So okay, here's a thread on the category of finite sets and a way in which it controls algebraic structure in symmetric monoidal categories. I think it's some really pretty stuff.
![](https://pbs.twimg.com/media/EoiH4ZdW8AAI21V.png)
![](https://pbs.twimg.com/media/EoiH4x2XYAAVVV5.png)
More from Math
Ok, it's time for a #FUNctionalAnalysis thread! Let's talk about Hilbert spaces. (I hope you like linear algebra, because that's what we're
If you want to impress people, you can just say a Hilbert space is just a complete infinite dimensional inner product space and leave it at that, but let's talk about what that actually means.
When you first learn about vectors, you talk about them as arrows in space; things with a magnitude and a direction. These are elements of R^n where n is the number of dimensions of the space you care about.
You also talk about the dot product (or inner product) as a way to tell when vectors are orthogonal. (I'm purposely saying "orthogonal" instead of "perpendicular" here, but when you actually think about arrows, it's the same thing.)
As my linear algebra students are about to see, R^n is far from the only interesting vector space. A classic example is the space of polynomials of dimension less than or equal to n
— syzygay (@syzygay1) August 9, 2020
If you want to impress people, you can just say a Hilbert space is just a complete infinite dimensional inner product space and leave it at that, but let's talk about what that actually means.
When you first learn about vectors, you talk about them as arrows in space; things with a magnitude and a direction. These are elements of R^n where n is the number of dimensions of the space you care about.
![](https://pbs.twimg.com/media/EtFblCnUcAQpqd4.png)
You also talk about the dot product (or inner product) as a way to tell when vectors are orthogonal. (I'm purposely saying "orthogonal" instead of "perpendicular" here, but when you actually think about arrows, it's the same thing.)
![](https://pbs.twimg.com/media/EtFcT7PUcAAlgt_.png)
As my linear algebra students are about to see, R^n is far from the only interesting vector space. A classic example is the space of polynomials of dimension less than or equal to n
![](https://pbs.twimg.com/media/EtFciZ0UYAAwT3Y.png)
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https://t.co/FBfXhUrH5d
Microorganisms in biofilms are enclosed by an extracellular matrix that confers protection and improves survival. Previous studies have shown that viruses can secondarily colonize preexisting biofilms, and viral biofilms have also been described.
...we raise the perspective that CoVs can persistently infect bats due to their association with biofilm structures. This phenomenon potentially provides an optimal environment for nonpathogenic & well-adapted viruses to interact with the host, as well as for viral recombination.
Biofilms can also enhance virion viability in extracellular environments, such as on fomites and in aquatic sediments, allowing viral persistence and dissemination.
Viruses and other pathogens are often studied as stand-alone entities, despite that, in nature, they mostly live in multispecies associations called biofilms—both externally and within the host.
https://t.co/FBfXhUrH5d
![](https://pbs.twimg.com/media/FEqKpQ8XoBM0rmU.jpg)
Microorganisms in biofilms are enclosed by an extracellular matrix that confers protection and improves survival. Previous studies have shown that viruses can secondarily colonize preexisting biofilms, and viral biofilms have also been described.
![](https://pbs.twimg.com/media/FEqLB4VX0AMsP1E.jpg)
...we raise the perspective that CoVs can persistently infect bats due to their association with biofilm structures. This phenomenon potentially provides an optimal environment for nonpathogenic & well-adapted viruses to interact with the host, as well as for viral recombination.
![](https://pbs.twimg.com/media/FEqLqY1XwAE2-ke.jpg)
Biofilms can also enhance virion viability in extracellular environments, such as on fomites and in aquatic sediments, allowing viral persistence and dissemination.
![](https://pbs.twimg.com/media/FEqMEUAWYAgfkSP.jpg)
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