Real Analysis
Latest update: 4/25
持续更新中，最后一次更新6/5
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最后一次编辑4/25
In many places, students must arrive very early in the morning to attend school. Some people believe that starting the school day early is the best approach to support learning, but others believe that starting the school day at a later time in the morning would be better for students. Which view do you agree with and why?
观点：I believe starting the school day early is the best approach to support learning.
First thing first, starting the school day early enables students to have more time to study. To be more specific, noting is more precious than time especially for students. Early morning is the period when studens are most energetic. Therefore, if the studying time in the morning is extended, the students' efficiency of learning can be improved accordingly. A typical example to the point might be my high school. During the first year, the first class in the moring began at 8:00 late in the morning, I could not to anything except listening to teachers because I could spare no time do something else. The next year our school started to make changes. We were supposed to take an extra self-study class that starts from 6:30 early in the morning. I started to make use of the time in the early morning, and I found that I could recite two more passages than before each day. Moreover, during the self-study class, I could even get prepared for the coming classes and have a better understanding of the knowledge I learned. As a result, my grades became better and better due to the new approach to start the school early. In the light of this example, we can see that starting the school day early is a effective way to support learning.
TPO 62 模考我的答案
Scaled score 22
raw score 3.0
Both the lecture and the passage discessed about the use of artificial reefs. However, their viewpoints are strongly contradictory for several reasons.
Firstly, the reading points out that artificial reefs may increase the populations of seveal species of fish, but the lecturer supposes that this do not mean the overall increasement of the fish. This is because the artifial reef is more likely to attract fish that live distant from the reef, so the fish have to move to the reef. When people catch the fish around the reef, the population of the kind of fish being caught will decrease beacause they cannot maintain more polulations.
Moreover, the author holds that artificial reefs can alse improve the economical competitiveness of small-scale fishers. The lecturer, however, believes that it is fatal to remain the reef positions secret. He gives us with the example of the use of large net. This approach would definitely destroy the reef and the only safe way is to make the position of the reef known by people, thus this would do no help to improve the economical competitiveness of small-scale fishers.
Besides, the professor also refutes the idea that the use of artificial reefs can recycle useless materials. He states that the implementation of artificial reefs is sure to cause environmental damage. He provides us with the example of osborne reef to illustrate the point. This kind of reef is made of car tyres. When a storm occurs, the reef will get loose and became loose parts which will couse environmental damage, and make many sea creatures leaving dead on the sea floor.
In sum, according to the discussion above, the speaker totally disagrees with the conclusions revealed in the reading passage.
raw score 4.0
Nowadays, there are thousands of students graduating from high scholl every year. Accordingly, the graduate shtdents are facing a choice of whether to take at least a year off to work or travel before studying at a university. The problem is causing polarized debate among those graduates and their parents. I, given the chance, would like to state it is necessary for students gratudated from high school to take at least a year off to work or travel before entering college.
First of all, sparing one's time to work before entering university helps him to gain work experience. To be specific, the working experience and skills acquired through the process helps oneself to quickly adapt to college life. A good example in case might be myself. After I graduated from high school, I decided to take a year off to do seveal part-time jobs. I worked as a waiter in a restaurant and I learned to be concentrated on what I am doing. Besides, I also served as a programmer doing data analysis in an IT company, from which I learned a lot techniques to deal with large amounts of data. As a result, everything seemed to be easy for me when I entered college. The working experience I gained helped me do get straight A in my first semester. In the light of this, I suppose it is crutial to take at least one year off to work after high school.
Moreover, I believe that spending a year or more travelling before becoming a college student is of geat significance. That is to say, through travelling, one can get familiar with the characteristics of the city he gets by, which will help him to decide which place to work or stay after graduating from college. To exemplify that, I would like to take myself as an example again. During the one year off, I also travelled to some popular city in China like Beijing or Shanghai. I discovered from my travelling experience that Beijing is an energetic city but it is too cold to stay at during winter. Besides, I found that Shanghai is full of opportunities especially chances for IT engineers like me. I finally decided to stay in Shanghai after college and thanks to my one year off travelling experience, I think I have made the right choice because I have just got a decent job in Shanghai. Therefore, spending a year travelling is important as well.
In a word, I certainly agree with the idea that students gratudated from high school should take at least a year off to work or travel before entering college, for it helps to make oneself more adaptive to the college and can also let the student make the relatively correct choice of working place after graduation.
A university has money in its budget to do ONE thing to improve its facilities for students: it can either improve the quality of the technology (for example, computers and printers that it provides for students, OR it can redesign spaces where students hold club meetings and gather with friends in their spare time to make these areas more comfortable and appealing. Which idea do you prefer, and why?
Universities are trying their best to promote its facilities for students, while the budget is usually limited. It seems unpractical for universities to spend money on every aspects of the school. Some people believe that improving the quality of the technology should be put in the first place. While others, in contrast, insist that it is better for school to redesign the leisure areas to make these areas more comfortable and appealing. I, given the chance, would like to state that it is better to spend money to improve the technology standard of the school.
First thing first, the student will have a better working efficiency with higher quality technological devices. To be more specific, technological devices is being used nearly everywhere inside the campus. There are printers inside the office, computers in the IT center, and online teaching system placed in every dormitory. Therefore, with the promotion of these technological devices, students' quality of life and study will be improved accordingly. A good example in case might be my college. When I was in the last semester of school, I was busy doing a machine learning project to finish my thesis. The time-consuming trainig process usually takes about two days to run. However, our school used their fundings and provided our laboratory with better computer CPUs and GPUs, hence my programming project could be done within 4 hours, which profoundly improved my research effeciency. From this case we can conclude that it is worth investing on the technology inside the school.
Moreover, funding on the quality of the technology can save students' money. That is to say, oceans of internet resources and technological tools are being used by college students especially those who major in science. Only a small part of those resources are free of charge, which means it is common for students to pay for these resources. For instance, a student major in data science usually use calculating tools like MATLAB and Mathematica. Those tools charges about 100 dollars for individual each year which is too expensive for a student. Consider if the school pay for those fees, it will certainly lighten students' financial burden. Some people may think that the school should fund in entertainment first, this opinion does make sense to some extent, for students need to relax themselves. However, there are already enough entertainment like cafeteria or shopping mall outside the campus, there is no need to pay extra money to redesign the leisure areas. Therefore, I firmly hold the opinion that it is better to spend money improving the technology standard of the school first.
In a word, I believe that universities should improve the quality of the technology first, for it can both technically and financially benefit the students.
(To exemplify that, I would like to take my college as an example again. Being a student major in statistics and data science, I usually use calculating tools like MATLAB and Mathematica. Those tools charges about 100 dollars for individual each year which is too expensive for a student like me. Luckily, my college had signed a contract with MATLAB and all the students can use the product for free. Thanks to the school's policy, I could save roughly over a thousand dollars throughout my college life. In the light of this, )
内积空间：
赋范空间：
定义内积诱导的范数
定义范数诱导的度量
完备度量空间：每个柯西列都收敛.
连续函数的
矩阵范数：对于
命题：
常见的完备度量空间：
注意：有限维赋范线性空间都是完备的，赋范线性空间里，完备等价于闭。
紧致：度量空间
结论：
定义（连续函数） 以下是
压缩映射：
若
定义（一致连续）
定义（连通）：度量空间
定义（弧连通）：度量空间
定理：
定理 线性变换
定理 复方阵
主要思路，首先是由各个特征子空间的和是直和，构造多项式
定理 矩阵
定理 复数域上的任何
思路均为归纳法.
定理（牛顿公式） 对
定理（根子空间分解） 设
思路：设
此外，设
定义（循环子空间）
定理（多项式矩阵相关）
定理
定理（
定义（有限维实线性空间的内积） 任取
定理（柯西不等式）
定理（三角不等式）
定理 同一个内积在两组基
定理 任意实方阵都可以正交相似为准上三角阵，对角元为所有实特征值和成对复特征值构成的
例子
右推左，
推论 对任意实对称方阵
思路：将原来的基做正交化，即正交矩阵
定理（QR分解）
欧式空间的内积 满足双线性，对称性和正定性.
酉空间的内积 满足共轭双线性，共轭对称性和正定性.
定理
定理（
思路：将
定理（奇异值分解）
定理（矩阵的极分解）
证明利用初等变换即可。
给出几道有价值的题目和我的做法。
3.设
是独立均匀地取自 维超立方体上的两点， 表示两点的欧氏距离.证明
证 令
remark 对于有二阶导数的函数
在这里，取
6.对某概率空间上的随机变量
， .若 且 与 独立，证明：
证 记
计算
部分笔记：Prob.pdf
引理 设随机变量
引理 对一切
记
设
证 取事件
设
证
推广：我们有
如果满足
证 利用切比雪夫不等式的推广，取
对于独立试验序列，事件
令
若
设
如果
任意一致有界的非降函数列
设
设随机变量
设
设
设
设
随机变量序列
In this ungraded lab, you will - explore the sigmoid function (also known as the logistic function) - explore logistic regression; which uses the sigmoid function
import numpy as np
%matplotlib widget
import matplotlib.pyplot as plt
from plt_one_addpt_onclick import plt_one_addpt_onclick
from lab_utils_common import draw_vthresh
plt.style.use('./deeplearning.mplstyle')
As discussed in the lecture videos, for a classification task, we can start by using our linear regression model,
Let's implement the sigmoid function and see this for ourselves.
The formula for a sigmoid function is as follows -
In the case of logistic regression, z (the input to the sigmoid function), is the output of a linear regression model. - In the case of a single example,
NumPy has a function called exp(), which offers a convenient way to calculate the exponential ( z).
It also works with a single number as an input, as shown below.
# Input is an array.
input_array = np.array([1,2,3])
exp_array = np.exp(input_array)
print("Input to exp:", input_array)
print("Output of exp:", exp_array)
# Input is a single number
input_val = 1
exp_val = np.exp(input_val)
print("Input to exp:", input_val)
print("Output of exp:", exp_val)Input to exp: [1 2 3]
Output of exp: [ 2.72 7.39 20.09]
Input to exp: 1
Output of exp: 2.718281828459045The sigmoid function is implemented in python as shown in the cell below.
def sigmoid(z):
"""
Compute the sigmoid of z
Args:
z (ndarray): A scalar, numpy array of any size.
Returns:
g (ndarray): sigmoid(z), with the same shape as z
"""
g = 1/(1+np.exp(-z))
return gLet's see what the output of this function is for various value of z
# Generate an array of evenly spaced values between -10 and 10
z_tmp = np.arange(-10,11)
# Use the function implemented above to get the sigmoid values
y = sigmoid(z_tmp)
# Code for pretty printing the two arrays next to each other
np.set_printoptions(precision=3)
print("Input (z), Output (sigmoid(z))")
print(np.c_[z_tmp, y])Input (z), Output (sigmoid(z))
[[-1.000e+01 4.540e-05]
[-9.000e+00 1.234e-04]
[-8.000e+00 3.354e-04]
[-7.000e+00 9.111e-04]
[-6.000e+00 2.473e-03]
[-5.000e+00 6.693e-03]
[-4.000e+00 1.799e-02]
[-3.000e+00 4.743e-02]
[-2.000e+00 1.192e-01]
[-1.000e+00 2.689e-01]
[ 0.000e+00 5.000e-01]
[ 1.000e+00 7.311e-01]
[ 2.000e+00 8.808e-01]
[ 3.000e+00 9.526e-01]
[ 4.000e+00 9.820e-01]
[ 5.000e+00 9.933e-01]
[ 6.000e+00 9.975e-01]
[ 7.000e+00 9.991e-01]
[ 8.000e+00 9.997e-01]
[ 9.000e+00 9.999e-01]
[ 1.000e+01 1.000e+00]]The values in the left column are z, and the values in the right column are sigmoid(z). As you can see, the input values to the sigmoid range from -10 to 10, and the output values range from 0 to 1.
Now, let's try to plot this function using the matplotlib library.
# Plot z vs sigmoid(z)
fig,ax = plt.subplots(1,1,figsize=(5,3))
ax.plot(z_tmp, y, c="b")
ax.set_title("Sigmoid function")
ax.set_ylabel('sigmoid(z)')
ax.set_xlabel('z')
draw_vthresh(ax,0)Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …As you can see, the sigmoid function approaches 0 as z goes to large negative values and approaches 1 as z goes to large positive values.
A logistic regression model applies the sigmoid to the familiar linear regression model as shown below:
where
Let's apply logistic regression to the categorical data example of tumor classification.
First, load the examples and initial values for the parameters.
x_train = np.array([0., 1, 2, 3, 4, 5])
y_train = np.array([0, 0, 0, 1, 1, 1])
w_in = np.zeros((1))
b_in = 0Try the following steps: - Click on 'Run Logistic Regression' to find the best logistic regression model for the given training data - Note the resulting model fits the data quite well. - Note, the orange line is '
plt.close('all')
addpt = plt_one_addpt_onclick( x_train,y_train, w_in, b_in, logistic=True)Canvas(toolbar=Toolbar(toolitems=[('Home', 'Reset original view', 'home', 'home'), ('Back', 'Back to previous …