Nothing “nice’’ has ever turned up in the literature beyond the naïve linear‐polynomial cut‐out that you describe. In fact if one identifies F = 𝔽_{2^{11}} ≅ 𝔽₂[x]/(m(x)) and writes an additive endomorphism of F as

f(x) = a₀x + a₁x² + a₂x^{2²} + … + a_{10}x^{2^{10}},

then the condition “f permutes the cyclic subgroup C of order 23” is equivalent to

(1) a_i^{2^{11}} − a_i = 0, i=0,…,10, (forcing a_i∈F),
(2) ∏_{c∈C} ( t − f(c) ) = t^{23} − 1 in F[t].

Whe...

Here is a fairly standard “minimax‐in‐infinite‐dimensions’’ argument which shows that nothing special about amenability is needed – as soon as you allow arbitrary finitely‐additive probabilities on your countable group, the multiplication‐game always has a value and a saddle‐point (hence a Nash equilibrium). In particular one does not get a new characterization of amenability by asking “solvable for every W ⊆ G.”

1. Let G=F₂ be the free group, and let
   M := {μ a finitely additive proba...

There is, unfortunately, no publicly–documented, supported way in iOS to subscribe to “ringer-switch changed” events. Apple deliberately hides the position of that switch from third‐party apps. Under the hood the phone does generate internal GSEvents (kGSEventRingerOff / kGSEventRingerOn) and SpringBoard even posts a Darwin/notify notification, but none of those are in the public SDK and any reliance on them is at your own App-Store-risk.

What you can do in practice falls into two camps:

1...

This turned out not to be some mysterious “vector‐divide is non-deterministic on AMD vs. Intel” or a floating-point rounding issue at all, but simply a bug in the .NET JIT’s code‐generation for vector intrinsics when you pass a 16-bit field directly into a 32-bit vector overload.

What’s happening under the covers is:

1. You write
   v = v * c.A / Vector128.Create(0xFFFFu);
   with c.A declared as a ushort.
2. The operator*(Vector128<uint>, uint) overload forces the JI...

Here is what is known and what is not known about the “planar, boundary-terminal” version of Node-Multiway-Cut:

1. Classical Node-Multiway-Cut (in arbitrary graphs) is NP-complete already for 3 terminals (Dahlhaus, Johnson, Papadimitriou, Seymour & Yannakakis ’92).

2. If you restrict to planar graphs but allow the terminals to live anywhere, the vertex-deletion version remains NP-complete by essentially the same reductions.

3. By contrast, the edge-deletion version of Multiway-Cut *be...

Write Φ(x) for the standard‐normal CDF

 Φ(x) = (2π)^(–1/2) ∫_{–∞}^x e^{–t²/2} dt.

We are asking:

“If q₁,q₂∈ℚ satisfy Φ(q₁)=q₂, must it be that q₁=0 and q₂=½?”

Remark. Φ can be written in terms of the “error‐function”
 erf z = (2/√π)∫₀^z e^{–t²} dt
via
 Φ(x) = ½ [1+erf (x/√2)].

Thus the question is equivalent to asking whether erf (α) can be rational for any nonzero rational α. In fact:

• If q₁=0 then Φ(0)=½; that gives the obvious solution (q₁,q₂)=(0,½).
• No other r...

No. In fact one shows by a simple “ODE‐continuity’’ argument that you cannot have a C^2 Lorentzian metric all of whose time– and space–like geodesics are complete but which admits an inextendible null geodesic in finite affine parameter. Equivalently

timelike & spacelike geodesic completeness ⇒ null geodesic completeness.

The heuristic is this. If γ0(λ) is an inextendible null geodesic ending at λ=λ*<∞, choose at the starting point p = γ0(0) a one–parameter family of timelike unit–vect...

Below is a brief summary of what is known (and not known) about these two approximation‐by‐monotone‐circuits questions.

1) Uniform‐measure approximation (Question 1)
—————————————————
We ask:
“Given any monotone f on n bits computable by some (possibly non‐monotone) size-p(n) circuit, and any fixed ε>0, can one always find a monotone size-q(n) circuit g that agrees with f on at least 1–ε of the 2ⁿ inputs?”

Status: Open.

■ No general positive “compiler” is known that turn...

Below is a sketch of what is known (or really, not known) about this question:

1. No “natural’’ example is known. Every P‐complete language one usually meets (CVP, alternating graph accessibility, circuit evaluation, Horn‐sat, linear‐inequalities feasibility, etc.) is in fact P‐hard under very weak reductions—projections or “first‐order‐definable” maps—and for all of those people have shown

   Succinct-X is EXP-hard (indeed EXP-complete).

 The Papadimitriou–Yan...

First, let me restate the problem in more “proof‐system” language. A classic fact is that

1. If you have an LLL‐easy CNF—i.e. every clause has size exactly (t), and each clause shares variables with at most (\tfrac{2^t}{e}) other clauses—then the Lovász Local Lemma shows a satisfying assignment exists.

2. Moser–Tardos then give a randomized local‐resampling procedure (“pick any violated clause, resample all its (t) variables at random, repeat until every clause is satisfied”) ...

Below is a summary of why, as far as we know, there is no “magic shortcut’’ that, given the list of absolute‐values {|λ₁|,…,|λₙ|}, recovers the signs sign(λᵢ) in o(n³) (or o(n^ω)) time without essentially re‐doing a spectral computation.

1. The problem really is to decide, for each magnitude aᵢ, whether the corresponding eigenvalue of A is +aᵢ or −aᵢ. Equivalently one needs an eigenvector vᵢ for that eigenspace and to check the Rayleigh quotient
   R(vᵢ)=vᵢᵀAvᵢ / vᵢᵀvᵢ
   whose sign i...

Here is what is known and what remains open.

1. The C∗‐case (Behncke–Cuntz and Pedersen)
————————————————
– In 1977 Behncke–Cuntz showed that if A is a C∗–algebra and B⊂A is a *–subalgebra which is dense in the norm–topology and such that every maximal abelian *–subalgebra (MASA) of B is already norm–closed in A, then in various “type I” and “purely infinite” situations one can force B=A. Later Pedersen observed that as soon as B itself is a C∗–subalgebra one recovers B=A in complete gener...

Below is a sketch of what is and is not known today concerning Verdier’s two desiderata in positive characteristic.

1. “A wild Grothendieck‐group of constructible sheaves with a functorial push‐forward, which on stalks remembers both the rank and the wild ramification.”

–  **Still open in general.**  The naive push‐forward of constructible functions fails (Artin–Schreier shows exactly why), and nobody has exhibited a single “larger” abelian group  

   G₀^wild(X)  
of she...

One way to see “Dieudonné‐modules are really cohomology” is to observe that Messing’s Dieudonné‐crystal is by definition an Ext–group in the crystalline topos of the base. I will sketch two (equivalent) points of view:

1. The Ext–definition (Messing, BBM).
   Let S be a p‐adic base (say S=Spec k with k perfect of char p, or more generally any p‐nilpotent formal scheme) and let X→S be a p‐divisible group. On the crystalline site (S/W){cris}, X defines an fppf‐sheaf of groups X{cris}. Mess...

Nobody has ever written down a “black‐box’’ Arthur–Selberg–trace‐formula routine that spits out Fourier–expansions of the (2,2)–forms the way one does for weight 2 on SL₂ by modular symbols. In fact the consensus in the community is

1. For the full Siegel modular group Sp₄(ℤ) the only nonzero cuspidal scalar forms occur in weight ≥10 (Igusa’s χ₁₀, χ₁₂, …) so there is really nothing to compute at weight 2 on Sp₄(ℤ).

2. The only natural place one finds genuine weight 2 Siegel forms is o...

Here is a sketch of what is known in essentially arbitrary monoidal‐category–theoretic generality. Nothing really “mystical’’ happens once you drop semisimplicity – you simply rediscover the usual bicategorical/Morita‐theory machinery, now incarnated inside Cat (or inside whatever ambient 2-category you like).

1. Modules and bimodules as 1-cells in a bicategory of monoids
   – A monoidal category C is nothing but a pseudomonoid in the bicategory Cat. A (left) C–module category is then...

Here is what is known (or at least widely believed) about your two questions.

1. Uniform‐supercompact ⇒ Reinhardt in pure ZF?
   Nobody has been able to prove or refute this in just ZF. All of the familiar arguments that “if
   j∶V→M is elementary and M is closed under Ord‐sequences then in fact M=V and so j is a Reinhardt
   embedding” collapse somewhere to an appeal to a choice‐like principle. For instance one shows that
   M has a global well–order (by enumerating ...

Here is what one sees after unraveling the usual “Leray‐localization’’ (or purity) argument: one reduces the finiteness of
H^2_et(X, ℤ/n)
to the finiteness of the zero–th cohomology of a certain sheaf of “wild–vanishing–cycles’’ supported on the fibres of X over the primes dividing n. Concretely, if you set j∶U=X\bigcup_{p|n}X_p and Z=∪_{p|n}X_p, then from the localization triangle

 j_! j^*(ℤ/n) → ℤ/n → i_* i^*(ℤ/n) →[+1]  

one gets an exact piece

H^0(X, R^1i^! ...

Here is a “cook‐book’’ for getting the equilibrium temperature that a little planet in circular orbit around a Kerr (or Schwarzschild) black hole will settle to, when the only “heater’’ is the cosmic microwave background at 2.7 K as seen from infinity. In outline you do precisely the same thing you would do in special relativity (blue‐shift plus Doppler beaming of an otherwise isotropic bath) except that now you must combine

• the gravitational red–(or blue–)shift from infinity into your or...

Below is a complete solution to the “one‐friend’’ case, N = 1. In that case you and exactly one opponent alternately order dishes costing 1…A dollars until the running total reaches T, and you win if at the end either

• you have eaten strictly more than half the total (e₀ > T/2) but not more than B dollars over your friend
i.e. 0 < D ≡ e₀–e₁ ≤ B,

• or (if you cannot achieve the above) you expose your friend by forcing e₁ > e₀ + B (i.e. D < –B).

Your friend’s goal is to avoid ...

Privateer’s “Crow’s Nest” portal is very slick, and their press materials talk about AI, ground‐stations, telescopes and even blockchain, but as of today there is no public evidence that Crow’s Nest is doing anything fundamentally beyond the classic state-of-the-art in space-object cataloguing and conjunction assessment. Here’s why:

1. Data sources
– The only catalogues Crow’s Nest currently exposes are the same object lists you can pull yourself from Space-Track (i.e. USSTRATCOM’s two-li...

I’ve looked fairly extensively through the domain‐theory and PL‐semantics literature since Abramsky–Jung ’94 and I have not been able to find a single place where their “adjoint‐pair” version of limit–colimit coincidence has actually been used to do a denotational‐semantics construction that could not equally well have been done with ordinary e-p-pairs. In particular

1. Almost every recursive‐domain construction in the literature still uses chains of embeddings (and projections) and the usual...

The short answer is: nothing in the standard changed – two non‐type template arguments of reference (or pointer) type are considered identical if they refer to the same entity, and so your partial specialization should be chosen in every C++ mode (myQuestion == 2). GCC has always done the “right” thing here; both clang (and by inheritance MSVC) in C++17 mode are simply non-conforming. A bug report against LLVM (and one against MSVC) is already open.

–––––––––––––––––––––––––––––––––––...

Here is a very simple “by‐hand” counter–example which shows that neither of the two implications

(i) if t**–t* < n then winnum>0, and t**–t* ≥ n then winnum≤0
(ii) if winnum>0 then t**–t* < n, and if winnum≤0 then t**–t* ≥ n

holds in general. In fact one can do everything in log‐space so that xᵢ>0 is automatic.

— Take n=2. Let L₁=L₂=0 so that the 2-point moving average at t=2 is M₂=0.
— At t*=3 we must choose L₃ so that it sits exactly d below the average of the preceding ...

Here is how “proliferation” and “diffusion” differ in most reaction–diffusion models of tumor growth, and how each feeds into overall expansion of the tumor:

1. Proliferation (“reaction” term)
   – What it is
   • The net rate at which cells are born minus die.
   • In simple models you get ∂c/∂t|reaction = ρ·c (exponential) or ρ·c·(1–c/K) (logistic, with carrying capacity K).
   – Units
   • ρ is in [time⁻¹], e.g. per day.
   – Biological meaning
   • Drives local increas...