In dynamical systems, an equilibrium point is a point where the system can remain indefinitely if it starts there, assuming no external disturbances. An equilibrium point is classified based on its stability properties, which are determined by analyzing the behavior of the system near that point. A **hyperbolic equilibrium point** is a specific type of equilibrium point where the linearization of the system at that point has no eigenvalues with zero real parts.
Massera's lemma is a result in the field of differential equations and dynamical systems, particularly related to the stability of solutions to nonlinear differential equations. It is often applied in the context of the stability of solutions to the perturbed systems in the vicinity of an equilibrium point. The lemma provides a criterion for the asymptotic behavior of solutions to a nonlinear differential equation.
A Burns stanza, named after the Scottish poet Robert Burns, typically consists of a four-line stanza (quatrain) with a specific rhyme scheme of AABB. This format often emphasizes the rhythmic and musical quality of the verse, which is a hallmark of Burns's poetry. The meter is commonly iambic, making it suitable for singing. Burns’s use of the stanza is prominent in many of his works, contributing to their lyrical and folk-like nature.
"Laisser" is a French verb meaning "to let" or "to leave." However, if you are referring to "Laisse" specifically, it might not have a direct meaning in English and could depend on the context. In some contexts, "laisse" could refer to a type of poetic form, particularly in medieval literature, where it denotes a stanza or a verse. In other instances, it might be used in artistic or literary discussions.
A sestina is a structured form of poetry consisting of six stanzas of six lines each, followed by a three-line stanza called a tercet or envoi. The defining feature of a sestina is its intricate pattern of word repetition. Instead of traditional rhyme, the sestina employs a specific pattern in which the end words of the first stanza are used in a particular order in the following stanzas.
The Onegin stanza, also known as the "Pushkin sonnet," is a poetic form that consists of 14 lines arranged in a specific rhyme scheme and meter. It was popularized by the Russian poet Alexander Pushkin in his novel in verse, "Eugene Onegin." The form typically consists of a sequence of alternating rhymes and is written in iambic tetrameter.
Riding rhyme, often referred to as "riding rhyme" or "riding poetry," is a form of humorous verse that is traditionally associated with children's literature and nursery rhymes. It is characterized by its playful, rhythmic qualities and often includes elements of repetition, alliteration, and a simple storyline or theme. The term can also refer specifically to rhymes that accompany activities or games, particularly those involving horses or other forms of transportation.
Here is a list of some notable star systems located within 30 to 35 light-years from Earth: ### 1. **Alpha Centauri** - **Distance:** ~4.37 light-years - **Components:** Alpha Centauri A, Alpha Centauri B, Proxima Centauri (closest star to the Sun) ### 2. **Barnard's Star** - **Distance:** ~5.
One important quantum mechanics experiment, which using quantum effects explain the dependency of specific heat capacity on temperature, an effect which is not present in the Dulong-Petit law.
This is the solid-state analogue to the black-body radiation problem. It is also therefore a quantum mechanics-specific phenomenon.
It can be seen as the limit case of an Einstein solid at high temperatures. At lower temperatures, the heat capacity depends on temperature.
What makes lasers so special: Lasers vs other light sources.
How Lasers Work by Scientized (2017)
Source. An extremely good overview of how lasers work. Clearly explains the electron/photon exchange processes involved, notably spontaneous emission.
Talks about the importance of the metastable state to achieve population inversion.
Also briefly explains the imperfections that lead to the slightly imperfect non punctual spectrum seen in a real laser.
- youtu.be/_JOchLyNO_w?t=188 says LED is "also monochromatic", but that is not strictly true, it has way way larger frequency band than a laser. Only narrower compared to other sources such as incandescent light bulbs.
- youtu.be/_JOchLyNO_w?t=517 stimulated emission. This is the key to laser formation as it produces coherent photons.
- youtu.be/_JOchLyNO_w?t=581 spontaneous emission happens too fast (100 ns), which is not enough time for stimulated emission to happen. Metastable electrons to the rescue.
- youtu.be/_JOchLyNO_w?t=832 the parallel mirrors select perpendicular photons preferentially
The type of laser described at: Video "How Lasers Work by Scientized (2017)", notably youtu.be/_JOchLyNO_w?t=581. Mentioned at: youtu.be/_JOchLyNO_w?t=759 That point also mentions that 4-level lasers also exist and are more efficient. TODO dominance? Alternatives?
Bibliography:
Pinned article: Introduction to the OurBigBook Project
Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
Intro to OurBigBook
. Source. We have two killer features:
- topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculusArticles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.Figure 1. Screenshot of the "Derivative" topic page. View it live at: ourbigbook.com/go/topic/derivativeVideo 2. OurBigBook Web topics demo. Source. - local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
Figure 3. Visual Studio Code extension installation.Figure 4. Visual Studio Code extension tree navigation.Figure 5. Web editor. You can also edit articles on the Web editor without installing anything locally.Video 3. Edit locally and publish demo. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.Video 4. OurBigBook Visual Studio Code extension editing and navigation demo. Source. - Infinitely deep tables of contents:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact





