# Questions tagged [upper-lower-bounds]

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

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### Variational Lower Bound with latent SDE

In this paper https://arxiv.org/pdf/2007.06075.pdf, the authors give a formula in equations 13 and 14, for the ELBO for a specific VAE (latent variable governed by an SDE) that I have difficulty ...
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### Difficulty understanding some inequalities in S,A,I,R disease modelling

Consider this paper (reading the entire pdf is not required) on disease modelling. There are four real functions of interest, based on constant positive parameters $\mu,\nu,\delta,\beta$ and four ...
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### Number of elements in $Z^n$ with norm 2 less than some positive B [duplicate]

Is there any result or tight bound on the cardinal of : $\{\textbf{z}\in\mathbb{Z}^n / \lVert\textbf{z}\rVert_2 \leq B\}$ for some positive $B$. Did not find any topic on this, sorry if it is a dupe.....
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### Prove $\sum_{x\le Q} e(\alpha x)\ll \lVert \alpha\rVert^{-1}$ for $\alpha\not \in\mathbb Z$

I just want to know the proof that $\left\vert\sum_{x\le Q} e(\alpha x)\right\vert\ll \lVert \alpha\rVert^{-1}$ for $\alpha\not \in\mathbb Z$, for any positive integer $Q$. Here $\lVert x\rVert$ means ...
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### The lower bound about the binomial coefficient

From easy computation we can get \begin{align*} {n\choose{\ell}}=\frac{n}{\ell}\cdot\frac{n-1}{\ell-1}\cdots\frac{n-(\ell-1)}{1}\geq\frac{n^{\ell}}{\ell^{\ell}}, \end{align*} where the last inequality ...
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### Is a continuous function on an open or semi-open interval bounded?

Per sagemath.org: A function is said to be continuous on an interval when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks. If some function f(...
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### Looking for upper bound for a sum over compositions

Consider the matrix \begin{bmatrix} 1 & \alpha_1 & 0 & \dots & & & 0 \\ \alpha_1 & 1 & \alpha_2 & 0 & \dots & & 0 \\ 0 & \alpha_2 & 1 & \...
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### Bounding spectral radius of special matrix (extension of the extension)

This is an extension of Bounding spectral radius of special matrix (extension), which has been already solved. Let $A$ be an $n \times n$ matrix with all nonnegative entries and row sums strictly less ...
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### Thighter bound than Chebyshev for the error of an estimator

I have seen Chebyshev inequality applied to bound the error of an estimator wrt to its expected value. For example, the estimator of the mean of a set of values after they have been protected with a ...
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### Algorithmic Fine-Grained Space Lower Bounds

Given an algorithmic problem, theoretical computer science has powerful tools in order to give lower bounds on the number of required computation steps, based on the strong time hypothesis (SETH). For ...
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### Finding "growth" rate of a sum

Question Let $n$ be a positive integer. The sum in question is: $$S_n=\sum_{i=1}^{ n}\frac{1}{2^i}\bigg(1-\frac{1}{2^{i}}\bigg)^{n}.$$ Clearly this sum is convergent, and it seems like the sum goes to ...
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### The additive Chernoff bound for the absolute value.

I am trying to derive a generic, additive Chernoff bound for $\Pr[|X-\mu|\leq a]$ with $a>0$. By generic I mean a Chernoff bound in terms of the moment generating function instead of assuming a ...
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### Determine whether the family $\mathcal{F}=\{f \text{ holomorphic in } \mathbb{D} : f(0)=1 \text{ and } \mathfrak{R}f(z)>0 \}$ is normal

Let $\mathcal{F}=\{f \text{ holomorphic in } \mathbb{D} : f(0)=1 \text{ and } \mathfrak{R}f(z)>0 \: \forall z \in \mathbb{D}\}$. I want to determine whether this is a family of normal functions. In ...
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