# dot product (blogs) — RSS Amplifier Recent posts from the 4 feeds in the RSS Amplifier directory that cover dot product. Page: Feed: --- ## [Concurrent Servers: Part 7 - Rust](https://eli.thegreenplace.net/2026/concurrent-servers-part-7-rust/) _2026-08-15 · Eli Bendersky · Eli Bendersky's website_ This is part 7 in a series of posts on writing concurrent network servers. In this part, we discuss how the challenges described in earlier parts are tackled in the Rust programming language. All posts in the series: Part 1 - Introduction Part 2 - Threads Part 3 - Event-driven Part 4 - libuv … ## [Relative velocity and closing speed](https://eli.thegreenplace.net/2026/relative-velocity-and-closing-speed/) _2026-08-04 · Eli Bendersky · Eli Bendersky's website_ In Physics simulations or game engines it’s sometimes useful to determine the speed with which two objects are approaching each other. This post will discuss the concept of closing speed , which is the normal component of the relative velocity of two objects. Relative velocity and its components Suppose we … ## [Notes on the Fourier Transform](https://eli.thegreenplace.net/2026/notes-on-the-fourier-transform/) _2026-07-15 · Eli Bendersky · Eli Bendersky's website_ The Fourier series is a great tool for analyzing periodic functions. But what about functions that don’t repeat? We’ve seen that we can compute Fourier series for a non-periodic function defined on a finite interval, as long as we don’t care about its behavior beyond that interval … ## [Dot product: Component vs. Geometric definition](https://eli.thegreenplace.net/2026/dot-product-component-vs-geometric-definition/) _2026-07-11 · Eli Bendersky · Eli Bendersky's website_ The goal of this post is to answer a simple question: why are the following two definitions of the vector dot product in Euclidean space \[1\] equivalent for vectors \\vec{a} and \\vec{b} : Component definition: \\vec{a}\\cdot\\vec{b}=\\sum\_{i=1}^{n}a\_i b\_i Geometric definition: \\vec … ## [Summary of reading: April - June 2026](https://eli.thegreenplace.net/2026/summary-of-reading-april-june-2026/) _2026-07-01 · Eli Bendersky · Eli Bendersky's website_ "The Nuremberg Trial" by John Tusa and Ann Tusa - a detailed, meticulously researched account of the Nuremberg Trials. There's not a whole lot of side questing in this book - it's all focused on the trials themselves. Interesting read overall, though somewhat dry and academic. "Things Become Other Things: A Walking … ## [Counting operations instead of degree in polynomial approximation](http://www.shapeoperator.com/2026/03/27/polynomial-approximation-operation-count/) _2026-03-27 · Shape Operator_ A famous problem in approximation theory is to approximate the function \\left|x\\right| as accurately as possible on the interval -1 \\le x \\le 1 , measured by maximum absolute error, using either polynomials or rational functions of fixed degree. Perhaps it seems silly to approximate a function as simple as this, but it serves as a useful test problem for probing how well non-smooth functions can… ## [Matrix squaring can also rapidly sum the geometric series](http://www.shapeoperator.com/2026/03/22/matrix-squaring-geometric-series/) _2026-03-22 · Shape Operator_ The geometric series, S\_{n}(x) = \\sum\_{k=0}^{n} x^k, satisfies a linear recurrence S\_{n+1}(x) = x \\, S\_n(x) + 1 with constant (in n ) coefficients. This means it can also be written as a matrix power, S\_{n}(x) = \\begin{pmatrix} 1 & 0 \\end{pmatrix} M^{n} \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix}, with M = \\begin{pmatrix} x & 1 \\\\ 0 & 1 \\end{pmatrix}. Repeatedly squaring M can rapidly generate high… ## [Newton's method can rapidly sum the geometric series](http://www.shapeoperator.com/2026/03/21/newtons-method-geometric-series/) _2026-03-21 · Shape Operator_ Newton’s iteration for computing the reciprocal of a , y\_{n+1} = y\_n(2-ay\_n), can be derived by applying Newton’s method to the function f(y)=1/y-a . Substituting a=1-x gives y\_{n+1} = y\_n(2-(1-x)y\_n), an iteration for computing 1/(1-x) . Starting from y\_0=1 , the first few iterates are \\begin{aligned} y\_1 &= 1 \\cdot (1+x) = 1+x, \\\\ y\_2 &= (1+x)(1+x^2) = 1+x+x^2+x^3, \\\\ y\_3 &= (1+x+x^2+x^3)(1+x^4)… ## [A framework for making sense of metrics in technical organizations](https://mbernste.github.io/posts/metrics/) _2026-02-10 · Matthew N. Bernstein · Matthew N. Bernstein_ If you work in a quantitative or technical field, there is little doubt that you or your team has worked long and hard to define which metrics to measure and track. Using data-driven metrics is a critical practice for making rational decisions and deciphering truth in a complex and noisy world. However, as others have pointed out, an over-reliance on metrics can lead to poor outcomes. In an effort… ## [Understanding attention](https://mbernste.github.io/posts/attention/) _2025-12-21 · Matthew N. Bernstein · Matthew N. Bernstein_ Attention is a type of layer in a neural network that is widely regarded to be one of the most important breakthroughs that enabled the development of modern AI systems and large language models. At its heart, attention is a mechanism for explicitly drawing relationships between items in a set. In natural language processing, the set being processed are words (or tokens) and attention enables the… ## [Brisk AI News Briefings (Sponsored)](https://crawlproof.com/a/yNN4E9QfXJpU) _2025-12-21 · **Sponsored**_ Read AI summaries and listen to audio briefings for top stories. ## [Demystifying Euler’s number](https://mbernste.github.io/posts/eulers_number/) _2025-01-26 · Matthew N. Bernstein · Matthew N. Bernstein_ Euler’s number $e := 2.71828\\dots$ has, to me, always been a semi-mysterious number. While I understood many facts about $e$, I never felt I ever truly understood what it really was – it’s core essence so to speak. I believe that part of the reason for my confusion is that $e$ is often taught coming from two seemingly different perspectives: Either it is introduced in the context of compound… ## [Reproducing kernel Hilbert spaces and the kernel trick](https://mbernste.github.io/posts/rkhs/) _2024-12-14 · Matthew N. Bernstein · Matthew N. Bernstein_ If you’re a practitioner of machine learning, then there is little doubt you have seen or used an algorithm that falls into the general category of kernel methods. The premier example of such methods is the support vector machine. When introduced to these algorithms, one is taught that one must provide the algorithm with a kernel function that, intuitively, computes a degree of “similarity”… ## [Dot product](https://mbernste.github.io/posts/dot_product/) _2024-12-09 · Matthew N. Bernstein · Matthew N. Bernstein_ The dot product is a fundamental operation on two Euclidean vectors that captures a notion of similarity between the vectors. In this post, we’ll define the dot product and offer a number of angles for which to intuit the idea captured by this fundamental operation. ## [Intuiting biology (Part 1: Order and chaos in the crowded cell)](https://mbernste.github.io/posts/intuit_biology_goodsell/) _2024-11-24 · Matthew N. Bernstein · Matthew N. Bernstein_ Cells are crowded spaces packed with biomolecules colliding and interacting with one another. Despite this chaotic environment, biologists routinely describe intracellular functions using the clean mathematical language of networks. In this post I will attempt to reconcile these two seemingly contradictory perspectives of the cell. This post will serve as a first part in a series of blog posts I… ## [Notes on \_The Art of War\_ by Sun Tzu](https://mbernste.github.io/posts/art_of_war_part1/) _2024-11-16 · Matthew N. Bernstein · Matthew N. Bernstein_ I am currently reading Sun Tzu’s Art of War and am finding much wisdom in it. I have been taking notes during my reading and I thought I’d share them in this post. Here I cover Books 1 and 2. ## [Denoising diffusion probabilistic models (Part 2: Theoretical justification)](https://mbernste.github.io/posts/diffusion_part2/) _2024-10-20 · Matthew N. Bernstein · Matthew N. Bernstein_ In Part 1 of this series, we introduced the denoising diffusion probabilistic model for modeling and sampling from complex distributions. We described the diffusion model as a model that can generate new samples by learning how to reverse a diffusion process. In this post, we provide more theoretical justification for the objective function used to fit diffusion models and make connections between…