numerical-methods

Let $u$ solve the heat equation$$ u_t = u_{xx} \quad \text{in } (0,\pi)\times(0,\infty),$$ with Dirichlet boundary conditions $$ u(0,t)=u(\pi,t)=0,$$ and initial data $$ u(x,0)=f(x), $$ where $f \in ...

If you’ve spent any time in the open-source AI community recently, you’ve probably seen someone excitedly announce they are running a 70B parameter model locally, only to follow up an hour later asking why their system crashed with an OOM (Out of Memory) error. Deploying Large Language Models (LLMs) locally—whether for privacy, cost savings, or offline availability—is the new frontier for develop…

Has any one any idea to solve this equation J 1 (x) = (x 2 /10)*(J 1 (x) + J 3 (x)), in which J are spherical Bessel function normally write as $j_1 (x)$ and $j_3(x)$ Methods 1 serial expansion: $j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} -... Read more

Inside the nucleus of every cell, DNA doesn’t float freely. It’s wound, folded, bundled into a dense tangle of proteins and genetic material called chromatin, which compacts roughly two metres of DNA into a space just a few millionths of a metre across. Within that tangle are tiny domains, each about 100 nanometres wide (smaller than the wavelength of visible light, for context), and these domain…

Has any one any idea to solve this equation J[1, x] = (x^2/10)*(J[1, x] + J[3, x]), in which J are spherical Bessel function normally write as j_1 (x) and j_3(x) Methods 1 serial expansion: j_1(x) = \frac{\sin(x)}{(x)^2} - \frac{\cos(x)}{x} \approx \dfrac{x}{3} - \dfrac{(x)^3}{30} +... Read more

A couple days ago I wrote a post about turning a trick into a technique, finding another use for a clever way to construct simple, accurate approximations. I used as my example approximating the Bessel function J(x) with (1 + cos(x))/2. I learned via a helpful comment on Mathstodon that my approximation was the first-order […] The post Approximating even functions by powers of cosine first appear…

This paper presents an exhaustive historical and mathematical survey of the seventeen Permanent Axioms underlying the first machine-verified Coq formalisation of a global regularity result for the three-dimensional incompressible Navier-Stokes equations on the periodic torus T³. The formalisation establishes subcritical energy estimates for arbitrary smooth initial data in the Sobolev regularity …

Friends, another possible route to solution of above problem when explaining variables are correlated is to somehow orthogonalize the explaining variables and then use the orthogonalized variables in our analysis. I had thought about this yesterday but problem is that every hermite polynomial component has its own correlation matrix and there is no single correlation matrix that would have to be …

This paper proposes a structural reinterpretation of limits, derivatives, and partial differential equations based on the notion of convergence modes. In classical analysis, limits are defined through asymptotic conditions (e.g., ε–δ formulations), which characterize correctness but do not specify the operational structure by which convergence is realized. In practice, however—whether in numerica…

IntroductionThis simulation study evaluated how model fit in multilevel structural equation models (ML-SEM) is affected by within-person nonuniform measurement bias in intensive longitudinal data (ILD). This kind of bias would be given if item discrimination (i.e., their factor loadings) in multiple-item questionnaires varied within person across time. Prior simulation studies and ILD studies ten…

I have to solve this triple integral ∭ x y|z|³/(1+ (x²+y²)⁴)  dx  dy dz, with a domain T={(x, y, z) ∈ ℝ³: x ≤ 0, y ≥ 0, z² ≤ x²+y² ≤ 1}. Plotting with DESMOS 3D I see this: Actually I am not able to ...

The help page for ?Series and also online here under possible issues says this, and gives an example But when I try the same exact code in V 14.3 under windows 10, it works as is and no error is ...

I am writing a problem sheet for my students (I'm a math teacher) and I would like a way for my code to compute dynamically the solutions: The numerical variables are stored in a \def, so then I can ...

I want to use Parallelize with the creation and execution of dynamic pure functions with 3 parameters: #1, #2, #3. Every functions is created evaluating a couple of values contained into a list of ...

_Https://Doi.Org/10.5281/Zenodo.19605118_. 2026This paper defines operator stability within the Paton System and formalises how admissibility breaks down under sequential constraint. While the Paton Operator Calculus establishes how operators combine into admissible execution paths, this work examines how such paths approach instability and fail. The framework introduces cumulative strain as a st…

_Zenodo_. 2026Models are often treated as though they were generally valid or readily transferable to other contexts. This leads to misapplications, apparent contradictions, and forms of model overextension. By contrast, this paper develops a domain-based perspective within Epistemics as model management under finite conditions. Domains are not presupposed as given regions but are understood as f…

I have a function that takes in a parameter, uses it to define a system of differential equations, solves them uses NDSolve, and then returns a number based on that solution. I want to use NSolve to ...

I am trying to graph a system of 3 inequalities in 3 dimensions. These ineuqalities are: y-z>=1/2x, x-z>=0, and x+y>=3z. Currently I am struggling to get even a basic output. My input is: ...

Hi PF, I'm trying to find an approximate solution of a differential equation that can't be solved in exact form. The differential equation is of the form: $$ f'(t)=g(f(t),t) $$ I want to find the approximate solution in terms of a power series: $$ f(t) \approx f(0) + f'(0) \cdot t +... Read more

Numerical Dynamics　ー数力学ー Note: The full text is written in Japanese in order to preserve the precise definitions and logical structure specific to the language. DOI: 10.5281/zenodo.17772964 Abstract (English) Numerical Dynamics is not presented as an axiomatic system but as a theoretical framework for interpreting the dynamics of meaning in numerical and symbolic operations. The core of this theo…