Heterostasis is a concept from cybernetics that refers to the ability of a system to maintain stability through adaptation and change. Unlike homeostasis, which focuses on maintaining equilibrium or stability within a system by counteracting disturbances, heterostasis acknowledges that systems must sometimes adjust or change their functioning in response to varying external conditions in order to maintain overall stability. In cybernetic terms, heterostasis emphasizes the dynamic interactions and feedback loops that allow a system to respond effectively to external influences and changes.

Derrick's theorem is a result in the field of mathematical physics, particularly in the study of field theories and solitons. It concerns the stability of soliton solutions to certain field equations, specifically addressing the stability under small perturbations of the solutions. The theorem states that if a field configuration (such as a soliton) is localized and satisfies certain energy conditions, then it is stable against small perturbations if and only if its energy does not decrease under rescaling of the spatial variables.

Γ-convergence is a concept in the field of mathematical analysis, particularly in the study of functional analysis, calculus of variations, and optimization. It provides a way to analyze the convergence of functionals (typically a sequence of functions or energy functionals) in a manner that is particularly useful when studying minimization problems and variational methods.

The Sørensen–Dice coefficient (also known simply as the Dice coefficient or Dice similarity coefficient) is a statistical measure used to gauge the similarity between two sets. It is particularly useful in fields such as biology, natural language processing, and image analysis, where it helps in comparing the similarity and diversity of sample sets.

As of my last update in October 2023, "Gigapackets" isn't a widely recognized term in technology or networking. However, it can segment into two familiar concepts: "giga" which often refers to a billion (10^9) and is used in contexts related to data measurement (like gigabytes or gigabits), and "packets" which refers to units of data formatted for transmission over network protocols.

Computed tomography (CT) of the thyroid refers to the use of computed tomography imaging techniques to obtain detailed cross-sectional images of the thyroid gland and surrounding structures in the neck. This diagnostic imaging modality provides information about the anatomy and pathologies of the thyroid, such as nodules, tumors, cysts, and inflammatory conditions.

The Society of Cardiovascular Computed Tomography (SCCT) is a professional organization dedicated to advancing the field of cardiovascular imaging, particularly through the use of computed tomography (CT) technology. Founded in 2005, SCCT focuses on improving patient care and healthcare outcomes by promoting education, research, and collaboration in cardiovascular CT.

Alan Turing was a British mathematician, logician, cryptanalyst, and computer scientist, widely regarded as one of the fathers of computer science and artificial intelligence. Born on June 23, 1912, Turing made significant contributions to various fields, including mathematics, logic, and computer science. One of his most notable accomplishments during World War II was his work at Bletchley Park, where he played a crucial role in breaking the German Enigma code.

Anne Condon is a notable computer scientist known for her work in computational complexity theory, algorithms, and bioinformatics. She has made significant contributions to various areas of computer science, particularly in understanding the computational limits of problems and the design of efficient algorithms. Condon has held academic positions, including being a faculty member at institutions like the University of British Columbia. Her research often explores the intersection of computer science and biology, particularly in developing algorithms for analyzing biological data and understanding biological processes through a computational lens.

James D. Murray is a prominent figure in the field of applied mathematics and mathematical biology. He is best known for his contributions to mathematical modeling in biological systems, including ecology, epidemiology, and the spread of diseases. His work often involves using differential equations to describe dynamic systems in biology. Murray is also the author of the influential textbook "Mathematical Biology," which has been used widely in academia to teach the principles of applying mathematical techniques to biological problems.

Joel E. Cohen is a distinguished mathematician and researcher known for his work in various fields, including mathematical biology, ecological modeling, and applied mathematics. He is recognized for his contributions to understanding population dynamics, demographics, and resource management through quantitative methods. Cohen has held academic positions at institutions such as Columbia University, where he has engaged in interdisciplinary research that intersects science, mathematics, and social issues. His work often focuses on complex systems and how mathematical models can inform our understanding of biological and ecological processes.

The Schrödinger equation Hamiltonian has to be a Hermitian so we will have only positive energies I think: quantumcomputing.stackexchange.com/questions/12113/why-does-a-hamiltonian-have-to-be-hermitian

This is basically how quantum computing was first theorized by Richard Feynman: quantum computers as experiments that are hard to predict outcomes.

TODO answer that: quantumcomputing.stackexchange.com/questions/5005/why-it-is-hard-to-simulate-a-quantum-device-by-a-classical-devices. A good answer would be with a more physical example of quantum entanglement, e.g. on a photonic quantum computer.

This equation is a subcase of Equation "Schrödinger equation for a one dimensional particle" with $V(x)=x^2$.

We get the time-independent Schrödinger equation by substituting this $V$ into Equation "time-independent Schrödinger equation for a one dimensional particle":

Now, there are two ways to go about this.

The first is the stupid "here's a guess" + "hey this family of solutions forms a complete basis"! This is exactly how we solved the problem at Section "Solving partial differential equations with the Fourier series", except that now the complete basis are the Hermite functions.

The second is the much celebrated ladder operator method.

Examples:

Is the only atom that has a closed form solution, which allows for very good predictions, and gives awesome intuition about the orbitals in general.

It is arguably the most important solution of the Schrodinger equation.

In particular, it predicts:

The explicit solution can be written in terms of spherical harmonics.

Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.

Fixed quantum angular momentum in a given direction.

Can range between $\pm l$.

E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:

The z component of the quantum angular momentum is simply:so e.g. again for gallium:

Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure. $z$ is normally used by convention.