Prompt: Imagine you have a list of natural numbers (1, 2, 3, 4, ...). Riemann's zeta function takes a number 's' and sums the reciprocals of all the numbers in that list raised to the power of 's'.
For example, zeta(2) = 1/1^2 + 1/2^2 + 1/3^2 + ... (sum of reciprocals of squares of natural numbers).
This function is interesting because it helps solve many problems related to prime numbers, which are the building blocks of all natural numbers. However, the zeta function is not defined for all values of 's'. It gets a bit tricky at 's' = 1.
The trickiness at s = 1 in the Riemann zeta function can be visualized by imagining a teeter-totter:
Think about the sum for zeta(2) again: 1/1^2 + 1/2^2 + 1/3^2 + .... Each term gets smaller and smaller as you go further down the list. This makes the sum well-behaved and eventually converges to a specific value.
But now, imagine s = 1. The sum becomes 1/1 + 1/2 + 1/3 + .... Here, each term is either 1 or bigger than 1. It's like putting only heavy weights on one side of the teeter-totter. The sum keeps getting bigger and bigger, never settling on a specific value. Mathematicians call this situation "divergence."
There are ways to define the zeta function for other values of 's' even though the simple sum diverges at 1. It's like using advanced techniques to balance the teeter-totter so it makes sense even with those heavy weights. But understanding that initial imbalance at s = 1 is the first hurdle.
Prompt: The concept of a simplicial symmetric sphere spectrum into simple terms:
Spheres of differing dimensions. A 0-sphere is just two points, a 1-sphere is a circle, a 2-sphere is the surface of a ball, and so on.
A “spectrum” is like a sequence of these spheres, where each sphere is connected to the next one in a way that helps us study shapes and spaces in a stable manner.
The “sphere spectrum” is a particular sequence that starts with the 0-sphere, and includes all higher-dimensional spheres.
The "symmetric" part means we're keeping track of how these spheres can be rearranged or swapped around; important because it allows us to do more operations with our spectrum.
“Simplicial” refers to a way of building complex shapes from simple pieces, like Legos. Meaning that we're using a particular mathematical structure to describe how our spheres connect.
Thus a “simplicial symmetric sphere spectrum” is a sequence of spheres of increasing dimensions, built such that we can swap them around and perform various operations on them; this structure is useful for studying shapes and spaces in a way that's more flexible and powerful than just looking at individual spheres.
Thus the “simplicial symmetric sphere spectrum” is a powerful mathematical concept that bridges the gap between geometry and abstract algebra. It's a sequence of spheres of increasing dimensions, connected in a way that allows mathematicians to perform complex operations across dimensions. This is crucial in topology, the study of shapes and spaces. The "symmetric" aspect allows for rearrangements of spheres, enabling operations like addition and multiplication of entire spaces. The "simplicial" component allows building complex shapes from simple elements – the “Lego block” approach, making the concept computationally manageable, and also connecting it to fields like computer graphics and data science. Thus mathematicians gain insights into the very fabric of reality as we u
I shall conquer untruth by truth. And in resisting untruth, I shall put up with all suffering.
Model:
AIVision
Size:
1792 X 1008
(1.81 MP)
Used settings:
Prompt: Let the first act of every morning be to make the following resolve for the day:
- I shall not fear anyone on Earth.
- I shall fear only God.
- I shall not bear ill will toward anyone.
- I shall not submit to injustice from anyone.
- I shall conquer untruth by truth. And in resisting untruth, I shall put up with all suffering.
Prompt: A majestic tabby hairy Norwegian Forest Cat, lying on an old antique Victorian crimson couch in an steampunk style dark library room, in background many bookshelves with a lot of old books and brass steampunk artifacts like clocks, steampunk mechanical devices, clockwork instruments, steampunk weapons, whisky bottles, an old antique globe, some cobwebs, illuminated by candles, orange candlelight, in watercolor
Prompt: Additional prompt: Funny cartoon drawing of an angry bunny standing in front of a large carrot with his arms folded and an angry look: I won't give you that, it's my carrot!
Prompt: A majestic Holy Birman cat, lying on an old antique Victorian crimson couch in an steampunk style dark library room, in background many bookshelves with a lot of old books and brass steampunk artifacts like clocks, steampunk mechanical devices, clockwork instruments, steampunk weapons, whisky bottles, an old antique globe, some cobwebs, illuminated by candles, orange candlelight
Mysterious Black Cat in the Skull Field ** #CATURDAY
Model:
Realismo
Size:
1600 X 1200
(1.92 MP)
Used settings:
Prompt: A scary black cat with red glowing eyes, sitting on a pile of skulls, in an mythic enchanted tomb with cobwebs and spiders, illuminated by burning torches,
Prompt: A majestic Sacred Cat of Burma, sitting on an old antique Victorian crimson couch in an steampunk style dark dusty library room, in background bookshelf with many old books and brass steampunk artifacts like clocks, steampunk mechanical devices, steampunk weapons, an old antique globe, there are some cobwebs, the room was illuminated by candlelight and orange gas lamp,
Dream Level: is increased each time when you "Go Deeper" into the dream. Each new level is harder to achieve and
takes more iterations than the one before.
Rare Deep Dream: is any dream which went deeper than level 6.
Deep Dream
You cannot go deeper into someone else's dream. You must create your own.
Deep Dream
Currently going deeper is available only for Deep Dreams.
