Number Theory – Searching for example metrics on natural numbers

I am looking for metrics $ d_1, d_2 $ on natural numbers, such as:

1) For all $ a, b $ we have $ d_i (a, b) <$ 1

2) for all $ a, b $ we have $ d_i (a, b) $ is a rational number

3) if $ d = d_1 + d_2-d_1 cdot d_2 $ so for all $ a neq b, a neq c $ we have:
$ d (a, b) + d (a, c)> 1 $

4) for all $ a, b, m $ we have $ d_i (ma, mb) = d_i (a, b) $

Two of these examples are $ d_1 (a, b) = 1- frac { gcd (a, b) ^ 2} {ab} $ and $ d_2 (a, b) = 1- 2 frac { gcd (a, b)} {a + b} $

Thank you for your help.
The motivation for asking this question is given in the corresponding question:

A reinterpretation of the $ abc $ conjecture in terms of metric spaces?

smooth varieties – Family of Riemannian metrics all equal to a fixed metric

Consider a smooth ($ C ^ infty $multiple $ M $ and a family of Riemannian metrics $ {g ^ {(u)} } $ sure $ M $ it depends smoothly $ u in (a, b) $, in that in some coordinates $ x ^ i $ sure $ M $ we can write

$$ g ^ {(u)} = g_ {ij} (u, x) , text {d} x ^ i text {d} x ^ j $$

for smooth component functions $ g_ {ij} $.

Now the situation is the following. We are given another metric $ h_ {ij} (x) $ as for each fixed $ u = u_0 $, the metric $ g_ {ij} (u_0, x) $ can be transformed into $ h_ {ij} (x) $

My question is whether these transformations can be chosen to depend on $ u $ also, that is, there is a family of coordinate transformations $ phi_u: x mapsto tilde x $, $ u in (a, b) $ having the property that they transform $ g_ {ij} (u, x) $ in $ h_ {ij} (x) $, such as the map

$$ phi: (u, x) mapsto (u, phi_u (x)) $$

is smooth?

How will additional budget amounts affect metrics

I am trying to provide an analysis of what additional expenses of 1,000, 5,000 and 10,000 will have on my account and do not know how to calculate it together. I've extracted data from the last 4 months, configured them in an Excel sheet, and added additional columns for available impressions, incremental impressions, incremental clicks, incremental cost, and incremental conversions.

For example, the current account budget I work on every month is $ 22,780. I would like to know what the indicators and performance look like if we spend $ 22,780, $ 26,780, then $ 31,780 (increments of 1,000, 5,000 and 10,000) based on historical data from the last 4 months. I am currently using the following formulas:

Incremental Impressions = Available Prints * IS Lost (Budget)
Incremental Clicks = Incremental Impressions * CTR Stable
Incremental cost = incremental clicks * stable CPC
Incremental Conversions = Incremental Clicks * Stable Conversion Rates

Can someone help me and explain how I would determine incremental conversions, impressions, clicks, information system, etc.? I'm not even sure I can calculate a part of it. I know these are total estimates, but any idea about this would be extremely useful! Thank you!

Can I add custom metrics and dimensions to the Web property more than the app in Google Analytics?

I have a web property to which I could add custom measures and dimensions to track them on my web application via: Admin -> Property Settings -> Custom Definitions

But I've created a new web + app property in Google Analytics and I can not find a way to add custom metrics and dimensions.

Is it possible to add metrics and custom dimensions to the properties of the web application +?

dg.differential geometry – Express the Riemannian metric in extraction metrics

For a Riemannian collector $ M $ with an original metric $ g $, for every other possible measure $ g $ sure $ M $ is there a diffeomorphism $ f: (M, g) rightarrow (M, g) $ such as the withdrawal metric of $ f $ is $ g $? If this is not true in general, for what type of metric is it true and what properties do it not be true in general? I am particularly interested in the simple case where $ M $ is $ mathbb {R} ^ n $ and $ g $ is the usual Euclidean metric, but I am also interested in the general case.

postgresql – Detection of a Logical Replication Error on Postgres v10 on AWS RDS CloudWatch Metrics
"When the schema is changed on the editor and the replicated data begins to arrive at the subscriber but does not fit in the schema of the table, the replication will be wrong until the schema be updated "

How to detect / query these errors on Postgres?

I have tried

SELECT * FROM pg_stat_subscription;

but it shows that he has received the message from the publisher. It got lost when I changed the schema of the publisher and the subscriber stopped receiving updates.

On AWS, the associated RDS cloudwatch metrics are:

OldestReplicationSlotLag (The size of the replica that delays the most in terms of received WAL data, applies to PostgreSQL.)

but i do not think this is the right measure if i want to know if my updates have been corrected and applied to the subscribers replica database.

Do you have 50 pbn Backlinks with high domain metrics. for $ 5

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I will manually build 10 UNIQUE HOMEPAGE PNA backIinks on extremely domain (DA) and page (PA) domains. We also have a custom PBN package based on the keyword difficulty factor.


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autoscaling – Kubernetes: horizontal automatic scaling based on metrics from another namespace

I would like to configure horizontal automatic scaling for deployment based on input controller metrics deployed in another namespace.

I have a deployment (petclinic) deployed in a certain namespace (petclinic).

I have an input controller (nginx-ingress) deployed in another namespace (nginx-ingress).

I also have Prometheus and Prometheus-Adapter, so I have the API in the list of available APIs.

The metrics are collected and exposed, so the following command:

$ kubectl get --raw "/apis/" | jq .

gives the following result:

  "kind": "MetricValueList",
  "apiVersion": "",
  "metadata": {
    "selfLink": "/apis/"
  "items": (
      "describedObject": {
        "kind": "Ingress",
        "namespace": "nginx-ingress",
        "name": "petclinic",
        "apiVersion": "extensions/v1beta1"
      "metricName": "nginx_ingress_controller_requests",
      "timestamp": "2019-08-20T12:56:50Z",
      "value": "11"

Until here everything is fine, agree.

And what I need is to configure the HPA entity for my deployment. Do something like this:

apiVersion: autoscaling/v2beta1
kind: HorizontalPodAutoscaler
  name: petclinic
  namespace: petclinic
    apiVersion: apps/v1
    kind: Deployment
    name: petclinic-production
  minReplicas: 1
  maxReplicas: 10
  - type: Object
      metricName: nginx_ingress_controller_requests
        apiVersion: extensions/v1beta1
        kind: Ingress
        name: petclinic
      targetValue: 10k

Of course, this is incorrect because the nginx_ingress_controller_requests is related to the nginx-ingress namespace, so it does not work (well, as we expected):

    annotations: '({"type":"AbleToScale","status":"True","lastTransitionTime":"2019-08-19T18:43:42Z","reason":"SucceededGetScale","message":"the
        HPA controller was able to get the target''s current scale"},{"type":"ScalingActive","status":"False","lastTransitionTime":"2019-08-19T18:55:26Z","reason":"FailedGetObjectMetric","message":"the
        HPA was unable to compute the replica count: unable to get metric nginx_ingress_controller_requests:
        Ingress on petclinic petclinic/unable to fetch metrics
        from custom metrics API: the server could not find the metric nginx_ingress_controller_requests
        for ingresses.extensions petclinic"},{"type":"ScalingLimited","status":"False","lastTransitionTime":"2019-08-19T18:43:42Z","reason":"DesiredWithinRange","message":"the
        desired count is within the acceptable range"})' '({"type":""},{"type":"Resource","resource":{"name":"cpu","currentAverageUtilization":1,"currentAverageValue":"10m"}})' '({"type":"Object","object":{"target":{"kind":"Ingress","name":"petclinic","apiVersion":"extensions/v1beta1"},"metricName":"nginx_ingress_controller_requests","targetValue":"10k"}})' |

Alas, the autoscaling/v2beta1 API does not have the entity, so I can not "ask" to retrieve the value of another namespace. 🙁

Would anyone have the kindness to help me solve this puzzle? Is there a way to configure automatic scaling based on custom metrics belonging to another namespace?

Thanks in advance!

How to extract the use of Kubernetes' main metrics over time – metrics-server

I'm using metrics-server in Kubernetes. How can I retrieve data for these primary metrics:

  • Number of modules running for each horizontal module over a given period
  • CPU usage at times
  • Number of failures / restarts over a period of time

Which surfaces admit hyperbolic metrics delimited by geodesic arcs?

Consider a compact surface of the kind $ g $ with k $ limit components, ordered at $ 1 $ at k $. Next remove $ n_i geq 0 $ points of the $ i $-limit component, for $ n = 1, …, k. $ For which sequences $ (g, n_1, …, n_k) $, the surface above admits a hyperbolic metric of which $ i $-the component is delimited by $ n_i $ infinite geodesic arcs (go to $ n_i $ points to infinity)?
$ n_i = $ 0 means that the $ i $The third component is a geodesic loop (or a perforation).

The answer is yes for all $ n_i geq $ 3 but I am not sure of the other cases.