## description of the problem

Suppose we have an alphabet $$mathcal {X}$$, where for each letter $$X & # 39; in mathcal {X}$$ we have a corresponding distribution $$P_ {X & # 39;} in mathcal {P}$$.

A letter $$X in mathcal {X}$$ is sent through a channel, in which an opponent chooses $$k$$ random letters in $$mathcal {X}$$ (excluding the true $$X$$) to form a whole $$U = {X_1, ldots, X_k }$$and send $$N$$ random samples, taken i.i.d. of the distribution of the mixture $$bar {P} _U = frac {1} {K} (P_ {X_1} + ldots + P_ {X_k})$$, denoted $$Y_1, ldots, Y_N$$ (or $$Y ^ N$$ collectively). What is the minimum probability of error that the decoding receiver can reach?

### Target solution

As $$n rightarrow infty$$, we would expect the probability of error to be limited by $$frac {1} {| mathcal {X} | – K}$$, but I have a hard time proving that this is the case. Ideally, we would find a tight non-asymptotic bound.

### My attempt at proof

It seems like it should be a simple application of Fano's inequality. Given an estimator $$hat {X}$$ for $$X$$, we have,

$$P ( hat {X} neq X) geq 1 – frac {I (X; Y ^ N) + 1} {log | mathcal {X} |},$$

or $$I (X; Y ^ N)$$ means mutual information between $$X$$ and $$Y ^ N$$. I couldn't do better than using $$I (X; Y ^ N) leq nI (X; Y)$$and delimit mutual information in terms of maximum KL divergence per pair. Unfortunately this limits the error from zero as $$n$$ grows large enough.

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## what would be the ratio limit [closed]

I wonder what would be the answer of the following:

question 1

question 2

## A random central limit theorem

Let $$X_k$$, $$k = 1, 2, points$$, be a sequence of i.i.d. random variables with second finite moments. Also, let $$N_k geq 1$$, $$k = 1, 2, points$$, be a sequence of random variables taking integral values, such as $$lim_k N_k = infty$$ a.s .. In addition, suppose that each $$N_k$$ is independent of $$X_k$$& # 39; s.

Yes $$S_k: = sum_1 ^ {N_k} X_k$$, does that mean that $$(S_k – mu N_k) / sigma sqrt {N_k}$$ converges in the distribution to the standard normal variable
(or $$mu = mathbb {E} (X_k)$$ and $$sigma ^ 2 = mathbb {V} (X_k)$$) as $$k to infty$$?

## php – Limit number of regular expressions

If it is a question of limiting a number, the simplest would be:

``````if (\$numeo > 0 && \$numeo <= 1000) {
}
``````

You don't need a regular expression for something as simple as that, now if the goal is to apply the regular expression to another "group", you need to parse what you really want and what channel you want to match / marry, because I can give a regex answer that will actually break the logic of your existing regex that already exists.

## Limit the resolution of the image on the download file for the specific domain woocommerce

when i add a product, i have a form with two fields "upload file".
(product image field) and (product gallery image field).

when the user uploads an image for the first field (product image field), I want to limit the size of the image to (640 * 640 max).

And when it uploads an image for the second field (product gallery image field)
I want to limit the size of the image to (300 * 300 max).

Is it possible to define different sizes for two specific fields?

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## I know the limit of Euler, but how can we use it to solve this problem?

$$lim _ {n rightarrow infty} (1 + frac {2} {n}) ^ {n ^ {2}} e ^ {- 2 n}$$

## real analysis – Limit for a series of Bessel functions evaluated at zero

The following series arises in an electrostatic problem for a conducting cylinder:
$$V = sum_ {n = 1} ^ infty frac {J_0 (k_n rho) e ^ {- k_nz}} {k_nJ_1 (k_n) ^ 2}$$
or $$J_i$$ is the Bessel function $$i ^ {th}$$ order, and $$k_n$$ is the location of the $$n ^ {th}$$ zero of $$J_0$$. $$V$$ can converge for $$z> 0$$, and from numerical tests, conditionally converges $$z = 0$$ except for poles to $$rho = 0,2,4,6 …$$

Is there an analytical or asymptotic expression depending on $$rho$$ in the limit $$z rightarrow0$$, for $$rho> 2$$ in particular?

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## Limit of Python desktop applications – Software Engineering Stack Exchange

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## linux – Can I limit the data transfer on time by IP address?

I host relatively large files on my website, and it seemed to me that it would be trivial for someone to create a script that downloads them over and over again, nibbling my transfer of data and costing me money. Your typical home connection would be fast enough to cause me problems if the download was left 24 hours a day, 7 days a week, not to mention an ultra fast fiber connection or an appropriate remote server.

So I wonder if there is something available for Apache or even Ubuntu system wide which will impose restrictions by IP address? For example, 10 GB of transfer per 24 hours. When the limit is reached, the customer will receive a light page "Prohibited – quota reached" or will be refused the connection. I have looked around and I have found nothing other than various limiting solutions, which may help but will not solve the problem.

## optical – No need to turn left to get infinite focus (reach the limit of the lens)

For so-called unitary focusing lenses (many, but not all main lenses), bringing it closer to the sensor is what will make it focus endlessly, and that’s what mechanics of the lens DONE actually to focus.

Lens designs that focus only by bringing the front element or group closer to the rest of the lens can usually (if these are simple designs like tessars) be focused in the same way, the front element being as close as possible.

So-called internally focused lens designs that focus only or in addition by moving elements between the front and rear groups can also be focused this way, but this will cause problems. Most zooms are of this type, focusing them in this way can compromise the perfocality or sometimes even the quality of the image. Main designs using internal focus will most of the time have picture quality issues if the rear group is even at least the wrong distance from the sensor (ultra wide ones can give horrible results if ; they are at a fraction of mm!).