## lo.logic – Are there structures in a finite signature that are recursively categorically axiomatizable in SOL but not finitely categorically axiomatizable?

Recall that a structure $$mathcal{M} = langle M, I^sigma_M rangle$$ in a signature $$sigma$$ is categorically axiomatized by a second-order theory $$T$$ when, for any $$sigma$$-structure $$mathcal{N} = langle N, I^sigma_N rangle$$, $$langle N, mathcal{P}(N), I^sigma_N rangle vDash T$$ just in case $$mathcal{N}$$ is isomorphic to $$mathcal{M}$$.

It is fairly easy to find a structure in a finite signature that is categorically second-order axiomatizable but not finitely categorically second-order axiomatizable. Add a single function symbol $$f$$ to the language of second-order arithmetic, and choose a non-second-order-definable $$zeta: mathbb{N} rightarrow mathbb{N}$$. Then consider the theory $$T$$ that adds to the axioms of second-order arithmetic ($$mathsf{Z}^2$$) the sentence $$f(bar{n}) = overline{zeta(n)}$$ for each natural number $$n$$, where $$bar{m}$$ is the canonical numeral for $$m$$. (I owe the idea for this example to Andrew Bacon.)

This theory $$T$$, however, is not recursively axiomatizable. Is there a structure in a finite signature that has a recursive categorical second-order axiomatization but no finite categorical second-order axiomatization?

I believe that it is possible to find a recursively axiomatizable second-order theory $$T$$ whose spectrum (i.e., the set $${kappa in mathsf{Card}: exists mathcal{M} (mathcal{M} vDash T$$ and $$vert mathscr{M} vert = kappa)}$$) is shared by no finitely axiomatizable second-order theory, using partial truth predicates. (Consider the theory with $$mathsf{Z}^2$$ relativized to some predicate $$N$$ and $${$$“The cardinality of the non-$$N$$s is not $$Sigma^1_n$$-characterizable”$$: n in omega}$$.) But I cannot see how to turn this into a categorical theory.

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## digital signature – Understanding public key cryptography example

Frankly, I wasn’t sure whether this belongs in crypto.stackexchange.com or here, so forgive me. I’ve been going through a book where in one of the chapters the author tries to illustrate public key cryptography by imagining a “four-step business plan for Murder Incorporated” in a “world of strong privacy”.

1. Arrange for mystery billboards on major highways. Each contains a
single long number and the message “write this down.” Display ads with
the same message appear in major newspapers.

2. Put a full-page ad in the New York Times, apparently written in
gibberish.

3. Arrange a multiple assassination with high-profile targets, such as
film stars or major sports figures – perhaps a bomb at the Academy
Awards.

4. Send a message to all major media outlets telling them that the number
on all of those bulletin boards is a public key. If they use it to
decrypt the New York Times ad they will get a description of the
assassination, published the day before it happened.

I don’t quite understand the mechanics of how the outlets are able to decrypt the ad and get a description of the assassination. Is the ad encrypted with his private key and since both public and the private keys are mathematically related, whatever is encrypted with the private key can be decrypted with the public key (visible on the billboard) and vice versa?

## digital signature – gnu gcc source archives signed with expired GPG key?

The pedantic answer to your question is: when gcc-9.3.0 was released, the key was not yet expired:

$gpg --verify gcc-9.3.0.tar.gz.sig gcc-9.3.0.tar.gz gpg: Signature made Thu 12 Mar 2020 07:32:47 AM EDT gpg: using DSA key A328C3A2C3C45C06  The signature was made in March 2020, but the key expired in September 2020: gpg --list-keys A328C3A2C3C45C06 pub dsa1024/A328C3A2C3C45C06 2004-04-21 [SC] [expired: 2020-09-10] 33C235A34C46AA3FFB293709A328C3A2C3C45C06  So, the fact that it’s expired now is not a cause for concern. What is the cause for concern is that it’s a 1024/DSA key, which is probably not considered sufficiently strong these days. However, I can also see that the author has a newer key created in May, 2020: pub rsa4096/6C35B99309B5FA62 2020-05-28 [SC] D3A93CAD751C2AF4F8C7AD516C35B99309B5FA62  So, perhaps the next release of gcc will be signed with this key instead. ## directx – Specifying a root signature in the HLSL code of a DXR shader I’ve noticed that I cannot specify a root signature in the HLSL code of a DXR shader. For example, if I got a ray generation shader with the following declaration (rootsignature( "RootFlags(LOCAL_ROOT_SIGNATURE)," "DescriptorTable(" "UAV(u0, numDescriptors = 1)," "SRV(t0, numDescriptors = 1))")) (shader("raygeneration")) void RayGen() {}  CreateRootSignature yields the error message No root signature was found in the dxil library provided to CreateRootSignature. ( STATE_CREATION ERROR #696: CREATE_ROOT_SIGNATURE_BLOB_NOT_FOUND). I’ve noticed that even when I add a typo (for example, write roosignature instead of rootsignature), the compiler doesn’t complain about this typo. So, it seems like the whole attribute declaration is simply ignored. If I change the code to a simple rasterization shader, everything works as expected. So, is the specification of a root signature in the HLSL code of a DXR shader not supported? ## signature – Locally Sign transactions Blockcypher API I’m having a problem that I can’t solve since yesterday with the blockCypher APIs when I try to create a new transaction and submit it. https://www.blockcypher.com/dev/bitcoin/?shell#creating-transactions i have to sign locally but their php or cURL code don’t work (they give me several problems) I searched the internet but found nothing suitable. many signing tools found online tell me to enter a message in addition to the private key, what message should I enter? does not refer to any messages here: # next, you sign the data returned in the tosign array locally # any ECDSA secp256k1 signing tool should work #$PRIVATEKEY here is a hex-encoded private key corresponding to the input from address
CEztKBAYNoUEEaPYbkyFeXC5v8Jz9RoZH9

sign: 32b5ea64c253b6b466366647458cfd60de9cd29d7dc542293aa0b8b7300cd827 \$PRIVATEKEY

3045022100921fc36b911094280f07d8504a80fbab9b823a25f102e2bc69b14bcd369dfc7902200d07067d47f040e724b556e5bc3061af132d5a47bd96e901429d53c41e0f8cca


do you have any algorithm that i can use in php?

## Google Sheets: Exception: The parameters (number) don’t match the method signature for SpreadsheetApp.Range.setValues

With the following function:

function updateCells() {
const values = range.getDisplayValues()
const transposeCheck = values(0).map((_, iCol) => values.map(row => row(iCol)).some(cell => cell))
var countBoolean = transposeCheck.filter(Boolean).length
var summaryCommunityGroups = 'B'+countBoolean++
}


Exception: The parameters (number) don’t match the method signature

Help appreciated.

## marking a notebook with a special signature

Thanks for contributing an answer to Mathematica Stack Exchange!

But avoid

• Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. MathJax reference.

## multi signature – Taproot and Large multisigs

I have seen lot of people saying that taproot is only for large multisigs used in few projects or companies.

BlueMatt mentioned similar thing in last Taproot activation meeting:

13:01 < BlueMatt> only large multisigs care, really. plus whole new sig algorithm

http://gnusha.org/taproot-activation/2021-03-23.log

1. Is this true? Do you think only large multisigs care and it will benefit only large multisigs? What is a large multisig? (Main question)

Related questions which can be ignored.

1. If Taproot also involves a new signature algorithm: Schnorr, why would only large multisigs care?

2. Is usage of multisig in bitcoin projects an issue? Which projects already use multisig? Is everything possible without multisig?

## Gmail set to autoforward to another gmail, but signature is not showing

I have several gmails that are set to auto forward (and delete the original incoming email so as to keep the inbox empty) to my main gmail. I have a signature set up in each of them that is meant to tell me which gmail the original email was sent to (and then forwarded to my main gmail). The forwarding is working fine, but the signature is not being inserted. I have them set to insert the signatures in all new and reply/forward emails in settings.

Can anyone help?

Thanks!