Can every regular expression be written as sum of products?

I was trying to prove that Parikh Image of every regular language is semi-linear. Even though it is true for CFL, but this question was about regular languages. To prove this, I decided to proceed as follows:

Lemma 1: Every regular expression can be written as sum of products with star, i.e., $ R = R_1 + R_2+ … +R_n $ where each $ R_i$ is a regular expression not involving any $ +$ .

Lemma 2: The Parikh image of a language corresponding to a regular expression that doesn’t involve any $ +$ is linear.

I proved the lemma 2 by induction on the structure of this specific type of regular expression. (which was easy and correct, by the way).

After proving (allegedly) these, it was easy to state that the Parikh image of the language defined by $ R$ is just the union of linear sets, hence it is semi-linear.

I proved the lemma 1, again, using induction on the structure of the regular expression, but I am not sure if it is correct. Moreover, I am not sure if the Lemma 1 is even valid or not!

My proof was as follows:

  1. $ R = a $ where $ a \in \Sigma$ or $ a=\epsilon$ . This is already in the same form we required.

  2. $ R = R_1.R_2$ . This is also in required form.

  3. $ R = (R_1 + R_2).R_3$ . Replace by $ R_1.R_3 + R_2.R_3$ .

  4. $ R = (R_1 + R_2)^* $ Replace by $ (R_1^*.R_2^*)^*$

Do you think this proof is wrong, or even worse, the statement if wrong!

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Which D&D 3.0 & 3.5 Kingdoms of Kalamar books are considered officially licensed third-party products?

Hey I Can Chan’s answer to the question “How do you tell if a D&D book is 3.0 or 3.5?” describes a category of products as follows (emphasis mine):

officially licensed Kingdoms of Kalamar third party products—products bearing some combination of the Dungeons and Dragons logo and the Wizards of the Coast officially licensed product seal or including on their title pages that the product is officially licensed by Wizards of the Coast.

Which D&D 3.0 & 3.5 Kingdoms of Kalamar books are considered officially licensed third-party products?

What D&D 3.0 & 3.5 Dragonlance books are considered officially licensed third party products?

Best said by…

“Officially licensed Dragonlance third party products—products bearing some combination of the Dungeons and Dragons logo and the Wizards of the Coast officially licensed product seal or including on their title pages that the product is officially licensed by Wizards of the Coast.” …HeyICanChan

Emphasis mine.

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UX process for various products in different development phases

There are a lot of phases to go through in UX and UI design process. In research phase, there are various research methods available and many steps to take to achieve a meaningful result.

I would like to better understand, what UX research methods would be the most appropriate for projects in these cases?

1. Website

  • New website
  • Existing website – adding new pages, sections, updating any part, redesign

2. Web app

  • New web app

  • Existing web app – adding new pages, sections, updating any part, redesign

3. Mobile app

  • New mobile app

  • Existing mobile app – adding new pages, sections, updating any part, redesign

UX process for various products in different development phases

There are a lot of phases to go through in UX and UI design process. In research phase, there are various research methods available and many steps to take to achieve a meaningful result.

I would like to better understand, what UX research methods would be the most appropriate for projects in these cases?

1. Website

  • New website
  • Existing website – adding new pages, sections, updating any part, redesign

2. Web app

  • New web app

  • Existing web app – adding new pages, sections, updating any part, redesign

3. Mobile app

  • New mobile app

  • Existing mobile app – adding new pages, sections, updating any part, redesign