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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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Bounded from above integral with exponential

I am trying to bound from above the following integral $\int_{\left\{|u| \geq u_n\right\}} u^{4} e^{-\frac{u^{2}}{2n}} du$ where $u_n = 2 \sqrt{nlogn}$. Could you please give me any idea how I could ...
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Bound on the degree of a polynomial solution to a parametrized equation

Let $K$ be a field, $F = K(T)$ a rational function field over $K$. Let $G \in F[C, X_1, \ldots, X_n]$ be a polynomial with coefficients in $F$. We can consider the polynomial $G(c, X_1, \ldots, X_n)$ ...
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38 views

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that?

For three positive numbers verifying $xyz=2+x+y+z$, is there an upper bound on $x+y+z$? And, if so, which is that? I've managed to find only a lower bound, from $ xyz=2+x+y+z \le \frac{(x+y+z)^3}{27}...
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Partition a square with convex polygons. What is the maximal number of edges?

Given a square that is partitioned into convex polygons such that $n$ regions are created. What is the maxmimal number of edges in such a partition? Example. The two squares below both have $n=4$ ...
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26 views

Upper and lower bounds for a finite sum [duplicate]

Find upper and lower bound for the following finite sum: $\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$ My attempt: Using the integral test: we know that $\frac{1}{1} + \frac{...
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32 views

Upper and Lower bound of a finite sum

Find upper and lower bound for the following finite sum: $$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3} $$ My attempt is: Using the integral test: we know that $\frac{1}{1} + ...
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1answer
57 views

find the upper and lower bound for a finite sum

Find upper and lower bound for the following finite sum $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ My attempt: $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
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38 views

Quasi homomorphism from integers to reals.

Let $f:\mathbb{Z}\rightarrow \mathbb{R}$ be a quasi-homomorphism, i.e $|f(a+b)-f(a)-f(b)|\leq D$ $\forall$ $a$ and $b$ in $\mathbb{Z}$ ($\mathbb{R}$ and $\mathbb{Z}$ are here considered as additive ...
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finding upper and lower bounds of finite sums

Find upper and lower bound for the following finite sum $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ My attempt: $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
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28 views

Find upper and lower bounds for the finite sum

Find upper and lower bound for the following finite sum: $\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$ My attempt is: Using the integral test: we know that $\frac{1}{1} + \...
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1answer
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Upper and lower bounds for series and sequences [on hold]

Find upper and lower bounds for the following finite sum: $$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}$$
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A lower bound for Holder's inequality and an upper bound for the reverse Holder inequality!

I know that Holder's inequality already states, for non-negative sequences ${a(n)},{b(n)}$, that $$((∑a(n)b(n))/((∑a^{p}(n))^{1/p}(∑b^{q}(n))^{1/q}))≤1$$ where $p>1$ and $(1/p)+(1/q)=1$ and ...
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I want to prove that two norms are equivalent but I am struggling with an upper bound

I want to compare the usual norm on $L^2(-1,1)$ with the following: $$ \Vert f \Vert_H^{2} = \int_{-1}^1 \vert f(x) \vert^2 \frac{1}{1+x^2}dx $$ Now, for sure I have this:$$\Vert f \Vert_H \leq \...
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Asymptotics of the reciprocal Riemann Zeta Function

Assuming Riemanns hypothesis, I would like to obtain an upper bound on $$\left|\frac{1}{\zeta(\sigma+it)}\right|$$ for large $t$ and fixed $\sigma$. I believe it should be easy to show that it ...
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29 views

Log det of covariance and entropy

I understand log of determinant of covariance matrix bounds entropy for gaussian distributed data. Is this the case for non gaussian data as well and if so, why? What does Determinant of Covariance ...
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Bound for type of correlation measure

Assume you have a finite, discrete probability distribution for a joint random variable and such that $P(X=i,Y=j) = p_{i,j}$ for $i \in \{1, \dots, |X|\},j \in \{1, \dots, |Y|\}$. The marginal ...
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The classification of and description of the root near $1/2$ of $x^d +x^\left({d-1}\right) + x^\left({d-2}\right) + \dots + x^2 + x - 1=0$

In Arturas Dubickas paper "On the number of reducible polynomials of bounded naive height", manuscripta math. 144, 439–456 (2014) he discusses a bounding polynomial $x^d + x^\left({d-1}\right) + x^\...
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Bounding the dot product of two planar unit vectors.

Does there exist a continuous, monotone increasing function $f\colon[0,2]\to [0,1]$, satisfying $f(0)=0$ and $f(1)=1$, such that for all vectors $(a_1,b_1),(a_2,b_2)\in \mathbb{R}^2$ of unit length, i....
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How do I find the lower bound of $a_n:=n+\frac{100}{n}$ without inserting values?

How do I find the lower bound of $a_n:=n+\frac{100}{n}$ without inserting values? I already found out, that the sequence has no upper bound, because $\lim\limits_{n\to \infty}a_n\to\infty$. However, ...
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1answer
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Show $(N, ≤)$ is a partial order with least upper bounds (lubs) and greatest lower bounds (glbs) of all pairs.

does anyone have any solution or a good hint? Let $(\mathbb{N}, ≤)$ be the set of natural numbers with the relation $m ≤ n$, meaning $m$ divides $n$. Show $(\mathbb{N}, ≤)$ is a partial order with ...
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Tight Upper Bound for $\sum_{i=1}^ni \log i$

Can we find the tight upper bound for $\sum_{i=1}^ni\log i$ My approach: $$\sum_{i=1}^ni\log i\leq\sum_{i=1}^n\log {i^i}=\log\left(\prod_{i=1}^ni^i\right)\leq \log n^{n^2}= n^2\log n$$ My upper ...
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Show that if a partial order has all least upper bounds, then it necessarily also has all greatest lower bounds and vice versa.

any hints or solutions for question? Show that if a partial order has all least upper bounds, then it necessarily also has all greatest lower bounds and vice versa.
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Upper and lower bounds on difference of bounded variables

I have the following equations and inequalities: $1 = A' + B'$ $1 = A + B + C$ $A \le A'$ $B \le B'$ All variables are bounded below by zero and above by one. I wonder if I can find an analytic ...
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Upper bound for a division of two integrals

Consider two integrable functions $a(x)>b(x)>0$ for all $x$ and $$f(y)=\int_0^{\infty}a(x){\rm sinc}^2[(x_0-x)y]+b(x){\rm sinc}^2[(x_0+x)y] dx,$$ $$g(y)=\int_0^{\infty}b(x){\rm sinc}^2[(x_0-x)y]+...
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Proving the upper bound of the ratio of product of odd n numbers to even n numbers?

So question demands a proof that: Let $$x_n = \frac 12 * \frac34 * \frac56 * ... * \frac{2n-1}{2n} $$ Then show that; $$x_n \leq \frac 1{\sqrt{3n +1}} $$ So essentially what I have tried to do ...
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Can these bounds in terms of the abundancy index and deficiency functions be improved for deficient-perfect numbers?

Let $$\sigma(x) = \sum_{e \mid x}{e}$$ denote the sum of divisors of the positive integer $x$. Denote the abundancy index of $x$ by $I(x)=\sigma(x)/x$, and the deficiency of $x$ by $D(x)=2x-\sigma(x)$...
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Bounding double sum off diagonal

Let $\{a_n\}_{n\in\mathbb{N}}\subset\mathbb{C}$ and $N\in\mathbb{N}$. I have to prove the following bound $$ \sum_{n\leq N}\sum_{\substack{m\leq N\\ m\neq n}}|a_ma_n|\left(\log\frac{m}{n}\right)^{-2}\...
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Error upper bound of minimax polynomial approximation over $[a,b] \cup [c,d]$

I found that Jackson's inequality in approximation theory provides us a nice upper bound on the (infinite-norm) error of minimax polynomial over an interval $[a,b]$ as noted in [1, p. 16]. Can it be ...
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1answer
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Tight upper bound for the difference between two fractions in summations

I would like to compute the tight upper bound of an error term, $\epsilon$, defined as $\epsilon = \left |\frac{1}{k}\sum\limits_{i=1}^{k} \frac{x_i}{y_i} - \frac{\sum\limits_{i=1}^{k} x_i}{\sum\...
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1answer
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Find an upper bound for $|f(x)-P_4(x)|$, for $0 \le x \le 0.4$

So I'm stuck on something that is supposed to be fairly easy... I was able to work through the Taylor polynomial which I believe to be: $$x+x^3+\frac{x^5}{2!}+\frac{x^7}{3!}+\frac{x^9}{4!}$$ But I'...
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Upper bound for the reciprocal of a sum involving Möbius function

For $ n $ an integer greater than $ 6 $ , let $ Q(n)=\prod_{p\leq\sqrt{2n-3}}p $. Which upper bound in terms of $ n $ can we get for $ (\sum_{d\mid Q(n)}\frac{\mu(d)}{d})^{-1} $?
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1answer
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Improving the bound for $\sigma(q^k)/q^k$ where $q^k n^2$ is an odd perfect number given in Eulerian form

Let $x$ be a positive integer. (That is, let $x \in \mathbb{N}$.) We denote the sum of divisors of $x$ as $$\sigma(x) = \sum_{d \mid x}{d}.$$ We also denote the abundancy index of $x$ as $I(x)=\...
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Infimum of sum of two values.

Let the function $f_{n}:\mathbb{R}\rightarrow \mathbb{R_{\ge 0}}$ be defined as $$ f_{n}(x)=\sum_{i,j=1}^{n}\frac{(-1)^{i+j}\cos(\ln \frac{i}{j})}{(ij)^{x}}\quad \forall n\in\Bbb N $$ There is given ...
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Prove that these estimates cannot be improved

I proved that for a generic function $f:I\subset \mathbb{R}\to\mathbb{R}$ that is differentiable two times in the interval $I$ (open or closed, it makes no difference), we always have: $$\sup_{x\in I}...
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Boundedness of derivatives

I'm trying to show the following fact: Let $f:\mathbb{R}\to\mathbb{R}$ a function that is differentiable $p$ times in $\mathbb{R}$, and let $M_k=\sup_{x\in\mathbb{R}}|f^{(k)}(x)|$. Suppose that $M_0$ ...
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Show that lower semi-continuous function attains it's minimum. (Proof verification) (By contradiction)

Let $f: [0,1]\to \mathbb{R}$ be a lower semi-continuous function, then $$ \liminf_{x\to a} f(x) \geq f(a), \forall a \in [0,1]$$ I have to prove that $f$ attains its minimum on $[0,1]$, that is: $\...
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Proof of a lower bound on probability

How to prove that the probability of simultaneous occurrence of more than $\frac n 2$ events from $n$ independent Bernoulli trials is greater than or equal to:$$1-e^{-2n\left(p-\frac{1}{2}\right)^2}$$ ...
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Prove $\sum 3^k = O(3^n)$

Prove $\sum_{k=0}^n 3^k = O(3^n)$. Below there is a picture from my text that contains the proof. My question pertains to the notation and/or assumptions in the proof. I don't need help with basic ...
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Upper or lower bound of Wasserstein-2 metric on Gaussian distribution

I am constructing an iterative algorithm in which the Wasserstein-2 distance metric for continuous Gaussian distributions is being used. I am trying to find a general upper or lower bound of the ...
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I heard that the empty set $\emptyset$ is bounded. But I think this statement is not correct.

In Rudin's "Principles of Mathematical Analysis", there is the following definition of bounded. Definition: Suppose $S$ is an ordered set, and $E \subset S$. If there exists a $\beta \in S$ such ...
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Worst-Case Addition to Smallest Enclosing Circle

Imagine you have a convex polygon $p_1$ and put a smallest enclosing circle $SEC_1$ around it, the lengths of each side of the polygon $l_i$ can be anything and don't have to equal each other. Now, ...
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Bounded from below implies the existence of infimum?

There exists proposition that says that every sequence bounded from above admits a supremum. Is it also correct to say that every sequence bounded from below admits an infimum? Tks
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Can this inequality regarding odd perfect numbers be improved?

Let $\sigma(x)$ denote the sum of the divisors of $x$. Denote the deficiency of $x$ by $D(x)=2x-\sigma(x)$. If $N$ is odd and $\sigma(N)=2N$, then $N$ is called an odd perfect number. Euler showed ...
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On some inequalities relating the special/Euler prime and non-Euler part of odd perfect numbers

Let $N$ be an odd (positive) integer. If $\sigma(N)=2N$ where $\sigma(N)$ is the sum of the divisors of $N$, then $N$ is called an odd perfect number. Let $I(N)=\sigma(N)/N$ denote the abundancy ...
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Bounds on a convex function

Let $f(x)$ be an increasing and infinitely differentiable strictly convex function with $f(0)=0$, it is easy to show that $f(m)-\frac{m}{n}f(n)>0 $ for any $m> n> 0$ since $\frac{n}{m}f(m)+\...
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On Basak's “Bounds On Factors Of Odd Perfect Numbers”

Let $N = q^k n^2$ be an odd perfect number given in Eulerian form, i.e. $q$ is the special/Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. In what follows, we denote the ...
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1answer
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Upper bounds for number of $3$-flags in a $(2k, k^2)$-graph $G$

Let $G$ be an arbitrary simple graph on $2k$ vertices with $k^2$ edges, where $k \geq 2$. Let $F$ be a $3$-flag, i.e., three triangles that share a single edge (this graph has 5 vertices and 7 edges). ...
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1answer
54 views

Finding a upper bound (estimation) for the infinite series: $\frac{n^5!}{(n+1)^5!}+\frac{n^5!}{(n+2)^5!}+\frac{n^5!}{(n+3)^5!}+\cdots$

I have the infinite sum: \begin{align} S_n = \frac{n^5!}{(n+1)^5!}+\frac{n^5!}{(n+2)^5!}+\frac{n^5!}{(n+3)^5!}+\cdots \end{align} I must find a upper bound for $S_n$, that is, $0<S_n<f(n)$, ...
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2answers
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Riemann Integral: prove $\{U(P,f):P$ is a partition of $[a,b]\}$ is bounded below

[Defining the Riemann Integral: ] We consider a partition, $P=\{a=x_0<x_1<...<x_n=b\}$ of $[a,b]$ and a bounded function $f:[a,b]\rightarrow\mathbb{R}$. Next, we define- $$M_i=\sup\{f(x)|x\in[...
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60 views

“Trivial” assymptotic bound in Gallagher's paper

In his paper On the distribution of primes in short intervals, right before equation 9, Gallagher states that $$\sum_{d|D}\frac{\mu^2(d)*C^{\omega(d)}}{\phi(d)}\sum_{\substack{e\gt x/d \\ \text{(e,D)=...