Tag: Analysis

formal grammars – What is the relationship between language analysis and language verification?

There is a simple relationship between the two problems, but it only goes on one side: if you can analyze, then you can recognize. Indeed, if you run the analyzer and it generates a failure, the input is not in the language; if the analyzer produces an analysis tree, the entry is in the language.

Note, however, that an analyzer and an identifier not only solve different problems, but also receive different specifications. An identifier receives a language in abstract form. On the other hand, an analyzer receives a grammar, which not only implicitly defines a language, but also instructs the analyzer how to convert a valid entry into an analysis tree. Since the same language can have many equivalent grammars, it is unclear what it even means to convert an identifier to an analyzer – which grammar should the analyzer use?

Why are recognition software interesting, then? Precisely because of the one-sided connection, which states that an analyzer can be converted to an identifier. This relationship implies that if your language is difficult to recognize, you will not be able to analyze it (using a certain class of algorithms).

functional analysis – semigroups and Markov resolvers, difference in continuity

Let $ (E, d) $ be a locally compact separable metric space. We have a Markov process $ X = ( {X_t } _ {t ge 0}, {P_x } _ {x in E}) $ sure $ E $. For a limited measurable function $ f $ sure $ E $, we define
begin {align *}
P_tf (x) & = E_ {x} (f (X_t)), quad t> 0, \
R _ { alpha} f (x) & = int_ {0} ^ { infty} exp (- alpha t) P_tf (x) , dt, quad alpha> 0.
end {align *}
We assume that for any bounded measurable function $ f colon E à mathbb {R} $ and $ alpha> 0 $, function $ R _ { alpha} f $ East $ lambda $-Hölder continuous function activated $ (E, d) $. In other words, he contends that
begin {align *}
| R _ { alpha} f (x) -R _ { alpha} f (y) | le Cd (x, y) ^ { lambda}, quad x, y in E,
end {align *}
or $ lambda> 0 $ and C $> $ 0 are a positive constant independent of $ x, y $.

My question

We assume that the Markov process $ X $ is symmetrical with respect to a $ sigma $– finished measurement $ m $.
Under the above conditions, the semi-group $ P_tf (x) $ is also Hölder continues in $ x $?

If the semi-group is ultracontractive in the sense that $ P_t (L ^ 1 (E, m)) subset L ^ { infty} (E, m) $, for all $ t> 0 $ and $ f in L ^ 2 (E, m) $, we can find a bounded measurable function $ h colon E at mathbb {R} $ such as $ P_tf = R_ {1} h $. Without the ultracontractivity, can we prove the Hölder continuity of $ P_tf $?

About the definition of the convex from Rudin's analysis

This is from Rudin's mathematical analysis:

We call a set $ E subset R ^ k $ convex if $ lambda x + (1- lambda) y in E $ anytime $ x in E $, $ y in E $, and $ 0 < lambda <1 $.

What is the meaning of this definition? And how we come to the equation $ lambda x + (1- lambda) y in E $?

real analysis – a Bessel type integral

I meet the following integral while trying to find the inverse Fourier transform of the characteristic function of a certain sum of random variables. Here, $ 0 le lambda le1 $, $ p ge0 $, $ q ge0 $ are real, and $ n $ is an integer. I want to calculate the following integral:

$$ int_0 ^ {2 pi} e ^ {p cos ( lambda tau) + q cos ((1 – lambda) tau)}
cos (n tau) frac {d tau} {2 pi} $$

It is a generalization of a Bessel integral, in that for $ q = 0 $ and $ lambda = 1 $, I know it:

$$ int_0 ^ {2 pi} e ^ {p cos ( tau)} cos (n tau) frac {d tau} {2 pi}
= I_n (p) $$

or $ I_n (p) $ is the modified Bessel function of the first type.

fa.functional analysis – On the dense inlay of Banach spaces

Warning: When I asked this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake for several hours tonight … (I still hope, however, this is simply due to my ignorance.)

Question. Let $ E, F $ to be the Banach space and let's assume that $ F $ integrates densely and continuously into $ E $ (we therefore consider $ F $ as a subspace of $ E $ from now on). Suppose there is a constant $ M in (0, infty) $ with the following property:

For each $ e in E $ we can find a sequence $ (f_n) $ in $ F $ which converges towards $ e $ compared to $ | cdot | _E $ and who satisfies $ | f_n | _F le M | e | _E $.

Does it follow that $ F = E $?

Note. I first thought that the answer had to be yes due to some application of the open mapping theorem: clearly, it suffices to show that $ | cdot | _F $ and $ | cdot | _E $ are equivalent on $ F $, and by the open mapping theorem this is true if $ | cdot | _E $ is finished on $ F $; but I couldn't prove this last property.

Am I forgetting a simple argument or a simple counterexample?

real analysis – Is each function $ f: mathbb R to mathbb R $ differentiable at least one point on an everywhere dense subset of $ mathbb R $?

I was doing some pretty simple research a few hours ago and I almost asked a similar question with the word continued instead of differentiable in the title, but I found this question asked by Gro-Tsen where there was an affirmative answer to this question.

Apparently, this is the result of Blumberg, that for each $ f: mathbb R to mathbb R $ there is a dense subset $ D $ of $ mathbb R $ such as $ f | _D $ is continuous.

Blumberg's document can be found here and I did a little research of his arguments, however, I'm not sure they can be adapted to show that $ f $ is differentiable at least in one point when it is limited to an everywhere dense subset of $ mathbb R $.

Honestly i expect there to be $ f $´s which have the property that when they are limited to all dense subset everywhere possible of $ mathbb R $ are not differentiable everywhere on all these sets

However, I am not sure, and that is why I ask it here, as I think it is known, since the Blumberg result is established relatively long ago (1922).

Here's the question:

  • Is each function $ f: mathbb R to mathbb R $ differentiable at least one point on an everywhere dense subset of $ mathbb R $?

In other words:

  • Is it true that for each function $ f: mathbb R to mathbb R $ there is at least one dense set everywhere $ D subseteq mathbb R $ such as $ f | _D $ is at least one point differentiable?

Is there photo analysis software that pre-sorts images by identifying potential technical problems?

It's pretty easy to do if you can write in Python. Here is a good article on using an open-source computer vision package to detect global image blur:

Blur detection with OpenCV

Here is a quick script that will sort the images in fuzzy / ok directories:

#
# Sorts pictures in current directory into two subdirs, blurred and ok
#

import os
import shutil
import cv2

FOCUS_THRESHOLD = 80
BLURRED_DIR = 'blurred'
OK_DIR = 'ok'

blur_count = 0
files = (f for f in os.listdir('.') if f.endswith('.jpg'))

try:
   os.makedirs(BLURRED_DIR)
   os.makedirs(OK_DIR)
except:
   pass

for infile in files:

   print('Processing file %s ...' % (infile))
   cv_image = cv2.imread(infile)

   # Covert to grayscale
   gray = cv2.cvtColor(cv_image, cv2.COLOR_BGR2GRAY)

   # Compute the Laplacian of the image and then the focus
   #     measure is simply the variance of the Laplacian
   variance_of_laplacian = cv2.Laplacian(gray, cv2.CV_64F).var()

   # If below threshold, it's blurry
   if variance_of_laplacian < FOCUS_THRESHOLD:
      shutil.move(infile, BLURRED_DIR)
      blur_count += 1
   else:
      shutil.move(infile, OK_DIR)

print('Done.  Processed %d files into %d blurred, and %d ok.' % (len(files), blur_count, len(files)-blur_count))

Your most difficult problem will be to install python and opencv in your system. Google python3 for your operating system and how to install pip with, you can use pip3 to install opencv. Or, there are also python + opencv pre-construction installations. You do not need the latest version of opencv to run this script.

The script works very well and it measures the overall blurring of the image. It’s good for most photos. However, the overall measurement of the image means that these background photographs filled with one side and one bokeh will be placed in the blurred directory, and you will have to sort them. Either way, you should browse the blurry images to make sure there aren't any lost guards in there.

I hope this script speeds up your workflow.

A big improvement in this script is to include face detection, calculate the blur on the biggest faces in the photo and use these values ​​for the blur threshold, by default the general blur if no face is detected. I leave you this improvement!

calculation and analysis – Limit to infinity of arbitrary functions

Here I have the code that takes the limit of an expression

Limit((-I E^(I x) f1(y))/(g2^(Prime)(Prime))(y), x -> (Infinity) )

and the returned output is INDETERMINATE while the desired output is $ infty $. Or if I had to do it instead

Limit((g2^(Prime)(Prime))(y)/(-I E^(I x) f1(y)), x -> (Infinity) )

I would like to get 0 and not INDETERMINATE.

How would I let Mathematica know that $ f $ and $ g $ the functions are not relevant when assessing the limit?

Thanks for any help.

python 3.x – Simple Json log analysis application

I had a coding assignment (interview) task for a role of senior data engineer, validation of JSON data. Each record in the json entry indicates a client record, I need to write a script to validate that all of the records confirm to few business rules and some type / field length restrictions.

I used an object oriented and metadata based approach to write this parser in Python. However, I have received negative comments for the task and hope to get comments on the code. Here is the git repository with an additional Readme file for context and business rules – https://github.com/YADAAB/jsonparser

I'm looking for inputs on coding style, overall design, object oriented approach, etc.

An example of email validation since the stackXchange wants me to add code in the question 🙂

        email_regex = '^w+((.-)?w+)*@w+((.-)?w+)*(.w{2,3})+$'
    try:
        if(re.search(email_regex,self.line_dict(field_name))):
            return True
        else:
            self.error_dict(field_name) = str(self.error_dict.get(field_name, '')) + 'Invalid email'

Would like to answer specific questions. Thanks for checking!

calculation and analysis – How to get the step by step derivative of this?

I am very new to Mathematica and wanted to use it to help me solve some questions that I don't have math support to help me solve on paper. I would like to take this equation and derive it partially. However, I would also like to take a step by step to help me hold my hand so that I know what's going on. Instead, what I get is this error message. Could someone help me?

WolframAlpha("D(-T0/ ProductLog(-E^(-1+dS/dC), dS)")

WolframAlpha::timeout: The call to WolframAlpha(D(-T0/ ProductLog(-E^(-1+dS/dC), dS)) has exceeded 30. seconds. Increasing the value of the TimeConstraint option may improve the result.

Thank you!