Understand Convolution, Singularity, Kernel, etc: Math Reading Guide

In summary, the conversation discusses the topic of vortex methods and the reader's struggle to understand related mathematical terms. They express a desire to learn more about singularities, convolution, singularity, and kernels, but are unsure of where to start. The expert suggests that these concepts are related to calculus and recommends searching for individual papers and lecture notes online for a better understanding. They also provide tips for finding this information.
  • #1
vktsn0303
30
0
I'm reading a book on vortex methods and I came across the above mentioned terms, however, I don't understand what they mean in mathematical terms. The book seems to be quite valuable with its content and therefore I would like to understand what the author is trying to say using the above mentioned terms in his mathematics. Can someone please tell me what branch of mathematics or what book of mathematics I should read in order to understand these terms? If it helps, I would like to point out that I have a decent understanding of college level calculus.

Thanks in advance!
 
Mathematics news on Phys.org
  • #2
This is a rather broad question and depends very likely on the context of vortex theory, as I assume you don't want to hear a standard answer on what a singularity is. Could you narrow it down by some context, examples or further explanations? Otherwise it's almost impossible to answer without complaints about what is meant where and by whom.
 
  • #3
fresh_42 said:
This is a rather broad question and depends very likely on the context of vortex theory, as I assume you don't want to hear a standard answer on what a singularity is. Could you narrow it down by some context, examples or further explanations? Otherwise it's almost impossible to answer without complaints about what is meant where and by whom.
Yes, I agree it's a rather broad question. Sorry about that.
I would actually like to learn about singularities in a strict mathematical sense. So, if I have to learn about convolution, singularity and kernels in particular where should I start looking?

I did google about them a bit, found some information in wikipedia. But I think it would be much better if I knew the basics of whatever it is I have to know to understand convolution, singularity and kernel. That way I would be able to understand the context of application of these concepts.
 
  • #4
Well, all these belong to calculus in a way. E.g. I've found all terms in the book from Hewitt and Stromberg
https://www.amazon.com/dp/0387901388/?tag=pfamazon01-20
but this isn't quite easy to read as it is mainly based on measure theory, whereas usual college courses proceed along the lines real analysis - vector analysis - complex analysis and maybe followed by function theory and functional analysis. In addition to understand vortex theory, even some basics on differential geometry and differential equations might be needed. So in order to deal with vortex theory in special, it might be more promising to look out for individual papers, that deal with certain questions. Google often leads to lecture notes on certain topics, that can be read in a reasonable amount of time. Many universities provide such notes on the internet. But as a tip: it's better to search via Google rather than on the universities' homepages, as you normally cannot get through to the individual papers by starting on their homepages.
 
  • #5
fresh_42 said:
Well, all these belong to calculus in a way. E.g. I've found all terms in the book from Hewitt and Stromberg
https://www.amazon.com/dp/0387901388/?tag=pfamazon01-20
but this isn't quite easy to read as it is mainly based on measure theory, whereas usual college courses proceed along the lines real analysis - vector analysis - complex analysis and maybe followed by function theory and functional analysis. In addition to understand vortex theory, even some basics on differential geometry and differential equations might be needed. So in order to deal with vortex theory in special, it might be more promising to look out for individual papers, that deal with certain questions. Google often leads to lecture notes on certain topics, that can be read in a reasonable amount of time. Many universities provide such notes on the internet. But as a tip: it's better to search via Google rather than on the universities' homepages, as you normally cannot get through to the individual papers by starting on their homepages.
Thanks a lot for the information, fresh_42. I'll try to look up lecture notes that are made available online. I think that's the easier way to learn too.
 

FAQ: Understand Convolution, Singularity, Kernel, etc: Math Reading Guide

1. What is convolution and why is it important in mathematics?

Convolution is a mathematical operation that combines two functions to produce a third function. It is important in mathematics because it allows us to analyze and model complex systems by breaking them down into simpler components.

2. What is a singularity and how does it relate to convolution?

In mathematics, a singularity is a point where a function or equation becomes undefined or infinite. In convolution, singularities can occur when the two functions being convolved have singularities at the same point.

3. How is a kernel used in convolution?

A kernel is a small matrix used in convolution to perform mathematical operations on an image or signal. It is convolved with the input data to produce a filtered output.

4. What is the difference between discrete and continuous convolution?

Discrete convolution is used for processing discrete signals such as digital images, while continuous convolution is used for processing continuous signals such as analog signals. In discrete convolution, the kernel is applied to each pixel individually, while in continuous convolution, the kernel is applied to the entire signal at once.

5. How is convolution used in real-world applications?

Convolution has many practical applications in fields such as signal processing, image processing, and machine learning. It is used for tasks such as image filtering, noise reduction, and pattern recognition. It is also an important concept in the field of deep learning, where convolutional neural networks use convolution to extract features from images or signals.

Back
Top