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MindMap Gallery Knowing It in Mind Mathematical Thinking in Life

Knowing It in Mind Mathematical Thinking in Life

This is a mind map about "Knowing Things in Mind: Mathematical Thinking in Life". This book combines life cases with mathematical formulas and is dedicated to teaching readers how to use rigorous science and engineering thinking to look at the current world. Various phenomena. The content of the book is mainly divided into three parts: thinking, methods and learning.

Edited at 2024-11-22 19:28:07

Mason·Carter

Recent works View more works>>

Knowing It in Mind Mathematical Thinking in Life

Mason·Carter

Recent works View more works>>

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Outline

"Knowing It in Mind: Mathematical Thinking in Life"

Learning: How to learn to express

How to make faster progress in reading

Proactively predict and learn from gaps

Supervised learning in machine learning

Read academic papers

Read a book quickly

good learning methodology

single task learning

Multi-task learning: Multi-task learning based on the correlation between tasks

Transfer learning: focusing on basic abilities that can be transferred to other fields

Meta-learning: more focused on the future, with more tasks and less training data

How to express something clearly

Incremental expression from primary to secondary

Two modes of image transmission

The singular value decomposition of a matrix can achieve incremental expression from primary to secondary

Examples: describing numbers, English language, etc.

Methods: Strategies and techniques for solving difficult problems

Stability and transition: negative feedback and positive feedback

negative feedback

Such as weight loss exercise

Driven by the gap between reality and goals

The goal is stability

PID control: proportional control at the moment, integral control to summarize the history, and differential control to judge the future.

positive feedback

For example, "Howling", the success of Microsoft and Apple

Function: self-reinforcement, "snowball" effect, achieving leaps

There is a fine line between good and bad positive feedback: When doing something, you must first be keen to find some good positive feedback, and then persist in being patient. Once the flywheel starts spinning, good results will naturally follow.

Good design: find the bottom layer

Take designing a rice cooker, forgetting to take your ID card when picking up a high-speed rail ticket, etc. as examples

Two stages of design: design a suitable bottom layer; perform appropriate optimization based on the bottom layer

The role of upper-level optimization is limited, the key lies in the underlying design

Imitation: seize the essence and get rid of limitations

One of the effective ways to innovate: seize the essence and get rid of limitations

Taking Fresnel lens as an example

Usual process: top-down first, then bottom-up

Take the development direction of aircraft, steam sailing ships and artificial intelligence as examples

Basic probability determines whether to maintain success or take risks

Law of Large Numbers: As long as something happens enough times, the frequency of a certain result is equal to its probability

Casinos and claw machines, players always lose to the dealer

Enlightenment

Work hard to improve your base probability

If the base probability is high, then repetitions are your best friend

If the base probability is low, then increase the base probability or jump to a game where the probability is in your favor.

Doing both: explanations given by least squares estimation

system of equations without solution

In actual engineering, the situation where a system of equations has no solution is far more common and important than the situation where a system of equations has a solution.

Reason: Each measurement will generate an equation. In order to eliminate the influence of noise, the number of measurements must far exceed the number of unknowns included.

solution

delete some equations

Directly ignoring some points of view in pursuit of perfection, that is, not compromising or being moderate

Least squares method: the solution that minimizes the average error of all equations

Striving for excellence and taking care of yourself step by step

Two ways of solving problems

For example, finding the extreme value of a function: derivation method to obtain a perfect analytical solution; numerical method; product development model: waterfall model and agile model

Step-by-step: Divide the solution into perfect steps and complete one before doing the next

Strive for excellence: Complete a complete process quickly, then iterate repeatedly to continuously improve

done is better than perfect, also known as growth mindset

changing thinking

If it is not easy to operate something directly, you can change it into a form that can be easily manipulated, and then change it back to its original form.

Such as high-voltage transmission lines, signal modulation

simulated annealing algorithm

Take the function to find the maximum value as an example

Analytical solution

The most perfect, but often unavailable

Numerical solution: gradient method

Core: Find the right direction and strive for excellence

Problem: Sometimes it is difficult to determine the gradient or the function is in a black box; it may also get stuck in a local maximum

Numerical solution: hill climbing method

Advantages: no gradient required

Disadvantages: Paying a price, you may need to try many times, which reduces search efficiency; when the function is complex, it is easy to fall into the local highest point

The solution to getting stuck in the local highest point: accept temporary imperfections in a random way

Simulated annealing algorithm: The annealing process represents the process of attenuation of randomness from high to low; gradually reducing the acceptance of imperfect probabilities

Inspiration: Life is the process of seeking the optimal solution

When you are young, you should try more, accept the higher probability of imperfection, and try not to fall into the local highest point; as you grow older and experience, find your own direction and then dig deeper.

Thinking: See the world with rational thinking

Calmly accept reality and work hard to change the probability

three worldviews

Depends on human effort

fatalism

It’s up to people to plan things, but it’s up to God to make things happen

Through hard work, you can increase the probability of success; since probability is used as the measurement standard, success cannot be guaranteed no matter how well you do it.

Don’t overestimate “explanation” and underestimate “prediction”

It is easy to explain but of little value; it is difficult to predict but of great value

The evaluation criterion of a good model is its predictive ability

Taking polynomial fitting as an example, fitting all known data will result in noise, making the model unable to capture the underlying real data trend.

The key to a good theory is its ability to predict the future

observe

Propose theoretical hypotheses

Use this theoretical hypothesis to make a prediction

Use experiments to verify whether the prediction is successful

Analyze the results you get

diversity dividend

Everyone has cognitive blind spots. The consensus reached when people from different perspectives discuss it together is often closer to the truth.

The nature of systems of equations

An equation is a conclusion obtained from observation from one angle; observation from multiple angles results in multiple systems of equations, and solving a system of equations means observing from multiple angles to find the inner essence.

ill-conditioned equations

It is extremely sensitive to noise, and slightly changing the independent variables will cause a huge difference in the results.

The reason for sensitivity is that the slopes of the straight lines represented by each equation are too close to each other.

Only conclusions drawn from observations from a larger perspective are meaningful.

"Convolution" in life

The system's response to a shock sequence is actually the superposition of the response to each individual shock.

Frequent "little blessings" can bring about a greater response than occasional big happiness

Life is about every moment now, not about success one day in the future

Convolution theory helps us make decisions: recite words with high frequency; buy a small house in the city center instead of a large house in the suburbs, etc.

An in-depth analysis of the pros and cons

Advantages and disadvantages of CPU, TPU, and GPU: if there are advantages, there must be disadvantages; if there are disadvantages, there must be advantages

The story of the "Five Stone Gourd" in "Xiaoyaoyou": Everything has advantages and disadvantages. It is necessary to find a position where the strengths can be used, that is, to make the best use of things and to make the best use of people's talents.

NP-hard problem (traveling salesman problem as an example): trading disadvantages (degraded performance or long distance) for advantages (faster speed)

Take the initiative to exchange controllable disadvantages for greater benefits

The world is sparse: simple laws behind complex phenomena

Mathematical explanation of sparsity: If a time signal is sparse, then the value of most positions in the signal is zero

Examples from real life

Pickup locker passwords are sparse: sparseness is the key to security

Many things after Fourier transform (sine wave superposition) are sparse: such as the field of image compression

Emergence: A phenomenon in which simple rules among individuals lead to the emergence of "advanced" activities as a whole, such as ant colonies: the simple rules behind them are also sparse

Seemingly related, but actually independent: inspiration brought by conditional independence

The core idea of conditional independence: If event A and event B are conditionally independent with respect to event C, then knowing whether event A or event B occurs will not help us better infer the other if event C occurs. probability of event occurring

Two events seem to be related, but in fact, if the other event is conditionally independent, it is easy for people to mistake "correlation" for "causality"

For example: buying vanilla ice cream and the car not starting; wearing a jacket and the incidence of car accidents; logical problems that arise again in the spring breeze; the ABC theory of emotions

Kalman filter

Filter Principle: Combining the equation of state with observational data results in a better estimate of the final state.

Practical applications: Conversations in noisy markets; display data of air purifiers; vehicle positioning results, etc.