Competitive equilibrium refers to a state in an economic market where supply equals demand, and no individual buyer or seller can influence the market price. In this scenario, the prices of goods and services are determined by the interplay of supply and demand, and every participant (consumers, firms, etc.) in the market makes choices that maximize their utility (consumers) or profits (producers) given the existing market prices.

A spectrum auction is a process used by governments or regulatory authorities to allocate radio frequency spectrum rights to telecommunications companies and wireless service providers. Radio frequency spectrum is a limited natural resource that enables wireless communication, including mobile phone services, radio and television broadcasting, and various forms of wireless data transmission.

The Australian Capital Territory (ACT) is known for several notable attractions, landmarks, and features. Here are some of the "big things" associated with the ACT: 1. **Canberra**: As the capital city of Australia, Canberra is central to the country's politics and culture. It is home to important national institutions.

Western Australia is known for several "big things" that are popular tourist attractions. These oversized structures often celebrate local culture, industry, or natural features. Here are some of the notable "big things" in Western Australia: 1. **The Big Banana** - Located in Coffs Harbour, although not technically in Western Australia, it is a well-known example of the "big things" phenomenon found throughout Australia.

In New South Wales (NSW), Australia, "big things" typically refer to large, often quirky monuments or structures that are tourist attractions throughout the state. Here are some notable examples: 1. **The Big Banana** - Located in Coffs Harbour, it's one of the first and most famous big things in Australia. It features a banana-themed park with attractions like water slides and a mini-golf course.

William Compston could refer to a specific individual, but it appears that there isn't a widely recognized or notable figure by that name in public records, literature, or media up to October 2023. It’s possible that he could be a private individual, a fictional character, or someone emerging in prominence after my last update.

"Big things" in Victoria, Australia, refer to a collection of oversized structures or sculptures that are often found along highways and in tourist areas. These quirky attractions are usually named after local industries, landmarks, or wildlife, and they serve as fun photo opportunities for travelers.

Martin Pope is a prominent figure in the field of solid-state physics and material science. He is particularly known for his contributions to the understanding of organic semiconductors and the development of organic electronic devices, such as organic light-emitting diodes (OLEDs) and organic solar cells. His research has significantly advanced the field and helped in the commercialization of various organic electronic technologies.

Australian astrophysicists are scientists who study the properties and interactions of celestial bodies and the universe as a whole, based in Australia. They work in various fields within astrophysics, including cosmology, stellar dynamics, planetary science, and observational astronomy. These researchers often collaborate with international teams and contribute to large-scale projects, such as those involving space telescopes, observatories, and theoretical research.

Australian women physicists have made significant contributions to the field of physics across various sub-disciplines, including astrophysics, condensed matter physics, and particle physics, among others. Some notable Australian women physicists include: 1. **Lisa Kewley** - An astrophysicist known for her work on the evolution of galaxies and the interstellar medium. 2. **Elizabeth Gillies** - A physicist specializing in quantum mechanics and quantum computing.

Massimiliano Di Ventra is a theoretical physicist known for his work in the fields of condensed matter physics, nanotechnology, and computational physics. His research often focuses on quantum transport, electron transport in nanostructures, and the development of theoretical frameworks for understanding complex systems at the nanoscale. He is affiliated with a university or research institution, where he may also engage in teaching and mentoring students.

"Alan Head" does not appear to refer to a widely recognized concept, individual, or term as of my last update in October 2023. It’s possible that you may be referring to something more specific or niche that has emerged since then, or it could be a misspelling or miscommunication regarding a different term or name.

Andrew G. White could refer to a variety of individuals or entities, depending on the context. However, without specific information, it's difficult to pinpoint exactly who or what you're referring to. If you mean a person, it could refer to an academic, professional, or a public figure with that name. If you mean an organization or a company, it could refer to a business or a brand that includes that name.

Hope is a functional programming language that was designed in the early 1980s, primarily by the computer scientist Gordon P. A. S. Morrison and others, at the University of Cambridge. It was created as a research tool to explore concepts related to functional programming, type systems, and polymorphism.

Ben Eggleton is a notable figure in the field of physics and engineering, particularly known for his work in optical physics and nanotechnology. He has been involved in research focused on photonics and advanced materials. Additionally, he has held academic positions and contributed to various scientific publications.

Let's start with the one dimensional case. Let the $q:\mathbb{R}\to \mathbb{R}$ and a Functional $I$ defined by a function of three variables $F:{\mathbb{R}}^3\to \mathbb{R}$:

Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given $F$, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that $F$ such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

By $\frac{\partial F}{\partial q}$ and $\frac{\partial F}{\partial \dot{q}}$ we simply mean "the partial derivative of $F$ with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where $q$ is the unknown, since:

Now let's think about the multi-dimensional case. Instead of having $q:\mathbb{R}\to \mathbb{R}$, we now have $q:\mathbb{R}\to {\mathbb{R}}^n$. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case, $F$ is going to be a function of 5 variables $F:{\mathbb{R}}^5\to \mathbb{R}$ rather than 3 as in the one dimensional case, and the functional looks like:

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions $q_1$ and $q_2$ of order up to 2 in both variables:At this point, notation is getting a bit clunky, so people will often condense the $q_i$ vectoror just omit the arguments of $F$ entirely: