gr.group theory: some quantities associated with Hopf algebras of finite dimensions

leave $ (H, Δ, m, s) $ Be an algebra of Hopf. you this hopf algebra can be associated two obvious linear maps $ T_H, S_H: H a H $ with $ T_H = m circ Delta, S_H = s $.

Are there two Hopf algebras of finite dimensions? $ H, H & # 39; $ which are not isomorphic Hopf algebras but are equivalent in the following sense:

Each characteristic polynomial coefficient $ Det (T_H- lambda I) $ is equal to the corresponding coefficient of $ Det (T_ {H & # 39;} – lambda I) $. Similarly, each coefficient of $ Det (S_H- lambda I) $ is equal to the corresponding coefficients in $ Det (S_ {H & # 39;} – lambda I) $.

In other words, to what extent the set of these coefficients can determine the nature of Hopf algebras.

Are there two finite groups? $ G, G & # 39; $ of the same orders that are not isomorphic groups but their corresponding Hopf algebras $ CG, CG & # 39; $ Do you have the same amounts described above?

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I am considering setting up my own photo printing application, I see a lack of competitive services that offer an API to use, so I am investigating doing it myself.
I have seen this printer: https://www.amazon.co.uk/Mitsubishi-CP-D90DW-CP-D-90-DW-P/dp/B01NANPU8D I am not sure what alternatives exist or even what should be looking for. Basically, I want a printer that can print large runs of good quality photos at a competitive price.

r – It is necessary to separate the records with a column of QUANTITIES in individual records where QUANTITY is always 1

I have a data set with 82147 obs and 36 variables and I need to find the 200 main levels of the "Description" column with the highest frequency, but there is a column of AMOUNTS in the data set, so even though I am finding the highest levels "Description:" I do not get a real image of the elements that are repeated more frequently, since each record also includes a QUANTITY that is not 1.

Top20InvDesc <- names (sort (summary (as.factor (Inventory $ Description)), decreasing = T)[1:20])
Top20InvDesc

I have tried this and continue browsing the Internet to know how to do it, but I also do not know how to correctly formulate this question, so I am seeing many similar things, but none of that is what I need.

Top20InvDesc <- names (sort (summary (as.factor (Inventory $ Description)), decreasing = T)[1:20])
Top20InvDesc

Y

library (dplyr)
Inventory%>%
group_by (Description)%>%
top_n (5, quantity)

Let's say that a "syringe" is one of the levels in the "Description" column and is the most repeated level, but each record has a AMOUNT OF 5. There is also a level of "gloves" in the "Description" column and it is the fifth most repeated level, but the AMOUNT is 1000 for each one. I know that "gloves" should be the first item in the new data framework I'm trying to make, but I can not figure out how to get my code to do this. The easiest way that I can think of to solve my problem is to create a new data frame in which each item appears as QUANTITY 1 and only use the 20 main elements.

What I am receiving

Description

INTRAVENOUS ARM NEONATAL 4X1.5 IN SEMIFLEXIBLE FOAM DISPOSABLE LATEX BG / 24EA FREE

Quantity

32

What I want to get and the Armboard will now appear in the list 32 times.

Description

INTRAVENOUS ARM NEONATAL 4X1.5 IN SEMIFLEXIBLE FOAM DISPOSABLE LATEX BG / 24EA FREE

Quantity

one

My laptop has a 32 GB memory and a 180-watt power supply, so I thought I'll have to deal with the persistence processing time, but this will also make it much easier to work with the data.

Theory of the index: Can we define the spin structures and the invariants of Arf in terms of local quantities, such as the Chern classes and the Chern numbers?

I am interested if it is possible to represent a spin structure and the invariant Arf associated with it in terms of some kind of local fields.

For example, the first Chern class of a group of complex lines in a variety is an element of $ H ^ 2 (X, mathbb {Z}) $, that in a variety 2d equivalent to the specification of a whole number, the number of Chern $ c_1 in mathbb {Z} $.

All these concepts have local interpretations after putting a $ U (1) $ connection in the package, in terms of the 1-way connection $ A_ dx ^ mu $ and the curvature field of 2 shapes in the collector, $ F = ( partial_ mu A_ nu) dx ^ mu wedge dx ^ nu $. In this case, Chern's class can be considered as $ frac {1} {2 pi} F in H ^ 2 (X, mathbb {Z}) subset H ^ 2_ text {(from Rham)} (X, mathbb {R}) $. And Chern's number is given as the integral of this local amount, $ c_1 = frac {1} {2 pi} int_X F $.

I was curious to know whether or not there was an analogous interpretation of a spin structure and the invariant Arf.

From my point of view (pedestrian, physical) (as explained in this document), a turn structure is an element $ rho in H_1 (X, mathbb {Z} _2) $ that specifies periodic or antiperiodic boundary conditions in each non-trivial cycle, and the invariant Arf is $ (- 1) ^ {ind (D_ rho)} $, where $ D_ rho $ is the Dirac operator associated with the spin structure and $ ind (D_ rho) $ It is the index of the operator.

My question could also be formulated as if the index of a Dirac operator is an integral part of some local quantity that is "canonically" associated with the spin structure.

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