The Boy or Girl paradox is a thought experiment in probability that involves a seemingly counterintuitive scenario regarding gender. The classic version goes like this: A family has two children. We know that at least one of the children is a boy. What is the probability that both children are boys? Intuitively, many people might think the probability is 1/2, as there are two equally possible scenarios: either the children are (boy, boy) or (boy, girl).
For this sub-case, we can define the Lie algebra of a Lie group as the set of all matrices such that for all :If we fix a given and vary , we obtain a subgroup of . This type of subgroup is known as a one parameter subgroup.
The immediate question is then if every element of can be reached in a unique way (i.e. is the exponential map a bijection). By looking at the matrix logarithm however we conclude that this is not the case for real matrices, but it is for complex matrices.
TODO example it can be seen that the Lie algebra is not closed matrix multiplication, even though the corresponding group is by definition. But it is closed under the Lie bracket operation.
The Dirac equation can be derived basically "directly" from the Representation theory of the Lorentz group for the spin half representation, this is shown for example at Physics from Symmetry by Jakob Schwichtenberg (2015) 6.3 "Dirac Equation".
The Diract equation is the spacetime symmetry part of the quantum electrodynamics Lagrangian, i.e. is describes how spin half particles behave without interactions. The full quantum electrodynamics Lagrangian can then be reached by adding the internal symmetry.
As mentioned at spin comes naturally when adding relativity to quantum mechanics, this same method allows us to analogously derive the equations for other spin numbers.
Bibliography:
Deriving The Dirac equation by Andrew Dotson (2019)
Source. NCBI online tool to find and view all open reading frames in a given FASTA: www.ncbi.nlm.nih.gov/orffinder/
DNA replication is a key limiting factor of bacterial replication time by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16TODO confirm, but looks like it, e.g. E. Coli starts DNA replication before the previous one finished.
If you live in the relatively food abundant environment of another cell, then you don't have to be able to digest every single food source in existence, of defend against a wide range of predators.
So because DNA replication is a key limiting factor of bacterial replication time, you just reduce your genome to a minimum.
Power, Sex, Suicide by Nick Lane (2006) section "Gene loss as an evolutionary trajectory" puts it well:and also section "How to lose the cell wall without dying" page 184 has some related mentions:
One of the most extreme examples of gene loss is Rickettsia prowazekii, the cause of typhus. [...] Over evolutionary time Rickettsia has lost most of its genes, and now has a mere protein-coding genes left. [...] Rickettsia is a tiny bacterium, almost as small as a virus, which lives as a parasite inside other cells. It is so well adapted to this lifestyle that it can no longer survive outside its host cells. [...] It was able to lose most of its genes in this way simply because they were not needed: life inside other cells, if you can survive there at all, is a spoonfed existence.
While many types of bacteria do lose their cell wall during parts of their life cycle only two groups of prokaryotes have succeeded in losing their cell walls permanently, yet lived to tell the tale. It's interesting to consider the extenuating circumstances that permitted them to do so.[...]One group, the Mycoplasma, comprises mostly parasites, many of which live inside other cells. Mycoplasma cells are tiny, with very small genomes. M. genitalium, discovered in 1981, has the smallest known genome of any bacterial cell, encoding fewer than 500 genes. M. genitalium, discovered in 1981, has the smallest known genome of any bacterial cell, encoding fewer than 500 genes. [...] Like Rickettsia, Mycoplasma have lost virtually all the genes required for making nucleotides, amino acids, and so forth.
Caller ID spoofing is the technique of artificially manipulating the phone number that appears on the recipient's caller ID display. This can be done for a variety of reasons, both legitimate and illegitimate. ### How It Works When a call is made, the phone network typically transmits the calling party's number, which the receiving party sees on their caller ID.
Carl August von Schmidt (1822–1892) was a German architect best known for his significant contributions to the architectural landscape of the 19th century, particularly in Germany. His work encompassed a variety of building types, including churches, civic buildings, and residential structures. He often utilized elements of the Neoclassical and Historicist styles, which were popular during his time.
Trouton's rule is a principle in physical chemistry that provides an estimate for the entropy of vaporization of a liquid. It states that the entropy of vaporization (\( \Delta S_{vap} \)) of many liquids at their normal boiling points is approximately equal to a constant value, which is about 88 to 100 J/mol·K. This rule holds true for a variety of organic liquids, particularly those that are non-polar or weakly polar.
Cather Simpson is a renowned physicist and a professor known for her work in ultrafast optics and photonics. She has conducted significant research in the field of laser science and its applications, particularly in studying light-matter interactions on extremely short timescales. In addition to her scientific contributions, Cather Simpson is recognized for her commitment to science communication and education, aiming to inspire the next generation of scientists.
Athanase Dupré is a notable figure in the field of French art, particularly recognized for his contributions as a painter and artist in the 19th century. He is often associated with the Barbizon School, which emphasized naturalism and landscape painting. His work is characterized by a focus on the beauty of nature and the depiction of rural life.
"Causality" is a book by Judea Pearl, published in 2000, that presents a comprehensive analysis of causal reasoning and its implications in various fields such as statistics, artificial intelligence, and philosophy. Pearl, a prominent figure in the field of artificial intelligence, introduces a framework for understanding causation that goes beyond traditional correlation-based approaches. In the book, Pearl discusses the importance of distinguishing between correlation and causation, providing tools and methodologies for reasoning about causality.
The term "Centered World" might refer to various concepts depending on the context in which it is used. Without more context, it's difficult to provide a definitive answer, as "Centered World" doesn't correspond to a widely recognized concept in fields like psychology, sociology, philosophy, or geography. However, here are a few potential interpretations of the term: 1. **Philosophical or Psychological Context**: It could refer to a state of mental or emotional equilibrium.
Public transport by mode refers to the various types of transportation systems available for public use, categorized based on the mode of travel. Here’s an overview of the primary modes of public transport: 1. **Buses:** - Buses are a common mode of public transport that operate on fixed routes and schedules. They can serve cities, towns, and rural areas and are often cost-effective and accessible.
The Čech-to-derived functor spectral sequence is a tool in homological algebra and sheaf theory that relates Čech cohomology to derived functors, particularly sheaf cohomology. This spectral sequence emerges in contexts where one is interested in understanding the relationship between local properties, codified by Čech cohomology, and global properties captured by derived functors like the derived functors of sheaf cohomology. ### Overview of the Components Involved 1.
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
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