## Importance of the principal bundle in Chern-Simons theory

This is a very basic beginners question about Chern-Simons theory. The configurations that we sum over to get the partition function are given by a Lie-algebra valued 1-form $$A$$ on a topological 3-manifold, called the connection. More precisely, $$A$$ is a connection of a principal Lie-group bundle over the manifold.

What is the role of this principal bundle?

I didn’t find this spelled out explicitly anywhere, but I assume one does not sum over different choices of principle bundles when calculating the partition function? Or are different choices of principle bundle already encoded in different choices of $$A$$?

So is Chern-Simons theory not only a topological field theory that one can put on any topological manifold, but one that we can additionally put on any topological manifold with a Lie-group principle bundle? Can one restrict to trivial principle bundles without loosing any of the physical interpretation? If one talks about the “3-manifold invariants” of the Chern-Simons theory, does that refer to the partition function on the trivial principle bundle over those manifolds? Or is the partition function independent of the choice of bundle?

## Como puedo distribuir los elementos de un arreglo principal a otros 2 arreglos lo mas balanceada posible

Tengo un conjunto de 2N jugadores cada uno de los cuales tiene asociados un indicador numérico mayor que cero que define la calidad como jugador. A partir de este conjunto de jugadores, debo crear dos equipos de N jugadores, A y B, tales que la suma de los indicadores de calidad de sus componentes sea lo más similar posible.

import java.util.Arrays;  class Main {   public static void main(String[] args) {     //Arreglo de jugadores         int[] equipoDeJuego = {8,8,9,6,1,7,1,3,7,5};          int fuerzaA=0;         int fuerzaB=0;          //Equipo A         int[] A=new int[10];          //Equipo A         int[] B= new int[10];          //Se imprime la cantidad de los jugadores         //System.out.println("Cantidad de jugadores: "+equipoDeJuego.length);          //Ordenamos el arreglo de manera creciente         Arrays.sort(equipoDeJuego);            for(int i=0; i<(equipoDeJuego.length)/2; i++) {              //Integrantes del primer arreglo             A[i]=equipoDeJuego[i];             System.out.print("Equipo A:"+A[i]+" ");              fuerzaA+=A[i];         }         System.out.println("Fuerza equipo A: "+fuerzaA);         System.out.println();         for(int i=5; i<10; i++) {              //Integrantes del segundo arreglo             B[i]=equipoDeJuego[i];             System.out.print("Equipo B:"+B[i]+" ");             fuerzaB+=B[i];          }         System.out.println("Fuerza equipo B: "+fuerzaB);            int dF= fuerzaB-fuerzaA;         System.out.println("La diferencia es: "+dF);     } } 

Este código que hice me di cuenta que no resuelve el problema eficientemente, porque abra un equipo que tendrá A=(1,1,1,1,1,1) y el equipo B=(6), se balancea pero se que no es la mejor practica.

Como puedo crear un código que pueda distribuir la fuerza entre los equipos, es decir, ir lanzado a los mejores en ambos equipos basado en su diferencia hasta que ese equipo sea mejor que el otro.

Ejemplo:

int[] equipoDeJuego = {8,8,9,6};  Equipo A={8,9};  Equipo B={8,6};  

## Noncommutatove Leary Hirsch theorem in the context of noncommutative principal bundles

In the literature, are there some research on non commutative analogy of Leary Hitsch theorem in the context of non commutative Principal bundles?

## Windows Server 2016 “Failed to register the service principal name”

I have a Windows Server 2016 virtual host which is hosting a virtual domain controller, and a few additional servers. When the physical host needs to reboot for scheduled patching, upon startup the server receives the following errors:

“Failed to register the service principal name ‘Hyper-V Replica Service’. Failed to register the service principal name ‘Microsoft Virtual System Migration Service’. Failed to register the service principal name ‘Microsoft Virtual Console Service’.”

Service principal names are properly set in the attributes of AD of the host for each of the respective SPN’s, and I’m unsure how to trace this out any further. Does anyone have any first hand experience or recommendations in regards to this? There are also no NTDS port restrictions in place.

## ¿Por qué mi aplicación hecha en React me da error 404 en todas las páginas salvo la principal?

Estoy intentando probar la versión de producción de una web que he hecho utilizando React, pero cuando ejecuto serve -s build y abro mi web, todas las páginas menos la principal me lanzan un error 404.

En el archivo Index.js envuelvo la llamada a App.js utilizando BrowserRouter:

  <BrowserRouter>     <App />   </BrowserRouter>, 

Y la navegación está definida en el archivo App.js utilizando react-router-dom. Se ve así:

          <Route exact path="/" component={Home} />           <Route path="/ilustraciones" component={Ilustraciones} />           <Route path="/galeria" component={Galeria} />           <Route path="/login" component={Login} />           <Route path="/visor" component={Loginvisor} />           <Route path="/visorstl" component={LoginvisorSTL} />           <Route path="/profile" component={Profile} />           <Route path="/DICOM" component={LoginVisorDICOM} /> 

Mi web, si es necesario mirar en otro lado, está aquí: https://github.com/rgomez96/Tecnolab (en la rama Develop)

¿Cómo puedo solucionar esto? No es mandatorio desplegar la web utilizando serve así que podría cambiarlo si fuera necesario.

## Spring Security Principal null

метод контроллера:

@GetMapping("/postauth") public void postAuth(Principal principal, HttpSession session){ // principal = null Authentication authentication = SecurityContextHolder.getContext().getAuthentication(); // = null  SecurityContext context = (SecurityContext)session.getAttribute("SPRING_SECURITY_CONTEXT"); String  login = ((org.springframework.security.core.userdetails.User) context.getAuthentication().getPrincipal()).getUsername(); // = "user" } 

security.xml

<?xml version="1.0" encoding="UTF-8"?> <beans:beans xmlns="http://www.springframework.org/schema/security"         xmlns:beans="http://www.springframework.org/schema/beans"        xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"        xmlns:security="http://www.springframework.org/schema/security"        xsi:schemaLocation="http://www.springframework.org/schema/beans       http://www.springframework.org/schema/beans/spring-beans.xsd       http://www.springframework.org/schema/security       http://www.springframework.org/schema/security/spring-security-4.2.xsd">      <http pattern="/resources/**" security="none"/>     <http pattern="/auth" security="none"/>     <http pattern="/views/service/**" security="none"/>    <http auto-config="true" use-expressions="true">       <intercept-url pattern="favicon.ico" access="permitAll" />       <intercept-url pattern="/**"  access="hasRole('USER')"/>       <form-login authentication-failure-url="/views/service/error_auth.jsp" default-target-url="/postauth" always-use-default-target="true"/>        <remember-me key="ffw4334r2" token-validity-seconds="259200"/>        <anonymous username="guest" granted-authority="ROLE_ANONYMOUSLY"/>    </http>     <authentication-manager>         <authentication-provider>             <jdbc-user-service id="userService" data-source-ref="dataSourceMySQL"             users-by-username-query="select login, password, true from users where login = ?"             authorities-by-username-query="select login, authority from users where login = ?"/>             <password-encoder ref="passwordEncoder"/>         </authentication-provider>     </authentication-manager>      <beans:bean id="passwordEncoder" class="org.springframework.security.crypto.bcrypt.BCryptPasswordEncoder" >     </beans:bean>  </beans:beans> 

Почему в параметре метода контроллера principal всегда null и Authentification, полученный выше тоже. Приходится вытягивать имя авторизованного пользователя из сессии.

## Let R be the ring obtained by taking the quotient of Z6[x] by the principal ideal (2X+4).Then

A R has infinitely many elements .

B R is field.

C (5 )is unit in R.

D(4) is unit in R.

## An alternative representation of the principal symbol of the Laplace operator

Assume that $$(M,g)$$ is a $$n$$ dimensional Riemannian manifold. We denote by $$\Delta$$, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent?

1)There are globally defined vector fields $$X_i’s,\quad i=1,2,\ldots,n$$ such that $$\Delta=\sum_{i=1}^n \partial^2/\partial{X_i^2}$$

2)There exsits a first order differential operator $$D$$ on $$\chi^{\infty}(M)$$ such that at each cotangent vector in $$T^*(M)$$ we have $$(*)\;\;\;Det(P(D))= {P(\Delta)}^n$$ where $$“P”$$ stand for the principal symbol of the corresponding operator.

If the answer to the question is negative, what kind of obstruction appear to have the second condition $$“2”$$? Does every Riemannian manifold admit such a differential operator $$D$$ on $$\chi^{\infty}(M)$$? Apart from obvious examples in dimension $$1,2,4,8$$, are there some other examples(in other dimension)?

What about if we replace $$(*)$$ with $$trace (P(D))=nP(\Delta)$$

## What is the meaning of ‘components’ in principal component regression?

I’m learning principal component regression and I don’t understand the result I get from PCR method. My goal of using PCR is to reduce the number of predictors.

For example:

# Load the data data("Boston", package = "MASS") # Split the data into training and test set set.seed(123) training.samples <- Boston$medv %>% createDataPartition(p = 0.8, list = FALSE) train.data <- Boston[training.samples, ] test.data <- Boston[-training.samples, ] # Build the model on training set set.seed(123) model <- train( medv~., data = train.data, method = "pcr", scale = TRUE, trControl = trainControl("cv", number = 10), tuneLength = 10 ) # Print the best tuning parameter ncomp that # minimize the cross-validation error, RMSE summary(model) model$  bestTune  

I get:

Data:   X dimension: 407 13      Y dimension: 407 1 Fit method: svdpc Number of components considered: 5 TRAINING: % variance explained           1 comps  2 comps  3 comps  4 comps  5 comps X           47.48    58.40    68.00    74.75    80.94 .outcome    38.10    51.02    64.43    65.24    71.17 

and

  ncomp 5     5 
1. May I ask what does 1 comps...5 comps means?
2. What does theresult ncomp 5 from model\$ bestTune mean?
3. Where can I find the reduced model from those results? (My final goal is to select essential predictors.)

Thanks.

-C.T

## When are principal lines of curvature geodesics?

Let $$S$$ be a smooth surface embedded in $$\mathbb{R}^3$$. When are (some of) the principal lines of curvature geodesics on $$S$$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal line, is a geodesic? And perhaps the (red) min principal line connecting the two umbilical points is a geodesic?

Image from Jorge Sotomayor.1

Is there any $$S$$ all of whose principal lines of curvature are geodesics?

1Sotomayor, Jorge. “Historical Comments on Monge’s Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in $${\mathbb R}^ 3$$.” arXiv Abstract (2004). São Paulo Journal of Mathematical Sciences 2, 1 (2008), 99–143.