mathematics

Numerical integration is one of the central tools of scientific computing. Whenever a physical, biological, chemical, mechanical, electrical, or control system is modeled by differential equations, the next practical question is usually not only “what is the equation?” but also “which numerical method should be trusted to simulate it?” This question is more subtle than it first appears. A numeric…

By algebra of similarity type $\beta=\{0,1,\neg,\wedge,\vee\}$ we mean a set where the operations in $\beta$ are defined (arbitrarily). Analogously, by algebra of similarity type ...

superalgebra and (synthetic ) supergeometry A supermanifold is a space locally modeled on Cartesian spaces and superpoints. There are different approaches to the definition and theory of supermanifolds in the literature. The definition is popular. The definition has been argued to have advantages, see also the references at super ∞-groupoid. See at geometry of physics – supergeometry the section …

Let $\mathcal{P}$ be a polygon, and we add CCW direction to its boundary $\partial\mathcal{P}$. Let us measure the angle $\theta_i$ along each edge of the polygon for $i=1,...,N$ where N is the number ...

$$\text{Evaluate:}\quad3 + \cfrac{1}{4 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{3 + \cdots}}}}$$ A standard approach to solve this would be assuming it to be a variable ...

Problem Statement Given a binary array nums , return the maximum number of consecutive 1's present in the array. Example Input: nums = [1,1,0,1,1,1] Output: 3 Explanation: The longest sequence of consecutive 1's is [1,1,1] Length = 3 Brute Force Intuition (Interview Explanation) For every index, if the element is 1 , start moving forward and count how many consecutive 1's exist. Keep updating the…

Problem Statement Given a sorted integer array nums , remove the duplicates in-place such that each unique element appears only once. Return the number of unique elements k . The first k elements of the array should contain the unique elements in their original order. Example Input: nums = [1,1,2] Output: 2 Modified Array: [1,2,_] Brute Force Intuition (Interview Explanation) One straightforward …

Let $X$ be a random variable in $\mathbb{R}^d$ with law $\mu$. We denote by $\mathcal{P}(\mathbb{R}^d)$ the set of all Borel probability measures on $\mathbb{R}^d$. Assume that $\mu \in ...

We propose a family of weighted statistics based on the CUSUM process of the WLS residuals for the online detection of changepoints in a Random Coefficient Autoregressive model, using both the standard CUSUM and the Page-CUSUM process. We derive the asymptotics under the null of no changepoint for all possible weighing schemes, including the case of the standardized CUSUM, for which we derive a D…

This study proposes two novel time-varying model-averaging methods for time-varying parameter regression models. When the number of predictors is small, we propose a novel time-varying complete subset-averaging (TVCSA) procedure, where the optimal time-varying subset size is obtained by minimizing the local leave- h -out cross-validation criterion. The TVCSA method is asymptotically optimal for a…

We study the uniform convergence rates of nonparametric estimators for a probability density function and its derivatives when the density has a known pole. Such situations arise in some structural microeconometric models, for example, in auction, labor, and consumer search, where uniform convergence rates of density functions are important for nonparametric and semiparametric estimation. Existin…

I have a question about what is required in order to simultaneously diagonalize two linear operators. I'm reading through Shankar's Principles of Quantum Mechanics book and he states the following ...

"Do countries with higher GDP per capita also have longer life expectancy?" I built a tool that lets you explore questions like that across 48 countries by picking any two of five metrics as scatter-plot axes. Two implementation hinges: (1) metrics that span orders of magnitude (population: Singapore 5.6M to India 1,417M, a 250× range) must be plotted and correlated on a log scale or every point …

In the figure below of a semi-circle, show that ##h=\sqrt{ab}##. In the special case where ##a=1##, then ##h=\sqrt{b}##. A complete discussion can be found in chapter 2 of the book An Imaginary Tale by Paul J. Nahin.

synthetic differential geometry Introductions geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry Differentials Tangency The magic algebraic facts Theorems Axiomatics Models smooth algebra (-ring) differential equations, variational calculus Chern-Weil theory, ∞-Chern-Weil theory Cartan geometry (super, higher) Given a vector space and an elemen…

As generative AI makes the mass production of technically reasonable work effortless, traditional publication metrics will face (and probably already) are facing a massive inflation. For applied ...

Can someone take a look at this and explain to me like a 5 year old what is happening from node to node in this chain. I understand what it does, but I'd like to intuitively get the 'why' as well. The ...

Let's practice data science thinking through a probability problem The post Solving the 3Blue1Brown String Probability Problem (Without AI) appeared first on Towards Data Science .

I am learning about stochastic processes. Let $X_i \sim \mathrm{Bin}(1,\frac12)$ and independent $M$ and $N$ be the stopping times for the first Tail (i.e. $0$) and first Head (i.e. $1$). My task is ...

Mathematicians have considered how to watch every corner of a space—but soccer adds moving players, blocked views and constant action