathrm{d}y \leq 4$

Show that $$ \int_0^2 \int_0^2 \frac{x}{1+\ln(x^2y^2)} \,\mathrm{d}x \,\mathrm{d}y \leq 4. $$ When I think somewhat outside the box, I can see that when I sketch the boundaries in a one-...
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0answers
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Upper bound for the eigenvalues of a matrix

$A_n$ is a real symmetric $n \times n$ matrix defined by $$ A_n = \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n} \\ \frac{1}{2} & \frac{1}{2} & \...
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1answer
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Boundedness of a subset using boundedness of linear functionals

Let $S\subset X$ be a subset of a normed linear space such that $\sup_{x\in S} |f(x)|<\infty$ for all $f\in X^*$, the continuous dual of $X$. Prove that the set $S$ is bounded. By definition $S$ ...
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26 views

Estimating upper bounds of coloring problem in integer grid

The following is a "proof" of this problem: We color a grid with integer coordinates with $k$ colors. For which $k$ can we always find a monochromatic isosceles right triangle with its legs parallel ...
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26 views

Upper and Lower bound for LogIntegral function of inverse $x$

I know that $\frac{1}{x \ln x} \left(-\frac{1}{\ln x} +\frac{2}{\ln^2 x} -\frac{7}{\ln ^3 x} \right) \leq -li(\frac{1}{x})-\frac{1}{x \ln x} \leq \frac{1}{x \ln x} \left(-\frac{1}{\ln x} +\frac{2}{\...
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1answer
29 views

Tools for finding bounds on power series

Suppose someone hands you a series where the terms are some function of x and your goal is to find some bounds for the series for a given set of x-values (I'm thinking of power series in particular). ...
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1answer
21 views

Off-diagonal part of PSD matrix Eigenvalue Bound

Let $A\succeq 0$ and $M = A - \operatorname{diag}(A)$ be $M$ modified by setting the diagonal terms to zero. While $\operatorname{diag}(A)\succeq 0$, $M$ need not be positive or negative definite (...
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0answers
49 views

A sum that includes power of binomials: Possible limit?

I have the following sum: $$ \sum_{k=1}^{V} (-1)^{k-1} \frac{{V \choose k}^A}{{DV \choose k}^{A-1}} $$ Is there an approximation available for this sum? I computed this sum with python for different ...
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1answer
29 views

Lower bound on Sum of squared differences

Consider $n+2$ real numbers $x_i$ with $0 \leq x_i \leq \frac{1}{2}$. Additionally, not all $x_i$ are the same. Now define two quantities $\Phi = 4\sum_{i=0}^{n+1}x_i^2$ $\Phi' = \sum_{i=1}^{n}(x_{...
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1answer
63 views

A lower bound for $(1-e^x)^n$

I want to find a lower bound for $$(1-e^x)^n$$ $n$ integer, $x$ real, and $1-e^x\geq 0$. One lower bound is (Bernoulli's inequality) $$(1-e^x)^n\geq 1-ne^x$$ But I need a tighter lower bound that is ...
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1answer
47 views

Bounded operators versus bounded sequences

Let $X$ be a Banach space, $Y$ normed, $A_n\in B(X,Y)$. Prove that $(\|A_n\|)$ is bounded if and only if for all $x\in X$ and for all $f\in Y^*$: $(|f(A_n(x))|)$ is bounded. Any hints on that? $A_n$ ...