## What is the right way to make digital representations of fingerprints?

I am mainly a film and paper photographer: I take photos, process the film, and then make prints in the darkroom. The end product of what I do is most certainly a little paper.

But I would like to be able to show representations of these fingerprints in digital form (for example on the Internet …). I would be interested to know what is the best way to proceed, and how it is done professionally.

There are three obvious approaches:

• scan the neg (I can do it) and then process the digital copy so that it looks like the impression I would have made;
• take a very neat photo of the print by controlling the white balance and so on. (for a good representation of the color of the paper), and use it as an image;
• scan the impression with a flatbed scanner (this is really a variation of the previous approach).

The first is hard and unattractive: it takes a lot of work that does not interest me much to reproduce what I already do in the darkroom, and may or may not do a good job of representing what the l & # 39; impression really looks like.

The second I can do, and it should be easy enough. It's hard to keep prints flat, but I can matify them if need be.

I can not do the third one, but I could buy a flatbed scanner if that's clearly the best approach.

I would be interested to know two things.

• What do other people who have the same problem do but do not have access to the resources available, for example, to the creators of expensive photo books?
• How is it done professionally? If a good photo book is designed by a person whose product is a physical print and for whom the qualities of printing are important, how are these images currently being transformed into images on the page?

## group theory – What irreducible representations of the symmetric group are proper spaces of class sums?

In the context of complex representations of finite groups, a sum of class $$1_C = sum_ {g in C} g$$ acts on an irreducible representation $$V$$ as $$lambda (C, V) operatorname {Id}$$, or $$lambda (C, V) = | C | chi_V (C) / dim (V)$$ with $$chi_V$$ the character of $$V$$.

My question is which irreducible representations $$V$$ symmetric group are clean spaces of a class sum $$1_C$$. That is, when is there a class of conjugation? $$C$$ such as $$lambda (C, W) neq lambda (C, V)$$ for everyone $$W neq V$$?

If this is true, then the isotype component corresponding to $$V$$ in any representation is a clean space of $$1_C$$. A simple example is when $$V$$ is the $$2$$three-dimensional irreducible $$mathrm S_3$$ and $$C$$ the class of $$3$$-cycles. then $$V$$ is the clean space of $$1_C$$ with eigenvalue $$-1$$, or equivalently the nucleus of $$operatorname {id} + (123) + (132)$$. If we leave $$mathrm S_3$$ on trilinear forms, this means that the corresponding isotype subspace consists of satisfying forms
$$f (u, v, w) + f (v, w, u) + f (w, u, v) = 0.$$
Similarly, if we can do it for an irreducible representation $$V$$we can then describe the corresponding Schur functor on multilinear forms as the solution space of a single equation (rather than a system of equations).

I have checked with Maple that all irreducible representations of $$mathrm S_n$$ for $$n leq 8$$ are clean spaces in the sense explained. When $$n leq 5$$ and $$n = 7$$ there is even a class of conjugation $$C$$ it works for everyone $$V$$, but for $$n = 6$$ and $$n = 8$$ This is not the case.

## Classical groups, representations and associated sets.

I feel close relations between the classical compact Lie groups G (in particular, $$U (n)$$ and $$SO (n)$$), their representations $$V$$, their cohomologies and associated bundles on many $$M$$.

The last object requires some clarification: I mean a smooth variety $$M$$ with its principle $$G$$-package $$P_G$$ that associates with the tangent package $$TM$$ (ie its frame packet), we can form the associated package to $$V$$ by changing fiber: $$P_V: = P_G times _G V.$$

I can not find any reference or literature. The most relevant thing that I have found is probably that of Atiyah's classical groups and classical differential operators, but I do not have access to it.

### Questions

1. Given a representation $$V$$ of a classic compact Lie group $$G$$What can we say about cohomology (Lie group)? For example, is there a natural cohomology course that corresponds to $$V$$?
2. What can we say about the package associated with $$V$$? What are his Chern classes and roots? Are they related to the character of $$V$$? How his characterization map $$M to BG$$ look like?
3. Is $${P_V | V in Rep (G) }$$ exhaust all possible G-bundles on $$M$$ until isomorphism?

## Find all unit representations of the connected Poincaré group

I study the theory of the representation of Lie groups and its association with theoretical physics, and I worry about the following points. Is there an exhaustive way to find all the unit representations of the connected Poincaré group? $$SO (1,3) _e rtimes mathbb {R} ^ 4$$. I think Mackey's theory does the trick, but does anybody know of any other options?

## lie algebras – Weyl's theorem and representations

Let $$L$$ to be a semi-simple Lie algebra and let $$(V, varphi)$$ to be a finite dimension $$L-$$representation of the module. Our main goal is to prove that $$varphi$$ is completely reducible. Consider a $$L-$$sub-module of $$V$$ of codimension one, let $$0 longrightarrow W longrightarrow V longrightarrow F longrightarrow 0$$ to be an exact sequence (Where $$F$$ is a $$L-$$
module). From James Humphreys' book "Introduction to Lie Algebras and Representation Theory", I understood the following steps:

1. We take another own submodule of W noted W 'such that the exact sequence $$0 longrightarrow W / W longrightarrow V / W & # 39; longrightarrow F longrightarrow 0$$ split, so there is a dimension $$L-$$sub-module of $$V / W '$$ (say $$tilde {W} / W '$$) complementary to $$W / W '$$.

2.We proceed by induction on the dimension of $$W$$, so we get an exact sequence $$0 longrightarrow W longrightarrow tilde {W} longrightarrow F longrightarrow 0$$ who divides. It follows easily that $$V = W oplus X$$ where X is a complementary sub-module to $$W '$$ in $$tilde {W}$$.

3. We assume that $$W$$ is irreducible, so we can use Schur's lemma on $$c green_ {W}$$ say that $$Ker ; c$$ is a $$L-$$ sub-module of $$V$$, or $$c$$ is an endomorphism of $$V$$ defined in 6.2.

The other parts of the evidence are very difficult, I did not understand them. Can someone help me understand these parts? If there is another method understandable, can anyone share it with us?

## theory of representation – Extension of irreducible tempered representations of \$ GL_n ( mathbb Q_p) \$

Let $$V_1, V_2$$ be two irreducible temperate representations permissible to $$GL_n ( mathbb Q_p)$$, how to calculate all the extensions $$V$$ (as smooth representations of $$GL_n ( mathbb Q_p)$$) of $$V_1$$ by $$V_2$$ ?

Must such a representation $$V$$ to be eligible and temperate?

## Nt.number Theory – Eccentricity in the number of representations for sets too big to be sets of Sidon

Let $$A = {a_1 to be a set of integers. Let $$r_A (n) = # {(a_i, a_j): a_i + a_j = n }$$ the number of representations of $$n$$ as a sum of two elements of $$A$$. In the typical language, $$A$$ is a Sidon set (or $$B_2$$ define) if $$r_A (n) the 2$$ for everyone $$n$$. We know that the maximum size of a Sidon set that is a subset of $${1,2, dots, N }$$ is $$sqrt {N} (1 + o (1))$$.

My question, in general terms, is to ask if we can measure how often (and how much) $$r_A (n)$$ must exceed $$2$$, if $$A$$ contains at least $$(1+ epsilon) sqrt {N}$$ elements, for some $$epsilon> 0$$?

More concretely, let $$E (A)$$ denote the "eccentricity" of $$A$$, given by
$$E (A) = sum_n max {r_A (n) -2,0 }.$$ Yes $$| A |> (1+ epsilon) sqrt {N}$$ for some people $$epsilon> 0$$, then there must be $$delta> 0$$ such as $$E (A)> delta N$$?

My impetus for asking this question comes from my attempts to understand the binary numbers of $$sqrt {2}$$. It is currently known that the number of $$1$$is in the first $$N$$ binary digits of $$sqrt {2}$$ is $$ge sqrt {2N} (1 + o (1))$$, and for an infinite sequence of $$N$$This can be improved for $$ge sqrt {8N / pi} (1 + o (1))$$. However, this limitation comes in part from the assumption that the set of indices of the $$1$$Behaves like a Sidon set, which is too big to actually be. If we could show that $$E (A)> delta N$$, so I think a stronger lower limit could be proven.

## induced representations – application of Mackey's theory – abelian group

In order to advance my research, I am supposed to understand this fact:

Let $$A$$ to be an abelian group and $$S$$ a finite group acting on $$A$$.
This defines the semi-direct product $$A S$$:

Let $$chi$$ to be a character of $$A$$.
To define $$S_ chi = {s in S mid s chi = chi }$$
(Recall: $$s chi = chi fs forall a in A, chi (sas ^ {- 1}) = chi (a)$$ ).

Note that this defines a subgroup of $$A rtimes S> A rtimes S_ chi$$

We can expand each representation of $$A$$ to a representation of $$A rents S_ chi$$ by:
$$chi: A rightarrow mathbb {C}, chi (as): = chi (a)$$.

Let $$( sigma, V) in Irr (S_ chi)$$.
Note $$chi otimes sigma$$ is a representation of $$A rents S_ chi$$.
Then:

1. $$Ind_ {A rtimes S_ chi} ^ {A rtimes S} chi otimes sigma$$ is irreducible

2. For any irreducible representation of $$A S$$, there is $$chi in Irr (A)$$ and $$sigma in Irr (S _ { chi})$$ s.t. the representation is:
$$Ind_ {A rtimes S_ chi} ^ {A rtimes S} chi otimes sigma$$

If I understand correctly, the proof of this statement should be straightforward with Mackey's theory, but I can not see the picture.
My instructor calls this statement the Mackey's theory – which differs from the statement I see in other manuals (it seems like it's an app – correct me if I'm wrong).

## Scenario

Let's say that there is a `Shop` entity in my application. And two types of users: a `Customer` And one `Admin`. When a customer buys something from a store, his visit is recorded in the system (there is a counter of visits by customer shop).

Suppose, then, that a store is modeled as follows:

``````class Shop {
id: number;
name: string;
categories: Category();
}
``````

but, from the customer's point of view, the property `numberOfVisits: number` the `Shop` entity would also make a lot of sense:

``````class Shop {
id: number;
name: string;
categories: Category();
numberOfVisits: number; // this makes sense for customers
}
``````

This depends on the use case being run:

Use case: A: As an administrator, I would like to list all the stores that
exist in the system.

In this case, dealing with `Shop` entities without a visit desk would be fine.

Use cases: B & # 39; As a customer user, I would like to list all the stores that I have
visited.

In this case, the customer expects certain stores to be listed and the number of visits interested. So, deal with `Shop` entities that have the `numberOfVisits` the property would be important.

## Design approaches

I came across several design approaches, but none of them have convinced me yet.

### Approach # 1

Have a unique `Shop` entity with the counter counter property for each case:

``````class Shop {
id: number;
name: string;
categories: Category();
numberOfVisits: number;
}
``````

What I do not like in this approach, is to have a single entity with fields that would make no sense in some use cases or contexts. application (even if they would be settled on `null`). I do not think it's a very clean solution.

### Approach # 2

Have several `Shop` entities:

``````class Shop {
id: number;
name: string;
categories: Category();
}
``````
``````class ShopWithVisits {
id: number;
name: string;
categories: Category();
numberOfVisits: number;
}
``````

or even:

``````class Shop {
id: number;
name: string;
categories: Category();
numberOfVisits: number;
}
``````
``````class ShopWithVisits extends Shop {
numberOfVisits: number;
}
``````

I find this somehow better than # 1 because it is about different representations of the same entity in different use cases or contexts. But still, not sure if that could somehow be improved.

## Questions

What is your opinion about this? Which approach do you find the best and why? Is there another approach I could take to improve my design?

## topology algebra – Higher categorical analog of the equivalence between the category of representations of a monoid and the category of the monoid ring of the monoid

In classical algebra, there is a notion of "monoid rings" such that the functor taking monoids to monoid rings is the left adjoint of the forgetful functor from the category of commutative rings to that of commutative monoids forgetting l & # 39; addition. In addition, there is an equivalence between the category of representations of a monoid and the category of the monoid ring of the monoid.

Question 1:
Leave a functor $$F$$ of $$infty$$-category of $$E_ infty$$rings of specters at $$infty$$-category of $$E_ infty$$-spaces is the functor post-compositing functor infinite loop $$Omega ^ infty$$(we see algebras like $$infty$$cards and $$Omega ^ infty: Sp rightarrow S$$ is a monoidal laxal functor). So, is there a deputy commissioner right to $$F$$? ($$Sp$$ is the $$infty$$-category of spectra and $$S$$ is the $$infty$$-category of spaces.)

Question 2:
If question 1 is true, then we will designate the functor as $$mathbb {S} (-)$$. Let $$M$$ bean $$E_ infty$$-space. Is there an equivalence between the $$infty$$-Category $$Amusement (BM, Sp)$$ representations of $$M$$ and the $$infty$$-Category $$Mod _ { mathbb {S} (M)} (Sp)$$ modules of the monoid ring $$mathbb {S} (M)$$? ($$BM$$ is a $$infty$$-category that has a single object $$*$$ and $$Card (*, *)$$ = $$M$$ as $$A_ infty$$-the spaces.)