riemannian geometry – Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds

Consider the elliptic operator $Lu = – Delta u + langle nabla u , X rangle + c , u $ acting on functions on a closed Riemannian manifold $M$. Here $Delta$ denotes the Laplace-Beltrami operator, $X$ is an arbitrary smooth vector field, and $c geq 0$ is a smooth function on $M$ which does not vanish identically. Does $L$ have a so-called `principal eigenvalue’ $lambda_1 > 0$, whose corresponding (unique up to scaling) eigenfunction does not change sign?

A similar statement holds for smooth domains in $mathbb{R}^n$, as shown for instance in Evans’ PDE book, chapter 6. Moreover, in this paper it is sated that this fact is equivalent to the operator satisfying a maximum principle (which is indeed the case for the above $L$).

algebraic curves – A question about principal divisors and poles

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ct.category theory – What is the correct notion of Principal bundle over a category?

Motivation for my question:

It is a well-known fact that there exists a bijection between the set of isomorphism class of
principal $G$ bundles over a nice topological space $X$ and the set $(X,B’G)$ of homotopy class of continuous maps from $X$ to the classifying space $B’G$(using the different notation than conventional for convenience) of the Principal $G$ bundles.

Now let $X$ be a topological space and let $U=cup_{alpha in I} U_{alpha}$ be a covering of $X$. Now it is also well known that the functor $phi:C(U) rightarrow BG$ from the Cech Groupoid $C(U)$ of the cover $U$ of $X$ to the delooping groupoid $BG$ of the topological group $G$ can be considered as a principal $G$ bundle over the space $X$. (For example see definition 3.2 in https://arxiv.org/pdf/1403.7185.pdf).

If we move one step higher, that is a weak 2-functor from the Cech 2-groupoid $C^2(U)$ to the deoopolng $B^2G$ of a Weak 2 group $G$ (For definition of Cech 2 groupoid and delooping groupoid of weak 2 group please check example 2.20 and section 3.2 of https://arxiv.org/pdf/1403.7185.pdf and the definition of Weak 2-group is found in https://arxiv.org/abs/math/0307200 )

then we arrive at the definition of Principal 2-bundle over the space $X$ where the structure 2-group is the weak 2 group $G$ (see definition 3.8 in https://arxiv.org/pdf/1403.7185.pdf) which I guess will be equivalent to the local description of Christoph Wockel’s definition of Principal 2 bundles in the definition 1.8 in https://arxiv.org/pdf/0803.3692. ( Though I did not check rigorously that they are indeed same)

Now motivated from the observations above ,

My question is the following:

(1) Is a weak 2 functor $F:C rightarrow B^2G$ from a category $C$ to the delooping groupoid $B^2G$ of a weak 2-group $G$ can be a good choice of definition of Principal bundle over a category where the structure group is the 2-group $G$?


(2) To get an appropriate notion of local trivialisation of a principal bundle over a category we have to somehow appropriately define the notion of Cech Gropoid $tilde{Ch}(U)$of a “cover $U$ on the category $C$ (may be coming from some Grothendieck pretopology on Cat, the category of small categories) and then consider the functor $tilde{F}:tilde{C}h(U) rightarrow B^2G$ as a definition of locally trivializable Principal 2-bundles over a category?

I could not find any literature where a notion of locally trivializable Principal bundle over a general category is explicitly mentioned. So any suggestion of literature in this direction will also be very helpful.

Also I am curious to know about it’s corresponding notion in higher categories and in the context of infinity category .

Thank you.

Active Directory – Maximum expiration date of an Azure service principal credential password?

If you are assigning an Azure AD service principal to Azure (for example, Azure Container Registry), what is the maximum expiration date for the credential password for that service principal?

Is it 1 year? Can you set it to 10 years?

Any help in this regard is greatly appreciated!

centos – Cpanel license / ovh change ip principal

I am trying to change the outgoing ip in centos 7 in ovh.

I have this file in / etc / sysconfig / network-scripts / ifcfg-eth0

# Created by cloud-init on instance boot automatically, do not edit.

When I try to change eth0 to an IP failover, the server stops and I need to restart. When finished, the ifcfg-eth0 file returns by default.

Principal component analysis with different constraint

I would like to know if there are any books on the following problem:

begin {equation}
max limits_ {|| x || _p = 1} x ^ t A x
end {equation}

For $ p = 1 $ or $ p = infty $,
or $ x in mathbb {R} ^ n $ and $ A $ is a symmetric matrix $ A in mathbb {R} ^ {n times n} $.

Note that for $ p = $ 2 we recover the usual problem of the PCA. As I struggle to find a useful literature dealing with the problem for $ p = 1, p = infty $ it would be great if you could tell me a related literature. Thank you in advance.

Active Directory – Windows Server 2016 "Failed to Register the Service Principal Name"

I have a Windows Server 2016 virtual host that hosts a virtual domain controller and some additional servers. When the physical host must restart to apply a scheduled fix, the server receives the following errors at startup:

"Failed to register the service principal name & # 39; Hyper-V Replica Service".
Failed to register the service principal name & quot; Microsoft Virtual System Migration Service & quot;
Failed to register the service principal name & # 39; Microsoft Virtual Console Service & # 39; "

The names of the service principals are correctly defined in the AD attributes of the host for each of the respective SPNs, and I am not sure how to follow it further. Does anyone have first-hand experience or recommendations in this regard? There is also no NTDS port restriction in place.

mongodb after restoring the set of replicas from a backup, the principal is not master

after retrieving a replica set from a snapshot (for the test)
when I run commands on the primary, I had an error

not master and slaveOk = false

primary is not equal to master?
I have this output for the command rs.isMaster () on the primary.

shard2-replset: PRIMARY> rs.isMaster ()
"Guests": [
"setName": "shard2-replset",
"setVersion": 17,
"ismaster": false,
"secondary": true,
"primary": "mongo-stage-rs-01-b.prod:27018",
"me": "mongo-stage-rs-01-b.prod:27018",
"electionId": ObjectId ("7fffffff0000000000000028"),
"maxBsonObjectSize": 16777216,
"maxMessageSizeBytes": 48000000,
"maxWriteBatchSize": 1000,
"localTime": ISODate ("2019-05-15T17: 09: 13.275Z"),
"maxWireVersion": 4,
"minWireVersion": 0,
"ok": 1

is it true that the "primary" and the "me" are the same, yet "secondary": true and "ismaster": false,

                "ismaster": false,
"secondary": true,
"primary": "mongo-stage-rs-01-b.prod:27018",
"me": "mongo-stage-rs-01-b.prod:27018",

Mongodb version 3.2

replication – change of domain name for Mongodb principal

When attempting to initialize my replica set, the primary node has a specific domain: port matching. what I want is:

"vagrant-ubuntu-trusty-64: 27022"

but what I receive is:

"localhost: 27022"

This is the command I use to start the mongo instance:

sudo mongod –port 27022 –dbpath / db / config / data –configsvr –replSet config

That's what I use to access the mongo instance:

mongo –port 27022

I tried modifying my host file to add the domain name to the default domain of, but it did not work. localhost vagrant-ubuntu-trusty-64
# The following lines are desirable for IPv6-compatible hosts
:: 1 ip6-localhost ip6-loopback
fe00 :: 0 ip6-localnet
ff00 :: 0 ip6-mcastprefix
ff02 :: 1 ip6-allnodes
ff02 :: 2 ip6-allrouters
ff02 :: 3 ip6-allhosts

How can I get the specific torque "vagrant-ubuntu-trusty-64: 27022" when I initialize the mongo instance?

Is principal principal $ 1 billion of principal amount? On the one hand, a capital of $ H $?

If no, is there an easy counterexample? Are there any restrictions on $ G $ and $ H $ as it is true?
If so, does anyone know a reference?