Band-limited expansion
The band-limited expansion of the function f(x) is given by where sinc(x) = sin(πx)/πx. This is also called the sinc expansion, or the Whittaker cardinal after its discoverer E. T. Whittaker [1]. This is called the band-limited expansion of f because each term in the infinite sum is band-limited, i.e.
Read moreDelay differential equations
Sometimes the future state of a system depends not only on the current state (position, velocity, acceleration, etc.) but also on the previous state. Equations for modeling such systems are known as delay differential equations (DDEs), difference differential equations, retarded equations, etc. In a system with hysteresis, it matters not
Read moreLaplace transform inversion theorems
The way Laplace transforms, as presented in a typical differential equation course, are not very useful. Laplace transforms are useful, but not as presented. The use of Laplace transforms is presented is as follows: Transform your differential equation into an algebraic equation. Solve the algebraic equation. Invert the transform to obtain your
Read moreMellin transform and Riemann zeta
The Mellin transform of a function f is defined as For example, it follows directly from the definition that the gamma function Γ(s) is the Mellin transform of the function e−x. I ran across an exercise that states an impressive-looking theorem about the Mellin transform, namely that where F(s) denotes
Read moreTragedy of the (data) commons
Ben chats with Shayne Longpre and Robert Mahari of the Data Provenance Initiative about what GenAI means for the data commons. They discuss the decline of public datasets, the complexities of fair use in AI training, the challenges researchers face in accessing data, potential applications for synthetic data, and the
Read moreSawtooth waves
I woke up around 3:00 this morning to some sort of alarm outside. It did not sound like a car alarm; it sounded like a sawtooth wave. The pattern was like a Morse code O. Not SOS, or I would have gotten up to see if anyone needed help. Just
Read moreSales hiring just got smarter: Meet AI Interviewer for Sales
Today, sales professionals spend a staggering amount of their time on tasks that don’t directly contribute to selling—approximately 70%, according to the latest Salesforce ”State of Sales” report. One of the most time-consuming of these non-selling tasks is recruiting and hiring new talent, which often pulls sales leaders away from
Read moreHow to improve the developer experience in today’s ecommerce world
We walk through the process of designing an SDK that makes it easy for developers to integrate new technology into their e-commerce stack and is flexible enough to serve the needs of small businesses and large enterprises.
Read moreEnterprise 2024.7: Empower your subject matter experts to contribute
This release introduces Subject Matter Expert (SME) Auto-Assign to the Stack Overflow for Teams experience so expert knowledge is automatically captured, verified, and distributed to users.
Read morePioneering work is ugly
“A mathematician’s reputation rests on the number of bad proofs he has given. (Pioneer work is clumsy.)” — A. S. Besicovitch I’m sure I’ve written about this quote somewhere, but I can’t find where. The quote comes from A Mathematician’s Miscellany by J. E. Littlewood, citing Besicovitch. I’ve more often
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