First is the axiom of extensionality, if two sets contain exactly the same elements, then they are the same set. The second one is axiom of empty set. Third, axiom of pairing, if you have two sets, you can put them together into a new set. axiom of unition meaning to merge sets together, axiom … Continue reading The Nine Axioms of Set Theory

For centuries, mathematics education has been anchored in Euclidean geometry and Cartesian coordinates. From the rigid constructs of points, lines, and planes to the static nature of axes and fixed coordinate systems, students have been taught an outdated framework that does not reflect the deep, dynamic structures that govern modern physics and engineering. It is … Continue reading Rethinking Math Education: From Euclidean Geometry to Differential Geometry

First of all, Geometry, bundles, differential manifolds, topological manifolds, topology, set theory and underneath the logic to build the mathematic edifice to understand truly law of nature: special relativity, classical physics, quantum physics, general relativity, chemistry and electromagnetism etc. The paper's content is as follows: the first filed theory on James Clerk Maxwell, Herman Weyl, … Continue reading Fiber bundles, Yang and the geometry of spacetime: paper by Federico Pasinato 01

Core Architecture OpenManus is built on an agent-based architecture with a hierarchical inheritance pattern: BaseAgent (app/agent/base.py): The foundation class that provides: Memory management State transitions Execution loop control Stuck state detection ReActAgent (app/agent/react.py): Extends BaseAgent with: Think-Act cycle pattern Abstract methods for thinking and acting ToolCallAgent (app/agent/toolcall.py): Extends ReActAgent with: Tool/function calling capabilities Tool execution handling Response processing … Continue reading OpenManus Project Structure

The Proca Lagrangian is a fundamental concept in Quantum Field Theory (QFT) that describes the dynamics of a massive vector field (a field associated with a particle of spin-1 that has mass). It is named after the Romanian physicist Alexandru Proca, who first formulated it. The Proca Lagrangian is particularly important in the study of massive gauge bosons, such as … Continue reading Proca Lagrangian 

When undertaking the transition to advanced studies of curved space, in contrast to Euclidean space, it is essential to comprehend the notion of covariant derivatives, while also further exploring our established understanding of directional derivatives to effectively distinguish between the two. Key Differences Curvature Effects The directional derivative does not consider curvature—it is simply the … Continue reading Compare Directional Derivative and Covariant Derivative

A comprehensive understanding of the ordinary derivative in flat space is well established; nevertheless, the requirement for the covariant derivative emerges from the characteristics of curved space. In our universe, specifically within the framework of space-time, spaces are intrinsically curved as opposed to flat. Accordingly, the derivative must incorporate the consideration that the basis vectors … Continue reading What is Covariant Derivative in Flat and Curved Space

If we need to measure arc length, the rigorous way is as the following: The part in the middle is tensor metric. but it is from point of extrinsic view, what if we were an ant on the surface made up of u and v, how to calculate the arc length? it's similar as below: … Continue reading How Is Metric Tensor Deduced?

To truly understand how gauge field is required in the 4d version of the electromagnetism equation, However when asking what and how covariant derivative is defined and derived, AI hardly can be helpful other reciting dull and dry textbook words. The best brains of human beings are irreplicable by AI. So here I'd revisit eigenchris's … Continue reading What is Covariant Derivative in EM?

It's always an effective strategy to build from the simplest form for anything, so is applied to build MCP server. Thanks to Matthew Berman, he already posted a clip detailing how he created a "count number of Rs" in a word MCP server and run it in Claude chat box: Then go to the codes … Continue reading Build a Very Simple “Count Number of R” MCP Server