This article explains the chain rule of conditional probability, also known as the multiplication rule, by illustrating how successive events' probabilities combine, providing a clear, intuitive example that demonstrates the step‑by‑step derivation of the formula for calculating joint probabilities.
This article compiles a comprehensive list of links to past posts from the "Model Perspective" public account covering probability and statistics, calculus, linear algebra, and mathematical modeling, while noting upcoming collections on Python programming, academic writing, and competition analysis.
This article compiles a series of WeChat posts covering calculus fundamentals—including limits, derivatives, integration techniques, gradient descent, and multivariable concepts—providing direct links to each detailed tutorial for learners to explore.
This article explains the concepts of neighborhoods, interior, exterior, and boundary points of sets, introduces accumulation points, defines open and closed regions, and describes n‑dimensional Euclidean spaces and multivariable functions, illustrating each with diagrams.
This article explains the integration by parts technique, derived from the product rule, and walks through five detailed examples that show how to rewrite integrals as products of functions, apply the formula, and solve the resulting equations.
This article outlines the fundamental properties of integrals—including the sum rule, constant factor rule, interval splitting, and limit‑based computation—and introduces differentials, their basic properties, and how they relate to integral calculus.
This article explains how the Fundamental Theorem of Calculus bridges derivatives and integrals, provides a straightforward method for evaluating integrals via antiderivatives, illustrates the concept with telescopic sums, and walks through several concrete examples to demonstrate its practical use.
This article traces the pivotal role of mathematics from the 18th‑century golden age of classical mechanics through Maxwell's equations, Einstein's relativity, biological modeling, game‑theoretic economics, the birth of modern computers, and the emergence of chaos and fractal theory, highlighting key figures and breakthroughs that linked abstract theory to real‑world applications.
This article introduces the concept of sets, their definitions, cardinality, and key properties, then explores various common sets and provides a comprehensive overview of functions—including definitions, classifications such as constant, linear, polynomial, rational, exponential, logarithmic, and trigonometric functions—along with illustrative diagrams.
This article introduces the concept of numerical sequences, explains the definition of sequence limits, demonstrates how to prove limits through detailed examples, and outlines the basic arithmetic operations on limits, providing a concise summary of these fundamental mathematical principles.
This article explains the classic Nim game where two players alternately remove 1‑3 stones from a pile, presents the LeetCode‑style problem of determining if the first player can force a win, analyzes the pattern of losing positions, provides a concise Go solution, and offers a formal proof based on game theory.
The PHP pi() function returns the approximate value of the mathematical constant π as a float, with precision set by the php.ini precision directive (default 14), and can also be accessed via the predefined M_PI constant, illustrated with a simple code example.
An AI‑driven project called the Ramanujan Machine generates novel mathematical formulas—including continued‑fraction representations of constants like π, e, and the Catalan constant—sparking interest in automated conjecture making, while highlighting both the promise and limits of machine‑assisted mathematics.
The PHP log() function computes the natural logarithm of a number or, when a base is supplied, the logarithm with that base, and the article explains its signature, parameters, return value, and provides example code with expected output.
The article explains PHP’s log1p() function, which accurately computes log(1 + number) even for values near zero, details its syntax, parameters, return value, and provides example code with sample outputs demonstrating its precision across different inputs.
The PHP log10() function returns the base‑10 logarithm of a floating‑point argument, detailing its signature, parameter description, return value, and providing example code with expected output for various inputs.
The PHP expm1() function returns the value of exp(number) − 1, providing accurate results even when the argument is close to zero, and includes details on its syntax, parameters, return value, example usage, and sample output.
Learn how to translate common mathematical symbols such as summation, product, factorial, conditional expressions, and matrix multiplication into clear Python code, revealing the underlying computations and helping data scientists and ML practitioners deepen their mathematical intuition through practical examples.
Former Huawei strategist Xu Wenwei outlined ten critical mathematical problems—from lossy compression limits to DNN interpretability—that must be solved to advance communication technologies beyond Shannon’s limits, highlighting the pivotal role of mathematics in the emerging intelligent, fully connected society.
From Archimedes’ early approximation to modern record‑breaking memorisations, the article chronicles key mathematicians—Liu Hui, Zu Chongzhi, Van Ceulen, Jones, Lambert, Lindemann—and dates such as March 14, “ultimate” Pi dates, and Einstein’s birthday, highlighting global Pi Day traditions and celebrations.
This article introduces basic mathematical functions—including linear, quadratic, exponential, and step functions—explains sequences and their formulas, and shows how these concepts underpin neural‑network computations such as weighted inputs, activation functions like sigmoid, and error‑backpropagation, providing clear examples and visual illustrations.
This article explains how the golden ratio emerges in plant phyllotaxis, compares different angular ratios and their visual patterns, introduces continued fractions as a rigorous way to evaluate rational approximations, and links the mathematical findings to auxin‑driven growth mechanisms in plants.
This article introduces vector norms—including L0, L1, L2, L∞, and general Lp—explaining their mathematical definitions, geometric interpretations, and roles in machine learning such as compressive sensing and distance‑cosine equivalence after L2 normalization, supplemented with illustrative formulas and reference links.
This article provides a comprehensive overview of deep learning, covering essential mathematics and machine learning fundamentals, modern deep network architectures and regularization techniques, advanced research topics such as structured probabilistic models and generative methods, and a curated reading list for practitioners.
The article explains how basic mathematical principles—starting with simple predictors and linear classifiers, then extending to multi‑classifier systems, activation functions, and weight adjustments—underpin neural network architecture, illustrating each step with concrete examples to show how mathematics drives AI model training and performance.
This article explains the Python abs() and fabs() functions, detailing their usage, syntax, parameters, return values, and examples for calculating absolute values of numbers, including handling complex numbers with abs() and float/integer types with fabs().
This article explains the definitions, formulas, and step‑by‑step derivations for permutations, combinations, and repeated combinations, illustrating how A(n,m)=n!/(n‑m)! and C(n,m)=A(n,m)/m! are obtained using the multiplication principle and ball‑and‑separator models, including visual formula images and practical examples.
This article introduces the fundamental counting principles—classification addition and stepwise multiplication—explaining their concepts, formulas, conditions, and real-world examples such as travel route choices, while comparing their similarities and differences and highlighting their importance in mathematics and computer science.
The article outlines key mathematical areas—probability & statistics, linear algebra, calculus, information theory, and convex optimization—essential for deep learning, and recommends specific lecture notes and textbooks for each topic to guide learners toward effective study.
This article breaks down the essential mathematics—linear algebra, probability, calculus, and optimization—required for mastering deep learning, explains how each topic maps to core deep‑learning concepts, and outlines six progressive learning stages with concrete examples and recommended textbooks.
This article explores how fundamental geometric principles influence diverse designs, covering the honeycomb conjecture, slime‑mold network experiments, the physics of vinyl records, never‑loose nuts, Fibonacci sequences in nature, the golden ratio myth, and hidden geometric secrets in famous logos.
This article curates over 20 machine‑learning cheat sheets—including algorithm overviews, Python libraries, and essential mathematics—compiled from various online sources and illustrated with downloadable diagrams, offering a comprehensive reference as of June 1 2017.
The article reviews the surge of AI interest sparked by AlphaGo and Master, explains why strong mathematics—especially linear algebra, probability, statistics, calculus, and optimization—is crucial for AI practitioners, and provides curated free online courses, textbooks, and resources to help beginners master these subjects.
The article explores why mathematically equal expressions like 5×3 and 5+5+5 are not necessarily equivalent, explains the distinction between equality and equivalence, illustrates with real‑world and programming examples such as JavaScript’s == versus === and matrix multiplication rules, and discusses teaching implications.
The perceived difficulty of mathematical fields varies because each branch has its own language, foundational concepts, and required tools, making areas like algebraic geometry seem daunting while others such as number theory or combinatorics appear simpler yet still demand deep insight and advanced techniques.
The article explores how mathematical thinking—ideas, techniques, and abstraction—underpins software architecture, shaping problem‑solving, modeling, and design, and argues that a solid mathematical foundation can elevate an architect’s ability to create elegant, scalable systems.
This article explores how mathematical concepts and discrete math underpin software development, illustrating their impact on algorithm design, code quality, and problem‑solving through theory, real‑world examples, and a C program that enumerates ball‑selection combinations.
This tutorial explains the geometric derivation and step‑by‑step algorithm for calculating the tangent points of two circles of different radii and using them to construct quadratic Bézier curves that create the popular one‑click dismiss animation on mobile interfaces.
The article explores the history and principle of the Fourier Transform, illustrating how this mathematical prism decomposes complex waveforms into simple sine components and underpins technologies such as audio compression, image JPEG, speech recognition, scientific imaging, and even space research.
This article reflects a computer science student's journey into deep mathematics, explaining why studying set theory, analysis, algebra, topology, measure theory, and functional analysis provides essential tools for advanced research, linking abstract concepts to practical applications in vision, learning, and modern scientific fields.
The article recounts the intertwined lives of John von Neumann and Alan Turing, describing their early education, move to Princeton, personal quirks, and the poignant friendship that shaped early computer science and artificial intelligence, supported by historical photographs and references.
The article examines whether understanding lecture material ensures problem‑solving ability, proposing that the logic of listening and solving are inversely related, and uses combinatorial reasoning to illustrate how many distinct propositions can arise from a set of theorems, conditions, and scenarios.
After completing three years of university mathematics courses, the author visualises and explains a personal classification of major mathematical branches, discussing their relationships, cross‑disciplinary links, and reflections on ancient Chinese mathematics and future educational considerations.
