[MAST10005] 1. The Language of Mathematics

This is a personal note collection for MAST10005 (Calculus 1) from the University of Melbourne.


1.1 Mathematical Statements

  • Mathematical Statement: a sentence or expression that is unambiguously true of false.
    • e.g. let p be the statement1+1=3, p is an valid statement. p is false.
    • counter case: f(x) is continuous is not a M.S. without further information.
  • Conjunction: true for p and q. (let p and q be M.S.) Denoted as p \wedge q
  • Disjunction: true for p or q. (let p and q be M.S.) Denoted as p \vee q
  • Negation: true for not p. (let p be M.S.) Denoted as ~ p
  • Condition: let D be a set. Condition is a statement that is true or false depending on the choice of x \in D. Denoted with notation p(x).
    • e.g. Consider x \in N. Let p(x) be the condition “x is even”
  • Existential Quantifier: Let D be a set and p(x) be a condition over D. When at least onex \in D such that p(x) is true we say “there exists x \in D such that p(x)”. Denoted as \exists x \in D \; p(x)
  • Universal Quantifier: Let D be a set and p(x) be a condition over D. When p(x) is true for every x \in D we say “for all x \in D p(x)”. Denoted as \forall x \in D \; p(x)
  • Set of Numbers:
    • \mathbb{N} set of natural numbers. {1, 2, 3, 4, …}
    • \mathbb{Z} set of integers.
    • \mathbb{Q} set of rational numbers
    • \mathbb{R} set of real numbers
  • Set: A collection of unique objects.
    • e.g. {0, 0, 0, 1, 2, 7, 9} = {0, 1, 2, 7, 9}
    • Let A and B be sets. A and B are equal when every element of A is an element of B and every element of B is an element of A. Denoted A = B
  • Expressing Sets: Let D be set and let p(x) be a condition over D. The set of elements of D for which p(x) is true is denoted {x \in D | p(x)} (set-builder notation)
  • Subset: Let A and B be sets. When every element of A is an element of B, A is a subset of B. Denoted A \sqsubseteq B
  • Union: Let A and B be subsets of a set U. Union of A and B is the set of elements that are in at least one of A and B. Denoted A \cup B. That is: x \in A \cup B \; if \; and \; only \; if \; x \in A \; or \; x \in B
  • Intersection: Let A and B be subsets of a set U. Intersection of A and B is the set of elements that are both in A and B. Denoted A \cap B. That is: A \cap B = \{x \in A \cup B | x \in A \; and \; x \in B\}
  • Empty Set: \emptyset = \{\}
  • Cartesian Product: Let A and B be sets. Cartesian product of A and B is the set of all poosible ordered pairs we can build using elements of A as the first element and elements of B as the second element. Denoted as A \times B = \{(a, b)|a \in A, b \in B\}
    • e.g. \{0,1\} \times \{u,v,w\} = \{(0,u),(0,v),(0,w),(1,u),(1,v),(1,w)\}

为”不兼容”的设备安装Windows Subsystem for Android

最近微软正式推出了Android的子系统,尝试在自己的6代i3机器上安装却被告知不兼容。一番搜寻后寻找出如下解决方案。

首先去这个网站

https://store.rg-adguard.net/

输入WSA的商店地址,https://www.microsoft.com/en-us/p/windows-subsystem-for-android-with-amazon-appstore/9p3395vx91nr

并选择Slow(如图)

进入后下载Microsoft.UI.Xaml和WSA本体。其中Xaml视自己的架构下载:

下载后缀名为appx的文件

最后下载页面内最大的文件:WSA本体

下载完成后先去Microsoft Store更新电脑安装的所有应用,完成后双击安装Microsoft.UI.Xaml。

最后以管理员运行终端,输入 Add-AppxPackage WSA文件路径 ,回车即可安装。WSA文件路径为以msixbundle结尾的WSA安装文件

Zuk Z2 Plus 退出FFBM模式

今天不小心使手中的Zuk z2进入了FFBM模式,怎样操作都无法退出。搜寻一番后找到如下解决方案:

  • 同时按住音量+/音量-/电源键
  • 待手机震动后松开电源键
  • 继续按住两个音量键,直到菜单出现
  • 选择进入bootloader,然后选择重启

记一次显卡黑屏排查及解决过程

去年在闲鱼收了一块980ti打算换掉原来的RX470,当时买回来就有个奇怪的问题:只插一张980ti电脑点不亮,必须同时插470(甚至无需通电)才能正常开机。因为不是太影响使用,就也没太在意。

最近显卡涨价打算把980ti卖掉,突然想起还有这问题。出于销售的角度考虑,再次复现了只插一张卡的场景,问题依旧。

太奇怪了,为何一定要多插一张不通电的卡才能工作…尝试了彻底卸载AMD显卡驱动、重置主板,均无效。

最后尝试把980ti插到第二条pcie x16槽(速度是x8的)上,问题解决,可以正常开机了。

这下真相大白,原来这980ti只能运行在pcie x8的速度下…那为什么之前多插一张470就能运行了呢?查阅主板参数后发现,当x8的槽插入东西时,x16的槽自动降速为x8,所以980ti才能正常工作…

解决Windows 10卡任务栏小图标/窗口控件

最近电脑遇到很奇怪的问题:每次开机几分钟后便会出现各种操作问题:很多窗口的控件无法点击/点击后需要十几秒才有反应(包括任务管理器)、任务栏的更多图标全部变为空白。

一番折腾后找到如下解决方案:

打开任务管理器,结束桌面窗口管理器

此时便会恢复正常,但考虑到可能有系统文件损坏。修复损坏文件:以管理员运行Powershell后输入 Sfc /scannow 即可。

修复效果

尝试在Windows 10下使用SkyAR更换天空

最近了解到这个有趣的项目,遂尝试自己部署并记录下其中遇到的一些坑。

项目地址:https://github.com/jiupinjia/SkyAR

训练模型:https://drive.google.com/file/d/1COMROzwR4R_7mym6DL9LXhHQlJmJaV0J/view?usp=sharing


首先clone项目到本地,并下载模型文件。将模型带目录解压到项目目录内。

安装Python3, Anaconda:https://www.anaconda.com/products/individual#windows

安装项目目录内Requirements.txt内的依赖,但有几个需要注意/单独安装:

  • pytorch 使用conda安装,11.0是cuda的版本: conda install pytorch torchvision torchaudio cudatoolkit=11.0 -c pytorch
  • numpy 只能安装1.19.3,否则会报错: pip3 install numpy==1.19.3
  • 不装opencv-python,只装opencv-contrib-python: pip3 install opencv-contrib-python

安装完成后执行 python .\skymagic.py --path .\config\config-canyon-district9ship.json 即可看到效果。

附一个自己拍的视频,效果貌似不是很好…?

Ubuntu 20.04编译安装Deepin深度显卡驱动管理器

获取源码:https://github.com/linuxdeepin/deepin-graphics-driver-manager

安装依赖:

sudo apt install cmake qttools5-dev-tools qtbase5-dev freeglut3-dev libdtkwidget-dev libpci-dev 

编译:

cmake -DCMAKE_INSTALL_PREFIX:PATH=/usr .. # Install to /usr
make # Add -jx for parallel

安装:

sudo make install
sudo systemctl daemon-reload # refresh systemd
sudo systemctl enable driver-installer.service # enable installer service
sudo systemctl start driver-installer.service # start installer service

使用:

sudo deepin-graphics-driver-manager

OpenWRT小米路由器Mini – 不支持所上传的文件格式请确认选择的文件无误

今天尝试给小米路由器mini(MT7620a)上的自编译openwrt刷入原版镜像,后台升级出现上述错误。

遂尝试sysupgrade,依然无法刷入:

Device xiaomi,miwifi-mini not supported by this image
Supported devices: miwifi-mini
Image check ‘fwtool_check_image’ failed.

解决方案也很简单。编辑/lib/upgrade/fwtool.sh

将46行的device="$(cat /tmp/sysinfo/board_name)"修改为device="miwifi-mini"即可。