fa.functional analysis – The set of bounded lipschitz functions over a compact is barrelled but not a neighborhood of zero?

I recently learned that Banach spaces are barrelled, i.e any convex, balanced, absorbing and closed subset is a neighborhood of zero (wikipedia). I’m having trouble understanding why the following example fails.

The space $C((0,1))$ of real-valued (uniformly) continuous functions over the compact $(0,1)$ is a Banach for the sup norm. Fix $M$ and $L>0$. Let $X$ the subset consisting of $L$-lipschitz and $M$-bounded functions.

On the one hand, I checked that (and evidently made a mistake somewhere)

  • $X$ is convex: since convex combination of $L$-lipschitz is $L$-lipschitz, and convex combination of $M$-bounded is $M$-bounded
  • $X$ is balanced: since $-X = X$
  • $X$ is absorbing: for $f$ (uniformly) continuous over $(0,1)$, shrinking $f$ makes it more and more lipschitz, and more and more bounded, so $lambda f in X$ for some $lambda>0$
  • $X$ is closed: since uniform limit of $L$-lipschitz is $L$-lipschitz, and uniform limit of $M$-bounded is $M$-bounded

On the other hand, $X$ is compact (because bounded and equicontinuous, by Arzela-Ascoli theorem), so it cannot contain an open ball (open balls of $C((0,1))$ are non-compact by Riesz theorem). So it cannot be a neighborhood of zero.

So $X$ is a barrelled subset of a Banach space, yet it cannot be a neighborhood of zero… Where is the mistake?

calculus and analysis – Assumptions for Definite Integral of Log[]

The problems are related to the fact that the integral has several parameters. Depending on their values the integral may exist, or not.

Let us see.

This is your function

sol=(ri t)/(-b + v) + a/(Sqrt(t) (2 b v + 2 v^2));

Let us solve first an undetermined integral, calculating the terms of your function separately:

Integrate(#, v, Assumptions -> {ri > 0, t > 0, b > 0}) & /@ sol

(*  ri t Log(-b + v) + (a (Log(v)/(2 b) - Log(b + v)/(2 b)))/Sqrt(t)   *)

Now, one can see that in order for the integral to exist one should require v-b>0 v+b>0 and t!=0. Besides, it is useful to define that v1>0 and v2>v1. Of course, if the integration along the real axis corresponds to your intention. With this

AbsoluteTiming(
 Integrate(sol, {v, v1, v2}, 
  Assumptions -> {ri > 0, t > 0, b > 0, v1 > 0, v2 > v1, v1 > b}))

in 0.6s yields the result:

(*  {0.619135, 
 ri t Log((b - v2)/(b - v1)) + (a Log(((b + v1) v2)/(v1 (b + v2))))/(
  2 b Sqrt(t))}  *)

Have fun!

real analysis – Help with a limit involving incomplete beta integral

In trying to prove that the limit of a certain function approaches 1 as the positive integer parameter $n$ approaches infinity, I have ended up with the following intermediate expressions:
$$f(n)=2^{1+2n}B_{1/2}(n,n+2)$$
$$g(n)=4^nB_{1/2}(n+1,n)$$
$$ h(n)=n(n-1)/2 left( frac{f(n)}{n+1}-frac{g(n)}{n-1}right)$$
Can somebody kindly help me with the evaluation of $ lim_{n to infty} h(n)$?
If somebody could also plug it in Mathematica, I would be highly obliged. Thanks for any help in advance.
P.S.: In the above the notation $B_z(a,b)$ stands for the incomplete beta function defined by:
$$B_z(a,b)=intlimits_0^z u^{a-1}(1-u)^{b-1} mathrm{d}u.$$

redirects – Some questions about Screaming Frog analysis

I crawled my WordPress site using the Screaming Frog program and noticed a few issues which might have SEO implications and was hoping someone could help identify the problem (if there is one).

N.B: I have obscured the domain name in the image as I do not want to make it public.

The canonical URL for my domain is prefixed by https://www but as you can see from the tree graph, there are a number of URLs accessible under the http protocol and the non-www version of the domain. Those URLs are non-indexable but I’m curious as to why they’re accessible at all. I was thinking that shouldn’t happen if the proper redirects were in place.

In the WordPress admin settings I have entered the correct version of my domain for the WordPress and site addresses (i.e., https://www)

It’s possible my .htaccess file might require editing so I have pasted it below:

# BEGIN LSCACHE
## LITESPEED WP CACHE PLUGIN - Do not edit the contents of this block! ##
<IfModule LiteSpeed>
RewriteEngine on
CacheLookup on
RewriteRule .* - (E=Cache-Control:no-autoflush)
RewriteRule .object-cache.ini - (F,L)

### marker CACHE RESOURCE start ###
RewriteRule wp-content/.*/(^/)*(responsive|css|js|dynamic|loader|fonts).php - (E=cache-control:max-age=3600)
### marker CACHE RESOURCE end ###

### marker FAVICON start ###
RewriteRule favicon.ico$ - (E=cache-control:max-age=86400)
### marker FAVICON end ###

### marker DROPQS start ###
CacheKeyModify -qs:fbclid
CacheKeyModify -qs:gclid
CacheKeyModify -qs:utm*
CacheKeyModify -qs:_ga
### marker DROPQS end ###

</IfModule>
## LITESPEED WP CACHE PLUGIN - Do not edit the contents of this block! ##
# END LSCACHE
# BEGIN NON_LSCACHE
## LITESPEED WP CACHE PLUGIN - Do not edit the contents of this block! ##
### marker BROWSER CACHE start ###
<IfModule mod_expires.c>
ExpiresActive on
ExpiresByType application/pdf A31557600
ExpiresByType image/x-icon A31557600
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ExpiresByType image/svg+xml A31557600

ExpiresByType image/jpg A31557600
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ExpiresByType video/ogg A31557600
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ExpiresByType application/x-javascript A31557600

ExpiresByType application/x-font-ttf A31557600
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ExpiresByType application/font-woff2 A31557600
ExpiresByType application/vnd.ms-fontobject A31557600
ExpiresByType font/ttf A31557600
ExpiresByType font/otf A31557600
ExpiresByType font/woff A31557600
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</IfModule>
### marker BROWSER CACHE end ###

## LITESPEED WP CACHE PLUGIN - Do not edit the contents of this block! ##
# END NON_LSCACHE
#This Apache config file was created by Duplicator Installer on 2021-02-17 10:08:29.
#The original can be found in archived file with the name .htaccess__(HASH)

# BEGIN WordPress
# The directives (lines) between "BEGIN WordPress" and "END WordPress" are
# dynamically generated, and should only be modified via WordPress filters.
# Any changes to the directives between these markers will be overwritten.
<IfModule mod_rewrite.c>
RewriteEngine On
RewriteRule .* - (E=HTTP_AUTHORIZATION:%{HTTP:Authorization})
RewriteBase /
RewriteRule ^index.php$ - (L)
RewriteCond %{REQUEST_FILENAME} !-f
RewriteCond %{REQUEST_FILENAME} !-d
RewriteRule . /index.php (L)
</IfModule>

# END WordPress

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fa.functional analysis – Lipschitz estimate for compact operators

Let $T,S$ be two positive compact operators with zero null space.

When can I expect an estimate of the form

$$Vert T^{1/2}
S^{-1}T^{1/2}-1Vert le C Vert T-SVert_Y?$$

I intentionally write $Y$ on the right hand side to indicate that for such a bound to hold, one probably has to penalize small eigenvalues of $S$?

Is there perhaps a better way to quantify that if $T$ and $S$ are close, then $Vert T^{1/2}
S^{-1}T^{1/2}-1Vert$
is close to zero?

calculus and analysis – Series with ArcTan gives wrong symbolic answer in Wolfram Language

Recently, I have found a very uncomfortable problem when using Wolfram Language.
When calculating

Series(ArcTan(A + 1/x), {x, 0, 2},  Assumptions -> A > 0 && x > 0)

I get the wrong answer (tested in wolfram cloud and in Mathematica 12.1.1.0)

wrong answer

I am absolutely sure that the correct answer should be

correct answer

There is a plot of these functions to demonstrate the issue

plot demonstration

How to get rid of the problem?

calculus and analysis – Asymmetric multivariable Taylor expansion

I want to expand a two-variable function up to asymmetric orders in two expansion variables, i.e.

$$f(x,y) = T(f(x,y)) + mathcal{O}(x^2,y^3,xy,xy^2).$$

Note that, while quadratic terms in $y$ are retained, the (also!) quadratic terms in $xy$ are contained in the higher-order terms. Hence the asymmetry (and the ultimate origin of the problem).

The intuitive two-variable expansion

Series(f(x,y),{x,x0,1},{y,y0,2})

includes some of the undesired terms, namely $xy$ and $xy^2$.

I’ve seen the (maybe) related post Multivariable Taylor expansion does not work as expected. However, in this post the issue with the asymmetry in the two expansion variables is not commented, so the answers there are not directly applicable (in the best case).

Is there a simple way/function to perform this calculation?

fa.functional analysis – Almost commuting matrices, one a projection, is there a nearby projection that commutes?

Suppose that $P, A, Q in mathbb{M}^{n times n}(mathbb{R})$ (I’m still interested if it must be done over $mathbb{C}$), $P$ an orthogonal projection, and $lvert lvert PA-AP rvert rvert < delta$, is there some $Q$ an orthogonal projection such that $lvertlvert P-Q rvert rvert < epsilon(delta)$ (such that as $delta to 0$, so does $epsilon$) and $QA = AQ$? Obviously if there’s a broader result than just the finite-dimensional case, then that’s good too.

I’ve seen results vaguely like this, e.g. Lin’s Theorem is a result about almost-commuting normal matrices. However I haven’t seen a result about projections of this kind.

It would be nice if this were true, I hope to learn more about this type of result, but if the good people here could point me to this result or a similar result from which I could jump off, I’d very much appreciate it.

functional analysis – contraction in five dimensional Euclidean space

$T(u,v,w,x,y)=(v,0.5w,x,y,u)$

$S(u,v,w,x,y)=(y,u,v,0.5w,x)$

Could any one tell me whether S and T are contraction map on any matrix norm/induced vector norm? Are their composition both way i.e ST and TS, contraction? What can we say about Contraction of SSSSTTTT or TTTTSSSS? Thank you for your response.

python – Optimize Yahoo Finance Code for Analysis

I am trying to analyze a number of companies using financial data I gathered from Yahoo Finance. I am also using the yfinance API to get some more details about the company using functions. Since I am trying to do this for a number of companies Each Iteration needs to be quick. Currently, 1 Company takes about 3 seconds.

Is it because of the API calls or requests? Can I increase the below code efficiency?

import pandas as pd
import numpy as np
from pandas import ExcelWriter
import requests
import timeit
import yfinance as yf

def get_industry(stock):
    tickerdata = yf.Ticker(stock) 
    return tickerdata.info('industry')

def get_symbol(symbol):
    url = "http://d.yimg.com/autoc.finance.yahoo.com/autoc?query={}&region=1&lang=en".format(symbol)
    result = requests.get(url).json()
    for x in result('ResultSet')('Result'):
        if x('symbol') == symbol:
            return x('name')
def get_currency(stock):
    tickerdata = yf.Ticker(stock) 
    return tickerdata.info('currency')


quarters = ('Q4-2020','Q3-2020','Q2-2020','Q1-2020','Q4-2019','Q3-2019','Q2-2019','Q1-2019','Q4-2018','Q3-2018','Q2-2018','Q1-2018')
start_time = timeit.default_timer()
r = ()
for i in stock(:1):
    try:
        df = pd.read_csv(i+'_quarterly_balance-sheet.csv')
        df = df.drop('ttm',axis=1,errors='ignore')
        df('name') = df('name').str.replace('t','')
        df = df.iloc(:,:13).T
        df.columns = df.iloc(0)
        df = df(1:)
        df.insert(0,'Q',quarters)
        try:
            df.insert(0,'Currency',get_currency(i))
        except:
            df.insert(0,'Currency','Not Found')
        df.insert(0,'Ticker',i)
        df.insert(0,'Company',get_symbol(i))
        try:
            df.insert(0,'Industry',get_industry(i))
        except:
            df.insert(0,'Industry','Unknown')
        df = df.loc(:,df.columns.isin(('Q','Currency','Ticker','Company','Industry','TotalRevenue')))
        r.append(df)
    except:
        continue
df = pd.concat(r)
# code you want to evaluate
elapsed = timeit.default_timer() - start_time
elapsed

3.041297 Seconds