choosing finite subsets of natural numbers

Let $ t>0$ and $ \delta\in\big(0,\frac12\big)$ be fixed. For any $ k\in\mathbb{N}$ let $ I_k,J_k\in\mathbb{N}$ be finite subsets of natural numbers with cardinalities denoted as $ |I_k|,|J_k|$ , respectively. Now define numbers

$ \hspace{70pt}C_k:=\max\Big\{\max\limits_{i\in I_k}i^{2t}\sum\limits_{j\in J_k}j^{2t},\,\,\max\limits_{j\in J_k}j^{2t}\sum\limits_{i\in I_k}i^{2t}\Big\}$

and

$ \hspace{70pt}D_k:=\max\Big\{|I_k|\max\limits_{j\in J_k}j^{4t+2\delta},\,\,|J_k|\max\limits_{i\in I_k}i^{4t+2\delta}\Big\}$ .

Aim: choose the subsets $ |I_k,|J_k|$ so that

$ \hspace{100pt}R_k:=\frac{C_k}{D_k}\to\infty\,\,\,\,$ as$ \,\,\,\,k\to\infty$ .

How does DC work with natural 20?

To keep it short, the D&D Dungeon Master’s Screen has certain Difficulty Levels with the respective DC. The issue is that there are two Difficulty Levels namely Very Hard and Nearly Impossible whose DC is greater than 20.

My question is; If you are trying to, for example, lockpick a very high-difficulty lock that requires a DC 25 and you get natural 20, do you lockpick it as it is “Natural Success” or do you have to have a +5 or higher modifier with thieve’s tools as well? (My question applies for anything that may require a DC higher than 20, I set an example with the lockpick)

How does DC work with natural 20?

To keep it short, the D&D Dungeon Master’s Screen has certain Difficulty Levels with the respective DC. The issue is that there are two Difficulty Levels namely Very Hard and Nearly Impossible whose DC is greater than 20.

My question is; If you are trying to, for example, lockpick a very high-difficulty lock that requires a DC 25 and you get natural 20, do you lockpick it as it is “Natural Success” or do you have to have a +5 or higher modifier with thieve’s tools as well? (My question applies for anything that may require a DC higher than 20, I set an example with the lockpick)

Alternate notation for natural numbers and their addition?

The conventional way of expressing natural numbers uses base 10 notation which according to https://matheducators.stackexchange.com/questions/4367/how-to-teach-binary-numbers-to-5th-graders, some people don’t really fully understand how works. Not only that but the notation 2 + 2 + 2 is ambiguous. It could mean (2 + 2) + 2 or 2 + (2 + 2). I know that since addition is associative, all ways of bracketing any addition expression will always give the same answer but that doesn’t change the fact that the expression still has 2 different meanings. Some people refuse to take for granted that both meaning give the same answer just because people sometimes write it without brackets. To add to the extra confusion, we write 2 + 2 and not (2 + 2) and be like “How can I figure out what 2 + (2 + 2) means when I don’t even know what (2 + 2) means?”

The expression 2 + 2 $ \times$ 2 on the other hand can have a different meaning depending on the interpretation unless you use PEDMAS to decide that it can only mean one of those meanings. To make matters worse, some people may get confused by the type of calculator that I think treats all expressions as meaning the left associative expression and rely on its answer being the correct answer using the PEDMAS rule when it’s actually not. I think that because of many confusions similar to those ones, some mathematicians have a demand for formalization and understanding why statements are true. Some mathematicians might insist on having a simple to describe unambiguous notation for natural numbers and all addition expressions of them.

I have an idea for on that I’m wondering if is good. Let 0 denote the natural number 0 and $ S$ denote the successor operation. Now we define the notation for 0 to be $ 0$ , the notation for 1 to be $ S0$ , the notation for 2 to be $ SS0$ and so on. Next to express a sum of 2 natural numbers, we write + followed by the notations for each natural number so 2 + 2 can be represented as $ +SS0SS0$ . For any 2 notations you already constructed, we can also denote their sum as + followed by the notation for the first expression and then the notation for the second expression so (2 + 2) + 2 can be denoted $ ++SS0SS0SS0$ and 2 + (2 + 2) can be denoted $ +SS0+SS0SS0$ . Furthermore, it can be shown that no string of the characters +, $ S$ , and 0 has more than one meaning. Not only that but it can also be shown that given any meaningful expression, sticking more characters onto the end will never give you a meaningful expression. Also when ever you start with the empty string and keep sticking another character onto the end, there’s a nice simple way to compute whether or not you have yet completed a meaningful expression.

Por qué cuando hago scroll en el TableView las imágenes hacen AutoSize y vuelven a su tamaño natural?

Cuando cargo la aplicación por primera vez se ven todas las imagenes del tableview de un mismo tamaño y sin problemas, de esta manera.

introducir la descripción de la imagen aquí

Pero cuando hago scroll sobre el tableview, las imagenes vuelven a su estado natural, no entiendo por qué ocurre esto… Quedan así. introducir la descripción de la imagen aquí

Este es el código donde agrego la imagen en CELL en el tableview

let reference = user?["Logo"] as? String ?? ""         let fileUrl = NSURL(string: reference)          let placeholderImage = #imageLiteral(resourceName: "negociophoto") //placeholder if wanted          cell.imageView?.sd_setImage(with: fileUrl as URL?, placeholderImage: placeholderImage)         cell.imageView?.layer.cornerRadius = 5         cell.imageView?.layer.borderWidth = 1.0         cell.imageView?.layer.borderColor = UIColor.gray.cgColor         cell.imageView?.layer.masksToBounds = true         cell.imageView?.frame = CGRect(x: 10,y: 0,width: 10,height: 10)         cell.imageView?.contentMode = UIView.ContentMode.scaleAspectFit 

Espero alguien sepa por qué ocurre esto, muchas gracias por todo.

What does it mean for two natural numbers to be *approximately equal*?

This is related to this other question of mine about a paper of Colin and Honda.

I’m trying to follow the proofs line by line. I found the following piece of notation that is not explained in the text so i ask for its meaning or a reference where it is used other than this. I quote the beginning of page 6 in the cited paper

Since $ d(h(\gamma_0), \gamma_0) = N$ , it follows that $ d(h(\alpha), h(\gamma_0)) \approx N$ for every $ \alpha$ . (Here $ \approx$ means approximately equal to.)

For some context: $ \gamma_0, \alpha$ are simple closed curves in a surface, $ h$ is an automorphism of the surface, $ d(\cdot, \cdot)$ is the distance in the complex of curves and $ N$ is a natural number. Hence, we are saying that two natural numbers are approximately equal?

Since $ N$ is a number that, in that context, is arbitrarily large one could think that it means that $ \lim_{N\to \infty} d(h(\gamma_0), \gamma_0) – N = 0$ ? But later on the text a similar exppresion appears changing $ N$ by $ 0$ (end of page $ 7$ ).

So I would appreciate someone explaining to me what it means.

Reference request: Gauge natural bundles, and calculus of variation via the equivariant bundle approach

Let $ P\rightarrow M$ be a principal fibre bundle with structure group $ G$ , $ F$ a manifold and $ \alpha: G\times F\rightarrow F$ a smooth left action.

There is an associated fibre bundle $ E\rightarrow M$ with $ E=P\times_\alpha F=(P\times F)/G$ .

As it is well known, one may either treat sections of the associated fibre bundle “directly”, or consider maps $ \psi:P\rightarrow F$ which satisfy the equivariance property $ \psi(pg)=g^{-1}\cdot\psi(p)$ , where $ \cdot$ denotes the left action. Let us refer to this latter method as the “equivariant bundle approach”.

I am interested in describing the gauge field theories of physics using global language with appropriate rigour. However, most references I know treat this topic using the “direct approach”, and not with the equivariant approach, the chief exception being Gauge Theory and Variational Principles by David Bleecker.

Bleecker’s book however doesn’t go far enough for my present needs.

  • Bleecker only uses linear matter fields, eg. the case where $ F$ is a vector space and $ \alpha$ is a linear representation. Some things are easy to generalize, others appear to be highly nontrivial to me.
  • Bleecker treats only first-order Lagrangians. The connection between a higher-order variational calculus based on the equivariant bundle approach and between the more “standard” one built on the jet manifolds $ J^k(E)$ of the associated bundle is highly unclear to me. Example: If $ \bar\psi:M\rightarrow E$ is a section of an associated bundle, its $ k$ -th order behaviour is represented by the jet prolongation $ j^k\bar\psi:M\rightarrow J^k(E)$ , but if instead I use the equivariant map $ \psi:P\rightarrow F$ , what represents its $ k$ -th order behaviour? I assume it is related to something like $ J^k(P\times F)/G$ , but the specifics are unclear to me.

  • In Bleecker’s approach, connections are $ \mathfrak g$ -valued, $ \text{Ad}$ -equivariant 1-forms on $ P$ , however I am interested in treating them on the same footing as matter fields. Connections however are higher order associated objects in the sense that they are associated to $ J^1P$ . Bleecker absolutely doesn’t treat higher order principal bundles.

In short, I am interested in references that consider gauge theories, gauge natural bundles, including nonlinear and higher-order associated bundles and calculus of variations/Lagrangian field theory from the point of view where fields are fixed space-valued objects defined on the principal bundle (equivariant bundle approach), rather than using associated bundles directly.