Flash photolysis is a technique used in spectroscopy and photochemistry to study rapid chemical reactions and dynamics. It involves the use of a brief, intense flash of light (typically ultraviolet or visible light) to initiate a chemical reaction or to excite molecules from a ground state to an excited state. The general procedure includes the following steps: 1. **Preparation**: A sample containing the chemical species of interest is prepared in a suitable medium, such as a gas or liquid.

Bayesian Structural Time Series (BSTS) is a framework used for modeling and forecasting time series data that incorporates both structural components and Bayesian methods. The BSTS framework is particularly useful for analyzing data with complex patterns, such as trends, seasonality, and irregularities, while also allowing for the incorporation of various types of uncertainty. ### Key Components of Bayesian Structural Time Series: 1. **Structural Components**: - **Trend**: Captures long-term movements in the data.

A deflator is an economic measure used to adjust nominal economic indicators, such as Gross Domestic Product (GDP), to account for changes in price levels over time. It allows for the differentiation between real growth (adjusted for inflation) and nominal growth (not adjusted for inflation).

The Hodrick-Prescott (HP) filter is a mathematical tool used in macroeconomics and time series analysis to decompose a time series into a trend component and a cyclical component. It is particularly useful for analyzing economic data, such as GDP or other macroeconomic indicators, to separate the long-term trend from short-term fluctuations.

"Songs & More Songs by Tom Lehrer" is a compilation album by American musician and satirist Tom Lehrer. Released in 1965, the album features a collection of Lehrer's humorous and clever songs, showcasing his unique style of combining wit with musical talent. Lehrer is known for his satirical take on a variety of topics, including politics, education, and social issues, often using a blend of classical music influences and catchy melodies.

Secular variation refers to the long-term changes or trends observed in a particular phenomenon over an extended period, typically spanning decades to centuries. This term is commonly used in various fields, such as geology, paleoclimatology, and even economics, to describe gradual changes that are not tied to periodic cycles (like seasonal or annual changes). In the context of geology and geomagnetism, secular variation may refer to the gradual changes in the Earth's magnetic field intensity and direction over time.

A **stationary distribution** is a concept primarily used in the context of Markov chains and stochastic processes. It refers to a probability distribution that remains unchanged as time progresses. In other words, if the system is in the stationary distribution, the probabilities of being in each state do not change over time.

The crossing number of a graph is a classic concept in graph theory that refers to the minimum number of edge crossings in a drawing of the graph in the plane. When a graph is drawn on a two-dimensional surface (like a piece of paper), edges can sometimes cross over each other. The goal is to find a layout of the graph that minimizes these crossings. Here's a more detailed explanation: 1. **Graph**: A graph consists of vertices (or nodes) connected by edges (or links).

A **graph manifold** is a class of 3-dimensional manifolds characterized by their geometric structure, specifically how they can be decomposed into pieces that look like typical geometric shapes (in this case, they resemble a torus and other types of three-manifolds).

Chabauty topology is a concept used in algebraic geometry and arithmetic geometry, specifically in the study of the spaces of subvarieties of algebraic varieties. It is named after the mathematician Claude Chabauty, who developed this topology in the context of algebraic varieties and their rational points. In the Chabauty topology, one can think about the space of closed subsets of a given topological space (often within a certain context such as algebraic varieties).

A toroidal graph is a type of graph that can be embedded on the surface of a torus without any edges crossing. In other words, it can be drawn on the surface of a doughnut-shaped surface (a torus) in such a way that no two edges intersect except at their endpoints.

In the context of topology and abstract algebra, an **extension** of a topological group refers to a way of constructing a new topological group from a known one by incorporating additional structure. This often involves creating a new group whose structure represents a combination of an existing group and a simpler group.

A homogeneous space is a mathematical structure that exhibits a high degree of symmetry. More formally, in the context of geometry and algebra, a homogeneous space can be defined as follows: 1. **Definition**: A space \(X\) is called a homogeneous space if for any two points \(x, y \in X\), there exists a symmetry operation (usually described by a group action) that maps \(x\) to \(y\).

A **locally compact group** is a type of topological group that has the property of local compactness in addition to the group structure. Let's break down the definitions: 1. **Topological Group**: A group \( G \) is equipped with a topology such that both the group operation (multiplication) and the inverse operation are continuous.

A **monothetic group** is a term used in the context of taxonomy and systematics, particularly in the classification of organisms. It refers to a group of organisms that are united by a single common characteristic or a single attribute that defines that group. This characteristic is often a specific trait or combination of traits that all members of the group share, distinguishing them from organisms outside the group.

Hodge theory is a central area in differential geometry and algebraic geometry that studies the relationship between the topology of a manifold and its differential forms. It is particularly concerned with the decomposition of differential forms on a compact, oriented Riemannian manifold and the study of their cohomology groups. The key concepts in Hodge theory are: 1. **Differential Forms**: These are generalized functions that can be integrated over manifolds.

Cartan's theorems A and B are fundamental results in the theory of differential forms and the classification of certain types of differential equations, particularly within the context of differential geometry and the theory of distributions.

In algebraic geometry and related fields, a **coherent sheaf** is a specific type of sheaf that combines the properties of sheaves with certain algebraic conditions that make them suitable for studying geometric objects.

In algebraic geometry, a *motive* is a concept that originates from the desire to unify various cohomological theories and establish connections between them. It is part of the broader framework known as **motivic homotopy theory**, which aims to study algebraic varieties using techniques and tools from homotopy theory and algebraic topology.

The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that provides a powerful tool for calculating dimensions of certain spaces of sections of line bundles on smooth projective curves.