An implicit function is a function that is defined implicitly rather than explicitly. In other words, it is not given in the form \( y = f(x) \). Instead, an implicit function is defined by an equation that relates the variables \( x \) and \( y \) through an equation of the form \( F(x, y) = 0 \), where \( F \) is a function of both \( x \) and \( y \).

Regular homotopy is a concept from algebraic topology, specifically in the field of differential topology. It relates to the study of two smooth maps from one manifold to another and the idea of deforming one map into another through smooth transformations. In a more precise sense, let \( M \) and \( N \) be smooth manifolds.

In the context of topology and differential geometry, a **section** of a fiber bundle is a continuous function that assigns to each point in the base space exactly one point in the fiber. More formally, let's break this down: ### Fiber Bundle A **fiber bundle** consists of the following components: 1. **Base Space** \( B \): A topological space where the "fibers" are defined.

A **symplectic manifold** is a smooth manifold \( M \) equipped with a closed non-degenerate differential 2-form called the **symplectic form**, typically denoted by \( \omega \). Formally, a symplectic manifold is defined as follows: 1. **Manifold**: \( M \) is a differentiable manifold of even dimension, usually denoted as \( 2n \), where \( n \) is a positive integer.

The Whitney conditions refer to certain criteria in differential topology, specifically regarding the behavior of certain mappings and the properties of manifolds. There are two primary types of Whitney conditions: the Whitney condition for embeddings and the Whitney condition for stratifications of topological spaces. 1. **Whitney Condition for Embeddings:** This condition is concerned with the behavior of smooth maps between manifolds. Specifically, it provides conditions under which a smooth map between manifolds is an embedding.

Causality conditions refer to the criteria or principles that must be met in order to establish a causal relationship between two or more variables. In various fields such as statistics, philosophy, and science, causality is a foundational concept that helps in understanding how one event (the cause) can influence another event (the effect). Here are some key aspects typically associated with causality conditions: 1. **Temporal Precedence**: The cause must precede the effect in time.

The McVittie metric is a solution to the Einstein field equations in the context of general relativity that describes a specific type of spacetime geometry. It is named after the physicist William P. McVittie, who introduced it in the context of cosmology and gravitational theory. The McVittie metric represents a static, spherically symmetric gravitational field that can be considered as a black hole surrounded by a cosmological constant, which accounts for the effects of the expanding universe.

A Penrose diagram, also known as a conformal diagram, is a two-dimensional depiction of the causal structure of spacetime in the context of general relativity. It is named after the physicist Roger Penrose, who developed this diagrammatic representation to help visualize complex features of spacetime, especially in the vicinity of black holes and cosmological models.

Canonical quantum gravity is a theoretical framework that seeks to quantize the gravitational field using the canonical approach, which is derived from Hamiltonian mechanics. This approach is distinctive because it aims to reconcile general relativity, the classical theory of gravitation, with quantum mechanics, providing insights into how gravity behaves at the quantum scale. The key features of canonical quantum gravity include: 1. **Hamiltonian Formulation**: It begins by expressing general relativity in a Hamiltonian framework.

Linearized gravity is an approximation of general relativity that simplifies the complex equations describing the gravitational field. It is based on the idea that the gravitational field can be treated as a small perturbation around a flat spacetime, typically Minkowski spacetime, which describes a region of spacetime without significant gravitational effects. In the framework of general relativity, the gravitational field is represented by the geometry of spacetime, which is described by the Einstein field equations.

The geodesic equations describe the paths taken by particles moving under the influence of gravity in a curved spacetime, such as that described by Einstein's theory of general relativity. A geodesic represents the shortest path between two points in a curved space, analogous to a straight line in flat (Euclidean) space. In mathematical terms, the geodesic equations can be derived from the principle of least action or variational principles and are expressed in the form of a second-order differential equation.

As of my last knowledge update in October 2021, there isn't a widely known figure named Andrew Neitzke who is prominent in public discourse, media, or scholarship. It's possible that he could be a private individual or a relatively unknown person in a specific field. If Andrew Neitzke has since gained prominence or relevance in any particular context after this date, I wouldn't have that information.

Anne Schilling could refer to a number of individuals, and without more context, it's difficult to provide a specific answer. If you're referring to a person, it could be beneficial to include additional details, such as their profession, achievements, or any specific context in which you've heard the name. For instance, Anne Schilling might refer to a researcher, artist, or another professional in a particular field.

Eric Zaslow is a name that doesn't point to a widely recognized public figure or concept as of my last knowledge update in October 2023. If he is a person relevant in a specific context, such as a professional field or a recent news story, please provide more details, and I'll do my best to assist you. Otherwise, it's possible that he might not be a prominent figure within the commonly available sources.

Feza Gürsey (1921-1992) was a prominent Turkish physicist known for his contributions to theoretical physics, particularly in the fields of general relativity and quantum mechanics. He made significant contributions to the understanding of various physical phenomena and was involved in the development of theoretical frameworks that linked ideas in particle physics and cosmology. Gürsey was also dedicated to science education and played an important role in promoting scientific research in Turkey.

George Henry Livens was a notable British inventor and soldier who is best known for developing the Livens Projector, a form of artillery designed during World War I. The Livens Projector was essentially a large, muzzle-loading device that could launch canisters filled with gas or other chemical agents, enabling the delivery of chemical warfare over a significant distance. This innovation was part of a broader trend during the war where various nations explored the use of chemical weapons.

A Halo orbit is a type of orbital path that an object can take around a point in space, specifically around a Lagrangian point in the Earth-Moon system or any other two-body system. Lagrangian points are positions in space where the gravitational forces of two large bodies, like the Earth and the Moon, balance out the centrifugal force felt by a smaller object. There are five such points, denoted as L1, L2, L3, L4, and L5.

Muneer Ahmad Rashid appears to be a name that may refer to a specific individual, but as of my last knowledge update in October 2023, there isn’t widely available information on a person by that name.

N. V. V. J. Swamy could refer to a specific individual, acronym, or terminology that may not have widespread recognition beyond certain contexts. Without additional information or context, it’s difficult to provide a precise answer.

Peter Tait was a Scottish physicist and mathematician known for his work in the field of mathematical physics during the 19th century. He was born on November 27, 1831, and passed away on December 10, 1901. Tait is particularly recognized for his contributions to the study of knots and linkages, which are fundamental concepts in topology.