Before The Second Sex, the sexed/gendered body was not an object of phenomenological investigation. Beauvoir changed that. Her argument for sexual equality takes two directions. First, it exposes the ways that masculine ideology exploits the sexual difference to create systems of inequality. Second, it identifies the ways that arguments for equality erase the sexual difference in order to establish the masculine subject as the absolute human type. Here Plato is her target. Plato, beginning with the premise that sex is an accidental quality, concludes that women and men are equally qualified to become members of the guardian class. The price of women’s admission to this privileged class, however, is that they must train and live like men. Thus the discriminatory sexual difference remains in play. Only men or those who emulate them may rule. Beauvoir’s argument for equality does not fall into this trap. She insists that women and men treat each other as equals and that such treatment requires that their sexual differences be validated. Equality is not a synonym for sameness.

She develops the concept of freedom as transcendence (the movement toward an open future and indeterminate possibilities) to argue that we cannot be determined by the present. The essence of freedom as transcendence aligns freedom with uncertainty and risk. To be free is to be radically contingent. Though I find myself in a world of value and meaning, these values and meanings were brought into the world by others. I am free to reject, alter or endorse them for the meaning of the world is determined by human choices. Whatever choice I make, however, I cannot support it without the help of others. My values will find a home in the world only if others embrace them; only if I persuade others to make my values theirs.

If a definition that introduces other variables, check the assumptions about those variables. example: why does need to both be in and be a limit point of . (see remark 9.3.11 for another example)

because 𝜇μ\mu will give that support no consideration.

we adopt specific computational definitions that have been proposed to capture the two different structures of learning.

For those in charge, the act passed to keep more Chinese people from emigrating to the United States wasn’t enough. Fear of the Chinese population taking white middle class jobs caused those who had made it to the USA to be relegated to laundries and restaurants. For men not interested in either of those professions, there weren’t many other options besides gang life.

Soon — in the larger context of white America’s pervasive racism and xenophobia — Chinatown was labeled, even in the mainstream press, as a hopeless opium-and-prostitute-ridden slum. As The New York Times wrote about Chinatown in 1880: “There are some streets in New-York that begin with a very fair outlook, but grow so much worse with every block, as you walk through them, that there is no telling what they might come to if they were long enough.”

At the time, the Chinese population in America was a bachelor society of men who had worked the cross-country railroads and California gold mines. Chinese women never even had a chance to make it to the states, due to policymakers who had started to fear the influx of male Chinese immigrants and enacted the Chinese Exclusion Act of 1882. With the resulting abnormally high male-to-female ratio, Chinatown became known as a hotbed of masculine vice.

Measure theory was initially created to provide a useful abstraction of the notion of length of subsets of the real line—and, more generally, area and volume of subsets of Euclidean spaces. In particular, it provided a systematic answer to the question of which subsets of ℝ have a length. As later set theory developments showed (see non-measurable set), it is actually impossible to assign a length to all subsets of ℝ in a way that preserves some natural additivity and translation invariance properties. This suggests that picking out a suitable class of measurable subsets is an essential prerequisite.

Intuitive interpretation

This means that a measure is any function μ defined on a certain class X  of subsets of a set E, which satisfies a certain list of properties. These properties can be shown to hold in many different cases.

integrals on these functions

当空间维度是无穷而且不可数的时候（没有一个可数的基底），无法运用有限维或可数维度空间的办法来定义范数，但对于可积函数空间，仍然能够定义类似的概念

two-dimensional shapes must be scaled by area rather than radius or by side length. Linear scaling for a two-dimensional shape results in exaggerated differences.

Drawing symbols on a map is a lot like drawing points on a map. There are two main differences. The first is that you use symbols() instead of points(). The former lets you add scaled circles, squares, rectangles, stars, and other symbols. The second difference is that you must scale the two-dimensional symbols correctly. Whereas points only represent location, scaled shapes both represent location and a second metric that corresponds to the location.

Oftentimes, other files accompany the .shp file with extensions .dbf and .prj. The first is called the shapefile attribute format, and it stores data that is associated to the geographic features encoded in the main shapefile. For example, you might have counties in your shapefile (.shp), and the unemployment rate for each county would be in the attribute file format (.dbf).

There are many ways to add geographic boundaries in R, but there are two main ways. The first uses the maps package and relies on boundary data loaded by the package. The second brings in external data in the form of shapefiles, which is more flexible but requires a bit more care.

The third package, rgdal, relies on a library called GDAL, which stands for Geospatial Data Abstraction Library. It helps you with map projections and geographic math.

the meaning of a function should be independent of the parameter names chosen by its author -- has important consequences for programming languages. The simplest consequence is that the parameter names of a function must remain local to the body of the function.

“As the number of identically distributed, randomly generated variables increases, their sample mean (average) approaches their theoretical mean.”Besides being easily one of the most important laws of statistics, this is the basis for Monte Carlo simulations and allows us to build a stochastic model by the method of statistical trials.

He used the tools of random sampling and inferential statistics to model likelihoods of outcomes, originally applied to a card game (Monte Carlo Solitaire).

There are a broad spectrum of Monte Carlo methods, but they all share the commonality that they rely on random number generation to solve deterministic problems.

Monte Carlo (MC) methods are a subset of computational algorithms that use the process of repeated random sampling to make numerical estimations of unknown parameters.

The difference between the two is that machine learning emphasizes optimization and performance over statistical inference. Statistics is also concerned about performance but would like to calculate the uncertainty associated with parameters of the model. It will try to model the population statistics from the sample data points to assess that uncertainty.

The aim of inference is to find statistical properties of the underlying data and to estimate the uncertainty about those properties.

Even ignoring the advantage of avoiding a notational disagreement between countable and uncountable cases, just because something is countable doesn't mean that there is a canonical bijection between it and ℕ

For example, some intervals of the function on the top map to smaller intervals on the substituted function indicating "a higher density".

It can be seen from the plot that the areas under the curves differ a lot

Gaussian processes are non-parametric (although kernel hyperparameters blur the picture) they need to take into account the whole training data each time they make a prediction.

Gaussian processes are a non-parametric method. Parametric approaches distill knowledge about the training data into a set of numbers. For linear regression this is just two numbers, the slope and the intercept, whereas other approaches like neural networks may have 10s of millions. This means that after they are trained the cost of making predictions is dependent only on the number of parameters.

how Gaussian processes are ever supposed to generalize beyond their training data given the uncertainty property discussed above. Well the answer is that the generalization properties of GPs rest almost entirely within the choice of kernel.

When you’re using a GP to model your problem you can shape your prior belief via the choice of kernel

A key benefit is that the uncertainty of a fitted GP increases away from the training data — this is a direct consequence of GPs roots in probability and Bayesian inference.

Similarly to the narrowed distribution of possible heights of Obama what you can see is a narrower distribution of functions. The updated Gaussian process is constrained to the possible functions that fit our training data —the mean of our function intercepts all training points and so does every sampled function. We can also see that the standard deviation is higher away from our training data which reflects our lack of knowledge about these areas.

use Bayes’ rule to update our belief about the function to get the posterior Gaussian process AKA our updated belief about the function we’re trying to fit.

Instead of observing some photos of Obama we will instead observe some outputs of the unknown function at various points. For Gaussian processes our evidence is the training data.

A Gaussian process is a probability distribution over possible functions.

We can see that Obama is definitely taller than average, coming slightly above several other world leaders, however we can’t be quite sure how tall exactly. The probability distribution shown still reflects the small chance that Obama is average height and everyone else in the photo is unusually short.

Let’s consider that we’ve never heard of Barack Obama (bear with me), or at least we have no idea what his height is. However we do know he’s a male human being resident in the USA. Hence our belief about Obama’s height before seeing any evidence (in Bayesian terms this is our prior belief) should just be the distribution of heights of American males.

Bayesian inference might be an intimidating phrase but it boils down to just a method for updating our beliefs about the world based on evidence that we observe. In Bayesian inference our beliefs about the world are typically represented as probability distributions and Bayes’ rule tells us how to update these probability distributions.

sampling from a probability distribution. This means going from a set of possible outcomes to just one real outcome — rolling the dice in this example.

In many real world scenarios a continuous probability distribution is more appropriate as the outcome could be any real number and example of one is explored in the next section.

discrete probability distributions as there are a finite number of possible outcomes

Some uncertainty is due to our lack of knowledge is intrinsic to the world no matter how much knowledge we have. Since we are unable to completely remove uncertainty from the universe we best have a good way of dealing with it. Probability distributions are exactly that and it turns out that these are the key to understanding Gaussian processes.

Gaussian processes are another of these methods and their primary distinction is their relation to uncertainty.

Machine learning is an extension of linear regression in a few ways. Firstly is that modern ML

Machine learning is using data we have (known as training data) to learn a function that we can use to make predictions about data we don’t have yet.

Mathematical explanations are fundamentally different because no part of a mathematical system can be otherwise than it is given without changing the entire system as a whole.

Mathematical explanations are not causalFrom someone whose primary professional activity is to teach the mathematical modelling of causal relations, this might seem like a controversial, not to say contrary statement. But while mathematics can be used to describe causal relations — that is to say mathematical systems can be put into correspondence with real-world relationships — those systems, in contrast to the theoretical systems of many other disciplines, are not themselves causally constructed.

Everyone who has studied mathematics has experienced the chill of mathematical limbo, but my experience is that graduates of other disciplines, accustomed to prompt acquisition of perhaps only provisional apprehension of a theoretical whole, find the cold all the more acutely uncomfortable on account of its unfamiliarity. For them, that long interlude without even a glimpse of the big picture can be intensely disconcerting if not downright terrifying.

In the course of this instruction, I have noticed a number of fundamental differences in perspective and disposition between graduates from different disciplinary backgrounds. The identification and recognition of these paradigmatic distinctions has been instrumental in helping those individuals master those foundational mathematical concepts that were otherwise so elusive to them.

I will argue that high-functioning in the literary humanities (as well as the social sciences) can actually present a substantial impediment to the mastery of the mathematical language that mediates scientific knowledge.

intellectual activity into two distinct “cultures”: science, engineering and mathematics on the one hand, and what he called literary humanities on the other

The Taylor series of a real or complex-valued function f (x) that is infinitely differentiable at a real or complex number a is the power series f ( a ) + f ′ ( a ) 1 ! ( x − a ) + f ″ ( a ) 2 ! ( x − a ) 2 + f ‴ ( a ) 3 ! ( x − a ) 3 + ⋯ , {\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots ,}

one of Julia's most important benefits is its ability to achieve C-like performance while keeping the syntax elegant.

对实数集 中的一些子集定义测度，就是为它们定义长度。从概率的角度来看，确定区间 的某个子集的测度，就是确定在区间 中随机取一点，它落在这个子集内的概率。

He skillfully takes the focus off from matrices and shifts the reader’s attention more towards linear mappings. This makes his proofs elegant, simple, and pleasing. Conscious of the reader’s possible unfamiliarity as well as time frame, Axler does a fine job of preparing and developing readers’ understanding rather than fully detailing application methods and formulas.

sigma代数涵盖了“一切可能事件的一切可能组合”

我们说是随机变量Z所携带的信息

所以这个sigma代数越大，那么信息就越多

滤链（filtration）

所以为什么说sigma代数表示信息？其实当我们给定某个sigma代数的时候，就是假设这个sigma代数的每个事件都能确切知道其有没有发生，自然就是我们对于样本空间中什么事件发生的一部分知识啊！

illness, she proposed, had a startling ability to renew our awareness of the worlds inside and outside of us.

Woolf achieves two things: she argues that illness has been unfairly dismissed as unworthy of representation in literature, and, before she has even made this argument, she has already proved it by showing the vast range of experiences that illness comprises, the way sickness makes even an otherwise mundane experience seem tinged with Bardic drama, or the rainy-night fire of noir; illness, in other words, contains the grand battles and unmapped tundras and emotions bright-dark as Picasso’s harlequins present in so many books labeled “important,” yet it rarely appears as a main theme.

“Considering how common illness is,” Woolf writes, how tremendous the spiritual change that it brings, how astonishing, when the lights of health go down, the undiscovered countries that are then disclosed, what wastes and deserts of the soul a slight attack of influenza brings to view, what precipices and lawns sprinkled with bright flowers a little rise of temperature reveals, what ancient and obdurate oaks are uprooted in us by the act of sickness, how we go down into the pit of death and feel the waters of annihilation close above our heads and wake thinking to find ourselves in the presence of the angels and the harpers when we have a tooth out and come to the surface in the dentist’s arm-chair and confuse his “Rinse the mouth—rinse the mouth”

Still, despite her macabre history with influenza, Woolf still managed to scoff at the pandemic in its early days. In a diary entry from 1918, she off-handedly recorded a neighbor’s succumbing to influenza along with the weather, as if both were equally mundane and unimportant: “Rain for the first time for weeks today, & a funeral next door; dead of influenza.” A few months later, she remarked sarcastically, upon noting that her writerly friend Lytton Strachey was avoiding London due to the pandemic, that “we are, by the way, in the midst of a plague unmatched since the Black Death, according to the Times, who seem to tremble lest it settle upon Lord Northcliffe, & thus precipitate us into peace.” The sardonic tone suggests that Woolf initially viewed the pandemic as a bit of an overblown joke, the comparisons to the plague histrionic.

到了成熟的资本主义时期，一切都被商品化——按照马克思的说法，旧时代“一切固定的东西都烟消云散了”。

它的核心是，个人乃最高的价值，个人及其权利是社会的法律、政治、经济和文化的根基。

This, in turn, leads us to explore our own selves more deeply, drifting through our Escherian staircases, our orca-dotted seas. In this way, illness becomes a bridge to writing about both ourselves and the world around us more sharply. Illness is itself novelistic, epic, lyric, if we allow ourselves to express its contours.