The term "Curves" can refer to different concepts depending on the context in which it's used. Here are some of the common interpretations: 1. **Mathematics**: In mathematics, a curve is a continuous and smooth flowing line without sharp angles. Curves can be defined in different dimensions and can represent various functions or relationships in geometry and calculus. 2. **Statistics and Data Analysis**: In statistics, curves can represent distributions, trends, or relationships between variables.

Smooth manifolds are a fundamental concept in differential geometry and provide a framework for studying shapes and spaces that can be modeled in a way similar to Euclidean spaces. Here’s a more detailed explanation: ### Definition A **smooth manifold** is a topological manifold equipped with a global smooth structure.

In differential geometry, theorems are statements that have been proven to be true based on definitions, axioms, and previously established theorems within the field. Differential geometry itself is the study of curves, surfaces, and more generally, smooth manifolds using the techniques of differential calculus and linear algebra. It combines elements of geometry, calculus, and algebra.

A \((G, X)\)-manifold is a mathematical structure that arises in the context of differential geometry and group theory. In particular, it generalizes the notion of manifolds by introducing a group action on a manifold in a structured way. Here’s a breakdown of the components: 1. **Manifold \(X\)**: This is a topological space that locally resembles Euclidean space and allows for the definition of concepts such as continuity, differentiability, and integration.

Affine differential geometry is a branch of mathematics that studies the properties and structures of affine manifolds, which are manifolds equipped with an affine connection. Unlike Riemannian geometry, which relies on the notion of a metric to define geometric properties like lengths and angles, affine differential geometry primarily focuses on the properties that are invariant under affine transformations, such as parallel transport and affine curvature.

Analytic torsion is a concept in mathematical analysis, particularly in the fields of differential geometry and topology, relating to the behavior of certain types of Riemannian manifolds. It arises in the context of studying the spectral properties of differential operators, especially the Laplace operator.

A developable surface is a type of surface in geometry that can be flattened into a two-dimensional plane without distortion. This means that the surface can be "unfolded" or "rolled out" in such a way that there is no stretching, tearing, or compressing involved. Developable surfaces include shapes like: 1. **Planes**: Flat surfaces are obviously developable as they are already two-dimensional.

Cartan's equivalence method is a powerful mathematical framework developed by the French mathematician Henri Cartan in the early 20th century. It is primarily used in the field of differential geometry and the theory of differential equations, particularly for understanding the equivalence of geometric structures and their associated systems of differential equations.

Clairaut's relation, also known as Clairaut's theorem, is a fundamental result in differential geometry that relates the curvature of a surface to the derivatives of the surface's height function. Specifically, it applies to surfaces of revolution, which are surfaces generated by rotating a curve about an axis.

Clifford analysis is a branch of mathematical analysis that extends classical complex analysis to higher-dimensional spaces using the framework of Clifford algebras. It focuses on functions that operate in spaces equipped with a geometric structure defined by Clifford algebras, which generalize the concept of complex numbers to higher dimensions. In Clifford analysis, the primary objects of interest are functions that are defined on domains in Euclidean spaces and take values in a Clifford algebra.

A **closed geodesic** is a type of curve on a manifold that has several important properties in differential geometry and topology. Here are the key characteristics: 1. **Geodesic**: A geodesic is a curve that locally minimizes distance and is a generalization of the concept of a "straight line" to curved spaces. It can be defined as a curve whose tangent vector is parallel transported along the curve itself.

For Ciro Santilli's campaign for freedom of speech in China: Section "github.com/cirosantilli/china-dictatorship".

Ciro has the radical opinion that absolute freedom of speech must be guaranteed by law for anyone to talk about absolutely anything, anonymously if they wish, with the exception only of copyright-related infringement.

And Ciro believes that there should be no age restriction of access to any information.

People should be only be punished for actions that they actually do in the real world. Not even purportedly planning those actions must be punished. Access and ability to publish information must be completely and totally free.

If you don't like someone, you should just block them, or start your own campaign to prepare a counter for whatever it is that they are want to do.

This freedom does not need to apply to citizens and organizations of other countries, only to citizens of the country in question, since foreign governments can create influence campaigns to affect the rights of your citizens. More info at: cirosantilli.com/china-dictatorship/mark-government-controlled-social-media

Limiting foreign influence therefore requires some kind of nationality check, which could harm anonymity. But Ciro believes that almost certainly such checks can be carried out in anonymous blockchain consensus based mechanisms. Governments would issues nationality tokens, and tokens are used for anonymous confirmations of rights in a way that only the token owner, not even the government, can determine who used the token. E.g. something a bit like what Monero does. Rights could be checked on a once per account basis, or yearly basis, so transaction costs should not be a big issue. Maybe expensive proof-of-work systems can be completely bypassed to the existence of this central token authority?

Some people believe that freedom of speech means "freedom of speech that I agree with". Those people should move to China or some other dictatorship.

Related: cirosantilli.com/china-dictatorship/hate-speech

A **closed manifold** is a type of manifold that is both compact and without boundary. More specifically, a manifold \( M \) is called closed if it satisfies the following conditions: 1. **Compact**: This means that the manifold is a bounded space that is also complete, meaning that every open cover of the manifold has a finite subcover. In simple terms, a compact manifold is one that is "finite" in a sense and can be covered by a finite number of open sets.

A **differential invariant** is a property or quantity in differential geometry that remains unchanged under particular types of transformations, usually involving differentiable functions or mappings. These invariants play a crucial role in studying geometric objects and their properties without being affected by the coordinate system or parameterization used to describe them.

Dynamic fluid film equations are mathematical formulations that describe the behavior of thin films of fluid that flow under the influence of various forces, such as gravity, surface tension, and viscous forces. These equations are crucial in understanding phenomena in various fields, including materials science, engineering, and fluid dynamics. In general, a fluid film can be considered a thin layer of fluid with a small thickness compared to its lateral dimensions.

In differential geometry, the concept of the Laplace operator, often denoted as \(\Delta\) or \(\nabla^2\), is a generalization of the Laplacian from classical analysis to manifolds. It plays a significant role in understanding the geometric and analytical properties of functions defined on a manifold.

"Discoveries" by Josep Comas Solà is not a widely known work or publication, so it may not have significant recognition in mainstream literature or academic contexts.

A **G-fibration** is a concept in the field of algebraic topology, particularly in relation to homotopy theory and the study of fiber spaces. It is a generalization of the notion of a fibration, and it is typically associated with certain kinds of structured spaces and diagrams. In a broad sense, a G-fibration is a fibration where the fibers are not just sets but are equipped with a group action, typically from a topological group \( G \).