## rt.representation theory – Under what conditions representations of reductive Lie group in Banach space and in its Garding space have the same length?

Let $$G$$ be a real reductive Lie group (e.g. $$G=GL(n,mathbb{R})$$). Let $$rho$$ be a continuous representation of $$G$$ in a Banach space $$V$$. Let $$V^inftysubset V$$ be the subspace of smooth vectors equipped with the Garding topology. Let $$rho^infty$$ be the natural representation of $$G$$ in $$V^infty$$.

Under what precise technical conditions the representations $$rho$$ and $$rho^infty$$ have the same length?

A reference would be very helpful.

## combinatorics – How to generate all the combinations with repetition and another conditions?

I want to generate all the combinations with repetition for k variables with values from a set of n elements.

There are some ways, I like this formula, which I found on this forum (it is for n = 2 and k = 3, which is not a problem).

`````` With({n = 2, k = 3},
Join @@ Table(IntegerPartitions(s, {k}, Range(n)), {s, k, n k}))
``````

My question is, How can I add these conditions:

First: every value of n must be used at least once,
Second: every value of n must be used at most 10 times

Is there a way, how to improve the code above or should I try another method?

Thank you!

## How do I copy sharepoint list entries to another list when certain conditions are fulfilled (Status = publish & Audience = all)

We want to publish a FAQ list on a sharepoint site. We have another list that people work on, in which the question is logged, a person that should answer it is tagged and so on. I would like to copy the list entries to another list when the “Choice column for the audience” is set to “all” (which is done when the question is first logged) and the “Sign off person” sets the “Choice column for the status” to “publish”. How can I do this?

I checked out Power Automate/Flow but I can’t find anything that lets me put the conditions as described above.

Thank you very much for your help!
Franziska

## Pricing conditions in WooCommerce – WordPress Development Stack Exchange

Pricing conditions in WooCommerce – WordPress Development Stack Exchange

## real analysis – Sufficient conditions for \$dfrac{1-x^a}{1-x^b} > 1/2\$ for all \$x>0\$

What are the sufficient conditions for the following inequality to be true all $$x>0.$$

$$dfrac{1-x^a}{1-x^b} > frac{1}{2},,$$ where $$0 and $$bgeq 1$$.

Of course, the case is trivial when $$a=b.$$

Also, is it possible to state the necessary conditions?.

## python – Filtering a DataFrame based on two logical conditions, first one numpy array values, second one current day based

The following is just a working example.

I have a DataFrame containings a monotonous growing function.
Some of the values are actuals, some are forecasted.

I need to filter specific actuals values based on a set of milestones and I have to avoid to take forecasted values from the dataframe

I created this following script.
It works, but I think is not so much pythoninc.

I am a self taught and my working eviroment is `Google Colab`

Expected Output

• I would avoid the `for loop` and the `if` condition

• Understand if there is room of improvement in the code quality

``````#importing libraries
import pandas as pd
import numpy as np
import datetime

#working code mock-up
th_array = np.arange(0, 11000, 1000)
cumulated_array = np.arange(0, 5000, 185)

df_index = pd.date_range(end = "20/04/2021",
periods = len(cumulated_array))
df = pd.DataFrame(data = cumulated_array,index = df_index,
columns = ("cumulated"))

df_filtered = pd.DataFrame()
current_day =  pd.to_datetime(datetime.date.today())

#filtering loop
for y in th_array:
x = df((df('cumulated') > y) & (df.index < current_day))
if x.empty is False:
df_filtered = df_filtered.append(x.iloc(0))
``````

## algorithm – How to obtain Contiguous Subarrays with these conditions JavaScript

So we are given an array arr of N integers. For each index i, you are required to determine the number of contiguous subarrays that fulfill the following conditions:

The value at index i must be the maximum element in the contiguous subarrays, and
These contiguous subarrays must either start from or end on index i.

What I have been trying to do is create a prefix sum with a default of (0, -1) within a hash map, but I am not obtaining the answer I need. I can do it brute force but I need to solve it in O(n)

Input
Array arr is a non-empty list of unique integers that range between 1 to 1,000,000,000
Size N is between 1 and 1,000,000
Output
An array where each index i contains an integer denoting the maximum number of contiguous subarrays of arr(i)

Example:

``````arr = (3, 4, 1, 6, 2)
output = (1, 3, 1, 5, 1)
/* Explanation:

For index 0 - (3) is the only contiguous subarray that starts (or ends) with 3, and the maximum value in this subarray is 3.
For index 1 - (4), (3, 4), (4, 1)
For index 2 - (1)
For index 3 - (6), (6, 2), (1, 6), (4, 1, 6), (3, 4, 1, 6)
For index 4 - (2)
So, the answer for the above input is (1, 3, 1, 5, 1)

*/``````

## ordinary differential equations – ODE eigenvalue problem with unusual boundary conditions

I am given:

y”+λy=0, y(0)=0, (1−λ)y(1)+y′(1)=0

As usual we are looking for not trivial solutions.
Looks like a standard eigenvalue problem and yet I am totally stuck.
The case when lambda = 0 is rather obvious. A=B=0. Not much fun.
But when I start trying for lambda greater or smaller than zero, I get to this:

1. λ < 0, the solution is of the form:
B((1+ω^2)sinh(ω)+ωcosh(ω))=0 where λ=-ω^2

2. λ > 0, the solution is of the form:

B((1-ω^2)sin(ω)+ωcos(ω))=0 where λ=ω^2

The question states:Find the nontrivial stationary paths,
stating clearly the eigenfunctions y. In the case 1) I cant see any non trivial solutions but… well in the second case I cant see either. I know there are solutions.
Any help would be highly appreciated

## solving pde equation with four boundary conditions

how I can solve this PDE equation with four boundary conditions mentioned in the attached file via FEM or FDM or another methods?

It seems that this pde needs four boundary conditions. One of them is defined on a curved domain that its function is provided.

Data related to function phi in boundary condition is attached as an excel file.