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Finding a Prime Ideal in the Ring of $C^\infty$ Functions. <p>Let $R$ be the ring of infinitely differentiable real-valued functions on $(-1, 1)$ under pointwise addition and multiplication, and let $$F(x) = \left\{ \begin{array}{lr} e^{-1/x^4} & \text{if } x\neq 0\\ 0 & \text{if } x=0 \end{array} \right.$$</p> <p>We are given that $F$ is infinitely differentiable and that all its derivatives vanish at $0$. Then $(F)$ is the principal ideal generated by $F$, and let $A=\sqrt{(F)}$ be the radical of $(F)$. Let $M$ be the maximal ideal of all functions that vanish at $0$, and let $P=\cap_{n=1}^\infty M^n$.</p> <p>I've showed that $P$ consists of the functions all of whose derivatives vanish at $0$, and that $P$ is a prime ideal. From this, it follows that $F\in P$ and that $A\subseteq P$. I've also shown that this containment is proper, since $$F(x) = \left\{ \begin{array}{lr} e^{-1/x^2} & \text{if } x\neq 0\\ 0 & \text{if } x=0 \end{array} \right.$$ is in $P$ and not in $A$.</p> <p>Now, I need to find a prime ideal containing $(F)$ that's not $P$ or $M$, and also show that it's properly contained in $P$. From the previous part of the problem, I'm assuming that $A$ is in fact the prime ideal I'm looking for, but I'm having trouble proving that $A$ is prime:</p> <p>Assume $f$ and $g$ are infinitely differentiable functions such that $fg\in A$. Then, since $A=\sqrt{(F)}$, there is a positive integer $n$, and an infinitely differentiable function $h$ such that $f^ng^n=hF$. So now I need to show that some power of $f$ or $g$ is a multiple of $F$, but this is difficult. I tried dividing both sides by $g^n$, which would write $f^n$ as a multiple of $F$ if I can extend the function $\dfrac{h}{g^n}$ to not have any holes. Now, since $F$ only vanishes at $0$, it follows that if $g$ vanishes at some nonzero point, so does $h$. I guess we can assume that $g^n$ is not a multiple of $F$, though I'm not sure how to use that. </p> <p>So if $\dfrac{h}{g^n}$ vanishes at $a$, i want to extend this function to $a$ by defining it to be $\displaystyle\lim_{x\to a} \dfrac{h(x)}{g^n(x)}$. Since both numerator and denominator vanish, I can use L'Hopital's Rule to write it as $\dfrac{h'(x)}{ng'(x)g^{n-1}(x)}$. Presumably, I can keep doing this, since the quotient shouldn't go off to infinity, and take $n$ derivatives, whence the numerator becomes $h^{(n)}(x)$, and the denominator becomes the sum of terms which all have a $g(x)$ factor, except for one, which is $g'(x)^n$. If this doesn't vanish, I can get a limit, and if it does, I guess I can do this again for $g''$, etc. The only problematic case is if all of $g$'s derivatives at $a$ vanish. This also means that $h$'s derivatives all vanish at $a$, but I don't think this gives a contradiction, since neither $g$ not $h$ was required to even be in $P$, and even if either of them were, it would be possible for a function's derivatives to all vanish at two different points, right?</p>	0non-cybersec	Stackexchange
Had to work on labor day. Boss thanked me with beef. Pan seared, and roasted with butter and herbs. Keep it simple people..	0non-cybersec	Reddit
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arXiv:math/0001179v1 [math.KT] 31 Jan 2000 ar X iv :m at h/ 00 01 17 9v 1 [ m at h. K T ] 3 1 Ja n 20 00 ON THE DERIVED FUNCTOR ANALOGY IN THE CUNTZ-QUILLEN FRAMEWORK FOR CYCLIC HOMOLOGY by Guillermo Cortiñas Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas , Universidad de La Plata. Abstract. Cuntz and Quillen have shown that for algebras over a field k with char(k) = 0, periodic cyclic homology may be regarded, in some sense, as the derived functor of (non-commutative) de Rham (co-)homology. The purpose of this paper is to formalize this derived functor analogy. We show that the localization Def−1PA of the category PA of countable pro-algebras at the class of (infinitesimal) defor- mations exists (in any characteristic) (Theorem 3.2) and that, in characteristic zero, periodic cyclic homology is the derived functor of de Rham cohomology with respect to this localization (Corollary 5.4). We also compute the derived functor of rational K-theory for algebras over Q, which we show is essentially the fiber of the Chern character to negative cyclic homology (Theorem 6.2). 0. Introduction. In their paper [CQ2], Cuntz and Quillen show that, if char(k) = 0, then pe- riodic cyclic homology may be regarded, in some sense, as the derived functor of (non-commutative) de Rham (co-)homology. The purpose of this paper is to for- malize this derived functor analogy. We show that the localization Def −1 PA of the category PA of countable pro-algebras at the class of (infinitesimal) deforma- tions exists (in any characteristic) (Theorem 3.2) and that, in characteristic zero, periodic cyclic homology is the derived functor of de Rham cohomology with re- spect to this localization (Corollary 5.4). We also compute the derived functor of rational K-theory for algebras over Q, which we show is essentially the fiber of the Chern character to negative cyclic homology (Theorem 6.2). For the construction of Def−1PA, we equip PA with the analogy of a closed model category struc- ture, where the analogy of cofibrant objects are the quasi-free pro-algebras and the analogy of trivial fibrations are the deformations. Further, we define notions of strong and weak nil-homotopy between pro-algebra homomorphisms such that –as is the case with “real” model categories ([Q])– Def −1 PA turns out to be iso- morphic to the localization of PA at the class of weak nil-homotopy equivalences, and equivalent to the localization of the subcategory of quasi-free algebras (i.e. the cofibrant objects) at the class of strong nil-homotopy equivalences (cf. 3.2). Of 1991 Mathematics Subject Classification. 19D55, 18G10, 18G55. Key words and phrases. Periodic cyclic homology, K-theory, quasi-free algebra. Typeset by AMS-TEX 1 http://arxiv.org/abs/math/0001179v1 2 GUILLERMO CORTIÑAS course this result would be automatic if the structure we put on PA were a model category (cf [Q]), which we prove it is not (3.6). However the analogy we have is sufficient to prove those localization properties and to consider derived functors therefrom. Quillen proves (in [Q]) that a functor between model categories which maps weak equivalences between cofibrant objects into weak equivalences admits a derived functor. The analogy of this result also holds in our setting; it says roughly that if a functor PA −→ C remains invariant under pro-power series extensions of quasi-free pro-algebras (i.e. F (A{X}/ < X >∞) ∼= FA), then its left derived functor exists (Theorem 5.2). Functors satisfying the latter condition are called Poincaré functors, as the condition that defines them is precisely a Poincaré lemma for (non commutative) power series. For example if F satisfies the stronger condi- tion FA = FA[t] then it is Poincaré; such is the case of de Rham cohomology in characteristic zero. Unless explicitly mentioned, all results in this paper hold over any characteristic. The notion of nil-homotopy used here (although related to) is different from the usual notion of polynomial (or pol-) homotopy, as used for example in Karoubi- Villamayor K-theory (see Section 4 below). In fact, a typical homotopy equivalence under pol-homotopy is the inclusion into the polynomial pro-algebra B →֒ B[t] which is not an equivalence under nil-homotopy. Instead, the inclusion into the power series pro-algebra B →֒ B[t]/ < t >∞ is a nil-homotopy equivalence. Un- der nil-homotopy, quasi-free pro-algebras are precisely those having the homotopy extension property; other properties of quasi-free pro-algebras proven in [CQ1] are shown here to have a natural interpretation in terms of homotopy (Theorem 2.1). The rest of this paper is organized as follows. In section 1, the notion of (strong) nil-homotopy is introduced, and its first properties are proved. Section 2 is devoted to the interpretation of quasi-free pro-algebras as cofibrant objects with respect to the setting of the previous section (Theorem 2.1). The notion of weak nil-homotopy is introduced in section 3, where the existence of the localized category Def−1PA is proved (Theorem 3.2). Section 2 is devoted to the comparison between our notion of nil-homotopy and the usual, polynomial homotopy. We prove that the localization at the union of the classes of nil-deformations and graded deformations exists and can be calculated as a homotopy category (Theorem 4.1). Section 5 deals with the formalization of the derived functor analogy of [CQ2]. We establish sufficient conditions for the existence of left derived functors (Theorem 5.2) and prove that, in characteristic zero, these conditions are met by the de Rham supercomplex functor A 7→ XA of Cuntz-Quillen (Corollary 5.4). In section 6 we compute the derived functor of the rational K-theory of rational pro-algebras, (Theorem 6.2) and of the negative cyclic homology of pro-algebras over any field (Corollary 6.9). Note on Notation. We use most of the notations and notions established in [CQ 1,2,3]. However, some notations do differ: we write ∂i (i = 0, 1) for the natural inclusions 1 ∗ 0, 0 ∗ 1 : A −→ QA = A ∗A, and qa = ∂0a− ∂1a. Thus our qa is twice Cuntz-Quillen’s. Also our curvature is minus theirs; here ω(a, b) = ρaρb−ρ(ab). In this paper, the superscript B+ on a graded algebra B denotes the terms of positive degree, and not the even degree part as in op. cit.. The even and odd terms are indicated by Beven and Bodd. If A is a pro-algebra indexed by N, then the map An+1 −→ An is referred to as the structure map and is named σ or τ (subscripts are mostly omitted). Since for the most part we make no assumptions on chark, DERIVED FUNCTOR ANALOGY 3 none of the results of op. cit. which involve dividing by arbitrary integers holds. Such is the case of the isomorphism between QA and the de Rham algebra with Fedosov product ([CQ1]), – as it assumes 2 6= 0– which we do not use. We do use the fact that qAn/qAn+1 ∼= ΩnA as A-bimodules, which does hold even if 2 is not invertible. On the other hand the isomorphism between the tensor algebra TA and the algebra of even differential forms holds in any characteristic with the same proof as in [CQ1]. 1. A Closed Model Category Analogy. 1.0 We consider associative, non-necessarily unital algebras over a fixed ground field k. We write A and V for the categories of algebras and vector spaces and PA and PV for the corresponding pro-categories. As in [CQ3] we consider only countably indexed pro-objects. A map f ∈ PA(A,B) is called a fibration if it admits a right inverse as a map of pro-vector spaces, i.e. there exists s ∈ PV(B,A) such that fs = 1. Fibrations are denoted by a double headed arrow ։. By a (nil-) deformation (։̃) of a pro-algebra A we mean a fibration onto A which is isomorphic to one of the form P/K∞։̃P/K. Equivalently, p : B։̃A is a deformation iff it is a fibration and for K = Ker(p) we have K∞ = 0. For example the map: UA := TA/JA∞ πA ։̃A is a deformation, and is initial among all deformations with values in A. That is if p : B։̃A is a deformation then there exists a map f : UA −→ B with pf = πA. In particular if A is quasi-free in the sense of [CQ3] then p is split in PA (because πA is). Deformations admitting a right inverse shall be called deformation retractions; thus A is quasi-free iff every deformation B։̃A is a retraction (or A is a retract of every deformation onto it). It follows that quasi- free pro-algebras are precisely those pro-algebras A such that the map 0 ֌ A has the left lifting property (LLP) with respect to deformations. Thus we have the analogy of closed model category ([Q]) where fibrations are as above, trivial fibrations are deformations, and cofibrant objects are quasi-free algebras. To pursue this analogy a step further, we define our weak nil-equivalences (or wne’s) as follows. We say that a map f ∈ PA is a wne if any functor defined on PA and taking values in some category C which inverts (i.e. maps to isomorphisms) all nil-deformations also inverts f . Functors which invert wne’s are called nil- invariant. We shall show that the localization of PA with respect to deformations exists, whence f is a wne iff it is inverted upon localizing. For completeness, we call a map f quasi-free if it has the LLP with respect to deformations. Thus quasi-free maps play the rôle of cofibrations. I hurry to point out that the above notions of fibration, cofibration, and weak equivalence DO NOT make PA into a closed model or even into a model category. Indeed, if the map 0 −→ A factors as a weak equivalence followed by a fibration then A is weak equivalent to 0 (3.5). As there are pro-algebras which are not equivalent to zero, axiom M2 for a model category ([Q]) does not hold. The latter problem would be solved if we allowed free maps of the form A −→ A ∗ TV to be weak equivalences; in fact 4 GUILLERMO CORTIÑAS any map A −→ B factors as A −→ A ∗ TB followed by a 7→ f(a), ρb 7→ b. This simply means that there are nil-invariant functors which do not invert free maps. The notion of weak equivalence defined above may be expressed as the weak homotopy relation associated to a notion of strong homotopy between pro-algebra homomorphisms. The definition of this strong homotopy is the subject of the next subsection. Cylinders and nil-homotopy 1.1. The cylinder of a pro-algebra A is the following pro-algebra: (1) Cyl(A) := QA/qA∞ Here QA = A∗A is the free product (or coproduct, or sum) and qA = Ker(QA −→ A) is the kernel of the folding map. We write ∂0 = 1 ∗ 0 and ∂1 = 0 ∗ 1 for the canonical inclusions A −→ QA, ∼ ∂0 ∗ ∼ ∂1 : QA −→ CylA for the completion map, and p = pA : CylA։̃A for the the completion of the folding map µ : QA −→ A. We have a commutative diagram: (2) QA µ �� ∼ ∂0∗ ∼ ∂1 ""F FF FF FF F A CylA ∼oo One checks that ∼ ∂0 ∗ ∼ ∂1 is quasi-free if A is, whence CylA is a cylinder object in the sense of [Q, 1.5. Def. 4]. Given homomorphisms f, g : A −→ B, we write f ≡ g if there exists a map h : CylA −→ B making the following diagram commute: (3) QA f∗g // ∼ ∂0∗ ∼ ∂1 �� B CylA h == z z z z z z z z Note that as QA −→ CylA is an epimorphism (although not a fibration), if a homotopy (i.e. a factorization through CylA ) exists, it must be unique. For example if A and B are algebras, then f ≡ g iff there exists n such that for all a1, . . . , an ∈ A, we have (f(a1)− g(a1)) . . . (f(an)− g(an)) = 0 and the homotopy is the map sending the class of qa to f(a)−g(a). One checks that ≡ is a reflexive and symmetric relation, and that it is compatible with composition on the left: f0 ≡ f1 ⇒ f2f0 ≡ f2f1 (whenever composition makes sense). It follows that the equivalence relation ∼ generated by ≡ is compatible with composition on both sides. We say that f and g are (nil-) homotopic if f ∼ g. We write [PA] for the category having the same objects as PA and as morphisms the sets of equivalence classes: [A,B] := PA(A,B)/ ∼ DERIVED FUNCTOR ANALOGY 5 A map f ∈ PA is called a strong nil-homotopy equivalence if its class is an isomor- phism in [PA]. Remark 1.2. The homotopy relation defined above may also be defined in terms of n-fold cylinders. Set Cyl1A := CylA, ∼ ∂1i = ∼ ∂i and define the n-fold cylinder inductively by the pushout diagram: A ∼ ∂0 −−−−→ Cyl1A ∼ ∂ n−1 1 y y Cyln−1A −−−−→ CylnA Define ∼ ∂n0 as the composite map A ∼ ∂ n−1 0 −−−→ Cyln−1A −→ CylnA and ∼ ∂n1 as the composite A ∼ ∂11 −→ CylA −→ CylnA. One checks that two maps f, g : A −→ B are homotopic iff there exist n and h : CylnA −→ B such that the following diagram commutes: QA ∼ ∂n0 ∗ ∼ ∂n1 �� f∗g // B CylnA h << yyyyyyyyy The map h in the diagram above will be called a homotopy between f and g. The following lemma establishes a relation between the nil-homotopy equiva- lences just defined and the weak nil-equivalences of 1.0. above. Lemma 1.3. Let f : A։̃B be a deformation retraction. Then f is a strong nil- homotopy equivalence. Proof. We have to prove that g = sf ∼ 1. Upon re-indexing, we can assume f = {fn : An −→ Bn}, s = {sn : Bn −→ An} are inverse systems of maps commuting with the structure maps σ = σn, that σfnsn = σ and that for Kn = Ker fn we have Knn = 0. Then for a ∈ An, we have f(σ(gn ∗ 1)qa) = σ(fsfa− fa) = 0, from which σ(gn ∗ 1(qa)) ∈ Kn−1. Thus σ(gn ∗ 1)(qAn) n = 0 whence g ∗ 1 : QA −→ B factors through CylA, and g ≡ 1. � 2. Quasi-free Algebras and the Homotopy Extension Property. An interesting feature of nil-homotopy is that quasi-free algebras are precisely those having the homotopy extension property with respect to deformations. This fact is proven in Theorem 2.1 below. First we need: Power pro-algebras, power spans and power deformations 2.0. By a graded pro-algebra we mean a non-negatively graded object in PA, i.e. a pro-algebra B together with a direct sum decomposition of pro-vector spaces: B = ⊕∞ n=0 B n such that the multiplication map B ⊗ B = ⊕ Bn ⊗ Bm −→ B maps Bn ⊗ Bm 6 GUILLERMO CORTIÑAS into Bn+m. Thus B+ = ⊕∞ n=1 B n is a two-sided ideal in B, in the sense that multiplication maps B+ ⊗ B and B ⊗ B+ into B+. It is straightforward to show that every graded pro-algebra is isomorphic–by a homogeneous isomorphism– to an inverse system of graded algebras and homogeneous maps. The power pro- algebra associated with B is the pro-algebra B̂ := B/B+ ∞ . Thus a power pro- algebra is a particular kind of graded algebra. For instance if A is an algebra then the power pro-algebra associated to the polynomials in a set X is the pro- algebra {A{X}/ < X >n}, whose completion is the power series algebra in the non-commutative variables X . More generally, one considers the tensor algebra TÃ(M) = T0(A) ⊕ T 1(A) ⊕ T 2(A) · · · = A⊕M ⊕M ⊗Ã M ⊕ . . . whose associated power algebra is T̂Ã(M) = { ⊕n i=1 T i(M) : n ∈ N} and when M is the free module on a set X one recovers the polynomial and power series algebras. These construc- tions can be copied for pro-algebras, pro-sets and pro-modules with the obvious definitions. However in general the free pro-module associated with a pro-set is not proyective, as it doesn’t have the LLP with respect to all epimorphisms, but only with respect to fibrations (cf.[CQ3]). We use the following special notations. If V is a pro-vector space and I ⊳ A is an ideal in a pro-algebra, we write PA(V ) for the power algebra associated with TÃ(Ã ⊗ V ⊗ Ã) and GI(A) and ĜI(A) for the graded pro-algebra A⊕ I/I2⊕ I2/I3⊕ . . . and its associated power algebra. If B is a graded pro-algebra and u : A −→ B0 is a homomorphism, then by a power span of u we mean a k-linear map T = ∑∞ n=1 Dn : A −→ B̂ + such that the following diagram commutes: (4) A⊗A multiplication −−−−−−−−→ A u⊗T+T⊗u−T⊗T y yT B̂ ⊗ B̂ ⊕ B̂ ⊗ B̂ ⊕ B̂ ⊗ B̂ sum+multiplication −−−−−−−−−−−−→ B̂+ Briefly, we write (4’) T (xy) = uxTy + Txuy − TxTy to indicate the diagram above –even if A and B are not algebras. For example the ordinary Taylor span: k[x] −→ k[x][[y]] = {k[x, y]/ < y >n}, f(x) 7→ { n∑ i=0 f (i)(y) i! } is a power span of the canonical inclusion. Note that the image of f(x) in k[x, y]/ < y >n is just the class of f(x)−f(y) and is therefore defined in any characteristic; if f(x) = ∑n i=0 aix i then f (i)(y) i! is just short for ∑n−i j=0 ( n j ) ai+j which is defined every- where. Note also that any power span T induces a homomorphism h : CylA −→ B̂ with h ∼ ∂0 = u, which is a homotopy between u and h ∼ ∂1. Conversely if h is a homo- topy starting at u, then T : A q −→ qA −→ qA/qA∞ h −→ B̂ is a power span. Thus a power span is a special kind of homotopy where the target is a power algebra. By an n-truncated span we mean a linear map Tn : A −→ B/B +n+1 satisfying (4’). For example if T is a power span then Tn : A T −→ B/B+ ∞ −→ B/B+ n+1 is an n-truncated power span. Finally, by a power deformation retraction we mean a deformation retraction of the form B̂ → B0 where B is a graded algebra. DERIVED FUNCTOR ANALOGY 7 Theorem 2.1. (Compare [CQ1]). The following conditions are equivalent for a pro-algebra A. (i) (LLP) A is quasi-free. (ii) (Power Span Extension) If B is a graded algebra and u : A −→ B0 is a homomorphism then any truncated span Tn : A −→ B/B +n+1 lifts to a power span T : A −→ B̂. (iii) (Tubular Neighborhood) If f : B։̃A is a deformation with kernel I and B is quasi-free, then there is an isomorphism ι : B ∼= → ĜI(B) such that fι is the projection ĜI (B)։̃B/I = A. (iv) (Even Forms) There is a pro-algebra isomorphism UA ∼= ΩevenA/ Ωeven+ ∞ A which makes the following diagram commute: UA πA≀ �� ≃// ΩevenA/Ωeven+ ∞ A wwooo oo oo oo oo oo A (v) (de Rham Algebra) There is a pro-algebra isomorphism CylA ∼= ΩA/Ω+ ∞ A which makes the following diagram commute: CylA ≃ −−−−→ ΩA/Ω+ ∞ A y y A⊕ qA/qA2 ≃ −−−−→ A⊕ Ω1A Here the bottom arrow is the canonical isomorphism aqb 7→ adb. (vi) (Homotopy Extension) Given any commutative solid arrow diagram: A // ∼ ∂0 �� B ≀f �� CylA << // C where f is a deformation, the dotted arrow exists and makes it commute. Proof. (i)⇒(ii): Write Tn = ∑n i=1 Di where Di is the part of degree i; also let D0 = u. Thus un = u + Tn = ∑n i=0 Di is a homomorphism, from which the following identity follows: (5) −δDi = i∑ j=1 Dj ∪Di−j (0 ≤ i ≤ n) Here the maps Di are regarded as 1-cochains with values in B, the cup product is the composite of Dj ⊗Di−j with the multiplication map B⊗B −→ B and δ is the 8 GUILLERMO CORTIÑAS Hochschild co-boundary map—as defined by the appropriate arrow diagram. We must prove that a k-linear map Dn+1 : A −→ B n+1 exists so that (5’) −δDn+1 = n∑ i=1 Di ∪Dn+1−i holds. It is straightforward to check that the right hand side of (5’) is actually a cocycle, whence also a coboundary, as A is quasi-free. Explicitly, if g : Ω2(A) −→ Bn+1 is the bimodule homomorphism induced by the right hand side of (5’) and if f : A −→ Ω2(A) satisfies −δf = d ∪ d, then we can take Dn+1 = gf . (i) ⇐⇒ (iii): If (iii) holds then UA։̃A is a retraction, whence A is quasi-free. Supose conversely that (i) holds. Because A is quasi-free, we have direct sum decompositions B = A ⊕ I, and B/I2 = A ⊕ I/I2 = ĜI(B)/ĜI(B) + 2 . Write u : B։̃A →֒ ĜI(B) for the composite map, and p1 : B։̃ĜI(B)/ĜI(B) + 2 for the projection. Because B is quasifree, the truncated span T1 = p1 − u : B → ĜI(B) +/ĜI(B) + 2 extends to a power span T : A → ĜI(B) +/ĜI(B) + ∞ (by (ii)). It is clear that T induces the identity on I/I2; further, one checks –using (5)– that it also induces the identity on In/In+1. It follows that p : u+T is an isomorphism. (iii)⇒(iv): Applying (iii) to πA : UA։̃A, we get UA ∼= ĜJA/JA∞(UA) = Ω evenA/Ωeven+ ∞ A. (iv)⇒(i): Analogous to (iii)⇒(i). (ii)⇒(v): By (ii), we can lift the de Rham derivation d : A −→ Ω1A to a power span T : A −→ ΩA/Ω+ ∞ A of the identity map A = Ω0A. By the discussion above, 1+T induces a homomorphism h : CylA −→ ΩA/Ω+ ∞ A such that hq = T . In particular, h induces the canonical A-bimodule isomorphism qA/qA2 ∼= Ω1A mapping q to d. Thus we have hq = d+D, where D(A) ⊂ Ω≥2/Ω≥2 ∞ . It follows that the composite A⊗n hq⊗n −→ Ω+/Ω+ ∞ −→ Ω+/Ω+ n+1 is just the cocycle d∪n, whence the induced bimodule homomorphism qAn/qAn+1 ∼= ΩnA is the canonical isomorphism, and the proof ensues. (v)⇒(i): By virtue of (5), if T2 = d + D2 : A −→ Ω 1A ⊕ Ω2A is the 2-span induced by ∼ ∂1, then −δD2 = d ∪ d, whence A is quasi-free. (vi)⇒(v): Since ΩA/Ω+A ∞ ։̃Ω0A ⊕ Ω1A is a deformation, there exists a ho- momotopy h : CylA −→ ΩA/Ω+ ∞ A lifting the homotopy 1 ≡ 1 + d. The same argument as in the proof of (ii)⇒(v) shows that h is an isomorphism. (i)⇒(vi): As 0 −→ A is quasi-free, so are ∂0 and ∼ ∂0. � Example 2.2. Let A be an algebra, and let UA = TA/JA∞ its universal quasi- free model. By the theorem above, we have CylUA ∼= ΩUA/Ω+UA ∞ . We want to give an explicit isomorphism CylUA ∼= ΩUA/Ω+UA ∞ as well as to show that in this particular case, we also have an isomorphism ΩUA/Ω+UA ∞ ∼= PUA(A) First of all, we observe that given a vector space V , we have isomorphisms: QTV ∼= T (V ⊕ V ) ∼= T (V ⊕ qV ) ∼= T (V ) ∗ T (qV ) ∼= TT̃ V (T̃ V ⊗ V ⊗ T̃ V ) ∼= ΩTV DERIVED FUNCTOR ANALOGY 9 Here qV = {(v,−v) : v ∈ V } and the isomorphism V ⊕ V ∼= V ⊕ qV maps (v, 0) = ∂0v to itself while ∂1v 7→ qv. Thus the composite isomorphism α : QTV ~∼=ΩTV maps qv to dv and ∂0x to x (v ∈ V, x ∈ TV ). In particular this holds when V = A; in this case α maps the ideal < JA >⊂ QTA generated by JA (which we identify with its image through ∼ ∂0) into the ideal < JA >⊂ ΩTA, and qTA into Ω +TA. It follows that α induces an isomorphism QTA/F∞ ∼= ΩTA/G∞, where F and G are respectively the < JA > +qTA and < JA > +Ω+TA-adic filtrations. On the other hand we have CylUA = QTA/F ′ ∞ and ΩUA/Ω+UA ∞ = ΩTA/G′ where F ′ =< JAn > + < q(JAn) > +(qTA)n and G′ =< JAn > + < dJAn > +(Ω+TA)n. We have inclusions: Fn ⊃ F ′ n ⊃ F”n =< JAn > +(qTA)n and Gn ⊃ G′ n ⊃ G”n =< JAn > +(Ω+TA)n Lemma 2.3. below shows that for N sufficiently large, we also have inclu- sions F”n ⊃ FN and G”n ⊃ GN . It follows that α induces the isomorphism CylUA~∼=ΩUA/Ω+UA ∞ and that ΩUA/Ω+UA ∞ = ΩTA/G”∞ = PUA(A) Lemma 2.3. Let A ⊂ B be algebras and let ǫ : B → A be a homomorphism such that ǫa = a, (a ∈ A). Set I = Ker ǫ, and let J ⊂ A be an ideal. Consider the following filtration in B: B ⊃ Fn =< Jn > +In Then there is an isomorphism: B/F∞ ∼= B/(< J > +I) ∞ Proof. Let Gn =< J >n +In. It is straightforward to check that (< J > +I)2n ⊂ Gn, whence B/(< J > +I)∞ ∼= B/G∞. Thus we must prove that B/G∞ ∼= B/F∞. It is clear that Gn ⊃ Fn. I claim that for N = n2 + n− 1, we also have GN ⊃ Fn. To prove the claim –and the lemma– it suffices to show that < J >N⊂ Fn. Every element of < J >N is a sum of products of the form: (j1 + i1) . . . (jN + iN ) (jr ∈ J, ir ∈ I) After fully expanding the product above, we get a large sum in which those terms not in In have at most n-1 i’s and at least n2 j’s. Therefore, in each such term, at least n of the j’s must appear side by side, forming a string. Hence the term in question lives in < Jn >. � Remark 2.4. The de Rham pro-algebra ΩA/Ω+A ∞ = { ⊕n r=0 Ω rAn+1}, of a pro- algebra A = {An}, together with the natural differentials b and d and the Karoubi operator κ, can be regarded as a pro-truncated mixed DGA in the sense of [Kar]. Indeed, the identity: bdω + dbω = ω − κω holds in Ωr(An+1) for r < n and in Ω n ♮ An+1 = Ω nAn+1/[Ω 0A,ΩnA] for r = n. Thus: θΩ(A) = ((⊕n−1r=0Ω rAn+1)⊕ Ω n ♮ An+1, B + b) 10 GUILLERMO CORTIÑAS is a pro-differential graded vector space, equipped with an even-odd gradation. This is the pro-complex of [CQ-2]; if k ⊃ Q, it is homotopy equivalent to the (short) de Rham pro-complex: XUA : Ω0UA d ♮ −→ ←− b Ω1♮UA In any characteristic, we still have θΩUA ≈ XUA for every algebra A and θΩR ≈ XR for every quasi-free algebra R. In particular CylR carries all the relevant information for the cyclic homology of R. 3. The Homotopy Category. Weak nil-homotopy 3.0. We write [UPA] for the category having the same objects as PA and where the set of maps from A to B is [UA,UB]. We have a functor γ : PA → [UPA], A 7→ A, f 7→ [Uf ]. Two maps f, g ∈ PA(A,B) shall be called weakly nil homotopic if γf = γg; by a weak nil homotopy equivalence we shall mean a map f ∈ PA such that γf is an isomorphism. We show below that the class of weak nil homotopy equivalences is precisely the class of weak nil equivalences as defined in 1.0 above, and that γ is the localization of PA at this class. Further, we show that [UPA] is equivalent to the strong homotopy category [PAQ] of quasi-free algebras. First we need: Lemma 3.1. The functor U : PA → PAQ carries fibrations to fibrations and deformations to deformations. Proof. Let f = {fn : An ։ Bn} be a fibration, and let t = tn : Bn → An be a section of f in PV . Upon re-indexing, we can assume that ftτ = τ for the structure map of B. We want to construct a linear section t̂ of Uf lifting t. Consider the following composite of linear maps: sn : TBn JBnn ∼ → n−1⊕ i=0 Ω2iBn →֒ ∞⊕ i=0 Ω2iBn ∼= TBn Note that sn is a linear section of TBn ։ TBn/JB n. Consider the composite t̂n : TBn/JB n sn−→ TBn Ttn −−→ TAn −→ TAn/JA n n; then t̂n commutes with τ and t̂n(ρb0ω(b1, b2) . . . ω(b2l−1, b2l) = = ρtn(b0)(ω(tnb1, tnb2) + ρωtn(b1, b2)) . . . (ω(tnb2l−1, tnb2l) + ρωtn(b2l−1, b2l)) for 0 ≤ l ≤ n − 1. Here ρ : A → TA is the canonical section, ω(a, b) = a ⊗ b − ab is the curvature of ρ and ωtn is the curvature of tn. Now since ftτ = τ , we have ωtn(b, b ′) ∈ Ker τBfn (b, b ′ ∈ Bn) and ρωtn(b, b ′) ∈ Ker τTBTfn. It follows that Ufnt̂nτ UB n = τ UB n , whence Uf is a fibration. Suppose further that f is also a deformation, and let K = Ker f ; we can assume Knn = 0. Let L = KerUf ; if l ∈ Lnn then π A n l ∈ K n n = 0, hence L n n ⊂ JAn/JA n n, and L n2 n = 0. � DERIVED FUNCTOR ANALOGY 11 Theorem 3.2. (Compare [Qui, 1.13, Th.1]) (i) Strong nil-homotopy equivalences are precisely those maps which are in- verted by every functor which inverts deformation retractions. Weak nil- homotopy equivalences are precisely those maps in PA that are inverted by every nil-invariant functor, i.e. every functor which inverts all deforma- tions. (ii) The functor PA → [PA] is the localization of PA at the class of deformation retractions, the functor PAQ → [PAQ] is the localization at the class of power deformation retractions, and the functor γ : PA → [UPA] is the localization at the class of all deformations. There is a category equivalence: [UPA] ≈ [PAQ]. Proof. (i) Let se be the class of maps inverted by every functor which inverts deformation retractions and let se′ be the class of strong homotopy equivalences. By virtue of Lemma 1.3, the functor PA → [PA] inverts deformation retractions, whence se ⊂ se′. Conversely, if F inverts deformation retractions then it inverts CylA։̃A, and also ∼ ∂i, i = 0, 1. Thus F maps congruent maps to the same map; further, since f F ∼ g ⇐⇒ Ff = Fg is an equivalence relation, F also maps nil- homotopic maps to the same map, and strong nil-equivalences to isomorphisms. This proves the first assertion of (i). Next, write ω and ω′ for the classes of weak nil-equivalences (as defined in 1.0 above) and weak nil-homotopy equivalences. We have to prove that ω = ω′. In view of Lemmas 1.3 and 3.1, the functor γ is nil- invariant, whence ω ⊂ ω′. Now let F : PA → C be a nil-invariant functor, and let f ∈ ω′(A,B). Because FπA and FπB are isomorphisms in C, Ff will be an isomorphism iff FUf is. By definition, the fact that f ∈ ω′ means that Uf is a strong equivalence, and therefore is inverted by F . Thus ω = ω′. (ii) The first assertion of (ii) is immediate from the proof of the first assertion of (i). The second assertion follows similarly, in view of 2.1-iii). Now let F be a nil invariant functor as above. We have to show that F factors as F = F̃ γ for some F̃ : [UPA] → C, and that such F̃ is unique. We put F̃ (A) = F (A) and for [f ] ∈ [UPA](A,B), we set F̃ [f ] = FπBFf(FπA)−1. It is clear that F̃ is well-defined and that F = F̃ γ. Now suppose G is another functor with the same property as F̃ . Then GA = A on objects and if f ∈ PA(A,B) then G must map [Uf ] onto FUf = F̃ [Uf ]. Since any map [g] ∈ [UPA](UA,UB) factors as [g] = [πUB][Ug][πUA]−1, it suffices to prove that [πUA] = [UπA]. But both πUA and UπA are left inverse to the same map ι : UA→ U2A induced by Tρ : TA→ T 2A, whence (by Lemma 1.3) [πUA] = [ι]−1 = [UπA]. This proves the third assertion. By the proof of (i), the functor γ : PAQ → [UPA] induces a functor γ : [PAQ]→ [UPA]. Let γ′ : [UPA]→ [PAQ], A 7→ UA, [f ] 7→ [f ]. Then [πR] : γ′γ(R) = UR→ R and [πUA] : γγ′(A) = UA → A are natural isomorphisms γγ′ ∼= → 1 and γ′γ ∼= → 1. This concludes the proof. � Corollary 3.3. Let f, g : A→ B be pro-algebra homomorphisms. We have: (i) Strong ⇒ Weak: If f is a strong equivalence then it is also a weak equiv- alence. If f and g are strongly nil-homotopic then they are also weakly homotopic. (ii) Weak ⇒ Strong: The converse of (i) holds if A and B are quasi-free. 12 GUILLERMO CORTIÑAS Proof. As CylA→ A is a deformation, any nil invariant functor maps strong equiv- alences into isomorphisms and homotopic maps to the same map. In particular, this happens with the localization functor γ, proving (i). Part (ii) follows from the identities: [A,B] = [PAQ](A,B) = [UPA](A,B) = [UA,UB]. � By defintion, the class Def of deformations sits into the intersection of the class we of weak equivalences and the class Fib of fibrations. The proposition below shows that in fact Def = we ∩ Fib. In particular this proves that quasi-free maps are precisely those having the LLP with respect to those fibrations which are weak equivalences. Proposition 3.4. A fibration is a deformation iff it is a weak equivalence. Proof. If f is deformation then it is a weak equivalence by definition of the latter. Suppose now f : A ։ B is a fibration and a weak equivalence, and write K = Ker f . Upon re-indexing, we can assume f is an inverse system of epimorphisms {fn : An ։ Bn} commuting with structure maps. We must prove K ∞ = 0. I claim it suffices to check this for the particular case when f is a strong equivalence. For if f is a weak equivalence and a fibration then Uf is both a strong equivalence (by 3.3) and a fibration (by 3.1). Whence, if we know the proposition for strong equivalences, we have KerUf∞ = 0. Now a little diagram chasing shows that KerUfn ։ Kn is an epimorphism (n ≥ 1), whence also K ∞ = 0, proving the claim. Assume then that there exists g ∈ PA(B,A) with β := gf ∼ 1, and that g = {gn : Bn → An} is an inverse system of homomorphisms commuting with the structure maps. By definition of homotopy, there exist r ≥ 1 and αi ∈ PA(A,A) with 1 = α0 ≡ α1 ≡ · · · ≡ αr = β. Because α := α1 ≡ 1, for every n ∈ N there exists m0 ≥ n such that for m ≥ m0, τmn(α ∗ 1) factors as follows: QAm α∗1 −−−−→ Am y yτmn QAm qAmm h −−−−→ An Therefore, given a1, . . . , am ∈ Am, we have: 0 = τ(α ∗ 1)(qa1 . . . qam) = τ((αa1 − a1) . . . (αam − am)) ≡ (−1)mτ(a1 . . . am) mod < ταa1, . . . αam > Thus if a1, . . . , am ∈ Ker(τα), we have τ(a1 . . . am) = 0. We have proven the following statement: (6) (∀n ≥ 1)(∃m0 ≥ n) and for each m ≥ m0 an N = Nm ≥ m such that (Ker τmnαn) N = 0 We are going to show next that if α satisfies (6) and γ ≡ α, then γ satisfies (6) too. It will follow that β –and then also f– satisfies (6), whence K∞ = 0 as we had to DERIVED FUNCTOR ANALOGY 13 prove. So assume (6) holds for α and let γ : A → A with γ ≡ α. Proceeding as above, we can find, for each n, an m1 ≥ m0 ≥ n such that if m ≥ m1, then 0 ≡ (−1)mτα(a1 . . . am) mod < τγa1, . . . γam > In particular τmn(Ker γm) m ⊂ Ker τmnα whence for N as in (6) we have (Ker τγm) mN = 0. � Corollary 3.5. A pro-algebra A is weak equivalent to zero iff A∞ = 0. Proof. If A ∼ 0 then UA։̃0 is a deformation by 3.2-i) and 3.4. Therefore UA∞ = 0, whence A∞ = 0. The converse is trivial. � Remark 3.6. We can now see how far PA is from being a closed model category. Indeed: by 3.5 above, if 0→ A factors as a weak equivalence followed by a fibration, then A ∼ 0. On the other hand, if TV is a tensor algebra then clearly TV∞ 6= 0, despite the fact that the map 0 ֌ TV has the LLP with respect to all fibrations. 4. Nil-homotopy v. Polinomial homotopy. 4.0. We want to compare our nil-homotopy relation with the more usual notion of homotopy defined via polynomial homotopies, as used for example to define Karoubi-VillamayorK-theory ([KV]). Given two homomorphisms f, g ∈ PA(A,B), we shall write f pol ≡ g if there exists a homomorphism h : A→ B[t], with values in the polynomial ring on the commuting variable t, such that the following diagram commutes: A (f,g) �� h // B[t] (ǫ0,ǫ1){{ww ww ww ww w B ×B Here ǫi stands for “evaluation at i” (i = 0, 1). Note ǫ1 is defined even if B is not unital, in which case t /∈ B[t]; we set ǫ1( ∑n i=0 ait i) = ∑n i=0 ai. Also note that the map (ǫ0, ǫ1) is a fibration; a natural linear section is given by (b0, b1) → b0 + b1t. We observe that pol ≡ is a reflexive and symmetric relation, and that if f pol ≡ g then fh pol ≡ gh (whenever the composition makes sense). It follows that the equivalence relation pol ∼ generated by pol ≡ is preserved by composition on both sides. Thus B[t] plays the rôle the free path space of a topological space plays in ordinary topological homotopy. We showed in 1.2 above that nil-homotopy can be described in terms of higher fold cylinders. Analogously, polynomial homotopy (or simply pol-homotopy) can be defined in terms of higher free path spaces. Set BI = B[t], and define BI n inductively by the pull-back square: BI n −−−−−→ BI n−1 y yǫn−10 BI −−−−→ B 14 GUILLERMO CORTIÑAS We write ǫn0 and ǫ n 1 for the composite maps B In → BI ǫ0 → B and BI n → BI n−1 ǫ n−1 1 → B. Thus (ǫn0 , ǫ n 1 ) : B In → B × B is a fibration, and two maps f0, f1 : A → B are pol ∼ iff there exist n and h : A→ BI n such that hǫni = fi. We write [PA]pol for the (strong) polynomial homotopy category, and call a map f ∈ PA(A,B) a polynomial equivalence if its class [f ]pol is an isomorphism in [PA]pol. A typical polynomial equivalence is the projection B = ⊕∞ n=0 Bn ։ B0 of a graded algebra or pro-algebra onto the part of degree zero, which is homotopy inverse to the inclusion B0 →֒ B. A homotopy between the composite B ։ B0 →֒ B and the identity map is given by h : B → B[t], h(b) = btdeg(b). Projections of the form ⊕∞ n=0 Bn pol ։̃B0 shall be called graded deformations. For example power deformations are graded, because power algebras are. We also consider the category [UPA]pol having as objects those of PA and as homomorphisms from A to B the homotopy classes [UA,UB]pol. The relation between nil-homotopy and polynomial homotopy is established by the following: Theorem 4.1. (i) The functor U : PA → PAQ carries pol-homotopic maps to pol-homotopic maps. (ii) If f, g : A→ B are nil-homotopic and if A is quasi-free, then they are also pol-homotopic. (iii) The functor PA → [PA]pol is the localization at the class of graded de- formations, and the functor γ′ : PA → [UPA]pol is the localization at the union of the classes of nil-deformations and graded-deformations. There is a category equivalence [PAQ]pol ≈ [UPA]pol. Proof. (i) It suffices to show that if f, g : A→ B ∈ PA satisfy f pol ≡ g, then Uf pol ≡ Ug. Let H : A → B[t] be a homotopy from f to g. Then H ′ = HπA : UA → B[t] is a homotopy from fπA to gπA and Uf, Ug are liftings of fπA, gπA to UB. Hence by [CQ-2, Lemma 9.1], we have Uf pol ≡ Ug. (ii)By Theorem 2.1, the map CylA→ A is a power deformation retraction, hence a graded deformation. It follows that ∼ ∂0 pol ≡ ∼ ∂1, and then f pol ∼ g. (iii) The proof of the first assertion is analogous to the proof of the first asser- tion of Theorem 3.2-ii). Next, we must show that γ′ inverts both nil and graded deformations and is initial among functors with such property. That γ′ inverts graded deformations follows from (i), and that it inverts nil-deformations from (ii) and 3.2. If F : PA → C inverts both types of deformation, then F̃ : [UPA]pol → C, A 7→ FA, [UPA](A,B) ∋ [f ] 7→ FπBFf(FπA)−1 stisfies F̃ γ′ = F and is the only such functor. � 5. Derived Functors. Notations 5.0. Recall from [Q] that if F :M→M′ is a functor between model categories, then the total (left) derived functor LF : HoM → HoM′ is the (left) derived functor of the composite Γ′F : C → HoM′ with respect to the localization Γ :M−→ HoM. Similarly, given a category C together with a functor Γ : C → C′, DERIVED FUNCTOR ANALOGY 15 and a functor F : PA → C, we may (and do) consider the total left and right derived functors of F with respect to Γ and to γ : PA −→ [UPA] and γ′ : PA −→ [UPA]pol. Motivation 5.1. The following proposition generalizes a common procedure for deriving functors. As a motivation, recall the way crystalline (or infinitesimal) coho- mology is defined for commutative algebras of finite type over a field of characteristic zero. Given an algebra A one chooses a smooth k-algebra R and an epimorphism p : R ։ A and defines H∗crisA as the cohomology of the (commutative) de Rham pro-complex ΩR/I∞ where I = Ker p (cf. [H], [I]). The essential step in proving that H∗cris is well-defined is the observation that if A above is quasi-free, then R̂I is an algebra of power series over A, and that (continuous) H∗dR satisfies the Poincaré Lemma: HdR(A) ∼= H ∗ dRA[[t]] ([H]). Here H ∗ dRA[[t]] def = H∗(lim←−Ω ∗(A[t]/ < tn >). Actually Poincaré Lemma is derived from the stronger fact that Ω∗(A) ∼ → Ω∗A[t] is a homotopy equivalence of pro-complexes ([H]). A non-commutative analogue of this construction was given by Cuntz and Quillen in [CQ2]. They showed that the non-commutative de Rham pro-complex XUA associated to an associative algebra A has the homotopy type of the periodic cyclic complex θΩ(A). In the framework of this paper, we interpret these results as saying that crystalline and periodic (co)-homology are respectively the derived functors of commutative and of non- commutative de Rham cohomology (see 5.4 below). The next proposition gives sufficient and necessary conditions so that when the construction above is applied to an arbitrary functor F , the result represents the left derived functor LF. We call this condition the Poincaré condition because it resembles the Poincaré Lemma quoted above. In both the commutative and non-commutative cases, one uses the fact that, in characteristic zero, de Rham cohomology is invariant under polynomial equivalence. Thus the Poincaré condition is automatic (see 5.3). However there are Poincaré functors which are not pol-homotopy invariant. For instance the Grothen- dieck group K0 is nil-invariant (and therefore represents its derived functor) despite the fact that in general, K0(A[t]) 6= K0(A). Theorem-Definition 5.2. (Poincaré Functors) Let F : PA → C and Γ : C → C′ be functors. The following are equivalent: (i) FU represents the derived functor of F with respect to Γ and to γ : PA → [UPA]. (ii) ΓFU is nil-invariant. (iii) Given any commutative diagram: R0 p0 �� f // R1 p1 }}\|\| \|\| \|\| \|\| A where pi is a nil-deformation and Ri is quasi-free (i = 0, 1), the map ΓFf is an isomorphism in C′. (iv) Given any pro-vector space V and any quasi-free pro-algebra R, the map ΓF (R →֒ PR(V )) is an isomorphism in C ′. (v) Condition (iv) holds for V = A and R = UA (A ∈ PA). 16 GUILLERMO CORTIÑAS We call F a Poincaré functor if it satisfies the equivalent conditions above. Proof. We mimic the proof of the fact that a functor between model categories which preserves homotopy equivalences between cofibrant objects admits a derived functor ([Qui 1.4.1]). (i) ⇐⇒ (ii) That (i)⇒(ii) is clear. Assume now ΓFU is nil-invariant, and let F̂ : [UPA] → C′ be the induced functor. We have to prove that F̂ = LF, i.e. that ΓFU = F̂ γ is equipped with a natural map α : ΓFU −→ ΓF such that if Ĝ : [UPA] −→ C′ is another functor and β : G := Ĝγ −→ F is a natural map then β factors uniquely through α. Let α = ΓF (πA) : ΓFUA −→ ΓFA and set β = (βU)(GπA)−1 : GA −→ ΓFUA. Then β satisfies β = αβ and is the only such map. (ii)⇒(iii) The map f is a strong equivalence because each pi is a deformation and Ri is quasi-free. Therefore ΓFUf is an isomorphism. On the other hand we have πR1Uf = fπR0 where each πRi is a deformation retraction; thus it is enough to show that each ΓFπRi is an isomorphism. But if ιi : Ri → URi is a right inverse for πRi , then ιiπ Ri : URi → URi is a nil-equivalence, whence the proof reduces to showing that if g : UB −→ UB is a strong equivalence, then ΓFg is an isomorphism. We know by hypothesis that ΓFUg is an isomorphism, and we have πUBFUg = gπUB. But ΓFπUB must be an isomorphism, because ΓFUπB is, and both πUB and UπB have a right inverse in common; namely the map induced by Tρ : TB −→ T 2B. (iii)⇒(iv) Let r : PR(V )։̃R be the projection map. Then r is a deformation and is a retraction of the canonical inclusion. Thus (iv) is a particular case of (iii), with R0 = A = R and R1 = PR(V ). (v) is logically weaker than (iv). (v)⇒(ii) By virtue of Example 2.2, if (v) holds, then ΓF sends homotopy equiv- alences UA → UB to isomorphisms, whence ΓFU sends weak nil-equivalences to isomorphisms. � Corollary 5.3. If F preserves either nil-deformation retractions or graded defor- mations, then it is Poincaré. In the latter case FU represents the left derived functor with respect to both γ : PA −→ [UPA] and to γ′ : PA −→ [UPA]pol. Proof. That F is Poincaré means that its restriction to PAQ preserves nil homotopy (cf. 3.2). Such is the case if F preserves either nil-homotopy or, by 4.1-ii), pol- homotopy of arbitrary pro-algebras. The same argument as in the proof of the theorem shows that, in the latter case, FU also represents the derived functor with respect to γ′. � Corollary 5.4. Let X : PA → PS :=((Pro-Supercomplexes)) be the functor which assigns to every pro-algebra A the de Rham pro-super complex XA of 2.4 above. Let Γ : PS −→ HoPS be the localization at the class of homotopy equivalences and let γ and γ′ be as above. If the ground field k has char(k) = 0 then the functor X is Poincaré (relative to Γ and to γ), and its left derived functor with respect to both γ and γ′ is represented by the periodic cyclic pro-complex θΩ of 2.4 above. Proof. In characteristic zero, the functor X preserves polynomial homotopy (e.g. by [Kas], or by [CQ2&3]), whence it is Poincaré and XU represents LX (by 5.3). DERIVED FUNCTOR ANALOGY 17 On the other hand, in any characteristic, XUA is homotopy equivalent to θΩUA, because UA is quasi-free (e.g. by [P]). In characteristic zero, by virtue of Good- willie’s theorem ([G1], [CQ2]), θΩUA has the homotopy type of θΩA. Summing up, if char(k) = 0 then FU ≈ θΩ represents LX . � Remark 5.5. In characteristic p > 0, the lemma above fails to hold. Indeed, if X were Poincaré then –by 5.2–the homology of the periodic cyclic complex CP (P0(k)) = Hom(Xk,XP0(k)) should be zero, which –as a straightforward calculation shows– it is not. See also Lemma 6.6 below. 6. The derived functors of rational K-theory and Cyclic Homology. The purpose of this section is to show that the functor which assigns to every Q-pro-algebra its rational K-theory space is (almost) a Poincaré functor, and that its left derived functor is essentially the fiber of the Chern character with values in negative cyclic homology. See Theorem 6.2 below for a precise statement. The proof of Theorem 6.2 has two main ingredients. The first ingredient is Goodwillie’s isomorphism (7) KQ∗ (A, I) ∼= HN∗(A, I) between the relative rational K-group of a nilpotent ideal and its analogue in neg- ative cyclic homology [G2]. Actually Goodwillie’s result is stated and proven for unital algebras; we shall use an adaptation of this that holds for arbitrary pro- algebras, which is obtained in 6.1 below. This adaptation says that the relative K-group of an infinitesimal deformation is isomorphic to the corresponding nega- tive cyclic homology group, and essentially reduces the question of the Poincaréness of K to that of HN . The second ingredient is the calculation of relative HN for a power deformation. This calculation is carried out without any hypothesis on the characteristic of k (Proposition 6.8). 6.0. The derived functor of rational K-theory. We use the following model for the rational K-theory of a unital algebra or ring: KQ(A) := Q∞BGlA Here Gl is the general linear group, and B denotes the simplicial set associated to the category of Gl. Thus for us KQ(A) is a simplicial set; note that its homotopy groups are precisely Quillen’s rationalK-groups. For general, non-necessarily unital algebras over the ground field k we set: KQ(A) := fiber(KQ(Ã) −→ KQ(k)) Thus in general KQ(A) depends on k, and coincides with the usual rational K- group if A is unital or more generally if it is excisive for KQ. Now we extend this 18 GUILLERMO CORTIÑAS definition to the case of pro-algebras, by taking homotopy inverse limits, as follows. If A = {Aλ : λ ∈ Λ} we put: KQ(A) := holim ← Λ KQ(Aλ) Next we generalize Goodwillie’s isomorphism to the pro-algebra case; we assume throughout that chark = 0. Recall from [G2] that the isomorphism (7) is induced by a natural Chern character KQ∗ (A) −→ HN∗(A) := HN∗(A/k) which is defined for every unital algebra A. By [W] this character may be realized as a simplicial map ch : KQ(A) −→ SN(A), where SN is constructed as follows. First truncate the total chain complex for negative cyclic homology to obtain a complex CN t such that Hn(CN t) = HNn(A) (n ≥ 1) and Hn(CN t) = 0 if n ≤ 0. Next define SN as the result of applying the Dold-Kan correspondence to CN t. Hence SN is a connected, fibrant simplicial set with πnSN(A) = HNnA (n ≥ 1), and the isomorphism (7) says that the map between fibers KQ(A, I) −→ SN(A, I) is a weak equivalence. If now A is any –non necessarily unital– algebra, and I ⊳ A is a nilpotent ideal, then we have weak equivalences: (8) KQ(A, I) ∼= K Q(Ã, I) ch ∼ −→ SN(Ã, I) ∼= SN(A, I) He have thus extended (7) to non-unital algebras. If now A = {Aλ : λ ∈ Λ} is a pro-algebra, we set SN(A) = holim ← Λ SN(Aλ), and write ch : K Q(A/k) −→ SN(A) for the map induced by passage to holim ← . As holim ← preserves fibers, fibrations and weak equivalences of fibrant s. sets, (cf. [BK]) it follows that the weak equivalences (8) hold for arbitrary deformations and pro-algebras. We have proven: Lemma 6.1. With the notations and definitions of 6.0 above, there is a natural map of fibrant simplicial sets ch : KQ(A) −→ SN(A) which is defined for all pro- algebras A, and coincides with Goodwillie’s character in the case of unital algebras. If f : A։̃B is a deformation, then the induced map KQ(f) ≈ SN(f) is a weak equivalence. Proof. See the discussion above. � 6.1.1. In particular the lemma above holds if f is a power deformation of quasi-free pro-algebras, whence –by Theorem 5.2-iv)– KQ will be Poincaré iff SN is. In the next subsection we compute the homotopy groups of SN(f) for power deformations of quasi-free pro-algebras and show that these are all zero except for π1, which is nonzero. Thus the simplicial set SN ′ obtained from the complex CN by truncating in degree 2, so that πn(SN ′) = HNn if n ≥ 2 and zero otherwise is a Poincaré functor; further, its derived functor is null-homotopic, cf. 6.9 below. It follows that the K-theory space obtained by the same process as above using the elementary group instead of the general linear group is a Poincaré functor. Explicitly, the functor: (9) KEQ(A) := holim ← Λ fiber(Q∞(EÃ −→ Q∞Ek) is Poincaré. DERIVED FUNCTOR ANALOGY 19 Theorem 6.2. (The derived functor of K-theory) The functor A 7→ KQ(A) is not Poincaré. However, the functor A 7→ KEQ(A) of (9) above is, and therefore it has a left derived functor LKEQ. Set LKQn (A) := πnLKE Q; then there is an exact sequence: . . . −→ HNn+1A −→ LK Q n (A) −→ K Q n (A) −→ Hn(A) −→ . . . −→ HN3(A) −→ LK Q 2 (A) −→ K Q 2 (A) −→ HN2(A) Proof. The first two assertions follow from the discussion above and 6.9 below. To prove the third assertion consider the exact sequence of K-groups associated with the universal deformation πA : UA։̃A. Then LKQn (A) = K Q n (UA) (n ≥ 2) (by 5.2) and Kn(π A) ∼= HNn(π A) (n ≥ 1) (by 6.1). Because UA is quasi-free, HNn(UA) = 0 for n ≥ 2, and therefore HNn(π A) ∼= HNn+1(A), for n ≥ 2. This proves that the sequence is exact at LK Q 2 (A) and to the left. By the same argument, the natural map HN2(A) →֒ HN1(π A) is injective, whence K Q 2 (A) −→ K Q 1 (π A) factors through ch2. It follows that the sequence is exact also at K Q 2 (A), completing the proof. � 6.3. The derived functor of negative cyclic homology. The purpose of this subsection is to compute the homotopy type of the relative space SN(PA(V ) −→ A) associated with a power deformation retraction of a quasi- free pro-algebra A over a field. We do not make any assumptions with regards to chark. The calculation uses two lemmas (6.4 and 6.6) which show the patologies that appear in characteristic p > 0. In particular, 6.6 gives a different proof of the fact that the de Rham pro-complex X is Poincaré iff chark = 0. In Lemma 6.4 we give a formula for the homotopy type of the X pro-complex of a free product. Recall that if A and B are algebras, then there is an isomorphism of vector spaces: A ∗B = A⊕B ⊕ T (A⊗B)⊕ T (B ⊗ A)⊕ T (A⊗B)⊗ A⊕ T (B ⊗A)⊗B In particular, the natural inclusion T (A⊗B) →֒ A∗B is an algebra homomorphism. Putting this map together with the natural inclusions A →֒ A ∗B and B →֒ A ∗B, we get map of super complexes: XA⊕XB ⊕XT (A⊗B) ι →֒ X(A ∗B) As all the maps in the above discussion are natural, all of this generalizes imme- diately to the case of pro-algebras. The following lemma may be regarded as a particular, easy case of [FT, 3.2.1]. We give an independent proof in this particular case. 20 GUILLERMO CORTIÑAS Lemma 6.4. (Compare [FT, 3.2.1]) Let A, B be pro-algebras. There exist a natural map of pro-mixed complexes: π : X(A ∗ B) → XA ⊕XB ⊕XT (A⊗ B) such that πι = 1 and a natural homotopy h : 1 ∼ ιπ. Proof. By naturality, we may assume A and B are algebras. The map A ∗ B → A × B, a 7→ (a, 0), b 7→ (0, b) induces a retraction XA ∗ B → XA ⊕ XB. Write XA ∗ B = XA ⊕XB ⊕ Y . Thus Y0 = U ⊕ V := T (A ⊗ B) ⊕ T (B ⊗ A) ⊕ T (A ⊗ B) ⊗ A ⊕ T (B ⊗ A) ⊗ B. where U is the sum of the first two terms and V is the sum of the last two. Further, one checks that: Y1 ∼= T (A⊗B)dA⊕ T (B ⊗ A)dB ⊕ ˜T (A⊗B)⊗ AdB ⊕ ˜T (B ⊗A)⊗BdA ∼= Y0 Consider the maps: α : U → U , x0y0 . . . xnyn 7→ ynx0 . . . yn−1xn, and µ : V → U , x0y0 . . . xnynx 7→ xx0y0 . . . xnyn. Under the identifications above, the map ι1 sends x ∈ T (A ⊗ B) ∼= Ω1T (A ⊗ B)♮ onto x + αx ∈ U . Define a mixed complex map π : Y → XT (A⊗B), π0(u0, u1, v0, v1) = x0+µy0+αx1+αµy1, π1(u0, u1, v0, v1) = u0, ui ∈ U , vi ∈ V ; 0 denotes the alphabetical order, and 1 denotes the inverse order. One checks that πι = 1. Further the map h : Y0 → Y1, h(x0, x1, y0, y1) = (0, x1 + µy1, y0, y1) verifies ι1π1 = hb and ι0π0 = bh. � Corollary 6.5. (Compare [CQ-3, 7.3]) If chark = 0, then there is a homotopy equivalence of supercomplexes: X(A ∗B) ≈ XA⊕XB. Proof. Immediate from the well-known calculation of the cyclic homology of a ten- sor algebra (e.g. [FT, 2.3.1]). � Lemma 6.6. Let A be an algebra, V a vector space and PA(V ) the power pro- algebra. Give TV and T (A⊗TV ) a gradation by setting deg(a) = 0 and deg(v) = 1 (a ∈ A, v ∈ V ). Then there exists a natural homotopy equivalence of pro-mixed complexes XPA(V ) ≈ XA⊕ {X deg≤nTV ⊕Xdeg≤nT (A⊗ TV ) : n ≥ 1} Proof. By definition the power pro-algebra PA(V ) is graded, and the gradation is given by the prescription of the lemma. This gradation is reflected by the X- complex; we have a degree decomposition: C := X(PA(V )) = {⊕ 2n i=0X deg=i(PA(V )n+1)} We observe that for i ≤ n the direct summand subcomplexes corresponding to degree i in the X complex of PA(V )n = A∗TV/ < V > n and of A∗TV are isomor- phic. Further the pro-complex D := {⊕2ni≥n+1C deg=i n+1 } is the zero pro-complex, as the structure maps τDn,2n are all zero. Therefore C is isomorphic to the pro-complex {Xdeg≤n(A ∗ TV )}. Now the lemma is immediate from 6.4. � Remark 6.7. As the homotopy equivalence in the lemma above is natural, it ex- tends automatically to pro-algebras. Since on the other hand the Hochschild, cyclic DERIVED FUNCTOR ANALOGY 21 and related homology groups of a tensor algebra are well known, one could conceiv- ably write down explicitly all the relative pro-homology groups for the projection PA(V ) −→ A in any characteristic. In the next proposition we calculate the negative cyclic group for the particular case when A is an algebra and V is a vector space. Since in characteristic zero HH0(TV ) ∼= HH1(TV ), our calculation can also be derived from [G2] in this particular case. Proposition 6.8. Let k be a field of characteristic p ≥ 0, and let A be a quasi-free k-pro-algebra. If V is a pro-vector space and f : PA(V ) −→ A is the natural pro- jection, then SN(f) is an Eilenberg-Maclane space E(Υ(A, V ), 1), where Υ(A, V ) is an abelian group which depends functorially on A and V . Explicitly if A is a quasi-free algebra and V is a vector space, then Υ(A, V ) = Π∞n=0(Cn ⊕ ⊕ r≥0 Dn,r) is the infinite product of the following co-invariant spaces: Cn = (T nV )Z/n and Dn,r = ( ⊕ i1+···+ir=n A⊗ T i1V ⊗ · · · ⊗ A⊗ T irV )Z/r Here Z/n and Z/r act by v1 ⊗ · · · ⊗ vn 7→ vn ⊗ v1 ⊗ · · · ⊗ vn−1 and by a1 ⊗ x1 ⊗ · · · ⊗ ar ⊗ xr 7→ ar ⊗ xr ⊗ a1 ⊗ x1 ⊗ · · · ⊗ ar−1 ⊗ xr−1 Proof. By the cofinality theorem for holim ← ([BK]), we may assume A and V are indexed by N. Thus for n ≥ 1 we have an exact sequence: (10) 0 −→ lim←− 1HNn(fi) −→ πn(SN(f)) −→ lim←−Hn(fi) −→ 0 Since PA(V ) is quasi-free, the inverse system {HNn(fi) : i ∈ N} is isomorphic to the inverse system {HNn(X(f)) : i ∈ N} (here X is regarded as a mixed complex). Thus both ends in the exact sequence above are zero for n ≥ 2. Furthermore SN(f) is connected by definition; this concludes the proof of the first assertion. Assume now A is a quasi-free algebra and V is a vector space. It follows form 6.6 that we have an isomorphism of pro-vector spaces (11) {HN1(fn) : n ∈ N} ∼= { n⊕ i=0 T iVZi}⊕ n⊕ i=0 ⊕ r≥0 ⊕ j1+...jr=i (A⊗ T i1V ⊗ · · · ⊗A⊗ V ir)Z/r : n ∈ N} As every map in the pro-vector space of the right hand of (11) is a surjection, the lim←− 1 term in (10) is zero, and the second assertion of the proposition follows. � Corollary 6.9. The functor A 7→ SN(A) is not Poincaré, regardless of the char- acteristic of k. The functor A 7→ SN ′(A) of 6.1.1 above is Poincaré (in any characteristic) and its left derived functor is null homotopic. � 22 GUILLERMO CORTIÑAS References [BK] A. Bousfield, D. Kan, Homotopy limits, completions and localizations, Springer Lecture Notes in Math. 304. [CQ1] J. Cuntz, D. Quillen, Algebra extensions and non-singularity, J. Amer. Math. Soc. 8 (1995), 251-290. [CQ2] , Cyclic homology and non-singularity, J. Amer. Math. Soc. 8 (1995), 373-442. [CQ3] , Excision in bivariant periodic cyclic cohomology, Preprint (1995). [FT] B.L. Feigin, B.L. Tsygan, Additive K-theory, Springer Lecture Notes in Math. 1289, 67- 209. [G1] T. Goodwillie, Cyclic homology, derivations and the free loopspace, Topology 24 (1985), 187-215. [G2] , Relative algebraic K-theory and cyclic homology, Annals of Math. 124 (1986), 347-402. [H] R. Hartshorne, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 5-99. [I] L. Illusie, Report on crystalline cohomology, Proc. Symp. Pure Math. XXIX (1975), 459- 478. [Kar] M. Karoubi, Algèbres graduées mixtes, Preprint (1995). [KV] M. Karoubi, O.E. Villamayor, K-thèorie algebrique et K-thèorie topologique, Math. Scand 28 (1971), 265-307. [Kas] C. Kassel, Cyclic homology, comodules and mixed complexes, J. Algebra 107 (1987), 195- 216. [P] M. Puschnigg, Explicit product structures in cyclic homology theories, Preprint (1996). [Q] D. Quillen, Homotopical algebra, Springer Lecture Notes in Math 43 (1967). [W] C. Weibel, Nil K-theory maps to cyclic homology, Trans. Amer. Math. Soc. 303 (1987), 541-558. Guillermo Cortiñas, Departamento de Matemática, Facultad de Ciencias Exactas, Calles 50 y 115, (1900) La Plata, Argentina. E-mail address: [email protected]	0non-cybersec	arXiv
Google wins trial against Oracle as jury finds Android is “fair use”.	0non-cybersec	Reddit
GitHub account unavailable on Xcode 10.2. <p>I noticed a strange behaviour in my Xcode (v10.2, running on a 2016 MacBook Pro with macOS 10.14.4). I have most of my projects under source control with a local Git repository. I then create a remote to GitHub via Xcode and commit and push from the source control menu.</p> <p>Since yesterday every time I committed I got asked for my username and password. At first I thought it was something normal due to some connectivity issue but then the issue persisted and, going to Xcode Preferences > accounts I noticed that my GitHub account was greyed out and, upon selecting it (or trying to) the right side of the window showed this message:</p> <p>Account Unavailable Your account details could not be loaded because Accountname support is currently unavailable</p> <p>At this point it is impossible to remove my GitHub account from Xcode in order to add it once more and it is also impossible to add any source control account, as if something with version control in general would be very broken. I thought it an issue with my SSH key for GitHub so I went on and scrapped the old one and generated a new one. The account stayed on "Loading" for a good minute and then the same message presented itself once more.</p> <p>I really have no idea what to do here.</p> <p>What is causing this and how do I get out of it?</p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
Saw Henry Winkler retweet this gold.	0non-cybersec	Reddit
Visual Studio & C++: Use filters as directories. <p>I Use <strong>Visual Studio 2012 Professional</strong> and <strong>C++</strong>. When creating so called 'filters' and adding source files to them, the files in the solution explorer are divided into sub directories, while on the file system they are all in the same directory (the project directory)</p> <p><strong>Question:</strong> I want to add a file to a filter while at the same time adding it to the corresponding directory on the file system. How can this be accomplished other than manually moving files around?</p>	0non-cybersec	Stackexchange
A FIRST gen GTR Skyline at the local BMW meet! (KPGC10). http://imgur.com/a/Rbwol	0non-cybersec	Reddit
Beatboxing Recorder.	0non-cybersec	Reddit
Best feature of our hotel in Minori, Italy. 5/5 paws..	0non-cybersec	Reddit
Ent teaches how to light a fire using only a battery and a gum wrapper!.	0non-cybersec	Reddit
Sobolev estimation of second derivative against Laplacian and higher terms. <p>Given $u \in H^2(\Omega)$ (and $\Omega \subseteq \mathbb{R}^n$ with appropriate properties) is there a way to estimate the norm of the second derivative $\Vert D^2 u\Vert_{L^2(\Omega)}^2$ against the Laplace norm $\Vert \Delta u\Vert^2_{L^2(\Omega)}$, first order norm $\Vert \nabla u\Vert^2_{L^2(\Omega)}$ and the boundary values $\Vert u\|_{\partial \Omega}\Vert_{L^2(\partial\Omega)}$, $\Vert \partial_n u\Vert_{L^2(\Omega)}$?</p> <p>So does there exist a $c\in \mathbb{R}_{>0}$ so that $$ \Vert D^2 u\Vert_{L^2(\Omega)}^2 \leq c\left( \Vert \Delta u\Vert^2_{L^2(\Omega)} + \Vert \nabla u\Vert^2_{L^2(\Omega)} + \Vert u\|_{\partial \Omega}\Vert_{L^2(\partial\Omega)}^2 + \Vert \partial_n u\Vert_{L^2(\Omega)}^2\right) $$ for all $u \in H^2(\Omega)$? Is there a simple proof or literature I could look at? (Sadly, most of my books don't cover more than $H^1(\Omega)$ really when it comes to these estimations.)</p> <p>I don't actually expect to need all of these values of $u$ but these are what I could afford to include if necessary. For a moment I was thinking about a "Poincaré-like" approach but this situation is maybe a little different.</p> <hr> <h2>Answer for the special case $(\Delta u)\|_{\partial\Omega} = 0$</h2> <p>One possibility to partially solve this problem for the case $u \in H^2(\Omega)$ with $(\Delta u)\|_{\partial\Omega} = 0$ would be to state that under certain assumptions there always exists an unique solution to the homogeneous Dirichlet problem $$ -\Delta w = f \\ w\|_{\partial \Omega} = 0 $$ with $w \in H^1(\Omega)$ and $f \in L^2(\Omega)$. Given a domain $\Omega$ with sufficient regularity (in the literature this seems to be $C^2$-boundary, however there might be generalizations on that) we get $w \in H^2(\Omega)$ as well as the existence of a constant $c\in \mathbb{R}_{>0}$ (independent of $w$ and $f$) so that $$ \Vert w\Vert_{H^2(\Omega)}^2 \leq c \Vert f\Vert_{L^2(\Omega)}^2 $$ (see for example the book of Gilbarg and Trudinger mentioned by <em>Behaviour</em> in the comments). If you now choose $f:=-\Delta u \in L^2(\Omega)$ you get that the solution of the above equation is uniquely given by $w=u$ (since we have zero boundary values as of $(\Delta u)\|_{\partial\Omega}=0$) and thus we have $$ \Vert u\Vert_{H^2(\Omega)}^2 = \Vert w\Vert_{H^2(\Omega)}^2 \leq c \Vert f\Vert_{L^2(\Omega)}^2 = c \Vert \Delta u\Vert_{L^2(\Omega)}^2. $$</p> <p>So this estimate is mostly based on the regularity results you have for the Laplace equation for your domain and boundary values. Since I didn't find a regularity result for convex polygon-domains and inhomogeneous boundary conditions I restricted myself to that version until I find further results.</p> <p>Also I want to mention that even if you had a regularity result for the inhomogeneous Dirichlet problem including a boundary term in form of $$ \Vert w\Vert_{H^2(\Omega)}^2 \leq c \left(\Vert f\Vert_{L^2(\Omega)}^2 + \Vert g\Vert_{L^2(\partial\Omega)}^2\right) $$ where $g \in L^2(\partial\Omega)$ describes the Dirichlet boundary condition on $\partial \Omega$, you'd end up with a result like $$ \Vert u\Vert_{H^2(\Omega)}^2 \leq c \left(\Vert \Delta u\Vert_{L^2(\Omega)}^2 + \Vert \Delta u \Vert_{L^2(\partial \Omega)}^2 \right), $$ which at least in my case includes information I don't have.</p> <p>Now the question is if there exists a generalization of this result to the whole space $H^2(\Omega)$ without including any other terms than specified in the inequality at the beginning. Since it's important for applications one would also want to have the domain $\Omega$ as convex polygon at most.</p>	0non-cybersec	Stackexchange
Error after upgrading Bugzilla from 4.0 to 4.0.2. <p>I had a good working new installation of Bugzilla 4.0.</p> <p>I have upgraded to version 4.0.2 using the <a href="http://www.bugzilla.org/docs/4.0/en/html/upgrade.html" rel="nofollow">patch</a>.</p> <p>Everything went ok until I had to clear the <strong>shutdownhtml</strong> field in the Params page</p> <pre><code>http://192.168.0.22/bugzilla/editparams.cgi </code></pre> <p>Doing that, gave the error on the screen:</p> <pre><code>Software error: Error reading ./data/params: Permission denied at Bugzilla/Config.pm line 323. Compilation failed in require at /var/www/bugzilla/editparams.cgi line 28. BEGIN failed--compilation aborted at /var/www/bugzilla/editparams.cgi line 28. For help, please send mail to the webmaster (webmaster@localhost), giving this error message and the time and date of the error. </code></pre> <p>Ok, it's definitely a permission error. I have set /var/www/bugzilla as follow:</p> <pre><code>drwxr-xr-x 16 www-data www-data 4096 2011-09-22 12:58 bugzilla/ </code></pre> <p>Q1: What user should have done the patch upgrade? I used Administrator with sudo. -- was that wrong? Q2: What should be the settings for the directory structure and files? isn't 755 and chown/chgrp www-data is enough?</p> <p>ANSWER: I had to change the following directory in order to fix the permission rights</p> <p>chmod 777 /var/www/bugzilla/data/template/template/en/default (for all changes to be written) chmod 777 /var/www/bugzilla/data/params</p> <p>This works, but still I don't know how it happened.</p>	0non-cybersec	Stackexchange
What are the world's greatest SOLVED mysteries?.	0non-cybersec	Reddit
Frog that caught a firefly.	0non-cybersec	Reddit
You can do this!.	0non-cybersec	Reddit
Create your own style. <p>I know this question has been asked many times, but I could not find a good answer for my question, yet...</p> <p>I wanted to develop my own thesis style i.e. <code>\usepackage{harthesis}</code>, which I did, but when I compile it, I got an error saying <code>harthesis.sty</code> is not found...</p> <p>What is the easiest and best way to create your own thesis style? I am sure this question would be very helpful for many people.</p> <p>These are part of my developed, but not limited to thesis styles:</p> <ol> <li><code>\permanentaddress</code></li> <li><code>\examiners</code></li> <li><code>\convocation</code></li> </ol> <p>and so on..</p> <p>These were intended to be my title page and other university requirements before I even start writing my Intro..</p> <p>Any suggestions? Thanks.</p>	0non-cybersec	Stackexchange
Reaper.	0non-cybersec	Reddit
Remsec driver analysis - Part 3.	1cybersec	Reddit
If I saw this while stoned, I would cry [pic].	0non-cybersec	Reddit
iPhone: once unlocked, always unlocked?. <p>I got an iPhone from a telco carrier two years ago, as at that time, I didn't really know whether I bought the phone unlocked or was it locked to that specific carrier.</p> <p>All I knew was, I change my telco afterwards, and I could use the new telco's data plan and voice and SMS service without trouble. </p> <p>As I want to travel to overseas soon, I am looking forward to buy local 3G SimCards to satisfy my appetite for Internet surfing. I wonder whether I can do so without performing extra steps on my iPhone?</p> <p>I think when I managed to change from one carrier to another previously, it indicated the phone was unlocked, at least among local carrier, but I am unsure whether this phone will remain unlocked when I travel overseas? Or is there a possibility that although you can change local carrier, but for you to change to an oversea carrier you would need to do some other steps? </p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
Facebook apologizes to drag queens for deleting accounts.	0non-cybersec	Reddit
Tunisian's Just Held a Music Festival at the Abandoned Star Wars Set in the Desert.	0non-cybersec	Reddit
Capacity of slices in Go. <p>I have a question about the below code </p> <pre><code>package main import "fmt" func main() { var a []int printSlice("a", a) // append works on nil slices. a = append(a, 0) printSlice("a", a) // the slice grows as needed. a = append(a, 1) printSlice("a", a) // we can add more than one element at a time. a = append(a, 2, 3, 4) printSlice("a", a) } func printSlice(s string, x []int) { fmt.Printf("%s len=%d cap=%d %v\n", s, len(x), cap(x), x) } </code></pre> <p>I always guess what the result of running a piece of code will look like then run the code and check if my guess is correct. But this code resulted a little bit different from my guess: </p> <p><strong>Result:</strong><br> <strong><em>On my local go tour server:</em></strong> </p> <pre><code>a len=0 cap=0 [] a len=1 cap=1 [0] a len=2 cap=2 [0 1] a len=5 cap=6 [0 1 2 3 4] </code></pre> <p>Everything is ok until the last line but I don't get </p> <pre><code>cap=6 </code></pre> <p>why not </p> <pre><code>cap=5 </code></pre> <p>My opinion is I did not create slice with explicit capacity therefore my system gave it this value of <strong>6</strong>. </p> <p><strong>2</strong>) But when I tried this same code on the golang tour server I get a little more diffferent result like this : </p> <pre><code>a len=0 cap=0 [] a len=1 cap=2 [0] a len=2 cap=2 [0 1] a len=5 cap=8 [0 1 2 3 4] </code></pre> <p>What about <strong>cap=2</strong> on the second line and <strong>cap=8</strong> on the last line?</p>	0non-cybersec	Stackexchange
These video times....	0non-cybersec	Reddit
How to execute a GROUP BY ... COUNT or SUM in Django ORM?. <p><strong>Prologue:</strong></p> <p>This is a question arising often in SO:</p> <ul> <li><a href="https://stackoverflow.com/questions/5947475/django-models-group-by">Django Models Group By</a></li> <li><a href="https://stackoverflow.com/questions/327807/django-equivalent-for-count-and-group-by">Django equivalent for count and group by</a></li> <li><a href="https://stackoverflow.com/questions/629551/how-to-query-as-group-by-in-django">How to query as GROUP BY in django?</a></li> <li><a href="https://stackoverflow.com/questions/40175134/how-to-use-the-orm-for-the-equivalent-of-a-sql-count-group-and-join-query/45563598#45563598">How to use the ORM for the equivalent of a SQL count, group and join query?</a></li> </ul> <p>I have composed an example on SO Documentation but since the Documentation will get shut down on August 8, 2017, I will follow the suggestion of <a href="https://meta.stackoverflow.com/questions/354217/sunsetting-documentation/354302#354302">this widely upvoted and discussed meta answer</a> and transform my example to a self-answered post.</p> <p>Of course, I would be more than happy to see any different approach as well!!</p> <hr> <p><strong>Question:</strong></p> <p>Assume the model:</p> <pre><code>class Books(models.Model): title = models.CharField() author = models.CharField() price = models.FloatField() </code></pre> <p>How can I perform the following queries on that model utilizing Django ORM:</p> <ul> <li><p><code>GROUP BY ... COUNT</code>:</p> <pre><code>SELECT author, COUNT(author) AS count FROM myapp_books GROUP BY author </code></pre></li> <li><p><code>GROUP BY ... SUM</code>:</p> <pre><code>SELECT author, SUM (price) AS total_price FROM myapp_books GROUP BY author </code></pre></li> </ul>	0non-cybersec	Stackexchange
Nintendo Switch set to Smash Sales Target [Comparative Console Sales & Traffic Data].	0non-cybersec	Reddit
[OC] Sweden divided into two equally populated areas.	0non-cybersec	Reddit
Is it possible to open GitHub code in Plunker?. <p>I have a GitHub account linked to <a href="http://plnkr.co/" rel="noreferrer">Plunker</a> but I don't see any way to open a code file or project through it. Is it possible to open GitHub files through a linked Plunker account?</p>	0non-cybersec	Stackexchange
Dean Elgar takes a stunning catch against Australia.	0non-cybersec	Reddit
[UPDATE]: I [24/f] am desperately falling in love with my roommate [28/m] and I have no idea how to tell him without ruining everything.. Original post: https://www.reddit.com/r/relationships/comments/6yp1ip/i_24f_am_desperately_falling_in_love_with_my/?st=J7G0G49H&sh=6a5a9d0b OKAY, SO. This issue has been driving me crazy lately to the point where I'm actually becoming stressed out about it pretty hardcore. I called one of my best girlfriends. I called my sister. I prepared an entire mental speech for him at my birthday party this Friday. I finally started to accept that it was time to lay my cards all out on the table. Then, last night, we were just hanging out like we usually do; drinking a bottle of wine, playing music, drawing, talking about life. We ended up under the desk and he goes "I'm gonna cuddle you", so we cuddled. Nothing crazy yet. But then we fell asleep cuddling, and the next thing I know he's sleepily pressing his face against my cheek and kissed it. I pulled back to look at him and see if he was for real and then he went in and kissed me. Boom. Fireworks. We talked a bunch after a really heavy makeout sesh, and it turns out that we have been exactly on the same page... for 3 years. This doesn't feel real. I was shaking and cried happy tears as he told me everything that was on his mind. I'm really excited to see where this goes... I've never felt this way about anybody before. :') --- tl;dr: Drank wine, cuddled, he made the first move :)	0non-cybersec	Reddit
Monitor USB HID events?. <p>I am looking for a way to monitor the raw USB HID events generated by devices such as keyboards, mice and other similar devices. There used to a IOUSBFamily kext with logging enabled that could be installed that would do this, but Apple hasn't updated it since OSX 10.9, so it doesn't seem to work anymore. </p>	0non-cybersec	Stackexchange
What's the "standard" way to express a zero/null morpheme in an interlinear gloss in LaTeX?. <p>I'm interested in what the "norm" is for the field.</p> <p>I tried both <code>$\oslash$</code> and <code>$\emptyset$</code>, but neither looks normal... or maybe I'm over thinking this. Is there a standard in linguistics publications?</p> <p>e.g. <a href="http://en.wikipedia.org/wiki/Null_morpheme" rel="nofollow">http://en.wikipedia.org/wiki/Null_morpheme</a></p>	0non-cybersec	Stackexchange
Delusional BRZ seller..	0non-cybersec	Reddit
MRW my friend tells me working extra hard during the prime of your life is worth it so you can enjoy the final 20 years of your life in luxury being old and gray..	0non-cybersec	Reddit
How to get the current year using typescript in angular6. <p>How to get the current year using typescript in angular6 </p> <pre><code>currentYear:Date; this.currentYear=new Date("YYYY"); alert(this.currentYear); </code></pre> <blockquote> <p>It shows Invalid Date</p> </blockquote>	0non-cybersec	Stackexchange
Earth Water Wind Fire collaborative makeup interpretation by /u/BNSquash, /u/Checkmate1234, /u/juntaoren, and /u/Catgirl007.	0non-cybersec	Reddit
Is there a specific reason why 3-way Facetime doesn't exist?. I mean, it's not like the technology doesn't exist when other companies offer this same service	0non-cybersec	Reddit
Man Convicted of Murder After Slitting Teacher’s Throat in Front of Students at Long Beach Park.	0non-cybersec	Reddit
Which team(s) will crash and burn this season? 2012/2013. Apart from the obvious candidate of Orlando, which other teams do you think will fall behind in the up and coming season? I personally feel that Houston Rockets will have a tough season ahead, as good as Lin is, he can't carry a franchise.	0non-cybersec	Reddit
Paw Patrol: Riley and Rocket Secure the Perimeter.	0non-cybersec	Reddit
Uniqueness of abelian group structure on a given set and recursive algorithms. <p>If we have some function $f$ under $\mathbb{Z}$ and $$f(a, f(b, c)) = f(f(a, b), c)$$ $$f(a, b) = f(b, a)$$ $$f(a, 0) = a$$ $$f(a, -a) = 0$$ meaning $f$ is an abelian group with an identity element of $0$. Is that enough to prove that $f$ is addition? In other words, is an abelian group with a specific identity element unique within a certain domain? Similarly, could multiplication be proved if we had an abelian group with an identity element of $1$?</p> <p>More generally, I'm trying to figure out if it is possible to deduce an algorithm (set of simple unambiguous recursive rules) from a set of axioms for some operation under a domain (axioms could perhaps be thought of as a form of ambiguous recursion). The first step is to figure out if a set of axioms can define the uniqueness of an operation. The more difficult step is to actually infer the specific recursive steps necessary.</p>	0non-cybersec	Stackexchange
ValueError: numpy.dtype has the wrong size, try recompiling. <p>I just installed pandas and statsmodels package on my python 2.7 When I tried "import pandas as pd", this error message comes out. Can anyone help? Thanks!!!</p> <pre><code>numpy.dtype has the wrong size, try recompiling Traceback (most recent call last): File "<stdin>", line 1, in <module> File "C:\analytics\ext\python27\lib\site-packages\statsmodels-0.5.0-py2.7-win32.egg\statsmodels\formula\__init__.py", line 4, in <module> from formulatools import handle_formula_data File "C:\analytics\ext\python27\lib\site-packages\statsmodels-0.5.0-py2.7-win32.egg\statsmodels\formula\formulatools.p y", line 1, in <module> import statsmodels.tools.data as data_util File "C:\analytics\ext\python27\lib\site-packages\statsmodels-0.5.0-py2.7-win32.egg\statsmodels\tools\__init__.py", li ne 1, in <module> from tools import add_constant, categorical File "C:\analytics\ext\python27\lib\site-packages\statsmodels-0.5.0-py2.7-win32.egg\statsmodels\tools\tools.py", line 14, in <module> from pandas import DataFrame File "C:\analytics\ext\python27\lib\site-packages\pandas\__init__.py", line 6, in <module> from . import hashtable, tslib, lib File "numpy.pxd", line 157, in init pandas.tslib (pandas\tslib.c:49133) ValueError: numpy.dtype has the wrong size, try recompiling </code></pre>	0non-cybersec	Stackexchange
Consecutive fullbody routines. Hello there, I've been working out for about 2 years now. I'm doing Foundation One. I always do fullbody routines. This week, I can exercise today, tomorrow and on Sunday, or just today and on Sunday. I was thinking about opting out for only 2 fullbody routines this week, since I'd like to rest a bit more and I'm also doing Wing Chun and running. What advice would you have for me?	0non-cybersec	Reddit
Differentiating an endomorphism. <p>Let $(M,\rho)$ be a symplectic $2$-dimensional manifold, and let $J$ be a compatible complex structure on $M$, i.e. the symmetric $(0,2)$-tensor $$g(,) = \rho(,J)$$ is a Riemannian metric. Denote by $\nabla$ the Levi-Civita connection. Now let $V\in\Gamma(TM)$ be a vector field (with some further assumptions which are probably not needed). I would like to compute $$\left.\frac{d}{dt}\right\|_{t=0}\left((d\varphi_V^{-t})_{\varphi_V^t(m)}J(\varphi_V^t(m))(d\varphi_V^t)_m\right),$$ where the map in the brackets is a map $T_mM\to T_mM$. Here $\varphi_V^t$ denotes the flow of $V$. If my computations are correct, I get the expression $$-\nabla_{J}V + (\nabla_VJ)() + J(\nabla_V)\tag{1},$$ which is again an endomorphism of $TM$. I would like to further simplify this expression. This would be easy if we had $$\nabla_{J}V = J(\nabla_V),$$ but I'm afraid that this is not true (at least without some more assumptions/structure). The only results I have about relations between $\nabla$ and $J$ are $$J(\nabla_J) = (\nabla_J)J\quad\mathrm{and}\quad\nabla_{J}J = -J(\nabla_*J)$$ (see e.g. lemma 4.1.14 in <a href="http://rads.stackoverflow.com/amzn/click/0198504519" rel="nofollow">this book</a>), but I haven't been able to get anywhere using them.</p> <p>My questions are:</p> <ol> <li>Is my computation $(1)$ correct?</li> <li>Is it possible to further simplify the expression $(1)$?</li> </ol> <p>If you think that further assumptions are needed to solve the problem, please let me know.</p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
"House of the rising sun" played on tesla coils. (Explanation of how it works in the video description)..	0non-cybersec	Reddit
How do we know what the gravity is on planets we've never been to?.	0non-cybersec	Reddit
"database locked" error in ios while updating query. <p>I am using the below code <strong>to update the query</strong> using <code>sqlite</code>.<br> But am getting <code>"database is locked error"</code>.<br> I tried searching some SO link and it was suggested to close database, but I did that again am getting the same error. I have mentioned where I am getting error in the code.</p> <pre><code>const char dbpath = [databasePath UTF8String]; if (sqlite3_open(dbpath, &database) == SQLITE_OK) { NSString locationNo =NULL; NSString querySQL = [NSString stringWithFormat:@"select count() from code"]; const char query_stmt = [querySQL UTF8String]; if (sqlite3_prepare_v2(database,query_stmt, -1, &statement, NULL) == SQLITE_OK) { if (sqlite3_step(statement) == SQLITE_ROW) { locationNo = [[NSString alloc] initWithUTF8String:(const char ) sqlite3_column_text(statement, 0)]; int count= [locationNo intValue]; sqlite3_close(database); NSLog(@"%@",locationNo); if(0==count) { NSString insertSQL = [NSString stringWithFormat:@"insert into favourite_code (code_id,code_1,code_2,code_3,code_4,code_5,code_6, status, record_status) VALUES (\"%d\",\"%@\", \"%@\", \"%@\", \"%@\", \"%@\", \"%@\", \"%@\", \"%@\")",1 ,code1,code2,code3,code4,code5,code6,@"Y", @"Y"]; const char insert_stmt = [insertSQL UTF8String]; sqlite3_prepare_v2(database, insert_stmt,-1, &statement, NULL); if (sqlite3_step(statement) == SQLITE_DONE) { return YES; } else { return NO; } sqlite3_reset(statement); sqlite3_close(database); } else{ =========================== Getting Error in the below lines ========================= const char *sq1l = "update code SET code_1=?, code_2=?, code_3=?, code_4=?, code_5=?,code_6=? WHERE code_id=1"; if (sqlite3_prepare_v2(database, sq1l, -1, &statement, NULL) != SQLITE_OK) { NSLog(@"Error: failed to prepare statement with message '%s'.", sqlite3_errmsg(database)); } else { sqlite3_bind_text(statement, 1, [code1 UTF8String], -1, SQLITE_TRANSIENT); sqlite3_bind_text(statement, 2, [code1 UTF8String], -1, SQLITE_TRANSIENT); sqlite3_bind_text(statement, 3, [code1 UTF8String], -1, SQLITE_TRANSIENT); sqlite3_bind_text(statement, 4, [code1 UTF8String], -1, SQLITE_TRANSIENT); sqlite3_bind_text(statement, 5, [code1 UTF8String], -1, SQLITE_TRANSIENT); sqlite3_bind_text(statement, 6, [code1 UTF8String], -1, SQLITE_TRANSIENT); } int success = sqlite3_step(statement); if (success != SQLITE_DONE) { NSLog(@"Error: failed to prepare statement with message '%s'.", sqlite3_errmsg(database)); //result = FALSE; } else { NSLog(@"Error: failed to prepare statement with message '%s'.", sqlite3_errmsg(database)); //result = TRUE; } =================================END========================================= } sqlite3_reset(statement); } else { NSLog(@"Not found"); locationNo=@"1"; } sqlite3_reset(statement); } } </code></pre>	0non-cybersec	Stackexchange
How can I get the dropbox short url from Dropbox Chooser?. <p>Is there a way to get the short url for sharing a document through the Dropbox Chooser?</p>	0non-cybersec	Stackexchange
Show that if $A$ and $B$ are commuting matrices, then $e^{A+B} = e^Ae^B$. <blockquote> <p>Let <span class="math-container">$A$</span>, <span class="math-container">$B$</span> <span class="math-container">$\in \mathbb{C}^{n \times n}$</span> such that <span class="math-container">$AB = BA$</span>. Consider <span class="math-container">$e^{A+B} = \sum_{k=0}^{\infty} \frac{(A+B)^k}{k!}$</span>. Since <span class="math-container">$A,B$</span> commute, <span class="math-container">$(A+B)^k = \sum_{j=0}^k {k \choose j} A^j B^{k-j}$</span> by the binomial formula.</p> <p>Hence, <span class="math-container">$$e^{A+B} = \sum_{k=0}^{\infty} \frac{\sum_{j=0}^k {k \choose j} A^j B^{k-j}}{k!}.$$</span></p> </blockquote> <p>I am having a tough time simplifying this to get the form <span class="math-container">$e^Ae^B$</span>. Any tips?</p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
Grammar for context free language. <p>I want to give a grammar for the following language:</p> <p><span class="math-container">$$L = \{x^r \# y \|x, y \in \{a, b, c\}^*\\ \text{ and }x\text{ is a contiguous sub-string of }y\}$$</span></p> <p>where <span class="math-container">$x ^ r$</span> denotes the backward written word <span class="math-container">$x$</span></p> <p>I have tried the grammar, is this right or have I made a mistake:</p> <p><span class="math-container">$S \to Sa \mid Sb \mid Sc \mid T$</span>,</p> <p><span class="math-container">$T \to aTa \mid bTb \mid cTc \mid U$</span>,</p> <p><span class="math-container">$U \to Ua \mid Ub \mid Uc \mid \#$</span></p>	0non-cybersec	Stackexchange
How to fix that Jet Lighter you love. I have this pretty old jet lighter i bought at a boot sale, and today - it broke ( gas goes on for too long and flame is less strong ) and upon opening, it seemed to be spraying gas out of the side of the small tube. Went online and found the following tutorial at [Instructables](http://www.instructables.com/id/Fixing-that-jet-lighter-you-love/), which helped alot - i'm not here yet, but i'm trying to find spare parts. Now i know there are a lot of different types, and i think us smokers - i mean jet lighter users, might like some links on how to fix our buddies. So /r/howto, do you know any more links or places to fix it?	0non-cybersec	Reddit
What if death is just a disease we haven't found a cure for yet?.	0non-cybersec	Reddit
Imagine if pee was yellow instead of red.. Edit: Haha ^yes. Hot baby ftw.	0non-cybersec	Reddit
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
Non overlapped Serial Port hangs at CloseHandle. <p>I wrote a serial port class that I developed and for simplicity I used blocking/synchronous/<strong>non-overlapped</strong>. I went through all MSDN documentations and it was strait forward for me.</p> <p>I don't have any problem with Opening, Transmitting or Receiving Bytes from the port. All operations are <strong>synchronous</strong> and there is no-threading complexity.</p> <pre><code>function TSerialPort.Open: Boolean; var h: THandle; port_timeouts: TCommTimeouts; dcb: TDCB; begin Result := False; if Assigned(FHandleStream) then begin // already open Exit(True); end; h := CreateFile(PChar('\\?\' + FComPort), GENERIC_WRITE or GENERIC_READ, 0, nil, OPEN_EXISTING, FILE_ATTRIBUTE_NORMAL, 0); // RaiseLastOSError(); if h <> INVALID_HANDLE_VALUE then begin { REMARKS at https://docs.microsoft.com/en-us/windows/desktop/api/winbase/ns-winbase-_commtimeouts If an application sets ReadIntervalTimeout and ReadTotalTimeoutMultiplier to MAXDWORD and sets ReadTotalTimeoutConstant to a value greater than zero and less than MAXDWORD, one of the following occurs when the ReadFile function is called: * If there are any bytes in the input buffer, ReadFile returns immediately with the bytes in the buffer. * If there are no bytes in the input buffer, ReadFile waits until a byte arrives and then returns immediately. * If no bytes arrive within the time specified by ReadTotalTimeoutConstant, ReadFile times out. } FillChar(port_timeouts, Sizeof(port_timeouts), 0); port_timeouts.ReadIntervalTimeout := MAXDWORD; port_timeouts.ReadTotalTimeoutMultiplier := MAXDWORD; port_timeouts.ReadTotalTimeoutConstant := 50; // in ms port_timeouts.WriteTotalTimeoutConstant := 2000; // in ms if SetCommTimeOuts(h, port_timeouts) then begin FillChar(dcb, Sizeof(dcb), 0); dcb.DCBlength := sizeof(dcb); if GetCommState(h, dcb) then begin dcb.BaudRate := FBaudRate; // baud rate dcb.ByteSize := StrToIntDef(FFrameType.Chars[0], 8); // data size dcb.StopBits := ONESTOPBIT; // 1 stop bit dcb.Parity := NOPARITY; case FFrameType.ToUpper.Chars[1] of 'E': dcb.Parity := EVENPARITY; 'O': dcb.Parity := ODDPARITY; end; dcb.Flags := dcb_Binary or dcb_Parity or dcb_ErrorChar or (DTR_CONTROL_ENABLE shl 4) or (RTS_CONTROL_ENABLE shl 12); dcb.ErrorChar := '?'; // parity error will be replaced with this char if SetCommState(h, dcb) then begin FHandleStream := THandleStream.Create(h); Result := True; end; end; end; if not Result then begin CloseHandle(h); end; end; end; function TSerialPort.Transmit(const s: TBytes): Boolean; var len: NativeInt; begin Result := False; len := Length(s); if Assigned(FHandleStream) and (len > 0) then begin // total timeout to transmit is 2sec!! Result := (FHandleStream.Write(s, Length(s)) = len); end; end; function TSerialPort.Receive(var r: Byte): Boolean; begin Result := False; if Assigned(FHandleStream) then begin // read timeout is 50ms Result := (FHandleStream.Read(r, 1) = 1); end; end; </code></pre> <p>My problem starts at closing the port. After all my communications, when I try to close the serial port, my Application totally hangs at CloseHandle() API. And that happens <strong>randomly</strong>. Which is meaningless to me since I use synchronous mode, there can not be any pending operations. When I request a close, It must simply close the handle.</p> <p>I searched the problem on the google and stack-overflow. There are many people who faced the similar problems but most of them are related with .NET serial port driver and their asynchronous mode operations which I don't have.</p> <p>And also some people forgot to set timeouts properly and they faced blocking issue at ReadFile and WriteFile API that is fully normal. But again this is not my problem, I've set CommTimeouts as it is indicated in MSDN remarks.</p> <pre><code>function TSerialPort.Close: Boolean; var h: THandle; begin Result := True; if Assigned(FHandleStream) then begin h := FHandleStream.Handle; FreeAndNil(FHandleStream); if h <> INVALID_HANDLE_VALUE then begin //PurgeComm(h, PURGE_TXABORT or PURGE_RXABORT or PURGE_TXCLEAR or PURGE_RXCLEAR); // didn't help //ClearCommError(h, PDWORD(nil)^, nil); // didn't help //CancelIO(h); // didn't help Result := CloseHandle(h); <------------ hangs here end; end; end; </code></pre> <p>Some people on Microsoft forum, suggest calling CloseHandle() in different thread. I have tried that as well. But that time it hangs while trying to free AnonymousThread that I created. Even I left FreeOnTerminate:=true as default, it hangs and I get memory leakage report by Delphi.</p> <p>Another bothering problem when it hangs, I have to close Delphi IDE fully and reopen. Otherwise I can't compile the code again since exe is still used.</p> <pre><code>function TSerialPort.Close: Boolean; var h: THandle; t: TThread; Event: TEvent; begin Result := True; if Assigned(FHandleStream) then begin h := FHandleStream.Handle; FreeAndNil(FHandleStream); if h <> INVALID_HANDLE_VALUE then begin PurgeComm(h, PURGE_TXABORT or PURGE_RXABORT or PURGE_TXCLEAR or PURGE_RXCLEAR); Event := TEvent.Create(nil, False, False, 'COM PORT CLOSE'); t := TThread.CreateAnonymousThread( procedure() begin CloseHandle(h); If Assigned(Event) then Event.SetEvent(); end); t.FreeOnTerminate := False; t.Start; Event.WaitFor(1000); FreeAndNil(t); // <---------- that time it hangs here, why??!! FreeAndNil(Event); end; end; end; </code></pre> <p>In my notebook I'm using USB to Serial Port converters from FTDI. Some people said that it is because of FTDI driver. But I'm using all microsoft drivers that is signed by Microsoft Windows Hardware Compatibility Publisher. There is no third party driver in my system. But when I disconnect the USB adapter, CloseHandle API unfreeze itself. Some people reports that, even native Serial Ports that are build in their motherboards have same issue.</p> <p>So far I couldn't solve the problem. Any help or workaround highly appreciated.</p> <p>Thanks.</p>	0non-cybersec	Stackexchange
That's enough Tinder for today.	0non-cybersec	Reddit
Making "brick" paths.	0non-cybersec	Reddit
Im about to get schwifty in another dimension..	0non-cybersec	Reddit
Projective and injective objects. <p>I'm trying to prove the following:</p> <p>If <span class="math-container">$\mathcal{A}$</span> is an Abelian Category, then every object in <span class="math-container">$\mathcal{A}$</span> is projective if and only if every object is injective.</p> <p>My attempt: Since every object in <span class="math-container">$\mathcal{A}$</span> is projective, we can find an epic morphism from any object to the others. Consequently, every object is isomorphic to 0 and so every object is injective. The converse is similar.</p> <p>However, I don't think that my proof is true. Can anyone help?</p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
Transferring ByteArray through Parcel returns NullPointerException. <pre><code>import android.os.Parcel; import android.os.Parcelable; public class MClass implements Parcelable { private byte[] _byte; public MClass() { } public MClass(Parcel in) { readFromParcel(in); } public byte[] get_byte() { return _byte; } public void set_byte(byte[] _byte) { this._byte = _byte; } public int describeContents() { return 0; } public void writeToParcel(Parcel dest, int flags) { dest.writeByteArray(_byte); } public void readFromParcel(Parcel in) { in.readByteArray(_byte); //LOE - Line Of Exception } public static final Parcelable.Creator CREATOR = new Parcelable.Creator() { public MClass createFromParcel(Parcel in) { return new MClass(in); } public MClass[] newArray(int size) { return new MClass[size]; } }; } </code></pre> <p>Whenever I am going to retrieve the bytes in my following array it is returning exception of <code>NullPointerException</code>. Can any one say what is the problem? What I am trying to do is to transfer a downloaded image bytes from one activity to another.</p>	0non-cybersec	Stackexchange
Apache2 Alias ignored: "File does not exist" despite no trailing slash and rewrite enabled. <p>In Ubuntu 2016.04's default Apache2, I tried to add a virtual host with an alias:</p> <pre><code>sudo mkdir /var/www/simplesamlphp echo "Hello" > /var/www/simplesamlphp/index.html (as root) sudo chown -R www-data:www-data /var/www/simplesamlphp sudo chmod -R 755 /var/www </code></pre> <p>Second, I created <code>/etc/apache2/sites-available/simplesamlphp.conf</code>:</p> <pre><code><VirtualHost *> ServerName simplesamlphp DocumentRoot /var/www/simplesamlphp SetEnv SIMPLESAMLPHP_CONFIG_DIR /var/simplesamlphp/config Alias /simplesaml /var/simplesamlphp/www <Directory /var/simplesamlphp/www> <IfModule !mod_authz_core.c> # For Apache 2.2: Order allow,deny Allow from all </IfModule> <IfModule mod_authz_core.c> # For Apache 2.4: Require all granted </IfModule> </Directory> </VirtualHost> </code></pre> <p>Third, I added this line to <code>/etc/hosts</code>:</p> <pre><code>127.0.0.1 simplesamlphp </code></pre> <p>Fourth, I ran:</p> <pre><code>sudo a2enmod rewrite sudo a2ensite simplesamlphp.conf sudo service apache2 restart </code></pre> <p>PROBLEM: Accessing <code>http://simplesamlphp/simplesaml</code> gives <code>The requested URL /simplesaml was not found on this server</code> and the following appears in <code>/var/log/apache2/error.log</code>:</p> <pre><code>AH00128: File does not exist: /var/www/html/simplesaml </code></pre> <p>What did I do wrong?<br> <em>By the way, I am following <a href="https://simplesamlphp.org/docs/stable/simplesamlphp-install#section_6" rel="nofollow noreferrer">these instructions</a>. Actually, I am not sure why a DocumentRoot is needed despite all web content being in <code>/var/simplesamlphp/www</code>.</em></p>	0non-cybersec	Stackexchange
Just finished it. I thought you might like it :).	0non-cybersec	Reddit
How to represent a 2D matrix in Java?. <p>I have to create in Java a 2D matrix (consisting of double values) as well as a 1D vector. It should be possible to access individual rows and columns as well as individual elements. Moreover, it should be thread-safe (threads writing at the same time). Perhaps later I need some matrix operations too.</p> <p>Which data structure is best for this? Just a 2D array or a TreeMap? Or is there any amazing external library?</p>	0non-cybersec	Stackexchange
Do We Enough Playgrounds?.	0non-cybersec	Reddit
I see your healthy breakfasts, and give you the full Scottish fry-up!.	0non-cybersec	Reddit
Man City Fans, do you feel dirty?. How do you support a team who's mantra is literally to buy success? I can understand if you were a fan pre-sheik, but nowadays, Man City stands for everything that is wrong in modern football. Every time I see a player linked with the club I cringe a little bit, knowing their main motivation for going would be a fat paycheck. You can't help but fear for their career usually. I can understand buying reinforcements in weak areas of the squad, but City will just throw money in the direction of whatever name is making rounds at the moment. What squad needs 8 or 9 DMs???? End Rant. I'm not dissing City fans, but your club shames professional sports.	0non-cybersec	Reddit
Pop's response to David Aldridge reminded me of this from awhile ago: "Gregg Popovich’s Wife Wants Him to Be Nicer to the Media".	0non-cybersec	Reddit
As I've grown older I've realized that Santa likes rich kids more than everyone else.	0non-cybersec	Reddit
Is it bad practice to override a deprecated method?. <p>Is this code bad practice as the method <code>show()</code> is deprecated? Is it okay to override here?</p> <pre class="lang-java prettyprint-override"><code>public class Window extends JFrame { public Window() { // Do things. } public void show() { // <- Comes up with a warning as deprecated code. // Do other things. } } </code></pre>	0non-cybersec	Stackexchange
Figured this belonged here. This a report original credits go to u/Brithynes.	0non-cybersec	Reddit
MySQL datatype for username of maximum 15 characters. <p>I currently have a database with the "user" table having username as one of the columns.</p> <p>I allow a maximum of 15 characters for the username in my system. Is it fine to have the username column as a varchar(15) datatype?</p>	0non-cybersec	Stackexchange
Using HaProxy environmental variables in haproxy.cfg not working. <p>Trying to figure out why environmental variables in haproxy.cfg aren't working in HA-Proxy version 1.5.2 </p> <p>on the command line, using Printenv I get a list of environmental variables like FE_PORT_8000_TCP_ADDR=172.17.0.4</p> <p>Which I need to use in the haproxy.cfg. According to this and the docs <a href="https://serverfault.com/questions/625705/how-can-i-use-environment-variables-in-haproxy-conf">How can I use environment variables in haproxy.conf</a> using $FE_PORT_8000_TCP_ADDR or ${FE_PORT_8000_TCP_ADDR} should work. However that is not working.</p> <p>In Haporxy.cfg hardcoding DOES work, and accessed in a browser it shows as expected:</p> <pre><code>backend FE # balance roundrobin server FE1 172.17.0.4:8000 maxconn 256 </code></pre> <p>But environmental variable with same supposed value doesn't, in the browser it gives 503 Service Unavailable.</p> <pre><code>backend FE # balance roundrobin server FE1 $FE_PORT_8000_TCP_ADDR:8000 maxconn 256 </code></pre> <p>Any ideas on what is being done wrong?</p> <p>UPDATE: This person has what looks like the same issue <a href="https://serverfault.com/questions/625705/how-can-i-use-environment-variables-in-haproxy-conf#new-answer?newreg=b89a86e2ccb64c96bcd92a771397e648">How can I use environment variables in haproxy.conf</a></p>	0non-cybersec	Stackexchange
What's the default value of an union in avro Idl?. <p>For unions in avro Idl something like below what would be the default values?<br> 1. union {null, string} var = null;<br> 2. union {string, null} = "xyz";<br> 3. union {null, string} = "xyz";<br> 4. union {null, string, array} = [];</p> <p>My assumption is the default values would be always the first item in the union. Is my understand correct?</p>	0non-cybersec	Stackexchange
Introduction to Analysis: Actually Constant. <p>I more or less understand the concept of locally constant. If a function, $f(x)$, is locally constant then there is a sufficiently small neighborhood $(a - \delta, a + \delta)$ about $a$ such that $f(x)$ is constant.</p> <p>However, what do they mean when they say Actually Constant? </p>	0non-cybersec	Stackexchange
How can I download and install fonts dynamically to iOS app. <p>A client would like to dynamically add fonts to an iOS app by downloading them with an API call. </p> <p>Is this possible? All the resources I've dredged up show how to manually drag the .ttf file to Xcode and add it to the plist. Is it possible to download a font and use it on the fly client side, programmatically?</p> <p>Thanks.</p>	0non-cybersec	Stackexchange
Method overloading - good or bad design?. <p>I like to overload methods to support more and more default cases. What is the performance impact of method overloading? From your experience, is it advisable to overload methods? What is the limit? What are the workarounds? </p>	0non-cybersec	Stackexchange
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
The Definitive Rundown of Every Upcoming Playstation 4 Exclusive.	0non-cybersec	Reddit
Best way to cook a thick steak? [instructions below].	0non-cybersec	Reddit
Just finished a Pencil drawing of John Wick (Keanu Reeves)..	0non-cybersec	Reddit
Mahoutsukai no Yome (The Ancient Magus' Bride) TV Preview #3.	0non-cybersec	Reddit
This pharmacy....	0non-cybersec	Reddit
Live Stream Audio in Wavesurfer.js. <p>I need to plot a streaming audio which comes as buffers of 2 seconds. Besides that, I need to plot the audio and have features like play, pause, zoom and span. Wavesurfer.js has most of these features except that it only plots for single/ static audio file. I need it to work for streaming audio too. What tweak should I do? Has anyone done anything like this before? Any other better alternatives? Specification: Plot a streaming wave file in colour. The user should be able to pause, play, zoom and span into the audio.</p>	0non-cybersec	Stackexchange
Yolo Bypass Wildlife Area, Davis, CA [OC] [3869x5803].	0non-cybersec	Reddit
How to use spot instance with amazon elastic beanstalk?. <p>I have one infra that use amazon elastic beanstalk to deploy my application. I need to scale my app adding some spot instances that EB do not support.</p> <p>So I create a second autoscaling from a launch configuration with spot instances. The autoscaling use the same load balancer created by beanstalk.</p> <p>To up instances with the last version of my app, I copy the user data from the original launch configuration (created with beanstalk) to the launch configuration with spot instances (created by me).</p> <p>This work fine, but:</p> <ol> <li><p>how to update spot instances that have come up from the second autoscaling when the beanstalk update instances managed by him with a new version of the app?</p> </li> <li><p>is there another way so easy as, and elegant, to use spot instances and enjoy the benefits of beanstalk?</p> </li> </ol> <p><strong>UPDATE</strong></p> <p>Elastic Beanstalk add support to spot instance since 2019... see: <a href="https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html" rel="nofollow noreferrer">https://docs.aws.amazon.com/elasticbeanstalk/latest/relnotes/release-2019-11-25-spot.html</a></p>	0non-cybersec	Stackexchange
Is accepting the current and the previous one-time password a bad practice?. <p>I often see two-factor authentication (2FA) methods using one-time passwords (OTP) implementations wherein the current (previous) and sometimes even 2 or 3 previous tokens are still valid. This is probably done for several reasons, I can think of:</p> <ul> <li>to overcome possible time sync issues between the client and server (user experience),</li> <li>for convenience of the sysadmins not to constantly have support calls about not being able to login.</li> </ul> <p>Is accepting the previous OTP a bad practice? And if so, what is the general recommendation regarding the implementation of such "features"?</p> <p><em>Correct me if I'm wrong but when the token is 6 digits, the number of possible combinations are 1.000.000 (000000 - 999999). So only 1 in a million tokens in accepted during a 30 second time-span. But when accepting the previous (or even more previous) tokens the chances are basically 2, 3, 4, or 5 in one million.</em></p>	0non-cybersec	Stackexchange
ERROR: Could not build wheels for pyopencl which use PEP 517 and cannot be installed directly. <p>I am trying to install pyopencl using <code>pip install pyopencl</code> but am getting the following error lines:</p> <pre><code>ERROR: Failed building wheel for pyopencl \ Failed to build pyopencl \ ERROR: Could not build wheels for pyopencl which use PEP 517 and cannot be installed directly \ </code></pre> <p>I've seen other posts about errors like this but those solutions did not seem to work.</p> <p>Full error log trace:</p> <p><a href="https://i.stack.imgur.com/Xpvpo.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/Xpvpo.png" alt="enter image description here" /></a></p>	0non-cybersec	Stackexchange
Ubuntu will not get latest version from PPA. <p><strong>NOTE:</strong> I know that I am using 15.10, an EOL release, but I am developing the PPA on 15.10, and all packages are for it. This also isn't affecting any other PPAs or packages, and I think it is not related to my release.</p> <hr> <p>I have created a PPA for my Ubuntu 15.10 system.<br> I added it and updated like so:</p> <pre><code>sudo add-apt-repository ppa:nann-sahibdeep/inverse-ppa sudo apt-get update </code></pre> <p>After some testing, turns out the first version, <code>4.1-0ubuntu1</code> had a packaging error, so I made a new package, <code>4.1-1ubuntu1</code>. </p> <p>Launchpad confirms I have uploaded it to the PPA. But, the problem arises when I try and install it.</p> <pre><code>sudo apt-get install inverse-dev </code></pre> <p>It's trying to fetch <code>4.1-0</code> and not <code>4.1-1</code>, which is why the install fails.</p> <p><strong>Is there any reason that Ubuntu is not getting the latest version on the PPA?</strong></p>	0non-cybersec	Stackexchange
GAME THREAD: Golden State Warriors (9-8) @ Sacramento Kings (4-10) - (Dec. 01, 2013). ##General Information TIME \|MEDIA \|LOCATION \|MISC :------------\|:------------------------------------\|:-------------------\|:------------------------- 06:00 Eastern \|TV: Away: CSN-Bay Area, Home: News 10 \| Sleep Train Arena, Sacramento, CA \| [Live chat](http://webchat.freenode.net/?channels=r/NBA&uio=MTE9MjQ255/) 05:00 Central \|Streaming: N/A \| Team Subreddits\| 04:00 Mountain\|Game Story: [NBA.com](http://www.nba.com/games/20131201/GSWSAC/gameinfo.html#nbaGIlive)\| [/r/warriors](http://reddit.com/r/warriors) \| 03:00 Pacific \|Box Score: [NBA.com](http://www.nba.com/games/20131201/GSWSAC/gameinfo.html#nbaGIboxscore) \| [/r/kings](http://reddit.com/r/kings) \| ----- [Reddit Stream](http://nba-gamethread.herokuapp.com/reddit-stream/) (You must click this link from the comment page.) Remember to upvote the Game Threads for maximum visibility.	0non-cybersec	Reddit
[50/50] Christian Louboutin Marie Antoinette shoes \| Fashionista killed live on catwalk (NSFW).	0non-cybersec	Reddit

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