No – a small amount of mutual information does not force the joint support to contain a large Cartesian product, and, in the equivalent graph‐theoretic language, a dense bipartite graph need not contain a large complete bipartite sub-graph (biclique).
A single probabilistic construction already shows this.

1. A convenient setting

Write
 N := 2ⁿ ...

The two things that are happening when you press the Home-button are

1. As soon as applicationDidEnterBackground: returns your process is not allowed to touch the GPU any more.
2. A fraction of a second after that, SpringBoard asks Core Animation for a bitmap (“app snapshot”) that it can show in the App- Switcher and while the app is relaunching.

Those two facts are not contradictory, because SpringBoard does not expect your code to render anything for it at that moment – it si...

Short answer
————————————————
Nobody knows. At the moment there is neither

• a proof that a (pure) DPLL procedure that always branches on a randomly chosen variable finishes in polynomial expected time on every instance that satisfies the Lovász-Local-Lemma (LLL) condition, nor

• an instance that satisfies this LLL condition for which the same DPLL procedure can be shown to need super-polynomial (let alone exponential) expected running time.

So the question you ask is open.

Why we...

Let

 I(k) = { u ∈ ℤ : 0 < u < p/k },

and let

 S(k) = { x ∈ {1,… ,q–1} : g^x mod p ∈ I(k) }.

Because g generates the subgroup of order q, the map

 x ↦ g^x (mod p)

is a permutation of that subgroup, hence for a uniformly random exponent x the value g^x is uniformly distributed among the q non-zero subgroup elements.
Therefore

 |S(k)| = q · |I(k)|/p ≈ q/k . (1)

Probability of hitting the target interval

For any algorithm that,...

Short answer
At present nobody knows the answer; no concrete pair
  F ⊂ ℓ₂ (finite) and X (infinite–dimensional Banach)
with “F does not embed isometrically into X’’ has been produced.
Equally, nobody has been able to prove that such a counter-example does not exist. The problem is therefore still open.

What is known

1. Positive results (existence of many embeddings).
   • Any finite metric space embeds isometrically in some ℓ∞ⁿ, hence every Banach space that contains arbitrar...

Below is a summary (with precise references) of what is presently known when one removes the words “semisimple” and even “fusion’’ from the usual theory of module-categories.

1. Module categories (= actegories) in complete generality

The notion of a left, right or bi-module category over a monoidal category 𝓒 goes back to Bénabou, Kelly and Street:

• J. Bénabou, “...

Yes – weight (2, 2) (often simply called “weight 2’’) Siegel modular forms of genus 2 have been, and are still being, computed, but not by the same cohomological / modular–symbol method that works so well for weight 2 elliptic modular forms.
The successful calculations use a mixture of

• explicit Fourier–Jacobi expansions,
• Jacobi-form and Gritsenko-lift technology,
• restriction to Humbert surfaces,
• point counting on moduli spaces, and
• ad-hoc linear algebra,

rather than the Sel...

The language you were shown inside IBM was the Vienna Development Method notation – internally it was usually called “META-IV”, and outside IBM it later became known as the VDM Specification Language or simply VDM-SL.

What it was
• A model-based formal specification language created in the IBM Vienna Laboratory in the early 1970s.
• Its vocabulary is typed set theory: sets, sequences, maps, Cartesian products, comprehensions, etc.; predicates are written in first-order logic.
• Operati...

Write the primes in increasing order

  P1 = 2 , P2 = 3 , P3 = 5 , … , Pn , Pn+1 , …

and put

  Sn = Pn + Pn+1 .

The problem asks for the least natural number k such that only finitely many Sn are divisible by k.

1. What we can prove at present
   • Every Sn with n ≥ 2 is even, so k = 1 or 2 certainly will not work.
   • Using the elementary argument you gave (it can be rewritten a little more cleanly in terms of residues) one sees that 3, 4 and 6 also fail: there are ...

Write
((x)) = x − ⌊x⌋ − 1/2 ( = 0 if x ∈ ℤ )
and
s(h , k) = Σ_{r=1}^{k-1} ((r/k)) ((hr/k)) ( Dedekind sum ).

1. A classical congruence (Meyer 1957)
   For coprime positive integers h , k

     12 k · s(h , k) ≡ h + h̄    (mod k),          (1)
   

where h̄ is the inverse of h modulo k.
See J. Meyer, J. Reine Angew. Math. 203 (1960), 143–157;
H. Rademacher – E. Grosswald, “Dedekind sums”, Th. 2.

Equation (1) is the only ingredient that is needed.

2. The case n =...

I’m pretty sure you’re remembering David Gerrold’s military-SF novel that has been published under several different titles:

• Yesterday’s Children (Ballantine, 1972, hardback & paperback)
• Star Hunt (Ballantine/Del Rey, 1977)
• Starhunt (Pocket Books, 1985 – the edition most people saw in the mid-’80s)

Gerrold later expanded the same material into the first volume of his “Star Wolf” series (Voyage of the Star Wolf, 1990), but the episode you describe is already pres...

Nothing even remotely close to the passage turns up in any of the usual places where printed SF/fantasy of the pre-Internet era can be searched.

• Google Books (which has most Anglophone magazines, paperbacks, hardbacks and fanzines from the 19-th century onward) returns zero matches for the distinctive strings
 “Forty thousand legions walked”
 “desert of the moon in search of the land of Narda”
 “fly space was crowded by their cars and travel logs”
 “cosmic acne of space”.

• The ...

The piece you remember is

SEGUNDA GÉNESIS
(“Second Genesis”), written by Carlos Trillo and drawn by Horacio Altuna.

What it is
• A one-off short story (14 pages) that the two Argentine masters produced in the mid-1970s for the weekly magazine Skorpio.
• It was reprinted many times afterwards, both in Argentina and in Spain, in the big anthology magazines 1984, Zona 84 and Cimoc; that is almost certainly the thick Spanish-language magazine you borrowed in the early 1990s.
• Heavy Met...

Short version
What you are asking for does not exists in the main-line Linux kernel, and for good technical reasons the kernel developers are not likely to add it.
After a resume the kernel already knows exactly which virtual pages are resident and which ones are still on the swap device – that information is part of the hibernate image.
What it does ​not​ do is read those swapped-out pages back into RAM “just in case”. If a process touches one of them the normal demand-paging machinery p...

The book you’re remembering is the second volume of J. B. Stephens’ post-apocalyptic young-adult series The Big Empty – specifically

• Paradise City (2005) – while several of the scenes you cite come from the opening volume, The Big Empty (2004).
(The series continues with Desolation Angels, 2006, and No Exit, 2007.)

Why it matches the scenes you recall

1. The walled town and escape through the sewers (The Big Empty)
   • Seven teen survivors (Jackson, “Maggie”/Magda, Clementine, Isa...

The story you are thinking of is “Transfusion” by Chad Oliver.
• first published in The Magazine of Fantasy & Science Fiction, January 1959
• very widely re-printed – for example in Robert Silverberg’s anthology “Worlds of Maybe” (1968) and in Isaac Asimov & Martin H. Greenberg’s “The Time-Travelers” (1986)

Outline that matches your memories

• The viewpoint character is a twentieth-/twenty-first-century scientist (an anthropologist in Oliver’s story) whose own future world has been blighte...

Short answer
Yes—if we formulate the null and the alternative inside the semiparametric proportional-odds (PO) model the Wilcoxon–Mann–Whitney (WMW) statistic is the efficient (hence, asymptotically most powerful) test for a two–sample comparison. No other regular test—rank-based or not—can have larger first-order (Pitman) power against contiguous PO alternatives. Outside that model, however, the WMW test is not uniformly most powerful; as soon as the PO assumption is violated another test c...

Short answer
There is no “Land of Shadu-kam” in the classical Persian sources. What Eco (2013) and before him Keightley (1828) took for the name of a fairy province is only the ordinary Persian adjective /​noun شادکام šâdkâm, literally ‘happy-and-satisfied, fortunate, blissful’. In the passages on which they were relying the word was being used descriptively, not as a place-name, and in the West it was misunderstood and re-transcribed as the exotic-sounding “Shad-u-kâm” or “Shadu-kam”.

...

Short version
• Nothing even remotely like the story is found in the Qurʾān, in the canonical ḥadīth collections or in any mediaeval Muslim, Byzantine or Georgian source.
• The tale appears for the first time in the 19th-century Russian medical-popular literature that became interested in the “Caucasian sour-milk drink” kefir (кефир).
• In those Russian and the slightly later French, German and English publications the words are always: “according to the belief of the local orthodox (= ...

Short version • In the 1930-50s a number of American military and Veterans-Administration hospitals, university radiology departments and a few private “hernia clinics” ran special outpatient units that dealt almost exclusively with symptomatic hiatal hernia.
• Their main activity was non-operative reduction of the hernia under fluoroscopy (sometimes called “pneumatic,” “syringe,” or “inflation” reduction) followed by corsets, diet, posture and physiotherapy.
• Results were modest and relaps...

В ВК действуют два типа ограничений:

1. «Глобальное» – не более 3-х обращений к API в секунду на один access-token (или IP, если запросы идут без токена).
2. Метод-специфические квоты. Для «тяжёлых» методов (поиск, wall.search, newsfeed.search и т.п.) дополнительно задаётся ещё и «количество запросов за интервал времени». Конкретные цифры компания не публикует — их можно только эмпирически оценивать.

Что происходит с users.search
• После превышения квоты поиск «глушится»: метод, а также об...

“Qubit fidelity” is a concept that comes from the gate-model world, where one asks “if I try to apply a π-pulse (or a CNOT) how likely am I to get the state I expect?”.
D-Wave processors do not apply gates; they realise an analogue, time-dependent Ising Hamiltonian

H(s)=A(s)∑iσi​x+B(s)⎛⎝∑i hiσi​z+∑i<j> Jijσi​zσj​z⎞⎠ ,   0≤s≤1,

and the question is therefore not “how accurate is a gate?” but

• How accurately do the programmed hi, Jij appear on the chip?
• How uniform is the transverse energ...

求めたい量
「区間 0 ≤ t ≤ T のあいだずっと，動作中（＝ダウンしていない）ワーカーが m 個以上ある確率」

は，次のようにモデル化すると計算できます。

• ワーカーの故障は独立に指数分布（故障率 λ）で起こる
• 故障したワーカーは一定時間 t だけ停止したのち必ず復帰する
（復帰時間 t が一定だとマルコフ性が壊れるので， 計算しやすい形にするため「平均 t の指数分布で復帰する」と近似するのが普通です。 一定時間にこだわる場合も，後述の方法をそのまま Erlang 分布（指数分布を k 本直列にしたもの）で近似すれば十分高精度が得られます。）

この仮定の下で

μ = 1/t …修復率
N …総ワーカー数
m …「動いていてほしいワーカー数」
F = N – m …同時にダウンしていても許される台数

と置きます。

────────────────────

1. 連続時間マルコフ過程を作る ───────────────────...

Short answer
Atari handed Digital Research an advance of about a quarter-million dollars and then paid a running royalty of roughly one U.S. dollar for every ST it shipped, with a contractual ceiling that put the total value of the deal a little above two million dollars. In other words, Jack Tramiel got the whole of GEM, CP/M-68 K (“GEMDOS”) and the desktop for about US $1 per machine.

Where the figure comes from
Contemporary trade-press stories covered the agreement while it was being ne...

Searching the literature in which the existential quantifier first appears (from Peano in the 1890’s up to the middle of the twentieth century) shows a fairly clear picture:

• Peano (from the Formulario, 1899-1908) used his “serif” ∃, but for a set–former, not for an existential quantifier, and he never put a bar through it.

• Whitehead & Russell, Principia Mathematica (1910–13) are the first to use Peano’s sign as an existential quantifier. When they want to deny existence they simply pref...

No – the statement you would like to have is already false in the very smallest non-trivial example.
We exhibit a single cyclic module for which

• the Littlewood–Richardson coefficient
c^{,n\lambda}{,n\mu , n\nu} = c^{(2n)}{(n),(n)} = 1

is non–zero,
• but no submodule (L) of type (\lambda) has the required behaviour with respect to a given submodule (N) of type (n\mu).

(The same construction works for any discrete valuation ring; to avoid notation write (\mathfrak o=\ma...

Yes – adding an axiom that identifies two coinductive values that are bisimilar with Leibniz equality does not lead to inconsistency. Just as with the more familiar “functional extensionality” axiom, the principle is independent of Coq’s core logic (it cannot be proved inside Coq), but it is consistent with it: there is a standard model of the Calculus of Inductive Constructions in which the axiom is true.

Why the axiom is not provable
————————————————— With an intensional equality, tw...

Answer to Question 1
Yes. In ZF a “uniformly supercompact’’ cardinal is automatically Reinhardt.
Suppose κ is uniformly supercompact and let

  j : V → M   (crit j = κ)

be the fixed elementary embedding with M transitive and

  ^λM ⊆ M   for every ordinal λ,

i.e. M is closed under all ordinal–length sequences.
Because every subset of an ordinal is such a sequence (identify a set A⊆λ with its characteristic function λ→2), we have

  𝒫(λ) ⊆ M for every ordinal λ, and in particular 𝒫...

Adobe adds the checksum merely to be able to find out later whether a particular XFA packet has been changed.
It is not a security feature – it is only a “did-anything-inside-this-packet change?” flag – and therefore it is not described in the public XFA specification.

1. Which algorithm is used?
   • It is SHA-1.
   The 20-byte digest is written as 40 lower-case hexadecimal digits.

2. What is hashed?

   • Exactly the byte sequence that is located between the start t...

No.
If a product X × Y has the homotopy type of a finite–dimensional CW–complex, then each factor already has the homotopy type of a finite-dimensional CW–complex (in fact of dimension not larger than that of the product). Hence no counter–example can exist.

Outline of the argument

Let

  f : X × Y → K, g : K → X × Y

be homot...

You’re thinking of

Knee-Deep in Thunder
by Sheila Moon (Macmillan, 1967; very widely re-issued by Scholastic in the 1970s-1980s).

Why it matches the points you remember

• Three children from different backgrounds are the protagonists.
– Maris (a girl from California),
– Hal (an English boy) and
– Raf (a boy whose father is from India)
are swept out of the ordinary world and magically reduced to insect size in the roots and leaf-litter of a redwood forest, which for them ...

It’s Gordon R. Dickson’s novelette “In the Bone”.

• first published in Analog Science Fact & Fiction, September 1963 (John W. Campbell, ed.)
• subsequently anthologised many times – e.g. Analog 4 (1966), The Best of Gordon R. Dickson (1978), The Analog Anthology #2 (1981) – so it would have been easy to encounter it in an ­-80s paperback collection.

Story outline (matching the details in the question)

– A small, all-male survey ship is trying to get back to Earth, but a navigation acciden...

Let

  Φ(x)= 1/√(2π)⋅∫_{-∞}^{x} e^{-t²/2},dt ,  x∈ℝ ,

and assume that q₁ , q₂ are rational numbers with

  Φ(q₁)=q₂. (1)

Question: must we have q₁=0 and q₂=1/2 ?

What is known

1. Relation with the error function
     Φ(x)=½(1+erf(x/√2)),  erf(z)= 2/√π ∫₀^{z} e^{-t²},dt .
     Hence (1) is equivalent to

  erf(α)=2q₂−1  with  α = q₁/√2 . (2)

  The point α is algebraic of degree 2 (it is rational times √2) as soon a...

Short answer
There is presently no peer-reviewed documentation showing that a non-human animal group systematically changes the way it signals (e.g. calls louder, approaches more closely, or switches to another sensory channel) in order to accommodate a congenitally or permanently deaf member. Searches of the main animal-behaviour, bio-acoustics and comparative-psychology databases (Web of Science, Scopus, PubMed, Google Scholar) using combinations of the terms deaf/deafness/hearing loss + vo...

Short answer
No. In the twenty-odd years since the handbook chapter appeared the “adjunction–version’’ of the limit/colimit–coincidence theorem has never been needed in published work on the denotational semantics of programming languages. All semantic applications that rely on limit/colimit coincidence – recursive‐domain equations for PCF, fully-abstract game models, models for references, concurrency etc. – continue to use the original form of the theorem for ω-chains of embedding–projecti...

Short answer
The earliest route that was deliberately laid out, put under the crown’s maintenance-obligation and kept open for wheeled traffic all the way from the Swedish side to the Finnish side was the road that the sources call Norrstigen / Kustlandsvägen and that, after the postal ordinance of 1636, became the Norra postvägen (the Northern Post Road).
It reached the Finnish town of Torneå (Fi. Tornio) – at the head of the Gulf of Bothnia – in the years 1638-1642. From Tornio a similar ...

At the moment nobody knows the answer.
What is known can be summed up as follows.

1. Reformulation in the Fourier–analytic language

Put
f := 1_A – 1_B (takes only the values ±1).

The requirement
λ(A∩D)=λ(B∩D)=½λ(D) for every unit Euclidean disc D

is exactly the same as

     (f * χ_D)(x) = 0                  for every x∈ℝ²,        (1)

where χ_D is the indicator of the unit disc and * denotes convolution.
Taking Fourier transforms g...

At the moment nobody knows the answer – neither for

S₁ = { x ↦ x+1 , x ↦ x³ } (with the two inverse maps added so that the graph is 4–regular)

nor for

S₂ = { x ↦ x+1 , x ↦ 3x , x ↦ x³ } (again with the inverses added).

In other words, it is completely open whether the family of graphs G(p,S₁) (resp. G(p,S₂)) obtained from the prime fields Fₚ (with p ≡ 2 (mod 3) so that x ↦ x³ is a permutation) forms an expander family.

Why we do not kn...

The inequality you want to prove

|K|^{\frac{n-1}{n}}\le \frac{1}{n,\omega _n^{1/n}};H^{,n-1}(\partial K) \qquad (1)

for every compact set K ⊂ ℝⁿ is nothing other than the usual isoperimetric inequality written with the geometric – i.e. Hausdorff – surface measure instead of the Minkowski (outer) perimeter μ₊(K).
At first sight it therefore looks as if one should only have to show

μ₊(K)\le H^{,n-1}(\partial K) (2)

and then inse...

No – beginning in dimension 3 the two conditions are independent.
There are smooth manifolds which have

 • no non–trivial smooth (i.e. ) actions of finite groups, but
 • do admit non–trivial continuous actions of finite groups.

Consequently the absence of smooth actions does ​not​ imply the absence of continuous actions.

1. Why the implication holds in low dimensions -----------------------------------------...

Short answer
The file format that the FrontEnd reads out of “ *.trie ” files is not documented and is not written by any public-facing Wolfram-Language function.
Only the internal “Code Assist” paclet that ships with the product knows how to turn a list of strings (or of rules) into the compact binary form that the FrontEnd loads. Consequently a compiled trie can only be produced with the same private code that Wolfram Research uses when it builds the product.

The good news is that this co...

Short version
There is no public- or semi-public way to know when the Ring/Silent switch is flipped, nor to query its current position.
Apple has designed the audio system so that you can (and should) let the OS do the muting for you, but you cannot examine or be notified of the switch itself. Any solution that does so depends on private framework symbols and will be rejected by App Review.

Detailed explanation

1. Public API
   • AVAudioSession, Core Audio, UIKit and the other public fra...

Short answer
────────────────
In the public-cloud Azure App Service the value that you put in siteConfig.loadBalancing is ignored; internally the platform always uses the same algorithm – a simple weighted round-robin (all instances have the same weight) with an optional ARR-affinity cookie for stickiness.
The other five algorithms that appear in the ARM schema exist only because the same SiteConfig object is also used by the on-premises versions of the service (Azure Pack / Azure Stack Hu...

Yes – for every positive integer k the two quantities coincide.

We first re-write the definition of T(k) a little more systematically and then show that the set that is being counted in T(k) can be split up into β(n,k) distinct elements for every n ≤ k!.
Because S(k) is the sum of these β(n,k)’s, the two numbers are equal.

1. A convenient notation

Through...