SQL 2019 Server Not accepting new ports, only using 58865

I’ve been trying to set up a db for a user list on my server. I am trying to link it to a C# Visual Studio, and I want to use the external port 31135, and I’ve set all of the IP- settings in the Server Configuration Manager correctly, but still uses 58865.

If it helps any, the only other main process running on the server is WAMP but I disabled the MySQL in there to test it and even with it disabled it still wants to use 58865.

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finite automata – Proof that for every $k > 1$, there exists a language $A_k subseteq {0, 1}^*$ s.t. a DFA accepting $A_k$ has $k$ states but no less

I am trying to prove that for every $k > 1$, there exists a language $A_k subseteq {0, 1}^*$ such that a DFA accepting $A_k$ has $k$ states but no less.

I thought about proving this in two ways: constructively, or by contradiction.

By Contradiction: For some $k > 1$, suppose there is no language $A_k subseteq {0, 1}^*$ where a DFA accepting $A_k$ exists with exactly $k$ states but no less. Then any language $L subseteq {0, 1}^*$, that has an accepting DFA with $n$ states, also has an accepting DFA with $n – 1$ states. Extending this reasoning downward, this implies that every $L$ has an accepting DFA with $1$ state. This is a contradiction.

By construction: Let $A_k = {w in {0, 1}^* | w text{ has } 0^{k – 1} text{ as a substring}}$. But now I need to prove that $A_k$ cannot be accepted by a DFA with less than $k$ states. I am unsure of how to do this even though it feels true intuitively.

Is my proof by contradiction valid and is the proof by construction doable? Maybe both are wrong and I need to use some other perspective.

scanning – In 2021, which film scanners accepting different film formats are compatible with macOS Big Sur?

With Apple’s recent rollout of their new iMac line, I have been considering upgrading my 2011 computer to something more modern.

However, I know that my scanner, Epson V500, doesn’t work with macOS Big Sur as of May 2021 (I have tried on another computer running Big Sur) – and what good would a new computer be if I can’t scan pictures with it?

So upgrading computers would also mean having to purchase a new scanner.

I have been looking for articles comparing currently available scanners that would be able to scan the formats I shoot – 110, 16mm, 35mm half-frame, 35mm, 120, and 6×10, and have been struggling to find a review that wasn’t mostly oriented towards 35mm film scanning or very expensive professional gear.

What are good, consumer-level film scanners that are Big Sur compatible and can scan a variety of formats?

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