winger

s = input()

prevs = dict()
a = []
for i in range(len(s)):
    if s[i] in prevs:
        a.append(i - prevs[s[i]])
    else:
        a.append(i + 1)
    prevs[s[i]] = i

winger

root = "cargo"

[packages]

[packages.bitflags]
dependencies = []
path = "/Users/winger/.cargo/registry/src/github.com-1ecc6299db9ec823/bitflags-0.1.1"
version = "0.1.1"

[packages.cargo]

winger

#pragma once
#include <iostream>
#include <algorithm>
#include <vector>
#include <unordered_map>
#include <string>
#include <variant>

using Key = std::string;
using Value = std::variant<std::string, double, bool>;

winger

import torch
import numpy as np
from matplotlib import pyplot as plt

torch.set_printoptions(precision=10)

class ResidualBlock(torch.nn.Module):
    def __init__(self, dims, bottleneck):
        super(ResidualBlock, self).__init__()
        self.linear1 = torch.nn.Linear(dims, bottleneck)

winger

import Mathlib.Data.List.Sort
import Mathlib.Data.List.Basic
import Mathlib.Data.Nat.Digits.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Finset.Sort
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.List

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import Mathlib.Data.Nat.Basic
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Order.Filter.AtTopBot.Basic

winger

A Gaussian Lower Bound for a Binomial Right Tail

We show that for all integers $k \ge 1$ and all $p \in (0, \tfrac{1}{2})$, if

$$n := 2k,\qquad \sigma^2 := p(1-p),\qquad z := \frac{(1/2 - p)\sqrt{2k}}{\sigma},$$

then

$$1 - \Phi(z) + \frac{1}{2} \binom{2k}{k},\sigma^{2k} ;\le; \mathbb{P}\bigl(\mathrm{Bin}(2k,p)\ge k\bigr),$$