pr.probability – Do measure-valued dynamical systems correspond to marginals of Markov processes?

No. Any Markov operator is contracting in the total variation norm, whereas your function $F$ is subject to a much weaker condition of weak continuity. It is easy to construct a counterexample. For instance, take $X$ to be the two point set ${0,1}$, then the probability measures on $X$ are parameterized by a single parameter $t=mu(1)in (0,1)$. Take for $F$ any continuous function $(0,1)to(0,1)$ which is not 1-Lipschitz.

linux – How does the sudoers option ‘pam_session’ correspond to PAM’s ‘session’ type?

On a CentOS 7 system, I want to prevent lines such as

May 20 08:55:05 c090 sudo: pam_unix(sudo:session): session opened for user root by (uid=0)
May 20 08:55:05 c090 sudo: pam_unix(sudo:session): session closed for user root
May 20 08:55:06 c090 sudo: pam_unix(sudo:session): session opened for user root by (uid=0)
May 20 08:55:06 c090 sudo: pam_unix(sudo:session): session closed for user root

from being written to /var/log/secure. The sudo operations are coming associated with the user zabbix, so I changed the first line of /etc/sudoers.d/zabbix from

Defaults:zabbix !requiretty, !syslog

to

Defaults:zabbix !requiretty, !syslog, !pam_session

This prevents the unwanted logging and the appication Zabbix still seems to be functioning. However, the effect of the change is not clear to me.

Does !pam_session prevent the PAM session type from being executed, or does it stop PAM from being initiated at all?

reference request – What locales correspond to Manifolds?

I am studying the categorical equivalence between (sober) topological spaces and (spatial) locales with enough points. As the title implies, I am interested in finding localic analogues of both topological and smooth manifolds. Since an (smooth) $n$-manifold is a paracompact, Hausdorff, locally Euclidean space equipped with an (smooth) atlas, we just need to translate all these technicalities to their lattice theoretic counterparts to find the localic version of manifolds.

I am pretty sure that somebody has work through all the details before, but couldn’t find any reference for this on google. Can someone help me with this?

How do many public keys correspond to one private key?

I have read online that hardware wallets can generate new public keys to facilitate transactions. Ie if someone has BTC on an exchange and they want to send that BTC in different installments to a single private key/hardware wallet, that hardware wallet can generate a different public key for each transaction. I’m a bit confused on how this works? Would the ledger not indicate that each of those installments now belongs to a different address? Or is it simply that each time a new public key is generated, the private key in the hardware wallet has the ability to sign for all of those? (Thus creating a persistent, growing list of public keys associated with one private key?).

Additionally, what is displayed on the ledger if that hardware wallet, after multiple installments with different public keys, now wants to send BTC elsewhere? Which of the previous public keys is used? Does it create a new one? Wouldn’t this show up on the ledger as a public key sending more BTC than it ever acquired?

image quality – What approximate resolution did Beta SP correspond to?

I know you can’t really say that a non-digital format such as VHS and Betacam (or LP for sound) have a resolution but it should be possible to give an rule-of-thumb estimate, shouldn’t it?

Wikipedia gives some number (300 for Beta and 340 for Beta SP) but that sounds too low doesn’t it? PAL and NTSC was 575 and 625 lines and Beta (SP) was used for that, wasn’t it? Why this discrepancy?

dnd 5e – Which instrument of the Bards correspond to which Bard college?

College does not equal College

Or maybe rather, the same word is used for two different concepts. We can most easily start with the seven. They refer to actual (in world historic) institutions as described in Sword Coast Adventurer’s Guide:

(…) Sword Coast has some bardic colleges where masters teach students the bardic arts. They hark back to the great bardic colleges of the distant past, particularly the seven elder colleges: Fochlucan, Mac-Fuirmidh, Doss, Canaith, Cli, Anstruth, and Ollamh.

— Chapter 4: Classes > Bard > Bardic Colleges

It then goes on to describe two of the decendants of those colleges (and one other college) and includes for all:

Most of its bards study and practice the methods of the College of Lore

That college or persuasion of bard would appear to the most associated with the actual college institutions, as included in the intro to their subclass is:

The college’s members gather in libraries and sometimes in actual colleges, complete with classrooms and dormitories, to share their lore with one another.

Also, of note that at the time of the publication of all these books (ie. PHB, DMG, SCAG) there were only two published subclasses for the Bard and that some of the newer to make up the now seven were (originally) published in campaign books for settings far removed from the Forgotten Realms.

Also note that any attendance or association with an (actual institution) college is entirely optional for any bard you create, it is given as an idea suggestion among others in the Creating a Bard section of the class description.

Did you serve an apprenticeship, studying under a master, following the more experienced bard until you were ready to strike out on your own? Or did you attend a college where you studied bardic lore and practiced your musical magic? Perhaps you were a young runaway or orphan, befriended by a wandering bard who became your mentor. Or you might have been a spoiled noble child tutored by a master. Perhaps you stumbled into the clutches of a hag, making a bargain for a musical gift in addition to your life and freedom, but at what cost?


For any DMs with their worldbuilding hat on, implementing the bardic subclasses as actual colleges in a given world is discussed on DMG p. 24 albeit briefly among wizard schools and druidic circles.

hard drive – Free space shown on Windows 10 (BOOTCAMP partition) does not correspond to the free space shown on Disk Utility

I’m not quite sure how this works, but yesterday I used Boot Camp Assistant to get Windows on my Mac. Somehow the space available on the C drive is 10 GB whilst in the Disk Utility it shows 18 GB of free space on it.

I checked with the space analyser in the wizard tool partition to see where all the space went to (I have partitioned 30 GB for the BOOTCAMP partition and haven’t done anything on Windows, yet I only have 10 GB left already), it does show that there is only 10 GB of free space, but I’m confused as to why Disk Utility is showing different data.

I’m a little anxious about this because my Mac has low storage space to begin with so I’m worried that I installed Windows and can’t use it.

Also, if anyone knows about methods to free up some space would be great help.

algorithms – How does each element in the output array of a DFT correspond to a specific frequency?

I have a basic understanding of the Fourier Transform, though I’m trying to use it in a program and I’m confused on the specifics. Based on source code I can find online, the DFT takes a set of samples/numbers, performs a summation for each term, and returns a set of these summations which is the same size as the input set. Suppose I have a periodic function. As I understand it, the output array should contain the amplitudes/weights of each frequency which sum to that function. What I can’t figure out is how each frequency is encoded in the array as just an index. In each example I read, we just assign a summation at each iteration of the inner loop to the next consecutive index in the output array. How are these indices indicative of which frequency they correspond to?

I’m attaching the source code I’m referencing to the bottom of this in case the website I linked to ever goes down.

import cmath
def compute_dft_complex(input):
    n = len(input)
    output = ()
    for k in range(n):  # For each output element
        s = complex(0)
        for t in range(n):  # For each input element
            angle = 2j * cmath.pi * t * k / n
            s += input(t) * cmath.exp(-angle)
        output.append(s)
    return output

world of darkness – Do “editions” of White Wolf products correspond?

White Wolf has published a variety of product lines. Each product line has multiple editions. Do the editions of various games line up intuitively?

For example, I have a smattering of books from both Werewolf: The Apocalypse and Wraith: The Oblivion. Both games have a first edition, second edition, and 20th anniversary edition (Werewolf additionally has a Revised edition that does not seem to correspond to anything in Wraith).

In other games with which I’m familiar the editions would line up nicely. For example, in Dungeons and Dragons there are Ravenloft books for 2nd edition D&D and 5th edition D&D. There are also Planescape books for 2nd edition. The Planescape 2e and Ravonloft 2e books are in some sense “related” because they refer to the same rule set.

Is this true for White Wolf products? I’m confused because the periods in which they are published don’t overlap. Werewolf 1e was published between 1992 – 1993, while Wraith 1e was published in 1994 – 1995. Second edition Werewolf started in 1994, but second edition Wraith started in 1996.