Questions tagged [upper-lower-bounds]

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

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How small is $\sum_{d \mid p_n\#}\mu(d)\sum_{r^2 = 1 \pmod{p_{n+1}d}}\frac{(x - r) \pmod {p_{n +1}d} + 1}{p_{n+1}d}$?

Rough Conjecture: Define $f(x) = \sum_{d \mid \sqrt{x + 1}\#}\mu(d)\sum_{r^2 = 1 \pmod d} \frac{(x - r) \pmod d}{d}$ where the modulus operation takes the least non-negative residue in $\Bbb{Z}$. ...
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Bound on the covariance of a sum

Problem Consider $N$ random variables $X_1,\, \dots,\, X_N \in \mathbb{R}^d$. They are assumed zero-mean, i.e, $\text{E} \left(X_i\right) = 0$, and their covariances are denoted $\text{E}\left[X_i X_i^...
Tasty's user avatar
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Lower bound for the formula

Is there any formula to find the lower bound or the approximation of the below? $$ \left(\frac{y+\frac{k}{y}}{x+\frac{k}{x}} \right)^2 - \left(\frac{y}{x}\right)^2 $$
hlm's user avatar
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Control the growth of an infinite series as a function of a parameter

I would like to control the growth of the series $$ \sum_{n=1}^{\infty} \frac{r^n}{\sqrt{n}} $$ as $r\nearrow 1$; i.e. a lower bound of the form $\sum_{n=1}^{\infty} \frac{r^n}{\sqrt{n}} \geq C f(r)$ ...
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3 answers
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Is the state norm of an asymptotically stable linear system always bounded?

Suppose I have the dynamical system $$x_{i+1} = A_{i+1} x_{i}$$ with state vector $x \in \mathbb{R}^{n}$. If its given that $\lim_{i \to \infty} \Vert x_{i} \Vert = 0$, does this imply $\Vert x_{i} \...
Bart Wolleswinkel's user avatar
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bounding the tail of $p$-series with trig factor

The tail of $p$-series ($p > 1$) can be upper bounded by (from e.g. here): \begin{equation} \sum_{j=n}^\infty \frac{1}{j^p} = O(n^{1-p}). \end{equation} I'm interested in finding the upper bound of ...
user185671631's user avatar
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finding a tight scaling bound (in terms of the Big-O notation) of a function of an infinite sum of $1/n^2$.

I have a real-valued function $f(x)=\sum_{n=x+1}^\infty \frac{1}{n^2}$ where $x \in \mathbb{N}$. I want to understand how $f(x)$ scales with respect to $x$. One thing I tried is as follows: Since it ...
Mohan's user avatar
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Upper bound on expectation given probability [closed]

Let $X$ be a positive random variable. I know that $\mathbb{P}[X \leq a] \geq q$. Any hints about how to find an upper bound on $E[X]$ in terms of $q$ and $a$? Using Markov inequality, I can obtain $E[...
Carlos Murguia's user avatar
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Lower bound on a term involving positive definite matrix

If $A$ is a positive definite matrix, can we derive a lower bound for the term $x^T A y$, where $x, y$ are two vectors of the following form: $$(x-z)^T A (x-y) \geq \alpha (x-z)^T (x-y), $$ where $\...
Ki Chao's user avatar
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2 answers
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Upper and lower bounds for $x/\ln x$

It is well known that $$ (x-1)-\frac{3}{2}\left(x-1\right)^{2}<\frac{\ln x}{x}<x-1 $$ for every $x>1$. This inequalities are good around $x=1$. I found that $$ \frac{2x}{x^2+1}<\frac{\ln x}...
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Smallest prime of the form $\frac{a^n + 1}{a+1}$ has $ 1 < n < a + 2$?

Consider primes of the form $$\frac{a^n + 1}{a+1}$$ for integer $a>1$ and integer $n>1$. Conjecture : (for any fixed $a$) The smallest prime of the form $\frac{a^n + 1}{a+1}$ has $ 1 < n < ...
mick's user avatar
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Lower and upper bounds on conditional expected value.

Suppose we have a random variable $X$ which can take negative values. I am wondering if it is possible to find exact or high-probable lower and bounds for $\mathbb{E}(X|X\ge a)$ based on $a$, $var(X)$ ...
Amin's user avatar
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"Nice" form/bounds for $\ln\prod_{i=1}^n \left(1 + \frac{x_i^2}{n}\right)$

Does the expression $$\ln\prod_{i = 1}^n\left(1 + \frac{x_i^2}{n}\right)$$ have either a "nicer" closed-form, or quantitative upper/lower bounds on such a form? For me, "nice" ...
Mark's user avatar
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2 votes
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Lower Bound of a Function for proving an Inequality

Actually, I am trying to solve a mathematical problem, which involves proving an inequality. For that I already know the bounds of LHS of the inequality and now I need the lower bound of the RHS of ...
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Seeking Lower Bound for Partition Probability in Random Variable Analysis

I am reaching out to seek assistance with a probability problem involving random variables. For each $p$ in $[1,\infty)$, consider positive random variables $X_{1,p}, X_{2,p}, \ldots, X_{n,p}$ such ...
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Bounds on solutions to $a\alpha-b\ln{\Gamma(\alpha)}-c=0$

What would give tight upper and lower bounds to the two $\alpha$ ($\alpha\in\mathbb{R}^+$) solutions of the following equation? $$a\alpha-b\ln{\Gamma(\alpha)}-c=0$$ Background I am working with a ...
jblood94's user avatar
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What is the lower bound of $\max\{a+b,c+d\}$? [closed]

We know that for any $a,b,c,d\in\mathbb{R}$, $\max\{a+b,c+d\}\leq \max\{a,c\}+\max\{b,d\}$. reference My question are (1) how tight is this inequality? and when does $=$ holds? I think it holds when $...
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Upper bound for the sum of the multiplication of adjacent items of a list

If $\sum_{i=1}^n x_i = X$, where $X$ is a positive constant and $\forall i, x_i$ is a positive integer, then what is the upper bound of $\sum_{i=1}^{n-1} (x_i x_{i+1})$? Intuitively, set $ x_1=\dots=...
white's user avatar
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Using the average of unknown bounded numeric data to find tighter bounds on the data [closed]

Some playing around with Desmos graphing calculator suggests that, given bounded numerical data whose actual values we do not know, we can derive tighter bounds on the data depending on the value of ...
FabrizzioMuzz's user avatar
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Maxima of the Sum of Divisors Function and Upper Bound of a similar Ratio

If we define a function (aka Gronwall’s function) as: $$F(n)=\frac{\sigma(n)}{n \log \log n}$$ Then for $n>15$, it does have an upper bound. I want to know what's that specific upper bound is? Also ...
Ok-Virus2237's user avatar
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Prove this property of rational upper bounds of $\{x\in\Bbb Q\mid x^2<2\}$ [duplicate]

Prove or disprove that if $\frac ab$ ($a$, $b\in\Bbb Z$) is in upper bound of $\{x\in\Bbb Q\mid x^2<2\}$, then $a^2>2b^2$. Do not introduce real numbers yet. I suppose this is true, but it ...
youthdoo's user avatar
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Upper bound on continuous bounded function with average value 1

Suppose $0 < r \leq a(t) \leq R < \infty$ is a continuous function on $[0, 1]$ and consider $$w(t) = \frac{1}{a(t)} \left( \int _0 ^1 \frac{1}{a(s)} ds \right)^{-1}.$$ Are there any attainable ...
A. L.'s user avatar
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Prove that $f(x,y) = \frac{h(xy)}{yh(x) + xh(y)} < 1$

I want to prove that $f(x,y) = \frac{h(xy)}{yh(x) + xh(y)} < 1$ in $(x,y) \in (0,1)^2$, where $h(x) = -x\log_2x - (1-x)\log_2(1-x)$ is the binary entropy function. Proving that it is bounded by ...
Nikita Dezhic's user avatar
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1 answer
51 views

Upper bound on $\sum_{k=n}^\infty \frac{1}{k(k - c)}$

I know that $\sum_{k=n}^\infty \frac{1}{k^2} < \frac{1}{n-1} = O(\frac{1}{n})$. I confronted a similar situation but now I have some finite constant $c \in \mathbb{R}$ in the denominator: \begin{...
Hailey Han's user avatar
5 votes
1 answer
90 views

Taking the limit in Holder's inequality

I have a standard normal random variable $X\sim\mathcal{N}(0,1)$ and an event $E$ with $\mathbb{P}(E)=p$, and this event is about $X$ and some other variables. I am interested in upper-bounding the ...
ba029188's user avatar
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2 answers
75 views

How to Prove n! is unbounded from above [closed]

I want to show that n! is an unbounded sequence from above. I have that the definition of an unbounded sequence is-- A sequence Xn is unbounded from above if for all m ∈ R, there exists an n ∈ N s.t. ...
math gurl's user avatar
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44 views

Upper bound of diagonal matrix multiplied with Hermitian matrix

Suppose that ${\bf R} \in \mathbb{C}^{n \times n}$ is a Hermitian matrix, and ${\bf D}$ is a diagonal matrix with main diagonal being ${\bf d} \in \mathbb{C}^{n \times 1}$. I am looking for the ...
H. H.'s user avatar
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1 vote
2 answers
34 views

Finding the bounds (if they exist) and the maximum and minimum of the group

I'm trying to find the supremum and infimum of a group $C={\{\frac pq| p \in Z , 1 \le q \in N, {p^2}\le 2 {q^2}\}}$ I started by using the ${p^2} \le 2{q^2}$ and got to $-\sqrt2 \le \frac pq \le \...
S. M's user avatar
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Let $f :$ [0:1] $\to \mathbb {R}$ continuously differentiable. Prove some inequation. [duplicate]

Let $f :$ [0:1] $\to \mathbb {R}$ continuously differentiable. Prove: $|\int_0^1 f(x)dx - {1 \over n} \sum_{k=0}^{n-1} f({k \over n})| \leq {M \over n}$ with $M = \int_0^1|f^{'}(x)|dx$. So I have ...
MathStudent101's user avatar
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1 answer
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Count bounded integers with bound on sum

Given two integers $x$ and $y$, each with lower and upper bounds ($x_{lb} \leq x \leq x_{ub}$ and $y_{lb} \leq y \leq y_{ub}$), count how many pairs have sum between $s_{lb}$ and $s_{ub}$. Of course ...
Stevineon's user avatar
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0 answers
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Proving an upper bound of a recurrence

So I want to prove that the recurrence $H(n)=\sqrt{n}H(\sqrt{n})+\log_2 n$ is bounded above by $O(n)$ or there exists $ c\in \mathbb{R}^{+}$: $$H(n)=\sqrt{n}H(\sqrt{n})+\log_2 n\leq cn$$ Now I couldn'...
shinny.dogma's user avatar
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1 answer
47 views

Tight Bound for log mean exponent under constraints

Given $m \leq x_1,\ldots,x_n \leq M$, what is a good upper bound for $n\log\left(\frac{\sum_i e^{x_i}}{n}\right)$? My approach: $e^{x_i} \leq 1 + \frac{e^M-1}{M}x$ for all $x \leq M$. Using this, we ...
Black Jack 21's user avatar
2 votes
1 answer
52 views

Proving that this sigmoid sequence converges

The problem Given $h_0=0.6$ I want to prove that the following iterative sequence converges. I also want to find the value it converges to. $$h_{t+1}= \sigma(3h_t-1)=\frac{1}{1+e^{-(3h_t-1)}}$$ My ...
John Katsantas's user avatar
3 votes
2 answers
131 views

Bounding $\|(I-A)^{-1}\|_2$ for $\rho(A)<1$

I have a large, right sub-stochastic, sparse matrix with spectral radius $\rho(A)<1$. I'm attempting to bound the spectral norm of $(I - A)^{-1}$ via its Neumann series, $$\|(I-A)^{-1}\|_2=\Big\|\...
Set's user avatar
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Upper bound for the expectation of a random variable times an indicator function

Suppose we have a random variable $X$. Is there any way of deriving an upper bound of the following expectation: $$E[X * \mathbf{1}_{X\ge x_0}],$$ where $\mathbf{1}_{()}$ is an 0-1 indicator function, ...
Vergil's user avatar
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Minimum number of edges for a tree that joins the $27$ nodes of a $3 \times 3 \times 3$ regular grid

In 2014, Dumitrescu and Tóth (see Covering Grids by Trees, Figure 2) proved the existence of an inside-the-box tree consisting of $13$ connected line segments covering all the $27$ nodes of the ...
Marco Ripà's user avatar
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1 vote
1 answer
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Decay bound on $\int_0^1 (1 - x^r) \cos(kx) dx$.

For $r \in (0, 2)$, I would like to estimate the decay of $f(k) = \int_0^1 (1 - x^r) \cos(kx) dx$ as $k \to \infty$. Indeed, according to Riemann–Lebesgue lemma, $\lim_{k \to \infty} f(k) = 0$. For $r ...
jvc's user avatar
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Calculation for a lower bound for the amount of primes

I know there are several bounds for $\pi(x)$, which is the prime counting function. But for my thesis I want to calculate a very rough estimation. I'd appreciate any feedback on my calculation. So I ...
Lereu's user avatar
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Upper bound for the largest coefficient of the Legendre Polynomial $2^n\sum_{k=0}^n x^k {n\choose k} { \frac{n+k-1}{2} \choose n}$

The legendre polynomial can be defined as $$P_n(x)=2^n\sum_{k=0}^n x^k {n\choose k} { \frac{n+k-1}{2} \choose n}.$$ For example: $$P_5(x)=1/8(63x^5-70x^3+15x)\\ P_6(x)=1/16(231x^6-315x^4+105x^2-5).$$ ...
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Boundedness of $\int_a^b e^{iF(x)} dx$ for $F''(x) \ge r > 0$

My question is regarding the last sentence of Lemma 4.4. of Titchmarsh's book The Theory of the Riemann Zeta-Function: First I approached by taking three intervals as the text has done but for new $\...
Ali's user avatar
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1 vote
1 answer
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Does this series converges fast enough to $e$ in order to guarantee this upper bound?

Define the integral $I_n$ as $$I_n = \int_0^1 \frac{d^n}{dx^n} \frac{(x-x^2)^n}{n!} e^xdx = a_ne+b_n $$ where $a_n$ and $b_n$ are integers. We can integrate $I_n$ by parts $n$ times $$I_n = (-1)^n \...
Pinteco's user avatar
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2 votes
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Estimate for a second order non-linear ODE

I am considering the following non-linear ODE \begin{cases} \ddot y(x)\left(\ln(x) - 2\ln(y(x))\right) - 2\frac{(\dot y(x))^2}{y(x)} = 0 &\text{in }[0,T]\\\\ y(0) = 0\\\\ \dot y(T) = c \end{cases} ...
Falcon's user avatar
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2 votes
2 answers
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Understanding Rudin's reasoning on why $|P(re^{i\theta})| > |P(0)|$

I am taking a stroll through Chapter 10 of Big Rudin, and I am stuck on the fundamental theorem of algebra: 10.25 Theorem If $n$ is a positive integer and $$ P(z) = z^n + a_{n-1}z^{n-1} + \cdots + ...
Atom's user avatar
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Limit of a function with an integral to have a variable in the upper bound

I am a condensed matter physicist and I am studying a function called Kubo-Toyabe function, the function depends on time $$t$$ and written as follows $$ P_{\mu}^{LF}\left(t\right)=1-\frac{2\Delta^{2}}{...
Jogja_papua's user avatar
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25 views

Optimal Piecewise Linear Upper Bound for a function

I am looking for the optimal piecewise linear upper bound for a single variable function $f(x)$ - specified as a list of $(x,f(x))$ pairs. So, I have $(x_1,f(x_1)),(x_2,f(x_2)), \ldots, (x_n,f(x_n))$. ...
dexter04's user avatar
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Showing bound - probability question

From Stirzaker's elementary probability Chapter 1, Q21 c) There are $x\geq 2$ red balls and $y\geq 1$ yellow balls in an urn. Two balls are drawn without replacement. Let $p$ be the probability that ...
b.b.89's user avatar
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Dirichlet's test with unimodal coefficients

Briefly: If we modify the hypotheses of Dirichlet's test to require a unimodal sequence of coefficients, not necessarily a monotonic sequence, then do we still get the same quantitative bound on $\sum ...
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Loop Bounds vs. Iteration Domain in Polyhedral optimization

Context: I was reading a tutorial on polyhedral optimization. But got confused while trying to translate the iteration domain (i.e. loop bound) to set builder notation. Problem Description: A code ...
F.C. Akhi's user avatar
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How do I finish proving that this set does not have a lower bound?

I am having some troubles on a real analysis exercise where I had to prove that this set has no lower bound. Here's the problem and how I tackled the problem but I don't know what to do next to prove ...
The Hokkanen's user avatar
2 votes
1 answer
361 views

Sharp bounding of a sum involving Möbius function

I am trying to bound as sharply as possible the partial sum $$S(n)=\sum_{k=1}^{\frac{n}{p_{\pi\left(\sqrt{n}\right)}}} \mu(k) \left(\pi\left(\frac{n}{k}\right) + f(n,k)\right)$$ Where $\pi(x)$ is the ...
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