gm.general mathematics – Examples of conjectures whose major versions were disproved later with a different proof with stronger consequences

Mathematics several times has conjectures of form
$$mathsf{Statement A}impliesmathsf{Statement B}.$$

In such cases falsity of $mathsf{Statement B}$ implies falsity of $mathsf{Statement A}$. However since falsity of $mathsf{Statement A}$ does not imply falsity of $mathsf{Statement B}$ it might be that disproving $mathsf{Statement B}$ might be the easiest route to disproving $mathsf{Statement A}$. However a direct disproof of $mathsf{Statement A}$ might reveal something else not directly revealed by $mathsf{Statement B}$ without falsifying $mathsf{Statement B}$. Are there known good examples?

For example Merten’s conjecture implies Riemann Hypothesis. However I am not sure if falsifying Merten’s conjecture revealed something different from a disproof of Riemann Hypothesis since the latter’s status remains unsolved.

dnd 5e – Are there any examples of “level requirements” that can be ignored with Use Magic Device?

The Thief Rogue’s 13th level feature allows them to ignore requirements on the use of magic items:

Use Magic Device. By 13th level, you have learned enough about the workings of magic that you can improvise the use of items even when they are not intended for you. You ignore all class, race, and level requirements on the use of magic items.

I’ve found examples of items that have class requirements (e.g. a Rod of the Pact Keeper) and of items that have race requirements (e.g. a Moonblade). However, I couldn’t find any examples of items that have level-based requirements on their use.

Are there any, or is this a redundant (/future-proofing) clause in the feature?

Items from any published 5th edition material would be welcome as answers; if the only examples are in UA that would be useful information as well.

lo.logic – Concrete examples of statements not provable in PRA + $epsilon_0$-induction that are provable in PA?

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cryptography – Examples of SSH key exchange algorithms requiring encryption-capable host keys

In the SSH spec, Section 7.1, key exchange algorithms are distinguished based on whether they require an “encryption-capable” or a “signature-capable” host key algorithm.

If I understood their details correctly, the well-known DH-based key exchanges algorithms such as curve25519-sha256, diffie-hellman-group14-sha256 and ecdh-sha2-nistp256 all require a signature-capable host key algorithm. What are examples of SSH key exchange algorithms that instead require an encryption-capable host key algorithm?

Examples of “non equivalent” algebras that are derived equivalent?

One can define different equivalence relations between algebras depending on what one want to study, but also these definitions may have their own life and not result as one expected at first.

My interest is in derived equivalence. Which is your favorite example of two algebras A and B such that D(A) is triangulated equivalent to D(B) but one feel that they are really ”different”? Maybe another way to formulate the question is: what is your favorite invariant that ”should be” derived invariant but it is not?

My personal example is an algebra A with finite global dimension, and B another algebra derived equivalent to A but with different global dimension. Do someone has any other ”example” of ”non-equivalent” algebras A and B that are still derived equivalent?

Tips And Tricks For Restaurant Website Design & Inspiring Examples

The top restaurant websites have beautiful presentations. But great design isn’t just about aesthetics; usability matters too. To show you how to get the maximum benefit from both aesthetics and usability and to inspire you and help you understand what it takes to craft a great restaurant website design we’ll start with tips and hints from our designers and conclude with examples of great restaurant website designs